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Convection driven geodynamo modelsof varying Ekman numberG. R. Sarson a , C. A. Jones a & A. W. Longbottom ba Department of Mathematics , University of Exeter , Exeter,EX4 4QE, UKb Department of Mathematical and Computational Sciences ,University of St. Andrews , St. Andrews, KY16 9SS, UKPublished online: 01 Dec 2006.

To cite this article: G. R. Sarson , C. A. Jones & A. W. Longbottom (1998) Convection drivengeodynamo models of varying Ekman number, Geophysical & Astrophysical Fluid Dynamics,88:1-4, 225-259, DOI: 10.1080/03091929808245475

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CONVECTION DRIVEN GEODYNAMO MODELS OF VARYING EKMAN NUMBER

G. R. SARSON ’,*, C. A. JONES a and A. W. LONGBOTTOM

a Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK; Department of Mathematical and Computational Sciences, University

of St. Andrews, St. Andrews KY16 PSS, UK

(Received 10 June 1997; In final form 21 January 1998)

We investigate the dynamo action arising from convection in a rapidly-rotating spherical shell. A single mode of the non-axisymmetric field is solved for, in addition to the axi- symmetric (mean) field. This allows dynamo action to be obtained without any imposed parameterisation, yet results in a system tractable enough that a range of physical regimes can be investigated. We describe the different types of dynamo obtained for varying Ekman and Roberts numbers. For the smaller values of these parameters, hyper- diffusivities have been used to model the effccts of small lengthscale turbulent diffusion.

At relatively high Ekman numbers (c. lop3) dynamo action is obtained for moderately supercritical convective flows, with the flow in the form of travelling-wave convective rolls. At lower (hyperdiffusive) Ekman numbers, magnetic field maintenance occurs only for strongly supercritical flows. The resultant dynamos are temporally chaotic and dominated by strong fields and flows in the vicinity of the inner core, resembling the fully three-dimensional solutions of Glatzmaier and Roberts (1995b, 1996a). The inner core plays a critical role in these solutions, and rotates progradely at of order 1” per year.

Calculations are conducted with ‘dipolar’ equatorial symmetry imposed and with no imposed symmetry. The imposed symmetry constraint proves an unphysical restriction for all our solutions. At lower Ekman numbers, the equatorially symmetric magnetic field is only intermittently significant; fluctuations in this component remain a plausible ‘trigger’ for reversals of the dominant dipole field, however, and so potentially important for the geodynamo.

Taylor torque ‘integrals are calculated to quantify the adjustment of our solutions to the low viscosity regime. Our low Ekman number solutions satisfy the Taylor constraint appropriate to this regime more poorly than those a t high Ekman number, however. Here the Lorentz torque on coaxial cylinders is balanced by the viscous torque associated with the hyperdiffusively-affected short wavelength velocities. Thus these solutions also remain viscously controlled, and might be expected to depend somewhat on the form of hyperviscosity assumed. At neither high nor low Ekman numbers does

* Corresponding author.

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removing the imposed symmetry constraint reliably assist towards satisfying Taylor’s constraint.

Keywords: Dynamos; convection; hyperdiffusivity; Taylor’s constraint

1. INTRODUCTION

Although the general mechanism by which the geomagnetic field is generated is now the subject of broad consensus - convection in the Earth’s conducting fluid outer core maintaining the magnetic field by dynamo action - the details of this process remain far from well understood. Observational data is limited by the long timescales over which the dynamo process occurs, and we cannot rely upon significant amounts of new or improved data becoming available in the near future. Experimental work cannot truly duplicate the conditions under which dynamo action occurs on planetary scales. Analytical studies have no such limitations, but are by necessity restricted to idealised simplifications of the true problem pertaining to the Earth; and whilst the concepts arising from such work [the (Y and w effects resulting from helical flow and differential rotation, for example (e.g., Roberts, 1994)] are undoubtedly of help in understanding the behaviour of more complex dynamo systems, the complicated interaction of field and flow in the latter cannot be encompassed by such ideas alone.

Thus, whilst both observational and theoretical approaches remain crucial for a true understanding of the dynamo - and whilst agree- ment with the observations must of course remain the ultimate test of any dynamo model - numerical simulation is arguably the most promising mode of enquiry for the geodynamo at present, with detailed three-dimensional models now within the limits of compu- tational capability. Such studies remain formidable undertakings however; for a model retaining most of the features thought to be important for the Earth, at parameter values even approaching those of relevance, essentially only isolated runs are possible. There therefore remains a place for simpler models, broadly replicating the behaviour of the full system, but sufficiently simple that a wide range of parameter space, and the effect of a number of factors, can be investigated in detail.

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CONVECTION DRIVEN GEODYNAMO MODELS 227

Our ‘self-consistent mean-field’ dynamo model is intended to fulfil this need. In this model, attention is focussed largely on the axi- symmetric (mean) field. Unlike traditional mean-field models however, we do not invoke an artificial prescription for the action of the non- axisymmetric components of the field; rather, we treat a single azimuthal mode (wavenumber) of the non-axisymmetric field rigor- ously, fully retaining its interaction with the mean field. Hence internally self-consistent dynamo action is possible, in the sense that no processes beyond the scope of the model need be postulated.

The model remains incomplete in that the non-axisymmetric wavenumber used is chosen essentially arbitrarily, and all interactions involving other wavenumbers are neglected. Nevertheless, we remain capable of accurately modelling the processes by which non-axisym- metric ‘MAC’ waves (‘Magnetic-Archimedean-Coriolis’, after the three dominant forces) can be excited in the presence of an axisym- metric field, and can also act to sustain the field on which they ‘ride’. Thus the two idealised models most commonly studied in magnetohy- drodynamics - magnetoconvective instabilities and mean-field dyna- mos - are spliced together, and their nonlinear interaction can be investigated, retaining the kernel of the fully three-dimensional problem we ultimately aim to model.

