Shock waves are usually considered as a sharp transition between a
macroscopic supersonic (and super-Alfvénic) upstream flow (state 1)
and slowed down to a subsonic velocity downstream flow (state 2),
providing a mass flow jn through the shock surface. It
is assumed that a gas particle (or an elementary macroscopic fluid cell)
is at any instant of time in the local thermodynamic equilibrium
state corresponding to the instantaneous values of the macroscopic
parameters. The Maxwellian distribution of all species is ensured
after a few molecular (or Coulomb) collisions have occurred. The
macroscopic parameters characterising the state of the gas, such as
density, specific internal energy, or temperature, change slowly in
comparison with the rates of the relaxation processes leading to
thermodynamic equilibrium. We consider here a single-fluid plasma
model assuming complete electron-ion relaxation. Under these
conditions, in a frame moving with the shock front, with the matter
flux across the shock surface jn
0, the conservation laws for
mass (in non-relativistic flows), momentum and energy can be written
as follows:
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(1) |
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
Here U = (un, ut) is the bulk
velocity, w =
+ P /
is the gas
enthalpy,
,
P,
are the internal gas energy, pressure and density respectively.
The subscripts n and t are used for the normal and transverse
components respectively. We used the standard notations [A] =
A2 - A1 for the jump of a function
A between the downstream and upstream regions. In the MHD case
the relations equivalent to Eq. 1-4 were obtained by
de Hoffmann
& Teller (1950).
The equations are valid in the case of a
magnetic field frozen-into moving plasma, where E =
-U / c × B. A specific feature of MHD shock
waves is the so-called coplanarity theorem (e.g.
Landau &
Lifshitz 1984)
saying that the upstream and downstream magnetic fields
B1 and
B2 and the shock normal all lie in the same
plane as it is illustrated in Fig. 1. It is
important to note that if Bn
0 there is an especial
reference frame where local
velocity U and magnetic field B are parallel
both in the upstream and downstream, providing E = 0.
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Figure 1. A sketch illustrating the coplanarity theorem for a plane ideal MHD-shock. The upstream and downstream bulk velocities U1 and U2, magnetic fields B1 and B2 and the shock normal N all lie in the same plane. The shock is at rest in the reference frame where also ut1 = 0. The shock is of infinitesimal width in the sketch. Simulated structure of the transition region of a collisionless shock is shown in Fig. 2 and Fig. 3 where its finite width is apparent. |
From Eq. 1-4 one may obtain a generalised Rankine-Hugoniot (RH) adiabat
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(5) |
The RH adiabat connects the macroscopic parameters downstream of the flow once the upstream state is known. In a parallel shock (Bt = 0)
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(6) |
![]() |
(7) |
where g is the gas
adiabatic exponent. We restrict ourselves here to a fast mode
shock where cs1 < u1, and
va2 < u2 <
cs2, for va1 <
cs1. The phase velocity va2
is the Alfvén velocity in the downstream, cs1,
cs2 are the sound speeds in the upstream and downstream
respectively. We define here the shock Mach numbers as
s =
vsh / cs1 and
a =
vsh / va1.
In the case of a perpendicular shock (Bn = 0) the compression ratio is
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(8) |
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In a single fluid strong shock with
s >> 1 and
a >> 1 one
gets
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(9) |
for any magnetic field inclination (e.g. Draine & McKee 1993). The mass per particle µ was assumed to be [1.4/2.3] mH and vs8 is the shock velocity in 108 cm s-1.
The RH adiabat does not depend on the exact nature of the
dissipation mechanisms that provide the transition between the
states 1 and 2. It assumes a single-fluid motion in regular
electromagnetic fields. However, the dissipative effects control the
thickness of the shock transition layer. In the case of a weak shock
of Mach number s
- 1 << 1 the thickness is large
enough, allowing a macroscopic hydrodynamical description of the
fluid inside the shock transition layer (e.g.
Landau &
Lifshitz 1984).
The gas shock width
in collisional hydrodynamics without magnetic fields is given by
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(10) |
Following
Landau &
Lifshitz (1959)
the gas shock width in Eq. 10 can be expressed through the viscosities
and
, and
thermal conductivity
,
since
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Here cv and cp are specific heats at
constant volume and at constant pressure respectively. Extrapolating
Eq. 10 to a shock of finite strength where P2 -
P1 ~ P2, one may show that the gas
shock width is of the
order of the mean free path
.
It is instructive to note that the entropy is non-monotonic inside
the finite width of a weak gas shock
(s - 1) << 1
and the total RH jump of the entropy
s across the
shock is of the third order in
(
s - 1):
![]() |
(11) |
while the density, temperature and pressure jumps are
(
s - 1)
(Landau &
Lifshitz 1959).
In plasma shocks the shock structure is more complex because of a
relatively slow electron-ion temperature relaxation. Such a shock
consists of an ion viscous jump and an electron-ion thermal relaxation
zone. In the case of plasma shocks the structure of the ion viscous
jump is similar to the single fluid shock width structure discussed
above and can be studied accounting for the entropy of an isothermal
electron fluid. The shock ion viscous jump has a width of the order
of the ion mean free path. The scattering length (the mean free path
to /2 deflection)
p of a
proton of velocity v7
(measured in 100 km s-1) due to binary Coulomb collisions with
plasma protons of density n (measured in cm-3) can be
estimated as
p
7 ×
1014 v74 n-1 cm
(Spitzer 1962).
After the reionisation (z < 6) the Coulomb mean
free path in the WHIM of overdensity
is
p
3.5 ×
1021 v74
-1
(1 + z)-3
(
b
h2 / 0.02)-1 cm. Here and below
b is the
baryon density parameter.
The mean free path due to Coulomb collisions is typically
some orders of magnitude smaller than that for the charge-exchange
collisions in the WHIM after reionisation. The ion-electron thermal
relaxation occurs on scales about
e ×
(mp / Z me)1/2.
Since
e ~
p, the width
of the relaxation zone is substantially larger than the scale size
of the ion viscous jump. The application of the single fluid shock
model Eq. 1-4 to electron-ion
plasmas assumes full ion-electron temperature relaxation over the
shock width. For a discussion of the relaxation processes see e.g.
Bykov et al. 2008
- Chapter 8, this volume, and references therein.
In a rarefied hot cosmic plasma the Coulomb collisions are not sufficient to provide the viscous dissipation of the incoming flow, and collective effects due to the plasma flow instabilities play a major role, providing the collisionless shocks, as it is directly observed in the heliosphere. The observed structure of supernova remnants (e.g. Weisskopf & Hughes 2006) is consistent with that expected if their forward shocks are collisionless. Moreover, the non-thermal synchrotron emission seen in radio and X-rays is rather a strong argument for high energy particle acceleration by the shock that definitely favours its collisionless nature. That allows us to suggest that cosmological shocks in a rarefied highly ionised plasma (after the reionisation epoch) are likely to be collisionless. There are yet very few observational studies of cosmological shocks (e.g. Markevitch & Vikhlinin 2007). We review some basic principles of collisionless shock physics in the next section.