H.K. Moffatt- Cosmic Dynamos: From Alpha to Omega

8/3/2019 H.K. Moffatt- Cosmic Dynamos: From Alpha to Omega

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ets like the Earth. I n the absence of electromotiveand the magnetic fields they generate, willturn be

What m aintains the Ea rth's magnetic f ield? Why does the solarmagnetic field follow a 22-year cycle? How can we explain thesignificant magnetic fields on planets like Jupiter a nd Saturn?Dynam o theory is starting to answer these questions

indicate? Why does the dominant ingredient of the field,the dipole, oscillate with periods - 103-104 years? Andwhy does the North-South polarity of the geomagneticfield randomly reverse at intervals - 106years?

Cosmic dynamos:

104 years. (The decay time, T, for currents in ar, and resistivity, q, is T-?/rr'q, Therx3500 km and q x 3 m's-l,

other than through a dynamo mechanism.One of the most remarkable features of dynamo theory isthat, despite the extreme disparity of physical conditions inthe Earth and Sun, the same basic principles arepotentially capable of explaining the observed magnetic

A fundamental challenge in

tically convert the kinetice r g y of fluid motion into

of currents, and

stivity of the fluid. Such aThe problem appears most

a magneticS o u r c e Of theis a is conserved if the effects of resistivity are negligible

ep interior of the Earth.

1 Magnetic lines of force, "6-lin es", are convected by the fluid flow, U,we know that the as if they were elastic strings embedded in the fluid. As the "strings"are stretched, the magnetic field becomes stronger. The magnetic fluxthrough a small closed loop, C, moving with the flow is cD=ABcosO. Flux

Satellite observations andmagnetographs reveal thatthe Sun also has a complexand ever-changing magneticfield, and that this is stronglycorrelated with the granula-tion pattern of upwellingsand downwellings on thesolar surface. The naturaldecay time for the solarmagnetic field is much long-er than for the geomagneticfield (the Sun has a muchlarger radius and a smallerresistivity). In fact this decaytime is of the order of the ageof the Sun ( - 4 x 109years).There is not, therefore, thesame pressing need for adynamo theory of the solarmagnetic field - it couldconceivably be a relic of amagnetic field trapped in theSun during its formation.However, the main solarfield exhibits periodic beha-viour with the same 22-yearperiod as the sunspot cycle; itis diEcult to comprehendsuch short-term variability

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I I2 Stretch-twist- fold cycle whereby the magne tic f lux through a f ixe d loop, CO , n a conduct ing f lu id may be doubled l ike the tension in an e last ic bandbehaviour of both. It is asmall step to extend thetheory to those planets withsignificant magnetic fields(Jupiter, Saturn, Uranus andNeptune); and to the count-less stars, like the Sun, whose

, magnetic fields can now bedetected by the Zeemansplitting of their spectrallines.Frozen fieldsSince the time of Faraday, ithas been illuminating tothink of magnetic fields interms of magnetic lines offorce that behave like elasticstrings threading their waythrough the conductingmedium. This picture isparticularly helpful in theelectrodynamics of conduct-ing fluids, known as magne-tohydrodynamics (MHD -see box); one of the funda-mental theorems of MHD isthat in a perfectly conducting1 (i.e. zero resistivity) fluid,particles that lie on a line offorce at any instant continueto lie on a line of force for allsubsequent times. This al-lows us to identify magneticlines of force with materiallines moving with the fluid:we say that the magnetic fieldis frozen in the fluid. Thevelocity field, U, of the fluidsimply carries the magnetic field lines, B, as if they wereelastic strings (figure 1) . Figure 1 also shows a small closedmaterial curve, C, bounding an area, A, through whichthere is a non-zero magnetic flux, 0 ABcose, where B isthe magnitude of B and 8 s the angle between B and n,the vector normal to C . As C moves with the fluid, this fluxis trapped and remains constant. If the projected area(Acose) decreases, then B must increase to keep the flux(0 ) constant. Therefore, for incompressible flow, stretch-ing the lines of force implies a proportionate increase in B(just as it would increase the tension in elastic strings).We can now see the seeds of dynamo action: if we candevise a flow which stretches the lines of force withoutlimit, then the field intensity will grow from an arbitrarilyweak level to a level that is not only detectable, but also

