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Lecture 34
Shocks

In the lecture on conservative form we rewrote ideal MHD as a set of local conservation laws. Once the equations are written in divergence form, discontinuous weak solutions are not a pathology but a mathematical possibility. In the Alfvén-wave lecture we also learned that a magnetized plasma supports several characteristic families—fast, Alfvén, and slow. MHD shocks are the nonlinear dissipative descendants of those same branches.

A shock forms when a driver outruns the ability of the surrounding plasma to respond smoothly. In an ordinary gas the key threshold is the sound speed. In a magnetized plasma the sharper statement is that the flow becomes super-characteristic with respect to one of the MHD wave families. Then the disturbance steepens, a thin internal layer forms, and the large-scale solution is described by jump conditions rather than by a smooth profile.

Overview

A shock in MHD is a thin non-ideal layer whose thickness is determined by viscosity, resistivity, heat conduction, or kinetic microphysics, but whose jump is determined almost entirely by the conservative fluxes outside the layer. Because MHD has several characteristic branches, it supports several shock families rather than only the single hydrodynamic compressive shock.

Historical Perspective

The modern theory of shocks begins with the Rankine–Hugoniot conditions of compressible gas dynamics. The extension to magnetized fluids was developed early in plasma physics by de Hoffmann and Teller, who showed in 1950 that ideal MHD admits multiple shock families rather than only the single hydrodynamic compressive jump de Hoffmann and Teller (1950). That richer structure is not an accident: it follows directly from the fact that MHD has several characteristic wave branches. Since then, shocks have become central in heliophysics, astrophysics, and laboratory plasma physics because they mediate the conversion of directed flow energy into heat, magnetic compression, and energetic particles Draine and McKee (1993).

34.1 Why shocks matter

Localized conversion layers. Shocks are important because they are the natural interfaces between a fast plasma flow and a more slowly moving or nearly stationary ambient medium. Examples include the solar wind striking a planetary magnetosphere, a coronal mass ejection plowing into the interplanetary medium, supernova ejecta sweeping through the interstellar medium, reconnection outflows terminating against surrounding plasma, and laboratory pistons propagating into a target plasma Draine and McKee (1993).

Across a shock, density, pressure, magnetic-field direction, and flow speed can all change on a length scale that is tiny compared with the macroscopic flow scale. The shock therefore acts as a highly localized conversion layer: ram pressure is turned into thermal pressure, magnetic pressure, and entropy. In a collisional fluid the required irreversibility can be supplied by viscosity and heat conduction. In a weakly collisional or collisionless plasma the same role is played by wave–particle interactions, ion reflection, phase mixing, and kinetic microstructure rather than by binary collisions alone.

Outer and inner problems. It helps to think of a shock as two coupled problems. The outer problem determines which upstream and downstream states are compatible with conservation laws. The inner problem determines how the plasma actually makes the transition between those two states inside the thin layer. Ideal MHD is already enough for the outer problem, but not for the inner one. That split is why one can derive the jump conditions in a purely fluid framework while still needing kinetic physics to understand the detailed structure of many space and laboratory shocks.

Caution

Conceptual warning. Ideal MHD does not create entropy. The macroscopic jump conditions therefore do not by themselves explain why a shock is irreversible. A physical shock is the singular limit of a thin internal layer in which some small non-ideal effect—viscosity, resistivity, heat transport, ion reflection, or wave–particle scattering—breaks time reversibility.

34.2 Jump conditions from conservative form

A one-line derivation. Suppose a conserved variable \(U\) satisfies

\[\pp {U}{t} + \divergence \vect {F}(U) = 0. \tag{34.1}\]
Let a planar discontinuity move with speed \(V_s\) along the unit normal \(\vect {n}\). If we integrate (34.1) across a thin pillbox that straddles the layer, the side-surface contributions vanish as the pillbox thickness goes to zero and we obtain the jump rule
\[\bigl [\vect {n}\cdot \vect {F} - V_s U\bigr ] = 0. \tag{34.2}\]
In the shock rest frame, \(V_s=0\), so each conserved normal flux must be continuous:
\[\bigl [\vect {n}\cdot \vect {F}\bigr ]=0. \tag{34.3}\]
This is the whole logic behind the Rankine–Hugoniot conditions.

