J. Plasma Fusion Res. SERIES, Vol.5 (2002) 274-277

Spatiotemporal Structure of Magnetic Field and convestion
Vortices Generated in a Rotating Spherical Shell
ISHIHARA Norio and KIDA Shigeo'
Faculty of Science, Nagoya University, Nagoya 464-8602, JapanI National InstituteFusion Science, Toki 509-5292,forJapan(Received: 1l December 2OOl I Accepted: 20 June 2002)

Abstract
Intensification mechanisms of magnetic field by thermal convection in a rotating spherical shell areinvestigated by the direct numerical simulation analysis of the MHD Boussinesq equation. It is found thatan axial magnetic dipole field whose energy is 15 times as large as the kinetic energy is generated.
Anticyclonic vortices play a key role in sustaining the structure.

Keywords:
dynamo, MHD, vortex, spherical shell, thermal convection

1. Introduction

2. Formulation

In recent years, much progress has been made in
our understanding of the origin of the geomagnetic field

We consider a dynamo action driven by thermal
convection in a rotating spherical shell. It is assumed
that an electrically conducting fluid is confined between
two concentric spheres of radii /i and /o which arewhich is believed to be generated by thermal convective
motion of an electrically conducting fluid in the outercore. Many research groups have discussed themechanisms from the point of view of the formation ofrotating at a common constant angular velocity.

loops in magnetic lines of the mean structure averaged

in space and constant in time, and the former is higher

over the longitudinal direction (see ref. [] for a recent

than the latter. There are vacuum outside the outersphere and a perfect insulator inside the inner sphere.
All the physical quantities of the fluid, such as thedensity (except in the buoyancy term), the thermal

Temperature on the inner and outer spheres is uniform

review). Unfortunately, however, this mean-fieldapproach misses an instantaneous localized structure andtime-dependence of dynamo magnetic field. It is notwell understood how the spatial structure of magnetic
field develops if it starts with a weak seed of magnetic
field because of complexity of the dynamo magnetic

diffusivity, the volume expansion coefficient, thekinematic viscosity, the magnetic diffusivity and themagnetic permeability are assumed to be constant andfield. It is indispensable to clarify these dynamicaluniform. The gravity force acts on the fluid in thebehavior in understanding the fundamental mechanism.

direction to the center of the sphere. Under the MHD
and Boussinesq approximation, dynamics of this system

In order to capture the behavior, spatiotemporal
structure of magnetic field is investigated, withis described by the Navier-Stokes equation, theparticular attention to the region of maximal field

induction equation and the thermal conduction equation.

strensth.

The boundary condition is imposed on the inner andouter spheres as such that the velocity vanishes, theCorresponding author' s e-mail: ishihara@ toki.theory.nifs.ac jp
11A@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchIshihara N. er al., Spatiotemporal Structure of Magnetic Field and Convection Vortices

temperature is uniform and constant, and the magneticx104

field is connected continuously with the outside
potential fields. We solve the governing equations using
spectral method based on the Chebyshev polynomial/spherical harmonic expansion as an initial value
problem. Time integration is performed by the use of
the Crank-Nicolson/Adams-Bashforth scheme. TheChebyshev tau method is employed in order to satisfy
the boundary conditions (see ref. [2] for more details of020406080

the numerical scheme).

Fig. 1 Temporal evolution of total kinetic energy (solidline) and total magnetic energy (dashed line) integrated over the entire space.

