## 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<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.

Original language | English |
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Pages (from-to) | 1531-1555 |

Number of pages | 25 |

Journal | Communications in Mathematical Sciences |

Volume | 17 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2019 |

### Bibliographical note

Publisher Copyright:© 2019 International Press.

### Funding

Acknowledgments. The work of Quan Wang was supported by the National Nature Science Foundation of China (NSFC), Grant No. 11901408.

Funders | Funder number |
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National Natural Science Foundation of China | 11901408 |

## Keywords

- Dynamical transition theory
- Linear stability
- Magnetohydrodynamics convection
- Principle of exchange of stabilities
- Spherical harmonics