LETTERS

Formation of current coils in geodynamo simulationsAkira Kageyama1, Takehiro Miyagoshi1 & Tetsuya Sato1

Computer simulations have been playing an important role in thedevelopment of our understanding of the geodynamo1–3, butdirect numerical simulation of the geodynamo with a realisticparameter regime is still beyond the power of today’s supercom-puters. Difficulties in simulating the geodynamo arise from theextreme conditions of the core, which are characterized by verylarge or very small values of the non-dimensional parameters ofthe system. Among them, the Ekman number, E, has beenadopted as a barometer of the distance of simulations from realcore conditions, in which E is of the order of 10215. Following theinitial computer simulations of the geodynamo4,5, the Ekmannumber achieved has been steadily decreasing, with recent geo-dynamo simulations6–8 performed with E of the order of 1026.Here we present a geodynamo simulation with an Ekman numberof the order of 1027—the highest-resolution simulation yetachieved, making use of 4,096 processors of the EarthSimulator. We have found that both the convection flow andmagnetic field structures are qualitatively different from thosefound in larger-Ekman-number dynamos. The convection takesthe form of sheet plumes or radial sheet jets9, rather than thecolumnar cell structures10 that are usually found. We have foundthat this sheet plume convection is an effective dynamo and thegenerated current is organized as a set of coils in the shape ofhelical springs or at times as a torus.

The Earth’s outer core is liquid iron in convective motion. Tomodel the dynamo process in the outer core, we have performed athree-dimensional numerical simulation of a magnetohydrodynamic(MHD) dynamo in a rotating spherical shell: an electrically conduct-ing fluid is confined between two concentric and co-rotating sphereswith inner and outer radii of ri 5 0.3 and ro 5 1.0, respectively. Thetwo spheres rotate with the same angular velocity V. The tempera-tures of the inner and outer spheres are kept hot and cold, respect-ively. Gravity works towards the centre of the spheres. We apply weakperturbations to the temperature and magnetic fields of the initial(unstable) condition. Thermal convection sets in, and the flowgenerates the magnetic field through the so-called MHD dynamoprocess. The Rayleigh number, Ra (ref. 11), measured at the bottomof the shell is 1.5 3 1010, which is 300–1,000 times larger than thecritical value for the onset of convection. Both the Prandtl and mag-netic Prandtl numbers are unity. The Ekman number3 defined byE~n

�2Vr2

o is 2.3 3 1027 in this simulation, where n is viscosity.We believe that this is the lowest Ekman number achieved to datein geodynamo simulations.

The smaller the Ekman number, the more difficult it is to performthe simulation, as it requires higher resolution and a higher paralle-lization rate of the code12. By making use of a newly developed spher-ical grid system, the ‘Yin-Yang’ grid13,14, we have achieved a highresolution geodynamo simulation for a full spherical shell region.The Yin-Yang grid is a kind of overset grid15, applied to the sphericalgeometry. Two identical grids, the Yin grid and the Yang grid, arecombined with partial overlap to cover the full spherical shell region.The grid size is 511 (in r) 3 514 (in h) 3 1,538 (in w) 3 2 (Yin and

Yang), with r radius (0.3 # r # 1.0), h co-latitude (p/4 # h # 3p/4),and w longitude (23p/4 # w # 3p/4). For this simulation, we haveused 512 nodes or 4,096 processors of the Earth Simulator, which isthe maximum size allowed for a calculation.

It is broadly accepted that the convection flow in a rapidly rotatingspherical shell may be described as a set of (time dependent) colum-nar convection cells16. The convection columns are straight and par-allel to the spherical rotation axis, due to the strong constraint of theCoriolis force (Taylor-Proudman’s theorem). However, in our low-Eregime (of the order of 1027), we have found that the convection ismade of a rather different basic structure—fine-scale thin jet sheetsor sheet plumes.

