Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations

Abstract

We investigate the long-wave linear stability and instability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations, in vorticity-current formulation, on the periodic domain ${\mathbb{T}}_{\alpha }\times \mathbb{T}=(\mathbb{R}/(\frac{2\pi }{\alpha }\mathbb{Z})\times \mathbb{R}/(2\pi \mathbb{Z}))$, around a periodic shear flow $(U(y),0)$ coupled with a constant background magnetic field $\mathbf{b}=({\mathrm{b}}_1,{\mathrm{b}}_2)$. It is a non-trivial extension of a recent paper for the Navier-Stokes equations by Colombo, Dolce, Montalto & Ventura to the MHD setting in the spirit of the classical works of Kolmogorov, Meshalkin, Sinai and Yudovich. We establish explicit conditions on the shear flow profile $U(y)$ involving the viscosity $\nu$, the resistivity $\eta$ and the components of the background magnetic field $\mathbf{b}$ to obtain linear long-wave stability and instability in the regime $\alpha \ll 1$. The proof combines a non-perturbative normal form transformation decoupling the zero Fourier mode from the non-zero modes with sharp asymptotic expansions of the eigenvalues bifurcating from the zero unperturbed eigenvalue with respect to the parameter $\alpha$. As a dynamical consequence, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.