# 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 self-similar discontinuous solutions and all continuous centered self-similar 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., E-mail: klimchakovdmitry@gmail.com, and Petrosyan, A. S., E-mail: 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., E-mail: klimchakovdmitry@gmail.com, & Petrosyan, A. S., E-mail: 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., E-mail: klimchakovdmitry@gmail.com, and Petrosyan, A. S., E-mail: apetrosy@iki.rssi.ru. Thu .
"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., E-mail: klimchakovdmitry@gmail.com and Petrosyan, A. S., E-mail: 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 self-similar discontinuous solutions and all continuous centered self-similar 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. Self-similar 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 = {Thu Sep 15 00:00:00 EDT 2016},

month = {Thu Sep 15 00:00:00 EDT 2016}

}