Dipole and octapole field reversals in a rotating spherical shell: Magnetohydrodynamicdynamo simulationMárcia M. Ochi, Akira Kageyama, and Tetsuya Sato Citation: Physics of Plasmas (1994-present) 6, 777 (1999); doi: 10.1063/1.873317 View online: http://dx.doi.org/10.1063/1.873317 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/6/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extended magnetohydrodynamic simulations of field reversed configuration formation and sustainment withrotating magnetic field current drive Phys. Plasmas 17, 062502 (2010); 10.1063/1.3436630 Simulation study of the symmetry-breaking instability and the dipole field reversal in a rotating spherical shelldynamo Phys. Plasmas 15, 082903 (2008); 10.1063/1.2959120 Magnetohydrodynamic simulations of turbulent magnetic reconnection Phys. Plasmas 11, 5605 (2004); 10.1063/1.1806827 Reynolds-averaged turbulence model for magnetohydrodynamic dynamo in a rotating spherical shell Phys. Plasmas 11, 5316 (2004); 10.1063/1.1792285 Electron acceleration in the dynamic magnetotail: Test particle orbits in three-dimensional magnetohydrodynamicsimulation fields Phys. Plasmas 11, 1825 (2004); 10.1063/1.1704641

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Dipole and octapole field reversals in a rotating spherical shell:Magnetohydrodynamic dynamo simulation

Marcia M. Ochi, Akira Kageyama, and Tetsuya SatoTheory and Computer Simulation Center, National Institute for Fusion Science, Toki 509-5292, Japan

~Received 29 July 1998; accepted 8 December 1998!

Computer simulation results of a compressible magnetohydrodynamic~MHD! dynamo in a rotatingspherical shell are presented. The reversal of either the dipole or the octapole field polarity isobserved accompanying flip–flop magnetic energy transitions. Three energy transitions areobserved. The dipole reversal occurs in the first transition. In the initial stage of the dipole reversal,the magnetic energy is about five times larger than the kinetic energy, and the convection patternconsists essentially of six straight cyclonic and anticyclonic vortex column pairs aligned to therotation axis. In the intermediate stage, the magnetic energy decreases and the field is observed tochange direction in some regions. This is accompanied by reorganization of the vortex columns anda transition to a new energy level. In the final stage, the field is completely reversed, the number ofconvection columns is increased and the magnetic energy is lower than in the second stage. In theother two magnetic energy transitions, the octapole component reverses polarity. ©1999American Institute of Physics.@S1070-664X~99!04303-7#

I. INTRODUCTION

The magnetic field generation by a magnetohydrody-namic ~MHD! dynamo is an important and fascinating re-search topic and finds many applications in planetary andastrophysical objects. It is well known that the Sun and theEarth have magnetic fields dominantly dipolar and that thesefields may reverse polarity. The magnetic field is believed tobe sustained by dynamo action in the convection regions.Fluid motions of the highly conducting medium of the con-vection regions in the presence of a magnetic field induceelectric currents, which themselves generate the magneticfield.1 Much computer simulation effort has been done in thisresearch and important steps toward a satisfactory under-standing of the magnetic field generation mechanism inspherical shell geometry have been achieved.2–13 The essen-tial three-dimensional, nonlinear, and self-consistent natureof the dynamo problem makes computer simulation a veryfascinating and challenging tool in the MHD dynamo.

Interesting research has been developed by Glatzmaierand Roberts.2–5 In Refs. 2 and 3, they study the MHD dy-namo in a rotating spherical shell using the Boussinesq ap-proximation in the regime of parameters, which are consid-ered to be appropriate for modeling the Earth’s field.However, it is not clear whether viscous effects are suffi-ciently small in their model for geodynamo applications. Thedipole field generation in this model occurs exclusively nearthe inner–outer core boundary. The dipole field reversal isobserved to occur once in their simulation. In their recentpapers,4,5 these authors show the results obtained for anelas-tic approximation and show the effects when compositionalbuoyancy, in addition to thermal buoyancy, is introduced asthe driving mechanism for convection in the shell region.However, dipole reversals are not observed in this new ap-proach. Independently and simultaneously, Kageyama andSato11–13 exhibit the dipole field generation in a rotating

