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Convection-driven geodynamo models
By C. A. J o n e s
School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, UK
There has been signi­ cant progress in the development of numerical geodynamo models over the last ­ ve years. Advances in computer technology have made it possible to perform three-dimensional simulations, with thermal or compositional convection as the driving mechanism. These numerical simulations give reasonable results for the morphology and strength of the ­ eld at the core{mantle boundary, and the models are also capable of giving reversals and excursions which can be compared with palaeomagnetic observations; they also predict di¬erential rotation between the inner core and the mantle.
However, there are still a number of fundamental problems associated with the simulations, which are proving hard to overcome. Despite the advances in computing power, the models are still expensive and take a long time to run. This problem may diminish as faster machines become available, and new numerical methods exploit parallelization e¬ectively, but currently there are no practical schemes available which work at low Ekman number.
Even with turbulent values of the di¬usivities (and the question of whether isotropic di¬usivities are appropriate is still unresolved), the appropriate dynamical regime has not yet been reached. In consequence, modelling assumptions about the nature of the ®ow near the boundaries have to be made, and di¬erent choices can have profound e¬ects on the dynamics. The nature of large-scale magnetoconvection at small E is still not well understood, and until we have more understanding of this issue, it will be di¯ cult to have a great deal of con­ dence in the predictions of the numerical models.
Keywords: dynam o; Earth’s core; rotating convection

1. Introduction
In the past ten years there has been considerable activity in geodynamo modelling, the main e¬ort being directed towards convection-driven geodynamo models. Very signi­ cant progress has been achieved; models constructed from solutions of the Navier{Stokes and Maxwell equations can now be sensibly compared with geophysical observations of the actual geomagnetic ­ eld. Results obtained from palaeomagnetic data, such as the nature of the reversals of the main ­ eld, can also be modelled directly from the fundamental equations (Sarson & Jones 1999; Glatzmaier et al . 1999). Despite this undoubted progress, there are still a number of fundamental problems which remain unresolved. The parameter regime in which the current generation of numerical models can be run is very far from the regime of geophysical parameter values; so far, indeed, that the strong similarity between the model outputs and the geodynamo is quite surprising.

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The rotating spherical shell model is a natural problem to study as a geodynamo model, and a considerable number of papers have been based on it; we refer to it here as the `zero-order model’ because it is a clear-cut mathematical problem, although it should be emphasized that physical e¬ects not included in the zero-order model may be very signi­ cant for the real geodynamo. In the zero-order model there is just one source of buoyancy, whereas in the geodynamo there are two, thermal and compositional buoyancy. We also assume homogeneous spherically symmetric boundary conditions, whereas in the geodynamo mantle convection is spatially inhomogeneous. Heat ®ux absorbed by the mantle will be a function of the spherical polar coordinates
and , which is determined mainly by the dynamics of mantle convection (Zhang & Gubbins 1993; Olson & Glatzmaier 1995), and this can a¬ect dynamo action (Sarson et al . 1997a; Glatzmaier et al . 1999). There is also a considerable density variation between the inner-core boundary (ICB) and the core{mantle boundary (CMB), but this is ignored in the Boussinesq zero-order model.
Even within the zero-order model, there are a number of choices to be made; the buoyancy source can be distributed uniformly throughout the inner core (the uniform heating model) or put entirely in the inner core through a ®ux at the ICB. The boundary conditions can be taken as no-slip or stress-free; various options can be taken to model the inner core. Sometimes it is neglected altogether, and convection in a whole sphere is considered; sometimes the inner core is given its current size (about 0.35 times the radius of the outer core) but it is assumed to be electrically insulating; and sometimes it is allowed to be electrically conducting, usually with a similar conductivity to that of the outer core. The resulting dynamo behaviour is strongly a¬ected by which of these choices is made.
A strong motivation for the activity in geodynamo modelling is the possibility of using the models to compare with geophysical measurements. Secular variation studies and palaeomagnetic observations are two important examples, but dynamo models might also help us to understand the physical conditions at the CMB. Another exciting application is to apply our understanding of the geodynamo to the magnetic ­ elds of other planets; the recently discovered internal ­ elds on the Jovian moons Ganymede and Io (Schubert et al. 1996; Sarson et al. 1997b) are prime candidates to test dynamo models.
In this paper, we concentrate on interpreting and understanding results from the zero-order models, and on trying to relate these results to the actual geodynamo. The models need to be fully validated before we can reliably use them as a tool for further geophysical studies.
Previous reviews on the dynamics and magnetohydrodynamics of the Earth’s core include Fearn (1998), Glatzmaier & Roberts (1997), Hollerbach (1996) and Braginsky (1994). Dynamo theory was reviewed by Roberts & Soward (1992) and Roberts (1994) and more background information can be found in Roberts & Gubbins (1987).

2. The zero-order model and governing equations
We consider an electrically conducting Boussinesq ®uid layer con­ ned between two concentric spherical surfaces, at r = ri (corresponding to the ICB) and at r = ro (corresponding to the CMB) in a frame rotating at angular velocity 0, which we identify with the mantle frame. The inner core is taken as a solid electrically conducting sphere concentric with the outer core, which is free to rotate about the z-axis at
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angular velocity i relative to the mantle frame. We denote the density de­ cit due to compositional or thermal variations as C, the codensity. The codensity source, which
drives the convection, is denoted by S. This source leads to a codensity di¬erence ¢C
across the ®uid outer core. The distance between the surfaces d = ro ri = 2260 km is taken as the length-scale, the time-scale is the magnetic di¬usion time d2= 80 kyr, the magnetic ­ eld unit is (2 0 )1=2 20 Gauss (where is the ®uid density, is the permeability and is the magnetic di¬usivity) and the codensity unit is ¢C. The
governing MHD equations for the velocity u, the magnetic ­ eld B and the codensity
C are

E qPr

Du Dt

+

z^

u = rp + j

B + Er2u + qRaCr ;

(2.1)

@B = r2B + r (u B ); @t

@C @t

=

qr2C

u rC + S;

r B = 0;

(2.2) (2.3) (2.4)

r u = 0:

(2.5)

Here j = r B is the current, and the dimensionless parameters are the Ekman number E = =2 0d2, the Prandtl number Pr = = , the Roberts number q = = and the Modi­ ed Rayleigh number R = g ¢T d=2 0 . In these de­ nitions, is the kinematic viscosity and is the codensity di¬usivity. Other combinations of the
parameters are sometimes used, notably the magnetic Prandtl number Prm = = = qPr and non-rotating Rayleigh number Ra = g ¢T d3= = R=E.
In addition we have two further dimensionless parameters, the radius ratio ri=ro, which is usually taken to be around 1=3 to model the geodynamo, and the magnetic
di¬usivity ratio of the outer to the inner core, o= i. The most popular choice is o= i = 1, which we call a conducting inner core.
In the conducting-inner-core case, we need to solve for the magnetic ­ eld Bi and the angular velocity i. The inner-core magnetic ­ eld is governed by

@B +
@t

@B i@ =

i r2B :
o

(2.6)

The mechanical boundary conditions are that there should be no ®ow through the ICB and CMB surfaces, and that there should be no-slip at these surfaces, so u = 0 at the CMB and u = iz^ r at the ICB. Alternatively, stress-free conditions can be applied at the ICB, CMB or both (Kuang & Bloxham 1997).

