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

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 ${\mu }^2\lesssim {\mathrm{Pr}}_{\mathrm{m}}=\nu /\mu \le 1$. We establish that small perturbations around this steady state remain close to it, provided their size is of order $\epsilon \ll {\nu }^{2/3}$ 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.