W.P. Wood and H.K. Moffatt- Large and Small Scale Motions in Kinematic Dynamos

8/3/2019 W.P. Wood and H.K. Moffatt- Large and Small Scale Motions in Kinematic Dynamos

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Geophys. Astrophys. Fluid Dynamics, 1985, Vol. 32, pp. 135-1610309-1929/85/3202-0135%lS.SO/O0 985 Gordon and Breach Science Publishers Inc. and OPA LtdPrinted in Great Britain

Large and Small Scale Motionsin Kinematic DynamosW. P. W O O D t and H. K. MOFFATTDAMTP, Silver Street, Cambridge, U.K.(Received July 24, 1984; infinal form December 6, 1984)

Kinematic, axisymm etric mean field dyn amo s are exam ined for a num ber of models inorder to study the coop eration between small scale turbulent motions (a-affect) an dthe large scale motions (a-effect and meridional flow). Fo r a spherical dynam o weshow that it is more difficult to excite an u*-dynamo with dipolar parity in thepresence of differential rotation, which increases with increasing depth, when themagnitude of the toroidal shearing effect is too small. Meridional streaming in theabsence of differential rotation ha s an inhibitory effect on a2-dynamos. For dynamosin spheres the ease with which a dynamo is excited when the combined effects ofdifferential rotation and meridional flow are present depends on the model for a(r, ).For a model where the maximum in the toroidal shearing is towards the edge of theregion enclosing the streamlines a small amount of meridional flow enhances dynamoaction.For dynamos confined to a spherical shell the effect of a small amount of single cellmeridional flow is to enhance dynamo action when the toroidal shearing is confinedto a narrow region near the lower boundary of the shell. If the shearing is locatedaway from the boundary the meridional flow inhibits dynamo action. For speeds offlow considered here qu adr up ola r fields are mo re easily excited.For circulation with a double cell structure dipolar oscillatory dynamos are morereadily excited independent of the location of the maximum in the toroidal shearingeffect.

?Permanent address: Department of Mathematics, Statistics and Computer Science,University of N ewcastle, New South Wales, 2308, Australia.135

www.moffatt.tc


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1361. INTRODUCTION

W. P. WOOD AND H. K. MOFFATT

The theory of the turbulent dynamo provides a framework withinwhich one attempts to account for various types of magnetic fields incosmical bodies. Essentially two types of dynamo action are en-visaged within the theory when it is considered in its simplest form.Either the magnetic field is produced by turbulent motions aloneand these turbulent motions are parameterized by means of the a-effect (a2-dynamos) or else there is the combined effect of turbulentmotion and differential rotation (the o-effect) giving rise to an g o -dynamo. In the literature (e.g. Moffatt, 1978; Krause and Radler,1980) the steady fields characteristic of a2 -dy namos have normallybeen associated with the earth's magnetic field while the am-dynamoshave been invoked to model many of the properties of the solar field.

A nu mbe r of numerical studies (Roberts, 1972; Roberts and Stix,1972; Jepps, 1975 and Radler, 1975) have been undertaken tosimulate dynamo processes within the earth and the convection zoneof the sun. In these studies plausible models were used for both the CIand o effects. Some of the arbitrariness for the solar field wasreduced in a study of Belvedere, Pa te rno a nd Stix (1980) in whichthe a-effect was modelled while the non-uniform rotation wascalculated independently by Belvedere and Paterno (1977) on theassumption that latitude dependent heat transport sets up ameridional circulation which in turn causes the non-uniform rota-tion. In all of the studies mentioned above meridional circulation hasbeen introduced and it has been shown that the inclusion of thiscirculation can greatly affect the critical parameters in the models.Roberts (1972) showed that for a particular model of the a and oeffects the presence of meridional flow can transform oscillatorydyna mo s into steady dynamos.Another intriguing aspect of the previous studies is that in the caseof axisymmetric systems steady solutions are associated with smallvalues of the dynamo number, N , while for larger values of N thefield is oscillatory.In this article we consider axisymmetric dynamos of a 2 0 typewhen meridional circulation is present. The effect of meridionalmotion may play a crucial role in the solar dynamo since Schmidt(1982) has shown that for certain models of the convection zone thecirculation speeds can be quite substantial with the 6 component of


