A quasi-linear model of plasma shock structure in a longitudinal magnetic field

It is pointed out that interpenetration of two plasma streams along a uniform magnetic field is prevented by the hose instability. Thus the shock thickness resulting from supersonic streaming of a tenuous plasma along a uniform magnetic field may be very much less than the collision length. We suggest that an actual stationary shock front under these conditions is formed by the hose interaction between the incoming supersonic stream and a precursor, consisting of ions evaporating forward from the non-linear disordering which results when the hose instability reaches large amplitude. Treating the growth of the hose instability with the WKB approximation applied to the linearized equations, we show that the scale of the shock front is characterized by the ion and electron Larmor radii, and that the density in the precursor relaxes to a small but non-vanishing value, proportional to one over the square of the Mach number, infinitely far ahead of the shock.