The preceding analysis leads to a compact space-time description for the particle or mass density. Of particular interest to us now is a characterization of collective motion in a plasma. In such a description, the identity of individual particles is put aside and the plasma is characterized by the space-time changes of macroscopic variables such as bulk speed, temperature, and pressure defined in a fluidic context.

To begin with we take the Continuity Equation, Eq.(6.12), as a necessary equation for the particle density everywhere in space and time. That is, if another equation is developed in which Nj(r, t) appears as an independent function, both equations must hold simultaneously and any solution methodology is expected to involve both equations.

To be specific, we consider a space-time ensemble of Nj particles each of
mass rrij and charge qj to which we assign an average velocity Vj over an infinitesimal volume containing Nj. A simplified one-dimensional analogy is the case of freeway traffic with cars moving at different speeds, all together amounting to some average speed of traffic motion which, nonetheless, may vary with time and location.

In order to provide some continuity to our preceding analysis, we suppose that an electric field E and magnetic field В act on this moving space-time ensemble of Nj particles, referred to as a fluid element. Its equation of motion is

d у, / ч

Njmj~dt = NjqJE+ Ni4j{VixB)’ J = i’e■ (6.21)

Recall that Nj used here is determined by Eq.(6.12) everywhere in space and time.

The two terms on the right hand side possess an easily established interpretation; clearly, the second term c)Vj/c)t is simply the acceleration of the Nj ensemble at a fixed coordinate point; then, the first term incorporates the process of spatial variation in the ensemble velocity much as in our one-dimensional analogy the traffic in one section of the freeway may move faster than that in another section; the term convection is often used for this kind of fluid motion. We insert

Eq.(6.23b) into Eq.(6.21) and again use Pj = Nj mj to obtain a corresponding equation for the ensemble’s motion

= P-E + P-(V, XB)

where the mass density Pj satisfies the Continuity Equation, Eq.(6.20), and pjC(r, t) = Nj(r, t)qj is the respective charge density. Note that, because of electromagnetic interactions in a plasma, PjC, E and В are all related via Maxwell’s Equation which, simultaneously, must also hold with Eq.(6.24).

At this juncture, we recognize another process which is generally important in a hot plasma: the thermal motion of particles makes them randomly enter and exit the fluid element considered, leading to local pressure gradients which act as an additional force in Eq.(6.24). While, in general, this force can be obtained

wherein the local particle pressures are related to the respective mass density via the Equation of State

where C is a constant and 7j is the ratio of specific heats for constant pressure and constant volume (7j = Cp/Cv) referring to the j-th species.

An important point must be emphasized that concerns the electric and magnetic fields, E and B, when the above set of equations is evaluated for ions and electrons. As noted previously, it is required that Maxwell’s Equations be satisfied for the medium of interest. The complete set of resultant equations for Pi, Pe, Vj, Ve, pi, pe, E, В-consisting of 16 simultaneous scalar equations — represents what is known as a self-consistent description of the plasma fluid approximation by the so called magnetohydrodynamic (MHD) equations. The imposition of initial and boundary conditions clearly leads to a nontrivial problem description.