Dynamos Driven by Poloidal Flow Exist

GEOPHYSICAL RESEARCH LETTERS, VOL. 23, NO. 8, PAGES 857-860, APRIL 15, 1996

Dynamos driven by poloidal flow exist J. J. Love &: David

Gubbins

Department of Earth Sciences,University of Leeds, United Kingdoan

Abstract. We have discovereda class of dynamos driven by purely poloidal fluid motion, thereby demonstrating that toroidM motion is not essentialfor dynamo action and that there is no counterpart to the antidynamo theorem that purely toroidal motions cannot generatemagnetic fields. The fully three-dimensional

(3D) dynamoaction of thesemodelsresultsfrom the

rems concerning the nature of the velocity field. El-

sasset[1946]and Bul!ardand Gellman[1954]showed that poloidM motion is essentialfor dynamoaction, i.e. that

toroidM

motion

alone is insufficient

to sustain

a

magneticfield, someradial poloidalmotion is necessary. Other analyses have bounded the minimum amount

of radial motion necessary for dynamoaction [Busse,

pushing and twisting of magnetic field lines by heli- 1975],and haveboundedthe minimumsizeof the magcal motion in a manner akin to the t•2 mechanism of neticReynoldsnumber[Backus,1958;Childtess, 1969; of dynamoactionsusmean-field electrodynamics, No dynamo with dipole Proctor,1977]. Our discovery symmetry was found for an axisymmetric distribution tained by purely poloidM motion demonstrates that of hellcity; indeed, some azimuthal variation in helicity there is no counterpartto the toroidM flow anti-dynamo is required. Among the suite of dynamosthat we have theorem. investigated, the poloidM flow dynamo with the smallest critical magnetic Reynoldsnumber is very nearly the Method dynamo with the most nonaxisymmetric distribution of helicity. We considera conductingfluid spheresurroundedby

a stationaryelectricalinsulator. Equation(1) is then solvedby the Bullard-Gellmanmethod [Bullard and Gellman,1954]:discretizing the equationby expanding

Introduction

Dynamo action in the Sun and the Earth is sustained by the motion of electricallyconductingfluid. In a kinematic dynamo analysis,suchas that discussedhere, one investigatesthe types of fluid motion which sustain dynamo action by solving the magnetic induction equa-

both the velocity field and magnetic field in terms of

a

tion,

OrB----RmVX (U X B)•- V2B,

(1)

where B denotesthe magneticfield and u is the dimensionlessfluid velocity. The magneticReynoldsnumber, R,•, givesa measureof the effectivenessby which fluid motion acts to amplify the magnetic field comparedto diffusive decay due to electTicalresistance. Early work on the dynamo resultedin somenegative results, the so-cMled anti-dynamo theorems; the most

famousbeingthat dueto Cowling[1934],whoshowed that fluid motion:' cannotgenerateax!symmetric magnetic fields, a result which holds for time-dependent

fieldsandtime-dependent compressible fluids[Hideand Palmer, 1982]. And recentlyit has been shownthat

fluidmotion cannot sustain purelytoroidal magnetic fields [Kaiser, Schmitt andBusse, 1994].In addition to theorems concerning the magnetic fieldare theo-

Figure 1.

Copyright1996by theAmericanGeophysical Union.

Poloidalfluid velocitycomponents:(a)

meridional section showing streamlines of meridional

circulation, sø•,(b) equatorial sectionshowing streamlinesof convective motion,s•s+ s• 2c.

Papernumber96GL00846 0094-8534/96/96GL-00846505.00 857

858

LOVE AND GUBBINS: DYNAMOS

motion,t•ø, is differentialrotation,whilstthe poloidal component, %0,is meridionalmotionwhichhas been foundto promotesteadysolutions in ctw-dynamos [Roberts, 1972]. The sectoralpolotrialharmonics, s22s and s22c, contribute convective overturning withnmlti-cellular motion three-deep alongthe radius.In Fig. 1 weshow the formof %0ands2a s+ s22c. Previousstudiesusing(2) addressed the Braginsky limit, ½0-• oc, andthuswereconcerned with dynamo actiondominatedby differentialrotation [Kumar and Roberts,1975]. In this study we fix ½0equalto zero,

