Convection-driven geodynamo models

By C A Jones

School of Mathematical Sciences University of Exeter Exeter EX4 4QE UK

There has been signishy cant progress in the development of numerical geodynamo mod-els over the last shy ve years Advances in computer technology have made it possibleto perform three-dimensional simulations with thermal or compositional convectionas the driving mechanism These numerical simulations give reasonable results forthe morphology and strength of the shy eld at the coremantle boundary and the mod-els are also capable of giving reversals and excursions which can be compared withpalaeomagnetic observations they also predict dinoterential rotation between the innercore and the mantle

However there are still a number of fundamental problems associated with thesimulations which are proving hard to overcome Despite the advances in computingpower the models are still expensive and take a long time to run This problem maydiminish as faster machines become available and new numerical methods exploitparallelization enotectively but currently there are no practical schemes available whichwork at low Ekman number

Even with turbulent values of the dinotusivities (and the question of whether iso-tropic dinotusivities are appropriate is still unresolved) the appropriate dynamicalregime has not yet been reached In consequence modelling assumptions about thenature of the regow near the boundaries have to be made and dinoterent choices canhave profound enotects on the dynamics The nature of large-scale magnetoconvectionat small E is still not well understood and until we have more understanding of thisissue it will be dimacr cult to have a great deal of conshy dence in the predictions of thenumerical models

Keywords dynamo Earthrsquos core rotating convection

1 Introduction

In the past ten years there has been considerable activity in geodynamo modellingthe main enotort being directed towards convection-driven geodynamo models Verysignishy cant progress has been achieved models constructed from solutions of theNavierStokes and Maxwell equations can now be sensibly compared with geophys-ical observations of the actual geomagnetic shy eld Results obtained from palaeomag-netic data such as the nature of the reversals of the main shy eld can also be modelleddirectly from the fundamental equations (Sarson amp Jones 1999 Glatzmaier et al 1999) Despite this undoubted progress there are still a number of fundamentalproblems which remain unresolved The parameter regime in which the current gen-eration of numerical models can be run is very far from the regime of geophysicalparameter values so far indeed that the strong similarity between the model out-puts and the geodynamo is quite surprising

Phil Trans R Soc Lond A (2000) 358 873897

873

c 2000 The Royal Society

874 C A Jones

The rotating spherical shell model is a natural problem to study as a geodynamomodel and a considerable number of papers have been based on it we refer to it hereas the `zero-order modelrsquo because it is a clear-cut mathematical problem althoughit should be emphasized that physical enotects not included in the zero-order modelmay be very signishy cant for the real geodynamo In the zero-order model there is justone source of buoyancy whereas in the geodynamo there are two thermal and com-positional buoyancy We also assume homogeneous spherically symmetric boundaryconditions whereas in the geodynamo mantle convection is spatially inhomogeneousHeat regux absorbed by the mantle will be a function of the spherical polar coordinates

and which is determined mainly by the dynamics of mantle convection (Zhang ampGubbins 1993 Olson amp Glatzmaier 1995) and this can anotect dynamo action (Sarsonet al 1997a Glatzmaier et al 1999) There is also a considerable density variationbetween the inner-core boundary (ICB) and the coremantle boundary (CMB) butthis is ignored in the Boussinesq zero-order model

Even within the zero-order model there are a number of choices to be madethe buoyancy source can be distributed uniformly throughout the inner core (theuniform heating model) or put entirely in the inner core through a regux at the ICBThe boundary conditions can be taken as no-slip or stress-free various options can betaken to model the inner core Sometimes it is neglected altogether and convectionin a whole sphere is considered sometimes the inner core is given its current size(about 035 times the radius of the outer core) but it is assumed to be electricallyinsulating and sometimes it is allowed to be electrically conducting usually with asimilar conductivity to that of the outer core The resulting dynamo behaviour isstrongly anotected by which of these choices is made

A strong motivation for the activity in geodynamo modelling is the possibilityof using the models to compare with geophysical measurements Secular variationstudies and palaeomagnetic observations are two important examples but dynamomodels might also help us to understand the physical conditions at the CMB Anotherexciting application is to apply our understanding of the geodynamo to the magneticshy elds of other planets the recently discovered internal shy elds on the Jovian moonsGanymede and Io (Schubert et al 1996 Sarson et al 1997b) are prime candidatesto test dynamo models

In this paper we concentrate on interpreting and understanding results from thezero-order models and on trying to relate these results to the actual geodynamoThe models need to be fully validated before we can reliably use them as a tool forfurther geophysical studies

Previous reviews on the dynamics and magnetohydrodynamics of the Earthrsquos coreinclude Fearn (1998) Glatzmaier amp Roberts (1997) Hollerbach (1996) and Braginsky(1994) Dynamo theory was reviewed by Roberts amp Soward (1992) and Roberts(1994) and more background information can be found in Roberts amp Gubbins (1987)

2 The zero-order model and governing equations

We consider an electrically conducting Boussinesq reguid layer conshy ned between twoconcentric spherical surfaces at r = ri (corresponding to the ICB) and at r = ro

(corresponding to the CMB) in a frame rotating at angular velocity 0 which weidentify with the mantle frame The inner core is taken as a solid electrically conduct-ing sphere concentric with the outer core which is free to rotate about the z-axis at

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 875

angular velocity i relative to the mantle frame We denote the density deshy cit due tocompositional or thermal variations as C the codensity The codensity source whichdrives the convection is denoted by S This source leads to a codensity dinoterence centCacross the reguid outer core The distance between the surfaces d = ro ri = 2260 km istaken as the length-scale the time-scale is the magnetic dinotusion time d2= 80 kyrthe magnetic shy eld unit is (2 0 )1=2 20 Gauss (where is the reguid density isthe permeability and is the magnetic dinotusivity) and the codensity unit is centC Thegoverning MHD equations for the velocity u the magnetic shy eld B and the codensityC are

E

qPr

Du

Dt+ z u = rp+ j B + Er2u + qRaCr (21)

B

t= r2B + r (u B ) (22)

C

t= qr2C u rC + S (23)

r B = 0 (24)

r u = 0 (25)

Here j = r B is the current and the dimensionless parameters are the Ekmannumber E = =2 0d

2 the Prandtl number Pr = = the Roberts number q = =and the Modishy ed Rayleigh number R = g centTd=2 0 In these deshy nitions isthe kinematic viscosity and is the codensity dinotusivity Other combinations of theparameters are sometimes used notably the magnetic Prandtl number Prm = = =qPr and non-rotating Rayleigh number Ra = g centTd3= = R=E

In addition we have two further dimensionless parameters the radius ratio ri=rowhich is usually taken to be around 1=3 to model the geodynamo and the magneticdinotusivity ratio of the outer to the inner core o= i The most popular choice is

o= i = 1 which we call a conducting inner coreIn the conducting-inner-core case we need to solve for the magnetic shy eld B i and

the angular velocity i The inner-core magnetic shy eld is governed by

B

t+ i

B

=

i

or2B (26)

The mechanical boundary conditions are that there should be no regow through theICB and CMB surfaces and that there should be no-slip at these surfaces so u = 0at the CMB and u = iz r at the ICB Alternatively stress-free conditions canbe applied at the ICB CMB or both (Kuang amp Bloxham 1997)

Thermal boundary conditions must also be applied Popular choices are C constanton r = ri and r = ro or shy xed regux boundary conditions or some combination of thetwo Another common choice is to assume a uniform source in the outer core togetherwith shy xed codensity boundaries The non-magnetic problem with a shy xed codensity(usually heat) source has been extensively studied

The magnetic boundary conditions are that the normal component of B and thetangential components of the electric shy eld E are continuous The latter conditionscan easily be converted into conditions on the current using the MHD form of Ohmrsquoslaw Note that if stress-free rather than no-slip conditions are used at a conduct-ing boundary then the velocity on the boundary will enter the magnetic boundaryconditions through Ohmrsquos law

Phil Trans R Soc Lond A (2000)

876 C A Jones

The angular velocity i is determined by the torque balance at the ICB In generalthe rate of change of the angular momentum of the inner core is determined byadding the magnetic torque the viscous torque and any external torque such as agravitational torque Bunotett (1997) has argued that asymmetries in the mantle giverise to asymmetries in the shape of the inner core and that the resulting gravitationaltorque can be even larger than the magnetic torque In the stress-free case the viscoustorque is zero so the rate of change of angular momentum of the inner core is thendetermined solely by magnetic and external torque Estimates of the moment ofinertia of the inner core and the likely order of magnitude of the magnetic torquesuggest that the inner core can spin up due to magnetic torque in a few days onthe dynamo time-scale the inner core will therefore be in torque equilibrium If theviscous torque is small this means that the total magnetic and external torque mustbe zero any torque supplied to one region of the ICB must be balanced by an oppositetorque somewhere else so that the torque surface integral adds up to zero this hasbeen called the `inner-core Taylor constraintrsquo

With this scaling the dimensionless strength of the magnetic shy eld gives a localmeasure of the Elsasser number = B2=2 0 Although the Elsasser number isonly a derived quantity and not an input parameter it is useful for comparison withmagnetoconvection models where is an input parameter Similarly the dimension-less strength of the velocity shy eld gives a local measure of the magnetic Reynoldsnumber Rm = ud= Again this is a derived quantity but it is very useful to com-pare its value with the results of kinematic dynamo models where Rm is an inputparameter

3 Numerical methods

The most popular technique for solving the dynamo equations numerically has beenthe pseudo-spectral method recently described by Cox amp Matthews (1997) andHollerbach (2000) and applied to the geodynamo by Glatzmaier amp Roberts (1995)and Kuang amp Bloxham (1999) The vector shy elds u and B are expanded into toroidaland poloidal parts and these scalar shy elds together with the C shy eld are expanded inspherical harmonics In the radial direction either a shy nite dinoterence scheme or expan-sion in Chebyshev polynomials is used These expansions are inserted into (21)(25)and time-dependent equations for the coemacr cients are obtained The linear dinotu-sion operators are handled using an implicit scheme such as the CrankNicholsonmethod while the nonlinear terms are handled explicitly by converting from spectralspace to physical space so that the required multiplications can be handled easilyThis procedure leads to accurate solutions provided moderate parameter values areused it does however require large computing resources In addition to the obvi-ous problem associated with computing fully three-dimensional numerical solutionsthere are two additional problems which make the execution times even longer shy rstthere is no emacr cient fast Legendre transform as there is for Fourier transforms andsecond at small E the time-step has to be very small (see Walker et al (1998) fordetails)

Motivated by these dimacr culties a number of ways of reducing the CPU timerequired have been tried One method has been to truncate the equations severelyin the azimuthal direction sometimes leaving only the axisymmetric and one non-axisymmetric eim mode this is known as the 2 1