Thermal boundary conditions must also be applied. Popular choices are C constant on r = ri and r = ro, or ­ xed ®ux boundary conditions, or some combination of the two. Another common choice is to assume a uniform source in the outer core, together with ­ xed codensity boundaries. The non-magnetic problem with a ­ xed codensity

(usually heat) source has been extensively studied. The magnetic boundary conditions are that the normal component of B and the

tangential components of the electric ­ eld E are continuous. The latter conditions can easily be converted into conditions on the current using the MHD form of Ohm’s law. Note that if stress-free rather than no-slip conditions are used at a conducting boundary, then the velocity on the boundary will enter the magnetic boundary conditions through Ohm’s law.

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C. A. Jones

The angular velocity i is determined by the torque balance at the ICB. In general, the rate of change of the angular momentum of the inner core is determined by adding the magnetic torque, the viscous torque and any external torque, such as a gravitational torque. Bu¬ett (1997) has argued that asymmetries in the mantle give rise to asymmetries in the shape of the inner core, and that the resulting gravitational torque can be even larger than the magnetic torque. In the stress-free case, the viscous torque is zero, so the rate of change of angular momentum of the inner core is then determined solely by magnetic and external torque. Estimates of the moment of inertia of the inner core and the likely order of magnitude of the magnetic torque suggest that the inner core can spin up due to magnetic torque in a few days; on the dynamo time-scale the inner core will therefore be in torque equilibrium. If the viscous torque is small, this means that the total magnetic and external torque must be zero; any torque supplied to one region of the ICB must be balanced by an opposite torque somewhere else, so that the torque surface integral adds up to zero; this has been called the `inner-core Taylor constraint’.
With this scaling, the dimensionless strength of the magnetic ­ eld gives a local measure of the Elsasser number, = B2=2 0 . Although the Elsasser number is only a derived quantity and not an input parameter, it is useful for comparison with magnetoconvection models where is an input parameter. Similarly, the dimensionless strength of the velocity ­ eld gives a local measure of the magnetic Reynolds number Rm = ud= . Again, this is a derived quantity, but it is very useful to compare its value with the results of kinematic dynamo models, where Rm is an input parameter.

3. Numerical methods

The most popular technique for solving the dynamo equations numerically has been

the pseudo-spectral method, recently described by Cox & Matthews (1997) and

Hollerbach (2000), and applied to the geodynamo by Glatzmaier & Roberts (1995)

and Kuang & Bloxham (1999). The vector ­ elds u and B are expanded into toroidal

and poloidal parts, and these scalar ­ elds, together with the C ­ eld, are expanded in

spherical harmonics. In the radial direction either a ­ nite di¬erence scheme or expan-

sion in Chebyshev polynomials is used. These expansions are inserted into (2.1){(2.5)

and time-dependent equations for the coe¯ cients are obtained. The linear di¬u-

sion operators are handled using an implicit scheme, such as the Crank{Nicholson

method, while the nonlinear terms are handled explicitly by converting from spectral

space to physical space so that the required multiplications can be handled easily.

This procedure leads to accurate solutions provided moderate parameter values are

used; it does, however, require large computing resources. In addition to the obvi-

ous problem associated with computing fully three-dimensional numerical solutions,

there are two additional problems which make the execution times even longer: ­ rst,

there is no e¯ cient fast Legendre transform, as there is for Fourier transforms, and

second, at small E the time-step has to be very small (see Walker et al . (1998) for

details).

Motivated by these di¯ culties, a number of ways of reducing the CPU time

required have been tried. One method has been to truncate the equations severely

in the azimuthal direction, sometimes leaving only the axisymmetric and one non-

axisymmetric eim

mode;

this

is

known

as

the

2

1 2

-dimensional

method.

This

tech-

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Convection-driven geodynamo models

877

nique reduces the CPU time required by a large factor. The disadvantage of this

method is that the non-axisymmetric wavenumber m has to be chosen a priori,

rather than emerging from the calculation. Nevertheless, the axisymmetric ­ elds

generated

by

the

2

1 2

-dimensional

method

are

remarkably

close

to

those

computed

by

fully three-dimensional calculations provided the azimuthal wavenumber is chosen

sensibly: in the limited parameter regimes that can be explored numerically, choos-

ing m as the ­ rst mode to become linearly unstable as R is increased is normally an

adequate

choice.

The

2

1 2

-dimensional

approach

is

therefore

a

useful

tool

in

exploring

dynamo behaviour, but one can only have full con­ dence in the results if they are

calibrated by at least some fully three-dimensional simulations. The simulations used

for

­

gures

2

and

3

below

are

a

compromise

between

the

2

1 2

-dimensional

model

and

fully three-dimensional simulations; the r and directions are fully resolved, but a

small number of non-axisymmetric modes (typically four) are present, together with

the axisymmetric modes.

Another popular technique has been the use of hyperdi¬usion to reach lower

Ekman number solutions. Here the di¬usive operator r2 is replaced by some for-

mula such as (1 + n3)r2, where n is the order of the spherical harmonic in the

expansion of the ­ elds and is a constant. This method is very simple to imple-

ment if the pseudo-spectral method is used, and it stabilizes the numerical scheme

for values of

0:05, allowing small E to be reached with tolerable (but still

small!) time-steps. Signi­ cant advances have been made using this scheme (see, for

example, Glatzmaier & Roberts 1995, 1997), but there is a serious price to pay. As

pointed out by Zhang & Jones (1997), hyperdi¬usion damps out small azimuthal

wavenumbers, which can be preferred if the magnetic ­ eld is weak. To illustrate

this we reproduce some of the Zhang & Jones (1997) results in ­ gure 1a{c. In

­ gure 1a we show the ®ow pattern at small E at the onset of non-magnetic con-

vection; note the large preferred m visible from the tight roll con­ guration, and

the largely z-independent nature of the ®ow expected from the Proudman{Taylor

theorem. In ­ gure 1b we see the result of the same calculation but with hypervis-

cosity and no magnetic ­ eld, while in ­ gure 1c we see the equivalent picture for

the case of an imposed azimuthal magnetic ­ eld and no hyperviscosity. Note that

the hyperviscosity has produced a similar e¬ect to the magnetic ­ eld, in both cases

reducing the dominant azimuthal wavenumber. In consequence, it is very di¯ cult to

tell in dynamo simulations with hyperviscosity whether the azimuthal wavenumber

is being determined by the magnetic ­ eld or the hyperviscosity. In this respect, the

three-dimensional hyperdi¬usive dynamos su¬er from the same disadvantage as the

2

1 2

-dimensional

models

in

that

the

dominant

azimuthal

wavenumbers

are

selected

somewhat arbitrarily.

Another approach is the use of ­ nite di¬erences in all three directions (see, for

example, Kageyama & Sato 1997a). These methods became unfashionable because

of di¯ culties in treating the ®ow near the rotation axis, and in implementing the

correct magnetic boundary conditions, which are non-local. However, the develop-

ment of fast massively parallel machines and the failure to resolve the Legendre

transform problem (Lesur & Gubbins 1999) has stimulated new interest in ­ nite-

di¬erence techniques, which can take full advantage of parallel architecture. The

pseudo-spectral technique requires a redistribution of the arrays amongst the dif-

ferent processors at each time-step, and this makes it a less attractive scheme for

parallel architecture.

