It is easily seen from Equations (7.258)-(7.261) that there is no shock (i.e., no jump in plasma parameters across the shock front) when the upstream flow is exactly sonic: that is, when . In other words, when . However, if then the upstream and downstream plasma parameters become different (i.e., , ), and a true shock develops. In fact, it can be demonstrated that
The previous discussion seems to imply that a parallel shock can be either compressive (i.e., ) or expansive (i.e., ). However, there is one additional physics principle that needs to be factored into our analysis--namely, the second law of thermodynamics. This law states that the entropy of a closed system can spontaneously increase, but can never spontaneously decrease (Reif 1965). Now, in general, the entropy per particle is different on either side of a hydrodynamic shock front. Accordingly, the second law of thermodynamics mandates that the downstream entropy must exceed the upstream entropy, so as to ensure that the shock generates a net increase, rather than a net decrease, in the overall entropy of the system, as the plasma flows through it.
The (suitably normalized) entropy per particle of an ideal plasma takes the form [see Equation (4.51)]
The upstream Mach number, , is a good measure of shock strength: that is, if then there is no shock, if then the shock is weak, and if then the shock is strong. We can define an analogous downstream Mach number, . It is easily demonstrated from the jump conditions that if then . In other words, in the shock rest frame, the shock is associated with an irreversible (because the entropy suddenly increases) transition from supersonic to subsonic flow. Note that , whereas , in the limit . In other words, as the shock strength increases, the compression ratio, , asymptotes to a finite value, whereas the pressure ratio, , increases without limit. For a conventional plasma with , the limiting value of the compression ratio is 4: in other words, the downstream density can never be more than four times the upstream density. We conclude that, in the strong shock limit, , the large jump in the plasma pressure across the shock front must be predominately a consequence of a large jump in the plasma temperature, rather than the plasma density. In fact, Equations (7.260) and (7.261) imply that
As we have seen, the condition for the existence of a hydrodynamic shock is , or . In other words, in the shock frame, the upstream plasma velocity, , must be supersonic. However, by Galilean invariance, can also be interpreted as the propagation velocity of the shock through an initially stationary plasma. It follows that, in a stationary plasma, a parallel, or hydrodynamic, shock propagates along the magnetic field with a supersonic velocity.