Influence of the rippling on the collisionless ion and electron motion in the shock front: A model study

The only well-developed theory of collisionless shocks is the one-dimensional (1-D) stationary theory, which explicitly assumes that all variables in the shock front, including electric and magnetic fields, depend only on the coordinate along the shock normal and do not depend on time. These two assumptions are essential for establishing the Rankine-Hugoniot relations [see, e.g., Tidman and Krall, 1971] which connect the upstream and downstream states of plasma and magneti