Parallel Shocks

(939) | |||

(940) |

Substitution into the general jump conditions (933)-(938) yields

with

Here, , where is the upstream sound speed. Thus, the upstream flow is supersonic if , and subsonic if . Incidentally, as is clear from the above expressions, a parallel shock is

It is easily seen from Eqs. (941)-(944) that there is no shock (*i.e.*, no jump in plasma parameters across the shock front) when the upstream flow is exactly
sonic: *i.e.*, 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 is easily demonstrated that

Note that the upper and lower limits in the above inequalities are all attained simultaneously.

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 which 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. 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 Eq. (224)]

(950) |

(951) |

(952) |

The upstream Mach number, , is a good measure of shock strength:
*i.e.*, 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 (since 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: *i.e.*, 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, Eqs. (945)-(946) imply that

(953) |

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.