Fundamentals of Plasma Physics

2.5 Magnetohydrodynamic equations 51

whereP⊥^2 D is the 2-D perpendicular pressure, and has dimensions of energy per

unit area, whileρ 2 Dis the 2-D mass density and has dimensions of mass per unit

area. Thus,ρ 2 D∼ 1 /Aso thatP⊥^2 D∼ρ 2 DT⊥ ∼T⊥/Ain which case Eq.(2.88)

can be re-written as

const.=

P⊥^2 D

ρ^22 D

∼T⊥A∼

(

1

LA

)

T⊥

LA

B

(2.89)

or

P⊥

ρB

=const. (2.90)

Equations (2.87) and (2.90) are called the double adiabatic or CGL laws afterChew,

Goldberger and Low (1956) who first developed them using a Vlasov analysis).

B

A L

Figure 2.5: Magneticflux tube withfluxΦ =BA.

Single adiabatic law If collisions are sufficiently frequent to equilibrate the perpen-
dicular and parallel temperatures, then the pressure tensor becomes fullyisotropic and the
dimensionality of the system isN= 3so thatγ= 5/ 3 .There is now just one pressure and
temperature and the adiabatic relation becomes

P

ρ^5 /^3

=const. (2.91)

2.5.6 MHD approximations for Maxwell’s equations

The various assumptions contained in MHD lead to a simplifying approximation ofMaxwell’s
equations. In particular, the assumption of charge neutrality in MHD makes Poisson’s
equation superfluous because Poisson’s equation prescribes the relationship between non-
neutrality and the electrostatic component of the electric field. The assumption of charge
neutrality has implications for the current density also. To see this, the2-fluid continuity