Planar-shock geometry. Take a planar shock at rest, with unit normal \(\vect {n}\) directed from upstream state 1 to downstream state 2. Decompose the flow and field into normal and tangential pieces,

\[u_n = \uvec \cdot \vect {n}, \qquad \uvec _t = \uvec - u_n\vect {n}, \qquad B_n = \B \cdot \vect {n}, \qquad \B _t = \B - B_n\vect {n}. \tag{34.4}\]
If \([f]\equiv f_2-f_1\), then the conservative MHD laws give

\[[\rho u_n] = 0, \tag{34.5a}\]
\[\left [\rho u_n^2 + p + \frac {B_t^2-B_n^2}{2\muo }\right ] = 0, \tag{34.5b}\]
\[\left [\rho u_n\uvec _t - \frac {B_n\B _t}{\muo }\right ] = 0, \tag{34.5c}\]
\[[B_n] = 0, \tag{34.5d}\]
\[[u_n\B _t - B_n\uvec _t] = 0, \tag{34.5e}\]
\[\left [ u_n \left ( \frac {1}{2}\rho u^2 + \frac {\gamma }{\gamma -1}p + \frac {B^2}{\muo } \right ) - \frac {B_n(\uvec \cdot \B )}{\muo } \right ] = 0. \tag{34.5f}\]

What each condition means. Equation (34.5a) says that the mass flux

\[m \equiv \rho u_n \tag{34.6}\]
is constant across the layer. Equation (34.5d) is just the statement that magnetic field lines do not begin or end on the shock. The momentum jump conditions show explicitly how the Maxwell stress modifies the hydrodynamic pressure balance, while (34.5e) is the stationary form of the induction law and is equivalent to continuity of the tangential electric field.

These equations are the precise mathematical sense in which shocks tie back to the conservative-form lecture: a shock is not a failure of the equations, but a weak solution of those same conservation laws.

34.3 Entropy production and physical admissibility

Entropy must increase. The Rankine–Hugoniot conditions alone do not guarantee that a mathematical jump is a physical shock. A genuine shock must also satisfy the second law of thermodynamics. For specific entropy \(s\), one may write the entropy balance schematically as

\[\pp {(\rho s)}{t} + \divergence (\rho s\uvec ) = \sigma , \qquad \sigma \ge 0,\]
where \(\sigma \) is the irreversible entropy production rate. For a steady planar shock this becomes
\[m[s] = \int _{-\infty }^{+\infty } \sigma \, dn > 0. \tag{34.8}\]
So the downstream state must have higher entropy than the upstream state.

Why ideal MHD still knows about shocks. Ideal MHD is formally dissipationless, so the outer equations do not display the microscopic mechanism that creates the entropy. Nevertheless, the jump conditions are still correct outside the layer because they are just conservation laws. The shock should therefore be viewed as the singular limit of a thin internal structure in which small viscosity, resistivity, heat transport, or kinetic processes break reversibility. The entropy increase is then the surviving macroscopic memory of that small-scale physics.

Evolutionary conditions. In MHD, the entropy condition still does not completely determine which discontinuities are physically realizable because there are several characteristic branches. One also asks whether a discontinuity is evolutionary: loosely speaking, the correct number of characteristics must impinge on the layer so that the downstream state is determined by the upstream data together with the internal dissipative structure. In isotropic ideal MHD, the standard compressive evolutionary shocks are the fast and slow shocks. Intermediate shocks are more subtle, and their admissibility depends on the model and on the detailed regularization of the layer Markovskii and Somov (1996).