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878

C. A. Jones

(a)

(b)

(c)
Figure 1. The pattern of convection with the preferred azimuthal wavenumber. On the right-hand side are contours of azimuthal velocity in a meridian plane; on the left-hand side are contours of azimuthal velocity in the equatorial plane: (a) E = 2 10 5 with no magnetic ¯eld and no hyperviscosity; (b) E = 2 10 5 with no magnetic ¯eld, but with hyperviscosity = 0:1; (c) E = 2 10 5 with no hyperviscosity but with magnetic ¯eld = 1.
4. Numerical results in the Busse{Zhang regime The dimensionless parameter values for the Earth are rather extreme (E 10 15 and q 10 5; see also x 6), and no numerical simulation has even approached them. There is, however, one region of parameter space where solutions can be found with large, but not impossibly large, computing resources. This is the Busse{Zhang regime (Sarson et al . 1998), so called after the pioneering work of Busse (1976) and Zhang & Busse (1989) who ­ rst found solutions in this region of parameter space. Here dynamo action occurs at comparatively low Rayleigh number; typically a few times the critical value for the onset of convection itself. In this regime, the velocities are typically O(10) on the thermal di¬usion time-scale. Since velocities have to be at least of order O(102) on the magnetic di¬usion time-scale for dynamo action, i.e. the magnetic Reynolds number has to be O(102), the value of q required for a successful
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Convection-driven geodynamo models

879

dynamo is O(10). It is possible to obtain dynamo action at lower values of q by raising

the Rayleigh number; in the range 10Rc < R < 100Rc velocities are typically O(102) on the thermal time-scale, and then q O(1) is su¯ cient for dynamo action. We call

solutions at q O(1) and large R Glatzmaier{Roberts dynamos. In principle, Busse{

Zhang dynamos can occur over a wide range of Ekman numbers, down to very small

values; in practice, numerical instability makes it virtually impossible to go below

E 10 4 without using hyperviscosity, and, as mentioned above, hyperviscosity does

not allow us to capture the essential features of low Ekman number behaviour. Most

published Busse{Zhang solutions therefore have E in the numerically accessible range

10 4 < E < 10 3. However, within this rather limited range of parameter space, a

large number of calculations have been performed.

In addition to the pioneering papers of Busse and Zhang referred to above, recent

fully three-dimensional simulations in this regime have been performed by Busse et

al . (1998), Sakuraba & Kono (1999), Katayama et al . (1999), Kageyama & Sato

(1997a{c), Kitauchi & Kida (1998), Kida & Kitauchi (1998a; b) and Olson et al .

(1999)

(see

also

Christensen

et

al .

1999).

Fully

2

1 2

-dimensional

simulations

in

this

regime have been performed by Jones et al . (1995), Sarson et al. (1997a, 1998) and

Morrison & Fearn (2000).

Although the processes involved in convection-driven dynamos are very complex,

there are fortunately many points of similarity emerging from all this activity, and

some physical understanding of the basic processes is beginning to emerge. Figure 2

shows a snapshot of some results of a simulation with limited resolution in the

direction (modes m = 0; 2; 4; 6 and 8 are present). In ­ gure 2a; b a slice at constant

z is shown (i.e. a plane parallel to the Equator), ­ gure 2a showing the ®ow and

­ gure 2b showing the magnetic ­ eld. The parameter values are R = 35, q = 10,

E = 10 3 and Pr = 1; model details are as in Sarson et al . (1998). Figure 2c gives

the corresponding axisymmetric ­ elds plotted in a meridional slice. The pattern of

convection found in this regime consists of columnar rolls which drift around in a

prograde (eastward) direction. There is generally a small di¬erential rotation, but

not a di¬erential rotation strong enough to produce a powerful !-e¬ect; R is too low

to generate a strong thermal wind; the ®ows are not large enough (unless Pr is small)

for the nonlinear u ru Reynolds stress terms to generate di¬erential rotation, and

the Lorentz force is not strong enough to drive substantial ®ows. In consequence, the

dynamos are of the 2-type with toroidal and poloidal ­ elds being generated locally

in the convection rolls themselves (Olson et al. 1999). The toroidal and poloidal

components of magnetic ­ eld are of broadly similar strength, and the magnetic ­ eld

does not appear to have a very powerful e¬ect on the convection in this regime. At

E 10 3 the dominant m is typically around 4, while at E 10 4 the dominant

m is in the range 6{8, depending on the model details, in particular the codensity

source and the boundary conditions. Fixed ®ux boundary conditions, as used in the

­ gure 2 calculations, generally favour lower values of m. The preferred number of

rolls is usually close to that expected from linear non-magnetic convection at the

same Ekman number (Christensen et al . 1999); this is, perhaps, slightly surprising

as magnetoconvection studies indicate that lower values of m are preferred with

imposed uniform magnetic ­ elds when the Elsasser number is O(1). However, the

magnetic ­ eld in these solutions is very far from uniform and is strongly in®uenced

by ®ux expulsion: it therefore appears that it is the convection which selects the

wavenumber and the ­ eld responds to this choice.

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880

C. A. Jones

142.6 U

- 54.75

0

54.75

Figure 2. A snapshot of a solution in the Busse{Zhang regime, R = 35, q = 10, E = 10 3 . Azimuthal wavenumbers m = 0; 2; 4; 6; 8 were present in the calculation. (a) The velocity ¯eld in the plane z = 0:15 (where z = 0 is the Equator and z = 1:5 is the outer boundary North Pole; the plot is viewed from above) with imposed dipolar symmetry. The colours show the vertical velocity uz (with colour intervals one-tenth of the maximum amplitude stated), the arrows the horizontal velocity uh (the longest arrow corresponding to the maximum stated).

Hollerbach & Jones (1993) noted that the dynamical behaviour outside a cylinder enclosing the inner core (the tangent cylinder) was rather di¬erent from that of the polar regions inside the tangent cylinder. The two polar regions are usually fairly quiescent in the Busse{Zhang regime. Convection is more easily excited outside the tangent cylinder, where the convection rolls disturb the geostrophic constraint least. At mildly supercritical R, the polar regions are e¬ectively convectively stable.
As noted by Busse (1976) the convection roll pattern is e¬ective at generating magnetic ­ eld, and magnetic Reynolds numbers of O(100) are all that is needed. It is necessary to have motion along the rolls as well as around the rolls for dynamo action to occur. There are a number of possible sources for this, which contribute in di¬erent proportions according to the model details. The convection itself induces a z-velocity antisymmetric about the equatorial plane, which is signi­ cant in all the models. With no-slip boundaries, Ekman suction at the boundaries can also drive a z-velocity, thus enhancing the convective ®ow. Finally, if an inner core is present, the interaction of the rolls with the inner core near the tangent cylinder can generate z-velocity. Particularly in models where the driving is strongest near the inner core, this can be a powerful e¬ect (see ­ gure 2a).
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Convection-driven geodynamo models

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3.8 B

- 5.272

0

5.272

Figure 2. (Cont.) (b) As ¯gure 2a, but for magnetic ¯elds, Bz and Bh .