2-dimensional method This tech-

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Convection-driven geodynamo models 877

nique reduces the CPU time required by a large factor The disadvantage of thismethod is that the non-axisymmetric wavenumber m has to be chosen a priorirather than emerging from the calculation Nevertheless the axisymmetric shy eldsgenerated by the 2 1

2-dimensional method are remarkably close to those computed by

fully three-dimensional calculations provided the azimuthal wavenumber is chosensensibly in the limited parameter regimes that can be explored numerically choos-ing m as the shy rst mode to become linearly unstable as R is increased is normally anadequate choice The 21

2-dimensional approach is therefore a useful tool in exploring

dynamo behaviour but one can only have full conshy dence in the results if they arecalibrated by at least some fully three-dimensional simulations The simulations usedfor shy gures 2 and 3 below are a compromise between the 21

2-dimensional model and

fully three-dimensional simulations the r and directions are fully resolved but asmall number of non-axisymmetric modes (typically four) are present together withthe axisymmetric modes

Another popular technique has been the use of hyperdinotusion to reach lowerEkman number solutions Here the dinotusive operator r2 is replaced by some for-mula such as (1 + n3)r2 where n is the order of the spherical harmonic in theexpansion of the shy elds and is a constant This method is very simple to imple-ment if the pseudo-spectral method is used and it stabilizes the numerical schemefor values of 005 allowing small E to be reached with tolerable (but stillsmall) time-steps Signishy cant advances have been made using this scheme (see forexample Glatzmaier amp Roberts 1995 1997) but there is a serious price to pay Aspointed out by Zhang amp Jones (1997) hyperdinotusion damps out small azimuthalwavenumbers which can be preferred if the magnetic shy eld is weak To illustratethis we reproduce some of the Zhang amp Jones (1997) results in shy gure 1ac Inshy gure 1a we show the regow pattern at small E at the onset of non-magnetic con-vection note the large preferred m visible from the tight roll conshy guration andthe largely z-independent nature of the regow expected from the ProudmanTaylortheorem In shy gure 1b we see the result of the same calculation but with hypervis-cosity and no magnetic shy eld while in shy gure 1c we see the equivalent picture forthe case of an imposed azimuthal magnetic shy eld and no hyperviscosity Note thatthe hyperviscosity has produced a similar enotect to the magnetic shy eld in both casesreducing the dominant azimuthal wavenumber In consequence it is very dimacr cult totell in dynamo simulations with hyperviscosity whether the azimuthal wavenumberis being determined by the magnetic shy eld or the hyperviscosity In this respect thethree-dimensional hyperdinotusive dynamos sunoter from the same disadvantage as the2 1

2-dimensional models in that the dominant azimuthal wavenumbers are selected

somewhat arbitrarilyAnother approach is the use of shy nite dinoterences in all three directions (see for

example Kageyama amp Sato 1997a) These methods became unfashionable becauseof dimacr culties in treating the regow near the rotation axis and in implementing thecorrect magnetic boundary conditions which are non-local However the develop-ment of fast massively parallel machines and the failure to resolve the Legendretransform problem (Lesur amp Gubbins 1999) has stimulated new interest in shy nite-dinoterence techniques which can take full advantage of parallel architecture Thepseudo-spectral technique requires a redistribution of the arrays amongst the dif-ferent processors at each time-step and this makes it a less attractive scheme forparallel architecture

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

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Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

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880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

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Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

874 C A Jones

The rotating spherical shell model is a natural problem to study as a geodynamomodel and a considerable number of papers have been based on it we refer to it hereas the `zero-order modelrsquo because it is a clear-cut mathematical problem althoughit should be emphasized that physical enotects not included in the zero-order modelmay be very signishy cant for the real geodynamo In the zero-order model there is justone source of buoyancy whereas in the geodynamo there are two thermal and com-positional buoyancy We also assume homogeneous spherically symmetric boundaryconditions whereas in the geodynamo mantle convection is spatially inhomogeneousHeat regux absorbed by the mantle will be a function of the spherical polar coordinates

and which is determined mainly by the dynamics of mantle convection (Zhang ampGubbins 1993 Olson amp Glatzmaier 1995) and this can anotect dynamo action (Sarsonet al 1997a Glatzmaier et al 1999) There is also a considerable density variationbetween the inner-core boundary (ICB) and the coremantle boundary (CMB) butthis is ignored in the Boussinesq zero-order model

Even within the zero-order model there are a number of choices to be madethe buoyancy source can be distributed uniformly throughout the inner core (theuniform heating model) or put entirely in the inner core through a regux at the ICBThe boundary conditions can be taken as no-slip or stress-free various options can betaken to model the inner core Sometimes it is neglected altogether and convectionin a whole sphere is considered sometimes the inner core is given its current size(about 035 times the radius of the outer core) but it is assumed to be electricallyinsulating and sometimes it is allowed to be electrically conducting usually with asimilar conductivity to that of the outer core The resulting dynamo behaviour isstrongly anotected by which of these choices is made

A strong motivation for the activity in geodynamo modelling is the possibilityof using the models to compare with geophysical measurements Secular variationstudies and palaeomagnetic observations are two important examples but dynamomodels might also help us to understand the physical conditions at the CMB Anotherexciting application is to apply our understanding of the geodynamo to the magneticshy elds of other planets the recently discovered internal shy elds on the Jovian moonsGanymede and Io (Schubert et al 1996 Sarson et al 1997b) are prime candidatesto test dynamo models

In this paper we concentrate on interpreting and understanding results from thezero-order models and on trying to relate these results to the actual geodynamoThe models need to be fully validated before we can reliably use them as a tool forfurther geophysical studies

Previous reviews on the dynamics and magnetohydrodynamics of the Earthrsquos coreinclude Fearn (1998) Glatzmaier amp Roberts (1997) Hollerbach (1996) and Braginsky(1994) Dynamo theory was reviewed by Roberts amp Soward (1992) and Roberts(1994) and more background information can be found in Roberts amp Gubbins (1987)

2 The zero-order model and governing equations

We consider an electrically conducting Boussinesq reguid layer conshy ned between twoconcentric spherical surfaces at r = ri (corresponding to the ICB) and at r = ro

(corresponding to the CMB) in a frame rotating at angular velocity 0 which weidentify with the mantle frame The inner core is taken as a solid electrically conduct-ing sphere concentric with the outer core which is free to rotate about the z-axis at

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 875

angular velocity i relative to the mantle frame We denote the density deshy cit due tocompositional or thermal variations as C the codensity The codensity source whichdrives the convection is denoted by S This source leads to a codensity dinoterence centCacross the reguid outer core The distance between the surfaces d = ro ri = 2260 km istaken as the length-scale the time-scale is the magnetic dinotusion time d2= 80 kyrthe magnetic shy eld unit is (2 0 )1=2 20 Gauss (where is the reguid density isthe permeability and is the magnetic dinotusivity) and the codensity unit is centC Thegoverning MHD equations for the velocity u the magnetic shy eld B and the codensityC are

E

qPr

Du

Dt+ z u = rp+ j B + Er2u + qRaCr (21)

B

t= r2B + r (u B ) (22)

C

t= qr2C u rC + S (23)

r B = 0 (24)

r u = 0 (25)

Here j = r B is the current and the dimensionless parameters are the Ekmannumber E = =2 0d

2 the Prandtl number Pr = = the Roberts number q = =and the Modishy ed Rayleigh number R = g centTd=2 0 In these deshy nitions isthe kinematic viscosity and is the codensity dinotusivity Other combinations of theparameters are sometimes used notably the magnetic Prandtl number Prm = = =qPr and non-rotating Rayleigh number Ra = g centTd3= = R=E

In addition we have two further dimensionless parameters the radius ratio ri=rowhich is usually taken to be around 1=3 to model the geodynamo and the magneticdinotusivity ratio of the outer to the inner core o= i The most popular choice is

o= i = 1 which we call a conducting inner coreIn the conducting-inner-core case we need to solve for the magnetic shy eld B i and

the angular velocity i The inner-core magnetic shy eld is governed by

B

t+ i

B

=

i

or2B (26)

The mechanical boundary conditions are that there should be no regow through theICB and CMB surfaces and that there should be no-slip at these surfaces so u = 0at the CMB and u = iz r at the ICB Alternatively stress-free conditions canbe applied at the ICB CMB or both (Kuang amp Bloxham 1997)

Thermal boundary conditions must also be applied Popular choices are C constanton r = ri and r = ro or shy xed regux boundary conditions or some combination of thetwo Another common choice is to assume a uniform source in the outer core togetherwith shy xed codensity boundaries The non-magnetic problem with a shy xed codensity(usually heat) source has been extensively studied

The magnetic boundary conditions are that the normal component of B and thetangential components of the electric shy eld E are continuous The latter conditionscan easily be converted into conditions on the current using the MHD form of Ohmrsquoslaw Note that if stress-free rather than no-slip conditions are used at a conduct-ing boundary then the velocity on the boundary will enter the magnetic boundaryconditions through Ohmrsquos law

Phil Trans R Soc Lond A (2000)

876 C A Jones

The angular velocity i is determined by the torque balance at the ICB In generalthe rate of change of the angular momentum of the inner core is determined byadding the magnetic torque the viscous torque and any external torque such as agravitational torque Bunotett (1997) has argued that asymmetries in the mantle giverise to asymmetries in the shape of the inner core and that the resulting gravitationaltorque can be even larger than the magnetic torque In the stress-free case the viscoustorque is zero so the rate of change of angular momentum of the inner core is thendetermined solely by magnetic and external torque Estimates of the moment ofinertia of the inner core and the likely order of magnitude of the magnetic torquesuggest that the inner core can spin up due to magnetic torque in a few days onthe dynamo time-scale the inner core will therefore be in torque equilibrium If theviscous torque is small this means that the total magnetic and external torque mustbe zero any torque supplied to one region of the ICB must be balanced by an oppositetorque somewhere else so that the torque surface integral adds up to zero this hasbeen called the `inner-core Taylor constraintrsquo

With this scaling the dimensionless strength of the magnetic shy eld gives a localmeasure of the Elsasser number = B2=2 0 Although the Elsasser number isonly a derived quantity and not an input parameter it is useful for comparison withmagnetoconvection models where is an input parameter Similarly the dimension-less strength of the velocity shy eld gives a local measure of the magnetic Reynoldsnumber Rm = ud= Again this is a derived quantity but it is very useful to com-pare its value with the results of kinematic dynamo models where Rm is an inputparameter

