Parametric instabilities in rotating space and astrophysical plasmas

Rotating magnetohydrodynamic shallow water equations are obtained from classical magnetohydrodynamic equations for incompressible inviscid heavy plasma layer with free surface in external magnetic field. We have investigated the interaction of wave packets in the magnetohydrodynamic shallow water flows in external magnetic field in f-plane and β-plane approximations. Linear analysis leads to magnetogravity waves (or magneto-Poincare waves) and magnetostrophic waves in f-plane approximation or magneto-Rossby waves in β-plane approximation. Magnetostrophic mode disappears without the external magnetic field. Using the asymptotic multiscale methods we received that on f-plane 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; on β-plane there are three magneto-Rossby waves interactions. 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 on f-plane: 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. Also four types of parametric growth were investigated on f-plane: 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. Simultaniously, on β-plane we found two types of instabilities: one magneto-Rossby wave decays into two magneto-Rossby waves and magneto-Rossby wave amplifies in field of two magneto-Rossby waves. For all types of instabilities the growth rates are found.