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KINEMATIC DYNAMOS 137the velocity in the range 1-10m/s a t the solar surface an d taking onlarger values deeper down. Here we investigate the effects such largespeeds will have on a2w-dynamos confined to an outer convectionzone. We have include the a-effect generation of toroidal field fromthe poloidal field since one obtains modified critical behaviour to thepurely aw situation. For dynamo action in general one question toconsider is how do the large scale motions contribute towards theproduction of a dynamo given that the presence of turbulent motionis always necessary. We show that a2w-dynamos, and consequentlyam-dynamos, are associated with larger values for N because thepresence of a small w-effect is inhibitory to dynamo action.Furthermore since all estimates of a and w for the sun yield amagnetic Reynolds number for the differential rotation greatly inexcess of the a magnetic Reynolds number then it is usual toneglect the a-effect in pfoducing the toroidal field. This neglect iscomputationally convenient since the equations may be scaled sothat only one dynamo number is present. The relative influence ofthe a and w effects is lost. However as we show below whenmeridional flow is present small changes in the magnitude of a canlead to qualitatively different dynamo action.

2. B A S I C E Q U A T I O N S A N D M E T H O D O F S O L U T I O NWe consider a sphere of radius R and assuming symmetry about therotation axis the mean field dynam o equations may be written as

aA/at = RJV, x v x A) +R,aBT - v x (vx A), (1)aBT/at= R,V X (V, X BT)+R,V X [VT x (V x A)]

+R,V x a(V x A)-V x (V x BT), ( 2 )where the velocity and magnetic fields are respectively, V = V, +V Tand B = B, +B, with B, = V x A . All quantities are in dimensionlessform and the magnetic Reynolds numbers R,, R, and R, arerespectively a,R/q, wR3/q and VR/q with the turbulent magneticdiffusivity, q , taken to be constant. Lengths a re ,scaled by R andtimes are measured in terms of the diffusion time, R2/q. The


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138 W . b. WOOD AND H. K . MOFFATTparameter go is a measure of the a-effect while 0 is a measure of thegradient for the differential rotation. R, parameterizes the meanmeridional flow and V is taken to be a typical flow speed in the 8direction. We identify the d ynam o number N as R,R,. Fo r pure am-dynamos the a term in equation ( 2 ) is neglected and V,=O. For thiscase equations (1) and (2) may be rescaled (Roberts, 1972) so thatonly one parameter, N , appears in the equations. To considera2w -dyna mos with V, # O we find that in the numerical study of thisproblem it is convenient to rescale (1) and (2) by replacing A by R,Aso that the parameters appearing are R,, N and R,.Following standard techniques (e.g. Krause and Radler, 1980)originally due to Bullard an d Gellman (1954), we expand thepoloidal and toroidal parts of the magnetic field in the form

( 3 41A ( x ,8, t )= O,O, e p f a , (x) P,(cos 8) ,{ n = l1B,(x, 8,t )= O,O, e p f1 , ( x ) P, (COS 0) ,{ n = l

with x = r / R .In the following we tak e, for simplicity,

and ~ ( x )s a function to be prescribed. Further, we suppose that themeridional circulation V, is solenoida l so that we may write

= {2 x - 1 f ( x ) P 2 ,4 - [d (xf / d x ]P i , O } , (6)and the exact form of f is discussed below. The toroidal velocityfield is written as

V,={O,O,xsin8[1 + w 1 ( x ) + w 2 ( x ) P 2 ] } , (7)where the form of w 1 and w2 will depend on the model adopted.Adopting the separation of variables as indicated by equations


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KINEMATIC DYNAMOS 139(3)-(7) we eliminate the 6 dependence by multiplying both sides ofequations ( 1 ) and ( 2 ) by P, nd integrating with respect to cos6between - and 1 . The resulting equations may be written as

+ R ~ { w ; ( c 1 3 a k - 1 +c14 ak + 1 ) w , (c 1 5 ak - 3 ) + C 16 a k - 1

where dashes are differentiation with respect to x and the ci ci(k)where k = 1,2, . . .,M . The ci are listed in the Appendix.The coupled equations (8) and (9) divide naturally into twofamilies: (i) the dipolar type ( a 2 k - 1 , b Z k )with the field anti-symmetricwith respect to the equator an d (ii) the qua drupo lar type, (c&, b 2 k - l )which are symmetric with respect to the equator. These equationsare to be solved subject to the boundary conditions.

i) For a sphere,

and


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140 W. P. W O O D A N D H. K. MO FFA T Tii) F o r a shell with a n inner radius of x = x u the boundaryconditions (10) is replaced by

