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Burgers' equation and cosmic ray shocks

Published online by Cambridge University Press:  13 March 2009

G. M. Webb  and
J. F. McKenzie
G. M. Webb
Affiliation:
Max-Planck-Institut für Kernphysik, Postfach 10 39 80, D-6900 Heidelberg, Federal Republic of Germany
J. F. McKenzie
Affiliation:
Department of Mathematics and Applied Mathematics, University of Natal, King George V Avenue, Durban, Natal, South Africa
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Abstract

The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a small-amplitude, long-wavelength, but nonlinear wave equation. The present paper derives Burgers' equation for three different hydrodynamical models of cosmic ray shocks. The main development of the paper concerns the hydrodynamical model in which Alfvénic effects are neglected. For this model it is shown that the steady-state solution of Burgers' equation is equivalent to the weak, but smooth, transition solutions of the shock structure equation obtained previously. It is shown that Burgers' equation may be regarded as a wave energy equation, in which the interaction of the wave with the background medium is taken into account. Burgers' equation is also derived for Alfvénic models in which the Alfvén waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfvén velocity relative to the background fluid.

Type
Research Article
Information
Journal of Plasma Physics , Volume 31 , Issue 3 , June 1984 , pp. 337 - 368
Copyright
Copyright © Cambridge University Press 1984

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