In previous papers (Jones et al., 1995; Sarson et al., 1997), we obtained simple working dynamos with this system, and investigated the effect of some possible core-mantle boundary scenarios on these models. Both these studies had, for computational simplicity, assumed equatorial symmetry (antisymmetry) in the velocity and codensity (magnetic) fields. This is a reasonable first approximation, since the geomagnetic field is observed to be dominantly antisymmetric (dipolar) with respect to reflection in the equator, and convection in rapidly-rotating systems is known to be dominantly symmetric under this operation (Busse, 1970). It remains an artificial constraint, however. Moreover, previous authors have argued that the interaction between dipolar and quadrupolar (equatorially symmetric) magnetic fields is important in allowing the system to adapt to very low viscosity (Jault, 1996), or in triggering episodes of reversals of the dipole field (Galtzmaier and Roberts, 1995a). In the current work, we therefore relax this constraint, investigating the effect of the previously neglected symmetry components on the solutions obtained.

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The preceding studies had also, for computational reasons, been restricted to relatively high values of the Ekman and Roberts numbers (defined in Section 2 below), compared to the values anticipated for the Earth’s core (e.g., Gubbins and Roberts, 1987). Computational restrictions still prevent us from remedying this situation in an entirely satisfactory way. We attempt to approximate more closely the low Ekman and Roberts number regime, however, through the use of hyperdiffusive forms for these quantities. Hyperdiffusivities attempt crudely to model the preferential coupling that should occur between very short lengthscale features, excluded from our numerical model, and the shorter lengthscale features which we retain. These smaller wavelengths should be subject to stronger (turbulent or eddy) damp- ing than the larger wavelengths; imposing a wavenumber-dependent diffusion operator achieves this effect, and allows us to attain smaller values of the bulk Ekman and Roberts numbers.

Unfortunately, no truly satisfactory theory for this process is currently available, and we are forced to adopt an essentially arbitrarily prescribed form for the hyperdiffusivity function (zk., the function modifying the diffusion operator; see Section 2). Nevertheless we may hope to model the effect in a broad way. And whilst the use of hyperdiffusivities remains unsatisfactory, it does allow us further possi- bilities towards modelling the geodynamo; the resultant solutions, although somewhat artificial, may be closer to the true geodynamo than the relatively high Ekman and Roberts number solutions we are able to obtain without their use.

The introduction of hyperdiffusivities into our mean-field model also allows us simply to investigate their empirical effect on the dynamo system; given that they are routinely used in fully three- dimensional dynamo models (Glatzmaier and Roberts, 1995a, 1995b, 1996a; Kuang and Bloxham, 1997, 1998), their importance is of interest per se. Such an investigation, prohibitively expensive in more complex systems, remains practical for our simplified model.

2. MATHEMATICAL FORMALISM

The method used remains largely unchanged from that of Jones ef al. (1995) and Sarson et al. (1997), and so we present only a general

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CONVECTION DRIVEN GEODYNAMO MODELS 229

outline here, giving specific details only where we differ from the previous works.

We consider the magnetostrophic approximation of the magneto- hydrodynamic (MHD) equations - for velocity U, magnetic field B, codensity C, and pressure P - as appropriate to a buoyant, Boussinesq fluid, rapidly rotating about the polar (2) axis in spherical geometry ( I , O,$J). The codensity can be considered as the temperature or as the fractional component of lighter constituent (of an assumed binary fluid), for the cases of thermal or compositional buoyancy respectively. (Both types of buoyancy are thought to contribute to convection in the Earth's core; e.g., Braginsky and Roberts, 1995.)

Thus we have

e,xU = -VP+ qRaCr + (VxB)xB + EV2U (1)

dB - = VX(UXB) + V2B at

dC - = -1J. VC + qV 2C + S, at (3)

V U = V * B = 0, (4)

where the non-dimensional Roberts, modified Rayleigh and Ekman numbers arising are respectively given by

in terms of the codensity diffusitivity r;, magnetic diffusivity rl, gravitational acceleration g , coefficient of expansion a, mean (back- ground) radial codensity gradient p, rotation rate R, kinematic viscosity Y, and length scale L. Here e, is a unit vector in the z- direction. Also S represents the sources and sinks of codensity appropriate to the buoyancy regime we wish to model; these are entirely taken up by the static, spherically-symmetric base state, C&), however (and in practice it is CO, rather than S = - qV 'C0, that we specify), and we henceforth concentrate on the perturbation to this state due to the convection.

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In the mean-field approximation we split all quantities into axi- symmetric (overbarred) and non-axisymmetric (lower case) parts

U = D + U , B = B + b , C = ~ + C . ( 6 )

We restrict the non-axisymmetric parts to a single mode, i.e., to a single wavenumber m in an expansion in azimuth of the form C, exp (imd), neglecting the quadratic interactions which give rise to higher modes. This permits self-consistent dynamo action to be studied at relatively low computational cost. In the current work we consider m = 2 exclusively. (Similar results have been obtained with m = 4; higher m have not yet been thoroughly investigated. For comparison, the three-dimensional solutions of Glatzmaier and Roberts show a broad peak of energy in wavenumbers m = 6 to 13; Glatzmaier, personal communication.)

We consider a fluid shell of unit width and radius ratio I , given ri = C/(1 - c) 5 r 5 1/(1 - Q = r,. For the present, we fix C = 1/3, so that ri = 1/2, and ro = 3/2. No slip boundary conditions are used on U, so that U = 0 applies at the outer boundary r = r,. The inner core is free to rotate about the z-axis, its rotation rate being determined by the balance of the z-components of the viscous and electromagnetic torques on the surface r = Ti. The magnetic field B is matched to an external potential field at r,, but is solved explicitly within the rotating inner core, assuming a conductivity equal to that of the outer core. To model thermal convection, we apply a fixed flux condition to the codensity at ri; constant C is applied at r,. Consistent with this model, the background codensity distribution, Co, is that appropriate to an influx of buoyant material at the inner core boundary,

c, = -1 +--. 3 4r (7)

The same codensity scenario was used in Glatzmaier and Roberts’ (1995a, b) studies, although they later (1996a, b) adopted mixed thermal and compositional convection, with the latter employing a zero-flux outer boundary condition.

We expand all vector fields as sums of toroidal and poloidal parts,

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CONVECTION DRIVEN GEODYNAMO MODELS 23 1

All quantities (v, $ , . . .) are expanded in associated Legendre functions, Pt(cos8) , in 8, with riZ fixed as appropriate to the quantity expanded. The expansion is carried out over N functions in fi (or over N/2 functions for solutions with equatorial symmetry imposed; see below). All quantities are further expanded in Chebyshev polynomials, Ti[x(i-)], in r , x (r ) being the linear mapping of Y onto (- 1, 1) within the fluid shell, or onto (0,l) within the inner core (where symmetry arguments allow us to consider only the Ti symmetric or antisym- metric in Y). These expansions are carried out over L + L' functions; L' is here a constant, equal to the number of boundary conditions the relevant quantity must satisfy, the boundary conditions being implemented as additional equations. The solutions reported in the present work used N = 48, L = 32, although the resolution of these solutions was also checked at the higher truncation N = 72, L = 48.