This condition was used by Belvedere, Pa ter no and Stix (1980) andis a special case of general boundary condition given by Roberts(1972). The use of this bo undary cond ition presupposes that a fieldwithin the shell (convection zone) is unable to penetrate into the(radiative) core which has a much higher conductivity. Whendynamo action is present equation (12) is strictly only valid whenthe a-effect vanishes on x , and the dynamo field is oscillatory. In hisnumerical studies on shells Roberts (1972) took bo th ak and bk to bezero on x, and conjectured that the error would be small. In ourcalculations we found that using the different conditions leads todifferences in the critical numbers of the order of 4 percent or lessfor the oscillatory modes. To determine the eigenvalue, p , in equations(8) and (9), the derivatives are replaced by finite differences and thesystem is solved as an algebraic eigenvalue problem

AX= PX,with the dimension of A determined by M , the truncation level forthe field expansions and m the number of grid points. For most ofthe results reported here M was taken to be 5 and 6 while m rangedfrom 20 t o 40. Th e convergence of so lutions for larger values of M iswell established. Furthermore the results for am-dynamos are stableto variations in m. Since we were interested in establishing generaltrends for arbitrary models it did not seem worthwhile in terms ofcomputing time to pursue the precise details of the convergenceproperties of the critical numbers. As a check on our numericalprocedures we compared our solutions with previous authors. In allcases the agreement was satisfying as will be indicated below.Since we shall be looking at the growth rates of the eigenvalueswith R,, the appropriate magnetic Reynolds number, it is pertinentto recall that for a sphere the free decay modes in the absence offluid motion may be found analytically (Moffatt, 1978). Th e slowestdecaying mode is p = -IT or the dipolar fields while the quad-rupolar fields have p = 20.19.


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KINEMATIC DYNAMOS 141For a shell with the boundary conditions given by (12) thepoloidal decay modes are given by the roots of

J k + l / 2 ( K X u ) J l / 2 - k ( K ) - J - ( k + 1 / 2 ) ( K X u ) J k - l / Z ( K ) = o ,

where K = - p 2 . F o r k = 1 this expression reduces to the non-trivialroots of t a n K(x u- )=Kx,. When x,= 0.7 then p - - 6.2. Thislarge value of IpI for a field confined to a shell implies that largerReynolds numbers will be needed to excite the field to its criticalstate than for the case of a field in a sphere.3. MODELS FOR A SPHEREFor the situations where dynamo action is possible throughout asphere we considered two models which have appeared in theliterature (Steenbeck and Krause, 1966; Roberts, 1972 and Jepps,1975).Model 1

a = 1, o1= -x, f=7168(2)1 2~6(1-~)2/243.Model 2a=729x8(1 - ~ ~ ) ~ / 1 6 ,1 1 9 6 8 3 ( 1 - ~ ~ ) ~ / 4 0 9 6 0 ,nd f as forModel 1.In both models the a-effect is positive in the northern hemispherewhile o; s negative. Model 1 is probably quite artificial, however itdoes provide a model in which the CI and o effects are uniformthroughout the sphere. In Model 2 the maxima for the twoinduction effects are separated. Stix (1973) concludes that theoscillating nature of a o- dy na m os is due to the geometry of theinduction effects. Since these two models have different geometrieswe conclude below that the oscillatory nature of ao-dynamos is dueto the driving mechanism of the o-effects.Fo r az-dynam os with R, =R, = 0 the field is maintained by theturbulent motions alone. For Models 1 and 2 the critical values ofR , (with M = 6 and m=40) are shown in Table I with the values


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142 W. P. WOOD AND H. K . MOFFATTTABLE I

Model Dipole Quadrupo le1 7.66 (7.64) 7.85 (7.80)2 13.08 (13.04) 13.43 (13.11)

obtained by Roberts (1972) in parentheses. F o r larger values of R,the values of Re p increase steadily away from zero with highermodes also having Re p >O , e.g. Model 1 with R,= 12 gives Rep=18.52 and 1.22 (dipolar); and Re p=18.54 and 1.43 (quadrupolar).Holding R,=O we introduced a meridional circulation in bothmodels. When R, was at its critical value then for IRpl#O thedynamo fails while for R, values above critical the values of Re pmonotonically decrease with increasing IR, I. To excite a2-dynamosin the presence of circulation requires larger values of R, than in thezero circulation situation. The inhibiting effect of circulation on Edynam os (with dipolar parity) was also found by H ellmich (1978) ina dynam ical calculation. O u r result differs in som e respects t o Radler(1975) where a different model was used. In term s of the param eterused here Radler finds that for the dipolar field the dynamo isinhibited for -


