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 $T_{\alpha }\times T=\left(R,/,\left(\frac{2\pi }{\alpha }Z\right),\times ,R,/,\left(2\pi Z\right)\right)$, around a periodic shear flow $\left(U\right(y),0)$ coupled with a constant background magnetic field $b=(b_1,b_2)$. It is a nontrivial extension of a recent paper for the Navier–Stokes equations by Colombo, Dolce, Montalto, and 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 $U\left(y\right)$ involving the viscosity $\nu$, the resistivity $\eta$, and the components of the background magnetic field $b$ to obtain linear long-wave stability and instability in the regime $\alpha \ll 1$. The proof combines a nonperturbative normal-form transformation decoupling the zero Fourier mode from the nonzero 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.