R

10 '2 98-

?-

6-

5-

,I

-16

I

I

I

I

I

I

-12,

,

0

• I•2

lOO

-

DRIVEN BY POLOIDAL FLOW EXIST

and therefore restrict ourselvesto purely polotrial motion; we explore dynamo action by varying the relative proportion of meridional and convective motions

as measuredby the ratio ½•/e2.We searchfor magnetic fields with dipole symmetry, i.e. antisymmetric upon reflection through the equatorial plane, as is geophysically and hellophysically relevant.

b Results

and

Discussion

8o

A suite of steady poloidal flow dynamos were found. All are dominated by strong meridional motion, with 6o

negative½•,meaningthat the senseof the meridional motion is one of upwellingalong the equator and downwelling along the geographicpoles. The critical mag-

4o

netic Reynoldsnumber,R•, as a functionof ex/e2,is shown in Fig. 2a. Dynamo efficiencyis optimum for model (9, where the magneticReynoldsnumber attains

2o

a minimum,R•m___ 44 for e•/e2 -6. O-

•

-16

i -12

I

• -8

•

•

•

-4

•;1/•;2

The magnetic field of (,9 is shown in Fig. 3, where it is revealed that meridional motion promotes the sustenance of poloidal axisymmetric fields; for the opti-

mum dynamo80% of the magneticenergyis axisymmetric and 78% is poloidal. The surfacefield of (.9 Figure 2. (a) Criticalmagnetic Reynolds number, R•, is extremely simple, Fig. 3b, consistingof two flux as a function of the ratio of meridional motion to conpatchesconcentratednear the geographicpoles, the revectivemotion, q/e2. Model (.9 is the dynamowith sult of strong meridional motion, which sweepspoloidal smallestR•. (b) The averagehelicity,(h)2, and the magneticfield lines towardsthe polesand concentrates nonaxisymmetric helicity,(h.)•a,asa function of •/•2. them by fluid downwelling. The magnetic field is fairly Note that model (.9is very nearly the model of maximal complicatedin the fluid interior, Fig. 3c,e,f, but the azimuthallyaveragedfield, Fig. 3a,d, is largescaleand does not strongly reflect the underlying multi-cellular convective flow. The numericalconvergenceof (.9is vertoroidal and poloidal vector harmonicsand radial grid ified by comparing magnetic spectra for two different points. For a prescribedvelocity field, u, the induction truncations of harmonic expansion;Fig. 4. equation is reducedto an algebraiceigenvalueproblem. In mean-field electrodynamics [Steenbeck, Krauseand Standard numericaltechniquesare usedto solvefor the Riidler, 1966], ct regeneration of magnetic field results magnetic field, B, and the critical magnetic Reynolds from the average induction sustained by short lengthnumber,R•,, at whichdynamoactionoccurs.R• is definedsothat (u) = 1, where(...) denotesa volumetric scale helical motion. Our dynamo models are fully 3D, there has been no averagingand they are not meanRMS average. NA"

Kumar andRoberts[1975]foundnumerically conver- field dynamos. However, it is instructive to consider

gent magnetic fields sustainedby a flow of the form

u-

ø+

+

+

the dynamo action of our models in terms of helicity,

h = u. (V x u), and its regenerative effects[Parker, 1979]. Helicity arises from the cross products of pairs (2)

ofvelocityharmonics. Theconvective termss22s ands22c

where{e0,e•, ½2}are adjustableparameters.Toroidal generateonly axisymmetric helicity,Ih}as, whilstthe

LOVE AND GUBBINS: DYNAMOS

DRIVEN

BY POLOIDAL

FLOW EXIST

859

90

b

Figure3. Themagnetic fieldformodel (.9:(a)meridional section showing azimuthally averaged magnetic field lines,(b)satellite viewshowing contours ofB,,,(c)equatorial section showing contours ofBo,(d)meridional section showing azimuthally average contours ofB,, (e)meridional section showing contours ofB, at longitude •b- 0, (f) meridional section showing contours ofB, at longitude •b- 90.

meridional circulation s• ø combines witheachconvective No dipole-symmetricdynamo was found with ex = 0, helicity,/h)NA -- 0, and no term to givenonaxisymmetric helicity,(h}NA,varying the caseof axisymmetric in azimuth as sin2•b,seePig. 5.

dipole-symmetric dynamo is sustained for e2 - 0, the

caseof no helicity,{h) = 0. Interestingly,the most efficient of our dynamo models, that with the small-

est R•, model (9, is the dynamowith very nearly the

largest/h)•,

seeFig. lb. This resultindicatesthat

the nonaxisymmetric distribution of helicity is an important constituentin dynamoswith dipole symmetry. That the spatial arrangementof helicity is an important factor in dynamo efficiencyhas been reported previously

10 ø 10-1

[Loveand Gubbins,1996].