Although the magnetic ­ eld does not appear to have a striking e¬ect on the ®ow, it must have some e¬ect. Since the nonlinear advection term in the equation of motion does not play a very important role, the Lorentz force is the only signi­ cant nonlinear term. Furthermore, the induction equation is linear in B , so the amplitude of the magnetic ­ eld can only be determined by the Lorentz force limiting the ®ow in some way. There appear to be a number of ways in which this limiting could take place. (i) In an ! dynamo, generating ­ eld principally by a powerful di¬erential rotation, the Lorentz force will act to brake the di¬erential rotation and hence control the dynamo process. The amplitude of the ­ eld is then determined by the strength at which the forces producing di¬erential rotation are su¯ ciently opposed by magnetic braking. However, this does not seem a likely mechanism in the Busse{Zhang regime, where di¬erential rotation does not appear to be very important. A variation on this, which can apply also to 2 models, is that the Lorentz force can generate a di¬erential rotation which disrupts the dynamo process, thus bringing about equilibrium; the Malkus{Proctor scenario (Malkus & Proctor 1975). Although it might well be important at very low E, this mechanism does not seem to be very active in the numerical simulations. Another possibility, (ii), mentioned by Olson et al. (1999), is that the magnetic ­ eld directly reduces the convective velocity and hence reduces the magnetic Reynolds number to its critical value. In this mechanism, the magnetic braking acts not on the axisymmetric di¬erential rotation, but on each individual roll. Possibility (iii) is that the magnetic ­ eld acts to reduce the stretching properties of the ®ow, i.e. magnetic ­ eld turns the generating process from an e¯ cient dynamo into a less e¯ cient dynamo. If this mechanism operates, the mean kinetic energy is not necessarily smaller with the saturated magnetic ­ eld than it is with zero ­ eld,

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882

C. A. Jones

B

A r sinq

T

v/ r sin q )

y r sinq

) )

B. . . . . . .

2.3712

A r sinq .

1.4393

T. . . . . . .

0.9132

v/ r sin q ) 47.4548

y r sin q .

2.3477

(contours are at max./10)

Figure 2. (Cont.) (c) A meridian slice of the axisymmetric components of various quantities: B, toroidal (zonal) magnetic ¯eld; Ar sin , lines of force of poloidal (meridional) magnetic ¯eld; C, codensity; v=(r sin ), the azimuthal angular velocity; r sin , streamlines of the meridional circulation. Colour intervals are one-tenth of the maximum stated, with purple corresponding to negative values. In plots of zonal ¯elds, positive contours denote eastward-directed ¯elds. In plots of meridional quantities, the sense of ¯eld is clockwise around positive contours and anti-clockwise around negative contours.

and the nonlinear magnetic Reynolds number could be quite di¬erent in di¬erent parameter regimes. Since the stretching properties of a ®ow depend on rather subtle features involving derivatives of the velocities themselves, two ®ows which look super­ cially rather similar can have very di¬erent dynamo action properties. We should therefore perhaps not be too surprised that the convection in the magnetically saturated regime looks pretty similar to that in the zero-­ eld regime. The di¬erences are in the stretching properties of the ­ eld, which do not show up well in a simple velocity ­ eld plot. It is probably too early to say whether mechanism (ii) or (iii) is the most important in the Busse{Zhang regime, but we note from the data available that while there is some evidence that magnetic ­ eld reduces mean kinetic energy and hence typical velocities, the reduction is not generally very great, while there is a considerable variation in the magnetic Reynolds number of the saturated ®ow, suggesting mechanism (iii) may be the more important.
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While the in®uences of the magnetic ­ eld on the ®ow are not generally large in this regime, as noted above, it is of interest to try to analyse those which do occur. Kagayema & Sato (1997c) noted that rolls can be divided into cyclonic and anticyclonic rolls, depending on whether the vorticity of the roll added to, or subtracted from, the planetary vorticity. Anticyclonic rolls have the z-velocity positive in the Northern Hemisphere and negative in the Southern Hemisphere. The z-velocity is therefore divergent near the Equator, and so by the continuity equation, the horizontal ®ow in an anticyclonic roll converges near the Equator. This horizontal converging ®ow tends to accumulate magnetic ­ eld near the centres of anticyclonic rolls in the equatorial region, but in cyclonic rolls the horizontal ®ow near the Equator is divergent, and so no ­ eld accumulates there. Note that in ­ gure 2b, the vertical magnetic ­ eld is strongest in the two anticyclonic rolls (clockwise ®ow), so the magnetic ­ eld produces a strong asymmetry between cyclonic and anticyclonic rolls. Sakuraba & Kono (1999) note that the magnetic ­ eld in the centres of anticyclonic rolls tends to expand them, so they become larger than the cyclonic rolls (note that the anticyclonic vortices in ­ gure 2a are indeed more pronounced than the cyclonic vortices). The average vorticity of the roll system is then negative, leading to westward ®ow near the CMB and eastward ®ow near the inner core.

5. Numerical results in the Glatzmaier{Roberts regime
The other regime that has been studied is the higher Rayleigh number regime, typically R O(100Rc). Since the velocities here are larger compared with the thermal di¬usion time than in the Busse{Zhang regime, one can a¬ord to reduce q to an O(1) value, and still have Rm large enough for dynamo action. Since at smaller values of E convection is more organized, reducing E allows larger R and so allows smaller q. Interestingly, increasing the Rayleigh number at ­ xed Ekman number can actually lead to a loss of dynamo action (Morrison & Fearn 2000; Christensen et al . 1999). The velocities may be larger, but the more disorganized ®ow is less e¯ cient as a dynamo. Christensen et al. (1999) ­ nd empirically qc 450E3=4 as the minimum possible value of q for dynamo action on the basis of their numerical experiments.
Some of the larger R calculations have been done with hyperviscosity, so it is not easy to estimate the e¬ective Ekman number in these cases. Nevertheless, the main features of these higher R dynamos have been found in a range of models and are probably robust.
In ­ gure 3 we show the same constant z slice and meridional slice as in ­ gure 2, but now the Rayleigh number is much larger and the Ekman and Roberts numbers are reduced by a factor of 10: the parameters are R = 45 000, E = 10 4, but with some hyperdi¬usivity, and q = 1. The most important feature is that the di¬erential rotation v=r sin becomes much stronger in this regime (cf. ­ gures 2c and 3c). The convection is more chaotic, and the rolls lose their clear-cut identity, typically lasting for only a short time. Nevertheless, the convection is highly e¯ cient, in the sense that the temperature gradients are con­ ned to thin boundary layers and the bulk of the region outside the tangent cylinder and outside of these boundary layers is nearly isothermal; note the much thinner boundary layers of C in ­ gure 3c than in ­ gure 2c. Because the surface area of the outer CMB boundary layer is so much greater than that of the ICB boundary, this isothermal region is at a temperature much closer to that of the CMB than the ICB, i.e. it is relatively cool. Convection
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C. A. Jones

