Parallel MHD Shocks

The first special case is the so-called parallel MHD shock, in which both the upstream and downstream plasma flows are parallel to the magnetic field, as well as perpendicular to the shock front. In other words,

$${\bf V}_1 = (V_1,\,0,\,0), {\bf V}_2 = (V_2,\,0,\,0),$$ (7.256)

$${\bf B}_1 = (B_1,\,0,\,0), {\bf B}_2 = (B_2,\,0,\,0).$$ (7.257)

Substitution into the general jump conditions (7.250)-(7.255) yields

$$\frac{B_2}{B_1} = 1, \frac{\rho_2}{\rho_1} = r,$$ (7.258)

$$\frac{V_2}{V_1} = r^{\,-1}, \frac{p_2}{p_1} = R,$$ (7.259)

with

$$r = \frac{({\mit \Gamma}+1)\,M_1^{\,2}}{2+({\mit\Gamma}-1)\,M_1^{\,2}},$$ (7.260)

$$R = 1+ {\mit \Gamma}\,M_1^{\,2}\,(1-r^{\,-1})= \frac{({\mit \Gamma}+1)\,r-({\mit\Gamma}-1)}{({\mit\Gamma}+1)-({\mit\Gamma}-1)\,r}.$$ (7.261)

Here, $M_1= V_1/V_{S\,1}$ , where $V_{S\,1}=({\mit\Gamma}\,p_1/\rho_1)^{1/2}$ is the upstream sound speed. Thus, the upstream flow is supersonic if $M_1>1$ , and subsonic if $M_1<1$ . Incidentally, as is clear from the previous expressions, a parallel shock is unaffected by the presence of a magnetic field. In fact, this type of shock is identical to that which occurs in neutral fluids, and is, therefore, usually called a hydrodynamic shock.

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 $M_1=1$ . In other words, $r=R=1$ when $M_1=1$ . However, if $M_1\neq 1$ then the upstream and downstream plasma parameters become different (i.e., $r\neq 1$ , $R\neq 1$ ), and a true shock develops. In fact, it can be demonstrated that

$$\frac{{\mit\Gamma}-1}{{\mit\Gamma}+1} \leq r \leq \frac{{\mit\Gamma}+1}{{\mit\Gamma}-1},$$ (7.262)

$$0\leq R \leq \infty,$$ (7.263)

$$\frac{{\mit\Gamma}-1}{2\,{\mit\Gamma}}\leq M_1^{\,2} \leq \infty.$$ (7.264)

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

The previous discussion seems to imply that a parallel shock can be either compressive (i.e., $r>1$ ) or expansive (i.e., $r<1$ ). 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)]

$$S = \ln\left(\frac{p}{\rho^{\,{\mit\Gamma}}}\right).$$ (7.265)

Hence, the difference between the upstream and downstream entropies is

$$[S]^2_1 =\ln R - {\mit\Gamma}\,\ln r.$$ (7.266)

Now, using (7.261),

$$r\,\frac{d[S]_1^2}{dr} = \frac{r}{R}\,\frac{dR}{dr}-{\mit\Gamma} ... ...({\mit\Gamma}+1)\,r-({\mit\Gamma}-1)]\,[({\mit\Gamma}+1)-({\mit\Gamma}-1)\,r]}.$$ (7.267)

Furthermore, it is easily seen from Equations (7.262)-(7.264) that $d[S]_1^2/dr\geq 0$ in all situations of physical interest. However, $[S]_1^2=0$ when $r=1$ , because, in this case, there is no discontinuity in plasma parameters across the shock front. We conclude that $[S]_1^2<0$ for $r<1$ , and $[S]_1^2>0$ for $r>1$ . It follows that the second law of thermodynamics requires hydrodynamic shocks to be compressive: that is, $r>1$ . In other words, the plasma density must always increase when a shock front is crossed in the direction of the relative plasma flow. It turns out that this is a general rule that applies to all three types of MHD shock (Boyd and Sanderson 2003).

The upstream Mach number, $M_1$ , is a good measure of shock strength: that is, if $M_1=1$ then there is no shock, if $M_1-1 \ll 1$ then the shock is weak, and if $M_1\gg 1$ then the shock is strong. We can define an analogous downstream Mach number, $M_2=V_2/({\mit\Gamma}\,p_2/\rho_2)^{1/2}$ . It is easily demonstrated from the jump conditions that if $M_1>1$ then $M_2 < 1$ . 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 $r\equiv \rho_2/\rho_1\rightarrow ({\mit\Gamma}+1)/({\mit\Gamma}-1)$ , whereas $R\equiv p_2/p_1\rightarrow\infty$ , in the limit $M_1\rightarrow \infty$ . In other words, as the shock strength increases, the compression ratio, $r$ , asymptotes to a finite value, whereas the pressure ratio, $R$ , increases without limit. For a conventional plasma with ${\mit\Gamma}=5/3$ , 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, $M_1\gg 1$ , 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

$$\frac{T_2}{T_1} \equiv \frac{R}{r}\rightarrow \frac{2\,{\mit\Gamma}\,({\mit\Gamma}-1)\,M_1^{\,2}}{({\mit\Gamma}+1)^{\,2}}\gg 1$$ (7.268)

as $M_1\rightarrow \infty$ . Thus, a strong parallel, or hydrodynamic, shock is associated with intense plasma heating.

As we have seen, the condition for the existence of a hydrodynamic shock is $M_1>1$ , or $V_1 > V_{S\,1}$ . In other words, in the shock frame, the upstream plasma velocity, $V_1$ , must be supersonic. However, by Galilean invariance, $V_1$ 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.