| Summary: | The purpose of plasma astrophysics is the study and description of the flow of rotating plasma in order to understand the evolution of various objects in the universe, from stars and planetary systems to galaxies and galaxy clusters. A number of new applications and observations have appeared in recent years and actualized the problem of studying large-scale magnetohydrodynamic flows, such as a thin layer under the convective zone of the sun (solar tachocline), propagation of accreting matter in neutron stars, accretion disks in astrophysics, dynamics of neutron star atmospheres, and magnetoactive atmospheres of exoplanets tidally locked with their host star. The article aims to discuss a fundamental problem in the description and study of multiscale astrophysical plasma flows by studying its general properties characterizing different objects in the universe. We are dealing with the development of geophysical hydrodynamic ideas concerning substantial differences in plasma flow behavior due to the presence of magnetic fields and stratification. We discuss shallow water magnetohydrodynamic equations (one-layer and two-layer models) and two-dimensional magnetohydrodynamic equations as a basis for studying large-scale flows in plasma astrophysics. We discuss the novel set of equations in the external magnetic field. The following topics will be addressed: Linear theory of magneto-Rossby waves, three-wave interactions and related parametric instabilities, zonal flows, and turbulence.
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