A pseudo-spectral method is adopted, nonlinear terms being evaluated on a grid of N/2 x L collocation points. (Separate treatment of the terms symmetric and antisymmetric with respect to the equator allows the number of points in 0 to be reduced from N to N/2.) The time-stepping is semi-implicit (Crank-Nicolson) for linear terms, explicit for nonlinear terms, and O(St ') accurate.

In an attempt to model the effects of turbulent diffusion occurring on lengthscales finer than our resolution - and to allow lower Ekman and Roberts numbers to be obtained - hyperdiffusivities are introduced, as discussed in Section 1. All diffusion terms, calculated in angular spectral space, are taken to vary with Legendre degree, ii, such that

v2 = (1 +Xfi3)V2. (10)

We use this form to facilitate comparisons with Glatzmaier and Roberts (1995a, b), who employed the identical functional form. We retain the hyperdiffusive parameter, A, as a variable, and may clearly remove the hyperdiffusive approximation by setting X = 0; hyper- diffusivity is used for the solutions in Section 4, but not for those in Section 3.

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In previous work we considered only velocity fields symmetric with respect to reflection in the equator; this allowed magnetic fields either antisymmetric (of dipolar symmetry) or symmetric (quadrupolar symmetry) under this operation to be considered in isolation. Because of our geophysical motivation, we chose to consider the dipolar symmetry magnetic fields. That is, we originally enforced

ur(e) = ur(n - el, ve(e) = -uO(. - e), u,(e) = u,(n - e), (11)

(the symmetry or antisymmetry of the varying components simply conforming to the intuitive concept of equatorial mirror-symmetry), through the use of restricted expansions in 8; e.g., for the axisymmetric magnetic field,

Although such solutions are internally self-consistent (e.g., Gubbins and Zhang, 1993), the imposition of this symmetry constitutes an un- physical restriction on the solution. We therefore now also consider solutions free of this imposed constraint, using the full expansions

N N

and investigate how the two different expansions affect the solutions obtained .

3. ‘BUSSE- ZHANG’ DYNAMOS

We initially consider solutions of essentially the strong field type described by Jones et al. (1995) and Sarson et al. (1997); the solutions described here differ from the former in the background codensity

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CONVECTION DRIVEN GEODYNAMO MODELS 233

distribution used, described in Section 2, and in the later relaxation of the imposed symmetry constraint. Broadly similar behaviour has been obtained for other buoyancy source distributions, including those used in the earlier studies.

The velocity fields of these solutions are dominated by azimuthally drifting rolls aligned with the rotation axis, outside the coaxial cylinder circumscribing the inner core. We refer to these dynamos as of Busse- Zhang type, after their similarity to the convective dynamo solutions studied in detail by those authors (e.g., Zhang and Busse, 1989, 1990, and earlier references therein), pursuing a line of inquiry initiated by Busse (1976). (Somewhat similar numerical solutions had previously been obtained by Gilman, 1981 and Glatzmaier, 1984.)

Figure 1 shows the time-variation of magnetic and kinetic energies for a dipole-imposed dynamo of this type, obtained for q = 10, Ra = 125, E = lop3. We have here calculated the energies as

where Vi and V, are the volumes of inner and outer cores respectively. We have also adopted dimensional units based on the length scale of the core, L N 2200 km, the magnetic diffusion time scale, 7 = L 2 / q - 60000yr, and the magnetic field strength, B = ( 2 R p p ~ ) " ~ - 2.2mT. The magnetic energy of this solution exceeds the kinetic by several orders of magnitude, and both lie within reasonable limits for the Earth (e.g., Gubbins and Roberts, 1987).

1'10"

5'1 0"

t ( Y 4 1-10' 2'1 0' 3'10' 4'10'

FIGURE I dynamo obtained for g = 10, Ru = 125, E =

Magnetic and kinetic energies with time for the dipole symmetry

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The time-dependence of this solution is somewhat chaotic, as with the strong field solution of Jones et al. (1995) (cf. the simple vacillatory behaviour obtained upon the addition of a stably-stratified layer; Sarson et al., 1997). The different source of convection (an in-flux of buoyancy at the inner boundary) does not appear to have drastically affected the dynamo action. The basic form of solution remains a travelling wave (albeit a chaotically modulated one) with velocity and magnetic field approximately co-rotating.

A snapshot of the axisymmetric state of this solution is shown in Figure 2. Figures 3 and 4 illustrate the travelling-wave convection, showing the velocity field in a slice of constant z (k, a plane parallel to the equator) and on a surface of constant r respectively. The relatively simple roll form is apparent, disturbed somewhat by the presence of the inner core, which induces a strong vertical flow where it impinges on the flow in the plane. Values of the magnetic Reynolds

B . . . . . A r sin@. . . c . . . . . . . v/r sine. . . 3 r sine. . .

(Contours are

8 1179

0.7012

0.8059

8 7 4.0 7 5 8

5.8888

a t Max./lO)

FIGURE 2 A snapshot of the axisymmetric state of the dipole symmetry dynamo with q = 10, Ra = 125, E = B is the toroidal (zonal) magnetic field; Ar sin 6' the lines of force of the poloidal (meridional) field; C is the codensity; v/r sin 6' the toroidal angular velocity; and $r sin 6' the streamlines of the poloidal velocity. As all quantities are purely symmetric or antisymmetric with respect to the equator, only one hemisphere is shown. Contour intervals are one-tenth of the maximum stated, with dashed lines corresponding to negative quantities. In plots of zonal fields, positive contours show eastward directed fields. In plots of streamlines or lines of force, the sense of field is clockwise around positive contours, anticlockwise around negative.

