Geoscience Reference
In-Depth Information
−
1
and length scales greater than
r
g
the guiding center perpendicular drift equation for particle motion in a mag-
netic field is given by
To summarize, for time scales greater than
E
B
2
M
qB
2
d
E
M
qB
2
g
B
W
D
=
×
/
+
/
/
+
/
×
B
dt
1
qB
2
MV
2
2
B
MV
2
qRB
2
B
n
+
/
⊥
/
(
B
×∇
B
)
+
||
/
׈
(2.54)
Since the particle motions just described are quite complex, it has proved
useful to develop an intuition based on parameters that are nearly conserved
in the motion. The most “rugged” of these adiabatic invariants is the particle
magnetic moment
μ
. Here, we prove that if the time scale for field changes is
τ
g
=
−
1
(the gyroperiod), then
μ
larger than
is conserved. We refer the reader
to Schmidt (1966) for proof that conservation also holds if the length scale for
changes in
B
is much greater than
r
gi
(the ion gyroradius).
If
B
is uniform in space but time varying,
K
B
2
∂μ/∂
t
=
∂/∂
t
(
K
⊥
/
B
)
=
(∂
K
⊥
/∂
t
)/
B
−
(∂
B
/∂
t
)
⊥
/
The rate of change of perpendicular energy in the guiding center approximation
can be estimated by the energy gained in one gyration divided by the time the
gyration takes. Thus,
=
K
g
/τ
g
K
⊥
/
t
and
q
K
g
=
F
·
d
l
=
(
E
+
V
×
B
)
·
d
l
where the integral is around one gyroloop of the particle motion. We note that
d
l
=
V
dt
so
(
V
×
B
)
·
d
l
=
(
V
×
B
)
·
V
dt
=
0
This is an example of the fact that magnetic forces do not change particle energy.
The energy change in one gyration is thus
K
g
=
q
E
·
d
l
We can transform this to a surface integral
q
K
g
=
(
∇×
E
)
·
d
a
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