In the growing phase of convection in our simulation, the flow isformed in multicellular columns piled in the radial direction17, butafter saturation, the convection takes a rather different form.Figure 1a shows the z or axial component of the vorticity, vz, inthe equatorial and meridional planes at time t 5 430 in our simu-lation unit, which is normalized by the sound wave crossing time ofro. The major flow is composed of many plumes elongated in theradial direction, rather than columns. Analysing the velocity vectorsin detail, we have found that the plume structure is composed of jetflow in the positive s-direction and in the negative s-direction, sideby side; here we use cylindrical coordinates (s, w, z) for descriptiononly. Supplementary Fig. 1 shows the velocity field in the equatorialplane by vector arrows. The width w of the jet plumes is very thin(w < 0.025), staying almost constant as radius s increases18. Theazimuthal Fourier mode number, m, of the flow is about 50 ats 5 0.4. The diffusion time td scaled by w is td 5 w2/g 5 34 (whereg is resistivity), which is much shorter than our calculation time of430.

1Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, Yokohama, 236-0001, Japan.

a b

Figure 1 | Equatorial and meridional cross-sections of the axial componentof the vorticity, vz. The Ekman number, E, is 2.3 3 1027 in a and 2.6 3 1026

in b. Convection plumes are evident in the equatorial cross-sections. Themeridional cross-sections show that the flow is nearly two-dimensional. Theconvection in these low-Ekman-number regimes is organized as a set of thinplume sheets, rather than columnar cells. It can be seen that the largerEkman number in b leads to thicker plume sheets.

Vol 454 | 28 August 2008 | doi:10.1038/nature07227

1106

©2008 Macmillan Publishers Limited. All rights reserved

As the meridional cross-section (Fig. 1a) shows, z-dependence ofthe velocity is very weak: the convection can be described as a set ofsheet-like plumes. These sheets are not stationary; the timescale of thepattern changes may be estimated (roughly) from the sheet width wand the flow velocity v (timescale < w/v). The typical magneticReynolds number Rm 5 dv/g 5 700, where d is shell depth(d 5 ro 2 ri). The Elsasser number L is of the order of 1–10. For acomparison, we show simulation results for a larger Ekman number,E 5 2.6 3 1026 (Fig. 1b); this figure suggests that a larger Ekmannumber leads to thicker plume sheets.

Sheet plume convection has already been discovered in laboratoryconvection experiments on water in a hemispherical shell by Sumitaand Olson9. Figure 2a shows a snapshot of a convection pattern oftheir experiment visualized by flakes, seen through the transparentequatorial plane. The Ekman number in their experiment wasE 5 1 3 1026. Figure 2b shows the distribution of vz of our simu-lation visualized by volume rendering viewed from the north. As theflakes are sensitive to velocity shear, the visualization of vz in Fig. 2bis a natural counterpart of Fig. 2a. Note that the sheet plumes branchoff, going in an outward (positive s) direction.

Another important characteristic of the flow observed in our simu-lation is the existence of zonal flow near the outer boundary. Thedirection of the zonal flow is westward. Figure 1a shows an indistinctspiral pattern with four arms, which is embedded in the zonal flowwith azimuthal mode number m 5 0. The power of the zonal flow,measured at the radius in the equator, is an order of magnitude largerthan other modes. The co-existence of the sheet plumes and the zonalflow is a new feature that was not observed in the experiments bySumita and Olson9. This zonal flow region near the outer sphericalboundary is absent when E 5 2.6 3 1026 (Fig. 1b). This could beexplained by a weak constraint imposed by the Coriolis force. Weare analysing the formation mechanism of the zonal flow, and it willbe reported in the near future.

By observing the plume pattern as a movie in the equatorial plane,we found that the plumes can be divided into two parts—a lower part(near the inner core) and an upper part (near and above the middledepth of the spherical shell). The lower part drifts eastward, andupper part drifts westward. We note that the convection velocitydevelops faster than the magnetic field. The sheet plume structureis formed before the magnetic energy becomes large enough to affectthe velocity field. Therefore, the sheet structure is not formed ormaintained by the magnetic field. This is in contrast with a dynamosimulation under a low-E regime in a box geometry19, in which smallscale flows are strongly affected by the Lorentz force.