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CONVECTION DRIVEN GEODYNAMO MODELS 235

-268.6 0 268.6

FIGURE 3 Thevelocityfieldin theplanez = 0.3(wherez = Oistheequator,andz = 1.5 the outer boundary North pole; the plot is viewed from above), for the dipole symmetry snapshot detailed above. The colour contours show the vertical velocity U, (with contour intervals one-tenth of the maximum amplitude stated), the arrows the horizontal velocity U,, (the longest arrow corresponding to the maximum stated). (See Color Plate I).

number, R, (equivalent to IUI in our non-dimensionalisation) of order several hundred are typical for dynamos of this type. (Our unit of U corresponds to .a dimensional velocity of order 1.2 x 10-6ms-* in the Earth.)

The morphology of this solution can vary appreciably with time, as might be expected from the chaotic variations apparent in Figure 1; the dominant features of both velocity and magnetic field oscillate appreciably, both in position and strength, from the snapshot shown. Nevertheless, the largest-scale features remain rather predictable

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U

G. R. SARSON et al. - 257.6

-21 2.2 0 212.2

FIGURE 4 The velocity field on the radial surface r = 1 (the half-width of the fluid shell), for the dipole symmetry snapshot detailed above. The colour contours show the radial velocity U,, the arrows the tangential velocity U,. A cylindrical equidistant projection has been used. Other plotting details are as before. (See Color Plate 11).

(zonal magnetic field remains consistently concentrated in the outer- most equatorial region and in the inner core tangent cylinder, for example) so that the underlying dynamo mechanism appears rather constant.

When the restriction to dipolar symmetry is removed, a significant (k, energetically comparable, or slightly dominant) component of quadrupolar (equatorially symmetric) magnetic field is excited. The velocity field remains predominantly symmetric. The time-evolution of energy, subdivided by equatorial symmetry, is shown in Figure 5. Figure 6 shows a snapshot of the axisymmetric solution, which some- what resembles the dipole-imposed solution (Fig. 2), although there is a significant component of quadrupolar magnetic field present. The nature of the convection is largely unchanged, as can be seen from a plot of the velocity on the mid-depth surface (Fig.7). Again, the velocity is predominantly non-axisymmetric (approximately 80 -.90% energetically), and predominantly toroidal (65 - 80% energetically); values of R,, are also relatively unchanged.

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CONVECTION DRIVEN GEODYNAMO MODELS 237

Em(S) ----- Em@) Em (J)

4

time (yrs) 0.O'1O0 I , , l l l , , l ~ l ' " ~ 1 ' 1 1 1 ' 1 1 1 ~ 1 ' ' ' ' 1 ' 1 1 ,

2.5'1 0'' -

1 '1 0' 2'10' 3'10' 4'10'

1 V O ' 2*10' 3'1 0' 4'10'

FIGURE 5 Magnetic and kinetic energies with time for the mixed symmetry snap- shot obtained for q = 10, Ra = 125, E = The energies are subdivided into equa- tonally symmetric (S) and antisymmetric ( A ) parts.

Figure 8 shows the azimuthal average of the helicity, H = U . V X U, an important quantity in the kinematic generation of magnetic fields. (Our unit of H corresponds to a dimensional value of 6 x msW2.) The helicity has significant maxima within the inner core tangent cylinder: in the vicinity of the poles (arising almost purely from the axisymmetric velocity), and on the inner core boundary (the result of both axisymmetric and non-axisymmetric components). There are additional, lesser, helicity maxima outside the inner core tangent cylinder, at the ends of the non-axisymmetric roll flow. This is reasonable, as the rolls show some degree of correlation between the vertical velocity, U,, and the vertical vorticity, w,, where o = V x U; the latter being associated with the centres of the rolls of horizontal flow visible in Figure 3. (Although relatively minor for this solution, this non-axisymmetric source of helicity is the dominant feature for the somewhat similar solutions obtained with the stably-stratified codensity distribution used by Sarson et al., 1997.)

The magnetic field on the outer boundary is shown in Figure 9. Although clearly no longer purely dipolar (equatorially antisymmetric),

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. .

are

7 1246

0 6196

0 6461

2300 5044

7 7047

at Max./lO)

FIGURE 6 A snapshot of the axisymmetric state of the mixed symmetry dynamo with q = 10, Ra = 125, E = In the absence of an imposed equatorial symmetry, both hemispheres must now be shown; other plotting details are as before, however.

and with a strong disparity of field strengths between northern and southern hemispheres, the surface field remains dominantly dipolar topologically. It is somewhat too strong for the geomagnetic field; scaling with the magnetic field strength l3 gives a maximum surface field of almost 4 mT, compared with the observed core-mantle boundary field of 1mT. Furthermore, reversals in the sense of solution (both dipolar and quadrupolar components) now occur relatively frequently, whereas the dipole-imposed solution was of fixed

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CONVECTION DRIVEN GEODYNAMO MODELS 239

U -.

367.5

-245.2 0 245.2

FIGURE 7 The velocity field on the surface r = 1 (the half-width), for the mixed symmetry snapshot detailed above. The colour contours show the radial velocity U,, the arrows show the tangential velocity U,. Other plotting details are as before. (See Color Plate 111).

polarity; this contrasts markedly with the intermittent reversal record of the geodynamo. The magnetic energy for this solution is relatively evenly split between axisymmetric and non-axisymmetric parts, as Figure 9 visually suggests.

Both our dipole-imposed and our mixed symmetry solutions are of strong field type and are strongly nonlinear, arising from a subcritical bifurcation of the type envisaged by Roberts (1978) (see e.g., Holler- bach, 1996). Unfortunately the details of this bifurcation picture could not be traced here; reaching the strong field solution branch via a series of bifurcations from non-magnetic and weak field solutions proved numerically impracticable. Isolating a solution of pure dipolar or quadrupolar symmetry on this strong field branch (without the imposed symmetry constraint) did not prove possible. The bifurcation to a mixed symmetry solution must therefore occur before the system attains the strong field branch (as might indeed be anticipated), and the dipole-imposed strong field solutions must be seen as non-physical.

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FIGURE 8 The azimuthally averaged helicity, H , in a meridional slice, for the mixed symmetry snapshot detailed above. The maximum absolute value is 9.5 x 10’; other plotting details are as before.