34.4 Shock families, geometry, and useful frames

Connection to the wave lecture. The classification of MHD shocks follows directly from the wave families derived in the Alfvén-wave lecture. For propagation along the shock normal, the fast and slow phase speeds are

\[c_{\mathrm {f},\mathrm {sl}}^2 = \frac {1}{2} \left [ c_s^2 + v_A^2 \pm \sqrt {\left (c_s^2+v_A^2\right )^2 - 4c_s^2 v_{A,n}^2} \right ], \qquad v_{A,n}^2 = \frac {B_n^2}{\muo \rho }, \tag{34.9}\]
where \(c_s^2=\gamma p/\rho \) and \(v_A^2=B^2/(\muo \rho )\). A shock is called fast, slow, or intermediate according to which characteristic branch is crossed when going from state 1 to state 2.

Geometry versus family. It is useful to separate geometry from family. Geometrically, one speaks of parallel, perpendicular, and oblique shocks according to the upstream angle between \(\B _1\) and the shock normal \(\vect {n}\). Dynamically, one speaks of fast, slow, and intermediate shocks according to the MHD characteristic family involved. A shock can therefore be, for example, an oblique fast shock or a quasi-parallel intermediate shock.

Type

Relation to wave branch

Typical signature

Fast shock

Nonlinear continuation of the fast magnetosonic branch

Compresses the plasma and typically increases \(|\B _t|\); common at bow shocks and piston-driven fronts.

Slow shock

Nonlinear continuation of the slow branch

Compresses the plasma while typically decreasing \(|\B _t|\); central in many reconnection-exhaust models.

Intermediate shock

Associated with the Alfvén branch

Rotates the tangential magnetic field and can involve reversal of one tangential component; admissibility is more subtle than for fast and slow shocks.

Switch-on shock

Special fast-shock limit

Upstream \(\B _{t1}=0\) but downstream \(\B _{t2}\neq 0\).

Switch-off shock

Special slow-shock limit

Downstream \(\B _{t2}=0\); often discussed in reconnection theory.

Rotational discontinuity

Alfvénic discontinuity, not a compressive shock

Field direction rotates while density, pressure, and \(|\B |\) change little.

Tangential/contact discontinuity

Non-shock interfaces

No mass flux through the layer; pressure balance replaces entropy production as the main constraint.


Table 34.1: Common MHD shock and discontinuity families.

Coplanarity. For a steady planar ideal-MHD shock with nonzero mass flux, the upstream field, the downstream field, and the shock normal all lie in one plane. The flow vectors also lie in that plane. This coplanarity is one reason oblique shocks can often be reduced to an effective two-dimensional geometry even when they are embedded in a fully three-dimensional plasma configuration.

The de Hoffmann–Teller frame. A particularly useful frame for analyzing oblique shocks is the de Hoffmann–Teller frame de Hoffmann and Teller (1950). From (34.5e) one sees that the combination

\[\vect {v}_{\mathrm {HT}} \equiv \uvec _t - \frac {u_n}{B_n}\B _t \tag{34.10}\]
is the same on both sides of the shock whenever \(B_n\neq 0\). Boosting into the frame moving tangentially with velocity \(\vect {v}_{\mathrm {HT}}\) gives
\[\uvec ' = \uvec - \vect {v}_{\mathrm {HT}} = \frac {u_n}{B_n}\B , \qquad \E ' = -\uvec '\times \B = 0. \tag{34.11}\]
So in the de Hoffmann–Teller frame the plasma flow is parallel to the magnetic field on both sides of the shock.

Why this frame helps. In the de Hoffmann–Teller frame, one can think geometrically rather than algebraically. Fast shocks correspond to stronger bending and amplification of the tangential field, while slow shocks reduce the tangential field and may approach the switch-off limit. Rotational discontinuities also look especially simple in this frame because they are basically Alfvénic rotations of \(\B _t\) with little compression.

Important limitation. A finite de Hoffmann–Teller frame requires \(B_n\neq 0\). For an exactly perpendicular shock, where the upstream field is tangent to the shock surface, the necessary tangential boost diverges and no finite de Hoffmann–Teller frame exists. The construction is thus most useful for oblique shocks and for quasi-parallel or quasi-perpendicular limits, rather than for the exactly perpendicular case.