3. Results
The present system is characterized by fivenondimensional parameters, that is, the ratio rilro of theradii of the inner and outer spheres, the Taylor numberTa, the Rayleigh number Ra, the Prandtl number Pr, theStrebor number Sr. A direct numerical simulation isperformed in two stepsl. In the first step, the problem of> 0) and white lines for anticyclone (a, < 0). In thelinear period, cigar-shaped regions of strong magnetic
field are observed between cyclones and their western

thermal convection without magnetic field is solved,
adding a uniform random small perturbation to a slightly
unstable thermal conduction state with parameters rilro =O.5, Ta = 1.6 x IO6, Ra = 3.2 x lO4, Pr = 1. Nine pairsof cyclonic and anticyclonic vortices appear alternatelyaround the rotation axis. A cyclonic (an anticyclonic)
vortex has the same (opposite) direction as (to) therotation of this shell.

neighboring anti-cyclones off the equatorial plane at

middle latitude [type A, Fig. 2(a)]. In the firstequilibrium period, strong magnetic field is generated
between anticyclone and outer sphere, inside of
anticyclones in addition to the place observed in thelinear period tFig. 2(b)1. Banana-shaped blobs arelocated between anticyclones and outer sphere (type B).

Cigar-shaped ones pararell to the rotation axis arelocated inside anticyclones (type C). Interestingly, theseblobs always appear together, as reported in [3]. In thesecond equilibrium period, the magnetic field is strong
inside anticyclones (type C), as well as between these
vortices and the outer sphere (Type B) and eastside ofanticylones (type D). See ref. [3] for magnetic field

In the second step, dynamo simulations areperformed, by adding small magnetic perturbations to
the thermal convection field for various values of St.
There is a critical Strebor number below which dynamo
action is effective. This critical Strebor number is 0.149

for the present set of parameters. In the following, wereport the results of St = 0.1. Figure I shows temporal
evolution of total kinetic energy and total magnetic

structure outside the outer sphere.

In order to capture the dynamics, we take an
overview of the most intensified locations of theenergy integrated over the entire space. There are three
distinct periods classified by the temporal variation of
the magnetic energy, that is, the linear growing (1 < t <26), the first equilibrium (28 < t < 38) and the second
equilibrium (t > 43) periods. The mechanism and theplace of magnetic field intensification are different in
the respective periods.
The typical structure of strong magnetic field isshown in Fig. 2. Gray and white surfaces represent
isosurface of magnitude of magnetic field and vorticity,respectively. The axial component of vorticity on theequatorial plane is drawn by black lines for cyclone (ar.

magnetic field. In Fig. 3, we plot (a) the latitudinal and(b) the radial positions of the maximums of lD.l over thewhole simulation period, respectively. The maximums
are searched in each hemisphere separately since thefield is nearly symmetric with respect to the equatorialplane. Different symbols distinguish the areas oflocations of the maximums. They are divided into three

in the temporal and latitudinal plane, namely, theregions of higher latitude than20" both in the northernand southern hemispheres at earlier times t < 40 and therest. The former two are distineuished with crosses xI There
are a couple of ways to carry out an MHD dynamo simulation. For example, it is simulated by adding perturbations in bothtemperature and magnetic fields to an unstable thermal conduction state. Another one is to start with a perturbed unstable thermal conduction state under a given magnetic field and to remove the imposed magnetic field after some temporal evolution.

275

Ishihara N. et al., Spatiotemporal Structure of Magnetic Field and Convection Vortices

Fig. 2 Snapshots of the vorticity and magnetic fields.

Gray and white surfaces represent isosurface ofmagnetic field and vorticity magnitude, respectively. The axial component of vorticity on theequatorial plane is shown by contours of black (ar.
> 0) and white (ar, < 0). (a) Linear period lt = 2Ol,(b) the first equilibrium period (t= 34), (c) the second equilibrium period (f = 60).

position is mainly scattered over the outer half of thespherical shell [Fig.3(b)]. Around t=28,the beginningof the first-equilibrium period, new maximums, shownwith solid circles, appear around the middle radialregion on the equatorial plane. They are localized in anarrow annulus throughout the first-equilibrium period.
This is nothing but the central region of anti-cyclones(type C) at which the magnetic field is intensified
through stretching of magnetic lines perpendicularly tothe equatorial plane. Coexistence of B and C are thecharacteristics of the first-equilibrium period. Later, inthe second-equilibrium period (r > 43) these maximum
positions C spread out more in both the latitudinal andradial directions, which is the manifestation of more
violent activity of the magnetic field. As reported in ref.

