Dynamic Transition for Magnetohydrodynamic Convection

TL;DR

This paper analyzes the dynamic transition behavior of incompressible magnetohydrodynamic (MHD) equations in a 3D rectangular domain, identifying conditions for transitions to multiple equilibria or periodic solutions.

Contribution

It provides a mathematical description of the transition types in MHD convection, distinguishing between continuous and jump transitions in different scenarios.

Findings

Transition to multiple equilibria is continuous (type-I).

Transition to periodic solutions can be either continuous or jump (type-II).

The analysis characterizes the conditions for each transition type.

Abstract

Main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a rectangular domain in $\mathbb{R}^{3}$. Our analysis shows that the system undergoes a first dynamic transition either to multiple equilibria or to periodic solutions. In the case of transition to multiple equilibria, the transition is a type-I (continuous) transition. In the case of transition to periodic solutions, the transition can be either type-I or type-II (jump).