Non-linear waves interactions in rotating shallow water magnetohydrodynamics

This study is devoted to the development of the nonlinear theory of the magneto-Poincare waves and magnetostrophic waves in rotating layers of astrophysical and space plasma in the shallow-water approximation. These waves determine the large-scale dynamics of the various astrophysical and space objects such as solar tachocline, as well as  magnetoactive atmospheres of exoplanets trapped by tides of a carrier star, neutron stars atmospheres and the flows in accretion disks of neutron stars. For this purpose we derived magnetohydrodynamic shallow water equations with a rotation and presence of an external vertical magnetic field. The system is obtained from conventional magnetohydrodynamic equations for incompressible inviscid heavy plasma layer with free surface in an external vertical magnetic field. The pressure is assumed to be hydrostatic, and the height of the plasma layer is considered to be much smaller than horizontal scales of the flow. The magnetohydrodynamic equations in the shallow-water approximation play equally important role in the space and astrophysical plasma flows like classical shallow-water equations in the fluid dynamics of a neutral fluid. The magnetohydrodynamic shallow water equations with an external vertical magnetic field are modified by supplementing them with the equation for the vertical component of the magnetic field and divergence-free condition for magnetic field contains its vertical component. Thus the velocity field remains two-dimensional while the magnetic field becomes three-dimensional. It is shown that the presence of a vertical magnetic field significantly changes the dynamics of the wave processes in astrophysical plasma compared to the neutral fluid and plasma layer in a horizontal magnetic field.  We have investigated the interaction of Magneto-Poincare waves and magnetostrophic waves in the magnetohydrodynamic shallow water flows in external vertical magnetic field and in horizontal (toroidal and poloidal) magnetic field. In the absence of the horizontal magnetic field the dynamics of plasma appears to be similar to the neutral fluid dynamics and it is shown that there are four-waves interactions in this case. Using the asymptotic multiscale method we obtained the non-linear amplitude equations for the three interacting Magneto-Poincare waves and magnetostrophic waves. The analysis of the amplitude equations shows that there are two types of instabilities for four different types of three-waves configurations. These instabilities occur in both cases: in the external vertical magnetic field and in the horizontal magnetic field. For all types of instabilities the growth rates are found. In the absence of the vertical magnetic field we obtained the non-linear amplitude equations for the four interacting waves. It is shown that the resulting system of equations has the first integrals that describe the mechanism of energy transfer among interacting waves of small amplitude. This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).