The sheet plume convection is an effective dynamo. The totalmagnetic energy at t 5 430 (Fig. 1a) is about 4 times larger thanthe total convection energy. The typical Alfven velocity is 0.04 inour units. The magnetic field is generated mainly at mid-latitudeand outside the tangential cylinder. This is confirmed by examiningthe distribution of the dynamo source term D ; 2v?(j 3 B) in thespherical shell, where v, j and B are respectively velocity, current andmagnetic field. By analysing distributions of D, v and B, we havefound that there are two kinds of mechanism by which the magneticfield can be generated in this simulation; in the first kind, the mag-netic energy is generated by downward (2vs) plume flows. In thiscase, typical B is perpendicular to v and the field lines are drawn bythe downward v against the field tension force. This dynamo mech-anism is essentially the same as observed in our laminar dynamosimulation20. The second kind of dynamo mechanism is very differ-ent: in this case, B is parallel to v. The magnetic field is generated byupward (vs . 0) plume flows. The flow is accelerating in the positive sdirection after a plume leaves from the inner hot boundary. Themagnetic field lines embedded in the plume are stretched in theparallel direction by this acceleration. From mass conservation,stretching of a magnetic flux tube causes reduction of the tubecross-section, and therefore, flux density amplification. As the totalvolume of the flux tube is nearly constant, the total magnetic energy isincreased by this stretching process. This magnetic field amplifica-tion by field line stretching is described as a classical picture of theMHD dynamo mechanism21, and it has recently observed in geo-dynamo simulations22.

Stretching by the parallel flow makes the magnetic field bothstrong and straight. Thus a straight flux tube is formed that isembedded in the upward-moving sheet plume. The localizedstraight flux tube leads to the formation of a current coil aroundthe tube. Figure 3 shows a whole current field structure visualized bycurrent lines, started from a sphere at r 5 0.46 with nearly uniformdistribution of starting points. We can see that there are manycurrent coils. The distribution of the current coils is relatively loca-lized in the north and south mid-latitude. By analysing the currentfield in detail by virtual reality visualization23,24, we have found thatmost of the current coils are in the form of a helical spring, but thatsome of them are in a torus configuration. Plotting the distributionof the dynamo source term D, we have confirmed that the currentcoils are located in regions of strong positive D. The magnetic field isgenerated at the current coils. Figure 4 shows typical current coilswith magnetic field lines; a helical spring at higher latitude and atorus at lower latitude. As Fig. 3 suggests, the axial dipole field is notdominant in this simulation. Almost all the spherical harmonicmodes with degrees from l 5 1 (dipole) to l < 10 are of the sameorder of magnitude. In this sense, this could be regarded as a kindof general dynamo simulation, rather than a reproduction of thegeodynamo.

To analyse the magnetic field of our simulation data, we first drew(many) magnetic field lines, but the resulting pictures were too com-plicated to understand. However, by drawing current lines, thehidden structure of the magnetic field was disclosed. We found thatthe current coil structure survives, though in a less distinctivefashion, in the higher-Ekman-number (E 5 2.6 3 1026) case; seeSupplementary Fig. 2.

Our simulation indicates that the convection and its dynamoenters a new regime when we decrease the Ekman number to theorder of 1027. Dynamo scalings7,25 established by simulations withE $ 1026 should be checked in this new dynamo regime. The currentcoils and straight magnetic flux tubes found in this simulation may berelevant to the solar dynamo, as sunspots are formed by emergingmagnetic flux tubes. Finally, we note that the magnetic Prandtl num-ber (Pm 5 1) in this simulation is far from the Earth’s outer-corevalue, which is of the order of 1026—the next challenge is to find theeffects of different values of Pm.

a b

Figure 2 | Sheet plumes observed in experiment and simulation. a, Theexperiment by Sumita and Olson9. b, Our simulation. In Sumita and Olson’sexperiment, a rotating hemispherical shell is filled with water. Thecentrifugal force plus gravity works in the radial direction. The cooled innersphere causes thermal convection. This is a snapshot, visualized by flakesseen through the transparent equatorial plane; the Ekman numberE 5 1 3 1026 in this experiment. Similar sheet plumes can be observed inour simulation (b; E 5 2.3 3 1027). This is a visualization by volumerendering of the axial vorticity.