spherical shell. The dipole field generation is clearly ex-plained in terms of the so-calledav mechanism,12 and it isshown to be closely related to the well-organized structure ofthe fluid motion in convection columns aligned to the rota-tion axis. The physical parameters are not chosen to model aspecific natural system, but rather to study the general prop-erties of MHD dynamos. Recently, Kuang and Bloxham6

presented an alternative approach for geodynamo studies us-ing a model where the effects of viscosity are made verysmall, and thus this model is more suitable for modeling theEarth’s magnetic field. They show that their magnetic fieldgeneration mechanism is very different to that presented bythe Glatzmaier and Roberts’ model. However, dipole fieldreversals are not observed.

Sakuraba and Kono7 also study the geodynamo problemusing the Boussinesq approximation. They investigate theeffects of the presence of the inner core by running twodifferent simulations: one where the inner core is present andthe other where the inner core is not present. The authorssuggest that the Earth’s magnetic field will be stabilized asthe inner core grows, and that the field generation mecha-nism is related to a strong velocity shear and helicity of thefluid near the top and the bottom shell boundaries.

Kitauchi and Kida8,9 also study the incompressible fluidcase using the Boussinesq approximation and study the mag-netic field generation mechanism. They emphasize that thespatial correlation between the magnetic field and velocityfield, which is organized in robust convection columnpairs,10 is important in understanding the field generationprocess.

In the present simulation, we chose parameter settings,and initial and boundary conditions that would result ingreater thermal convection compared to our previoussimulation.12 To achieve this, we increased the temperaturedifference between the inner and outer boundaries of the

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spherical shell. Many new features are observed in this case.During the simulation time interval, three flip–flop energytransitions occur. The main result of the present simulation isthat we observe either the reversal of the dipole or the octa-pole component at each energy transition. The reversal of thedipole occurs in the first transition. This is the first time thatthe dipole field reversal is reported in a simulation wherecompressible effects are taken into consideration. The othertwo energy transitions are not accompanied by the dipolereversal, but rather by octapole reversals. The octapole is thesecond main symmetric magnetic mode. The results of oursimulation will be presented, highlighting the main features.The physical mechanism for the dipole and octapole rever-sals is not completely understood at the present stage. Amore complete analysis is difficult and is the topic of ongo-ing research. We present the results to the reader to promotediscussions.

In Sec. II we present the physical model and the numeri-cal method. The set of equations used in the simulation ispresented, as well as the initial and boundary conditions.Sections III and IV contain the simulation results where weemphasize the flip–flop magnetic energy transitions and fieldreversals, the convection motion, and the magnetic fieldstructures. To conclude, in Sec. V, the final comments andsummary are presented.

II. THE MODEL AND THE NUMERICAL METHOD

The simulation model is simple but contains the essentialelements required to understand the fundamental physicalprocesses in the MHD dynamo. We study a system consist-ing of an inner spherical core, which has a heat source tokeep its surface (r 5r i) at a high temperature; an outerspherical surface (r 5r o) that is kept at a low temperature;and an intermediate conducting fluid between these twospherical boundaries. The fluid motions are driven by ther-mal convection. The system rotates eastward at a constantangular velocityV. We use a reference system rotating at thesame angular velocity. The conducting fluid is studied by theMHD equations~including gravity and Coriolis force terms!as follows:

]r

]t52“–~rv!, ~1!

rdv

dt5 j3B2“p1rg12rv3V1mS ¹2v

11

3“~“–v! D , ~2!

1

g21

dp

dt52

g

g21p“–v1K¹2T1h j21F, ~3!

]A

]t52E, ~4!

p5rT, ~5!

B5“3A, ~6!

E52v3B1h j , ~7!

j5“3B, ~8!

“–B50, ~9!

g52g0

r 2 r , ~10!

F52m@ei j ei j 213~“–v!2#, ~11!

ei j 51

2 S ]v i

]xj1

]v j

]xiD , ~12!

wherer is the mass density,p is the pressure,v is the ve-locity, andB is the magnetic field. HereA, j , andE are thevector potential, current density, and electric field, respec-tively. The ratio of specific heat at constant pressure to thatat constant volumeg~55

3!, the viscositym, the thermal con-ductivity K, and the electrical resistivityh are assumed to beconstant. Hereg is the gravitational acceleration,r is theradial unit vector, andg0 is a constant.

