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Parallel Shocks

The first special case is the so-called parallel 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,
$\displaystyle {\bf V}_1 = (V_1,\,0,\,0),$ $\textstyle \mbox{\hspace{1cm}}$ $\displaystyle {\bf V}_2 = (V_2,\,0,\,0),$ (939)
$\displaystyle {\bf B}_1 = (B_1,\,0,\,0),$ $\textstyle \mbox{\hspace{1cm}}$ $\displaystyle {\bf B}_2 = (B_2,\,0,\,0).$ (940)

Substitution into the general jump conditions (933)-(938) yields
$\displaystyle \frac{B_2}{B_1}$ $\textstyle =$ $\displaystyle 1,$ (941)
$\displaystyle \frac{\rho_2}{\rho_1}$ $\textstyle =$ $\displaystyle r,$ (942)
$\displaystyle \frac{V_2}{V_1}$ $\textstyle =$ $\displaystyle r^{-1},$ (943)
$\displaystyle \frac{p_2}{p_1}$ $\textstyle =$ $\displaystyle R,$ (944)

$\displaystyle r$ $\textstyle =$ $\displaystyle \frac{(\Gamma+1)\,M_1^{\,2}}{2+(\Gamma-1)\,M_1^{\,2}},$ (945)
$\displaystyle R$ $\textstyle =$ $\displaystyle 1+ \Gamma\,M_1^{\,2}\,(1-r^{-1})= \frac{(\Gamma+1)\,r-(\Gamma-1)}{(\Gamma+1)-(\Gamma-1)\,r}.$ (946)

Here, $M_1= V_1/V_{S\,1}$, where $V_{S\,1}=(\Gamma\,p_1/\rho_1)^{1/2}$ 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 above 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 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 $M_1=1$. 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 is easily demonstrated that

$\displaystyle \frac{\Gamma-1}{\Gamma+1} \leq$ $\textstyle r$ $\displaystyle \leq \frac{\Gamma+1}{\Gamma-1},$ (947)
$\displaystyle 0\leq$ $\textstyle R$ $\displaystyle \leq \infty,$ (948)
$\displaystyle \frac{\Gamma-1}{2\,\Gamma}\leq$ $\textstyle M_1^{\,2}$ $\displaystyle \leq \infty.$ (949)

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., $r>1$) or expansive (i.e., $r<1$). 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)]

S = \ln(p/\rho^{\Gamma}).
\end{displaymath} (950)

Hence, the difference between the upstream and downstream entropies is
\begin{displaymath}[S]^2_1 =\ln R - \Gamma\,\ln r.
\end{displaymath} (951)

Now, using (945),
r\,\frac{d[S]_1^2}{dr} = \frac{r}{R}\,\frac{dR}{dr}-\Gamma
\end{displaymath} (952)

Furthermore, it is easily seen from Eqs. (947)-(949) that $d[S]_1^2/dr\geq 0$ in all situations of physical interest. However, $[S]_1^2=0$ when $r=1$, since, 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: i.e., $r>1$. 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 which applies to all three types of MHD shock.

The upstream Mach number, $M_1$, is a good measure of shock strength: i.e., 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, $M_2=V_2/(\Gamma\,p_2/\rho_2)^{1/2}$. 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 (since the entropy suddenly increases) transition from supersonic to subsonic flow. Note that $r\equiv \rho_2/\rho_1\rightarrow (\Gamma+1)/(\Gamma-1)$, 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, $P$, increases without limit. For a conventional plasma with $\Gamma=5/3$, 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, $M_1\gg 1$, 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, Eqs. (945)-(946) imply that

\frac{T_2}{T_1} \equiv \frac{R}{r}\rightarrow \frac{2\,\Gamma\,(\Gamma-1)\,M_1^{\,2}}{(\Gamma+1)^2}\gg 1
\end{displaymath} (953)

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.

next up previous
Next: Perpendicular Shocks Up: Magnetohydrodynamic Fluids Previous: MHD Shocks
Richard Fitzpatrick 2011-03-31