capable of reacting back dy-namically (through the LoE-entz force) on the flowresponsible for this, other-wise unlimited, growth. Hereagain, the elastic string ana-logy is helpful. Just as we mayincrease the tension in anelastic band by stretching,twisting and folding it, sothe intensity of a magneticfield confined to a single fluxtube can be doubled by thesame stretch-twist-fold se-quence (figure 2 ) . But there isa price to pay if the stretch-twist-fold sequence is re-peated many times (which ispossible, in principle, with amagnetic field but physicallyimpossible with an elasticband). An increasingly com-plex structure builds up onthe scale of the flux tubecross-section and at somestage the resistivity of thefluid, no matter how small itmay be, will lead to Jouledissipation (the productionof heat from magnetic energyat a rate proportional to theresistivity times the currentsquared),The large magnetic fieldgradients in this exampleimply large currents andhence severe Joule dissipa-tion. The first problem ofdynamo theory is to establishthat the average value of themagnetic field can continue to grow despite the increasinginfluence of Joule dissipation. The stretch-twist-foldmechanism, devised originally in 1972 by the late JacovBorisovich Zeldovich of the Institute of Cosmic Research,Moscow, is only illustrative. However, two other, morerealistic, ways for increasing the magnetic field intensity doexist - differential rotation and the %-effect.

Alpha and omegaImagine a rotating sphere of fluid in which the rate ofrotation, R, about the axis of the sphere, 0 2 , increases asthe axis is approached (figure 3 ) . Such a state ofdifferential rotation tends to be established whenconvection currents, driven by buoyancy forces, are also


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The Earth and the Sun are depicted schematically in figure5 . In both, the convective region is a spherical annulus, theliquid outer core for the Earth, and the convection zone inthe Sun. There the similarity apparently ends, for theconvective velocities are -mm s- in the Earths core,-km - in the solar convection zone. And yet

nfluenced by Coriolis forces: conservation of angularso far as theh the fluid are frozenn, they move with the fluid and are cranked round thexis Oz. This leads to a large azimuthal component ofBv. ote that this component is antisymmetric aboutOz.

The other mechanism is more subtle and requires us too beyond the frozen-field principle. The a-effect allowsa straight line of force, along the x-axis say, to be convertednto a helix by a velocity field which is circularly polarised(in the yz plane in this case - see figure 4). The simplestU, of this form has the components 0,uocos(kx-ot), uosin(kx-at), representing a circularly

= o / k .This velocity field has the remarkable property thatts vorticity, o curlu, is parallel to itself (it can be easilyerified that o =-ku). t follows that the helicity of thislow, u . w= -ku2 =-ku& is uniform in space. Helicity,nlike vorticity, distinguishes right-handedness fromleft-handedness in fluid motion. It is this helicity that isresponsible for creating the helical lines of force.Now a line current flowing parallel to the B-line wouldlikewise have the effect of replacing the straight B-line by aelical B-line (the line current produces a circularmagnetic field around the conductor and the addition ofthis field to the straight B-field produces the helical field).It is tempting to interpret the creation of this helix in termsof an induced current due to the helicity of the flow field.This conclusion is in fact correct, although theargument is simplistic; the current only the same type of an dynamo mechanism- isappears if the effects of non-zero 7. believed to apply to both! However, there-resistivity shift the phase of b (the are still major gaps in our under-magnetic perturbation to B caused standing of the dynamics of the flowsby the flow) relative to the phase responsible for dynamo action (asof the velocity field, U. Th e opposed to the purely electro-resulting current, j, is parallel magnetic, or kinematic, con-to the initial field, j =aB; the sequences) in both the Earthconstant a s proportional to and the Sun. Some of thethe phase shift, and hence outstanding problems into the fluid resistivity (in the field were tackled lastfigure 4 a s positive). year by about 30 interna-At first it seems surpris- tional geophysicists, astro-ing that j and B are parallel phy si cis ts a n d fl ui dbecause the electromotive dynamicists who partici-force U x B associated with pated in the six-monthFaradays law of induction Dyn am o Th eor y Pro-is perpendicular to B; how- gramme at the new Isaacever, when U and b are not Newton Institute for Math-in phase, the space-averaged ematical Sciences in Cam-electromotive force bridge, UK.is indeed parallel to B, and Th e most widely acceptedthis is the a-effect revealed by scenario for the Earth is thatthis simple example. In 1966 originally formulated in 1963 byMax Steenbeck, Fritz Kmuse and Stanislav Braginsky, then at theKarl-Heinz Radler of the Potsdam Schmidt Institute of Physics of theInst itut fur Astrophysik in Germany Earth in Moscow - Braginsky suggestedshowed that the same effect occurs in any that slow solidification of the liquid at theturbulent flow field provided the mean 3 Differential rotation: if the inner core boundary (ICB) releases a slighthelicity, , is non-zero. However, the rotation rate, R, of the fluid excess of lighter ingredients (sulphur foractual evaluation of a presents a number of increases with depth, then a example) in the alloy which makes up themajor difficulties. magnetic field line Passing liquid outer core. The resulting buoyancyWe can now combine these two funda- through the sphere tend generates convection currents as the lightto be cranked around themental mechanisms to see how a magnetic axis, This generatesan fluid rises in blobs or plumes towards thefield may be generated in a spherical fluid azimuthal component of core-mantle boundary (CMB), mixing withconductor, by what is known as the an magnetic field which can be the surrounding fluid in the process. As yet,(alpha-omega) mechanism. The total quite large however, there is merely speculation concern-