The integration of the MHD equations in spherical coor-dinates (r ,u,f) is based on the finite difference method.Simulations are performed in the full spherical shell region(r i<r<r o,0<u<p,0<f,2p). The grid numbers(Nr ,Nu ,Nf) are chosen to be equal to~50,38,64! to achievesufficient accuracy at reasonable CPU time. The detailsabout the numerical techniques, especially those concerningthe singularity problem on the poles and the severe Courant–Friedrichs–Lewy~CFL! condition on the time step due to theconcentrated grid points near the poles, have already beendiscussed in Ref. 11.

The boundary conditions atr i and r o are v50 and thatthe magnetic field has only a radial component. This guaran-tees that Poynting vector,E3B, has no radial component onthe boundaries. This ensures that any magnetic field must bea consequence of the dynamo action in the bulk of thespherical shell.

The temperature on the inner boundary is 3.5 timeslarger than the temperature on the outer boundary. The innerand outer boundary temperatures are kept constant through-out the simulation.

The parameters for this simulation arer i50.3, K54.24331023, g051.0,V57.0,h52.831024. The Taylornumber is defined byT5@2Vr(r o2r i)

2/m#2 and the Ray-leigh number by Ra5g0r(bcq2g0)(r o2r i)

4/(Km), whereb is the temperature gradient coefficient~the initial tempera-ture profile is given byb/r 1const! and cq5g/(g21). Inthis case,T and Ra are 5.883106 and 3.363104, respec-tively. The temperature gradient coefficient, and thus theRayleigh number, was increased to give greater thermal con-vection compared to our previous simulation.12

Lengths and temperatures are normalized to the radiusr o

and the temperature on the outer boundary, respectively. Thenormalization time is the sound crossing time (5r o /vs),wherevs is the sound speed. In this simulation, the thermaldiffusion time and resistive diffusion time are, respectively,equal to 116 and 1750. The simulation goes until the timeequals to 27 000, which is 15.4 times the resistive diffusiontime. This simulation required approximately 300 hours ofCPU time in the NEC/SX-4 supercomputer.

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III. FLIP–FLOP ENERGY TRANSITIONS

The simulation starts from an unstable hydrostatic andthermal equilibrium state with no magnetic field. A tempera-ture perturbation is introduced that causes the convectionmotion to begin. After a short time~aroundt5150!, the sys-tem reaches a saturation energy level. The convection motionpattern att5572 can be seen in Fig. 1~a!, just before theinitial magnetic field seed is introduced. The convection mo-tion pattern is verified to be well organized in five pairs ofalternating cyclonic~yellow color! and anticyclonic~bluecolor! convection columns whose axes are parallel to andencircle the rotation axis. The fluid in the cyclonic~anticy-clonic! column rotates in the same~opposite! direction as therotation of the spherical shell. These columns are isosurfaces

of the axial vorticity withvz560.9. At t5577, a randomand weak magnetic field perturbation is introduced. In thelinear regime, the magnetic field increases exponentially asthe convection motion pattern remains constant.

Figure 2 shows the total thermal, kinetic, and magneticenergies~logarithmic scale! versus time~normalized units!.These energies are integrated over the spherical fluid shell.Before introducing the initial magnetic field seed and duringthe linear regime~from time equal to 572 until 1656!, we cansee that there are no fluctuations in the kinetic energy. In thenonlinear regime, when the magnetic energy becomes large,time fluctuations are observed and the average magnetic en-ergy alternates between two fluctuating energy levels. Thelower level is approximately equal to the kinetic energy, and

FIG. 1. The convection motion pattern by the isosurfaces of the axial vorticity at level equal to60.9 for four different times:~a! 572,~b! 4843,~c! 8825, and~d! 25 610. The yellow~blue! color represents the cyclonic~anticyclonic! convection columns.