34.5 Worked examples: computing the downstream state

Parallel and perpendicular limits. The fully oblique problem is algebraically richer, but the parallel and perpendicular limits already show the main logic. In the parallel case the magnetic field drops out, so one sees entropy selection in exactly the same way as for an ordinary gas-dynamic normal shock. In the perpendicular case the field is compressed with the plasma, so one sees explicitly how magnetic pressure modifies the downstream state and why the relevant threshold is the fast magnetosonic speed rather than the sound speed alone.

Example 1: parallel shock. Take the shock normal parallel to the upstream magnetic field, with \(\uvec _{t1}=\uvec _{t2}=0\) and \(\B _{t1}=\B _{t2}=0\). Then \(B_n=B\) is continuous and the magnetic terms cancel from the momentum and energy jumps. The Rankine–Hugoniot conditions reduce to the hydrodynamic normal-shock equations,

\[\begin{aligned}\rho _1 u_1 &= \rho _2 u_2 \equiv m, \\ \rho _1 u_1^2 + p_1 &= \rho _2 u_2^2 + p_2, \\ \frac {1}{2}u_1^2 + \frac {\gamma }{\gamma -1}\frac {p_1}{\rho _1} &= \frac {1}{2}u_2^2 + \frac {\gamma }{\gamma -1}\frac {p_2}{\rho _2},\end{aligned} \tag{34.14}\]

with \(B_2=B_1\).

Define the compression ratio and upstream sonic Mach number by

\[r \equiv \frac {\rho _2}{\rho _1}, \qquad M_{s1} \equiv \frac {u_1}{c_{s1}}, \qquad c_{s1}^2 = \frac {\gamma p_1}{\rho _1}. \tag{34.15}\]
From mass conservation,
\[\frac {u_2}{u_1} = \frac {1}{r}. \tag{34.16}\]
Substitute this into the momentum jump:
\[\rho _1 u_1^2 + p_1 = \rho _1 \frac {u_1^2}{r} + p_2,\]
which gives
\[\frac {p_2}{p_1} = 1 + \gamma M_{s1}^2\left (1-\frac {1}{r}\right ). \tag{34.18}\]
Now substitute (34.16) and (34.18) into the energy equation (34.14). After multiplying through by \(2(\gamma -1)/(u_1^2)\) and simplifying, one finds the factorized relation
\[(r-1) \left [ \bigl ((\gamma -1)M_{s1}^2+2\bigr )r - (\gamma +1)M_{s1}^2 \right ] = 0. \tag{34.19}\]
Thus there are two algebraic branches:
\[r = 1 \qquad \text {or}\qquad r = \frac {(\gamma +1)M_{s1}^2}{(\gamma -1)M_{s1}^2+2}. \tag{34.20}\]
The first is the trivial no-shock solution. The second is the compressive branch when \(M_{s1}>1\), but it becomes an expansion branch when \(M_{s1}<1\).

The downstream state then follows directly:

\[\begin{aligned}\frac {u_2}{u_1} &= \frac {1}{r}, \\ \frac {p_2}{p_1} &= \frac {2\gamma M_{s1}^2-(\gamma -1)}{\gamma +1}, \\ \frac {T_2}{T_1} &= \frac {p_2/p_1}{r}, \\ M_{s2}^2 &= \frac {1+\tfrac {1}{2}(\gamma -1)M_{s1}^2}{\gamma M_{s1}^2-\tfrac {1}{2}(\gamma -1)}.\end{aligned}\]

For an ideal gas the entropy jump is

\[\frac {\Delta s}{c_v} = \ln \left (\frac {p_2/p_1}{r^{\gamma }}\right ). \tag{34.25}\]
In the strong-shock limit \(M_{s1}\gg 1\), the compression ratio approaches the familiar maximum
\[r \to \frac {\gamma +1}{\gamma -1}. \tag{34.26}\]

For a concrete number, let \(\gamma =5/3\) and \(M_{s1}=3\). Then

\[r = 3, \qquad \frac {u_2}{u_1} = \frac {1}{3}, \qquad \frac {p_2}{p_1} = 11, \qquad \frac {T_2}{T_1} = \frac {11}{3}, \qquad M_{s2} = \sqrt {\frac {3}{11}} \simeq 0.522.\]
The entropy rises by
\[\frac {\Delta s}{c_v} = \ln \left (\frac {11}{3^{5/3}}\right ) \simeq 0.567 > 0,\]
so this is a physical shock.

