Collisional Theory of Shock and Nonlinear Waves in a Plasma

The unsteady state problem of weak nonlinear plasma waves in the presence of a perpendicular magnetic field is studied, based on a system of moment equations of Boltzmann equations for electrons and ions. This system, rigorously closed and obtained by proper scalings in time and space with respect to the wave strength, describes a fully ionized plasma with all possible dissipative mechanisms including electron and ion viscosity, coupled electron and ion heat conduction, thermal diffusion and resistivity, as well as all the distinct Hall effects. In all the plasma parameter ranges, it is found that the evolution of weak nonlinear waves is governed by the Burgers equation, subject to given initial and boundary conditions. The formation of a shock wave and the evolution of a compression pulse are studied in detail. The transit time and distance for a fully developed shock wave are obtained, both being complicated functions of plasma parameters. In some parameter ranges, the transit time is smaller than the electron self‐collision time while the transit distance is smaller than the mean free path. The time scale for the evolution of a compression pulse depends strongly on its initial width. A steady shock front will shortly appear if the initial width of the pulse is large compared with the thickness of a steady shock wave. On the other hand, the pulse will immediately decay if its initial width is small compared with the thickness of a steady shock wave. Depending on the parameter range, the thickness of a steady state shock wave can be larger than, comparable to, or smaller than the mean free path. Eventually, however, a compression pulse becomes self‐similar, with its amplitude decaying as ${\text{1/t}}^{\text{1/2}}$ and its width spreading as $t^{\text{1/2}}$ . The inclusion of the electron inertia effect leads to the Korteweg‐de Vries‐Burgers equation which covers a wide range of wave profiles.