Motion of a current-vortex sheet in the magnetic Kelvin-Helmholtz instability

In this paper, we consider the Kelvin-Helmholtz instability in the magnetohydrodynamic flow. The motion of the interface is described by a current-vortex sheet. We examine the linear stability of the current-vortex sheet model and determine the growth rate of the interface. The interface is linearly stable for M_{A}<2 where M_{A} represents the Alfvén Mach number. It is found that the interface is linearly unstable in the limit of the critical Alfvén Mach number M_{A}=2, due to resonance of eigenvalues. We perform numerical simulations for the current-vortex sheet for both regimes of M_{A}<2 and M_{A}>2. The numerical results show the stabilizing effects of the magnetic field on the evolution of the current-vortex sheet when the magnetic field is sufficiently large. For the regime M_{A}<2, the sheet oscillates both longitudinally and transversely and the transverse surface wave is pronounced for a large M_{A}. Remarkably, the interface is nonlinearly unstable for M_{A}≈2, for M_{A}<2, which may be due to the propagation of surface waves. For the regime M_{A}>2, the roll-up of the spiral is weakened and the spiral is more pinched and stretched for smaller M_{A}. A comparison of the unstable evolutions of large and small values of M_{A} shows significant differences of the magnetic field and vortex sheet strength.