Collisional vs. Collisionless Plasma - Not A real Problem!

The aurora over interior Alaska: The plasma here is primarily collisionless. 

 Much ado is often made in plasma physics about the need to distinguish "collisional" from "collisionless" plasmas. In large measure, I suspect much of the confusion originates because people aren't clear about how, for example, the simplified MHD (magneto-hydrodynamic) regime is found from simplfying two fluid theory then one-fluid theory. Detailed bases for the classification of differing plasma fluid regimes is far beyond the scope of this blog, but interested readers can find any number of good introductory plasma physics text that delve into details (e.g. 'Introduction to Plasma Physics' by Francis F. Chen). 

The bottom line overview is that one proceeds by taking moments of the Boltzmann equation. E.g. the Boltzmann eqn. is:

 ¶ f/ ¶ t + v Ñ z f + (F/ m ) Ñ v f = (¶ f/ ¶ t)_c

 The first moment, which yields a 'two-fluid' separated (e.g. electron-ion) medium is obtained by integrating the above eqn. with F = q/m (E + v X B). If one then assumes a sufficiently hot plasma so it's collisionless, the term on the RHS,  (¶ f/ ¶ t)_c -> 0. This is the Vlasov equation, with space position x included: 

¶ f (x, v, t)  +   v · Ñ x  f _o  +   (q/ m) (E + v/ c  x B) Ñ v · f _o   = 0

 The 2nd moment is obtained by multiplying the original eqn. (Boltzmann) by mv then integrating it over dv. Anyway, the progression by using this procedure is that one gets in succession:

 Two -fluid theory (e.g. ions and electrons treated as a separate fluids) 

With the normalization: n _o  (x, t) =  ò  dv f _o  (x, v, t)   from which the 2 fluid velocity equations are obtained., i.e.

1) (¶ r _a  / ¶ t)  +  Ñ (r _a v _a) = 0

where the index  a = e, i.

for electrons, ions.   

2)  r _a (¶  V _o / ¶ t)  +    r v a · Ñ ( v a) =   -  Ñ p _a +  e _a n _a  (E + v a x B)

3)  p _o  r _a  g    =  const.

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One fluid theory (introducing low frequency, long wave length and quasi -neutral approximations, e.g.   n _e=    n _i

  (¶ r  / ¶ t)  +  Ñ (r v) = 0

where no separate  index for  e, i, is needed.  Then, more quickly to...
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MHD Theory

Ideal MHD theory case revolves around a version of Ohms's law with J X B = 0 so can be written :

 J  = s (E + v X B)

where s  is the conductivity One reason I don't like the collisional v. collisionless presentations as typically given is that people often forget that one can have two forms of MHD: the more traditional(ideal - or collisionless) or kinetic with one dimensional particle motion along field lines with 2-dimensional fluid theory). Plasma motions for both types are confined to parallel displacements (e.g. to B-field) for what we call magnetosonic waves. While motion perpendicular to B incites Alfven waves.

 In this sense, only magnetosonic waves are of interest if one is considering the collisional regime. But it one is restricting interest to Alfven dynamics solely (i.e. perpendicular motion) then technically the MHD eqns. are a reasonable approximation to a collisionless plasma. (In any case, to test this is so, one will always compute the plasma beta = 

(½  r  v ²)/   B ²/2 m ₀ 

 where  m ₀  is the magnetic permeability of free space, B is the field strength in Tesla(T), and p is the pressure, in Pa. If then, the beta << 0, the Ohm's law becomes: E + v X B = 0, so the plasma is sufficiently "collisionless". (By contrast, if one has E + v X B = nJ, one has resistive MHD).sed at all? Maybe the authors of future plasma physics texts can examine these issues in more detail before the publication of their next (or first!) editions.