The relaxation of the dipole symmetry ansatz resulted in a more irregular time-dependence, as described above. (Similar results were found for solutions obtained with other background codensity dis- tributions.) Interactions with the quadrupolar field clearly permit instabilities which the dipole-imposed system (non-physically) pre- cludes; computationally, considerably smaller time-steps are at times required to track the mixed symmetry solution. This greater variability (in the mixed symmetry case) is reflected in the solution morphology. Nevertheless the largest-scale features of both field and flow remain similar, and again remain rather consistent with time; suggesting that the basic dynamo mechanism is rather constant, and that important aspects of this mechanism are contained within the imposed- symmetry problem.

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CONVECTION DRIVEN GEODYNAMO MODELS 24 1

B ---+

1.763

-3.282 0 3.282

FIGURE 9 The magnetic field on the surface r = 1.5 (the outer boundary), for the mixed symmetry snapshot detailed above. The colour contours show the radial field B,, the arrows the tangential field Bt. Other plotting details are as before. (See Color Plate IV).

4. ‘GLATZMAIER-ROBERTS’ DYNAMOS

The dynamos detailed above were by necessity restricted to the relatively high Ekman and Roberts numbers at which reliable solutions could be obtained. It is far from clear that these parameters are low enough to constitute a reasonable model of the geodynamo. Despite some reservations (discussed in Section 1) we therefore use hyperdiffusivities to attain lower nominal values of these parameters, which might approximate the physics of the Earth’s core more closely.

The solutions obtained at lower Ekman and Roberts numbers (with hyperdiffusivity adopted) are rather different from the Busse - Zhang dynamos, being temporally chaotic and dominated by strong zonal fields concentrated largely within the inner core tangent cylinder. The axisymmetric parts of these solutions are rather simple, in fact, for the most part varying only qualitatively from their time-averaged form. We will refer to these dynamos as being of Glatzmaier - Roberts type, after their similarities to the solutions reported in Glatzmaier and

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Roberts (1995a, by 1996a), and to highlight the important differences between these solutions and the Busse- Zhang type dynamos located at higher q and E.

Figure 10 shows the time-variation of magnetic and kinetic energies for a solution of this type with dipole symmetry imposed, obtained for q = 1, Ra = 3 x lo4, E = lod4, X = 0.05. We have here retained relatively high q and E to ensure that a well converged solution is obtained for the corresponding (more computationally expensive) mixed symmetry case. The essential character of the solution is not greatly changed by further decreasing q and E (and increasing A), however. In particular, a solution investigated for q = 0.1, E = 1 O-', X = 0.075 behaves broadly similarly, the tendencies noted here (towards concentration of features at the inner core surface and with- in the tangent cylinder, for example) simply being more pronounced in that case. A snapshot of the axisymmetric state of this solution is shown in Figure 11.

When the restriction to dipole symmetry is removed, appreciable energy can once more be found in the quadrupolar symmetry magnetic field, although the field here remains dominantly antisymmetric for significant periods of the time-evolution (Fig. 12), during which the di- pole and mixed symmetry solutions are rather similar. (Hence the dipole-imposed solution has not been discussed independently.) The velocity field again remains predominantly equatorially symmetric. The kinetic energy is now two orders of magnitude larger than for the Busse - Zhang dynamos, but remains three orders of magnitude smaller than the magnetic energy, which is similar to that obtained in

2.5'10'' 2'10'*

l ' lo's

0.O'1O0 t ( Y 4 1-10' 2'10' 3'10' 4.1 0'

FIGURE 10 Magnetic and kinematic energies with time for the dipole symmetry dynamo obtained for q = 1, Ra = 3 x lo4, E = X = 0.05.

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CONVECTION DRIVEN GEODYNAMO MODELS 243

B . . . _ . A r sine. . . c . . . . . . . v/r sine. , ,

9 r sine.

(Contours are

11.2184

0.2399

0.2098

6195.9443

14.6073

at Yax./lO)

FIGURE 1 1 A snapshot of the axisymmetric state of the dipole symmetry dynamo obtained for q = 1, Ra = 3 x lo4, E = X = 0.05. The codensity here varies rapidly across a boundary layer at the inner core, where a fixed-flux condition has been employed. The scale of the plot has been chosen to highlight the variation in the bulk of the core, so that the variations within this boundary layer are not evident.

I

1 5'lO2' 6 o ' loo time (yrs)

1.10' 2'10' 3'10'

5'1 l*lo'l 0"

1 .__. , -- ~A time (yrs) o*looj t 1 1 1 fl i 1 1 I 1 t - r t - i? I 'i i I i I 1 ' ) i 7 I i-i j I i i i 14 1 ~ 0 ' 2'10' 3'10'

FIGURE 12 Magnetic and kinetic energies with time for the mixed symmetry dynamo obtained for q = 1, Ra = 3 x lo4, E = X = 0.05. E(S) shows the equatorially sym- metric part, E(A ) the antisymmetric.

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Section 3. Although all quantities vary chaotically, for most of the time-evolution 60 - 80% of the magnetic energy is in the toroidal field, and a similar proportion in the axisymmetric field. Of order 90% of the kinetic energy is toroidal, with on average 65-85% of this energy being axisymmetric.

A snapshot of the axisymmetric field is shown in Figure 13. The axisymmetric velocity and codensity of these solutions remain consistent with a thermal wind mechanism throughout the time-

B

A r sine

C

v/r sine

$ r sine

(Contours are

12 ZQQD

0 2550

0 2426

7425 I172

14 4792

at Max / lo)

FIGURE13 A snapshot of the axisymmetric state of the mixed symmetry dynamo obtained for q = 1, Ra = 3 x lo4, E = X =0.05.

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CONVECTION DRIVEN GEODYNAMO MODELS 245

evolution; although the magnetic and kinetic energies fluctuate more erratically than for the Busse - Zhang solutions, the largest-scale features of the field and flow remain strikingly consistent. For the highly supercritical states modelled here [these solutions are at Rayleigh numbers of order one thousand times supercritical, with respect to the original (non-magnetic) convective instability], convec- tion outside the inner core tangent cylinder is highly efficient, the absence of the inner core in this region providing no obstacle to the formation of the largely z-independent flows preferred by rapidly- rotating fluids. As a result this outer region, outside of a thin layer at the inner core boundary, becomes well-mixed at a codensity near to that of the outer boundary. In contrast, the region inside the tangent cylinder remains relatively little perturbed from the static state, with aC/& remaining 0(1) across the outer core. Smoothly matching these two regions necessitates a gradient, aCjat? < 0, across the tangent cylinder in the Northern hemisphere. (The opposite signs pertain in the Southern hemisphere.) In the E< 1 limit to which we aspire, and neglecting for the moment magnetic effects, the +-component of the curl of the axisymmetric momentum Eq. (1) requires

dv l a c dz r ae' _ - - --

The codensity gradient noted above therefore requires a corresponding large-scale negative gradient in v as a function of z. Thus the fluid in the vicinity of the inner core must rotate progradely (eastwards) relative to that near the poles. The same large-scale codensity struc- ture also implies a buoyancy gradient dC/dr on the rotational axis, driving a flow radially outwards there, with an associated clockwise meridional circulation (positive streamfunction $). This overall circulation pattern is consistent with net conservation of angular momentum.

