Next: Closure in Collisionless Magnetized
Up: Plasma Fluid Theory
Previous: MHD Equations
Drift Equations
The drift equations take the form:
and
In the drift limit, the motions of the electron and ion fluids are
sufficiently different that there is little to be gained in rewriting the
drift equations in terms of the centre of mass velocity and the plasma
current.
Instead, let us consider the
components of Eqs. (392) and (396):
In the above equations, we
have neglected all terms for the
sake of simplicity.
Equations (397)(398) can be inverted to give
Here,
is the
velocity, whereas

(401) 
and

(402) 
are termed the electron diamagnetic velocity and the ion diamagnetic
velocity, respectively.
According to Eqs. (399)(400), in the drift approximation the velocity of the electron
fluid perpendicular to the magnetic field is the sum of the
velocity and the electron diamagnetic velocity. Similarly, for the ion fluid.
Note that in the MHD approximation the perpendicular velocities of the
two fluids consist of the
velocity alone, and are,
therefore, identical to lowest order. The main difference between the
two ordering lies in the assumed magnitude of the electric field. In the
MHD limit

(403) 
whereas in the drift limit

(404) 
Thus, the MHD ordering can be regarded as a strong (in the sense
used in Sect. 2) electric field
ordering, whereas the drift ordering corresponds to a weak electric
field ordering.
The diamagnetic velocities are so named because the diamagnetic
current,

(405) 
generally acts to reduce the magnitude of the magnetic field inside
the plasma.
The electron diamagnetic velocity can be written

(406) 
In order to account for this velocity, let us consider a simplified
case in which the electron temperature is uniform, there is a
uniform density gradient running along the direction, and the magnetic
field is parallel to the axissee Fig. 7.
The electrons gyrate in the  plane in circles of radius
. At a given point, coordinate , say, on the axis,
the electrons that come from the right and the left have traversed distances
of order . Thus, the electrons from the right originate from
regions where the particle density is of order
greater than the regions from which the electrons from the left originate.
It follows that the directed particle flux
is unbalanced, with slightly more particles moving in the direction
than in the direction. Thus, there is a net particle flux
in the direction: i.e., in the direction of
.
The magnitude of this flux is

(407) 
Note that there is no unbalanced particle flux in the direction, since the
directed fluxes are due to electrons which originate from regions
where . We have now accounted for the first term on the
righthand side of the above equation.
We can account for the second term using similar arguments.
The ion diamagnetic velocity is similar in magnitude to the electron
diamagnetic velocity, but is oppositely directed, since ions gyrate
in the opposite direction to electrons.
Figure 7:
Origin of the diamagnetic velocity in a magnetized plasma.

The most curious aspect of diamagnetic flows is that they represent fluid flows
for which there is no corresponding motion of the particle guiding
centres. Nevertheless, the diamagnetic velocities are real fluid
velocities, and the associated diamagnetic current is a real current.
For instance, the diamagnetic current contributes to force balance inside the plasma,
and also gives rise to ohmic heating.
Next: Closure in Collisionless Magnetized
Up: Plasma Fluid Theory
Previous: MHD Equations
Richard Fitzpatrick
20110331