Mathematical analysis of long-time behavior of magnetized fluid instabilities with shear flow

TL;DR

This paper analyzes a Ginzburg-Landau type model to understand how shear flow influences the long-term behavior of magnetized fluid instabilities, revealing stability conditions and the impact of shear strength.

Contribution

It provides a mathematical framework for predicting steady-states and stability in magnetized fluid models with shear flow, highlighting the effects of shear strength on system behavior.

Findings

Weak shear leads to a unique stable steady-state.

Strong shear results in only the trivial zero solution.

Global attractors help classify long-time dynamics.

Abstract

We study a complex Ginzburg-Landau (GL) type model related to fluid instabilities in the boundary of magnetized toroidal plasmas (called edge-localized modes) with a prescribed shear flow on the Neumann boundary condition. We obtain the following universal results for the model in a one-dimensional interval. First, if the shear is weak, there is a unique linearly stable steady-state perturbed from the nonzero constant steady-state corresponding to the zero shear case. Second, if the shear is strong, there is no plausible steady-state except the trivial zero solution in the interval. With the help of these results and the existence of global attractors, we can dramatically reduce the number of cases for the long-time behavior of a solution in the model.