10-2

In mean-field theory, a is proportional to the mean

helicityand for an a 2 dynamoone might expectthe 10-3

most efficientdynamo would be the one with the largest

10-4

although they do not possessany toroidal differential rotation and appear to operate through a mechanism

(h)•s. Howeverthesedynamosare macroscopic, and I

I

I

I

I

I

0

5

10

15

20

25

somewhatakin to the a 2 mechanism, they are fully 3D

and thus helicity cannot be simply related to a generation. Axisymmetric helicity distorts field lines unil*igure 4. Magneticenergyof (.9asa functionof spheri- formly around the axis, which subsequentlyleads to calharmonicdegree.The dashed(solid)line represents cancellation rather than reinforcement of the preexistthe spectrumof the magneticfield calculatedwith a ing field. In these dynamoscancellationis reducedby spherical harmonic expansiontruncated at degree 16 the meridional circulation, which imparts an azimuthal Sphet'ica,l harmonic degree

(28). The energydecreases with increasing harmonic variationto the helicity,(h)•, andan azimuthalvariadegree,and sincethe modelsare adequatelyconverged, tion in magnetic regenerationas required by Cowling's the inclusionof higher degreeterms, thoseabovedegree theorem. It is this 3D induction, not the helicity itself, 16, has little effect on the overall solutions. which, when spatially averaged,givesa net a-effect.

860

LOVE AND GUBBINS: DYNAMOS DRIVEN BY POLOIDAL FLOW EXIST

Bullard, E. C. and H. Gellman, Homogeneous dynamos and terrestrial magnetism,Phil. Trans. R. Soc. Lond., 247, 213-278, 1954.

Busse,F. H., A necessaryconditionfor the geodynamo, J. Geophys.Res., 80, 278-280, 1975. Childress,S., Th•orie magn•tohydrodynamique del'effet dynamo, Dep. Mec. Pac. Sci., Paris, 1969.

Cowling,T. G., The magneticfield of sunspots,Month. Not. R. Astr. Soc., 94, 39-48, 1934.

Elsasser,W. M., Induction effectsin terrestrial magnetism' Part I. Theory, Physical Review, 69, 106-116, 1946.

Hide, R. and T. N. Pahner, Generalizationof Cowling's theorem, Geophys.Astrophys.Fluid Dyn., 19, 301-309, 1982.

Kaiser, R., B. J. Schmitt and F. H. Busse,On the invisible dynamo, Geophys.Astrophys.Fluid Dyn., 77, 93-109, 1994. Kumar, S. and P. H. Roberts, A three-dimensionalkinematic dynamo, Proc. R. $oc. Loud., 344, 235258, 1975.

Love, J. J. and D. Gubbins, Optimized kinematic dynamos, Geophys.J. Int., 124, 787-800, 1996.

Parker, E. N., CosmicalMagnetic Fields: Their Origin and Their Activity, Clarendon Press, Oxford, 1979.

Proctor, M. R. E., On Backus' necessarycondition for dynamo action in a conducting sphere, Geophys. Astrophys. Fluid Dyn., 9, 89-93, 1977.

Steenbeck,M.. F. Krause and K. H. Riidler, Berechnung der mittleren

Lorentz-Feldstiirke

u x B fiir ein

elektrisch leitendes Medium in turblenter, durch

Figure 5. Contours of helicity for model O in merid-

Coriolis-Kriiftebeeinfiu•ter Bewegung,Z. Natur-

ional sections:(a) azimuthallyaveragedhelicity,(b) helicity at longitude05= 0, (b) helicityat longitude

œorsch.,21a, 369-376, 1966.

05= 90.

J. J. Love • David Gubbins, Department of Earth Sciences,University of Leeds, Leeds LS2 9JT, UK

References

Backus, G. E., A class of self-sustainingdissipative spherical dynamos, Ann. Phys., 4, 372-447, (Received January 3, 1996;revised February 25,2996; 1958.

accepted March7, 1996)