2753.2 U

- 959

0

959

Figure 3. A snapshot of a solution in the Glatzmaier{Roberts regime, R = 45 000, q = 1, E = 10 4 with hyperviscosity = 0:05. Azimuthal wavenumbers m = 0; 2; 4; 6; 8 were present
in the calculation. No symmetry about the Equator was imposed. (a){(c) are as in ¯gure 2.

also occurs now in the polar regions inside the tangent cylinder: here the geometry is such that the mean temperature of the ®uid is closer to the average of the ICB and CMB temperatures, much hotter than that of the bulk of the ®uid outside the tangent cylinder. In consequence, there are strong temperature gradients in the direction, so that there is a thermal wind

@v = qR @T ; @z r @

(5.1)

which is large since @T =@ is O(1) and R is large in this regime. In the Northern Hemisphere @T =@ must be negative, for the reasons given above, and so @v=@z is negative in the Northern Hemisphere, so there is a strong prograde (eastward) ®ow near the ICB. This ®ow is then coupled to the ICB itself by viscous and magnetic torques (Aurnou et al. 1996), and so the inner core has to rotate rapidly eastward in this regime if gravitational torques are neglected. The pattern of the di¬erential rotation (see ­ gure 3c) is a very constant feature of models in this regime, and it leads to the generation of strong toroidal ­ elds in the regions of high shear near the ICB (see ­ gure 3b). The helicity in this regime is produced mainly by the interaction of convection with the inner core, and so poloidal ­ eld also is strongest near the inner core. The Glatzmaier{Roberts regime dynamos are therefore more like ! dynamos with di¬erential rotation playing an important role in the production of toroidal ­ eld. It should be noted, however, that the peak poloidal ­ eld strength is not that much smaller than the peak toroidal ­ eld strength.

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13.4 B

- 4.998

0 Figure 3. (Cont.) (b).

4.998

Another key di¬erence from the Busse{Zhang model is that the polar regions inside the tangent cylinder are now very active, convecting vigorously and generating strong magnetic ­ elds there. This polar convection changes the form of the codensity contours, which in turn change the form of the thermal wind, given by (5.1). The di¬erential rotation in the Busse{Zhang models is therefore not only much weaker (because the thermal wind driving is proportional to ) but has a di¬erent characteristic form (compare ­ gure 2 with ­ gure 3 ). In the Busse{Zhang models the form of the di¬erential rotation is such that there is no strongly preferred direction of rotation of the inner core, in contrast to the Glatzmaier{Roberts models.
Another characteristic feature of the Glatzmaier{Roberts regime is the presence of a strong axisymmetric meridional circulation ( sin in ­ gure 3 ) in the polar regions inside the tangent cylinder. This circulation rises out from the ICB at the poles and falls back towards the ICB along the tangent cylinder. The ®ow returns from the tangent cylinder back to the poles along a thin Ekman boundary layer near the ICB, which controls the strength of the meridional circulation. A simple calculation of the Ekman-layer ®ow indicates that the strength of the meridional circulation as measured by the maximum value of on the magnetic di¬usion timescale gives
1=2 m ax
This will be small in the Busse{Zhang regime, since there is small and is not large. In the Glatzmaier{Roberts regime, cannot be made very small for stability reasons, but is quite large, so a strong meridional circulation is found. This meridional circulation does not seem to be strongly a¬ected by magnetic ­ eld in the regime

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C. A. Jones

B

A r sinq

C

) )
)

v/ r sin q )

y r sinq

B. . . . . . .

17.6388

v/ r sin q )

0.2123

C. . . . . . .

0.2196

v/ r sin q ) 8984.3652

y r sinq .

17.1077

(contours are at max./10)

Figure 3. (Cont.) (c).
in which these models are run. Considerations such as these indicate how important it is to ­ nd out more about the parameter regime the geodynamo is operating in. Everyone knows that in the geodynamo R is large and E is small, but is the key parameter RE1=2 large, small or O(1)? If the meridional circulation is signi­ cant, it may have very important consequences for the dynamo, in particular for the reversal mechanism (Sarson & Jones 1999; Glatzmaier et al. 1999).
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6. The geodynamo parameter regime

Since it is clear that there are a number of di¬erent types of dynamo that can operate

at low Ekman number, we need to consider the geophysical parameters carefully to

see which types, if any, are connected with the geodynamo. It is also possible that

qualitative di¬erences occur between the behaviour at the moderately small Ekman

numbers used in the simulations and the extremely low Ekman number found in the

Earth’s core.

Although it is likely that turbulence plays an important part in core dynamics

(Braginsky & Meytlis 1990), we start by considering the parameter values based on

molecular values (see, for example, Braginsky & Roberts 1995). We take the mag-

netic di¬usivity of the core as 2 m2 s 1, which is equivalent to a conductivity of

4 105 S m 1. The thermal di¬usivity is 10 5 m2 s 1 and the viscous di¬usivity

is near

10 6 m2 s 1 (DeWijs et al . 1998). Based on these molecular di¬usivi-

ties, q 5 10 5 and P r 0:1. The situation is complicated further because in the

core thermal convection is not the only driving mechanism; compositional convection

can occur, and may indeed be more important. The molecular value of the material

di¬usivity of light material in the core is even smaller,

10 8 m2 s 1. The equa-

tion governing the concentration of light material, the codensity equation (2.3), is

the same as the temperature equation, but the appropriate boundary conditions are

di¬erent. The radius of the outer core is ro = 3:48 106 m, the radius of the inner core ri = 1:22 106 m and 0 = 7:3 10 5 s 1. The molecular value of the Ekman number is then 10 15.

Estimating the molecular value of the Rayleigh number is more di¯ cult; there

are two Rayleigh numbers to consider, one for the thermal part of the convection

and one for the compositional part. We consider the thermal Rayleigh number ­ rst.

Since it is the heat ®ux which is ­ xed by mantle convection, the problem is slightly

di¬erent from the classical Rayleigh{B´enard problem. A further di¯ culty is that the

heat ®ux coming out of the core is not well known. This heat ®ux can be divided into

two parts, that carried by conduction down the adiabatic gradient and that carried

by convection down the superadiabatic gradient. It is only the convective part that

is of relevance to driving the outer core; this total convective heat ®owing out of

the core is unlikely to exceed ca. 5 1012 W, and we use this as our estimate for

Fcon v 3 10 2 W m 2 at the CMB. It is possible that most of the heat ®ux is carried down the adiabatic gradient by conduction, and Fcon v is much lower than this (Nusselt number close to unity).

