Astrophysical flows as discussed in the previous chapter are subject to steepening, especially when coming off a time-dependent or intermittent source. Steepening results in shocks, where energy in bulk motion is partially dissipated into heat. Strong shocks thereby produce radiation, as alluded to in the general scheme pointed out in Section 1.2, further accompanied by entropy creation. In this chapter, we elucidate the physical processes governing various types of shocks.

Nonlinear steepening of relativistic disturbances

The analysis of small-amplitude MHD waves outlined in Section 5.3 indicates that the speed at which a linear disturbance propagates, Eq. (5.43), depends on local conditions. It is naively expected that over sufficiently long times the wave will be distorted, as the phase speed itself changes over the course of the wave trajectory. Waves generated at some location will eventually steepen into shocks, at which point the fluid picture breaks down. Inside the shock transition the wave dissipates, converting bulk energy into heat. The transition occurs over kinetic scales, roughly the collision length in collisional shocks, the skin depth in collisionless shocks, and the Thomson mean free path in radiation mediated shocks.

Riemann invariants and characteristics

A convenient way to analyze wave steepening is to write the MHD equations in terms of the so-called Riemann invariants, developed in compressible fluid dynamics to understand the process of steepening and shock formation.