David R. Fearn

On the computation of steady, self - consistent spherical dynamos

Abstract In an earlier paper (Fearn and Proctor, 1984) we described results from a preliminary model of a spherical hydromagnetic dynamo driven by convection. An iterative approach was used. Starting from some guess for the mean toroidal field B we solved for the form of the convective instability in the presence of this field. The mean e.m.f. E [defined in (2.13)] associated with the convection was calculated, and from this, anα-effect was constructed (α=EΦ/B). We then solved a mean fieldαΩ-dynamo model to produce a new “B”. This cycle was repeated until B converged. For a preliminary investigation, there were good reasons for using anα-effect formalism. However, a more straightforward and physically more realistic approach is to use the e.m.f. EΦ directly to force the mean field dynamo. This “EΩ-dynamo” is used here. The converged results of Fearn and Proctor (1984) are successfully reproduced and in addition we have found converged steady dynamos in the absence of any poloidal flow (cf. Roberts, 1972)....

Hydromagnetic waves in a differentially rotating annulus IV. Insulating boundaries

Abstract The first three papers in this series (Fearn, 1983b, 1984, 1985) have investigated the stability of a strong toroidal magnetic field Bo =Bo(s∗)Φ [where (s∗. Φ, z∗) are cylindrical polars] in a rapidly rotating system. The application is to the cores of the Earth and the planets but a simpler cylindrical geometry was chosen to permit a detailed study of the instabilities present. A further simplification was the use of electrically perfectly conducting boundary conditions. Here, we replace these with the boundary conditions appropriate to an insulating container. As expected, we find the same instabilities as for a perfectly conducting container, with quantitative changes in the critical parameters but no qualitative differences except for some interesting mixing between the ideal (“field gradient”) and resistive modes for azimuthal wavenumber m=1. In addition to these modes, we have also found the “exceptional” slow mode of Roberts and Loper (1979) and we investigate the conditions required for i...

Hydromagnetic waves in rapidly rotating spherical shells generated by magnetic toroidal decay modes

Abstract Instabilities in the form of slow azimuthally travelling hydrodynamic waves in a rapidly rotating, stress-free, electrically conducting spherical fluid shell are investigated. The instabilities are generated by the toroidal decay mode of the lowest order or a combination of toroidal decay modes. It is found that the Elsasser number Λc at the onset of instability is always Λc = O(10) for various profiles of the basic magnetic field. It is also found that the hydromagnetic waves of the preferred instability propagate eastward [i.e. for solutions proportional to exp i(mφ + ωt), ω < 0] and are characterized by nearly two-dimensional columnar fluid motions attempting to satisfy the Proudman-Taylor theorem, indicating that the most rapidly growing magnetic disturbance arranges itself in such a way that the corresponding magnetic forces balance only the ageostrophic component of the Coriolis force. Except for the Stewartson-type velocity boundary layer at the equator of the inner core, the velocity and ...

Nonlinear magnetoconvection in a rapidly rotating sphere and Taylor's constraint

Abstract We investigate thermally driven convection in a rapidly rotating sphere in the presence of a prescribed azimuthal magnetic field B1φ. Earlier work has looked at the linear problem. Here, we include the most important nonlinear effect; the geostrophic flow V G (s)1 φ. This is determined through the standard condition that leads to Taylors (1963) constraint in the limit of vanishing viscosity. The present work therefore follows on from earlier work on both kinematic α- and αω-dynamos and magnetoconvection. Examples of the latter have so far been restricted to plane-layer, duct and cylindrical geometries. The present work uses a spherical geometry and makes a further step towards physical realism in that the contributions from both the axisymmetric and non-axisymmetric components of the magnetic field to the Taylor integral are included. (The earlier magnetoconvection work only included the non-axisymmetric contributions while the kinematic dynamo calculations involved only the axisymmetric contrib...

Magnetic instabilities in rapidly rotating spherical geometries. I, From cylinders to spheres

Abstract The solution of the full nonlinear hydromagnetic dynamo problem is a major numerical undertaking. While efforts continue, supplementary studies into various aspects of the dynamo process can greatly improve our understanding of the mechanisms involved. In the present study, the linear stability of an electrically conducting fluid in a rigid, electrically insulating spherical container in the presence of a toroidal magnetic field Bo(r,θ)l⊘ and toroidal velocity field Uo(r,θ)l⊘, [where (r,θ,⊘) are polar coordinates] is investigated. The system, a model for the Earths fluid core, is rapidly rotating, the magnetostrophic approximation is used and thermal effects are excluded. Earlier studies have adopted a cylindrical geometry in order to simplify the numerical analysis. Although the cylindrical geometry retains the fundamental physics, a spherical geometry is a more appropriate model for the Earth. Here, we use the results which have been found for cylindrical systems as guidelines for the more rea...