A ®ux Rayleigh number can be de­ ned by evaluating the (notional) superadiabatic

temperature gradient that would be required to carry this convective part of the heat

®ux out by conduction, giving

g Raf =

d4Fcon v 2 cp

1029

taking

104 kg m 3, cp 8 102 J kg 1 K 1,

10 5 K 1 and g 10 m s 2

(Braginsky & Roberts 1995). The corresponding modi­ ed ®ux Rayleigh number is

Rf

=

g 2

d2Fcon 2 cp

v

1014 :

The ®ux Rayleigh number is appropriate for considering the onset of convection, but in strongly supercritical convection the superadiabatic temperature gradient will be

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C. A. Jones

greatly reduced from the value

= Fcon v ; cp

which it has near onset. In Rayleigh{B´enard convection the main e¬ect of increasing Ra is to increase the heat ®ux for a given temperature di¬erence between the boundaries; in ­ xed ®ux convection the main e¬ect is to decrease the temperature di¬erence for a given heat ®ux. If we want to use the results of Rayleigh{B´enard convection at ­ xed temperature (where there has been more previous work), it is more appropriate to estimate the Rayleigh number by estimating the typical temperature ®uctuation and using this as an estimate of the superadiabatic temperature di¬erence between the ICB and CMB, ¢T . The convective heat ®ux is
Z

Fcon v =

cp urT ds:

S

(6.1)

We must assume that the integral (6.1) can be approximated by taking typical values of ur and ¢T ; this assumes that the radial velocity and temperature ®uctuations are well correlated, but we would expect this in a convection-driven ®uid. We must also assume that the temperature variations are fairly uniform across the interior of the ®uid outer core, and not concentrated into thin plumes. In the neighbourhood of the inner core we then estimate

4 ri2 cp ur¢T 5 1012 W

(6.2)

and then ¢T 10 4 K, slightly less than the estimate of ¢T 10 3 K of Braginsky & Roberts (1995). This seems a tiny temperature di¬erence compared with the actual temperature di¬erence between the ICB and CMB (ca. 1300 K), but it re®ects the fact that very large convected heat ®uxes can be carried by very small superadiabatic gradients, just as they are in stars. A similar estimate for ¢T can be obtained by assuming that the magnitude of the buoyancy g ¢T is similar to the magnitude of the Coriolis force 2 0u with u again estimated at 3 10 4 m2 s 1.
The modi­ ed B´enard Rayleigh number is then

g ¢T d RB = 2 The non-rotating Rayleigh number is

1:5 107:

RaB = R=E 1:5 1022:

The Rayleigh number for compositional convection can be evaluated in a similar way,

provided the mass ®ux is assumed, which is more or less equivalent to assuming the

age of the inner core is known. Taking the density di¬erence between the inner and outer core as 0:6 103 kg m 3 and assuming the rate of growth of the inner core has

been constant over the last 4:5 109 years, then the rate of mass increase inside a

sphere of ­ xed radius r > ri is 3:2 104 kg s 1. Assuming most of this mass ®ux is carried out by advection rather than di¬usion, then

Z

3:2 104 kg s 1

Cur ds
S

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889

from (2.3). Again taking ¢C as the typical di¬erence in C between upward and

downward moving ®uid gives ¢C 5 10 6 kg m 3, giving the modi­ ed Rayleigh

number RB = g¢C d=2 0

1010; the fact that the compositional Rayleigh num-

ber is larger than the thermal Rayleigh number is due to being considerably

smaller than .

(a) The dynamo catastrophe
A number of attempts to achieve a low-Ekman-number convection-driven dynamo using the magnetostrophic equations in the interior together with Ekman boundary layers have been made. All have so far failed, due to what might be termed the `dynamo catastrophe’.
The attempts all started with the idea that convection at low Ekman number but O(1) Elsasser number need not occur on the very small azimuthal and radial length-scales that occur in the non-magnetic problem. Indeed, O(1) length-scales are the preferred linear mode when there is an imposed (axisymmetric) magnetic ­ eld. Since large-scale convection has been found in all simulations to give rise to dynamo action, the hope was that the large-scale convection could generate the large-scale ­ eld required and hence maintain a self-consistent large-scale dynamo. The problem with this scenario is that the magnetic ­ eld needs to be su¯ ciently strong everywhere and at all times to sustain convection on a large length-scale (Zhang & Gubbins 2000). If, due to ®uctuations, the ­ eld becomes small over a signi­ cant domain for a signi­ cant period (e.g. at a reversal), the length-scale of convection becomes short, and the dynamo fails. The magnetic ­ eld then starts to decay, and so small-scale convection takes over everywhere, permanently. A dynamo catastrophe has occurred, and the ­ eld can never recover.
We might ask why the simulations referred to above do not show this catastrophic failure. The simulations that have been performed to date have not been in the regime where the magnetic ­ eld is playing a major role in selecting the preferred wavenumber: indeed, as remarked above, in most simulations the preferred azimuthal wavenumber is not very di¬erent to that expected with non-magnetic linear convection. None of the simulations have yet reached the very low Ekman regime where the dynamo catastrophe occurs.
The dynamo catastrophe can be understood in terms of the various bifurcations that occur in convection-driven dynamos. As the Rayleigh number is increased, there may be various non-magnetic bifurcations in the convective ®ow, but eventually, at large enough magnetic Reynolds number, there is likely to be a critical value of the Rayleigh number Rc at which dynamo action occurs; this bifurcation may be supercritical or subcritical. If it is supercritical, then a ­ nite-amplitude magnetic­ eld state is established for values of R > Rc. This state is generally known as the weak-­ eld dynamo regime (see, for example, Roberts & Soward 1992) and is characterized by the small wavelengths in the direction perpendicular to the rotation axis. This state was analysed in the plane layer dynamo case by Childress & Soward (1972). They found that the weak-­ eld regime only existed for a fairly restricted range of Rayleigh number, because the magnetic ­ eld enhanced convection, eventually leading to runaway growth; the weak-­ eld regime therefore terminates at some Rrway, so the weak-­ eld regime only exists for Rc < R < Rrway. Childress & Soward (1972) conjectured that a subcritical branch of strong-­ eld solutions also exists, with
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wavenumbers of O(1) and with magnetic-­ eld strength such that the Elsasser number is O(1), and that when R > Rrway the solution jumps onto this branch. This strong-­ eld regime is likely to exist for Rayleigh numbers much less even than Rc, although for the numerical reasons given above this has not yet been unambiguously demonstrated. If the Rayleigh number is decreased on the strong-­ eld branch, there must come a point at which the magnetic Reynolds number is too small to support dynamo action, say R = Rs , and below this no magnetic ­ eld can be sustained. The dynamo catastrophe arises when Rs Rc, the likely situation in the geodynamo, and when the actual Rayleigh number is in the range Rs < R < Rc; `safe’ dynamos are those where R > Rc.
In subcritical dynamos there are at least two stable attracting states, one with magnetic ­ eld and one without. In most of the simulations, the state with the magnetic ­ eld is chaotic, so it is subject to considerable ®uctuation. It is possible to wander on a chaotic attractor without ever falling into an attracting ­ xed point; an example is the Lorentz attractor, where for some parameter values there are two attracting ­ xed points, but some solutions wind around these attracting ­ xed points without ever falling into their basin of attraction. The geodynamo could perhaps behave in a similar fashion, though other planets which no longer have internal dynamos may have su¬ered a dynamo catastrophe in the past. Another issue that arises with subcritical models of the geodynamo is how the magnetic ­ eld started in the ­ rst place; convection can only occur in a core at a temperature well above the Curie point; even if the geodynamo is currently subcritical, it must have been supercritical some time in its past.
It is of interest to estimate the parameter regime where we expect the dynamo catastrophe to occur, and conversely to ­ nd where `safe’ (supercritical) dynamos, i.e. those where magnetic ­ eld can be regenerated from the non-magnetic convecting state, are located. At Prandtl numbers of O(1), non-magnetic convection in a rapidly rotating sphere has a preferred azimuthal wavenumber m E 1=3, and a similar radial length-scale. The planform will therefore be of convection roll type, not dissimilar to that found in the Busse{Zhang models, but with a much smaller radial and azimuthal length-scale.
Since the asymptotic theory of linear convection in a rapidly rotating sphere (or spherical shell) at small E has now been worked out (Jones et al. 2000), at least in the case of uniform heating, we can apply this theory to see what kind of convection we expect at these extreme parameter values. At Pr = 0:1 the critical modi­ ed Rayleigh number for a radius ratio of 0:35 is 0:27 E 1=3, which is Rc = 2:7 104 at E = 10 15. The thermal Rayleigh number is therefore supercritical, R=Rc 500, but perhaps surprisingly not enormously so. The critical azimuthal wavenumber is m = 0:24E 1=3 which gives m 2:4 104. If this many rolls have to ­ t into the Earth’s core, each roll would have a diameter of ca. 0.4 km. Linear theory shows that at onset the rolls only occur at a speci­ c distance (about 0:7r0) from the rotation axis, but fully nonlinear convection rolls would probably ­ ll the whole ®uid core.

