Non-linear interactions of magneto-Poincare and magnetostrophic waves in rotating shallow water magnetohydrodynamic

We have investigated the interaction of wave packets in the magnetohydrodynamic shallow water flows in external vertical magnetic field. Using the asymptotic multiscale methods we received that three magneto-Poincare waves interact, three magnetostrophic waves also interact, We obtained intermode interactions: two magnetostrophic waves and magneto-Poincare wave, two magneto-Poincare waves and magnetostrophic wave. In all cases we derived nonlinear equations of three waves interactions and showed 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.