Keke Zhang

A numerical dynamo benchmark

We present the results of a benchmark study for a convection-driven magnetohydrodynamic dynamo problem in a rotating spherical shell. The solutions are stationary aside from azimuthal drift. One case of non-magnetic convection and two dynamos that differ in the assumptions concerning the inner core are studied. Six groups contributed numerical solutions which show good agreement. This provides an accurate reference standard with high confidence.

Spiralling columnar convection in rapidly rotating spherical fluid shells

It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 10 9 T 12 ) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of ( R — R c )/ R c = O (0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at ( R — R c )/ R c = O (0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

Convection driven magnetohydrodynamic dynamos in rotating spherical shells

Abstract Finite amplitude solutions for magnetohydrodynamic dynamos driven by convection in rotating spherical fluid shells with a radius ratio of ηequals; 0.4 are obtained numerically by the Galerkin method. Solutions which are twice periodic in the azimuth (case m equals; 2) are emphasized, but a few cases with higher azimuthal wavenumber have also been considered. An electrically insulating space outside the fluid shell has been assumed. A comparison of the dynamo solutions of both, dipolar and quadrupolar, symmetries with the corresponding non-magnetic solutions shows a strong increase of the amplitude of convection owing to the release of the rotational constraint by the Lorentz force. In some cases at low Taylor number the amplitude of convection is decreased, however, owing to the competition of the magnetic degree of freedom for the same energy source. The strength of differential rotation is usually reduced by the Lorentz force, especially in the case of quadrupolar dynamos which differ in this r...

On the onset of convection in rotating spherical shells

Abstract The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r i/r o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convec...

Finite amplitude convection and magnetic field generation in a rotating spherical shell

Abstract Finite amplitude solutions for convection in a rotating spherical fluid shell with a radius ratio of η=0.4 are obtained numerically by the Galerkin method. The case of the azimuthal wavenumber m=2 is emphasized, but solutions with m=4 are also considered. The pronounced distinction between different modes at low Prandtl numbers found in a preceding linear analysis (Zhang and Busse, 1987) is also found with respect to nonlinear properties. Only the positive-ω-mode exhibits subcritical finite amplitude convection. The stability of the stationary drifting solutions with respect to hydrodynamic disturbances is analyzed and regions of stability are presented. A major part of the paper is concerned with the growth of magnetic disturbances. The critical magnetic Prandtl number for the onset of dynamo action has been determined as function of the Rayleigh and Taylor numbers for the Prandtl numbers P=0.1 and P=1.0. Stationary and oscillatory dynamos with both, dipolar and quadrupolar, symmetries are close...

The effect of hyperviscosity on geodynamo models

Some fundamental difficulties in the construction of an Earth-like dynamo model and the role of hyperviscosity in attempting to overcome these difficulties are discussed. An Earth-like convection model influenced by an imposed toroidal magnetic field is investigated both with and without the effect of hyperviscosity. It is shown that hyperviscosity affects the dynamics of convection significantly; the effect is similar in some respects to that of a strong magnetic field. The effective Ekman number, a key parameter for an Earthlike dynamo model, is substantially increased when hyperviscosity is employed.

On coupling between the Poincare equation and the heat equation

It has been suggested that in a rapidly rotating fluid sphere, convection would be in the form of slowly drifting columnar rolls with small azimuthal scale (Roberts 1968; Busse 1970). The results in this paper show that there are two alternative convection modes which are preferred at small Prandtl numbers. The two new convection modes are, at leading order, essentially those inertial oscillation modes of the Poincare equation with the simplest structure along the axis of rotation and equatorial symmetry: one propagates in the eastward direction and the other propagates in the westward direction; both are trapped in the equatorial region. Buoyancy forces appear at next order to drive the oscillation against the weak effects of viscous damping. On the basis of the perturbation of solutions of the Poincare equation, and taking into account the effects of the Ekman boundary layer, complete analytical convection solutions are obtained for the first time in rotating spherical fluid systems. The condition of an inner sphere exerts an insignificant influence on equatorially trapped convection. Full numerical analysis of the problem demonstrates a quantitative agreement between the analytical and numerical analyses.

Symmetry properties of the dynamo equations for palaeomagnetism and geomagnetism

Abstract Recently both palaeomagnetists and geomagnetists have searched for symmetries in their data which would give some guide to the nature of the Earths dynamo, most frequently quoting analyses of mean-field (αω) kinematic dynamos. The separable solutions of the fully nonlinear, convective dynamo with spherically symmetric buoyancy forces and boundary conditions arise from the group of symmetry operations that leave a rotating sphere unchanged; they are more general than the rather specialised solutions usually quoted in the geomagnetic literature. The full set of symmetry operations is an Abelian Lie group but two simple, finite subgroups contain all the symmetries we have found in the recent literature. The smaller subgroup contains both reflection in the equatorial plane, which gives rise to the so-called ‘dipole/quadrupole’ separation, and inversion, or field reversal. The full group also includes rotations about the polar axis; these rotations would not normally be significant but current interest in core-mantle interactions, which can make the core longitude sensitive, demands that we include them. This larger subgroup includes, in addition to field reversal and equatorial reflection, rotation by an angle π about the polar axis and reflection through the origin. We give the spherical harmonic expansions for each separable solution and indicate the type of data required to discriminate between different symmetries.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

On inertial waves in a rotating fluid sphere

Several new results are obtained for the classical problem of inertial waves in a rotating fluid sphere which was formulated by Poincare more than a century ago. Explicit general analytical expressions for solutions of the problem are found in a rotating sphere for the first time. It is also discovered that there exists a special class of three-dimensional inertial waves that are nearly geostrophic and always travel slowly in the prograde direction. On the basis of the explicit general expression we are able to show that the internal viscous dissipation of all the inertial waves vanishes identically for a rotating fluid sphere. The result contrasts with the finite values obtained for the internal viscous dissipation for all other cases in which inertial waves have been studied.