How strong is the invisible component of the magnetic field in the Earth's core

The maximum strength of toroidal magnetic field in the Earths core is estimated on the basis of the stability of the magnetic field. In our model we take our basic magnetic field BO to be composed of both toroidal and poloidal axisymmetric decay modes of lowest order. While the strength of the poloidal component, BP is taken consistent with observation, the maximum strength of the toroidal field, |BT|max, is regarded as a parameter of the model. By demonstrating that viscous dissipation is of secondary importance and therefore that the results are independent of the parameter associated with viscosity, our model is eventually dependent on only one parameter: the ratio A of the maximum strength of the invisible toroidal field to the strength of the poloidal field at the pole of the core-mantle boundary. It is shown that |BT|max < 10|BP(θ = 0, r = ro)| in order that the basic magnetic field BO = BP + BT is stable, giving an estimated upper bound on strength of the invisible toroidal field of order 50 gauss.

Differential rotation and thermal convection in a rapidly rotating hydromagnetic system

Abstract As a model for hydromagnetic waves in the Earths core, we study the linear stability of an electrically conducting fluid confined in a cylindrical annulus. The system is rotating rapidly about the axis of the cylinder with angular velocity ω0 equals; ω012. and the fluid is differentially rotating with velocity U0 equals;U0(s*)lo relative to the rotating frame of reference. [Here, (s*, o, z*) are cylindrical polar coordinates and 1 x is the unit vector in the direction of increasing x.] A magnetic field B0 equals; B 0(s*)lo and temperature distribution T0(s*) are imposed on the fluid. In an earlier series of papers (Fearn, 1983b, 1984, 1985, 1988a,b) we focused attention on instabilities driven by the field B0. Here we study two facets of buoyancy driven waves. The first is the role of differential rotation. It can act to inhibit convection but may also itself act as a source of energy to drive instability. For values of the Roberts number qequals;k/η ≧ O(1), (k and η are the thermal and magnetic...

Magnetostrophic balance in non-axisymmetric, non-standard dynamo models

Abstract We investigate solvability conditions for the magnetostrophic equation for dynamo models which are neither axisymmetric nor contained within an insulating sphere. Effects of topography and mantle conductivity are discussed. Simplifications that apply for axisymmetric fields contained in a perfectly insulating mantle no longer apply and we conclude that the standard manipulation of the Taylor integral is no longer helpful; it is best used in its original form ∫J×Bdzdφ. Electromagnetic and topographic core-mantle coupling are fundamentally different to viscous coupling. For the latter, the magnetostrophic equation always has a solution (due to the role of Ekman suction). For the former (in the absence of viscous coupling), a solution requires that Taylors condition be satisfied. For the case of electromagnetic coupling, we derive the appropriate magnetic boundary conditions for various models of iower mantle conductivity. Finally, we derive the solvability condition (analogous to Taylors conditio...

The influence of Rayleigh number, azimuthal wavenumber and inner core radius on 2- D hydromagnetic dynamos

Abstract This study assesses the influence of different prescribed parameters on the solutions of a 2- 1 2 D hydromagnetic dynamo model. The numerical solution is fully resolved in r and θ, but severely truncated in φ so that only a single, prescribed value of the azimuthal wavenumber, m, is included in addition to the axisymmetric (m=0) part of the problem. This model is ideally suited for such a study since it is a self-consistent, convectively driven dynamo, capable of reproducing qualitatively similar axisymmetric magnetic fields to those of a Boussinesq 3D model, but at considerably lower computational effort. We have chosen to vary the Rayleigh number, Ra, for m=2 and m=4, and we find that the solution is dependent on the choice of both Ra and m. This means that the 2- 1 2 D model is too severely truncated in φ, and suggests that caution should be exercised when interpreting the results from a single run of any convectively driven numerical dynamo model, at a particular value of Ra. For m=2, and all other parameters fixed, we have also investigated the effect of varying the inner core radius, giving some insight into possible effects of the growth of an inner core on magnetic field generation in planetary bodies. A stabilising effect on the magnetic field is unexpectedly observed for sufficiently small inner core radii. The anticipated stabilising effect is observed as our inner core radius increases from about its present value, until the dynamo shuts off for a radius ratio, χ, of about a half, for our fixed value of Ra.

Magnetic instabilities in rapidly rotating spherical geometries II. more realistic fields and resistive instabilities

Abstract Magnetic instabilities play an important role in the understanding of the dynamics of the Earths fluid core. In this paper we continue our study of the linear stability of an electrically conducting fluid in a rapidly rotating, rigid, electrically insulating spherical geometry in the presence of a toroidal basic state, comprising magnetic field BMB O(r, θ)1⊘ and flow UMU O(r, θ)1⊘ The magnetostrophic approximation is employed to numerically analyse the two classes of instability which are likely to be relevant to the Earth. These are the field gradient (or ideal) instability, which requires strong field gradients and strong fields, and the resistive instability, dependent on finite resistivity and the presence of a zero in the basic state B O(r,θ). Based on a local analysis and numerical results in a cylindrical geometry we have established the existence of the field gradient instability in a spherical geometry for very simple basic states in the first paper of this series. Here, we extend the c...