Overturning of nonlinear Alfvén waves in a collisionless plasma

Abstract

An analysis is made of the overturning of nonlinear Alfvén waves in a collisionless plasma. It is shown that overturning is followed by the appearance of a region which broadens with time and consists of two collisionless shock waves which can be joined at the point s²=1. If only one Riemann invariant changes in the region of the collisionless shock waves, the waves are simple. The structures of the collisionless shock waves are constructed for different initial conditions of the nonlinear wave. The Whitham averaging method is used for this purpose. Conditions are obtained which are similar to the Rankine-Hugoniot adiabats for passage through the collisionless shock waves. The effect of overturning one of the collisionless shock waves, involving the zeroing of the density at the soliton peak on its trailing edge, is treated as a bifurcation for which a discontinuity occurs in an analog of the hydrodynamic velocity and phase of the nonlinear Alfvén wave. The width of one of the collisionless shock waves decreases with an increase in the parametera which determines the magnitude of the field discontinuity when overturning occurs.