Burgers' equation and cosmic ray shocks

Details Burgers' equation and cosmic ray shocks Burgers' equation and cosmic ray shocks

Jun 1984

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1984jplph..31..337w&link_type=abstract

Journal of Plasma Physics (ISSN 0022-3778), vol. 31, June 1984, p. 337-368.

Physics

Plasma Physics

14

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

Scientific paper

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 Alfvenic 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 Alfvenic models in which the Alfven waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfven velocity relative to the background fluid.

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