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Figure 2.3 shows the path of a positively charged particle in this case. Assuming
that the gradient is on scale lengths long compared with the gyroradius, i.e.,
![\begin{displaymath}r_{L} \ll L = \left( \vert\nabla B\vert / \vert{\bf B}\vert \right)^{-1} \ ,
\end{displaymath}](img48.gif) |
(2.17) |
the orbit
is almost circular but does not quite close. Since
,
rL
will be slightly smaller where B is larger and slightly larger where B is smaller.
This causes the orbit to drift in the direction of its gyromotion where rL is
larger, perpendicular to both
and to
.
A negatively charged
particle drifts in the opposite direction. The physics is thus clear.
Quantitatively we write
![\begin{displaymath}{\bf B}({\bf r} = {\bf x}_{c}) = {\bf b} \left( B(0) + {\bf x}_{c} . \nabla B(0) \right) \ ,
\end{displaymath}](img50.gif) |
(2.18) |
where
represents the unperturbed gyromotion and
is the unit vector
of the magnetic field. The perpendicular equation
of motion is
![\begin{displaymath}\frac{d {\bf v}_{\perp}}{d t} = \Omega_{c} {\bf v}_{\perp} \t...
...b}
\left( 1 + \frac{{\bf x}_{c} . \nabla B}{B(0)} \right) \ .
\end{displaymath}](img53.gif) |
(2.19) |
The particle velocity is now written as
,
where
represents the unperturbed gyromotion and
is the sum of the drift
velocity and any first order perturbations to the gyromotion. Substituting into (2.19),
grouping the zeroth order terms and deleting them, and
ignoring the second order term
,
the
first order equation becomes
![\begin{displaymath}\frac{d {\bf v}_{\perp, 1}}{d t} = \Omega_{c} \left( {\bf v}_...
...bla B / B_(0)
+ {\bf v}_{\perp, 1} \times {\bf b} \right) \ .
\end{displaymath}](img58.gif) |
(2.20) |
This equation is next averaged over a gyroperiod and
is identified as the
constant drift velocity
,
so that the time derivative becomes zero and the drift
velocity obeys the equation
![\begin{displaymath}{\bf v}_{D} \times {\bf b} = - < {\bf v}_{c} \times {\bf b}\ \frac{ {\bf x}_{c} . \nabla B}{B(0)} > \ .
\end{displaymath}](img59.gif) |
(2.21) |
Now
and the time-average
of the righthand term simplifies considerably since the x component of the
term
becomes
with a similar result for the y component. That is,
![\begin{displaymath}{\bf V}_{D} \times {\bf b} = \frac{ r_{L}^{2} \Omega_{c}}{2}\ \frac{\nabla B}{B} \ .
\end{displaymath}](img66.gif) |
(2.24) |
Rearranging, the final result is
The
drift speed therefore depends on the charge, mass, and perpendicular energy of the
particle, as well as on the magnetic field strength and the scale length of the gradient. This
drift can therefore cause currents and charge separations in the plasma. Moreover, as seen below,
the combination of a convection electric field and a
plasma drift can lead to particle
acceleration.
Next: Curvature drifts
Up: Motion in non-uniform magnetic
Previous: Motion in non-uniform magnetic
Iver Cairns
1999-08-04