As the inner core is viscously coupled to the outer core fluid, it will tend to rotate in the same sense as the adjacent fluid; the zonal flow described above therefore causes the inner core to rotate eastwards relative to the mantle. (The inner core rotation is determined by the balance of viscous and electromagnetic torques on the inner core boundary. The viscous torque associated with the thermal wind velocity is the dominant factor for this particular solution, with the

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electromagnetic torque playing a secondary role, essentially effecting only a slight decrease of the rotation rate, as in the kinematic, thermal- wind driven study of Aurnou et al. (1996); this mechanism is described further in Sarson and Jones, 1998.)

This effect, arising simply from the greater efficiency of convection in the region outside the inner core tangent cylinder, might be expected to be common to all highly supercritical models with viscously coupled inner cores. (Although it is not inconceivable that significant Lorentz forces or bulk viscosities, neglected in this simple analysis, might in some cases destroy this straight-forward relationship.) That a ther- mal wind structure, and concomitant progradely rotating inner core, should be a general feature of convective models in this regime has also been noted by Glatzmaier and Roberts (1996b).

When translated into dimensional units, the prograde rotation of the inner core corresponds to approximately 1" per year. This agrees well with the rotation rate calculated by Song and Richards (1996), who obtained a measure of this order by fitting seismic travel time residuals from the past 30 years to a model of an anisotropic rotating inner core. A more recent observational study by Su et al. (1996) derived a figure of 3" per year, and the numerical calculations of Glatzmaier and Roberts (1995b, 1996a) produced rotation rates of 2- 3" per year. In contrast, the Busse-Zhang type dynamos detailed in Section 3 have inner cores rotating at the equivalent of approximately 0.1" per year, and in no fixed sense.

The dominance of the strong zonal flow in the vicinity of the inner core can be seen in Figure 14, which shows the velocity in a plane of constant z. The non-axisymmetric flow structure is by comparison little developed, although the vertical flows excited at the inner core boundary are of the same order as those found in the Busse-Zhang dynamos. The non-axisymmetric component of flow also fluctuates rapidly with time, with no strong features persisting at low latitudes. Figure 15 shows the velocity field on the mid-depth of the shell. The dominance of the axisymmetric flow is again clear, in the strong outwards flow at high latitudes associated with the thermal wind mechanism, as well as in the dominantly zonal tangential flow.

The axisymmetric components also dominate the magnetic field, and these are again concentrated in the vicinity of the inner core tangent cylinder (Fig. 13). As with the axisymmetric velocity field, the

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-448.6 0 440.6

FIGURE 14 The velocity field in the plane z = 0.3, for the mixed symmetry snapshot detailed above. The colour contours show the vertical velocity U,, the arrows the horizontal velocity Uh. Other details are as for previous plots. (See Color Plate V).

basic form of these features remains relatively stable as the solution evolves. In this, the magnetically stabilising effect of the inner core identified by Hollerbach and Jones (1993, 1995) must play a part. Both toroidal and poloidal axisymmetric fields extend into the inner core, where only diffusive processes, acting on relatively long timescales, can operate. These fields are therefore anchored against fluctuations occurring on the shorter convective timesclaes; a particularly strong or persistent fluctuation is required before the field within the inner core can be significantly disturbed.

The form of the magnetic field can to some extent be understood with reference to the associated velocity structure. The strong toroidal field at the inner core boundary, in which most of the magnetic energy lies, can be explained by the shearing of poloidal field by the strong zonal flow associated with the thermal wind; i.e., via the well-known

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U

G . R. SARSON et a1

1229.2

-696.9 0 696.9

FIGURE 15 The velocity field on the radial surface r = 1 (the half-width), for the mixed symmetry snapshot detailed above. The colour contours show the radial velocity Ur, the arrows the tangential velocity U,. Other details are as for previous plots. (See Color Plate VIL

w-effect mechanism (e.g., Roberts, 1994). The generation of both toroidal and poloidal field may also be associated with the helicity, which is also strongly concentrated at the inner core boundary, as shown in Figure 16.

The magnetic field at the outer boundary is shown in Figure 17. In contrast to the field in the interior, the surface field is rather variable; the relatively stable poloidal field threading the inner core does not always extend to the outer surface. Large fluctuations can therefore occur, and the surface field often bears little relation to the field in the interior. It is also relatively weak - for the snapshot shown, the maximum radial field, B,, corresponds to a dimensional field of order 0.5mT - and the non-axisymmetric features show no consistent pattern of movement, drifting both westwards and eastwards at dif- ferent times. This particular solution is therefore not an ideal model for the geodynamo. It may be that an inappropriate choice of azimuthal wavenumber in our two-mode approximation is responsible for this aspect of our solution; further work on this is required.

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CONVECTION DRIVEN GEODYNAMO MODELS 249

FIGURE 16 The azimuthally averaged helicity, H, in a meridional slice, for the mixed symmetry snapshot detailed above. The maximum absolute value is 7.5 x 10’; other plotting details are as before.

Nevertheless, the essential (largely axisymmetric) processes occurring within the inner core tangent cylinder remain a plausible model for many features of the geomagnetic field.

The intermittent presence of significant proportions of energy in the quadrupolar (equatorially symmetric) magnetic field (Fig. 12) is potentially of interest in light of the Earth’s irregular excursion and reversal record. Particularly large fluctuations in this component of field may be sufficient to disturb the field in the inner core, and so to trigger critical episodes for the dynamo, as suggested by Glatzmaier and Roberts (1995a). The relatively stable dipolar field will ultimately be re-established, but with a change in polarity perhaps having occurred in the intervening time. Reversals of the dipole field have

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0.219

-.2450 0 .2450

FIGURE 17 The magnetic field on the surface r = 1.5 (the outer boundary), for the mixed symmetry snapshot detailed above. The colour contours show the radial field B,, the arrows the tangential field B,. Other plotting details are as before. (See Color Plate VII).

been obtained for solutions similar to those discussed here, and are addressed in detail by Sarson and Jones (1998).

