A "phase transition" from a planar stationary profile to a rippled structure

It is common wisdom that collisionless shocks become nonplanar and nonstationary at sufficiently high Mach numbers. Whatever the shock structure, the upstream and downstream fluxes of the mass, momentum, and energy should be equal. These conservation laws are satisfied at low Mach numbers when the shock front is planar and stationary. When this becomes impossible, inhomogeneity and time dependence, presumably as rippling, develop. Using test particle analysis in a model shock profile, this study shows that the shock structure changes as a kind of "phase transition" when the Mach number is increased while the shock angle, the upstream beta, and the magnetic compression are kept constant.