3 Numerical methods

The most popular technique for solving the dynamo equations numerically has beenthe pseudo-spectral method recently described by Cox amp Matthews (1997) andHollerbach (2000) and applied to the geodynamo by Glatzmaier amp Roberts (1995)and Kuang amp Bloxham (1999) The vector shy elds u and B are expanded into toroidaland poloidal parts and these scalar shy elds together with the C shy eld are expanded inspherical harmonics In the radial direction either a shy nite dinoterence scheme or expan-sion in Chebyshev polynomials is used These expansions are inserted into (21)(25)and time-dependent equations for the coemacr cients are obtained The linear dinotu-sion operators are handled using an implicit scheme such as the CrankNicholsonmethod while the nonlinear terms are handled explicitly by converting from spectralspace to physical space so that the required multiplications can be handled easilyThis procedure leads to accurate solutions provided moderate parameter values areused it does however require large computing resources In addition to the obvi-ous problem associated with computing fully three-dimensional numerical solutionsthere are two additional problems which make the execution times even longer shy rstthere is no emacr cient fast Legendre transform as there is for Fourier transforms andsecond at small E the time-step has to be very small (see Walker et al (1998) fordetails)

Motivated by these dimacr culties a number of ways of reducing the CPU timerequired have been tried One method has been to truncate the equations severelyin the azimuthal direction sometimes leaving only the axisymmetric and one non-axisymmetric eim mode this is known as the 2 1

2-dimensional method This tech-

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 877

nique reduces the CPU time required by a large factor The disadvantage of thismethod is that the non-axisymmetric wavenumber m has to be chosen a priorirather than emerging from the calculation Nevertheless the axisymmetric shy eldsgenerated by the 2 1

2-dimensional method are remarkably close to those computed by

fully three-dimensional calculations provided the azimuthal wavenumber is chosensensibly in the limited parameter regimes that can be explored numerically choos-ing m as the shy rst mode to become linearly unstable as R is increased is normally anadequate choice The 21

2-dimensional approach is therefore a useful tool in exploring

dynamo behaviour but one can only have full conshy dence in the results if they arecalibrated by at least some fully three-dimensional simulations The simulations usedfor shy gures 2 and 3 below are a compromise between the 21

2-dimensional model and

fully three-dimensional simulations the r and directions are fully resolved but asmall number of non-axisymmetric modes (typically four) are present together withthe axisymmetric modes

Another popular technique has been the use of hyperdinotusion to reach lowerEkman number solutions Here the dinotusive operator r2 is replaced by some for-mula such as (1 + n3)r2 where n is the order of the spherical harmonic in theexpansion of the shy elds and is a constant This method is very simple to imple-ment if the pseudo-spectral method is used and it stabilizes the numerical schemefor values of 005 allowing small E to be reached with tolerable (but stillsmall) time-steps Signishy cant advances have been made using this scheme (see forexample Glatzmaier amp Roberts 1995 1997) but there is a serious price to pay Aspointed out by Zhang amp Jones (1997) hyperdinotusion damps out small azimuthalwavenumbers which can be preferred if the magnetic shy eld is weak To illustratethis we reproduce some of the Zhang amp Jones (1997) results in shy gure 1ac Inshy gure 1a we show the regow pattern at small E at the onset of non-magnetic con-vection note the large preferred m visible from the tight roll conshy guration andthe largely z-independent nature of the regow expected from the ProudmanTaylortheorem In shy gure 1b we see the result of the same calculation but with hypervis-cosity and no magnetic shy eld while in shy gure 1c we see the equivalent picture forthe case of an imposed azimuthal magnetic shy eld and no hyperviscosity Note thatthe hyperviscosity has produced a similar enotect to the magnetic shy eld in both casesreducing the dominant azimuthal wavenumber In consequence it is very dimacr cult totell in dynamo simulations with hyperviscosity whether the azimuthal wavenumberis being determined by the magnetic shy eld or the hyperviscosity In this respect thethree-dimensional hyperdinotusive dynamos sunoter from the same disadvantage as the2 1

2-dimensional models in that the dominant azimuthal wavenumbers are selected

somewhat arbitrarilyAnother approach is the use of shy nite dinoterences in all three directions (see for

example Kageyama amp Sato 1997a) These methods became unfashionable becauseof dimacr culties in treating the regow near the rotation axis and in implementing thecorrect magnetic boundary conditions which are non-local However the develop-ment of fast massively parallel machines and the failure to resolve the Legendretransform problem (Lesur amp Gubbins 1999) has stimulated new interest in shy nite-dinoterence techniques which can take full advantage of parallel architecture Thepseudo-spectral technique requires a redistribution of the arrays amongst the dif-ferent processors at each time-step and this makes it a less attractive scheme forparallel architecture

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 875

angular velocity i relative to the mantle frame We denote the density deshy cit due tocompositional or thermal variations as C the codensity The codensity source whichdrives the convection is denoted by S This source leads to a codensity dinoterence centCacross the reguid outer core The distance between the surfaces d = ro ri = 2260 km istaken as the length-scale the time-scale is the magnetic dinotusion time d2= 80 kyrthe magnetic shy eld unit is (2 0 )1=2 20 Gauss (where is the reguid density isthe permeability and is the magnetic dinotusivity) and the codensity unit is centC Thegoverning MHD equations for the velocity u the magnetic shy eld B and the codensityC are

E

qPr

Du

Dt+ z u = rp+ j B + Er2u + qRaCr (21)

B

t= r2B + r (u B ) (22)

C

t= qr2C u rC + S (23)

r B = 0 (24)

r u = 0 (25)

Here j = r B is the current and the dimensionless parameters are the Ekmannumber E = =2 0d

2 the Prandtl number Pr = = the Roberts number q = =and the Modishy ed Rayleigh number R = g centTd=2 0 In these deshy nitions isthe kinematic viscosity and is the codensity dinotusivity Other combinations of theparameters are sometimes used notably the magnetic Prandtl number Prm = = =qPr and non-rotating Rayleigh number Ra = g centTd3= = R=E

In addition we have two further dimensionless parameters the radius ratio ri=rowhich is usually taken to be around 1=3 to model the geodynamo and the magneticdinotusivity ratio of the outer to the inner core o= i The most popular choice is

o= i = 1 which we call a conducting inner coreIn the conducting-inner-core case we need to solve for the magnetic shy eld B i and

the angular velocity i The inner-core magnetic shy eld is governed by

B

t+ i

B

=

i

or2B (26)

The mechanical boundary conditions are that there should be no regow through theICB and CMB surfaces and that there should be no-slip at these surfaces so u = 0at the CMB and u = iz r at the ICB Alternatively stress-free conditions canbe applied at the ICB CMB or both (Kuang amp Bloxham 1997)

Thermal boundary conditions must also be applied Popular choices are C constanton r = ri and r = ro or shy xed regux boundary conditions or some combination of thetwo Another common choice is to assume a uniform source in the outer core togetherwith shy xed codensity boundaries The non-magnetic problem with a shy xed codensity(usually heat) source has been extensively studied

The magnetic boundary conditions are that the normal component of B and thetangential components of the electric shy eld E are continuous The latter conditionscan easily be converted into conditions on the current using the MHD form of Ohmrsquoslaw Note that if stress-free rather than no-slip conditions are used at a conduct-ing boundary then the velocity on the boundary will enter the magnetic boundaryconditions through Ohmrsquos law

Phil Trans R Soc Lond A (2000)

876 C A Jones

The angular velocity i is determined by the torque balance at the ICB In generalthe rate of change of the angular momentum of the inner core is determined byadding the magnetic torque the viscous torque and any external torque such as agravitational torque Bunotett (1997) has argued that asymmetries in the mantle giverise to asymmetries in the shape of the inner core and that the resulting gravitationaltorque can be even larger than the magnetic torque In the stress-free case the viscoustorque is zero so the rate of change of angular momentum of the inner core is thendetermined solely by magnetic and external torque Estimates of the moment ofinertia of the inner core and the likely order of magnitude of the magnetic torquesuggest that the inner core can spin up due to magnetic torque in a few days onthe dynamo time-scale the inner core will therefore be in torque equilibrium If theviscous torque is small this means that the total magnetic and external torque mustbe zero any torque supplied to one region of the ICB must be balanced by an oppositetorque somewhere else so that the torque surface integral adds up to zero this hasbeen called the `inner-core Taylor constraintrsquo

With this scaling the dimensionless strength of the magnetic shy eld gives a localmeasure of the Elsasser number = B2=2 0 Although the Elsasser number isonly a derived quantity and not an input parameter it is useful for comparison withmagnetoconvection models where is an input parameter Similarly the dimension-less strength of the velocity shy eld gives a local measure of the magnetic Reynoldsnumber Rm = ud= Again this is a derived quantity but it is very useful to com-pare its value with the results of kinematic dynamo models where Rm is an inputparameter

3 Numerical methods

The most popular technique for solving the dynamo equations numerically has beenthe pseudo-spectral method recently described by Cox amp Matthews (1997) andHollerbach (2000) and applied to the geodynamo by Glatzmaier amp Roberts (1995)and Kuang amp Bloxham (1999) The vector shy elds u and B are expanded into toroidaland poloidal parts and these scalar shy elds together with the C shy eld are expanded inspherical harmonics In the radial direction either a shy nite dinoterence scheme or expan-sion in Chebyshev polynomials is used These expansions are inserted into (21)(25)and time-dependent equations for the coemacr cients are obtained The linear dinotu-sion operators are handled using an implicit scheme such as the CrankNicholsonmethod while the nonlinear terms are handled explicitly by converting from spectralspace to physical space so that the required multiplications can be handled easilyThis procedure leads to accurate solutions provided moderate parameter values areused it does however require large computing resources In addition to the obvi-ous problem associated with computing fully three-dimensional numerical solutionsthere are two additional problems which make the execution times even longer shy rstthere is no emacr cient fast Legendre transform as there is for Fourier transforms andsecond at small E the time-step has to be very small (see Walker et al (1998) fordetails)

Motivated by these dimacr culties a number of ways of reducing the CPU timerequired have been tried One method has been to truncate the equations severelyin the azimuthal direction sometimes leaving only the axisymmetric and one non-axisymmetric eim mode this is known as the 2 1

2-dimensional method This tech-

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 877

nique reduces the CPU time required by a large factor The disadvantage of thismethod is that the non-axisymmetric wavenumber m has to be chosen a priorirather than emerging from the calculation Nevertheless the axisymmetric shy eldsgenerated by the 2 1

2-dimensional method are remarkably close to those computed by

fully three-dimensional calculations provided the azimuthal wavenumber is chosensensibly in the limited parameter regimes that can be explored numerically choos-ing m as the shy rst mode to become linearly unstable as R is increased is normally anadequate choice The 21

2-dimensional approach is therefore a useful tool in exploring

dynamo behaviour but one can only have full conshy dence in the results if they arecalibrated by at least some fully three-dimensional simulations The simulations usedfor shy gures 2 and 3 below are a compromise between the 21

2-dimensional model and

fully three-dimensional simulations the r and directions are fully resolved but asmall number of non-axisymmetric modes (typically four) are present together withthe axisymmetric modes