The axisymmetric part of our solution is in many respects similar to that reported by Glatzmaier and Roberts (1995b, 1996a), with fields of similar magnitudes and morphologies being obtained for similar physical parameters. As well as being reassuring in terms of the vera- city of our code, this correspondence is gratifying in establishing the usefulness of our mean-field approximation. We can now use our simpler model to investigate further aspects of this solution, with some confidence that the conclusions may apply to a truly three-dimensional model.

5. BEHAVIOUR AT LOW EKMAN NUMBER

Under the conditions pertaining to the Earth’s core, viscous effects should be important only within narrow layers at the fluid boundaries. Since an analytical treatment of such boundary layers is impractical

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for our model, we retain a bulk viscosity. The alternative approach, of neglecting viscosity altogether, changes the nature of the problem significantly and introduces non-trivial constraints on the solvability. (Zero viscosity might well, in any case, behave differently from asymptotically small viscosity we wish to model.) Given that we have retained the viscous term, we are unfortunately computationally unable to attain Ekman numbers as low as those expected for the Earth. In the current work we have adopted hyperdiffusivities as one way of attaining lower values of E. Both for this approximation and for our original, non-hyperdiffusive case, we would like some quantitative measure of the suitability of our solutions to the low Ekman number regime. Jault (1996) has suggested that the simulta- neous presence of equatorially symmetric and antisymmetric compo- nents of magnetic field is important in allowing the field to adapt to low viscosity; we can also examine this hypothesis for our solutions.

A useful measure in this context is the integrated Taylor torque. Taylor (1963) was the first to note that, in the limit of vanishing viscosity and vanishing inertia, the net magnetic torque on cylinders coaxial with the rotation vector must vanish, there then being no other term in the momentum equation capable of balancing this torque. Taylor’s constraint, as this is known, applies for all coaxial cylinders. In the inviscid case, T(s) = 0 must therefore be satisfied at all cylindrical radii, s, where

T(s ) = / [(VxB)xB] dz, 4

the limits of integration being defined so as to contain the whole of the fluid outer core. (The overbar denotes the azimuthal average, as before.) Here T(s) essentially measures how well the field morphology has adapted to allow cancellation of magnetic moment as a function of z and 4.

Whilst the form of T(s) as a function of s is of some interest, it is useful to have a single measure characterising the solution. We there- fore also integrate over s, and consider

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a form which is normalised by definition. In the asymptotic low E regime, T should be small and decrease with E, otherwise the magnetic field must remain viscously limited.

This quantity has been monitored for the solutions detailed above. For the Busse - Zhang dynamo with dipole symmetry imposed, Taylor's constraint is not particularly well satisfied, as Figure 18 shows. (The relatively jagged appearance of this plot is simply due to the large intervals between evaluations of T, this quantity being too time-consuming to, calculate more regularly.) The presence of the quadrupolar field in the mixed symmetry solution does not appear to assit towards the satisfaction of Taylor's constraint for these solutions (Fig. 18). It was however found to do so for the somewhat similar solutions obtained with a stably-stratified codensity layer, so that Jault's (1 996) hypothesis remains plausible for some dynamo systems, although it does not appear to be a general result. For these solutions T appears to decrease slightly with E, although significantly decreased values of E cannot currently be obtained. There are some signs, therefore, that Taylor's constraint plays some role in the determina- tion of the Busse-Zhang solutions.

The values of T obtained remain relatively large, however. Furthermore, the form of the drifting-roll convection is somewhat suggestive of an Ekman suction mechanism, with the maxima of U, showing a significant correlation with the centres of the rolls (the maxima of wJ; see Figure 3. As the z-component of the flow will clearly remain viscously limited if it is dependent upon Ekman suction [and as this must be an important component for dynamo action,

Mixed ----- Dipole T

1 '1 0' 2'10' 3'10' 4'10'

FIGURE 18 The integrated Taylor measure, T, as a function of time for the Busse- Zhang dynamos of both symmetry types (q = 10, Ra = 150, E =

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flows lacking true three-dimensionally being known to be incapable of sustaining dynamos (see e.g., Roberts, 1994)] this may be a serious problem in maintaining that these solutions are of relevance to the asymptotic low E regime. This need not be a critical factor, however. Strong vertical velocities are also produced at the inner core boundary by the simple geometrical effect of the presence of the inner core. Preliminary investigations suggest that broadly similar dynamo action can be obtained with stress-free outer boundaries, so that Ekman suction need not be playing a dominant role. Nevertheless, it remains far from clear that these solutions are of relevance to the low Ekman numbers applicable to the Earth.

For the Glatzmaier-Roberts dynamos the integrated torque measure, T, is in fact even higher, as Figure 19 shows. Once again, T is not significantly decreased through the interaction of symmetric and antisymmetric magnetic fields. Although at nominally lower E, these dynamos therefore appear to be even further from a Taylor state.

In the magnetostrophic approximation, the only net torque capable of balancing the Taylor torque on coaxial cylinders is the viscous torque. We can calculate this as a function of cylindrical radius, s,

For the nominal value of E used for our Glatzmaier-Roberts solu- tions, this torque remains small. If we calculate this measure making use of the &dependent form of E employed in the hyperdiffusive

T Mixed -I--- Dipole

time (yrs) 0.0 2.5'10' 5.O'1Os 7.5'10' l .O*lO' 1.25'10' 1.5'10'

FIGURE 19 The integrated Taylor measure, T, as a function of time for the Glatzmaier-Roberts dynamos of both symmetry types (q = 1, Ra = 3 x lo4, E = X = 0.05).

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approximation (lo), however, it remains capable of balancing the Taylor torque, as can be seen in Figure 20. (This plot was calculated for the dipole symmetry solution, but the same conclusion holds for the mixed symmetry solution.) This torque balance must of course be required of any valid solution of the MHD equations. In this respect the excellent balance obtained is a useful check on our numerical accuracy; particularly as the torques must be numerically evaluated on an (s, z) grid differing from the ( I , 8) grid on which the equations are solved.