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the upper level is about five times larger. Different nonlinearregimes occur. We will refer to the four different nonlinearregimes as NR1, NR2, NR3, and NR4. The regime NR1refers to the energy state where the average magnetic energyis about five times larger than the kinetic energy. In NR2, theaverage magnetic energy is about equal to the kinetic energy.In NR3, the magnetic energy recovers the level of NR1. Theregime NR4 follows NR3. The thermal energy is about threeorders of magnitude higher than the magnetic energy andshows almost no variations in different nonlinear regimes,though there is a relatively small variation~within 1.0%!, inaccordance with the energy transitions.

In Figs. 1~b!–1~d!, we show the convection column pat-terns for three different times, specifically equal to 4843~NR1!, 8825~NR2!, and 25,610~NR4!. We can see that theconvection motion pattern is distinct for different times. Thisset of pictures is taken as an example: it is not intended toindicate that the convection structure remains constant ineach of the nonlinear regimes. Reorganization of the convec-tion columns occurs several times during the simulation. InFig. 1~b!, the number of convection column pairs is six, andin Fig. 1~c! and Fig. 1~d!, it is equal to seven. The number ofconvection column pairs tends to be larger in NR2 and NR4than in NR1 and NR3. We can note that the convection mo-tion pattern in the nonlinear regime is much more compli-cated than in our previous simulations.11–13 In our previoussimulations there was a fixed number of convection columnpairs regularly displayed around the rotation axis. Here, al-though the convection motion is always organized in cy-clonic and anticyclonic columns, these columns are neitherregularly distributed around the rotation axis nor constant inshape and number.

In Fig. 3 we show the time evolution of the radial com-

ponent of the velocity,v r , at the equator on the radiusr50.5. Thex axis is the longitude and they axis is the nor-malized time. Here we limit ourselves to show the graph forregime NR1. This is enough to illustrate the general driftmotion behavior in the nonlinear regimes, as the drift motionbehavior in NR2, NR3, and NR4 is qualitatively similar. Theblack regions correspond tov r.0 ~rising fluid! and thewhite regions, tov r,0 ~sinking fluid!. In this figure it isclear that in the nonlinear regime the number of convectioncolumn pairs is not constant and that their drift motion di-rection ~east- or westward! changes several times.

In NR1 and NR3, the number of convection columnpairs fluctuates between four and six. In NR2 and NR4, thenumber of columns is higher, fluctuating between six andeight. The several breaks and recombinations of the columns,and the changes of the drift motion velocity and direction,make the convection motion structure, and consequently themagnetic field structure, very complicated.

IV. THE ASSOCIATED DIPOLE AND OCTAPOLEFIELD REVERSALS

The magnetic field generation process has already beenreported in our previous work.12 Here we restrict ourselves to

FIG. 2. The time dependence of the thermal, kinetic, and magnetic energies.The different regimes in the nonlinear phase are denoted by NR1, NR2,NR3, and NR4.

FIG. 3. The drift motion of the convection columns as a function of time forregime NR1. The black~white! represents the rising~sinking! fluid at theequator andr 50.5. The horizontal axis represents the longitude~f direc-tion!.

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show the new features of the present simulation. The mag-netic field on the outer boundary (r 5r o) is expanded inspherical harmonicsYl

m(u,f), Br(r o)5( l 51Sm52 ll al

mYlm .

The dipole~l 51, m50! is observed to be the main compo-nent. Figure 4 shows the time dependence of the dipole,quadrupole, and octapole contributions to the total magneticfield ~for instance, the dipole power is equal to the ratio ofua1

0u2 to the sum of all other components!. We can observe inFig. 4 that the dipole contribution can be as high as 60%.The octapole~l 53, m50! is the second main symmetricmode while the quadrupole~l 52, m50! is very small andstrongly fluctuating.

In Fig. 5~a!, the magnitude of the dipole and the octapolemoments on the outer boundary,ua1

0u and ua30u, respectively,

are presented as functions of time. In order to remove thefluctuations, a local average was taken and it is shown in Fig.5~b!. The smoothed dipole curve shows a reversal at timeequal to 7822, in the energy transition from NR1 to NR2, asteady increase from NR2 to NR3, then a dipole intensitytransition from NR3 to NR4. Although the smoothed dipolegraph does not show any difference between the NR2 andNR3 regions, Fig. 5~a! shows that the dipole moment fluc-tuation increases greatly from NR2 to NR3. The smoothedoctapole curve shows a reversal from NR2 to NR3 and fromNR3 to NR4.

