52 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD
equations is multiplied byqσand then summed over species to obtain the charge conserva-
tion equation
∂
∂t
(∑
nσqs
)
+∇·J= 0. (2.92)
Thus, charge neutrality implies
∇·J= 0. (2.93)
Let us now consider Ampere’s law
∇×B=μ 0 J+μ 0 ε 0
∂E
∂t
. (2.94)
Taking the divergence gives
∇·J+ε 0
∂∇·E
∂t
= 0
which is equivalent to Eq.(2.92) if Poisson’s equation is invoked.
Finally, MHD is restricted to phenomena having characteristic velocitiesVphslow com-
pared to the speed of light in vacuumc= (ε 0 μ 0 )−^1 /^2 .Againtis assumed to represent the
characteristic time scale for a given phenomenon andlis assumed to represent the cor-
responding characteristic length scale so thatVph ∼l/t. Faraday’s equation gives the
scaling
∇×E=−
∂B
∂t
=⇒E∼Bl/t. (2.95)
On comparing the magnitude of the displacement current term in Eq.(2.94) tothe left hand
side it is seen that
μ 0 ε 0
∣
∣
∣
∣
∂E
∂t
∣
∣
∣
∣
|∇×B|
∼
c−^2 E/t
B/l
∼
(
Vph
c
) 2
. (2.96)
Thus, ifVph << cthe displacement current term can be dropped from Ampere’s law
resulting in the so-called “pre-Maxwell” form
∇×B=μ 0 J. (2.97)
The divergence of Eq. (2.97) gives Eq.(2.93) so it is unnecessary to specifyEq.(2.93)
separately.
2.6 Summary of MHD equations
We may now summarize the MHD equations:
- Mass conservation
∂ρ
∂t
+∇·(ρU) = 0. (2.98)
- Equation of state and associated equation of motion
(a) Single adiabatic regime, collisions equilibrate perpendicular and parallel tem-
peratures so that both pressure and temperature are isotropic
P
ρ^5 /^3
=const. (2.99)