Download 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

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 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  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 y
dt
 c x
d z
0
dt
where  
eB
m
(1.1.2)
(1.1.3)
is the cyclotron frequency. The solutions are
 x    sin c t ,
 y    cos  c t
 z  const.
The equation (1.14) can be integrated,
(1.1.4)
(1.1.5)
x    cos c t
where  
y   s i nc t
(1.1.6)
  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

  
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
m
d x
 e y B ,
dt
m
d y
dt
 e( E   x B )
The solution of the equations can be written
 x    sin  c t 
E
B
is the
 
EB
E
,
B
 y    cos  c t
(1.2.2)
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
where
d y
dt
 e x B
(1.2.3)
B  B0  B y
 x   x0   d
(1.2.4)
The unperturbed motion of the particle is written by
 x 0    sin c t , y   sin c t
We assume that both the gradient
we have
B y
(1.2.5)
and drift  d are small, then,
m d y
  x 0 B0   x 0 B y   d B0
e dt
(1.2.6)
Taking the time average gives the drift

d 
where  
m 
eB
1 B  B


2
B2
(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，


d  m //2 

m

ic  e(   B)
dt
R
where

ic
（1.2.8）
is the unit vector outward along the radius of the curvature.
It is similar to Eq.(1.2.1) for the case of the
force eE replaced
m //2 / R .
 
EB
drift with the
By analogy the curvature drift is given
by
d 
 //2
c R
(1.2.9)
Since  c takes the sign of the charge, the electron and ion have


opposite drift, the drift direction is eic  B . For axisymmetric
system, B 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
m


 
d
 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 
B2

(1.2.11)
The equation of motion is then


  m dE 
d
m
 e  B  2
B
dt
B dt
(1.2.12)

This equation is similar to Eq.(1.2.1) with eE being replaced by

m dE 
 2
 B . The
B dt
polarization drift is therefore


m dE  
1 dE
 d   2 (  B)  B 
c B dt
eB dt

(1.2.13)
The polarization current, which play an important role in the
neoclassical tearing modes, is


 m dE
jp  2
B 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
L(q, q , t )  T (q )  U (q, t )
where the
q
and
q
(2.1.1)
are the vector position and velocity over the all
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, qi
d L L

0
dt q i q i
(2.1.2)
which is derived from a variation principle (   Ldt  0 ).
If we define the Hamiltonian by
H ( p, q, t )   q i pi  L(q, q , t )
(2.1.3)
i
where
p i 
L
qi
. According to Eq.(2.1.2), we get a form of equations
of motion by Hamiltonian,
p i  
q i 
H
q i
H
p i
(2.1.4)
(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 qi 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
H  [( PR  AR ) 2  ( PZ  AZ ) 2  ( P  RA ) 2 / R 2 ]c 2  m02 c 4  e
c
c
c
(2.2.1)
where AR, AZ, and A are the vector potential components of the
magnetic field,  is the electrical potential, m0 is the rest mass, and
e is the charge. PR PPZ, are the canonical momenta conjugate to
R, and Z respectively,
PR  m0 u R 
e
AR
c
(2.2.2)
e
P  Rm 0 u + RA
c
PZ  m0 u Z 
where
u  
(2.2.3)
e
AZ
c
(2.2.4)
and   (1  u 2 / c 2 )1 / 2 is the relativistic factor.
The magnetic field can be expressed as
B      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
AR  0 , AZ   I ln
R

, A  
R0
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]
We introduce a generating function [3, 4] for changing to the
guiding center variables,
m0  0 R02
X
R
X
F1  
exp(
)(ln

) 2 tg  ZX
2
m0  0 R0
R0 m0  0 R0
(2.2.7)
where
X m 0  0 R0 ln
RC
R0
(2.2.8)
and c is the toroidal gyro-frequency,  the Larmor radius,  the
gyro-phase, subscripts o and c refer to the values at the magnetic
axis and the guiding center respectively. X and  are the new
coordinates conjugate to the momenta
2
PX  Z   sin  
P 
sin 2
4 RC
(2.2.9)
1
m0  C  2
2
(2.2.10)
That the moment is turned to be coordinate often occurs during
area-conserved canonical transformation [3]. The other two
canonical variables
P
and  do not change in the new coordinates.
The old coordinates are connected with new ones through four
identical equations,

