MULTIPLE EQUILIBRIA AND TRANSITIONS IN SPHERICAL MHD EQUATIONS


Özer S. S. , ŞENGÜL M. T. , Wang Q.

COMMUNICATIONS IN MATHEMATICAL SCIENCES, vol.17, no.6, pp.1531-1555, 2019 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 17 Issue: 6
  • Publication Date: 2019
  • Doi Number: 10.4310/cms.2019.v17.n6.a3
  • Title of Journal : COMMUNICATIONS IN MATHEMATICAL SCIENCES
  • Page Numbers: pp.1531-1555

Abstract

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 R and loses its stability at a critical threshold 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 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 2l(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 2l(c) dimensional sphere which contains steady states of the system or a drastic transition accompanied by a repeller bifurcated on R