Non-linear Three-wave Interactions in Rotating Shallow Water Magnetohydrodynamics

Abstract:

Rotating magnetohydrodynamic shallow water equations are obtained from classical magnetohydrodynamic equations for incompressible inviscid heavy fluid layer with free surface in a non-inertial frame of reference, which is rotating with fluid layer by averaging over fluid layer. The pressure is assumed to be hydrostatic, and the water layer height is considered to be much smaller than horizontal scales.

We have investigated the interaction of wave packets in the magnetohydrodynamic shallow water flows in external magnetic field. Linear analisys lead to to magnetogravity waves (or magneto-Poincare waves) and magnetostrophic waves. Magnetostrophic mode disappears without the external magnetic field. Using the asymptotic multiscale methods we received that in magneto-Poincare mode three waves interact, in magnetostrophic mode there are three waves interactions, and also there are intermode interactions: two magnetostrophic waves and magneto-Poincare wave, two magneto-Poincare waves and magnetostrophic wave. In all cases we obtained the non-linear interaction equations for the waves amplitudes.

Analysis of obtained nonlinear equations of three waves interactions shows the existence of two types of instability mechanisms: decay instabilities and parametric growth. It has been found that there are four types of decay instabilities: magneto-Poincare wave decays into two magneto-Poincare waves, magnetostrophic wave decays into two magnetostrophic waves, magneto-Poincare decays into one magnetostrophic and one magneto-Poincare wave, and magnetostrophic wave decays into one magneto-Poincare and one magnetostrophic wave. The growth rates of decay instabilities were received. Also four types of parametric growth were investigated: magneto-Poincare waves amplification, magnetostrophic waves amplification, magneto-Poincare wave growth in field of magnetostrophic wave, and magnetostrophic wave growth in field of magneto-Poincare wave. For all types of parametric growth the growth rates were found.