The Shallow-Water Magnetohydrodynamic Theory of Stratified Rotating Astrophysical Plasma Flows: Beta-Plane Approximation and Magnetic Rossby Waves

Rotating magnetohydrodynamic flows of a thin stratified plasma layer in a gravitational field with a free boundary in an external vertical magnetic field were investigated. Magnetohydrodynamic equations were obtained in the two-layer shallow-water approximation in an external magnetic field when plasma is divided into two layers of different densities. A system of shallow-water equations for rotating stratified plasma in an external magnetic field was obtained in the beta-plane approximation. A linear theory was developed, and solutions in the form of magnetic Rossby waves, along with corrections that describe the effects of stratification, were obtained for stationary solutions in the form of vertical or horizontal magnetic field. Qualitative analysis of dispersion curves reveals the presence of three-wave nonlinear interactions of magnetic Rossby waves for each of the stationary states. The existence of parametric instabilities was demonstrated and their increments were found.