-
托卡马克磁场位形中带电粒子的运动
王中天
核工业西南物理研究院
(2007<
/p>
年核聚变与等离子体物理暑期讲习班
)
Particle Dynamics in Tokamak
Configuration
Ⅰ
.
Charged particle motion in a general magnetic
field
1. Larmor orbits
Particle in the magnetic field
satisfies the equation of motion,
?
?
?
d
?
m
?
e
?
< br>?
B
dt
(1.1.1)
?
?
where
m
is
the
mass,
e
is
the
charge,
is
the
velocity
?
B
is
the
magnetic
field.
If
the
magnetic
field
is
uniform
in
z-direction
the
components of the equation are
d
?
x
?
?
c
?
y
dt
d
?
z
?
0
dt
m
,
d
?
y
dt
?
?
?
c
?
x
(1.1.2)
(1.1.3)
eB
where
?
?
is
the cyclotron frequency. The solutions are
?
x
?
?
?
sin
?
c
t
,
?
y
?
?
?
c
os
?
c
t
(1.1.4)
?
z
?
const
.
(1.1.5)
The equation (1.14) can be integrated,
x
?
?
?
cos
?
c
t
y
?
?
s
i
n
c
t
(1.1.6)
where
?
?
?
?
m
?
?
is
the
Larmor
radius.
Thus
the
particle
has
a
?
?
c
eB
helical orbit composed of the circular
motion and a constant velocity
in the
direction of the magnetic field.
2. Particle drifts
The
particle
orbits
calculated
in
last
section
resulted
from
the
assumption
of
a
uniform
magnetic
field
and
no
electric
field.
The
charged
particles
gyrate
rapidly
about
the
guiding
centre
of
their
motion.
The
perpendicular
drifts
of
the
guiding
centre
arise
in
the
presence of any of the
following [1]:
1)
an electric field perpendicular to the
magnetic field;
2)
a gradient in magnetic field
perpendicular to the magnetic field;
3)
curvature of
the magnetic field;
4)
a time dependent electric
field.
The drift velocity for each case
is derived below.
?
?
E
?
B
drift
If
there
is
an
electric
field
perpendicular
to
the
magnetic
field
the
particle orbit undergo a drift
perpendicular to both fields. This is the
so-called
?
?
E
?
B
drift.
The equation of motion is <
/p>
?
?
?
?
d
?
m
?
e
(
E
?
?
?
B
)
dt
(1.2.1)
Choosing
the
z
coordinate
along
the
magnetic
field
and
the
y
coordinate
along
the
perpendicular
electrical
field,
the
components
of
Eq.(1.2.1) are
d
?
m
x
?
e
?
y
B
,
dt
m
d
?
y
dt
?
e
(
E
?
?
x
B
)
The
solution of the equations can be written
E
?
x
?
?
?
sin
?
c
t
?
,
?
y
?
?
?
cos
?
c
t
(1.2.2)
E
B
B
is the
?
?
< br>E
?
B
drift which is independent of the
charge, mass, and
energy of the
particle. The whole plasma is therefore subject to
the
drift.
?
B
drift
If
the
magnetic
field
has
a
transverse
gradient,
this
leads
to
a
drift perpendicular to both the
magnetic field and its gradient.
Taking
the
magnetic
field
in
z
direction
and
its
gradient
in
y
direction, the y component of the
equation of motion is
m
d
?
y
dt
?
?
e
?
x
B
(1.2.3)
where
B
?
B
0
?
B
?
y
?
x
?
?
x
0
?
?
d
(1.2.4)
The unperturbed motion of
the particle is written by
?
x
0
?
?
?<
/p>
sin
?
c
t<
/p>
,
y
?
?
sin
?
c
t
(1.2.5)
We assume that both
the gradient
we have
B
?
y
and drift
?
d
are
small, then,
m
d
?<
/p>
y
?
?
?
x
0
B
0
?
?
x
0
B
?
y
?
?
d
B
0
e
dt
(1.2.6)
Taking the time
average gives the drift
?
d
where
?
?
m
?
?
eB
?
?
1
B
?
?
B
?
?
?
2
B
2
(1.2.7)
, the ion and electron have opposite
drift because of
the charge.
Curvature drift
When a
particle
’
s guiding center
follow a curved magnetic field
line it
undergoes a centrifugal
force
，
?
2
?
?
d
?
?
m
?
//
?
m
?
i
c
?
e
(
?
?
?
B
)
dt
R
（
1.2.8
）
where
?
i
c
is the unit vector outward along the
radius of the curvature.
?
?<
/p>
E
?
B
It
is
similar
to
Eq.(1.2.1)
for
the
case
of
the
force
by
eE
drift
with
the
2
replaced
m
?
//
/
R
.
By
analogy
the
curvature
drift
is
given
2
?
//
(1.2.9)
?
d
?
?
c
R
Since
?
c
takes
the
sign
of
the
charge,
the
electron
and
ion
have
opposite
drift,
the
drift
direction
is
system,
?
