= 0)
Effects of an Inhomogeneous Magnetic Field (E
For some purposes the motion of the “guiding” centers can be taken as a good approximation
of that of the particles. But it must be recognized that during the particle’s Larmor gyra or E
may be different from what they are at the guiding
tions, it “samples” regions where B
center (GC). For the magnetic field, the effects are important, and they are to be studied here.
We decompose formally the particle’s velocity v into a “Guiding Center” part vGC and a
plus a component
Larmor part vL . In turn, the GC velocity has a component v|| along B,
(a “drift”) perpendicular to B:
v = vGC + vL
(1)
vGC = v|| + vD
The Larmor velocity is defined by,
m
dvL
GC
= qvL × B
dt
vL|| = 0
(2)
has been explicitly taken to be at the particle’s Guiding Center.
where B
We next assume the plasma is magnetized with respect to the particles in question, so that
the Larmor radius is “small”:
changes appreciably
RL << Distance over which B
(3)
We also assume (to be verified later) that the drift velocities are small compared to those
involved in Larmor motion:
vD << vL
(4)
r) (r being the instantaneous location of the particle, measured from the
Because of (3), B(
GC :
GC) can be expanded to first order about B
r) ∼
GC + (r · ∇)B
B(
= B
(5)
The full equation of motion is then,
m
d(v + vD + vL )
CG + (r · ∇))B
= q(v + vD + vL ) × (B
dt
(6)
GC = 0 (parallel vectors). The term vD × (r · ∇)B
Of the terms on the right, clearly v|| × B
is of second order (product of small terms), and will be neglected.
GC ) cancels the mdvL /dt on the left, according to (2).
The term q(vL × B
This leaves,
dv||
dvD
q + vD × B
GC + vL × (r · ∇)B
+
=
v|| × (r · ∇)B
dt
dt
m
1
(7)
GC
B
b
Rc n
line through the GC has a radius of curvature Rc ,
If the B
then,
v||2
dv||
dv||
=
b +
n
(8)
dt
dt
Rc
and along the
where b and n are unit vectors tangent to B
line, respectively.
inner normal to the B
We can now separate out the components of Eq.(7) along B
(⊥)
() and perpendicular to B
Along B:
dv||
q =
vL × (r · ∇)B
dt
m
||
(9)
q dvD
=
n +
v|| × (r · ∇)B + vD × BGC + [vL × (r · ∇B)]⊥
Rc
dt
m
(10)
Perpendicular to B:
v||2
The next step is to average these equations over each “quasi-orbit” about the GC (“quasi”,
because, in an inhomogeneous field the Larmor orbits may not quite close on themselves GC are regarded as constant, but r
we ignore that). During each such orbit, v|| , vD and B
and vL vary cyclically. Because of this, any term where r and vL appear linearly will average
to zero, but care must be taken when they appear in pairs. In particular,
v|| × (r · ∇)B
= 0
(11)
Larmor
in the formulation (even with some inhomogeneity), this averagNote: If we had retained E
(no ∇E
drift terms exist).
ing step would have eliminated any first order ∇E
z
x n
b
y
To see what the averaging results are for the other terms, we
now re-write (9) and (10) in a local Cartesian coordinate frame
as shown in the figure. The y direction completes a righthanded set. The corresponding unit vector is the bi-normal
unit vector
= b × n
bn
(12)
Writing out Eq. (9):
dvz
∂By
q
∂Bx
∂By
∂Bx
+y
=
vLx x
−vLy x
+y
∂x
dt
m
∂y
∂x
∂y
Larmor
(13)
Projecting Eq. (10) along the normal (x) direction,
vz2
∂Bz
q
dvDx
∂Bz
+y
=
vDy Bz + vLy x
+
∂x
Rc
dt
m
∂y
Larmor
2
(14)
Projecting Eq. (10) along the bi-normal (y),
dvDy
∂Bz
q
∂Bz
+ y
=
− vDx Bz −
vLx x
∂x
dt
m
dy
Larmor
(15)
The Larmor motion is shown in the figure for a positive q. If
the Larmor radius is rL , we have
y
B
ωc t
x
r
x = rL cos ωc t
y = −rL sin ωc t
(16)
vLx = −ωc rL sin ωc t
vLy = −ωc rL cos ωc t
(17)
where ωc = qB
is the gyro frequency. Therefore, when doing
m
the averaging,
vL
1
xvLy L = −ωc rL2 cos2 ωc t L = − ωc rL2
2
2
yvLy L = ωc rL sin ωc t cos ωc tL = 0
(18)
xvLx L = −ωc rL2 sin ωc t cos ωc tL = 0
1
yvLx L = ωc rL2 sin2 ωc t L = ωc rL2
2
Notice the occurrence of the group
q 1
ω r2 .