Figure 6 shows the longitudinally averaged toroidalmagnetic field on the meridian plane. The red color meanseastward directed and the blue, westward directed. Figure 6corresponds to the time interval between the end of the NR1regime and the beginning of the NR2 regime. The sequenceof snapshots starts from the magnetic field configuration be-fore the dipole field reversal, shows the process during re-versal, and the transition to the NR2 regime. Figure 6~z!shows a typical average toroidal configuration in the NR2

regime. In Fig. 7, we show the corresponding averaged po-loidal magnetic field on the meridian plane.

We observe that the dipole reversal occurs only once inthe time interval of our simulation. The dipole reversal oc-curs at the transition from NR1 to NR2. A natural questionthat may rise is why the dipole reversal is not observed in thetransition from NR3 to NR4. In Fig. 8, we show for com-parison’s sake the set of snapshots of the average toroidalfield on the meridian plane, for the time interval between18 163 and 18 602. We can observe that in Fig. 8~a!, thereare four spots of the toroidal magnetic field, similarly as inFig. 6~a!. The number of convection columns is six and thepoloidal magnetic field has a positive~south→north! direc-tion. Several changes in the average toroidal magnetic fieldare observed following the snapshot in Fig. 8~a!, but thesubsequent steps are different from those observed in Fig. 6.This suggests that the nonlinear regions defined by thechanges in the magnetic energy level do not completely de-scribe the complicated dipole reversal mechanism.

A more global picture on how the dipole field reversal

FIG. 4. The contribution of the symmetric modes (m50) to the total mag-netic field as a function of time. The black line represents the dipole contri-bution, the green line, the octapole contribution, and the red line, the quad-rupole contribution.

FIG. 5. ~a! The dipole and the octapole and~b! the smoothed dipole andoctapole moments on the outer boundary,r 0 , as functions of time. The redline represents the dipole moment and the green line, the octapole moment.

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FIG. 6. It shows the longitudinally averaged toroidal magnetic field configuration on the meridian plane for times equal to~a! 7612,~b! 7621,~c! 7630,~d!7639,~e! 7648,~f! 7653,~g! 7657,~h! 7666,~i! 7670,~j! 7675,~k! 7679,~l! 7688,~m! 7751,~n! 7804,~o! 7822,~p! 7853,~q! 7898,~r! 7942,~s! 7987,~t!8031,~u! 8076,~v! 8121,~x! 8165, and~z! 8210. The red and blue shades represent the contours of the averaged toroidal component of the magnetic field.The blue color indicates westward directed and red, eastward directed. This sequence corresponds to the time interval by the end of the NR1 regime and thebeginning of the NR2 regime.

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occurs is provided by Fig. 9, where we show the Mercatorprojection of the magnetic field radial component onr o forthree different times:~a! 7576,~b! 7711, and~c! 8156. Thissequence corresponds to times before, near, and after thedipole reversal. In Fig. 9~a!, the field direction is negative inall the magnetic field concentration regions. The concentra-tion regions correspond to those of the cyclonic convectioncolumns. In Fig. 9~b!, we can note that, in the reorganizationprocess of the convection columns, the field directionchanges to positive in two columns and is negative in theother columns. Finally, in Fig. 9~c!, the field is completelyreversed. In Fig. 10 we show the convection motion patternsfor three different times, which correspond exactly to thosein Fig. 9. The yellow and blue colors have the same meaningas in Fig. 1 and the columns are isosurfaces of the axialvorticity with vz560.8.

V. FINAL COMMENTS AND SUMMARY

It is the topic of ongoing research to understand the fun-damental cause of the field reversals, but presently we areunable to give a comprehensive interpretation of the results.To avoid presenting misleading conclusions we rather dis-cuss features of the results that we believe may be associatedwith the reversal process.

