The simplified discussion of the plasma evolution is often related to magnetohydrodynamics (MHD) which is a one-fluid plasma description holding under very specific assumptions. The goal of the first part of the present Section (together with the related Appendix A) will be to present the reduction of the common two-fluids plasma dynamics to the simplified case of one-fluid description provided by MHD.

**4.1. Qualitative aspects of plasma dynamics**

Consider, as a starting point, a globally neutral plasma of electrons
and ions (in the simplest case protons)
all at the same temperature *T*_{0} and with mean particle
density *n*_{0}.
Suppose, for simplicity, local thermodynamical equilibrium
^{(11)}.
The charge densities will be given, respectively, by

(4.1) |

where is the
electrostatic potential. In Eqs. (4.1) it has been
assumed that the plasma is weakly coupled, i.e.
|*e* /
*T*_{0}| << 1.
A test charge *q*_{t} located in the origin will then
experience the electrostatic potential following from the Poisson
equation,

(4.2) |

or, using Eqs. (4.1) into Eq. (4.2)

(4.3) |

where

(4.4) |

is the Debye length. For a test particle the Coulomb potential
will then have the usual Coulomb form but it will be suppressed, at large
distances by a Yukawa term, i.e.
*e*^{-r/D}.

In the interstellar medium there are three kinds of regions which
are conventionally defined, namely H_{2} regions (where the Hydrogen
is predominantly in molecular form), H^{0} regions (where
Hydrogen is in atomic form) and H^{+} regions (where Hydrogen is
ionized). Sometimes H^{+} regions are denoted with HI and
H_{2} regions with HII. In the H^{+} regions the typical
temperature *T*_{0} is of the order
of 10-20 eV while for *n*_{0} let us take, for instance,
*n*_{0} ~ 3 × 10^{-2} cm^{-3}. Then
_{D} ~ 30 km.

For *r* >>
_{D} the
Coulomb potential is screened by the global effect of
the other particles in the plasma. Suppose now that particles
exchange momentum through two-body interactions. Their cross
section will be of the order of
_{em}^{2}
/ *T*_{0}^{2} and the mean free path will be
_{mfp} ~
*T*_{0}^{2} /
(_{em}^{2}
*n*_{0}), i.e. recalling Eq. (4.4)
_{D} <<
_{mfp}. This means
that the plasma is a weakly collisional system which is, in general,
*not* in local thermodynamical
equilibrium. This observation can be made more explicit by defining another
important scale, namely the plasma frequency which, in the system
under discussion, is given by

(4.5) |

where *m*_{e} is the electron mass. The plasma frequency is
the oscillation frequency of the electrons when they are displaced
from their equilibrium configuration in a background of approximately
fixed ions. Recalling that
*v*_{ther}
(*T*_{0} / *m*_{e})^{1/2} is the thermal
velocity of the charge carriers, the collision frequency
_{c}
*v*_{ther}
/ _{mfp}
is always much smaller than
_{p}
*v*_{ther}
/ _{D}.
Thus, in the idealized system described so far, the following
hierarchy of scales holds:

(4.6) |

which means that before doing one collision the system undergoes
many oscillations, or, in other words, that the mean free path
is not the shortest scale in the problem.
Usually one defines also the *plasma parameter*
=
*n*_{0}^{-1} _{D}^{-3},
i.e. the number of particles in the Debye sphere. In the approximation
of weakly coupled plasma
<< 1 which also
imply that the mean kinetic energy of the
particles is larger than the mean inter-particle potential, i.e.
| *e*|
<< *T*_{0} in the language of Eq. (4.1).