The present study is concerned with dynamical processes in a rotating layer of electrically conducting incompressible liquid located in a magnetic field which is parallel to the normal vector to the boundary surfaces. We take into account not only the convective terms, but also the diffusion terms in the magnetic field induction equations. This problem, as well as the geophysical hydrodynamics problem, calls for the construction of approximate variants of the principal hydromagnetic equations and rigorous mathematical analysis of these approximate equations. Moreover, advancements in the above problems depend on the approximations to be introduced. To this end, by introducing characteristic scales of variation of the variables in the original equations and estimating the magnitude orders of the terms involved in the equations, one can single out the principal and secondary terms, simplify the equations, and build a model of the process under consideration. By introduction of auxiliary functions, the system of partial differential equations is reduced to one scalar equation. This suggests the conclusion about the analytical structure of magnetohydrodynamic characteristics. From the results obtained it follows that the magnetic field generation in an electrically conducting liquid stems from the instability characterized by the corresponding relations between the gravitation force, the Coriolis force, the magnetic force, and the peculiarities of the relief topography.
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.
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