Sharp nonlinear stability criterion of viscous non-resistive MHD internal waves in 3D

Abstract

We consider the dynamics of two layers of incompressible electrically conducting fluid interacting with the magnetic field, which are confined within a 3D horizontally infinite slab and separated by a free internal interface. We assume that the upper fluid is heavier than the lower fluid so that the fluids are susceptible to the Rayleigh-Taylor instability. Yet, we show that the viscous and non-resistive problem around the equilibrium is nonlinearly stable provided that the strength of the vertical component of the steady magnetic field, $|{\bar{B}}_3|$, is greater than the critical value, ${\mathcal{M}}_c$, which we identify explicitly. We also prove that the problem is nonlinearly unstable if $|{\bar{B}}_3|<{\mathcal{M}}_c$. Our results indicate that the non-horizontal magnetic field has strong stabilizing effect on the Rayleigh-Taylor instability but the horizontal one does not have in 3D.

47pp

References in corpus (1)

• The compressible viscous surface-internal wave problem: nonlinear Rayleigh-Taylor instability

Cited by in corpus (1)

• Incompressible inviscid resistive MHD surface waves in 2D