Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation
Abstract
Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All selfsimilar discontinuous solutions and all continuous centered selfsimilar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations thatmore »
 Authors:
 Russian Academy of Sciences, Space Research Institute (Russian Federation)
 Publication Date:
 OSTI Identifier:
 22617178
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 123; Journal Issue: 3; Other Information: Copyright (c) 2016 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPRESSIBILITY; COMPRESSIBLE FLOW; CONFIGURATION; DECAY; DENSITY; DISTURBANCES; EQUATIONS; FILTERS; FLUIDS; GRAVITATIONAL FIELDS; INCOMPRESSIBLE FLOW; MAGNETOHYDRODYNAMICS; MASS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PERTURBATION THEORY; ROTATING PLASMA; SOUND WAVES; THIN FILMS
Citation Formats
Klimachkov, D. A., Email: klimchakovdmitry@gmail.com, and Petrosyan, A. S., Email: apetrosy@iki.rssi.ru. Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation. United States: N. p., 2016.
Web. doi:10.1134/S1063776116070098.
Klimachkov, D. A., Email: klimchakovdmitry@gmail.com, & Petrosyan, A. S., Email: apetrosy@iki.rssi.ru. Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation. United States. doi:10.1134/S1063776116070098.
Klimachkov, D. A., Email: klimchakovdmitry@gmail.com, and Petrosyan, A. S., Email: apetrosy@iki.rssi.ru. 2016.
"Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation". United States.
doi:10.1134/S1063776116070098.
@article{osti_22617178,
title = {Nonlinear theory of magnetohydrodynamic flows of a compressible fluid in the shallow water approximation},
author = {Klimachkov, D. A., Email: klimchakovdmitry@gmail.com and Petrosyan, A. S., Email: apetrosy@iki.rssi.ru},
abstractNote = {Shallow water magnetohydrodynamic (MHD) theory describing incompressible flows of plasma is generalized to the case of compressible flows. A system of MHD equations is obtained that describes the flow of a thin layer of compressible rotating plasma in a gravitational field in the shallow water approximation. The system of quasilinear hyperbolic equations obtained admits a complete simple wave analysis and a solution to the initial discontinuity decay problem in the simplest version of nonrotating flows. In the new equations, sound waves are filtered out, and the dependence of density on pressure on large scales is taken into account that describes static compressibility phenomena. In the equations obtained, the mass conservation law is formulated for a variable that nontrivially depends on the shape of the lower boundary, the characteristic vertical scale of the flow, and the scale of heights at which the variation of density becomes significant. A simple wave theory is developed for the system of equations obtained. All selfsimilar discontinuous solutions and all continuous centered selfsimilar solutions of the system are obtained. The initial discontinuity decay problem is solved explicitly for compressible MHD equations in the shallow water approximation. It is shown that there exist five different configurations that provide a solution to the initial discontinuity decay problem. For each configuration, conditions are found that are necessary and sufficient for its implementation. Differences between incompressible and compressible cases are analyzed. In spite of the formal similarity between the solutions in the classical case of MHD flows of an incompressible and compressible fluids, the nonlinear dynamics described by the solutions are essentially different due to the difference in the expressions for the squared propagation velocity of weak perturbations. In addition, the solutions obtained describe new physical phenomena related to the dependence of the height of the free boundary on the density of the fluid. Selfsimilar continuous and discontinuous solutions are obtained for a system on a slope, and a solution is found to the initial discontinuity decay problem in this case.},
doi = {10.1134/S1063776116070098},
journal = {Journal of Experimental and Theoretical Physics},
number = 3,
volume = 123,
place = {United States},
year = 2016,
month = 9
}

A set of equations is derived for the motion of a compressible ideal gas over a nonuniform boundary in the gravitational field in the shallowwater approximation. Classical simple waves are shown not to be the solutions to this set of equations. Generalized simple waves are found to exist only in the case of a linear underlyingsurface profile. All continuous and discontinuous solutions are obtained in an explicit form for the case of the boundary in the form of an inclined plane, and an analytical solution is found for the problem of the decay of an arbitrary discontinuity. This solution consistsmore »

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid in the shallow water approximation
The system of the magnetohydrodynamic equations for a heavy fluid has been analyzed in the shallow water approximation. All discontinuous selfsimilar solutions and all continuous centered selfsimilar solutions have been found. It has been shown that magnetogravity compression waves are broken with the formation of a magnetogravity shock wave. The initial decay discontinuity problem for the magnetohydrodynamic equations has been solved in the explicit form in the shallow water approximation. The existence of five different configurations implementing the solution of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for the implementation of each configurationmore » 
Nonlinear dynamics of flows of a heavy compressible gas in the shallow water approximation
The system of the equations of motion for a compressible gas in the gravitational field over a smooth underlying surface has been analyzed in the shallow water approximation. All continuous centered selfsimilar solutions and all discontinuous selfsimilar solutions have been obtained. The problem of the decay of an arbitrary discontinuity for the equations of motion of the compressible gas has been solved in the explicit form in the shallow water approximation. The existence of four different configurations implementing the solution of the problem of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for themore » 
Nonlinear magnetohydrodynamic waves in a steady zonal circulation for a shallow fluid shell on the surface of a rotating sphere
This paper considers twodimensional nonlinear MHD waves of large horizontal spatial scales for a thin magnetofluid layer on the surface of a rotating sphere. The shallow fluid hydrodynamic equations are generalized to include the effects of magnetic fields, and it is shown that the resulting MHD equations can be reduced to a single scalar equation for a stream function involving several free functions. For special choices of these free functions, two kinds of finiteamplitude MHD waves are obtained, propagating in the azimuthal direction relative to the uniformly rotating background atmosphere in the presence of a background zonal magnetic field andmore » 
Contributions to the theory of the propagation of magnetohydrodynamic waves in an electrically conductive compressible fluid. II.(in Rumanian)
It has been shown in the first part of the article that by using Clebsch's theorem, the study of the slow motion of a compressible, electrically conducting fluid involves only the solution of certain equations with partial derivatives. Boundary conditions are presented for the case of a fluid which moves in a container with rigid walls, using the process to examine the propagation of magnetohydrodynamic waves within the space between 2 planes. Such a layer may represent the atmosphere in a spheric star with a large radius, such as the sun. The action of a constant magnetic fleld H/sub o/,more »