Now take \(\gamma =5/3\) but \(M_{s1}=0.8\). Equation (34.20) still returns a nontrivial algebraic root,

\[r \simeq 0.703, \qquad \frac {p_2}{p_1} \simeq 0.550, \qquad \frac {\Delta s}{c_v} \simeq -1.12\times 10^{-2}.\]
This is an expansion discontinuity rather than a shock. It satisfies the flux balances but violates the entropy condition, so it is not physically admissible. This is the clearest place to see why conservation laws alone are not enough: one needs entropy production to select the shock branch.

Example 2: perpendicular shock. Now take \(B_n=0\), \(\uvec _{t1}=\uvec _{t2}=0\), and let \(B_1\) and \(B_2\) denote the upstream and downstream tangential field strengths. The induction jump condition gives

\[u_1 B_1 = u_2 B_2,\]
so with \(r=\rho _2/\rho _1\) and \(\rho _1 u_1 = \rho _2 u_2\),
\[\frac {u_2}{u_1} = \frac {1}{r}, \qquad \frac {B_2}{B_1} = r. \tag{34.31}\]
Thus a perpendicular shock compresses magnetic flux and mass by the same factor.

It is convenient to use the upstream Alfvén Mach number and plasma beta,

\[M_{A1} \equiv \frac {u_1}{v_{A1}}, \qquad v_{A1}^2 = \frac {B_1^2}{\muo \rho _1}, \qquad \beta _1 \equiv \frac {2\muo p_1}{B_1^2}. \tag{34.32}\]
The normal momentum jump becomes
\[\frac {p_2}{p_1} = 1 + \frac {2M_{A1}^2}{\beta _1}\left (1-\frac {1}{r}\right ) + \frac {1-r^2}{\beta _1}. \tag{34.33}\]
Using the energy jump then yields
\[(r-1) \left [ (2-\gamma )r^2 + \Bigl ((\gamma -1)M_{A1}^2 + \gamma (\beta _1+1)\Bigr )r - (\gamma +1)M_{A1}^2 \right ] = 0. \tag{34.34}\]
Again \(r=1\) is the trivial branch, while the nontrivial root determines the shock compression. The relevant threshold is now the upstream fast magnetosonic speed. For strictly perpendicular propagation,
\[c_{f1}^2 = c_{s1}^2 + v_{A1}^2, \qquad M_{f1}^2 \equiv \frac {u_1^2}{c_{f1}^2} = \frac {M_{A1}^2}{1+\gamma \beta _1/2}. \tag{34.35}\]
A compressive perpendicular shock therefore requires \(M_{f1}>1\).

As an example, take \(\gamma =5/3\), \(\beta _1=1\), and \(M_{A1}=3\). Then

\[M_{f1} = \frac {3}{\sqrt {1+5/6}} \simeq 2.22,\]
so the upstream flow is super-fast. Solving (34.34) gives
\[r \simeq 2.371.\]
The downstream state follows immediately:
\[\frac {u_2}{u_1} \simeq 0.422, \qquad \frac {B_2}{B_1} \simeq 2.371, \qquad \frac {p_2}{p_1} \simeq 6.79, \qquad \frac {T_2}{T_1} \simeq 2.86.\]
The downstream fast Mach number is
\[M_{f2} = \frac {u_2}{\sqrt {c_{s2}^2+v_{A2}^2}} = \frac {M_{A1}}{\sqrt {r^3 + \tfrac {1}{2}\gamma \beta _1 (p_2/p_1) r}} \simeq 0.580,\]
so the shock has converted a super-fast upstream flow into a sub-fast downstream flow, as expected. The entropy rise is
\[\frac {\Delta s}{c_v} = \ln \left (\frac {6.79}{2.371^{5/3}}\right ) \simeq 0.476 > 0.\]

