Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables

TL;DR

This paper analyzes the stability of 2D Couette flow with a magnetic field, showing that small perturbations decay exponentially after transient growth, using symmetric variables to understand the dynamics.

Contribution

The study introduces a reformulation of the MHD system in symmetric variables, providing new insights into stability thresholds and transient behaviors of the flow.

Findings

Small perturbations remain close to steady state if sufficiently small.

Vorticity and current density exhibit transient growth of order f u^{-1/3}.

Perturbations decay exponentially after a time-scale of order f u^{-1/3}.

Abstract

We consider a 2D incompressible and electrically conducting fluid in the domain $\mathbb{T}\times\mathbb{R}$. The aim is to quantify stability properties of the Couette flow $(y,0)$ with a constant homogenous magnetic field $(\beta,0)$ when $|\beta|>1/2$. The focus lies on the regime with small fluid viscosity $\nu$, magnetic resistivity $\mu$ and we assume that the magnetic Prandtl number satisfies $\mu^2\lesssim\mathrm{Pr}_{\mathrm{m}}=\nu/\mu\leq 1$. We establish that small perturbations around this steady state remain close to it, provided their size is of order $\varepsilon\ll\nu^{2/3}$ in $H^N$ with $N$ large enough. Additionally, the vorticity and current density experience a transient growth of order $\nu^{-1/3}$ while converging exponentially fast to an $x$-independent state after a time-scale of order $\nu^{-1/3}$. The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system's dynamic behavior.