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This letter discusses rotating magnetohydrodynamics (MHD) of a thin layer ofastrophysical plasma. To describe a thin plasma layer with a free surface in avertical external magnetic field we use the shallow water ap- proximation. Thepresence of a vertical magnetic field essentially changed the wave processesdynamics in astrophysical plasma compared to the neu- tral uid and plasma layerin a thoroidal magnetic field. In present case thre are three-waves nonlinearinteractions. Using the asymptotic mul- tiscale we deduced nonlinear wavepackets interaction equations: three magneto-Poincare waves interaction, threemagnetostrophic waves inter- action, the interaction of two magneto-Poincareand one magnetostrophic wave and two magnetostrophic and one magneto-Poincarewave interac- tion. The existence of decay instabilities and parametricamplifications is predicted. We found following four types of decayinstabilities: magneto- Poincare wave decays into two magneto-Poincare waves,magnetostrophic wave decays into two magnetostrophic waves, magneto-Poincarewave de- cays into one magneto-Poincare wave and one magnetostrophic wave,magnetostrophic wave decays into one magnetostrophic wave and onemagneto-Poincare wave. Also following mechanisms of parametric amplificationswere found: parametric amplification of magneto-Poincare waves, parametricamplification of magnetostrophic waves, magneto-Poincare wave amplification inmagnetostrophic wave presence nd magnetostrophic wave amplification inmagneto-Poincare wave presence. The instabilities growth rates and parametricalamplifications factors were found respectively.

Parametric Instabilities in Shallow Water Magnetohydrodynamics Of Astrophysical Plasma in External Magnetic Field