# Shock Waves in the Thin Flux Tube Approximation

- C. T. Russell,
- E. R. Priest and
- L. C. Lee

Published Online: 21 MAR 2013

DOI: 10.1029/GM058p0107

Copyright 1990 by the American Geophysical Union.

Book Title

## Physics of Magnetic Flux Ropes

Additional Information

#### How to Cite

Ferriz-Mas, A. (1990) Shock Waves in the Thin Flux Tube Approximation, in Physics of Magnetic Flux Ropes (eds C. T. Russell, E. R. Priest and L. C. Lee), American Geophysical Union, Washington, D. C.. doi: 10.1029/GM058p0107

#### Publication History

- Published Online: 21 MAR 2013
- Published Print: 1 JAN 1990

#### Book Series:

#### ISBN Information

Print ISBN: 9780875900261

Online ISBN: 9781118663868

- Summary
- Chapter
- References

### Keywords:

- Solar photosphere;
- Magnetic flux;
- Astrophysics

### Summary

A discussion of the properties of magnetohydro-dynamic (MHD) shock waves in the framework of the thin fluxtube approximation (to zeroth order in the expansion) is presented. We consider several simplifying assumptions which make an analytical study possible: Radiative energy exchange is neglected; the equation of state is that of a polytropic ideal gas (i.e., with constant ratio of specific heats), so that ionization effects are ignored; further, the backreaction of the shock on the ambient medium is ignored, so that the theory can be accurate only for not too strong shocks. Despite the limitations introduced by these assumptions, an analytical study provides useful insight into the physics of the problem, thus complementing existing and future numerical investigations under more realistic conditions.

We show that the properties of shock waves confined to flux tubes, as described by the thin flux tube equations, exhibit many analogies with those of slow MHD shocks in extended media. That could be expected on physical grounds: the basic compressive tube wave in the framework of the (zeroth-order) thin flux tube approximation is a slow mode with phase speed given by the cusp speed *C _{T} * =

*C*

_{S}*V*/(

_{A}*C*+

^{2}_{S}*V*)

^{2}_{A}^{1/2}(with

*C*and

_{S}*V*the sound and Alfvén speeds, respectively), and it is precisely the shock resulting from the nonlinear evolution and breaking of this mode that is the subject of our investigation. The analogies and differences with purely hydrodynamic shocks are also pointed out. In particular, it can be shown that the sub- or supercritical character of the flow velocity with respect to the sound, Alfvén and cusp speeds is derivable from thermodynamic considerations only, as for HD shocks, in contrast to general MHD shocks, for which the evolutionary conditionshave to be applied.

_{A}The theory of shock waves in thin flux tubes is not only of interest in connection with concentrated magnetic structures in the stellar atmospheres. Its understanding is conceptually important from both physical and mathematical point of view since the flux tube provides one of the simplest forms of equations governing the dynamics of a magnetized plasma confined by an external pressure and subject to a permanent constraint (internal gas pressure variations are related to internal magnetic field variations).