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KINEMATIC DYNAMOS 143poloidal flux expulsion which determines the values of R, which maybe used.We now consider the o-effect by pu tting R,=O an d allow R, toincrease away from zero. As Roberts an d Stix (1972) have shown fora particular a and o model due to Steenbeck and Krause (1969) fo ra2 0- dy na m os there is a transition region for the ratio RJR, overwhich the dynamo changes from being steady to being oscillatory.Both R, an d R, tak e their critical values over the range theystudied. For R,/R, 10- the dy nam o is stationary and they concludedth at the cr2-effect was domina nt. Obv iously for values between thesetwo extremes the a-effect will assist in the production of toroidalfield to a great or lesser extent. It is well established that oscillatoryNo-dynamos are associated with large values of N while thesteady a2- dynam os ar e associated with modest values of magneticReynolds numbers. To see why this should be so we held R,constant at the critical values for the two models and increased R,.The effect on Re p is shown schematically in Figures 1 and 2 for thedipolar and quadrupolar modes respectively. For Figures 1 and 2the horizontal axis, R,, represents the relevant magnetic Reynoldsnum ber, i.e. either R, o r R,, each one varying as indicated in thediagrams. As can be seen from the sketches the behaviour of Re p

FIGURE 1 A schematic plot of the growth rate of Rep for the dipolar fields ofModels 1 and 2. R, is increased to obtain a critical R,, then holding R, constant R ,is increased until R, is again critical. The solid line corresponds to Imp=O.


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144 W. P. WOOD AND H. K. MOFFATTRo CONSTANTRW INCREASING

FIGURE 2 The same as for Figure 1 with dipolar replaced by quadrupolar

with R, for the dipolar and quadrupolar parities is different abovethe critical value for R,. However the fact the a2m and consequentlyam-dynamos are associated with large values of R , is clearlyillustrated. The m-effect is inhibitory to dynamo action for smallvalues of R , for fields of dipolar parity provided dw,/dx


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KINEMATIC DYNAMOS 145TABLE I1

Values of R , and I m p .Model Dipole Quadrupole

1 349.0 522.3(28.6) (27.4)

2 1705.2 1598.6(27.96) (17.6)

quadrupolar. It should be stressed however that these values of R,and R, are not necessarily optimal in establishing a dynamo.However what we have shown is that the size of the a-effect relativeto the w-effect must be crucial in determining the preferred mode.The field lines for the u2 an d cr2w-dynamos for M ode l 2 a re shownin Figures 3-6.Having established an a2w -dyn am o we again intro duc edmeridional circulation. The results here are qualitatively differentfrom those for the a 2-d yn am o. Th e results are m odel dependent inthat for Model 1, R, =2 causes the dyn am o to fail for both dipolarand quadrupolar parities, while for R,= - 2 the dy na m o is enhanced.For Model 2 the situation is reversed in that R,>O leads to anenhancement of the dynamo action with a decrease in the oscillationfrequency of the field. For R, = 20 we find R, = 1315.2, Im p = 7.8 forthe dipolar field and R, = 722.3 (I m p= 0) for the quadrupolar modewith R , held at the critical values found previously. For the singlecell circulation used here we find that the quadrupole mode is moreeasily excited and this was found to be the case in the shellcalculations reported in the next section. Assuming that the circu-lation is in the direction of from poles to equator (R,>O), whichwould be consistent with an w(r ,e ) increasing inward, [e.g. seeKohler (1974)l then the enhancement of dynamo action could bedue to a tendency for the flow to push the flux into the region ofmaximum toroidal shearing. For Model 2 this maximum is atx=O.33. In Model 1 the shearing is uniform throughout the sphere.

Roberts (1972) found that for R, sufficiently large then thepreferred solutions were always steady in the case of am-dynamos.For Model 2 we looked at the quadrupole mode and for R,= -20with R, constant we increased the value of R, and for R,= 15.4 we


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146 W. P. WOOD AND H. K. MOFFATT

FIGURE 3 Dipolar a dynamo for Model 2. The B, lines are shown on the rightand lines for B,=c onstan t are on the left. An encircled do t (cross) indicates a vectorpointing out (into) the plane of the paper. The contours mark the levels k i , . ,*2times the maximum values of the toroidal and poloidal fields.found that the dynamo could be restored and in this case it wassteady.