(b) The minimum roll size for dynamo action
If we (temporarily) accept that nonlinear non-magnetic convection in the rapidly rotating spherical shell takes the form of rolls of typical diameter somewhat less than 1 km with typical velocity u 3 10 4 m2 s 1, could this ®ow be a dynamo? The
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Convection-driven geodynamo models

kinematic dynamo problem for a related type of ®ow,

p u = 2 sin cos

p 2 cos sin 2 sin sin

891 (6.3)

has recently been considered by Tilgner (1997). The ®ow has the form (6.3) inside a

cylinder of height and radius , and is zero outside it. The diameter of each roll is ,

so that is approximately 2 , where is the number of rolls that ­ t across the

diameter of the cylinder. The length-scale for the ®ow is now , so the local magnetic

Reynolds number is

0 m

=

. If the mean ­ eld strength is , the ®uctuations

produced by the ®ow will be 0

0 m

. The large-scale EMF is then u

b0, which

will maintain the current on the long length-scale

provided

0 m

, or more

precisely

0 m

m =(

)

2 m

2 mc

(6.4)

for some critical m c. If and remain ­ xed, as decreases, m must increase as m ( ) 1=2 to maintain dynamo action. Tilgner (1997) ­ nds that with a
lattice of size = 8 rolls across a diameter, m must exceed m c 15, and that for larger , m c increases as 1=2, as expected from (6.4). We therefore have from Tilgner’s calculation that m c 15( 8)1=2. cells across a diameter will
correspond roughly to cells round the circumference, and so for convecting rolls

with azimuthal wavenumber ,

1 2

, giving m c

7 1=2 for large . It

should be noted that the velocity ­ eld here is independent of ; in convecting rolls,

z is antisymmetric in , and the dynamo will be less e¯ cient. Data from the dynamo calculations of Christensen et al. (1999) suggest that the critical magnetic Reynolds

number is at least a factor of three greater in convection cells than in the Tilgner

cells, so we estimate

m c 20 1=2

large

(6.5)

as the critical magnetic Reynolds number at large azimuthal wavenumber. Note that

we are assuming here that the Rayleigh number is large enough for the convection

rolls to e¬ectively ­ ll most of the outer core for this estimate. Taking 3 10 4 m s 1

as our standard velocity for the core, and = 2 m2 s 1 for the magnetic di¬usivity,

m 500 for the Earth’s core. Using our estimate (6.5) we then get

600 as the

maximum possible value of at which we can expect dynamo action. Although this

is a large value of , it is a lot less than the value expected at

10 15, which

was 24 000. The value of at which = 600 is preferred at = 0 1 is around

6 10 11.

The outcome of our estimates is therefore that `safe’ (supercritical) geodynamos

could exist at Ekman numbers down to about

10 10, but at Ekman numbers

below 10 10, the preferred roll size for non-magnetic convection is too small to give

a viable dynamo. For Ekman numbers below

10 10, the dynamo must be sub-

critical.

(c) Convective velocities
So far our estimates have all been based on the value of 3 10 4 m s 1 for the convective velocity. Would convection in this parameter regime actually give rise

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to velocities of 3 10 4 m s 1? If our underlying physical assumptions are correct, then the answer has to be yes, or else our picture is not self-consistent. It might be argued that we already know the answer is yes, because the velocities coming out of the simulations are of the right order of magnitude: the Busse{Zhang regime velocities are perhaps a little low, and the Glatzmaier{Roberts regime velocities a little high, but given the uncertainties involved in estimating and in interpreting secular variation data to estimate u, this is not a cause for concern. However, this really begs the question, because the simulations assume turbulent values of the thermal, compositional and viscous di¬usion which are tuned to give the right answers.
The usual velocity scale for mildly supercritical convection is the thermal velocity scale =d 3 10 12 m s 1, eight orders of magnitude too small. The velocity scale for compositional convection is a further three orders smaller. Clearly, for a convection-based geodynamo theory to work, we have to explain why the observed velocities are so large compared with the thermal velocity scale. Weakly nonlinear theory suggests that the velocities scale with (R=Rc)1=2, and Zhang (1991) gives
u 25[(R Rc)=Rc]1=2 =d

for in­ nite Prandtl number convection. With the B´enard estimates for R and Rc, the expected velocity goes up to 2 10 9 m s 1, still far too small. Using compositional convection estimates reduces the expected velocity.
One possibility is that the magnetic ­ eld enhances the velocity, but even if the critical Rayleigh number is reduced from 104 down to unity by magnetic ­ eld, the
Zhang formula still gives velocities three orders of magnitude too small. We need an improvement on weakly nonlinear theory to reach velocities as large as 10 4 m s 1.
Unfortunately, not much is known about nonlinear convection at very low Ekman number. Probably the best available theory is that of Bassom & Zhang (1994), which
is, however, based on non-magnetic Boussinesq convection in a plane layer, not with spherical geometry. Ignoring some logarithmic terms of the Rayleigh number, they
give the vertical velocity for Rayleigh numbers well above critical as

uz 0:3R =d;

(6.6)

which gives uz 10 5 m s 1, which is getting closer to the required velocity. Considerable caution is needed in interpreting this result, though, because not only could the spherical geometry make a signi­ cant di¬erence, at large R the ®ow will almost certainly be unstable, and this may reduce the velocities achieved. Nevertheless, it does suggest that because of the unusual nonlinear dynamics of rapidly rotating ®ows, the very large velocities (in terms of the thermal di¬usion velocity scale) might be explicable. It is perhaps worth noting that the Bassom{Zhang theory gives an unusually strong dependence of typical velocity and Nusselt number on Rayleigh number with u R and Nu R (again ignoring logarithmic terms). Weakly nonlinear theory gives only u R1=2 as mentioned above. Boundary-layer theories of non-rotating non-magnetic convection have dependencies which are a¬ected by the precise nature of the boundary conditions, but Nusselt number Nu R1=3 and u R2=3 are perhaps typical.