m2 c L
From
q 1
ωc r 2 =
m2 L
where μ is the magnetic moment μ =
dvDz
dt
1 2
v
2 ⊥
B
1
2
mv⊥
/B.
2
q
m
=
=
ωc
B
and ωc rL = v⊥ , this group is,
μ
m
(19)
We now have from (13), (14), and (15),
v 2 ∂Bz
dvDx
v2
= − z + ωc vDy − ⊥
dt
Rc
2B ∂x
dvDy
v 2 ∂Bz
= −ωc vDx − ⊥
dt
2B ∂y
2
∂Bx ∂By
v 2 ∂Bz
v
= − ⊥
= ⊥
+
∂y
2B ∂x
2B ∂z
(a)
(b)
(20)
(c)
= 0 in the last equation. Also, since B 2 = Bx2 + By2 + Bz2 and
Notice we have used ∇ · B
Bx = By = 0 due to the choice of axes, B = Bz . Furthermore, taking gradients,
2B∇B = 2Bx ∇Bx + 2By ∇By + 2Bz ∇Bz = 2B∇Bz
and so ∇Bz = ∇B. Hence the subscript z will be omitted from Bz from here on.
3
These averaged equations contain rich information. Let us first consider separately (20a)
and (20b), i.e. the ⊥ part. Following the procedure used before, we define a complex vD
vector,
vD = vDx + ivDy
(21)
and combine the equations as (a) + i(b):
v2
d
v2
vDx + vDy = −
+ ωc vDy − ivDx − ⊥ ∇⊥ B
dt
Rc
2B
2
2
v
dvD
v
+ iωc vD = −
− ⊥ ∇⊥ B
dt
Rc 2B
(22)
∂
∂
where ∇⊥ = ∂x
i + ∂y
j. The homogeneous solution of (22) is the Larmor motion, which we
disregard once again, since we are after the drifts. These drifts are given by the particular
solution,
v2
v2
vD = vD,part = i
+ i ⊥ ∇⊥ B
ωc Rc
2Bωc
vD
In vector terms,
vD =
mv2
2
mv⊥
= i
+i
∇⊥ B
qBRc
2qB 2
mv2
2
+ mv⊥ b × ∇⊥ B
bn
qBRc
2qB 2
(23)
(24)
Of these, the first term is the Inertia Drift due to the centrifugal force on a particle which
line. One mechanistic way to view this is the following: The
is sliding along a curved B
mv 2
centrifugal force is − Rc|| n. This can be countered by the magnetic force due to a drift along
chosen such that,
bn,
mv||2
× Bb
= qvD (bn)
n = qvDI × B
I
Rc
b
∇B
b × ∇B
B
B
B
2
mv||
qBRc
, as in (24).
B
× b = n, leaving vD =
and, from (12), (bn)
I
4
The second term in (24) is the
so-called ∇B - Drift (Notice that
b × ∇⊥ B = b × ∇B). This will
be later shown to be also related
- line curvature, since, due to
to B
= 0, the magnitude of B can
∇·B
only vary if the lines are curved.
This ∇B - drift is seen to be perpendicular to b and the ∇B and
A physidirected to the left of B.
cal picture is in shown in the figure.