Another popular technique has been the use of hyperdinotusion to reach lowerEkman number solutions Here the dinotusive operator r2 is replaced by some for-mula such as (1 + n3)r2 where n is the order of the spherical harmonic in theexpansion of the shy elds and is a constant This method is very simple to imple-ment if the pseudo-spectral method is used and it stabilizes the numerical schemefor values of 005 allowing small E to be reached with tolerable (but stillsmall) time-steps Signishy cant advances have been made using this scheme (see forexample Glatzmaier amp Roberts 1995 1997) but there is a serious price to pay Aspointed out by Zhang amp Jones (1997) hyperdinotusion damps out small azimuthalwavenumbers which can be preferred if the magnetic shy eld is weak To illustratethis we reproduce some of the Zhang amp Jones (1997) results in shy gure 1ac Inshy gure 1a we show the regow pattern at small E at the onset of non-magnetic con-vection note the large preferred m visible from the tight roll conshy guration andthe largely z-independent nature of the regow expected from the ProudmanTaylortheorem In shy gure 1b we see the result of the same calculation but with hypervis-cosity and no magnetic shy eld while in shy gure 1c we see the equivalent picture forthe case of an imposed azimuthal magnetic shy eld and no hyperviscosity Note thatthe hyperviscosity has produced a similar enotect to the magnetic shy eld in both casesreducing the dominant azimuthal wavenumber In consequence it is very dimacr cult totell in dynamo simulations with hyperviscosity whether the azimuthal wavenumberis being determined by the magnetic shy eld or the hyperviscosity In this respect thethree-dimensional hyperdinotusive dynamos sunoter from the same disadvantage as the2 1

2-dimensional models in that the dominant azimuthal wavenumbers are selected

somewhat arbitrarilyAnother approach is the use of shy nite dinoterences in all three directions (see for

example Kageyama amp Sato 1997a) These methods became unfashionable becauseof dimacr culties in treating the regow near the rotation axis and in implementing thecorrect magnetic boundary conditions which are non-local However the develop-ment of fast massively parallel machines and the failure to resolve the Legendretransform problem (Lesur amp Gubbins 1999) has stimulated new interest in shy nite-dinoterence techniques which can take full advantage of parallel architecture Thepseudo-spectral technique requires a redistribution of the arrays amongst the dif-ferent processors at each time-step and this makes it a less attractive scheme forparallel architecture

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

876 C A Jones

The angular velocity i is determined by the torque balance at the ICB In generalthe rate of change of the angular momentum of the inner core is determined byadding the magnetic torque the viscous torque and any external torque such as agravitational torque Bunotett (1997) has argued that asymmetries in the mantle giverise to asymmetries in the shape of the inner core and that the resulting gravitationaltorque can be even larger than the magnetic torque In the stress-free case the viscoustorque is zero so the rate of change of angular momentum of the inner core is thendetermined solely by magnetic and external torque Estimates of the moment ofinertia of the inner core and the likely order of magnitude of the magnetic torquesuggest that the inner core can spin up due to magnetic torque in a few days onthe dynamo time-scale the inner core will therefore be in torque equilibrium If theviscous torque is small this means that the total magnetic and external torque mustbe zero any torque supplied to one region of the ICB must be balanced by an oppositetorque somewhere else so that the torque surface integral adds up to zero this hasbeen called the `inner-core Taylor constraintrsquo

With this scaling the dimensionless strength of the magnetic shy eld gives a localmeasure of the Elsasser number = B2=2 0 Although the Elsasser number isonly a derived quantity and not an input parameter it is useful for comparison withmagnetoconvection models where is an input parameter Similarly the dimension-less strength of the velocity shy eld gives a local measure of the magnetic Reynoldsnumber Rm = ud= Again this is a derived quantity but it is very useful to com-pare its value with the results of kinematic dynamo models where Rm is an inputparameter

3 Numerical methods

The most popular technique for solving the dynamo equations numerically has beenthe pseudo-spectral method recently described by Cox amp Matthews (1997) andHollerbach (2000) and applied to the geodynamo by Glatzmaier amp Roberts (1995)and Kuang amp Bloxham (1999) The vector shy elds u and B are expanded into toroidaland poloidal parts and these scalar shy elds together with the C shy eld are expanded inspherical harmonics In the radial direction either a shy nite dinoterence scheme or expan-sion in Chebyshev polynomials is used These expansions are inserted into (21)(25)and time-dependent equations for the coemacr cients are obtained The linear dinotu-sion operators are handled using an implicit scheme such as the CrankNicholsonmethod while the nonlinear terms are handled explicitly by converting from spectralspace to physical space so that the required multiplications can be handled easilyThis procedure leads to accurate solutions provided moderate parameter values areused it does however require large computing resources In addition to the obvi-ous problem associated with computing fully three-dimensional numerical solutionsthere are two additional problems which make the execution times even longer shy rstthere is no emacr cient fast Legendre transform as there is for Fourier transforms andsecond at small E the time-step has to be very small (see Walker et al (1998) fordetails)

Motivated by these dimacr culties a number of ways of reducing the CPU timerequired have been tried One method has been to truncate the equations severelyin the azimuthal direction sometimes leaving only the axisymmetric and one non-axisymmetric eim mode this is known as the 2 1

2-dimensional method This tech-

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 877

nique reduces the CPU time required by a large factor The disadvantage of thismethod is that the non-axisymmetric wavenumber m has to be chosen a priorirather than emerging from the calculation Nevertheless the axisymmetric shy eldsgenerated by the 2 1

2-dimensional method are remarkably close to those computed by

fully three-dimensional calculations provided the azimuthal wavenumber is chosensensibly in the limited parameter regimes that can be explored numerically choos-ing m as the shy rst mode to become linearly unstable as R is increased is normally anadequate choice The 21

2-dimensional approach is therefore a useful tool in exploring

dynamo behaviour but one can only have full conshy dence in the results if they arecalibrated by at least some fully three-dimensional simulations The simulations usedfor shy gures 2 and 3 below are a compromise between the 21

2-dimensional model and

fully three-dimensional simulations the r and directions are fully resolved but asmall number of non-axisymmetric modes (typically four) are present together withthe axisymmetric modes

Another popular technique has been the use of hyperdinotusion to reach lowerEkman number solutions Here the dinotusive operator r2 is replaced by some for-mula such as (1 + n3)r2 where n is the order of the spherical harmonic in theexpansion of the shy elds and is a constant This method is very simple to imple-ment if the pseudo-spectral method is used and it stabilizes the numerical schemefor values of 005 allowing small E to be reached with tolerable (but stillsmall) time-steps Signishy cant advances have been made using this scheme (see forexample Glatzmaier amp Roberts 1995 1997) but there is a serious price to pay Aspointed out by Zhang amp Jones (1997) hyperdinotusion damps out small azimuthalwavenumbers which can be preferred if the magnetic shy eld is weak To illustratethis we reproduce some of the Zhang amp Jones (1997) results in shy gure 1ac Inshy gure 1a we show the regow pattern at small E at the onset of non-magnetic con-vection note the large preferred m visible from the tight roll conshy guration andthe largely z-independent nature of the regow expected from the ProudmanTaylortheorem In shy gure 1b we see the result of the same calculation but with hypervis-cosity and no magnetic shy eld while in shy gure 1c we see the equivalent picture forthe case of an imposed azimuthal magnetic shy eld and no hyperviscosity Note thatthe hyperviscosity has produced a similar enotect to the magnetic shy eld in both casesreducing the dominant azimuthal wavenumber In consequence it is very dimacr cult totell in dynamo simulations with hyperviscosity whether the azimuthal wavenumberis being determined by the magnetic shy eld or the hyperviscosity In this respect thethree-dimensional hyperdinotusive dynamos sunoter from the same disadvantage as the2 1

2-dimensional models in that the dominant azimuthal wavenumbers are selected

somewhat arbitrarilyAnother approach is the use of shy nite dinoterences in all three directions (see for

example Kageyama amp Sato 1997a) These methods became unfashionable becauseof dimacr culties in treating the regow near the rotation axis and in implementing thecorrect magnetic boundary conditions which are non-local However the develop-ment of fast massively parallel machines and the failure to resolve the Legendretransform problem (Lesur amp Gubbins 1999) has stimulated new interest in shy nite-dinoterence techniques which can take full advantage of parallel architecture Thepseudo-spectral technique requires a redistribution of the arrays amongst the dif-ferent processors at each time-step and this makes it a less attractive scheme forparallel architecture

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

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880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 877

nique reduces the CPU time required by a large factor The disadvantage of thismethod is that the non-axisymmetric wavenumber m has to be chosen a priorirather than emerging from the calculation Nevertheless the axisymmetric shy eldsgenerated by the 2 1

2-dimensional method are remarkably close to those computed by

fully three-dimensional calculations provided the azimuthal wavenumber is chosensensibly in the limited parameter regimes that can be explored numerically choos-ing m as the shy rst mode to become linearly unstable as R is increased is normally anadequate choice The 21

2-dimensional approach is therefore a useful tool in exploring

dynamo behaviour but one can only have full conshy dence in the results if they arecalibrated by at least some fully three-dimensional simulations The simulations usedfor shy gures 2 and 3 below are a compromise between the 21

2-dimensional model and

fully three-dimensional simulations the r and directions are fully resolved but asmall number of non-axisymmetric modes (typically four) are present together withthe axisymmetric modes

Another popular technique has been the use of hyperdinotusion to reach lowerEkman number solutions Here the dinotusive operator r2 is replaced by some for-mula such as (1 + n3)r2 where n is the order of the spherical harmonic in theexpansion of the shy elds and is a constant This method is very simple to imple-ment if the pseudo-spectral method is used and it stabilizes the numerical schemefor values of 005 allowing small E to be reached with tolerable (but stillsmall) time-steps Signishy cant advances have been made using this scheme (see forexample Glatzmaier amp Roberts 1995 1997) but there is a serious price to pay Aspointed out by Zhang amp Jones (1997) hyperdinotusion damps out small azimuthalwavenumbers which can be preferred if the magnetic shy eld is weak To illustratethis we reproduce some of the Zhang amp Jones (1997) results in shy gure 1ac Inshy gure 1a we show the regow pattern at small E at the onset of non-magnetic con-vection note the large preferred m visible from the tight roll conshy guration andthe largely z-independent nature of the regow expected from the ProudmanTaylortheorem In shy gure 1b we see the result of the same calculation but with hypervis-cosity and no magnetic shy eld while in shy gure 1c we see the equivalent picture forthe case of an imposed azimuthal magnetic shy eld and no hyperviscosity Note thatthe hyperviscosity has produced a similar enotect to the magnetic shy eld in both casesreducing the dominant azimuthal wavenumber In consequence it is very dimacr cult totell in dynamo simulations with hyperviscosity whether the azimuthal wavenumberis being determined by the magnetic shy eld or the hyperviscosity In this respect thethree-dimensional hyperdinotusive dynamos sunoter from the same disadvantage as the2 1