We observe that the nonlinear interaction between themagnetic field and fluid velocity causes many changes in theshape and number of the convection columns as shown inFigs. 1 and 3. During the linear growth regime, when the

magnetic field is still weak, the convection motion pattern ischaracterized by five pairs of cyclonic/anticyclonic vortexcolumns regularly distributed around the rotation axis andthey drift westward. This convection pattern is illustrated inFig. 1~a!. When the magnetic field grows and reaches satu-ration levels, the nonlinear effects become important, and asa result the well-organized convection pattern is substitutedfor a very complex picture. Figures 1~b!–~d! show some ex-amples of the convection configuration for different times indifferent regimes. Changes in the drift motion occur manytimes, as we can observe in Fig. 3. These changes allow thecolumns to experience sometimes breaks and sometimescombinations. Therefore the number of the columns does notremain constant. Drastic modifications in the convection mo-tion pattern are observed during time intervals correspondingto the transition from a nonlinear regime to the next one.Despite the complexity of the simulation data, we can saythat the structure of the convection motion in vortex columnsis kept, and the flow regime is laminar all the time.

In Fig. 2, it is noted that during the nonlinear phase themagnetic and kinetic energies are highly time dependent.Fluctuations are observed and the system state at saturationlevels is not stationary. The convection motion is constantlyexperiencing reorganization processes, with several breaksand combinations of the convection columns, as alreadymentioned. This feature can be identified as the main sourceof the fluctuating magnetic fields and may be partly relatedto the magnetic field reversals. The energy analysis showsthat, after some time, precisely after regime NR1, the mag-netic energy drops, being accompanied by a slight decreaseof the kinetic energy. During the magnetic energy reductionprocess~the transition from NR1 to NR2!, the reversal of thedipole field occurs. The average kinetic and magnetic ener-gies keep these levels for some time, which corresponds tothe regime NR2 and then the system recovers the previouskinetic and magnetic energy levels, characterizing regimeNR3, which is followed by the longest regime, NR4. Thisregime goes until time equals to 27 000, when we stoppedour simulation. In NR1 and NR3 regimes, the number ofpairs of vortex columns fluctuates among4–5–6,while inNR2 and NR4 regimes this number remains larger. On aver-age, we can say that the level of order in the convectionmotion is higher in NR1 and NR3 regimes when the mag-netic energy is higher than in NR2 and NR4 regimes.

The strong dipole contribution, which remains as themain mode during almost all the simulation, is shown in Fig.4. The reversal of either magnetic dipole or octapole is ob-served at each transition between different energy levels. Inthe second and third magnetic energy transitions, the dipolecomponent does not reverse polarity, but shows an increase~NR2→NR3! and a transition~NR3→NR4! in its intensity.The analysis in terms of reversals and transitions is madeclearer in Fig. 5~b!, where we presented the smoothed dipoleand octapole moments versus time. The smoothed dipolecurve shows a reversal from NR1 to NR2, a gradual increaseduring NR2 and NR3, and an intensity transition from NR3to NR4. The smoothed octapole curve shows a reversal fromNR2 to NR3 and from NR3 to NR4. To our knowledge, theobservation of transitions between different magnetic energy

FIG. 7. It shows the poloidal magnetic field for the corresponding timesequence in Fig. 6.

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levels, which is accompanied by either dipole or octapolefield reversals, is a new result for this kind of simulation.

In Figs. 6 and 7, the average total magnetic field isshown during the transition from NR1 to NR2. The dipolefield reversal accompanies this transition. We note that thepoloidal and toroidal components of the magnetic fieldchange direction in the dipole reversal process@for instance,compare the snapshots~a! and~o! in Figs. 6 and 7, note thatthe dipole field polarity is determined by the field orientationin the middle and low latitutes#. The reason why the direc-tions of the poloidal and toroidal components are related toeach other can be understood in the same fundamentals as inRef. 12. Basically, the toroidal component of the magneticfield is converted from the poloidal field component by dif-ferential rotation. Meanwhile, the poloidal component is gen-erated from the toroidal field by a process of swallowing~longitudinal convection motion! and stretching~axial flowalong the convection columns! of the field lines that occursbetween a cyclonic and an anticyclonic convection column.The poloidal field generation process was pointed out as be-ing very similar to that which occurs in the field-reversedconfiguration~FRC! in plasma experiments,14,15 when twocountermagnetic helicity spheromaks merge together, andthe resultant field is a FRC. The toroidal field in Ref. 12 is