Figure 20 shows that the small-wavelength features affected by the hyperdiffusivity retain a critical role in the coaxial torque-balance. It therefore remains unclear how well these solutions model the low Ekman number regime, if at low Ewe expect the viscous torque to be

FIGURE 20 Integral measures as a function of cylindrical radius, s, for the dipole- imposed Glatzmaier-Roberts dynamo obtained for q = 1, Ra = 3 x lo4, E = X = 0.05. T(s) shows the Taylor torque, V(s) the viscous torque calculated for the nominal (bulk) Ekman number, and V(X,s) the viscous torque calculated for the hyperdiffusive Ekman number. (The latter is indistinguishable from the Taylor torque on this plot.)

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negligible and Taylor’s constraint to prevail. At present the use of hyperdiffusivities allows the solution to evade this constraint; at any finite Ekman number a balance between electromagnetic and viscous torques remains possible through the short wavelength (hyperdiffu- sivity-affected) components of the azimuthal flow, and so a Taylor state need never be attained. This may represent a true physical effect [viscous torques arising from turbulent eddies may mean that Taylor’s constraint need never be satisfied by any physical (non-inviscid) fluid] or it may be quite artificial, model dependent feature.

We should therefore be extremely wary of employing hyperdiffusiv- ities for numerical convenience, and assuming that by making the high wavenumber features small in magnitude we are removing them from our solution with no other effect. If we are to attach physical import to these solutions, we must do so on the basis of the hyperdiffusive model constituting a valid approximation of turbulent diffusive effects. This is not wholly unreasonable; turbulent effects clearly must act, for energetic reasons, to enhance diffusion in some way. But the absence of a firm theoretical basis for the form of this enhanced diffusion makes the use of an arbitrarily imposed hyperdiffusive relation, such as we have employed, a matter for some concern. The manner in which it acts must be expected to be somewhat sensitive to the exact form used. Preliminary investigations with alternative forms suggest that this is indeed the case.

Further work is clearly required on the exact role played by hyperdiffusivities. Given that their use currently remains essential for the achievement of low Ekman numbers [which applies to the three- dimensional dynamos of Glatzmaier and Roberts (1995b, 1996a) and Kuang and Bloxham (1997, 1998), as well as to the mean-field dynamos reported here] this has not received the detailed considera- tion it deserves.

6. CONCLUSIONS

Using a self-consistent mean-field approximation, strong field dynamo solutions were obtained across a range of Ekman and Roberts numbers. Hyperdiffusivities were required to allow the smaller values of these parameters to be reached.

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At relatively high Ekman and Roberts numbers, dynamo action is obtained at moderately supercritical values of the Rayleigh number, with the flows in the travelling-wave, axially aligned roll form commonly associated with rapidly-rotating convective systems. Un- fortunately, it is unlikely that these Busse- Zhang type solutions are a good model for the geodynamo; the values of Ekman and Roberts numbers at which they are found are too high to be geophysically relevant, and the solutions appear to remain viscously limited.

At lower (hyperdiffusive) Ekman and Roberts numbers, magnetic field maintenance occurs only for more strongly supercritical flows. The resultant dynamos are dominated by strong fields and flows in the vicinity of the inner core tangent cylinder, strongly resembling the solutions obtained by Glatzmaier and Roberts (1995b, 1996a). Despite the chaotic nature of these solutions, the dominant axisymmetric part fluctuates around a rather stable form; the axisymmetric codensity and velocity remain consistent with a simple thermal wind mechanism throughout. The viscous coupling this flow exerts on the inner core, together with the magnetic torque exerted by the concomitant magnetic field, results in an inner core that rotates progradely, in agreement with the observational studies of Song and Richards (1 996) and Su et al. (1997), and also with the numerical models of Glatzmaier and Roberts (1996b).

For both solution types, the magnetic field is significantly altered by the removal of the (artificial) dipole equatorial symmetry constraint originally imposed. Indeed, for the Busse- Zhang dynamos, the quadrupolar symmetry often dominates the magnetic energy, making these solutions rather unsuitable as models for the Earth’s predomi- nantly dipolar field. For the Glatzmaier- Roberts type dynamos, the equatorially symmetric magnetic field is only intermittently significant. This may be important in triggering reversals of the dominant dipole field, an excessively large fluctuation in the quadrupolar field perhaps being sufficient to overcome the dipole field anchored in the inner core, as suggested by Glatzmaier and Roberts (1995a). Our mean-field model exhibits similar reversals, which are studied in detail by Sarson and Jones (1998).

Jault (1996) has hypothesised that the presence of both dipolar and quadrupolar magnetic fields should assist towards the satisfaction of Taylor’s constraint, as required for solutions at low Ekman number. Comparisons of solutions with and without dipolar symmetry imposed

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suggest that this is not a general result, although it remains a reasonable scenario for some styles of dynamo action. (It is alternatively possible that we have simply not attained sufficiently low Ekman number for Taylor’s condition yet to be an overriding constraint.)

For the Glatzmaier- Roberts solutions we have obtained, the use of hyperdiffusivities results in the small wavelength components of flow retaining a significant role in the torque balance on coaxial cylinders, so that Taylor’s constraint need not be satisfied to the extent one would expect from the nominal (headline) viscosity. This balance must be expected to be rather dependent on the form of the hyperdiffusivity used, and so great care must be taken in equating these solutions to true low Ekman number states. Nevertheless, some features of these solutions (such as the large-scale differential rotation and the asso- ciated rotation of the inner core, for which a good physical explanation exists) might be expected to be rather robust.

Further work is clearly required on the behaviour at low Ekman number however, with the role of hyperdiffusivities requiring parti- cular attention. The rather different nature of the solutions obtained here, and by Glatzmaier and Roberts, from others in the literature (e.g., Kuang and Bloxham, 1997; Wicht and Russe, 1997) must also be addressed. The roles of the fluid boundary conditions and of the inertial terms (both of which may be significant in the differences between the various models) are of considerable theoretical interest, and must be further elucidated in future.

Acknowledgements

This work was supported by the UK PPARC grant GRlK06495. The numerical calculations made use of significant amounts of code originally developed by Rainer Hollerbach. The authors would like to thank Gary Glatzmaier and Peter Olson for their thorough and thoughtful reviews.

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