2-dimensional models in that the dominant azimuthal wavenumbers are selected

somewhat arbitrarilyAnother approach is the use of shy nite dinoterences in all three directions (see for

example Kageyama amp Sato 1997a) These methods became unfashionable becauseof dimacr culties in treating the regow near the rotation axis and in implementing thecorrect magnetic boundary conditions which are non-local However the develop-ment of fast massively parallel machines and the failure to resolve the Legendretransform problem (Lesur amp Gubbins 1999) has stimulated new interest in shy nite-dinoterence techniques which can take full advantage of parallel architecture Thepseudo-spectral technique requires a redistribution of the arrays amongst the dif-ferent processors at each time-step and this makes it a less attractive scheme forparallel architecture

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

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Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

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886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

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Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

878 C A Jones

(a) (b)

(c)

Figure 1 The pattern of convection with the preferred azimuthal wavenumber On the right-handside are contours of azimuthal velocity in a meridian plane on the left-hand side are contoursof azimuthal velocity in the equatorial plane (a) E = 2 10 5 with no magnetic macreld andno hyperviscosity (b) E = 2 10 5 with no magnetic macreld but with hyperviscosity = 01(c) E = 2 10 5 with no hyperviscosity but with magnetic macreld = 1

4 Numerical results in the BusseZhang regime

The dimensionless parameter values for the Earth are rather extreme (E 10 15

and q 10 5 see also x 6) and no numerical simulation has even approached themThere is however one region of parameter space where solutions can be found withlarge but not impossibly large computing resources This is the BusseZhang regime(Sarson et al 1998) so called after the pioneering work of Busse (1976) and Zhangamp Busse (1989) who shy rst found solutions in this region of parameter space Heredynamo action occurs at comparatively low Rayleigh number typically a few timesthe critical value for the onset of convection itself In this regime the velocities aretypically O(10) on the thermal dinotusion time-scale Since velocities have to be atleast of order O(102) on the magnetic dinotusion time-scale for dynamo action ie themagnetic Reynolds number has to be O(102) the value of q required for a successful

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 879

dynamo is O(10) It is possible to obtain dynamo action at lower values of q by raisingthe Rayleigh number in the range 10Rc lt R lt 100Rc velocities are typically O(102)on the thermal time-scale and then q O(1) is sumacr cient for dynamo action We callsolutions at q O(1) and large R GlatzmaierRoberts dynamos In principle BusseZhang dynamos can occur over a wide range of Ekman numbers down to very smallvalues in practice numerical instability makes it virtually impossible to go belowE 10 4 without using hyperviscosity and as mentioned above hyperviscosity doesnot allow us to capture the essential features of low Ekman number behaviour Mostpublished BusseZhang solutions therefore have E in the numerically accessible range10 4 lt E lt 10 3 However within this rather limited range of parameter space alarge number of calculations have been performed

In addition to the pioneering papers of Busse and Zhang referred to above recentfully three-dimensional simulations in this regime have been performed by Busse etal (1998) Sakuraba amp Kono (1999) Katayama et al (1999) Kageyama amp Sato(1997ac) Kitauchi amp Kida (1998) Kida amp Kitauchi (1998a b) and Olson et al (1999) (see also Christensen et al 1999) Fully 2 1

2-dimensional simulations in this

regime have been performed by Jones et al (1995) Sarson et al (1997a 1998) andMorrison amp Fearn (2000)

Although the processes involved in convection-driven dynamos are very complexthere are fortunately many points of similarity emerging from all this activity andsome physical understanding of the basic processes is beginning to emerge Figure 2shows a snapshot of some results of a simulation with limited resolution in thedirection (modes m = 0 2 4 6 and 8 are present) In shy gure 2a b a slice at constantz is shown (ie a plane parallel to the Equator) shy gure 2a showing the regow andshy gure 2b showing the magnetic shy eld The parameter values are R = 35 q = 10E = 10 3 and Pr = 1 model details are as in Sarson et al (1998) Figure 2c givesthe corresponding axisymmetric shy elds plotted in a meridional slice The pattern ofconvection found in this regime consists of columnar rolls which drift around in aprograde (eastward) direction There is generally a small dinoterential rotation butnot a dinoterential rotation strong enough to produce a powerful -enotect R is too lowto generate a strong thermal wind the regows are not large enough (unless Pr is small)for the nonlinear u ru Reynolds stress terms to generate dinoterential rotation andthe Lorentz force is not strong enough to drive substantial regows In consequence thedynamos are of the 2-type with toroidal and poloidal shy elds being generated locallyin the convection rolls themselves (Olson et al 1999) The toroidal and poloidalcomponents of magnetic shy eld are of broadly similar strength and the magnetic shy elddoes not appear to have a very powerful enotect on the convection in this regime AtE 10 3 the dominant m is typically around 4 while at E 10 4 the dominantm is in the range 68 depending on the model details in particular the codensitysource and the boundary conditions Fixed regux boundary conditions as used in theshy gure 2 calculations generally favour lower values of m The preferred number ofrolls is usually close to that expected from linear non-magnetic convection at thesame Ekman number (Christensen et al 1999) this is perhaps slightly surprisingas magnetoconvection studies indicate that lower values of m are preferred withimposed uniform magnetic shy elds when the Elsasser number is O(1) However themagnetic shy eld in these solutions is very far from uniform and is strongly inreguencedby regux expulsion it therefore appears that it is the convection which selects thewavenumber and the shy eld responds to this choice

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

880 C A Jones

- 5475 54750

U1426

Figure 2 A snapshot of a solution in the BusseZhang regime R = 35 q = 10 E = 10 3 Azimuthal wavenumbers m = 0 2 4 6 8 were present in the calculation (a) The velocity macreldin the plane z = 015 (where z = 0 is the Equator and z = 15 is the outer boundary North Polethe plot is viewed from above) with imposed dipolar symmetry The colours show the verticalvelocity uz (with colour intervals one-tenth of the maximum amplitude stated) the arrows thehorizontal velocity uh (the longest arrow corresponding to the maximum stated)

Hollerbach amp Jones (1993) noted that the dynamical behaviour outside a cylinderenclosing the inner core (the tangent cylinder) was rather dinoterent from that of thepolar regions inside the tangent cylinder The two polar regions are usually fairlyquiescent in the BusseZhang regime Convection is more easily excited outside thetangent cylinder where the convection rolls disturb the geostrophic constraint leastAt mildly supercritical R the polar regions are enotectively convectively stable

As noted by Busse (1976) the convection roll pattern is enotective at generatingmagnetic shy eld and magnetic Reynolds numbers of O(100) are all that is needed Itis necessary to have motion along the rolls as well as around the rolls for dynamoaction to occur There are a number of possible sources for this which contribute indinoterent proportions according to the model details The convection itself induces az-velocity antisymmetric about the equatorial plane which is signishy cant in all themodels With no-slip boundaries Ekman suction at the boundaries can also drive az-velocity thus enhancing the convective regow Finally if an inner core is present theinteraction of the rolls with the inner core near the tangent cylinder can generatez-velocity Particularly in models where the driving is strongest near the inner corethis can be a powerful enotect (see shy gure 2a)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 881

- 5272 52720

B38

Figure 2 (Cont) (b) As macrgure 2a but for magnetic macrelds Bz and Bh

Although the magnetic shy eld does not appear to have a striking enotect on the regow itmust have some enotect Since the nonlinear advection term in the equation of motiondoes not play a very important role the Lorentz force is the only signishy cant nonlinearterm Furthermore the induction equation is linear in B so the amplitude of themagnetic shy eld can only be determined by the Lorentz force limiting the regow in someway There appear to be a number of ways in which this limiting could take place(i) In an dynamo generating shy eld principally by a powerful dinoterential rotationthe Lorentz force will act to brake the dinoterential rotation and hence control thedynamo process The amplitude of the shy eld is then determined by the strength atwhich the forces producing dinoterential rotation are sumacr ciently opposed by magneticbraking However this does not seem a likely mechanism in the BusseZhang regimewhere dinoterential rotation does not appear to be very important A variation on thiswhich can apply also to 2 models is that the Lorentz force can generate a dinoter-ential rotation which disrupts the dynamo process thus bringing about equilibriumthe MalkusProctor scenario (Malkus amp Proctor 1975) Although it might well beimportant at very low E this mechanism does not seem to be very active in thenumerical simulations Another possibility (ii) mentioned by Olson et al (1999) isthat the magnetic shy eld directly reduces the convective velocity and hence reducesthe magnetic Reynolds number to its critical value In this mechanism the magneticbraking acts not on the axisymmetric dinoterential rotation but on each individualroll Possibility (iii) is that the magnetic shy eld acts to reduce the stretching propertiesof the regow ie magnetic shy eld turns the generating process from an emacr cient dynamointo a less emacr cient dynamo If this mechanism operates the mean kinetic energy isnot necessarily smaller with the saturated magnetic shy eld than it is with zero shy eld

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

882 C A Jones

T

B A r sin q

v r sin )) q

v r sin )) q

r sin qy

T

(contours are at max10)23477

474548091321439323712

A r sin q

r sin qy

B

Figure 2 (Cont) (c) A meridian slice of the axisymmetric components of various quantities Btoroidal (zonal) magnetic macreld Ar sin lines of force of poloidal (meridional) magnetic macreldC codensity v=(r sin ) the azimuthal angular velocity r sin streamlines of the meridionalcirculation Colour intervals are one-tenth of the maximum stated with purple correspondingto negative values In plots of zonal macrelds positive contours denote eastward-directed macreldsIn plots of meridional quantities the sense of macreld is clockwise around positive contours andanti-clockwise around negative contours

and the nonlinear magnetic Reynolds number could be quite dinoterent in dinoterentparameter regimes Since the stretching properties of a regow depend on rather subtlefeatures involving derivatives of the velocities themselves two regows which look super-shy cially rather similar can have very dinoterent dynamo action properties We shouldtherefore perhaps not be too surprised that the convection in the magnetically sat-urated regime looks pretty similar to that in the zero-shy eld regime The dinoterencesare in the stretching properties of the shy eld which do not show up well in a simplevelocity shy eld plot It is probably too early to say whether mechanism (ii) or (iii) isthe most important in the BusseZhang regime but we note from the data availablethat while there is some evidence that magnetic shy eld reduces mean kinetic energyand hence typical velocities the reduction is not generally very great while thereis a considerable variation in the magnetic Reynolds number of the saturated regowsuggesting mechanism (iii) may be the more important

