Nonlinear Waves Interactions in Rotating Shallow Water Magnetohydrodynamics

Abstract:

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 the solar tachocline, as well as dynamics of magnetoactive atmospheres of exoplanets trapped by tides of a host star 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 modified magnetohydrodynamic shallow water equations are supplemented them with the equation for the vertical component of the magnetic field and divergence-free condition for magnetic field which contains its vertical component. It is shown that the velocity field remains two-dimensional while the magnetic field becomes three-dimensional and it significantly changes the dynamics of the wave processes. 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 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 interaction equations for the waves amplitudes of three waves. The analysis of the amplitudes equations shows that there are two types of instabilities. These instabilities occur 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 of motion that describe the mechanism of energy transfer among interacting waves of small amplitude.

This research was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).