Fundamentals of Plasma Physics

140 Chapter 4. Elementary plasma waves

the boxed terms (this mode is called theEzmode since it has finiteEz) and the other
distinct mode consists solely of interrelationships between the unboxed terms (this mode
is called theBzmode since it has finiteBz). Since the modes are decoupled, it is possible
to “turn off” theEzmode when considering theBzmode and vice versa. If the plasma is
non-uniform, theEzandBzmodes can become coupled.
The ideal MHD formalism sidesteps discussion of theEzmode. Instead, two discon-
nected assumptions are invoked in ideal MHD, namely (i) it is assumed thatEz 1 =0and
(ii) the parallel currentJz 1 is assumed to arrange itself spontaneously in such a way as
to always satisfy∇·J 1 =0.This pair of assumptions completes the system of equa-
tions, but omits the parallel dynamics associated with theEzmode and instead replaces
this dynamics with an assumption thatJz 1 is determined by some unspecified automatic
feedback mechanism. In contrast, the two-fluid equations describe how particle dynamics
determines the relationship betweenJz 1 andEz 1 .Thus, while MHD is both simpler and
self-consistent, it omits some vital physics.

The two-fluid model is based on the linearized equations of motion

mσn

∂uσ 1

∂t

=nqσ(E 1 +uσ 1 ×B)−∇·Pσ 1. (4.94)

Charge neutrality is assumed so thatni=ne=n.Also, the pressure term is

∇·Pσ 1 =∇·





Pσ⊥ 1 00

0 Pσ⊥ 1 0

00 Pσz 1



=∇⊥Pσ⊥ 1 +ˆz∂Pσz^1

∂z

. (4.95)

Assumingω << ωciimpliesω << ωcealso and so perpendicular motion can be
described by drift theory for both ions and electrons. However, here the drift approximation
is used for thefluid equations, rather than for a single particle. Following the drift method
of analysis, the left hand side of Eq.(4.94) is neglected to first approximation, resulting in

uσ 1 ×B≃−E 1 ⊥+∇⊥Pσ⊥ 1 /nqσ (4.96)

which may be solved foruσ⊥ 1 to give

uσ⊥ 1 =

E 1 ×B

B^2

−

∇Pσ⊥ 1 ×B

nqσB^2

. (4.97)

The first term is the single-particleE×Bdrift and the second term is called the
diamagnetic drift, afluid effect that does not exist for single-particle motion. Because
uσ⊥ 1 is time-dependent there is also a polarization drift. Recalling that the form of the
single-particle polarization drift for electric field only isvp = mE ̇ 1 ⊥/qB^2 and using
E 1 ⊥−∇⊥Pσ⊥ 1 /nqσfor thefluid model instead of justE 1 ⊥for single particles (cf. right
hand side of Eq.(4.96)) thefluid polarization drift is obtained. With the inclusion of this
higher order correction, the perpendicularfluid motion becomes

uσ⊥ 1 =

E 1 ×B

B^2

−

∇Pσ⊥ 1 ×B

nqσB^2

+

mσ

qσB^2

E ̇ 1 ⊥− mσ

nq^2 σB^2

∇⊥P ̇σ⊥ 1. (4.98)

The last two terms are smaller than the first two terms by the ratioω/ωcσand so may
be ignored when the electron and ionfluid velocities are considered separately. However,