NATURE | Vol 454 | 28 August 2008 LETTERS

1107

©2008 Macmillan Publishers Limited. All rights reserved

METHODS SUMMARY

The physical model, basic equation, and boundary conditions adopted in this

simulation are the same as used in our previous work4,20,26–28. The compressible

MHD equations in the spherical shell geometry are discretized by the second-

order finite difference method. The magnetic field has only a radial component

on the inner and outer spherical boundaries. The time integration method is

fourth-order Runge-Kutta. To avoid grid convergence near the poles of a spher-

ical coordinate system, we have made use of the Yin-Yang grid system13,14 in this

simulation. We have used virtual reality visualization software VFIVE23,24 in the

CAVE virtual reality system29 to obtain Fig. 4a and e. The current torus cell

structure was first found by this virtual reality visualization.

Received 2 May; accepted 27 June 2008.

1. Roberts, P. H. & Glatzmaier, G. A. Geodynamo theory and simulations. Rev. Mod.Phys. 72, 1081–1123 (2000).

2. Jones, C. A. Convection-driven geodynamo models. Phil. Trans. R. Soc. Lond. A 358,873–897 (2000).

3. Kono, M. & Roberts, P. H. Recent geodynamo simulations and observations of thegeomagnetic field. Rev. Geophys. 40, 1013, doi:10.1029/2000RG000102 (2002).

4. Kageyama, A. et al. Computer simulation of a magnetohydrodynamic dynamo. II.Phys. Plasmas 2, 1421–1431 (1995).

5. Glatzmaier, G. A. & Roberts, P. H. A three-dimensional self-consistent computersimulation of a geomagnetic field reversal. Nature 377, 203–209 (1995).

6. Takahashi, F., Matsushima, M. & Honkura, Y. Simulations of a quasi-Taylor stategeomagnetic field including polarity reversals on the Earth simulator. Science 309,459–461 (2005).

7. Christensen, U. & Aubert, J. Scaling properties of convection-driven dynamos inrotating spherical shells and applications to planetary magnetic fields. Geophys. J.Int. 166, 97–114 (2006).

8. Sakuraba, A. & Hamano, Y. Turbulent structure in Earth’s fluid core inferred fromtime series of geomagnetic dipole moment. Geophys. Res. Lett. 34, L15308,doi:10.1029/2007GL029898 (2007).

Figure 3 | Current field structure near the inner core at t 5 430. This is thesame simulation as shown in Fig. 1a. The blue lines are current field lines.The left image shows the distribution in the spherical shell. The tracings ofthe current lines start from a sphere at r 5 0.46. The orange sphere of

r 5 0.41 is placed to help the view. The current field is organized as a set of‘coils’, located mainly in the middle latitudes. The right panel is a magnifiedimage. Most of the current coils are in the form of a helical spring, but someof them form a torus.

a b

e

c d

Figure 4 | Detailed structure of two typical current coils (blue) with theirassociated magnetic field lines (pink). There are two helical coils in a. Theupper one is a helical spring and the lower one is a torus. The coloured planein a is a cross-section perpendicular to the z-axis in the northern hemisphere.(This is not the equatorial plane.) The colour in the plane denotes the massdensity; the fluid with high density (red) is sinking (vs , 0), while the fluidwith low density (blue) is rising. Panels b, c and d show the helical spring coilof a (the upper blue coil) viewed from different angles. This is a typical helical

spring shape of the current coils. Panel e shows the current torus of a (that is,the lower blue coil), with velocity fields (arrows). The velocity fields at thefield lines are parallel to the magnetic field. This flow, which is a part of thesheet plume, is accelerating along the field line and it stretches out to form astraight magnetic flux tube. The magnetic field lines are located in theoutward vs . 0 sheet plume. a and e are snapshots taken by the virtual realityvisualization software VFIVE23,24 in the CAVE system29.

LETTERS NATURE | Vol 454 | 28 August 2008

1108

©2008 Macmillan Publishers Limited. All rights reserved

9. Sumita, I. & Olson, P. Laboratory experiments on high Rayleigh number thermalconvection in a rapidly rotating hemispherical shell. Phys. Earth Planet. Inter. 117,153–170 (2000).

10. Busse, F. A model of the geodynamo. Geophys. J. R. Astron. Soc. 42, 437–459(1975).

11. Gilman, P. & Glatzmaier, G. Compressible convection in a rotating spherical shell.I. Anelastic equations. Astrophys. J. 45 (Suppl.), 335–349 (1981).