To show how ∇B is related to line curvature, we start from Frenet’s first formula (illustrated
in the figure):
n
· ∇)b
= (b
(25)
Rc
2
· ∇)B.
and use the vector identity B × (∇ × B) = ∇ B2 − (B
2
= b, ∇ B2 = B∇B ,
Dividing this by B, and remembering B
B
b
ds
n
b +
db = b+ (b · ∇
)bds
Rc
dθ
· ∇)B
= ∇B − b × (∇ × B)
(b
and so,
· ∇)b
= (b · ∇) B = 1 [∇B − b × (∇ × B)]
− B b · ∇B
(b
B
B
B2
which is also Rnc , according to (25). We now cross-multiply
n
= (Rbn)
:
times b, to from b×
Rc
c
b × ∇B
bn
1
=
− b × b × (∇ × B)
Rc
B
B
b b · (∇ × B) − ∇ × B
= −(∇ × B)
⊥
or
b × ∇B
⊥
(∇ × B)
bn
=
−
Rc
B
B
and substituting this into the expression for (vD )∇B (from (24)),
2
2
⊥
mv⊥
(∇
×
B)
bn
mv
⊥
b × ∇B =
(vD )∇B =
−
B
2qB 2
2qB Rc
(26)
In the absence of appreciable net current ⊥ to B, one would have ∇ × B
= 0, and
⊥
curvature (1/Rc ), and directed along the
(vD )∇B would be strictly proportional to B-line
binormal vector just as the inertia drift. We should therefore group these terms together.
Starting at Eq. (24),
1 2
2
m v|| + 2 v⊥
2
− mv⊥ (∇ × B)
⊥
vD =
(27)
bn
qBRc
2qB 2
Parallel Drifts
To complete the picture, we now return to Eq. (20c) and investigate the parallel drift effects.
We had,
dv
mv 2 ∂B
m
(28)
=− ⊥
dt
2B ∂z
5
d( 1 mv 2 )
dv
dv
In steady state, m dt|| = mv|| dz|| = 2dz || . Since the total kinetic energy is conserved (no
forces, and B
forces perpendicular to v ),
E
d
1
mv||2
2
= −
dz
Substituting in (28),
d
1
2
mv⊥
dz
2
1
2
mv⊥
2
=
dz
or
d
dz
1
d
1 2 1 dB
mv
2 ⊥ B dz
1
2
mv⊥
2
=0
B
(29)
mv 2
The quantity μ = 2 B ⊥ (the magnetic moment, μ, according to our previous definition)
while executing Larmor
is seen to be invariant for a particle which moves along a B-line
This is sometimes called the second adiabatic invariant (“adirotations perpendicular to B.
field).
abatic” refers to the absence of energy input, as from a time-varying B
There are now two constants of the motion:
1 2
1 2
mv|| + mv⊥
2
2
E =
and
μ =
2
Eliminating the perpendicular energy 12 mv⊥
(30)
1
2
mv⊥
2
B
= μB, we obtain
1 2
mv = E − μB
2 ||
(31)
(32)
Showing that the parallel energy decreases as the particle moves into a region with a stronger
B field. At the same time, of course, the perpendicular energy increases: the varying field
has the effect of transferring energy from (⊥) to (||) or viceversa.
If B increases enough, a particle of given energy and magnetic moment can be stopped in
its parallel motion, and reflected. Suppose the particle has a parallel velocity v||0 when the
field is B0 . It will be reflected at turning point T if 12 mv||20 + μB0 = μBT :
BT = B0 +
1
mv20
2
μ
or
BT
= 1+
B0
6
= B0 +
v
v⊥
v20
2
v⊥0
B0
2
(33)
0
Defining the “pitch angle” θ = atan v⊥ /v|| ,
1
BT
= 1 + cotθ02 =
B0
sin2 θ0
B0
(34)
Bmax
v⊥
θ 0 v⊥ 0
v0
v
Any particle for which sin θ0 > B0 /Bmax will
be reflected at some BT given by (34). But
those with small enough pitch angle (sin θ0 < B0 /Bmax ) will not, and will escape through
the magnetic bottleneck. If a “Magnetic Bottle” is used to confine plasma between solenoids,
this “leakage”of low-pitch particles creates a peculiar distribution, with particles within a
“loss-cone” in pitch being mostly absent.
7
MIT OpenCourseWare
http://ocw.mit.edu
16.55 Ionized Gases
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.