Fundamentals of Plasma Physics

3.6 Relation of Drift Equations to the Double Adiabatic MHD Equations 93

the dashed curve), then the two opposing currents do not cancel and a net macroscopic cur-
rent appears toflow around the dashed curve, even thoughnoactual particlesflow around
the dashed curve. Inequality of the numbers of inside and outside particles corresponds to
a density gradient and so we see that a radial density gradient of gyrating particles gives a
net macroscopic azimuthal current. Similarly, if there is a radial temperature gradient, the
velocities of the inner and outer groups differ, resulting again in an apparent macroscopic
azimuthal current. The combination of density and temperature gradients is such that the
net macroscopic current depends on the pressure gradient as given by Eq.(3.128).
Taking diamagnetic current into account is critical for establishing acorrespondence
between the single particle drifts and the MHD equations, and having recognized this,
we are now in a position to derive this correspondence. In order for the derivation to be
tractable yet non-trivial, it will be assumed that the magnetic field is time-independent, but
the electric field will be allowed to depend on time. It is also assumed that thedominant
cross-field particle motion is thevE=E×B/B^2 drift;this assumption is consistent with
the hierarchy of particle drifts (i.e., polarization drift is a higher-order correction tovE).
Because both species have the samevE, no macroscopic current results fromvE, and
so all cross-field currents must result from the other, smaller drifts, namelyv∇B,vc,and
vp.Let us now add the magnetization current to the currents associated withv∇B,vc,and
vpto obtain the total macroscopic current

Jtotal =JM+J∇B+Jc+Jp=JM+

∑

σ

nσqσ(u∇B,σ+uc,σ+up,σ) (3.129)

whereJ∇B,Jc,Jpare currents due to gradB,curvature, and polarization drifts respec-
tively andu∇B,σ,uc,σandup,σare the mean (i.e.,fluid) velocities associated with these
drifts. These currents are explicitly:

1. gradBcurrent

J∇B =

∑

σnσqσu∇B,σ

= −

∑

σ

mσnσqσ

〈

v^2 ⊥σ

〉

2 B

∇B×B

qσB^2

=−P⊥

∇B×B

B^3

(3.130)

2. curvature current

Jc =

∑

σnσqσuc,σ

= −

∑

σnσqσmσ

〈

v^2 ‖σ

〉Bˆ·∇Bˆ×B

qσB^2

=−P‖

Bˆ·∇Bˆ×B

B^2

(3.131)

3. polarization current

Jp=

∑

σ

nσqσup,σ=

∑

σ

nσqσ

(

mσ

qσB^2

dE⊥

dt

)

=

ρ

B^2

dE⊥

dt

. (3.132)

Because the magnetic field was assumed to be constant, the time derivative ofvEis the
only contributor to the polarization drift current.