Ideal Magnetohydrodynamics Around Couette Flow: Long Time Stability and Vorticity-Current Instability

Abstract

This article considers the ideal 2D magnetohydrodynamic equations on an infinite periodic channel close to a combination of an affine shear flow, called Couette flow, and a constant magnetic field. This setting combines important physical effects of mixing and coupling of velocity and magnetic field. We establish the existence and stability of the velocity and magnetic field for Gevrey-class perturbations of size $\varepsilon$, valid up to times $t \sim \varepsilon^{-1}$. Additionally, the vorticity and current grow as $O(t)$ and there is no inviscid damping of the velocity and magnetic field. This has parallels to the above threshold case for the $3D$ Navier-Stokes \cite{bedrossian2022dynamics} where growth in `streaks' leads to time scales of $t\sim \varepsilon^{-1}$. In particular, for the ideal MHD equations, our article suggests that for a wide range of initial data, the scenario ``dynamo effect $\Rightarrow $ vorticity and current growth $\Rightarrow $ vorticity and current breakdown'' leads to instability and possible turbulences.