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 883

While the inreguences of the magnetic shy eld on the regow are not generally large inthis regime as noted above it is of interest to try to analyse those which do occurKagayema amp Sato (1997c) noted that rolls can be divided into cyclonic and anticy-clonic rolls depending on whether the vorticity of the roll added to or subtractedfrom the planetary vorticity Anticyclonic rolls have the z-velocity positive in theNorthern Hemisphere and negative in the Southern Hemisphere The z-velocity istherefore divergent near the Equator and so by the continuity equation the horizon-tal regow in an anticyclonic roll converges near the Equator This horizontal convergingregow tends to accumulate magnetic shy eld near the centres of anticyclonic rolls in theequatorial region but in cyclonic rolls the horizontal regow near the Equator is diver-gent and so no shy eld accumulates there Note that in shy gure 2b the vertical magneticshy eld is strongest in the two anticyclonic rolls (clockwise regow) so the magnetic shy eldproduces a strong asymmetry between cyclonic and anticyclonic rolls Sakuraba ampKono (1999) note that the magnetic shy eld in the centres of anticyclonic rolls tends toexpand them so they become larger than the cyclonic rolls (note that the anticy-clonic vortices in shy gure 2a are indeed more pronounced than the cyclonic vortices)The average vorticity of the roll system is then negative leading to westward regownear the CMB and eastward regow near the inner core

5 Numerical results in the GlatzmaierRoberts regime

The other regime that has been studied is the higher Rayleigh number regime typi-cally R O(100Rc) Since the velocities here are larger compared with the thermaldinotusion time than in the BusseZhang regime one can anotord to reduce q to an O(1)value and still have Rm large enough for dynamo action Since at smaller values ofE convection is more organized reducing E allows larger R and so allows smaller qInterestingly increasing the Rayleigh number at shy xed Ekman number can actuallylead to a loss of dynamo action (Morrison amp Fearn 2000 Christensen et al 1999)The velocities may be larger but the more disorganized regow is less emacr cient as adynamo Christensen et al (1999) shy nd empirically qc 450E3=4 as the minimumpossible value of q for dynamo action on the basis of their numerical experiments

Some of the larger R calculations have been done with hyperviscosity so it is noteasy to estimate the enotective Ekman number in these cases Nevertheless the mainfeatures of these higher R dynamos have been found in a range of models and areprobably robust

In shy gure 3 we show the same constant z slice and meridional slice as in shy gure 2but now the Rayleigh number is much larger and the Ekman and Roberts numbersare reduced by a factor of 10 the parameters are R = 45 000 E = 10 4 but withsome hyperdinotusivity and q = 1 The most important feature is that the dinoterentialrotation v=r sin becomes much stronger in this regime (cf shy gures 2c and 3c) Theconvection is more chaotic and the rolls lose their clear-cut identity typically lastingfor only a short time Nevertheless the convection is highly emacr cient in the sensethat the temperature gradients are conshy ned to thin boundary layers and the bulkof the region outside the tangent cylinder and outside of these boundary layers isnearly isothermal note the much thinner boundary layers of C in shy gure 3c than inshy gure 2c Because the surface area of the outer CMB boundary layer is so muchgreater than that of the ICB boundary this isothermal region is at a temperaturemuch closer to that of the CMB than the ICB ie it is relatively cool Convection

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

884 C A Jones

- 959 959

27532

0

U

Figure 3 A snapshot of a solution in the GlatzmaierRoberts regime R = 45 000 q = 1E = 10 4 with hyperviscosity = 005 Azimuthal wavenumbers m = 0 2 4 6 8 were presentin the calculation No symmetry about the Equator was imposed (a)(c) are as in macrgure 2

also occurs now in the polar regions inside the tangent cylinder here the geometryis such that the mean temperature of the reguid is closer to the average of the ICBand CMB temperatures much hotter than that of the bulk of the reguid outside thetangent cylinder In consequence there are strong temperature gradients in thedirection so that there is a thermal wind

v

z=qR

r

T

(51)

which is large since T= is O(1) and R is large in this regime In the NorthernHemisphere T= must be negative for the reasons given above and so v=z isnegative in the Northern Hemisphere so there is a strong prograde (eastward) regownear the ICB This regow is then coupled to the ICB itself by viscous and magnetictorques (Aurnou et al 1996) and so the inner core has to rotate rapidly eastwardin this regime if gravitational torques are neglected The pattern of the dinoterentialrotation (see shy gure 3c) is a very constant feature of models in this regime and itleads to the generation of strong toroidal shy elds in the regions of high shear near theICB (see shy gure 3b) The helicity in this regime is produced mainly by the interactionof convection with the inner core and so poloidal shy eld also is strongest near the innercore The GlatzmaierRoberts regime dynamos are therefore more like dynamoswith dinoterential rotation playing an important role in the production of toroidalshy eld It should be noted however that the peak poloidal shy eld strength is not thatmuch smaller than the peak toroidal shy eld strength

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 885

- 4998 4998

134

0

B

Figure 3 (Cont) (b)

Another key dinoterence from the BusseZhang model is that the polar regionsinside the tangent cylinder are now very active convecting vigorously and generatingstrong magnetic shy elds there This polar convection changes the form of the codensitycontours which in turn change the form of the thermal wind given by (51) Thedinoterential rotation in the BusseZhang models is therefore not only much weaker(because the thermal wind driving is proportional to ) but has a dinoterent char-acteristic form (compare shy gure 2 with shy gure 3 ) In the BusseZhang models theform of the dinoterential rotation is such that there is no strongly preferred directionof rotation of the inner core in contrast to the GlatzmaierRoberts models

Another characteristic feature of the GlatzmaierRoberts regime is the presenceof a strong axisymmetric meridional circulation ( sin in shy gure 3 ) in the polarregions inside the tangent cylinder This circulation rises out from the ICB at thepoles and falls back towards the ICB along the tangent cylinder The regow returnsfrom the tangent cylinder back to the poles along a thin Ekman boundary layernear the ICB which controls the strength of the meridional circulation A simplecalculation of the Ekman-layer regow indicates that the strength of the meridionalcirculation as measured by the maximum value of on the magnetic dinotusion time-scale gives

m ax1=2

This will be small in the BusseZhang regime since there is small and is notlarge In the GlatzmaierRoberts regime cannot be made very small for stabilityreasons but is quite large so a strong meridional circulation is found This merid-ional circulation does not seem to be strongly anotected by magnetic shy eld in the regime

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

886 C A Jones

y

C

B A r sin q

qr siny

C

(contours are at max10)171077

898436520219602123

176388

r sin q

B v r sin )) q

v r sin )) q

v r sin )) q

Figure 3 (Cont) (c)

in which these models are run Considerations such as these indicate how importantit is to shy nd out more about the parameter regime the geodynamo is operating inEveryone knows that in the geodynamo R is large and E is small but is the keyparameter RE1=2 large small or O(1) If the meridional circulation is signishy cant itmay have very important consequences for the dynamo in particular for the reversalmechanism (Sarson amp Jones 1999 Glatzmaier et al 1999)

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 887

6 The geodynamo parameter regime

Since it is clear that there are a number of dinoterent types of dynamo that can operateat low Ekman number we need to consider the geophysical parameters carefully tosee which types if any are connected with the geodynamo It is also possible thatqualitative dinoterences occur between the behaviour at the moderately small Ekmannumbers used in the simulations and the extremely low Ekman number found in theEarthrsquos core

Although it is likely that turbulence plays an important part in core dynamics(Braginsky amp Meytlis 1990) we start by considering the parameter values based onmolecular values (see for example Braginsky amp Roberts 1995) We take the mag-netic dinotusivity of the core as 2 m2 s 1 which is equivalent to a conductivity of4 105 S m 1 The thermal dinotusivity is 10 5 m2 s 1 and the viscous dinotusivityis near 10 6 m2 s 1 (DeWijs et al 1998) Based on these molecular dinotusivi-ties q 5 10 5 and Pr 01 The situation is complicated further because in thecore thermal convection is not the only driving mechanism compositional convectioncan occur and may indeed be more important The molecular value of the materialdinotusivity of light material in the core is even smaller 10 8 m2 s 1 The equa-tion governing the concentration of light material the codensity equation (23) isthe same as the temperature equation but the appropriate boundary conditions aredinoterent The radius of the outer core is ro = 348 106 m the radius of the innercore ri = 122 106 m and 0 = 73 10 5 s 1 The molecular value of the Ekmannumber is then 10 15

Estimating the molecular value of the Rayleigh number is more dimacr cult thereare two Rayleigh numbers to consider one for the thermal part of the convectionand one for the compositional part We consider the thermal Rayleigh number shy rstSince it is the heat regux which is shy xed by mantle convection the problem is slightlydinoterent from the classical RayleighBenard problem A further dimacr culty is that theheat regux coming out of the core is not well known This heat regux can be divided intotwo parts that carried by conduction down the adiabatic gradient and that carriedby convection down the superadiabatic gradient It is only the convective part thatis of relevance to driving the outer core this total convective heat regowing out ofthe core is unlikely to exceed ca 5 1012 W and we use this as our estimate forFcon v 3 10 2 W m 2 at the CMB It is possible that most of the heat regux iscarried down the adiabatic gradient by conduction and Fcon v is much lower thanthis (Nusselt number close to unity)

A regux Rayleigh number can be deshy ned by evaluating the (notional) superadiabatictemperature gradient that would be required to carry this convective part of the heatregux out by conduction giving

Raf =g d4Fcon v

2 c p1029

taking 104 kg m 3 c p 8 102 J kg 1 K 1 10 5 K 1 and g 10 m s 2

(Braginsky amp Roberts 1995) The corresponding modishy ed regux Rayleigh number is

Rf =g d2Fcon v

2 2 c p

1014

The regux Rayleigh number is appropriate for considering the onset of convection butin strongly supercritical convection the superadiabatic temperature gradient will be

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

888 C A Jones

greatly reduced from the value

=Fcon v

c p

which it has near onset In RayleighBacuteenard convection the main enotect of increasingRa is to increase the heat regux for a given temperature dinoterence between the bound-aries in shy xed regux convection the main enotect is to decrease the temperature dinoterencefor a given heat regux If we want to use the results of RayleighBacuteenard convection atshy xed temperature (where there has been more previous work) it is more appropriateto estimate the Rayleigh number by estimating the typical temperature reguctuationand using this as an estimate of the superadiabatic temperature dinoterence betweenthe ICB and CMB centT The convective heat regux is

Fcon v =

Z

S

cp urT ds (61)

We must assume that the integral (61) can be approximated by taking typical valuesof ur and centT this assumes that the radial velocity and temperature reguctuations arewell correlated but we would expect this in a convection-driven reguid We must alsoassume that the temperature variations are fairly uniform across the interior of thereguid outer core and not concentrated into thin plumes In the neighbourhood of theinner core we then estimate

4 r2i c p urcentT 5 1012 W (62)

and then centT 10 4 K slightly less than the estimate of centT 10 3 K of Braginskyamp Roberts (1995) This seems a tiny temperature dinoterence compared with the actualtemperature dinoterence between the ICB and CMB (ca 1300 K) but it reregects thefact that very large convected heat reguxes can be carried by very small superadiabaticgradients just as they are in stars A similar estimate for centT can be obtained byassuming that the magnitude of the buoyancy g centT is similar to the magnitude ofthe Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1