4. M O D E L S FOR A SHELLWe now consider dynamo action which is confined to a shell ofradius x = x u = 0 . 7 . As a model of the solar convection zone weenvisage that fields which are produced in the turbulent layer do not


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KINEMATIC DYNAMOS 147

\

FIGURE 4 Quadrupolar a dynamo for Model 2. Otherwise as for Figure 3 .


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148 W. P.WOOD AND H. K . MOFFATT

penetrate into the radiative core and the boundary conditions givenby e quation (12) apply.Model 3

We considered the following models:

a = 1 , o l = l - x ,

where t,bp is the stream function for the meridional flow (Roberts an dStix, 1972; Krause and Radler, 1980). The streamlines a re show n inFigure 7.


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FIGURE 6Figure 3.Quadrupolar u2w dynamo for Model 2 with t =O . Otherwise as for


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150 W. P. WOOD A N D H. K. MO F F A T T

Y 0 7 1.0FIGURE 7 Streamlines for a single cell meridional velocity field. The flow isrestricted t o the region 0 . 7 < x < 1.The direction of the flow is for R,>O.

Model 4As for Model 3 with

w1 3{1 -Erf[(x-x,)/0.075]}.Model 5

As for Model 4 with w1 entred at x=O.8.Model 6

~ + 2 7 ( 1 - ~ ) ( ~ - ~ , ) ~ / 4 ( 1x,)~,w1 nd $ p as for Model 4.


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KINEMATIC DYNAMOS 151In Model 6 the variation of a with x is the model used byBelvedere, Paterno and Stix (1980) where a is small at both the topand bottom of the convection zone. Cowling (1981) has pointed ou tthat the a-effect should be present throughout the convection zone

so in the absence of any detailed knowledge of a, allowing a to actover the whole shell is a plausible assumption. The models studiedhere have this property. The form of w1 used in Models 4-6 isconsistent with one of the m odels of Schm idt (1982) where thedifferential rotation is independent of 8 and steadily decreases fromthe inner boundary of the shell. In Models 4 and 6 the influence ofthe shear is confined to a small region near the lower boundary. Ithas been conjectured by Schiissler (1982) th at dynam o fields areconcentrated in the overshoot layer of the convection zone whichoperates at the interface of the convective and radiative zones. Themagnitude of the toroidal field is far greater than the poloidalcomponent and Models 4 6 place the production of the toroidalfield in this overshoot layer.The critical numbers for the models when treated as am-dynamosare given in Table 111. The differences between Models 4 and 6 aredue to the variation in a with x. For fixed R , the increase in thecritical number is then due to an increased R , value and thisconclusion is in agreement with the findings of Belvedere, Paternoand Stix (1980) for a particular model for the w-effect.

TABLE I11Critical values for the dipolar modes

Model N Im P3 1.7 x 104 76.84 3.6 x 104 105.35 8.4 x 103 142.56 8.2 x 104 137.0

The differences between Models 4 and 5 is due to the location ofthe maximum in the toroidal shearing where in Model 5 thismaximum is away from the boundary and this leads to an increasein the oscillation of the field.For Model 6 we considered it in the context of an a2w-dynamoand varied R,. For R , = 8 the critical dynamo number for the dipolar


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152 W. P. W OOD AND H. K. MOFFATTmode was 7.5 x 104 with I m p = 131.7 while for R E =18, N = 6. 3 x 104with Imp=107.0. The increase in a is again associated with adecrease in the frequency of the field.For the am-dynamo using Model 5 we introduced meridionalcirculation. There is no agreement as to the speed of the meridionalflow in the convective zone however we may use some results foundby S chmidt (1982). In the models com puted by Schm idt the solardifferential rotation is driven by either latitude dependent heattransport or anisotropic viscosity. The results for both mechanismsdepend crucially on the boundary conditions and the value chosenfor the Prandtl number. Schmidt obtains speeds at the surface whichm ay then vary from v , < l m s - ' to G ~ l O r n s - ' .F or & 5 l m s - ' thenumb er R, in ou r calculations may vary from 5 1 0 to -35depending on the m agnitude of q.As an estimate of q we have assumed the values (2-5) x 1 0 7 m2 s - ' .Belvedere and Paterno (1977) have also performed similar calcu-lations for meridional flows driven by latitude dependent heattransport. In their models the circulation speeds were < l m s - ' atthe solar surface and the circulation consisted of a single cell. In themodels calculated by Schmidt the circulation was not confined to asingle cell. To account for this albeit in an ad hoc way we introduceda stream function given by

