Large-Scale Compressibility in Rotating Flows of Astrophysical Plasma in the Shallow Water Approximation

Two systems of magnetohydrodynamic equations in the shallow water approximation are proposed as a basis for studies in the field of plasma astrophysics: the system of equations with a full allowance for the Coriolis force and the system of equations on the β-plane in which the changes of the Coriolis parameter are linear in coordinate. Both systems of equations take into account such fundamentally important phenomena in plasma astrophysics as the compressibility and external magnetic field effects, increasing significantly the potential for applying these equations to study astrophysical objects. Compressibility in plasma astrophysics is shown to change significantly the dispersion laws for magneto-Poincare, magnetostrophic, and magneto-Rossby waves. The same nonlinear interactions as those in the absence of compressibility have been found to be realized in the case of a compressible rotating plasma. Three-wave equations in the weak nonlinearity approximation, in which the interaction coefficients depend on plasma large-scale compressibility and thermodynamic characteristics, have been derived by the method of multiscale expansions. Expressions for the growth rates of the parametric instabilities of three-wave interactions with large-scale compressibility have been derived.