Shocks and discontinuities (magnetohydrodynamics)

Shocks and discontinuities are transition layers where the plasma properties change from one equilibrium state to another. The relation between the plasma properties on both sides of a shock or a discontinuity can be obtained from the conservative form of the magnetohydrodynamic (MHD) equations, assuming conservation of mass, momentum, energy and of formula_1.
Rankine–Hugoniot jump conditions for MHD.
The jump conditions across a time-independent MHD shock or discontinuity are referred as the Rankine–Hugoniot equations for MHD. In the frame moving with the shock/discontinuity, those jump conditions can be written:
where formula_8, v, p, B are the plasma density, velocity, (thermal) pressure and magnetic field respectively. The subscripts t and n refer to the tangential and normal components of a vector (with respect to the shock/discontinuity front). The subscripts 1 and 2 refer to the two states of the plasma on each side of the shock/discontinuity
Contact and tangential discontinuities.
Contact and tangential discontinuities are transition layers across which there is no particle transport. Thus, in the frame moving with the discontinuity, formula_9.
Contact discontinuities are discontinuities for which the thermal pressure, the magnetic field and the velocity are continuous. Only the mass density and temperature change.
Tangential discontinuities are discontinuities for which the total pressure (sum of the thermal and magnetic pressures) is conserved. The normal component of the magnetic field is identically zero. The density, thermal pressure and tangential component of the magnetic field vector can be discontinuous across the layer.
Shocks.
Shocks are transition layers across which there is a transport of particles. There are three types of shocks in MHD: slow-mode, intermediate and fast-mode shocks.
Intermediate shocks are non-compressive (meaning that the plasma density does not change across the shock).
A special case of the intermediate shock is referred to as a rotational discontinuity. They are isentropic. All thermodynamic quantities are continuous across the shock, but the tangential component of the magnetic field can "rotate".
Intermediate shocks in general however, unlike rotational discontinuities, can have a discontinuity in the pressure.
Slow-mode and fast-mode shocks are compressive and are associated with an increase in entropy. Across slow-mode shock, the tangential component of the magnetic field decreases. Across fast-mode shock it increases.
The type of shocks depend on the relative magnitude of the upstream velocity in the frame moving with the shock with respect to some characteristic speed. Those characteristic speeds, the slow and fast magnetosonic speeds, are related to the Alfvén speed, formula_10 and the sonic speed, formula_11 as follows:
where formula_14 is the Alfvén speed and formula_15 is the angle between the incoming magnetic field and the shock normal vector.
The normal component of the slow shock propagates with velocity formula_16 in the frame moving with the upstream plasma, that of the intermediate shock with velocity formula_17 and that of the fast shock with velocity formula_18. The fast mode waves have higher phase velocities than the slow mode waves because the density and magnetic field are in phase, whereas the slow mode wave components are out of phase.
References.
The original research on MHD shock waves can be found in the following papers.
Textbook references.