3.5 Drift equations 77
fields, the Lorentz equation becomes
m
d[vgc(t) +vL(t)]
dt
= q
[
E(xgc(t)) + (rL(t)·∇)E
]
+q[vgc(t) +vL(t)]×
[
B(xgc(t)) + (rL(t)·∇)B
]
.
(3.62)
The gyromotion (i.e., the fast cyclotron motion) is defined to be the solution of the equa-
tion
m
dvL(t)
dt
=qvL(t)×B(xgc(t)); (3.63)
subtracting this fast motion equation from Eq.(3.62) leaves
m
dvgc(t)
dt
= q
[
E(xgc(t)) + (rL(t)·∇)E
]
+q
{
vgc(t)×
[
B(xgc(t)) + (rL(t)·∇)B
]
+vL(t)×(rL(t)·∇)B
}
.
(3.64)
Let us now average Eq.(3.64) over one gyroperiod in which case termslinearin gyromotion
average to zero. What remains is an equation describing the slow quantities, namely
m
dvgc(t)
dt
=q
{
E(xgc(t))+vgc(t)×B(xgc(t)) +〈vL(t)×(rL(t)·∇)B〉
}
(3.65)
where〈〉means averaged over a cyclotron period. The guiding center velocity cannow be
decomposed into components perpendicular and parallel toB,
vgc(t) =v⊥gc(t) +v‖gc(t)B̂ (3.66)
so that
dvgc(t)
dt
=
dv⊥gc(t)
dt
+
d
(
v‖gc(t)B̂
)
dt
=
dv⊥gc(t)
dt
+
dv‖gc(t)
dt
B̂+v‖gc(t)d
B̂
dt
.(3.67)
Denoting the distance along the magnetic field bys, the derivative of the magnetic field
unit vector can be written, to lowest order, as
dB̂
dt
=
∂Bˆ
∂s
ds
dt
=v‖gcB̂·∇B,̂ (3.68)
so Eq.(3.65) becomes
m
[
dv⊥gc(t)
dt
+
dv‖gc(t)
dt
B̂+v^2
‖gc
B̂·∇B̂
]
= qE(xgc(t))
+qvgc(t)×B(xgc(t))
+q〈vL(t)×(rL(t)·∇)B〉.
(3.69)
The component of this equation alongBis
m
dv‖gc(t)
dt
=q
[
E‖(xgc(t)) +〈vL(t)×(rL(t)·∇)B〉‖