In fluid description of
plasmas (MHD) one does not consider velocity distributions. It is about
number density, flow velocity and pressure. This is about moment or fluid
equations (as NSE and Boltzmann/Landau equations).

The
“mother" of all hydrodynamic models is the continuity equation treating
observations with macroscopic character, where fluids and gases are considered
as continua. The corresponding infinitesimal volume “element” is a volume,
which is small compared to the considered overall (volume) space, and large
compared to the distances of the molecules. The displacement of such a volume
(a fluid particle) then is a not a displacement of a molecule, but the whole
volume element containing multiple molecules, whereby in hydrodynamics this
fluid is interpreted as a mathematical point.

One of
the key differentiator between plasma to neutral gas of neutral fluid is
the fact that its electrically positively and negatively charged
particles are strongly influenced by electric and magnetic fields, while
neutral gas is not.

An ideal plasma is a non-dissipative flow of the incompressible charged particles (CaF).

The MHD equations are derived from continuum theory of
non-polar fluids with three kinds of balance laws:

(1) conservation of mass

(2) balance of linear momentum

(3) balance of angular momentum (Ampere law and Faraday law).

The
MHD equations consists of 10 equations with 10 parameters accompanied
with appropriate boundary conditions from the underlying Maxwell
equations (CaF).

In (EyG) it is proven that smooth solutions of
non-ideal (viscous and resistive) incompressible magneto-hydrodynamic (plasma
fluid) equations satisfy a stochastic (conservation) law of flux. It is shown
that the magnetic flux through the fixed Plasma is an ionized gas
consisting of approximately equal numbers of positively charged ions and
negatively charged electrons.

References (BrK) Braun K., 3D-NSE, YME, GUT solution, 2019

(CaF) Cap F., Lehrbuch der Plasmaphysik und Magnetohydrodynamik, Springer-Verlag, Wien, New York, 1994

(EyG) Eyink G. L., Stochastic Line-Motion and Stochastic
Conservation Laws for Non-Ideal Hydrodynamic Models. I. Incompressible Fluids
and Isotropic Transport Coefficients, arXiv:0812.0153v1, 30 Nov 2008

(HaW) Hayes W. D., An alternative proof of the circulation, Quart. Appl. Math. 7 (1949), 235-236

(SeW) Sears W. R., Resler E. L., Theory of thin airfoils in fluids of high electrical conductivity, Journal of Fluid Mechanics, Vol. 5, Issue 2 (1959), 257-273