plane

field can be decomposed into a poloidal (North-South) ingredient, Bp, whose lines of force lie inmeridian planes, and a toroidal (East-West) ingredi-ent, BT, whose lines of force are circles about the rotationaxis Oz. We have seen that differential rotation generatesBT from Bp. Likewise, the a-effect, which converts eachBT-line to a helix wrapped on a torus around 02,generates Bp from BT. Resistive diffusion allows thisnewly generated Bp to reinforce the field Bp that westarted with.We thus have a regenerative cycle: J

A ifferential rotation (Q)BP a-effect plus diffusionHBT

This cycle is believed to be at the heart of both thegeodynamo and the solar dynamo. It should be noted thatwithout some resistivity (and its associated diffusion offield), this regenerative cycle cannot operate. Thus the verymechanism uoule dissipation) that causes the decay ofmagnetic fields is also necessary for their regeneration.Outstanding problems


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m g the length-scale ofthe rising elements, thedegree of turbulence thatthey generate, and therate at which they mixwith the heavier materialin the liquid outer core(the turbulent diffusionrate). At one extreme,the convection currentsmay be laige eddies onthe scale (approximately2000 km) of the depthof the convection zone;this is the view advancedby Braginsky, now at theUniversity of Californiaat L o s Angeles, withinthe framework of hiscon rove rsi a1 MO d e1-Z in which the mag-netic lines of force in thecore are nearly alignedwith the axis of rotation(the direction Oz).At the other extreme,

z

f /IX

AI l r

4 (a) Avelocity field with components u,=O, u y = uocos(kx-ot), U , = uosin(kx-ot).This field Is circularly polari sed in the yz plane and can deform a magnetic line offorce along the x-axis into a helix (b)blobs on the scale of 10-100 km may retain their integrity,like rising bubbles, throughout their journey from the ICBto the CMB. In this scenario the blobs follow a helical pathon a paraboloid of revolution about the rotation axis.Detailed study of the variation of the Earths field withtime, and resulting reconstruction of the field (by inversetheory) at the CMB, is potentially capable of distinguish-ing which of these pictures (if either!) is nearer to the truth.In the meantime, theorists are striving to develop models

that are at least internallyconsistent, a task that isstill at the absolute limitsof the most powerfulsupercomputers.The problems are noless acute for the Sun,but they have a ratherdifferent character. Oneproblem relates to thephenomenon of mag-net c buoyancy thepressure in a fluid is thesum of the magneticpressure (which is pro-portional to the square oft h e m a g n e t i c f i e l dstrength) and the fluidpressure. The fluid pres-sure and density there-fore tend to be low wherethe magnetic field isstrong. This means thatmagnetic flux tubes gen-erated in the solar con-vection zone tend to riseand erupt through the surface. This would be fine if thetimescale of rise were of the order of the 22-year sunspotcycle period; but the simplest theory of magneticbuoyancy, first developed in 1955 by Eugene Parker ofthe University of Chicago, suggests a much shortertimescale of the order of months or less - so short that adynamo process located in the convection zone appears tobe quite at variance with observed solar activity.Attempts are currently being made to circumvent this

