On the Stabilizing Effect of the Magnetic Fields in the Magnetic Rayleigh--Taylor Problem

Abstract

We investigate the stabilizing effect of the vertical equilibrium magnetic field in the Rayleigh--Taylor (RT) problem for a nonhomogeneous incompressible viscous magnetohydrodynamic (MHD) fluid of zero resistivity in the presence of a uniform gravitational field in a horizontally periodic domain, in which the velocity of the fluid is nonslip on both upper and lower flat boundaries. When an initial perturbation around a magnetic RT equilibrium state satisfies some relations, and the strength $|m|$ of the vertical magnetic field of the equilibrium state is bigger than the critical number $m_C$, we can use the Bogovskii function in the standing-wave form and adapt a two-tier energy method in Lagrangian coordinates to show the existence of a unique global-in-time (perturbed) stability solution to the magnetic RT problem. For the case of $|m|<m_C$, we show that the nonlinear RT instability still occurs by developing a new analysis technique based on the method of bootstrap instability. The current result reveals from the mathematical point of view that the sufficiently large vertical equilibrium magnetic field has a stabilizing effect and can prevent the RT instability in MHD flows from occurring. Also, the nonslip boundary condition in the direction of the equilibrium magnetic field contributes to the stabilizing effect of magnetic fields. Similar conclusions can also be verified for the horizontal magnetic field when the domain is vertically periodic, which shows that the horizontal magnetic field has the same stabilizing effect as the vertical one.