It is also useful to see what happens below threshold. With the same \(\gamma =5/3\) and \(\beta _1=1\) but \(M_{A1}=1.2\), one has \(M_{f1}\simeq 0.886<1\). The nontrivial positive root of (34.34) is then

\[r \simeq 0.840, \qquad \frac {p_2}{p_1} \simeq 0.745, \qquad \frac {\Delta s}{c_v} \simeq -3.08\times 10^{-3}.\]
So the algebra again offers an expansion branch, but not a physical shock. The entropy condition and the requirement that the upstream flow be super-fast eliminate it.

Takeaway from the examples. These two limits show the logic that survives in the more general oblique problem. First, the conservative laws determine a small number of algebraic branches for the downstream state. Second, not all of those branches are physical. The admissible shock is the one that is compressive, raises the entropy, and moves the flow from super-characteristic upstream to sub-characteristic downstream for the appropriate MHD family.

34.6 Shock structure and what ideal MHD leaves out

Collisional versus collisionless layers. The jump conditions say nothing about the detailed width of the transition. In a collisional plasma, one expects a thickness controlled by viscosity, thermal transport, and possibly resistivity. In a collisionless plasma, the layer instead develops ion-scale or electron-scale substructure, reflected particles, and dispersive fields. The shock is still a large-scale MHD object in the sense that the upstream and downstream states obey Rankine–Hugoniot balance, but the actual dissipation is supplied by kinetic physics.

Why this matters conceptually. This split between outer jump conditions and inner layer physics is the same structural idea that appears in reconnection and tearing modes. In all three problems the large-scale solution determines the drive, while a thin non-ideal layer determines the irreversible outcome. In shocks, however, the layer is usually not centered on a rational surface or a neutral line; instead it is the narrow conversion region required to connect two bulk states that can no longer be joined smoothly.

Caution

Do not confuse a shock with every sharp front. A tangential discontinuity or contact discontinuity can also be thin, but it is not a shock because there is no compressive transition selected by entropy production. A rotational discontinuity can rotate the field sharply while remaining essentially Alfvénic. The word “shock” should be reserved for the irreversible compressive layers that satisfy both jump conditions and an admissibility condition.

34.7 Experimental perspective

Why shock experiments are hard. Laboratory studies of MHD shocks are challenging because one must produce a plasma that is simultaneously compressible, sufficiently magnetized, and driven fast enough to cross the relevant characteristic speed. In that sense shock experiments are more demanding than the classic liquid-metal Alfvén-wave experiments discussed earlier: wave propagation only requires magnetic tension and conductivity, whereas shock formation also requires a strong nonlinear drive and an internal dissipation mechanism. Modern shock experiments therefore span laser-produced plasmas, pulsed-power devices, and large magnetized plasma facilities.

Wisconsin programme. At Wisconsin, the WiPAL and WiPPL facilities described by Forest and collaborators provide a flexible setting for this programme Forest et al. (2015). On the Big Red Ball, Endrizzi, Egedal, Forest and co-workers resolved the structure of a supercritical perpendicular shock produced by a large-radius theta pinch. They observed magnetically driven supermagnetosonic perpendicular flows, a reflected-ion foot, Hall-current magnetic structure, and adiabatic electron heating, giving a particularly clear laboratory example of how a collisionless shock departs from the simplest single-fluid picture Endrizzi et al. (2021).

A broader account is given in Endrizzi’s dissertation, which treated the formation of both parallel and perpendicular collisionless shocks in the Big Red Ball and emphasized the bridge between MHD jump conditions and kinetic shock substructure Endrizzi (2021).

Shocks in reconnection-driven flows. On TREX, Olson, Egedal, Endrizzi, Forest and collaborators identified a shock interface between the far-upstream supersonic inflow and a region of magnetic-flux pileup during driven reconnection. There the shock is not merely incidental; it is part of the force balance that regulates the inflow and helps connect the measured layer structure to the Rankine–Hugoniot constraints derived above Olson et al. (2021).

