Multiple equilibria and transitions in spherical MHD equations

In this study, we aim to describe the first dynamic transitions of the MHD equations in a thin spherical shell. It is well known that the MHD equations admit a motionless steady state solution with constant vertically aligned magnetic field and linearly conducted temperature. This basic solution is stable for small Rayleigh numbers $\mathrm{R}$ and loses its stability at a critical threshold $\mathrm{R}_c$. There are two possible sources for this instability. Either a set of real eigenvalues or a set of non-real eigenvalues cross the imaginary axis at $\mathrm{R}_c$. We restrict ourselves to the study of the first case. In this case, by the center manifold reduction, we reduce the full PDE to a system of $2 l_c + 1$ ODE’s where $l_c$ is a positive integer. We exhibit the most general reduction equation regardless of $l_c$. Then, it is shown that for $l_c = 1, 2$, the system either exhibits a continuous transition accompanied by an attractor homeomorphic to $2 l_c$ dimensional sphere which contains steady states of the system or a drastic transition accompanied by a repeller bifurcated on $\mathrm{R} \lt \mathrm{R}_c$. We show that there are parameter regimes where both types of transitions are realized. Besides, several identities involving the triple products of gradients of spherical harmonics are derived, which are useful for the study of related problems.

Keywords

magnetohydrodynamics convection, dynamical transition theory, spherical harmonics, linear stability, principle of exchange of stabilities

2010 Mathematics Subject Classification

37G30, 37N20, 76E25, 76W05, 82D10

Full Text (PDF format)

Received 21 May 2018

Accepted 1 July 2019

Published 26 December 2019