$ = [64(x - J 3 ( 1- )'(xb - ) /(1- J6] s in2@os0 fsin 20 cos0.The streamlines for the flow when x b=0.86 are shown in Figure 8.For Model 5 as an am-dynamo Figure 9 shows the field patternwithout circulation. The introduction of a double cell circulationwith R, = 35 does not noticeably alter the field pattern and we foundtha t N 112= 88.5 while I m p = 145.9. F o larger values of R, thedynamo is excited for smaller values of N1I2and the frequency doesnot vary a great deal. For this model the circulation assists in theproduction of a dynamo in the sense that the dynamo is more easilyexcited in terms of smaller values of N112.Model 5 produces a fieldwith different structure in the case of the single cell circulation. ForR,>O dyn am o action is inhibited and consequently larger values ofN'12 are needed to excite the dynamo. In Figure 10 we show thecase for R,=10 an d the critical numbers are now N'"=111.8 withI m p = 169.6. F o r this situation the poloidal field lines follow the fluid


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X 0 .7 1,o

FIGURE 8 Streamlines for a double cell meridional velocity field. The direction ofthe flow is for R,>O.

lines as the field now has to act against the fluid flow which being asingle cell type h as a tendency to g p e l the field. Noticeably the fieldis concentrated beneath the surface. In Figure 11 we have the fieldlines for a supercritical field in the presence of double cell circulationwhere N= 91.5, R , = 100, R e p = 10.4 a n d I m p = 169.7. The fieldhas moved towards the poles while the field lines have been pulledbeneath the surface.In Model 5 the position of maximum toroidal shearing is locatedaway from the boundary and we find that we have somewhatdifferent results when we consider Model 6 where the toroidalshearing is confined to a region near the lower boundary of the shell.For Model 6 as an a2w-dynam o with R , = 8 we find that thepresence of a single cell circulation with R , < 3 enhances dynamoaction while lowering the value of Imp. This is true for both dipolarand quadrupolar parities. As R , is steadily increased then thecirculation destroys the dynamo and larger and larger values of N


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FIGURE 9R,=35 ( 2 cell); at the instan t t=O. Otherwise as for Figure 3.Dipolar aw dynam o for Model 5. (i) N1'2=91.5, R,=O and (ii) N'I2=88.5,

FIG UR E 10 Dipolar aw dynamo for Model 5 N'/' = 111.8, R, = 10 (1 cell), and t = 0.Otherwise as for Figure 3.154


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\FIG UR E 11 Supercritical G(W dynamo fo r Model 5. N1=91.5, R,=100 (2 cell)Rep = 10.42, Im p = 169.7, and t =O. Otherwise as fo r Figure 3.

are needed to restore the field to its critical state. When circulation ispresent the quadrupolar mode is the one which is marginally easierto excite. For Rp=10 the critical numbers are 11.9 x 104 ( Imp=124.5) and 12.4 x 104 ( I m p = 128.7) for the quadrupolar and thedipolar parities respectively.The quadrupolar field for R,= 8 has 12.2 x 104 and I m p = 142.1when the circulation is absent. For R,


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156 W. P. WOOD AND H. K. MOFFATT

FIGURE 12 Dipolar ci20dynamo for Model 6 for (i) R , = 8, RL12= 98 . 85 , R , = 0, (ii)R,= 8, Rk12= 124.5, R , = 10 (1 cell); and t = 0. Otherwise as for Figure 3 .

Model 6 with R , = 8 and Rk12=98.85 in the absence of circulation.After introducing a single cell circulation and setting R,= 10 weobtain an almost identical field pattern if the dynamo is restored byvarying R,. The relevant numbers are now R,=8, R,=10 andR;I2 = 124.5. This result is reasonable since for the single cell circula-tion models of Belvedere and P ate rn o (1977) the meridional circulationdrives the differential rotation and consequently a relative increase inthe w-effect is going to offset the inhibitory effect of the circulation.Holding RkI2=98.85 and allowing R, to increase we obtain thefield pa tte rn show n in F igure 13. The critical value for R, is now12.0. The poloidal field lines now have a tendency to follow thestreamlines of the circulation. Consistent with the findings fordynamos in a sphere the increase in the cx-effect is accompanied by adecrease in Imp( = 122.8).


