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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 T0 and with mean particle density n0. Suppose, for simplicity, local thermodynamical equilibrium (11). The charge densities will be given, respectively, by

Equation 4.1 (4.1)

where phi is the electrostatic potential. In Eqs. (4.1) it has been assumed that the plasma is weakly coupled, i.e. |e phi / T0| << 1. A test charge qt located in the origin will then experience the electrostatic potential following from the Poisson equation,

Equation 4.2 (4.2)

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

Equation 4.3 (4.3)


Equation 4.4 (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/lambdaD.

In the interstellar medium there are three kinds of regions which are conventionally defined, namely H2 regions (where the Hydrogen is predominantly in molecular form), H0 regions (where Hydrogen is in atomic form) and H+ regions (where Hydrogen is ionized). Sometimes H+ regions are denoted with HI and H2 regions with HII. In the H+ regions the typical temperature T0 is of the order of 10-20 eV while for n0 let us take, for instance, n0 ~ 3 × 10-2 cm-3. Then lambdaD ~ 30 km.

For r >> lambdaD 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 alphaem2 / T02 and the mean free path will be ellmfp ~ T02 / (alphaem2 n0), i.e. recalling Eq. (4.4) lambdaD << ellmfp. 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

Equation 4.5 (4.5)

where me 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 vther appeq (T0 / me)1/2 is the thermal velocity of the charge carriers, the collision frequency omegac appeq vther / ellmfp is always much smaller than omegap appeq vther / lambdaD. Thus, in the idealized system described so far, the following hierarchy of scales holds:

Equation 4.6 (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 = n0-1 lambdaD-3, i.e. the number of particles in the Debye sphere. In the approximation of weakly coupled plasma N << 1 which also imply that the mean kinetic energy of the particles is larger than the mean inter-particle potential, i.e. | ephi| << T0 in the language of Eq. (4.1).

11 This assumption is violated in the realistic situations. Back.

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