Nonlinear dynamics of magnetohydrodynamic flows of heavy fluid over an   arbitrary surface in shallow water approximation
  The magnetohydrodynamic equations system for heavy fluid over an arbitrary surface in shallow water approximation is studied in the present paper. It is shown that simple wave solutions exist only for underlying surfaces that are slopes of constant inclination. All self-similar discontinuous and continuous solutions are found. The exact explicit solutions of initial discontinuity decay problem over a flat plane and a slope are found. It is shown that the initial discontinuity decay solution is represented by one of five possible wave configurations. For each configuration the necessary and sufficient conditions for its realization are found. The change of dependent and independent variables transforming the initial equations over a slope to those over a flat plane is found. 
Kirill Karelsky, Arakel Petrosyan, Stepan Tarasevich
physics.flu-dyn astro-ph.SR physics.plasm-ph