Related Big Red Ball experiments have also characterized fast magnetosonic waves driven by compact-toroid injection along an applied field. Those measurements do not yet constitute a fully developed shock, but they isolate the fast branch that, at larger amplitude, steepens into one of the standard MHD shock families Chu et al. (2023).

Takeaways
  • The jump conditions for a shock come directly from the conservative fluxes of MHD.
  • Entropy production and evolutionary conditions are what distinguish a physical shock from a merely algebraic discontinuity.
  • MHD supports multiple shock families because it supports multiple wave families.
  • Ideal MHD determines the outer jump, but the internal shock thickness and irreversibility come from non-ideal transport or kinetic physics.

Bibliography

    Frederic de Hoffmann and Edward Teller. Magneto-hydrodynamic shocks. Physical Review, 80 (4):692–703, 1950. doi:10.1103/PhysRev.80.692.

    Bruce T. Draine and Christopher F. McKee. Theory of interstellar shocks. Annual Review of Astronomy and Astrophysics, 31:373–432, 1993. doi:10.1146/annurev.aa.31.090193.002105.

    S. A. Markovskii and B. V. Somov. Magnetohydrodynamic discontinuities in space plasmas: Interrelation between stability and structure. Space Science Reviews, 78:443–506, 1996. doi:10.1007/BF00171927.

    C. B. Forest, K. Flanagan, M. Brookhart, M. Clark, C. M. Cooper, V. Désangles, J. Egedal, D. Endrizzi, I. V. Khalzov, H. Li, et al. The wisconsin plasma astrophysics laboratory. Journal of Plasma Physics, 81(5):345810501, 2015. doi:10.1017/S0022377815000975.

    Douglass Endrizzi, J. Egedal, M. Clark, K. Flanagan, S. Greess, J. Milhone, A. Millet-Ayala, J. Olson, E. E. Peterson, J. Wallace, and C. B. Forest. Laboratory resolved structure of supercritical perpendicular shocks. Physical Review Letters, 126(14):145001, 2021. doi:10.1103/PhysRevLett.126.145001.

    Douglass A. Endrizzi. The Formation of Parallel and Perpendicular Collisionless Shocks in the Big Red Ball. PhD thesis, University of Wisconsin–Madison, 2021.

    Joseph Olson, Jan Egedal, Michael Clark, Douglass A. Endrizzi, Samuel Greess, Alexander Millet-Ayala, Rachel Myers, Ethan E. Peterson, John Wallace, and Cary B. Forest. Regulation of the normalized rate of driven magnetic reconnection through shocked flux pileup. Journal of Plasma Physics, 87(3):175870301, 2021. doi:10.1017/S0022377821000659.

    F. Chu, S. J. Langendorf, J. Olson, T. Byvank, D. A. Endrizzi, A. L. LaJoie, K. J. McCollam, and C. B. Forest. Characterization of fast magnetosonic waves driven by compact toroid plasma injection along a magnetic field. Physics of Plasmas, 30(12):122110, 2023. doi:10.1063/5.0174537.

Problems

Problem 34.1.
Starting from a generic conservation law (34.1), derive the jump rule (34.2) for a discontinuity moving with speed \(V_s\). Then show explicitly how the steady shock conditions follow in the shock rest frame.
Problem 34.2.
For the parallel shock, take the strong-shock limit \(M_{s1}\gg 1\) and show directly from (34.20) that the compression ratio approaches \((\gamma +1)/(\gamma -1)\). Evaluate this limit for \(\gamma =5/3\).
Problem 34.3.
For a perpendicular shock, use (34.31) to show that the magnetic pressure jump scales like \(r^2\). Explain physically why this makes the perpendicular problem different from the hydrodynamic one even though both are described by Rankine–Hugoniot balances.
Problem 34.4.
Verify the de Hoffmann–Teller construction by showing that continuity of the tangential electric field implies that the quantity \(\uvec _t-(u_n/B_n)\B _t\) is the same on both sides of an oblique shock.