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FIGURE 13 Dipolar CIW dynamo for Model 6 for R,= 12, R;=98.85 and R,= 10(1 cell), and t=O. Otherwise as for Figure 3.

When the double cell circulation is present the dipolar mode is theone more readily excited for R,>O. For the circulation consideredhere this means that the flow at the surface is towards the poles. Thepresence of the circulation leads to higher values for the frequency ofthe field as was found for Model 5 . If the direction of the flow isreversed then the quadrupolar mode is the one more easily excited.However all the models of Schmidt are equivalent to R,>O and sobased on the simple model considered here double cell flows wouldlead to dipolar fields being the dominant mode.For the models we have computed it was noticed that theoscillation frequencies for fields confined to shells are a good deallarger than the frequency for Model 1 of Steenbeck and Krause(1969) where the field occupies the whole sphere. We com puted thecritical values for this model and obtained N1I2= 144.6 (Imp= 31.1).When the field is confined to a shell we obtained N1l2=412.6 andI m p = 77.7 and the frequency of the field has increased.


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158 W. P. W O O D A N D H. K. M O F F A T TFor Models 4-6 and for this model also the period of theoscillatory field given by 2rcR2/q I m p lies in the range 15-45 yearsdepending on the model and the value of q chosen.For all of the models studied here we found that if R , is increasedbeyond some limit then the critical eigenvalue is real and con-sequently the field is steady. This result occurred using the modi-fied boundary conditions uk( x , )= bk(x,)= 0. The boundary conditionswere then altered to account for Imp=O, and the results weremodified by less than 3%. For the single cell circulation the value ofR P - 15 yielded an Im p= O . Larger values of R, could be attained forthe double cell flow before the field turned steady. We did not search

for the critical R , for each model though it appears that R,>150.For the single cell flow it may be argued that if the poloidal field isto be maintained against flux expulsion then it would be necessaryto increase the relative weighting of the cr-effect and, as we haveshown, this leads to steady fields.Finally we looked at Model 6 in the absence of circulation withthe differential rotation now containing a term proportional toP,(cosB). W e adopted the model of R oberts an d Stix (1972) in whichthe solar core rotates more rapidly than the convection zone. Thismodel has the formo1 + j + $ ( l$j) ( l Erf [(x-x,)/O.O75]}

andw2= -+p { 1+Erf [(x -x,)/O.O75]},

where j s a parameter which varies between 0 and 1. Fo r R, = 8 thedynamo is critical when R;I2 = 57.8 a n d I m p = 287.9. The frequencyof the field is larger than for the case when the differential rotation isa function of r only.When the dynamo is critical then the introduction of double cellcirculation leads to enhancement of dynamo action with a corre-sponding increase in Imp. For example when R , = 3 5 with R, andR, held a t the values above then Re p= 2.0 4 and Im p= 28 9.8 . Againthis type of circulation is the one which assists in the production of adynamo.


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KINEMATIC DYNAMOS 1595. R E M A R K SUsing the simple models studied here we may conclude that fordynamo action in shells meridional flows with a double cell structurewill assist in the production of an oscillatory dynamo of dipolartype. This result seems to be independent of the choice of location ofthe maximum in the toroidal shearing and whether the differentialrotation is a function of r and 0 or r only.For single cell flows the dynamo is enhanced for small numbers ofrevolutions per diffusion time provided the toroidal shearing islocated near the lower boundary of the convection zone. In this casethe direction of flow will determine which mode is favoured. Forincreasing R,, dynamo action is inhibited independent of the loca-tion of the maximum of 0. or this situation the field is expelledfrom the central regions and the study of the resulting boundarylayer problem would be necessary to analyse the behaviour of thedynamo action in this region.