distributed throughout the outer core, contrarily to Glatz-maier and Roberts’ simulation,2 where it is concentrated nearthe inner–outer core boundary. The magnetic field genera-tion mechanism is very different in these two models. In thepresent simulation, we observe two toroidal field spots ofopposite signs in each hemisphere, in contrast to only one inour previous simulations. This seems to be an importantpoint to understand why the dipole reversal was not observedin Ref. 12.

In Fig. 6~a! for time equal to 7612, there are essentiallyfour spots of the toroidal magnetic field: two big spots, thered one in the northern hemisphere and the blue one in thesouthern hemisphere; and two smaller ones near the equator,the blue one in the north and the red one in the south. We cansee that the configuration is approximately antisymmetricrelative to the equator. The dipole field polarity is negative~north → south direction! and is associated with the twoopposite sign toroidal field concentrations near the equator.In Fig. 6~b! at time equal to 7621, we can see that two smallspots move to the equator, leaving from the two big spots.For the sake of explanation, we will refer to them as mr1~thered spot! and mb2~the blue one!. In Fig. 6~c!, time equal to7630, some rearrangement of the two small spots near theequator is observed. In Figs. 6~d! and 6~e!, times equal to

FIG. 8. It shows the longitudinally averaged toroidal magnetic field configuration on the meridian plane near the end of the NR3 regime for times equal to~a! 18 163,~b! 18 198,~c! 18 259,~d! 18 268,~e! 18 294,~f! 18 308,~g! 18 321,~h! 18 387,~i! 18 404,~j! 18 426,~k! 18 479, and~l! 18 602. The blue colorindicates westward directed and red, eastward directed.

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7639 and 7648, respectively, we observe that there is a ten-dency for reconnection between mr1 and the red spot nearthe equator. However, in Figs. 6~f! and 6~g!, times equal to7653 and 7657, this tendency is not confirmed and the spotmr1 moves back to the northern hemisphere. During the timeinterval in Figs. 6~f! and 6~g!, the field polarity changes fromnegative to positive in the northern hemisphere and remainsnegative in the southern hemisphere, as observed in the cor-responding snapshots in Fig. 7.

In the sequence given by Figs. 6~h!–6~m!, times between7666 and 7688, the spot mr 1 becomes large in the northernhemisphere, the small red spot near the equator disappears,and the blue small spot in the northern hemisphere moves tothe southern hemisphere. In this process, the field polaritychanges also in the southern hemisphere from negative topositive. The total magnetic field direction is positive fromthe snapshot of Fig. 6~o! on. After some time, accompaniedby changes in the number of convection column pairs and inthe magnetic field configuration, the new regime NR2 starts.The dipole polarity remains positive.

During the time interval between Figs. 6~a! and 6~i!, thenumber of vortex column pairs remains equal to six all thetime. However, the shape and size of these columns are notconstant. These changes are responsible for modifications in

the magnetic field configuration observed in the figure. Attime equal to 7675@Fig. 6~j!#, the number of column pairschanges to five and then to four and during the transition toNR2, several modifications occur in the shape and the num-ber of column pairs increases to higher numbers. The regimeNR2 is characterized by a larger number of convection col-umns.

Figure 7 shows the average poloidal magnetic field tothe corresponding times in Fig. 6. The average poloidal mag-netic field has a negative~north→south! direction until thetime approximately equals 7648~see the direction of the ar-rows in the middle and low latitute regions!. Then the poloi-dal field reverses in the northern hemisphere first and later inthe southern hemisphere, as we can see in the sequence givenby Figs. 7~a!–7~m!. From time equal to 7822 on@Fig. 7~o!#,the poloidal field is positive.

