Nonlinear Kelvin-Helmholtz instability in hydromagnetics

By taking into account the temporal as well as the spatial effects, a weakly nonlinear theory of the propagation of wave packets in the Kelvin-Helmholtz instability problem in the presence of uniform magnetic fields, acting along the surface of separation of two moving superposed fluids, is presented. With the use of the method of multiple scaling, the evolution of the amplitude of the two-dimensional wave packets, which is governed by a nonlinear Klein-Gordon equation, is derived. The various stability criteria arising out of this equation are examined. The nonlinear cut-off wavenumber, which separates the region of stability from that of instability, is determined.