(d ) The e® ect of turbulence in the core
If our understanding of the e¬ect of magnetic ­ elds on strongly nonlinear low-E convection is limited, the situation as regards turbulence in the core is even more

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uncertain. Stevenson (1979) and Braginsky & Meytlis (1990) have considered the problem, and made tentative mainly qualitative estimates of the e¬ect of turbulence. It is generally believed that turbulence is important in the Earth’s core and will enhance the di¬usive processes. This can signi­ cantly a¬ect behaviour; for example, if turbulent di¬usion can enhance the e¬ective value of E to greater than 10 10 the dynamo could be changed from subcritical to supercritical. Unfortunately, as stressed by Braginsky & Meytlis (1990) and Braginsky & Roberts (1995), turbulence in the extreme conditions in the core is quite unlike `normal’ turbulence. In normal turbulence, the nonlinear advection terms u ru in the momentum equation play a key role; but these terms are small in the Earth’s core. High Prandtl number B´enard convection at large Rayleigh number is an example of a system which shares this property and is well studied; the analogy is imperfect, though, because in the core Coriolis and Lorentz forces will give rise to strong anisotropy absent in high-Pr convection.
Braginsky & Meytlis (1990) consider turbulence generated by local convection with strong rotation and azimuthal magnetic ­ eld. They note that the fastest growing modes are plate-like cells, with the short radial length-scale expected from nonmagnetic convection, but with the azimuthal length-scale increased by the azimuthal magnetic ­ eld. The duration of the cells is assumed to be the growth rate of the convective instability, ca. 6 yr 1, and their size in the azimuthal and z-directions is then determined to be ca. 50 km from the usual estimate of the velocity. The radial length-scale is much smaller at ca. 2 km. The resulting turbulent di¬usivities are then ca. 1 m2 s 1 in the azimuthal and z-directions, and much smaller in the radial direction. The axis of the plate-like cells will be in the direction perpendicular to the rotation axis and the magnetic ­ eld, so unless the azimuthal ­ eld dominates other components, the anisotropic di¬usion tensor will be a strong function of position.
Another di¯ culty is that a local theory, such as the Braginsky{Meytlis theory, is unlikely to be valid near the ICB and the CMB; but it is the boundary layers that control high Rayleigh number convection.

(e) Taylor’s constraint

Taylor (1963) observed that if the magnetic ­ eld in the Earth’s core was in magnetostrophic balance (Coriolis, Lorentz and pressure forces, with viscous and inertial forces negligible), the magnetic ­ eld has to satisfy the constraint
Z

(j B ) dS = 0;
S

(6.7)

where S is any cylinder concentric with the rotation axis. A large literature has developed concerning nonlinear {! dynamo models, and whether or not they satisfy Taylor’s constraint, recently reviewed by Fearn (1998). It is natural to enquire whether the convection-driven geodynamo models satisfy Taylor’s constraint. Sarson & Jones (1999) analysed dynamos in the Glatzmaier{Roberts regime, with low headline E but with hyperviscosity. They found that Taylor’s constraint was not well satis­ ed, suggesting that viscous e¬ects are playing an important part in these models. This is not perhaps surprising, as we know the current generation of dynamo models is still viscously controlled rather than magnetically controlled. As E is reduced from the currently available values of 10 4, initially we expect the roll diameter to

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decrease so that viscous forces can still control the convection, and the magnetic­ eld scale to decrease similarly. These small-scale ­ elds need not obey Taylor’s constraint, since on that scale Lorentz forces can be balanced by viscous forces. However, as E reduces further, magnetic di¬usion will prevent the magnetic-­ eld scales getting smaller, while viscous forces can only be signi­ cant on very small length-scales unless large velocities occur. At these values of E, which are considerably smaller than those currently attainable, the dynamo must choose between satisfying Taylor’s constraint (the Malkus{Proctor scenario) or maintaining a high-velocity zonal wind with which viscosity acting through the Ekman boundary layer can balance the Lorentz torque (Braginsky’s model-Z scenario). The current generation of models has not yet reached E su¯ ciently small to give any indication into which of these two scenarios convection-driven dynamos will jump.

7. Conclusions
The current generation of numerical dynamos has been remarkably successful in bringing theoretical dynamo models into contact with geophysical reality. As a result of a concerted international e¬ort, we are beginning to understand the physical mechanisms operating in these models, so that we can analyse why they behave as they do. We are also beginning to understand how the models vary as functions of the parameters, within the numerically accessible range. Fundamentally di¬erent regimes are found depending on how large the Rayleigh number is compared with the Ekman number. As computer technology advances there is every prospect that our understanding of the behaviour at moderate R and E will be enhanced.
It is clear, however, that the parameter regime operating in the Earth is far beyond that which can be simulated directly. The only way open to us to bridge the gap in the parameter space is by developing our understanding of the physical processes further, so that we can work out the asymptotic laws that govern these processes and hence extrapolate to the geophysical parameter range. At present, we do not know enough about the basic dynamics of rapidly rotating convection in spherical geometry in the presence of magnetic ­ eld for this to be possible. Some of the key questions we would like to answer are as follows.
(i) What is the dominant length-scale of convection rolls in the core? Is the dominant transverse length-scale of the order of the core radius, with perhaps m = 2 as suggested by some magnetic observations (Gubbins & Bloxham 1987), secular variation studies (Bloxham & Jackson 1991) and with some theoretical support (Longbottom et al . 1995); or is the transverse length-scale much smaller? The current models show very little evidence of the magnetic ­ eld increasing the transverse length-scales above those predicted by non-magnetic low-E convection, so maybe the strongly non-axisymmetric magnetic ­ eld plays a more subtle role in the pattern selection process than is at present understood.
From our studies, we have a minimum roll diameter of ca. 15 km below which a dynamo cannot operate; the actual roll diameter cannot be too small, or else Ohmic dissipation would be too large, but we still have a large range of uncertainty. Unfortunately, small-scale magnetic ­ elds at the CMB cannot easily be detected at the Earth’s surface, because their in®uence declines rapidly with distance.
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(ii) How do the smaller scales of motion interact with the larger scales? In particular, if the e¬ective eddy di¬usivity is anisotropic, how will that a¬ect the large-scale behaviour? A related question is what is the nature of the thermal and compositional boundary layers, and do they control the convection?
(iii) What is the e¬ect of the magnetic ­ eld on the convection? The current models show only a rather weak e¬ect, less than would have been expected on the basis of magnetoconvection calculations with axisymmetric ­ elds. However, it is hard to believe that the ­ eld will not have a major impact at low E; the question is perhaps when it kicks in.
These questions will not be easy to answer, but with a combination of process studies of low E magnetoconvection, laboratory experiments and detailed comparison of dynamo simulation outputs with geophysical observation hopefully progress will be made.
I thank Dr Graeme Sarson for many discussions, and in particular for computing the solutions shown in ¯gures 2 and 3, and Professor K. Zhang for computing the solutions shown in ¯gure 1, and for many discussions in which our ideas about the geodynamo emerged. This work was supported by the UK PPARC grant GR/K06495.

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