Burgers' equation and cosmic ray shocks
Abstract
The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a smallamplitude, longwavelength, 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 steadystate 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.
 Publication:

Journal of Plasma Physics
 Pub Date:
 June 1984
 DOI:
 10.1017/S0022377800001707
 Bibcode:
 1984JPlPh..31..337W
 Keywords:

 Burger Equation;
 Cosmic Rays;
 Energetic Particles;
 Magnetohydrodynamic Waves;
 Particle Acceleration;
 Shock Wave Propagation;
 Distribution Functions;
 Hydrodynamic Equations;
 Magnetohydrodynamic Stability;
 Plasma Turbulence;
 Shock Wave Interaction;
 Wave Equations;
 Astrophysics