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Evolution of the trans-Alfvénic normal shock in a gas of finite electrical conductivity

Published online by Cambridge University Press:  28 March 2006

L. Todd
L. Todd
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

The stability of a steady, plane, one-dimensional trans-Alfvénic shock to small transverse disturbances in the velocity and magnetic field is discussed. In the undisturbed flow the magnetic field and velocity are normal to the plane of the shock which is treated as a discontinuity in an inviscid gas of zero thermal conductivity. However, the electrical conductivity of the ambient gas is taken to be finite, i.e. the magnetic diffusivity is very much larger than both the viscous diffusivity and the thermal diffusivity.

A small uniform transverse perturbation of the magnetic field is imposed at zero time. A specific super-Alfvénic normal shock is chosen. Computed values of the magnetic field, in and near the shock, are given for various later times. Two diffusing Alfvén waves of the amplitudes predicted by infinite conductivity theory are shown to propagate away from the shock region, leaving behind the expected steady-state shock profile.

Similar computations are carried out for a specific trans-Alfvénic normal shock. An analytic asymptotic solution, valid for large times, is also obtained. This result agrees with the computations carried out. The shock profile of the transverse quantities is one which grows linearly with time. Outside this magneto-hydrodynamic shock region which surrounds the trans-Alfvénic hydrodynamic shock (discontinuity), steady states are reached in each of which the transverse velocity and transverse magnetic field are uniform. An incident Alfvén wave, consisting of a weak diffusing current sheet, produces these same effects. The resolution of an arbitrary transverse fluctuation, in which both the magnetic field and the velocity have limiting values at large distances from the shock, is discussed. The solution for large times is found.

It is shown that the trans-Alfvénic normal shock and its two limiting cases, the null switch-on and null switch-off shocks, are unstable to general transverse disturbances although there exist particular disturbances of this kind which will not destroy them. An integral condition is obtained which, together with the relevant boundary conditions, determines the profiles of the transverse quantities in the trans-Alfvénic normal shock whenever a steady state is reached. This removes the puzzling arbitrariness of these profiles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 18 , Issue 3 , March 1964 , pp. 321 - 336
Copyright
© 1964 Cambridge University Press

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References

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