The modishy ed Bacuteenard Rayleigh number is then

RB =g centTd

215 107

The non-rotating Rayleigh number is

RaB = R=E 15 1022

The Rayleigh number for compositional convection can be evaluated in a similar wayprovided the mass regux is assumed which is more or less equivalent to assuming theage of the inner core is known Taking the density dinoterence between the inner andouter core as 06 103 kg m 3 and assuming the rate of growth of the inner core hasbeen constant over the last 45 109 years then the rate of mass increase inside asphere of shy xed radius r gt ri is 32 104 kg s 1 Assuming most of this mass regux iscarried out by advection rather than dinotusion then

32 104 kg s 1

Z

S

Cur ds

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 889

from (23) Again taking centC as the typical dinoterence in C between upward anddownward moving reguid gives centC 5 10 6 kg m 3 giving the modishy ed Rayleighnumber RB = gcentCd=2 0 1010 the fact that the compositional Rayleigh num-ber is larger than the thermal Rayleigh number is due to being considerablysmaller than

(a) The dynamo catastrophe

A number of attempts to achieve a low-Ekman-number convection-driven dynamousing the magnetostrophic equations in the interior together with Ekman boundarylayers have been made All have so far failed due to what might be termed the`dynamo catastrophersquo

The attempts all started with the idea that convection at low Ekman numberbut O(1) Elsasser number need not occur on the very small azimuthal and radiallength-scales that occur in the non-magnetic problem Indeed O(1) length-scalesare the preferred linear mode when there is an imposed (axisymmetric) magneticshy eld Since large-scale convection has been found in all simulations to give rise todynamo action the hope was that the large-scale convection could generate thelarge-scale shy eld required and hence maintain a self-consistent large-scale dynamoThe problem with this scenario is that the magnetic shy eld needs to be sumacr cientlystrong everywhere and at all times to sustain convection on a large length-scale(Zhang amp Gubbins 2000) If due to reguctuations the shy eld becomes small over asignishy cant domain for a signishy cant period (eg at a reversal) the length-scale ofconvection becomes short and the dynamo fails The magnetic shy eld then starts todecay and so small-scale convection takes over everywhere permanently A dynamocatastrophe has occurred and the shy eld can never recover

We might ask why the simulations referred to above do not show this catastrophicfailure The simulations that have been performed to date have not been in theregime where the magnetic shy eld is playing a major role in selecting the preferredwavenumber indeed as remarked above in most simulations the preferred azimuthalwavenumber is not very dinoterent to that expected with non-magnetic linear convec-tion None of the simulations have yet reached the very low Ekman regime wherethe dynamo catastrophe occurs

The dynamo catastrophe can be understood in terms of the various bifurcationsthat occur in convection-driven dynamos As the Rayleigh number is increased theremay be various non-magnetic bifurcations in the convective regow but eventually atlarge enough magnetic Reynolds number there is likely to be a critical value ofthe Rayleigh number Rc at which dynamo action occurs this bifurcation may besupercritical or subcritical If it is supercritical then a shy nite-amplitude magnetic-shy eld state is established for values of R gt Rc This state is generally known as theweak-shy eld dynamo regime (see for example Roberts amp Soward 1992) and is char-acterized by the small wavelengths in the direction perpendicular to the rotationaxis This state was analysed in the plane layer dynamo case by Childress amp Soward(1972) They found that the weak-shy eld regime only existed for a fairly restrictedrange of Rayleigh number because the magnetic shy eld enhanced convection eventu-ally leading to runaway growth the weak-shy eld regime therefore terminates at someRrway so the weak-shy eld regime only exists for Rc lt R lt Rrway Childress amp Soward(1972) conjectured that a subcritical branch of strong-shy eld solutions also exists with

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

890 C A Jones

wavenumbers of O(1) and with magnetic-shy eld strength such that the Elsasser num-ber is O(1) and that when R gt Rrway the solution jumps onto this branch Thisstrong-shy eld regime is likely to exist for Rayleigh numbers much less even than Rcalthough for the numerical reasons given above this has not yet been unambiguouslydemonstrated If the Rayleigh number is decreased on the strong-shy eld branch theremust come a point at which the magnetic Reynolds number is too small to supportdynamo action say R = R s and below this no magnetic shy eld can be sustained Thedynamo catastrophe arises when R s Rc the likely situation in the geodynamoand when the actual Rayleigh number is in the range R s lt R lt Rc `safersquo dynamosare those where R gt Rc

In subcritical dynamos there are at least two stable attracting states one with mag-netic shy eld and one without In most of the simulations the state with the magneticshy eld is chaotic so it is subject to considerable reguctuation It is possible to wanderon a chaotic attractor without ever falling into an attracting shy xed point an exampleis the Lorentz attractor where for some parameter values there are two attractingshy xed points but some solutions wind around these attracting shy xed points withoutever falling into their basin of attraction The geodynamo could perhaps behave ina similar fashion though other planets which no longer have internal dynamos mayhave sunotered a dynamo catastrophe in the past Another issue that arises with sub-critical models of the geodynamo is how the magnetic shy eld started in the shy rst placeconvection can only occur in a core at a temperature well above the Curie pointeven if the geodynamo is currently subcritical it must have been supercritical sometime in its past

It is of interest to estimate the parameter regime where we expect the dynamocatastrophe to occur and conversely to shy nd where `safersquo (supercritical) dynamosie those where magnetic shy eld can be regenerated from the non-magnetic convect-ing state are located At Prandtl numbers of O(1) non-magnetic convection in arapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3 and asimilar radial length-scale The planform will therefore be of convection roll typenot dissimilar to that found in the BusseZhang models but with a much smallerradial and azimuthal length-scale

Since the asymptotic theory of linear convection in a rapidly rotating sphere (orspherical shell) at small E has now been worked out (Jones et al 2000) at least inthe case of uniform heating we can apply this theory to see what kind of convectionwe expect at these extreme parameter values At Pr = 01 the critical modishy edRayleigh number for a radius ratio of 035 is 027 E 1=3 which is Rc = 27 104

at E = 10 15 The thermal Rayleigh number is therefore supercritical R=Rc 500but perhaps surprisingly not enormously so The critical azimuthal wavenumber ism = 024E 1=3 which gives m 24 104 If this many rolls have to shy t into theEarthrsquos core each roll would have a diameter of ca 04 km Linear theory shows thatat onset the rolls only occur at a specishy c distance (about 07r0) from the rotationaxis but fully nonlinear convection rolls would probably shy ll the whole reguid core

(b) The minimum roll size for dynamo action

If we (temporarily) accept that nonlinear non-magnetic convection in the rapidlyrotating spherical shell takes the form of rolls of typical diameter somewhat less than1 km with typical velocity u 3 10 4 m2 s 1 could this regow be a dynamo The

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 891

kinematic dynamo problem for a related type of regow

u =p

2 sin cosp

2 cos sin 2 sin sin (63)

has recently been considered by Tilgner (1997) The regow has the form (63) inside acylinder of height and radius and is zero outside it The diameter of each roll is so that is approximately 2 where is the number of rolls that shy t across thediameter of the cylinder The length-scale for the regow is now so the local magneticReynolds number is 0

m = If the mean shy eld strength is the reguctuations

produced by the regow will be 0 0m The large-scale EMF is then u b0 which

will maintain the current on the long length-scale provided 0m or more

precisely

0m m = ( ) 2

m2m c (64)

for some critical m c If and remain shy xed as decreases m must increaseas m ( ) 1=2 to maintain dynamo action Tilgner (1997) shy nds that with alattice of size = 8 rolls across a diameter m must exceed m c 15 and thatfor larger m c increases as 1=2 as expected from (64) We therefore havefrom Tilgnerrsquos calculation that m c 15( 8)1=2 cells across a diameter willcorrespond roughly to cells round the circumference and so for convecting rollswith azimuthal wavenumber 1

2 giving m c 7 1=2 for large It

should be noted that the velocity shy eld here is independent of in convecting rolls

z is antisymmetric in and the dynamo will be less emacr cient Data from the dynamocalculations of Christensen et al (1999) suggest that the critical magnetic Reynoldsnumber is at least a factor of three greater in convection cells than in the Tilgnercells so we estimate

m c 20 1=2 large (65)

as the critical magnetic Reynolds number at large azimuthal wavenumber Note thatwe are assuming here that the Rayleigh number is large enough for the convectionrolls to enotectively shy ll most of the outer core for this estimate Taking 3 10 4 m s 1

as our standard velocity for the core and = 2 m2 s 1 for the magnetic dinotusivity

m 500 for the Earthrsquos core Using our estimate (65) we then get 600 as themaximum possible value of at which we can expect dynamo action Although thisis a large value of it is a lot less than the value expected at 10 15 whichwas 24 000 The value of at which = 600 is preferred at = 0 1 is around6 10 11

The outcome of our estimates is therefore that `safersquo (supercritical) geodynamoscould exist at Ekman numbers down to about 10 10 but at Ekman numbersbelow 10 10 the preferred roll size for non-magnetic convection is too small to givea viable dynamo For Ekman numbers below 10 10 the dynamo must be sub-critical

(c) Convective velocities

So far our estimates have all been based on the value of 3 10 4 m s 1 for theconvective velocity Would convection in this parameter regime actually give rise

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

892 C A Jones

to velocities of 3 10 4 m s 1 If our underlying physical assumptions are correctthen the answer has to be yes or else our picture is not self-consistent It might beargued that we already know the answer is yes because the velocities coming out ofthe simulations are of the right order of magnitude the BusseZhang regime veloc-ities are perhaps a little low and the GlatzmaierRoberts regime velocities a littlehigh but given the uncertainties involved in estimating and in interpreting secularvariation data to estimate u this is not a cause for concern However this reallybegs the question because the simulations assume turbulent values of the thermalcompositional and viscous dinotusion which are tuned to give the right answers

The usual velocity scale for mildly supercritical convection is the thermal velocityscale =d 3 10 12 m s 1 eight orders of magnitude too small The velocityscale for compositional convection is a further three orders smaller Clearly for aconvection-based geodynamo theory to work we have to explain why the observedvelocities are so large compared with the thermal velocity scale Weakly nonlineartheory suggests that the velocities scale with (R=Rc)

1=2 and Zhang (1991) gives

u 25[(R Rc)=Rc]1=2 =d

for inshy nite Prandtl number convection With the Bacuteenard estimates for R and Rc theexpected velocity goes up to 2 10 9 m s 1 still far too small Using compositionalconvection estimates reduces the expected velocity

One possibility is that the magnetic shy eld enhances the velocity but even if thecritical Rayleigh number is reduced from 104 down to unity by magnetic shy eld theZhang formula still gives velocities three orders of magnitude too small We need animprovement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1

