Onset of Inertial Magnetoconvection in Rotating Fluid Spheres

Radostin D Simitev,Friedrich H Busse

doi:10.3390/fluids6010041

Abstract

The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied in this paper. Thermo-inertial convection is a distinctive flow regime on the border between rotating thermal convection and wave propagation. It occurs in astrophysical and geophysical contexts where self-sustained or external magnetic fields are commonly present. To investigate the onset of motion, a perturbation method is used here with an inviscid balance in the leading order and a buoyancy force acting against weak viscous dissipation in the next order of approximation. Analytical evaluation of constituent integral quantities is enabled by applying a Green’s function method for the exact solution of the heat equation following our earlier non-magnetic analysis. Results for the case of thermally infinitely conducting boundaries and for the case of nearly thermally insulating boundaries are obtained. In both cases, explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. It is found that an imposed azimuthal magnetic field exerts a stabilizing influence on the onset of inertial convection and as a consequence magneto-inertial convection with azimuthal wave number of unity is generally preferred.

Highlights

Buoyancy-driven motions of rotating, electrically conducting fluids in the presence of magnetic fields represent a fundamental aspect of the dynamics of stellar and planetary interiors, see, e.g., in [1,2,3,4]
A rather detailed classification of magnetoconvection waves in a rotating cylindrical annulus has been recently attempted by Hori et al [13] and the authors of [14] who proceeded further to make useful comparisons with nonlinear spherical dynamo simulations and to provide estimates for the strength of the “hidden” azimuthal part of the magnetic field within the core
While the mode with the largest absolute value of ω is preferred as long as Θ and ur are in phase, the mode with the minimum absolute value of ω becomes preferred as the phase difference increases as the latter is detrimental to the work done by the buoyancy force

Summary

Introduction

Buoyancy-driven motions of rotating, electrically conducting fluids in the presence of magnetic fields represent a fundamental aspect of the dynamics of stellar and planetary interiors, see, e.g., in [1,2,3,4]. A rather detailed classification of magnetoconvection waves in a rotating cylindrical annulus has been recently attempted by Hori et al [13] and the authors of [14] who proceeded further to make useful comparisons with nonlinear spherical dynamo simulations and to provide estimates for the strength of the “hidden” azimuthal part of the magnetic field within the core These authors used the rotating annulus model of Busse [15,16] and only considered values of the Prandtl number of the order unity. At sufficiently small values of the Prandtl number, a different style of convection exists that is sometimes called inertial or equatorially-attached convection or thermo-inertial waves [18,19,20,21] In this limit, convection oscillates so fast that the viscous force does not enter the leading-order balance. It is important to understand how this regime of inertial convection is affected by an imposed magnetic field

Results

Discussion

Conclusion