ReferencesBelvedere, G. and Paterno, L., Convection in a rotating deep compressible sphericalBelvedere, G., Paterno, L. and Stix, M., Dynamo action of a mean flow caused byBullard, E. C. and Gellman, H., Homogeneous dynamos and terrestrial magnetism,Cowling, T. G., The present status of dynamo theory, Annu. Reu. Astron. &Deinzer, W., Kusserow, H.-U.v. & Stix, M., Steady and oscillatory alpha-omegaHellmich, R., Finite amplitude a-dynamos with large scale incompressible circu-Jepps, S . A., Numerical model of hydromagnetic models, J . Fluid Mech. 67, 625Kohler, H ., Differential rotation, Sol. Phys. 34, 11 (1974).Krause, F. and Radler, K. H., Mean-field Magnetohydrodynamics and Dynamo Theory,Moffatt, H. K., Magnetic Field Generation in Electrically Conducting Fluids, C U PMoffatt, H. K. acd Kamkar, H. in Proceedings of the Workshop on T h e T h eo r y of

shell-application to sun , Sol. Phys. 54, 289 (1977).latitude-dependent heat-transport, Astron. & Astrophys. 86, 40 (1980).Philos. Pans. R. Soc. London A 247,213 (1954).Astrophys. 19, 115 (1981).dynamos, Astron. & Astrophys. 36, 69 (1974).lation, Geophys. & Astrophys. Fluid Dyn. 10, 89 (1978).(1975).

Pergamon, Oxford (1980).(1978).Stellar and Planetary Magnetism, Budapest (1982).


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160 W. P. W O O D A N D H. K. MO F F A T TRadler, K. H., Some new results on the generation of magnetic fields by dynamo

action, Me m . Soc. R. Sci. Liege 6 serie VIII, 109 (1975 ).Roberts, P. H., Kinematic dynamo models, Philos. P a n s . R. Soc. London A 272, 663(1972).Roberts, P. H. and Stix, M., Alpha-effect dynamos, by Bullard-Gellman formalism,Astron. & Astrophys. 18, 453 (1972).Schm idt, W., Models of solar d ifferential rotation, Geophys. & Astrophys. Fluid Dyn.21, 27 (1982).Schiissler, M. in Solar and Stellar Magnetic Fields, Origins and Corona1 Effects,IA U Symp. 102, Zurich (1982).Steenbeck, M. and Krause, F., Erklarung stellarer und planetarer Magnetfelder durcheinen turbulenzbedingten Dynamomechanismus, Z . Naturforsch. 219, 1285(1966).

Steenbeck, M. and Krause, F., Zur Dynamotheorie stellarer und planetarer Magnet-felder I. Berechnung sonnenahnlicher Wechselfeldgeneratoren, Astron. Nachr.291, 49 (1969).Stix, M., Spherical alpha-omega dynamos by a variational method, Astron. &Astrophys. 24, 275 (1973).Weiss, N., Expulsion of magnetic flux by eddies, Proc. R. Soc. London Ser. A 293,310 (1966).

Appendixk + 2c1=- 2 k + 3 k -1c2 =-2k-13(k- l )(k-2)

2(2k- 3)(2k- 1)c3 =

k 2 + k - 3c -4-(2k+3)(2k-1)3( k + 3)(k + 2)

c 5 =2(2 k+ 5)(2k+ 3)-3(k- 1)2(k-2)c6 = (2k- 1)(2k-3)

c I 2= k2c2,

3(k- )(k- ) 2 ( k- 1)2(2k-5)(2k- 3)(2k- 1)1 5 =

-k2(k- l )(k-3)2(2k- 1)(2k-3)(2k+3)

1 6 =

(k + 1)2(k+ 2)(k +4)2(2k - )(2k+5)(2k +3)1 7 =


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- k(k+1)c -7 - ( 2 k - 1 ) ( 2 k + 3 ) 3 ( I ~ + 2 ) ~ ( k + 3 )

* - 2 k+3) (2k+ 5 ) c -

KINEMATIC DYNAMOS

- 3 ( k + l ) ( k + 2 ) ( k + 3 )(2k+ 3) (2k+ 5) g =

3 ( k - 2 ) ( k - l ) k( 2 k - 3 ) ( 2 k - 1 ) 10

=

c1 = k+ 1 ) 2 ~ 1 ,

- (k+ 3)2(k+ 4)(k+ 2 ) I 8 = 2 ( 2 k + 5 ) (2 k + 7 ) ( 2 k + 3 )

3 ( k - 3 ) ( k - 2 ) ( k - 1 )(2k- )(2k- )(2k- )19 =

- k2(k- )( 2 k - 1 ) ( 2 k - 3 ) ( 2 k + 3)20 =

- 3 ( k + l ) ( k + 2 )C 2 1 = ( 2 k + ) ( 2 k - 1) (2 k+ 5)

- 2c22 =-k + 3 C 1 8

161

W.P. Wood and H.K. Moffatt- Large and Small Scale Motions in Kinematic Dynamos

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