Nonlinear stability of Kelvin–Helmholtz waves in magnetic fluids stressed by a time-dependent acceleration and a tangential magnetic field

Abstract

The nonlinear stability of surface waves propagating between two superposed streaming magnetic fluids is investigated. The fluids are stressed by a constant tangential magnetic field and a vertical periodic acceleration. The solution employs the method of multiple scales. Owing to the periodicity, resonant cases appear. Two parametrically nonlinear Schrödinger equations are derived for the resonant cases to describe the elevation of weakly nonlinear capillary waves. The standard nonlinear Schrödinger equation is satisfied for the non resonant cases. Necessary and sufficient conditions for stability are obtained. A formula for the surface elevation is obtained in each case. It is found that the magnetic field, the velocities and the frequency of the applied periodic force play dual roles in the resonant region. Investigation of the stability criterion by nonlinear perturbation shows that an increase in the acceleration frequency has a stabilizing effect. The stabilizing role of the frequency is due to the destabilizing effect of the amplitude of the periodic acceleration.