Quasi-geostrophic motions in a rotating layer of an electrically conducting fluid

Large-scale nonlinear oscillations of an electrically conducting ideal fluid of varying depth are considered with the magnetic, Archimedean, and Coriolis forces taken into account. The main equations are derived from an analysis of the scales of quasi-geostrophic motions. Under the assumptions that the Rossby numbers (a measure of the ratio of the local and advective accelerations to the Coriolis acceleration) are of the same order, the problem is reduced to a system of three nonlinear equations for hydromagnetic pressure and two functions describing the magnetic field. For an infinitely long horizontal layer of an electrically conducting rotating fluid, the exact solution of the corresponding nonlinear equations and the dispersion relation are obtained under the assumption of an approximately constant slope of the upper boundary surface of the layer at a distance of the order of the wavelength.