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,

$${\bf V}_1 = (V_1,\,0,\,0), {\bf V}_2 = (V_2,\,0,\,0),$$ (8.186)

$${\bf B}_1 = (B_1,\,0,\,0), {\bf B}_2 = (B_2,\,0,\,0).$$ (8.187)

Substitution into the general jump conditions (8.180)–(8.185) yields

$$\frac{B_2}{B_1} = 1, \frac{\rho_2}{\rho_1} = r,$$ (8.188)

$$\frac{V_2}{V_1} = r^{-1}, \frac{p_2}{p_1} = R,$$ (8.189)

with

$$r = \frac{({\mit \Gamma}+1)\,M_1^{2}}{2+({\mit\Gamma}-1)\,M_1^{2}},$$ (8.190)

$$R = 1+ {\mit \Gamma}\,M_1^{2}\,(1-r^{-1})= \frac{({\mit \Gamma}+1)\,r-({\mit\Gamma}-1)}{({\mit\Gamma}+1)-({\mit\Gamma}-1)\,r}.$$ (8.191)

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 (Fitzpatrick 2017).

It is easily seen from Equations (8.188)–(8.191) 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

$$\frac{{\mit\Gamma}-1}{{\mit\Gamma}+1} \leq r \leq \frac{{\mit\Gamma}+1}{{\mit\Gamma}-1},$$ (8.192)

$$0\leq R \leq \infty,$$ (8.193)

$$\frac{{\mit\Gamma}-1}{2\,{\mit\Gamma}}\leq M_1^{2} \leq \infty.$$ (8.194)

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)]

$$S = \ln\left(\frac{p}{\rho^{{\mit\Gamma}}}\right).$$ (8.195)

Hence, the difference between the upstream and downstream entropies is

$$[S]^2_1 =\ln R - {\mit\Gamma}\,\ln r.$$ (8.196)

Now, using (8.191),

$$r\,\frac{d[S]_1^2}{dr} = \frac{r}{R}\,\frac{dR}{dr}-{\mit\Gamma} ... ...({\mit\Gamma}+1)\,r-({\mit\Gamma}-1)]\,[({\mit\Gamma}+1)-({\mit\Gamma}-1)\,r]}.$$ (8.197)

Furthermore, it is easily seen from Equations (8.192)–(8.194) 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 (8.190) and (8.191) imply that

$$\frac{T_2}{T_1} \equiv \frac{R}{r}\rightarrow \frac{2\,{\mit\Gamma}\,({\mit\Gamma}-1)\,M_1^{2}}{({\mit\Gamma}+1)^{2}}\gg 1$$ (8.198)

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.