Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 55

Now let

V

⊥ =

W D + u

where

qB 2

W D = ( F

⊥ ×

B )/ (2.53)

This substitution is motivated by the result we already have that W D =
(

B 2 when F

× )/

⊥ =

E B q E . Substituting (2.53) into (2.52) yields

F

B 2 M ∂ W D /∂ t + M ∂ u /∂ t =

⊥ ×

B / × B + q u × B + F

⊥

The first term on the left vanishes, while the first term on the right-hand side
equals

− F and thus cancels the other F term. This leaves

⊥ ⊥

M ∂ u /∂ t = q ( u × B )

The solution to this, of course, is just the gyromotion at frequency

= qB / M

The interpretation we make is that in a frame moving at W D , the particle
motion is pure gyration. This yields the concept of a guiding center motion, since
W D gives the velocity of the center of gyromotion. Some examples of guiding
center drifts due to various forces are as follows. For an electric field, F = q E and

qB 2 B 2 W D = q E × B / = E × B /

For the gravitational field, F = M g and

qB 2 W D = M g × B /

B 2 , For the inertial force F =− M ∂ W D /∂ t and, letting W D = E × B /

qB 2 M t E B 2 B qB 2

W D = ( F × B )/ = − 1 / ∂/∂ × B / ×

M

qB 2

W D =− / (∂ E /∂ t × B ) × B

M

qB 2

W D = / (∂ E /∂ t )

Notice that this expression can be related to a displacement current in the plasma,
since using J = ( ne W Di − ne W De ) and M m yields

ne M eB 2

nM

B 2

J = / ∂ E /∂ t = / ∂ E /∂ t