B
?
?
e
i
c
?
B
.
For
axisymmetric
drift
and
curvature
drift
are
in
same
direction.
I
will
show you later.
Polarization drift
When an
electric field perpendicular to the magnetic field
varies
in time it results in what is
called the polarization drift. The name is
given
from
the
fact
the
ion
and
electron
drifts
are
in
opposite
direction and give
rise to a polarization current.
The equation of motion is
?<
/p>
?
?
?
d
?
m
?
e
[
E
(
t
)
?
?
?
B
]
dt
(1.2.10)
The
electric
field
can
be
transformed
away
by
moving
to
an
accelerated frame having
a velocity
?
?
?
E
?
B
?
f
?
B
2
(1.2.11)
The equation of motion is then
< br>?
?
?
d
?
m
d
E
?
m
?
e
?
?
B
?
2
?
B
dt
B
dt
?
(1.2.12)
?
e
E
This
equation
is
similar
to
Eq.(1.2.1)
with
?
m
d
E
?
?
2
?
B
. The polarization drift is therefore
B
dt
being
replaced
by
?
?
?
?
m
d
E
1
d
E
?
d
< br>?
?
2
(
?
B
)
?
B
?
?
c
B
dt
eB
dt
?<
/p>
(1.2.13)
The
polarization
current,
which
play
an
important
role
in
the
neoclassical tearing
modes, is
?
?
?
d
E
j
p
?
m
B
2
dt
(1.2.14)
where
?
m
is
the mass density.
Ⅱ
. Charged
Particle motion in Tokamak Configuration
1. Hamiltonian Euations
Various
equivalent
forms
of
equations
of
motion
may
be
obtained by coordinate transformations.
One such form is obtained
by
introducing a lagrangian function
?
,
t
)
?
T
(
q
?
< br>)
?
U
(
q
,
t
)
L
(
q
,
q
(2.1.1)
?
are the vector position and velocity
over the all
where the
q
and
q
degrees of freedom, T is
the kinetic energy, U is the potential energy,
and
any
constraints
are
assumed
to
be
time
independent.
The
equations of motion are,
for each coordinates,
q
i
d
?
L
?
L
?
?
0
?
i
?
q
i
< br>dt
?
q
(2.1.2)
which is
derived from a variation principle (
?
?
Ldt
?
0
).
If we define the Hamiltonian by
?
i
p
i
?
L
(
q
,
q
?
,
t<
/p>
)
H
(
p
,
q
,
t
)
?
?
q
i
(2.1.3)
where
?
i
?
p
?
L
?
q
i
. According to Eq.(2.1.2), we get a form of
equations
of motion by Hamiltonian,
?
i
?
?
p
?
H
?
q
i
(2.1.4)
?
i
?
q
?
H
?
p
i
(2.1.5)
The set
of p and q is known as generalized momenta and
coordinates.
Eqs. (2.1.4) and (2.1.5)
are Hamiltonian equations. Any set variables
p and q whose time evolution is given
by Eqs. (2.1.4) and (2.1.5) is
said to
be canonical with
p
i
and
q
i
said to be conjugate variables.
2. Canonical transformation
In
tokamak
configuration,
the
relativistic
Hamiltonian
of
a
charged particle can be expressed as
e
e
e
2
4
H
?
[(
P
R
?
A
R
)
2
?
(
P
Z
?
A
Z
)
2
?<
/p>
(
P
?
?
RA
?
)
2
/
R
2
]
c
2
?
m
< br>0
c
?
e
?
c
c
c
(2.2.1)
where
A
R
,
A
Z
,
and
A
?
are
the
vector
potential
components
of
the
magnetic
field,
?
is the electrical
potential,
m
0
is the rest mass, and
e is
the charge. P
R
??
P
????
P
Z,
??
are the canonical momenta
conjugate to
R,
???
and Z respectively,
P
R
?
m
0
u
R
?
e
A
R
c
(2.2.2)
(2.2.3)
e
P
?
?
Rm
< br>0
u
?
+
R
A
?
c
P
Z
?
m
0
u
Z
?
A
Z
(2.2.4)
where
u
?
??
e
c
and
?
?
(
1
?<
/p>
u
2
/
c
2
)
1
/
2
is the relativistic
factor.
The magnetic field can be
expressed as
B
?<
/p>
?
?
?
?
?
?
I
?
?
(2.2.5)
where
??
is
related
to
the
poloidal
flux
of
the
magnetic
field,
I
is
related
to
the
poloidal
current,
R
is
the
major
radius.
Then,
in
tokamaks
we have
A
R
?
0
,
A
Z<
/p>
?
?
I
ln
R
?
,
A
?<
/p>
?
?
R
0
R
(2.2.6)
“There
has
been
a
gradual
evolution
over
the
years
away
from
the
averaging
approach
and
towards
the
transformation
approach”
said Littlejohn
[2]