5 (a) The geodynamo process is located in the liquid outercore of the Earth. Helical convection is driven bya process of gravitational segregation andthere is an associated slow growth of the solid inner core. ( b )The solar dynamo is driven by turbulence and differential rotation in the convectionzone or, possibly, by instabilities confined to a layer at the bottom of the convection zone. Magnetic flux erupts periodically through the surface(photosphere) giving the well-known 22-year cycle of magnetic activityb

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location of a possiblelayer dynamo

turbulent convectionzone

Earth Sun


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difficulty: the solar rotation (with a period of about 27days) undoubtedly impedes the eruption process, butperhaps not sufficiently; alternatively, the true seat ofdynamo action may be in a layer near the bottom of theconvection zone where a stable density gradient may moreeffectively impede the rise of magnetic flux tubes - muchas if they were imbedded in a layer of treacle!The new science ,of solar seismology (see Deep secretsof the Sun Physics World March 1991 p40), which usesthe increasingly detailed observations of oscillations andwaves on the solar surface to infer the density, rotation andmagnetic field in the interior, has the potential to resolvethis issue, but the clues obtained so far do not yet fit easilyinto any pattern consistent with the alluring an-dynamomechanism.Turbulence aheadA second problem which attracted heated discussion at theIsaac Newton Institute, a discussion that Newton himselfwould have relished, related to the role of turbulentdiffusion associated with random motion of fluid elementsin the solar dynamo. Molecular diffusion is far too weak tocatalyse the dynamo process on the required 22-year time-scale, making turbulent diffusion an essential ingredient.However, a dynamo-generated magnetic field suppressesthe very turbulence on which it feeds; does thissuppression effect kill the dynamo, or does it merelynudge it into another mode of operation (e.g. one in whichregions of strong turbulence and strong magnetic field areseparate but still able to interact across their commonfrontier)?The Earth and the Sun are the tw o most familiarexamples of cosmic bodies containing self-generatedmagnetic fields. But the generality of the dynamomechanism suggests that any self-gravitating rotating massof conducting fluid - whether planet, star or galaxy - willexhibit a magnetic field generated by the an, or similar,mechanism. The general theory is robust, but reconcilia-tion of detailed theory and observation in each individualcase is certain to present challenges of exceptionaldifficulty.

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4Further readingS I Braginsky 1991 Towards a realistic theory of the geodynamoGeophys. Astrophys. Fluid Dyn. 60 89-134D Gough and J Toomre 1991 Seismic observations of the solarinterior Ann . Rev. Astron . Astrophys. 29 627-684H K Moffatt and A Tsinober 1992 Helicity in laminar andturbulent flow Ann . Rev. Fluid Mech. 24 281-312P H Roberts and A M Soward 1992 Dynamo theory A n n . Rev.Fluid Mech. 24 459-512R Rosner and N 0 Weiss 1992 The origins of the solar cycle inThe Solar Cycle K L Harvey ed (Astronomy Society of the Pacific,San Francisco) 27 5 1 1-531The following books arising from the Dynamo TheoryProgramme of the Isaac Newton Institute in Cambridge will bepublished soon:M R E Proctor and A D Gilbert eds 1993 Lectures on Solar andPlanetay Dynamos (Cambridge University Press, Cambridge)M R E Proctor, P Matthews and A Rucklidge eds 1993 T heoy ofSolar and Planetay Dynamos (Cambridge University Press,Cambridge)

K I U I Moff~Ms in the, Departmentof Applied Mathematicsand Theoretical Physics, University of Cambridge, UK

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H.K. Moffatt- Cosmic Dynamos: From Alpha to Omega

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