12. Zhang, K. & Schubert, G. Comparison of terrestrial and solar dynamos. Rep. Prog.Phys. 69, 1581–1605 (2006).

13. Kageyama, A. & Sato, T. ‘‘Yin-Yang grid’’: An overset grid in spherical geometry.Geochem. Geophys. Geosyst. 5, doi:10.1029/2004GC000734 (2004).

14. Kageyama, A. et al. A 15.2 TFlops simulation of geodynamo on the earth simulator.Proc. ACM/IEEE SC2004 Conf.–35–43 (2004).

15. Chesshire, G. & Henshaw, W. Composite overlapping meshes for the solution ofpartial differential equations. J. Comput. Phys. 90, 1–64 (1990).

16. Busse, F. Convective flows in rapidly rotating spheres and their dynamo action.Phys. Fluids 14, 1301–1314 (2002).

17. Plaut, E. & Busse, F. Multicellular convection in rotating annuli. J. Fluid Mech. 528,119–133 (2005).

18. Aubert, J., Brito, D., Nataf, H.-C., Cardin, P. & Masson, J.-P. A systematicexperimental study of rapidly rotating spherical convection in water and liquidgallium. Phys. Earth Planet. Inter. 128, 51–74 (2001).

19. Stellmach, S. & Hansen, U. Cartesian convection driven dynamos at low Ekmannumber. Phys. Rev. E 70, 056312 (2004).

20. Kageyama, A. & Sato, T. Generation mechanism of a dipole field by amagnetohydrodynamic dynamo. Phys. Rev. E 55, 4617–4626 (1997).

21. Davidson, P. An Introduction to Magnetohydrodynamics (Cambridge Univ. Press,2001).

22. Aubert, J., Aurnou, J. & Wicht, J. The magnetic structure of convection-drivennumerical dynamos. Geophys. J. Int. 172, 945–956 (2008).

23. Kageyama, A., Tamura, Y. & Sato, T. Visualization of vector field by virtual reality.Prog. Theor. Phys. 138 (Suppl.), 665–673 (2000).

24. Ohno, N. & Kageyama, A. Scientific visualization of geophysical simulation databy the CAVE VR system with volume rendering. Phys. Earth Planet. Inter. 163,305–311 (2007).

25. Olson, P. & Christensen, U. R. Dipole moment scaling for convection-drivenplanetary dynamos. Earth Planet. Sci. Lett. 250, 561–571 (2006).

26. Ochi, M. M., Kageyama, A. & Sato, T. Dipole and octupole field reversals in arotating spherical shell: Magnetohydrodynamic dynamo simulation. Phys. Plasmas6, 777–787 (1999).

27. Kageyama, A., Ochi, M. M. & Sato, T. Flip-flop transitions of the magneticintensity and polarity reversals in the magnetohydrodynamic dynamo. Phys. Rev.Lett. 82, 5409–5412 (1999).

28. Li, J., Sato, T. & Kageyama, A. Repeated and sudden reversals of the dipole fieldgenerated by a spherical dynamo action. Science 295, 1887–1890 (2002).

29. Cruz-Neira, C., Sandin, D. & DeFanti, T. Surrounded-screen projection-basedvirtual reality. Proc. SIGGRAPH 93, 135–142 (1993).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements We thank P. Olson and J. Aurnou for pointing out the closeconnection between the sheet plume structure observed in this simulation and theexperiments of ref. 9. We thank I. Sumita for a detailed explanation of his laboratoryexperiments and comments, and for providing Fig. 2a. We thank N. Ohno forhelping with the visualization. This work was supported by KAKENHI (17540404)and The Mitsubishi Foundation.

Author Contributions A.K. was involved in project planning, simulation codedevelopment, simulation runs, data analysis and manuscript preparation. T.M. wasinvolved in a part of the simulation runs, data analysis and manuscript preparation.T.S. was involved in project planning and manuscript preparation.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. Correspondence and requests for materials should beaddressed to A.K. (kage@jamstec.go.jp).

NATURE | Vol 454 | 28 August 2008 LETTERS

1109

©2008 Macmillan Publishers Limited. All rights reserved