The existence of two toroidal magnetic field concentra-tions in each hemisphere that are antisymmetric relative tothe equator seems to play a role in the dipole reversal pro-cess. The complete understanding of the dipole reversal phe-nomenon will require additional analysis. In the presentwork, we limit ourselves to present the results and discussthem to some extent. A configuration with four spots of thetoroidal magnetic field occurs several times during NR1 andalso during NR3. In particular, in Fig 8~a! we show that asimilar configuration occurs at the end of the NR3 regime.When this configuration occurs in our simulation, it is al-ways associated with a well-organized convection motionconfiguration. In other words, this magnetic field configura-tion occurs when the convection columns are regularly dis-tributed around the rotation axis, and they are almost of thesame size. But as it was already mentioned, the configurationof the columns is very time dependent and new magneticfield configurations are constantly being produced. The tran-sition from NR1 to NR2 provides a favorable configurationto reversal, but the transition from NR3 to NR4 does not. Onthe other hand, the octapole that is the second strongest sym-metric mode reverses polarity in the last two transitions be-tween different magnetic energy levels.

Figure 10 illustrates how the convection motion struc-ture changes during the dipole reversal process, while Fig. 9shows the reversal process from the Mereator projectionpoint of view; the radial component of the magnetic fieldwas analyzed on the outer boundary. It is clear from bothfigures that the reversal process occurs in three steps. Thefield concentration regions in Fig. 9 correspond to those ofthe cyclonic convection columns. In Fig. 9~a!, in all the con-centration regions, the magnetic field is negative. This pic-ture corresponds to a certain time in the regime NR1. In Fig.9~b!, the field changes sign in two columns. This correspondsto a configuration in the middle of the transition from NR1 toNR2. The reversal process is accompanied by a reorganiza-tion process of the convection columns. We can see that thenumber of columns decreases from Figs. 10~a! to 10~b!. InFig. 9~c!, the reversal process is already completed and thefield is positive in all the columns. The number of columns ishigher and the size of the columns smaller, as it is shown inFig. 10~c!. The magnetic energy is even lower and the sys-

FIG. 9. The Mercator projection of the radial component of the magneticfield on the outer boundary,r 0 , for three different times:~a! 7576,~b! 7711,and ~c! 8156. The solid lines represent the positive values and the dashedlines, the negative ones.

785Phys. Plasmas, Vol. 6, No. 3, March 1999 Ochi, Kageyama, and Sato

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tem reaches already the regime NR2. In conclusion, Figs.9~a!–9~c! show that the dipole reversal occurs in few steps:first in some of the columns and then in the others, until theglobal dipole polarity reversal process is finally completed.

We presented the simulation results on the compressibleMHD fluid in a rotating spherical shell for a set of param-eters where the dipole polarity reversal is successfully ob-served. This is the first time that the dipole reversal result isreported in a compressible MHD fluid. To understand whatparameters are crucial for reversal process to occur, a com-plete parameter survey is required. However, as each simu-lation requires about 300 CPU hours on a supercomputer,clearly unrealistic computational resources would then be re-

quired. The first results on the dipole field reversal in a three-dimensional self-consistent numerical solution of the MHDequations were obtained by Glatzmaier and Roberts in Ref.2. Many differences are observed between their model andour model. The set of equations, the numerical method, therange of physical parameters, and the boundary conditionsare among these differences. We adopt a simple picture tostudy the MHD dynamo, and we successfully obtain the di-pole field generation and its reversal. This suggest that theintermittent dipole reversal of the Earth’s magnetic field ispart of the intrinsic nature of a MHD open system. Specificphysical conditions are not required to observe the dipolereversal in our model.

FIG. 10. The convection motion pattern, by the isosurfaces of the axial vorticity at a level equal to60.8, for three different times:~a! 7486,~b! 7711, and~c!8156. The yellow~blue! color represents the cyclonic~anticyclonic! convection columns.

786 Phys. Plasmas, Vol. 6, No. 3, March 1999 Ochi, Kageyama, and Sato

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In summary, a MHD dynamo simulation has shown~a!the generation of a strong magnetic field in a rotating spheri-cal shell, mainly the dipole;~b! the flip–flop transitions be-tween different magnetic energy levels; and~c! the magneticdipole or octapole field reversal at each energy transition.

ACKNOWLEDGMENT

This work is supported by the Ministry of Education,Science, and Culture of Japan.

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