and open circles O, respectively. The rest is divided
further into two by the radial coordinate, i.e. the middleregion (solid circle O) and the outer-sphere side (squaren).
It is seen in Fig. 3 that the maximums are localized
at the middle latitude (37" + 6", where * denotes thestandard deviation) from the beginning through the endof the first-equilibrium period. The maximum distributes
symmetrically, in the statistical sense, in the northernand southern hemispheres. These are the strong
magnetic field intensified between cyclones and theirwestem neighboring anti-cyclones (type A). Their radial[3], a recurrent interaction between convection vortices
and magnetic field is observed in this period. While thecrosses and open circles disappear in this period, anothergroup of maximums (shown by squares) appear near theouter boundary and at low latitude (30" + 3o), whichcorresponds to the banana-shaped blobs of strong
magnetic field intensified along accelerating streamlines276

Ishihara N. et aI., Spatiotemporal Structure of Magnetic Field and Convection Vorticeso(a)90:rl'i.i:i*n'ir*:x,x ^.Jxx x'-i.-,-':.x x,lf,xdl-xx xryri!,iiYrxx:o<xro<6.o< x a-'-gotrr;;;lu*tr'**+iJ' t' 3 r i rt'grrrr'Cqtt}Q$f rFt +S;.So--a-a--o o^o-- oloooooqooloo\oof o34eo,$qofo|"s-o";"*o^o-att'to to].totato'.: "- at"

o-90

402006080ft)
xxlr(ntrx ooax
oa xxoDo(xxxoxtrtr(Do oxx xoxo oxx (Do(Doxo ao,oxoxxoo)!ilqr)<D(x xxxoo xx 0.5

DOODi)aooa,aoX(OXOOXeOaOXIOoa)tc(o(tx xo20.
aoooanaa -oa aao-ao0!Ctr tro- Oa--otrtroo.

- ooa-..aooa o -oataaaa-a a -aa oo- ao a--ooaa
t-axa aaaa a-4060

tr.
.
aoo -Ooo-eoa la a-.
o-oao
aa oo80Fig. 3 The locations of the maximums of lb,l are plotted with symbols on (a) the latitudinal and (b) radial coordinates
versus time plane. The maximums are searched in each hemisphere separately. Symbols distinguish the areas oflocations of the maximums: Cross x and open circle o refer to higher latitude than 20" in the northern and southern hemispheres, respectively, at early times (t< 40), whereas square o and solid circle o refer to the outer-sphereside and the middle radial regions, respectively.

through the stagnation point just outside of the Ekman
boundary layer along the outer sphere (type B).
Incidentally, type D is not observed here because thesimulation period, that is, between cyclones and theirwestern neighboring anticylones at middle latitude as

magnitude of magnetic field is not so large. It is

anticyclones, and between anticyclones and the outerboundary in the first and second equilibrium periods.

well as low latitude in the linear period, insideinteresting to note that these open squares appear morein the periods of lower magnetic energy (50 < r < 60 and
70 < t < 80), while they are rare when the activity of themagnetic field is higher (40 < t < 50 and 60 < t <70).

References
P.H.
Robertsand
G.A. Glatzmaier, Rev. Mod.
[1]In this paper, we have investigated intensification

Phys.72,1081 (2000).
l2l H. Kitauchi and S. Kida, Phys. Fluids 10, 457(l998).

mechanism of magnetic field by thermal convection in a[3] N. Ishihara and S. Kida, J. Phys. Soc. Jpn. 69,1582

4. Summary(2000).

rotating spherical shell with particular attention to theregion of maximal field strength. An axial magnetic
dipole field whose energy is 15 times as large as thekinetic energy is generated. The magnetic field ist4l S. Kida and N. Ishihara, J. Plasma Fusion Res. 77,35s (2001).

t5l N. Ishihara and S. Kida, Research Report NIFS,

intensified in four particular places over all theNIFS-708 (2001).
277