Unfortunately not much is known about nonlinear convection at very low Ekmannumber Probably the best available theory is that of Bassom amp Zhang (1994) whichis however based on non-magnetic Boussinesq convection in a plane layer not withspherical geometry Ignoring some logarithmic terms of the Rayleigh number theygive the vertical velocity for Rayleigh numbers well above critical as

uz 03R =d (66)

which gives uz 10 5 m s 1 which is getting closer to the required velocity Consid-erable caution is needed in interpreting this result though because not only could thespherical geometry make a signishy cant dinoterence at large R the regow will almost cer-tainly be unstable and this may reduce the velocities achieved Nevertheless it doessuggest that because of the unusual nonlinear dynamics of rapidly rotating regows thevery large velocities (in terms of the thermal dinotusion velocity scale) might be expli-cable It is perhaps worth noting that the BassomZhang theory gives an unusuallystrong dependence of typical velocity and Nusselt number on Rayleigh number withu R and Nu R (again ignoring logarithmic terms) Weakly nonlinear theorygives only u R1=2 as mentioned above Boundary-layer theories of non-rotatingnon-magnetic convection have dependencies which are anotected by the precise natureof the boundary conditions but Nusselt number Nu R1=3 and u R2=3 areperhaps typical

(d ) The ereg ect of turbulence in the core

If our understanding of the enotect of magnetic shy elds on strongly nonlinear low-Econvection is limited the situation as regards turbulence in the core is even more

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 893

uncertain Stevenson (1979) and Braginsky amp Meytlis (1990) have considered theproblem and made tentative mainly qualitative estimates of the enotect of turbulenceIt is generally believed that turbulence is important in the Earthrsquos core and willenhance the dinotusive processes This can signishy cantly anotect behaviour for exampleif turbulent dinotusion can enhance the enotective value of E to greater than 10 10

the dynamo could be changed from subcritical to supercritical Unfortunately asstressed by Braginsky amp Meytlis (1990) and Braginsky amp Roberts (1995) turbulencein the extreme conditions in the core is quite unlike `normalrsquo turbulence In normalturbulence the nonlinear advection terms u ru in the momentum equation playa key role but these terms are small in the Earthrsquos core High Prandtl numberBenard convection at large Rayleigh number is an example of a system which sharesthis property and is well studied the analogy is imperfect though because in thecore Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Prconvection

Braginsky amp Meytlis (1990) consider turbulence generated by local convection withstrong rotation and azimuthal magnetic shy eld They note that the fastest growingmodes are plate-like cells with the short radial length-scale expected from non-magnetic convection but with the azimuthal length-scale increased by the azimuthalmagnetic shy eld The duration of the cells is assumed to be the growth rate of theconvective instability ca 6 yr 1 and their size in the azimuthal and z-directions isthen determined to be ca 50 km from the usual estimate of the velocity The radiallength-scale is much smaller at ca 2 km The resulting turbulent dinotusivities arethen ca 1 m2 s 1 in the azimuthal and z-directions and much smaller in the radialdirection The axis of the plate-like cells will be in the direction perpendicular to therotation axis and the magnetic shy eld so unless the azimuthal shy eld dominates othercomponents the anisotropic dinotusion tensor will be a strong function of position

Another dimacr culty is that a local theory such as the BraginskyMeytlis theory isunlikely to be valid near the ICB and the CMB but it is the boundary layers thatcontrol high Rayleigh number convection

(e) Taylorrsquos constraint

Taylor (1963) observed that if the magnetic shy eld in the Earthrsquos core was in mag-netostrophic balance (Coriolis Lorentz and pressure forces with viscous and inertialforces negligible) the magnetic shy eld has to satisfy the constraint

Z

S

(j B ) dS = 0 (67)

where S is any cylinder concentric with the rotation axis A large literature hasdeveloped concerning nonlinear dynamo models and whether or not they sat-isfy Taylorrsquos constraint recently reviewed by Fearn (1998) It is natural to enquirewhether the convection-driven geodynamo models satisfy Taylorrsquos constraint Sarsonamp Jones (1999) analysed dynamos in the GlatzmaierRoberts regime with low head-line E but with hyperviscosity They found that Taylorrsquos constraint was not well sat-isshy ed suggesting that viscous enotects are playing an important part in these modelsThis is not perhaps surprising as we know the current generation of dynamo mod-els is still viscously controlled rather than magnetically controlled As E is reducedfrom the currently available values of 10 4 initially we expect the roll diameter to

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

894 C A Jones

decrease so that viscous forces can still control the convection and the magnetic-shy eld scale to decrease similarly These small-scale shy elds need not obey Taylorrsquos con-straint since on that scale Lorentz forces can be balanced by viscous forces Howeveras E reduces further magnetic dinotusion will prevent the magnetic-shy eld scales get-ting smaller while viscous forces can only be signishy cant on very small length-scalesunless large velocities occur At these values of E which are considerably smallerthan those currently attainable the dynamo must choose between satisfying Tay-lorrsquos constraint (the MalkusProctor scenario) or maintaining a high-velocity zonalwind with which viscosity acting through the Ekman boundary layer can balance theLorentz torque (Braginskyrsquos model-Z scenario) The current generation of models hasnot yet reached E sumacr ciently small to give any indication into which of these twoscenarios convection-driven dynamos will jump

7 Conclusions

The current generation of numerical dynamos has been remarkably successful inbringing theoretical dynamo models into contact with geophysical reality As a resultof a concerted international enotort we are beginning to understand the physical mech-anisms operating in these models so that we can analyse why they behave as theydo We are also beginning to understand how the models vary as functions of theparameters within the numerically accessible range Fundamentally dinoterent regimesare found depending on how large the Rayleigh number is compared with the Ekmannumber As computer technology advances there is every prospect that our under-standing of the behaviour at moderate R and E will be enhanced

It is clear however that the parameter regime operating in the Earth is far beyondthat which can be simulated directly The only way open to us to bridge the gapin the parameter space is by developing our understanding of the physical processesfurther so that we can work out the asymptotic laws that govern these processesand hence extrapolate to the geophysical parameter range At present we do notknow enough about the basic dynamics of rapidly rotating convection in sphericalgeometry in the presence of magnetic shy eld for this to be possible Some of the keyquestions we would like to answer are as follows

(i) What is the dominant length-scale of convection rolls in the core Is the domi-nant transverse length-scale of the order of the core radius with perhaps m = 2as suggested by some magnetic observations (Gubbins amp Bloxham 1987) secu-lar variation studies (Bloxham amp Jackson 1991) and with some theoretical sup-port (Longbottom et al 1995) or is the transverse length-scale much smallerThe current models show very little evidence of the magnetic shy eld increasingthe transverse length-scales above those predicted by non-magnetic low-E con-vection so maybe the strongly non-axisymmetric magnetic shy eld plays a moresubtle role in the pattern selection process than is at present understood

From our studies we have a minimum roll diameter of ca 15 km below whicha dynamo cannot operate the actual roll diameter cannot be too small or elseOhmic dissipation would be too large but we still have a large range of uncer-tainty Unfortunately small-scale magnetic shy elds at the CMB cannot easily bedetected at the Earthrsquos surface because their inreguence declines rapidly withdistance

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

Convection-driven geodynamo models 895

(ii) How do the smaller scales of motion interact with the larger scales In par-ticular if the enotective eddy dinotusivity is anisotropic how will that anotect thelarge-scale behaviour A related question is what is the nature of the thermaland compositional boundary layers and do they control the convection

(iii) What is the enotect of the magnetic shy eld on the convection The current modelsshow only a rather weak enotect less than would have been expected on thebasis of magnetoconvection calculations with axisymmetric shy elds However itis hard to believe that the shy eld will not have a major impact at low E thequestion is perhaps when it kicks in

These questions will not be easy to answer but with a combination of processstudies of low E magnetoconvection laboratory experiments and detailed comparisonof dynamo simulation outputs with geophysical observation hopefully progress willbe made

I thank Dr Graeme Sarson for many discussions and in particular for computing the solutionsshown in macrgures 2 and 3 and Professor K Zhang for computing the solutions shown in macrgure 1and for many discussions in which our ideas about the geodynamo emerged This work wassupported by the UK PPARC grant GRK06495

References

Aurnou J Brito D amp Olson P 1996 Mechanics of inner core super-rotation Geophys ResLett 23 34013404

Bassom A P amp Zhang K 1994 Strongly nonlinear convection cells in a rapidly rotating deg uidlayer Geophys Astrophys Fluid Dynam 76 223238

Bloxham J amp Jackson A 1991 Fluid deg ow near the surface of the Earthrsquo s outer core RevGeophys 29 97120

Braginsky S I 1994 The nonlinear dynamo and model-Z Lectures on solar and planetarydynamos (ed M R E Proctor amp A D Gilbert) pp 267304 Cambridge University Press

Braginsky S I amp Meytlis V P 1990 Local turbulence in the Earthrsquo s core Geophys AstrophysFluid Dynam 55 7187

Braginsky S I amp Roberts P H 1995 Equations governing convection in Earthrsquo s core and thegeodynamo Geophys Astrophys Fluid Dyn 79 197

Bureg ett B A 1997 Geodynamic estimates of the viscosity of the Earthrsquo s inner core Nature 388571573

Busse F H 1976 Generation of planetary magnetism by convection Phys Earth Planet Inte-riors 11 241268

Busse F H Grote E amp Tilgner A 1998 On convection driven dynamos in rotating sphericalshells Studia Geoph Geod 42 16

Childress S amp Soward A M 1972 Convection driven hydromagnetic dynamo Phys Rev Lett29 837839

Christensen U Olson P amp Glatzmaier G A 1999 Numerical modelling of the geodynamo asystematic parameter study Geophys Jl Int 138 393409

Cox S M amp Matthews P C 1997 A pseudo-spectral code for convection with an analyti-calnumerical implementation of horizontal boundary conditions Int Jl Numer Meth Fluids25 151166

DeWijs G A Kresse G Vocadlo L Dobson D Alfe D Gillan M J amp Price G D1998 The viscosity of liquid iron at the physical conditions of the Earthrsquo s core Nature 392805807

Phil Trans R Soc Lond A (2000)

896 C A Jones

Fearn D R 1998 Hydromagnetic deg ow in planetary cores Rep Prog Phys 61 175235

Glatzmaier G A amp Roberts P H 1995 A three-dimensional convective dynamo solution withrotating and macrnitely conducting inner core and mantle Phys Earth Planet Interiors 916375

Glatzmaier G A amp Roberts P H 1997 Simulating the geodynamo Contemp Phys 38 269288

Glatzmaier G A Coe R S Hongre L amp Roberts P H 1999 The role of the Earthrsquo s mantlein controlling the frequency of geomagnetic reversals Nature 401 885890

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