where is the plasma mass density, and is the ratio of specific heats.

It is often observed that the above set of equations are identical to the equations governing the motion of an inviscid, adiabatic, perfectly conducting, electrically neutral liquid. Indeed, this observation is sometimes used as the sole justification for the MHD equations. After all, a hot, tenuous, quasi-neutral plasma is highly conducting, and if the motion is sufficiently fast then both viscosity and heat conduction can be plausibly neglected. However, we can appreciate, from Sect. 3, that this is a highly oversimplified and misleading argument. The problem is, of course, that a weakly coupled plasma is a far more complicated dynamical system than a conducting liquid.

According to the discussion in
Sect. 3, the MHD equations are only valid when

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If the plasma dynamics becomes too fast (*i.e.*,
)
then resonances occur with the motions of individual particles
(*e.g.*, the cyclotron resonances) which
invalidate the MHD equations. Furthermore, effects, such as electron
inertia and the Hall effect, which are not taken into account in the
MHD equations, become important.

MHD is essentially a *single-fluid* plasma theory. A single-fluid
approach is justified because the perpendicular motion is dominated by
drifts, which are the same for both
plasma species. Furthermore,
the relative streaming velocity, , of
both species parallel to the magnetic field is strongly constrained by the
fundamental MHD ordering (see Sect. 3.9)

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Broadly speaking, the MHD equations describe relatively *violent*,
*large-scale*
motions of highly *magnetized* plasmas.

Strictly speaking, the MHD equations are only valid in *collisional*
plasmas (*i.e.*, plasmas in which the mean-free-path is much smaller
than the typical variation scale-length). However, as discussed in Sect. 3.13,
the MHD equations also fairly well describe the *perpendicular* (but not
the *parallel*!) motions of collisionless
plasmas.

Assuming that the MHD equations are valid, let us now investigate their properties.