Waves in a rotating layer of an ideal electrically conducting incompressible fluid with allowance effects of diffusion of magnetic field

Abstract

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces and diffusion of magnetic field. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.The system of nonlinear partial differential equations describing the dynamics of a rotating layer of an ideal conducting incompressible fluid is difficult to investigate because of its vector character. Therefore, it is natural to try to reduce it to equivalent scalar equations for auxiliary functions.We consider the nonlinear system of partial differential equations that model perturbations in a layer of an ideal conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces and diffusions of magnetic field. The purpose of this study is to reduce this system to a scalar equation and to construct analytical solutions to the corresponding boundary value problems.The accounting of diffusive members is necessary when studying dynamics of waves of more local character, i.e., when the horizontal scale of change of hydromagnetic sizes much less than a radius of a considered layer, and also at very great time scales. It would be desirable to see influence of diffusion of a magnetic field on its generation. Whether there will be able to be a magnetic field as much as long time and whether it will exist at shutdown of an inoculating field.The motion of a conducting fluid in a magnetic field causes electric currents. These currents change the magnetic field. At the same time, the forces acting on the currents in the magnetic field can change the character of the fluid motion. Hence, hydrodynamic motion and electromagnetic phenomena are interrelated. This relation is described by the joint system of field equations and the equations of motion of a fluid. According to the works by the well-known Swedish physicist and astrophysicist G. Alfven, the interrelation between electromagnetic and hydrodynamic phenomena strengthens as the linear scale of a phenomenon increases. For large-scale phenomena, this interrelation can be rather strong. For example, this is true of star interiors and the Earth's liquid core.Large-scale motions of an electrically conducting fluid have been intensively studied. In particular, we note, which investigate a model constructed in the approximation of fast rotation. Within the framework of this theory, the inertial force is ignored in the equation of motion. As a result, the inertial, Alfven, and Rossby waves are filtered out. Furthermore, in the limit of fast rotation, the velocity v is not determined uniquely but rather up to a...