Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 42

neutral wind U . We use the subscript notation again for brevity, where j stands
for i (the single-ion gas) or e (the electron gas). Since V j = V j − U , (2.28)

becomes

nq j V j × B

nq j E + nM j v jn V j 0 =− k B T j ∇ n + nM j g + − (2.33)

where everything is expressed in the moving reference frame (note that n and

g are unchanged in a nonrelativistic transformation). If we divide through by
nM j v jn and gather terms, this can be written
V j − κ j V j × B

∇

b j E + D j / H j ˆ

W j =− D j ∇ n / n + g ≡ (2.34)

B is a unit vector in the B direction,

where g is a unit vector in the g ˆ

direction, M j v jn ),

which has the same sign as the particle charge, D j is the diffusion coefficient
( k B T j /

κ j is the ratio of gyrofrequency to collision frequency ( q j B /

M j v jn ), which also has the algebraic sign
of q j , and H j is the scale height ( k B T j /

M j v jn ) , b j is the mobility ( q j /

. Notice that the velocity W j is the
fluid velocity that would arise in an unmagnetized plasma subject to the same
forces.
The absolute value of

M j g )

κ j determines whether a particle does or does not make a
cycle about the magnetic field before a collision takes place. For a small absolute
value of κ j , many collisions occur and the particle basically moves parallel to the

applied forces as if there were no magnetic field. This is illustrated in Fig. 2.4a
for ions and electrons subject to an electric field. The collisionless case (

infinite)
is shown in Fig. 2.4b for particles initially at rest. After about one gyroperiod
the particles are moving at right angles to the electric field. In this important case
the final velocity is identical for ions and electrons and equal to E

κ

B 2 . For × B /

1 (Fig. 2.4c), the net motion is at a 45 ◦ angle to the an absolute value of κ =

electric field.
These results can be seen analytically from (2.34). Consider first the case of a
very high collision frequency ( κ j 1). Then the first term on the left-hand side

dominates and

+ D j / H j ˆ

V j = W j = b j E − D j ∇ n / n g (2.35)

which is the same as the fluid velocity that would arise in the case of an unmag-
netized plasma. The velocity is parallel to the forces. For

j very large (a “colli-
sionless” plasma), the component of (2.34) parallel to B is unchanged:

κ

V j

|| = b j E −

+ D j / H j ˆ g · B

D j ∇ n / n (2.36a)