PR  m0  c e
Rc
cos 
sin 
PZ   X
R  RC exp( 
(2.2.11)
(2.2.12)
 cos 
RC
)
(2.2.13)
Z  PX   sin  
2
4 RC
sin 2
(2.2.14)
The Jacobian in the area-conserved transformation is unity [3], that
is,
（2.2.15）
d  JdP dPx dP ddXd
J 1
The
exact
Hamiltonian
H  {2m0  c P [(
(2.2.16)
for
the
relativistic
particles
is
Rc 2
1
) sin 2   cos 2  ]  2 [ P  e]2 }c 2  m02 c 4  e
R
R
(2.2.17)
It is suitable for particle simulation. The equations of motion and
Vlasov’s equation could be derived from the Hamiltonian.
For the gyro-kinetics the Hamiltonian in Eq.(2.2.17) could be
averaged with the gyro-phase;
H  (2m0  c P  m02 u2 )c 2  m02 c 4  e
(2.2.18)
We form a new invariant [2],

1
Px dX
2 
New momenta
, P , P
to , ,  and
d
d
 b ,
 
dt
dt
(2.2.19)
,which are the three invariants, are conjugate
,
e
P   0   
c
which is actually
the position variable [5]. The bounce frequency and the precession
frequency are obtained from Hamiltonian equations,
d H
 1

(
)
dt 
H
d
H


 

 (  ) /(
)

dt 
H

b 
(2.2.20)
(2.2.21)
For the trapped particles in the large aspect ratio configuration, that
is, the inversed aspect ratio   1 , we get
t 
8qR0 m0 ( 0 P / m0 ) 0.5

[ E (k1 )  (1  k1 ) K (k1 )]
2
(2.2.22)
which is the toroidal magnetic flux enclosed by drift surface.
According to Eq.(2.2.20) and (2.2.21) the bounce frequency and the
precession frequency are obtained [5，6, 7],
 ( 0 P / m0 ) 0.5
bt 
2qR0 K (k1 )
t 
(2.2.23)
2 0 P E (k1 ) 1
4 0 P s E (k1 )
2
[
 ]
[
 (1  k1 )]
2
2
 p m0 R0 K (k1 ) 2  p m0 R0 K (k1 )
(2.2.24)
2 0 P

Gt
 p m0 R02
where
  (1 
K (k1 )
p
is the poloidal gyro-frequency,   r / R0 , k 
2
1
2 C P / m0  u20
c2
m0 u20
4 0 P
,
1
)2
, and s is the magnetic shear, E(k1 ) and
are complete elliptic functions, Gt the normalized precession
velocity of the trapped particle seen in Fig.1,
For the circulating particles,
c 
 0 m0 r 2 2qR0 m0 u 0

E (k )
2

bc 
u 0
2qR0 K (k )
(2.2.25)
(2.2.26)
u20
u20 s E (k )
E (k )
k2
c  q bc 
[
 (1  )] 
2 p rR0 K (k )
2
 p R02 K (k )
 q bc 
u20
2 p rR0
(2.2.27)
Gc  q bc   0
where q is safety factor, Gc is the normalized precession velocity of
the circulating particle seen in Fig.2,  represents direction of the
circulating particle and k 2  k12 . It is found for first time that the
precession of all the circulating electrons is in ion diamagnetic
direction if magnetic shear is neglected. It is easy to understand.
Deep-trapped particles experience in the low field site. The
precession is in one direction. Barely trapped particles experience
both high field site and low field site. The precession is reversed.
The circulating particles, which play very important role in the
electron fishbone modes [8], like the barely trapped particles
experience both high field site and low field site, therefore, the
precession reversal is reasonable.
3. Non-relativistic scenario
The Hamiltonian of the charged particle is
H 0   C P 0 
1
[ P 0  e0 ( X , Px )] 2  e ,
2
2mRC
(2.2.28)
Equations of motion are
dR BR u Ru E

( 
),
dt
B R R02
(2.2.29)
dZ BZ u Ru E

( 
)  d ,
dt
B R R02
(2.2.30)
where
u E   R02
 P

E
1
u2
= R02 E  R0
, d =
( c   2 ) . The

Bp
 0 R0
m
R
 
EB
Ru
drift, B drift, Curvature drift are recovered. If u  2E
R
R0
is
smaller than  d , particle will continuously drift along z-direction
until it is lost. Not all particles could be confined in tokamak
configuration. There is somewhat of “ loss cone” like in the mirror
machine.
The velocities in R and Z directions are easy to change to the
radial and poloidal directions through rotating the coordinates. For
any tokamak configuration, the particle guiding-center equations of
motion are reduced in (R,Z coordinates,
 
p 
where
BR
d ,
Bp
(2.2.31)
B p u Ru E
B
(  2 )  z d ,
B R R0
Bp
u  R ，
(2.2.32)
is in radial direction, while p is in poloidal
direction. The Eqs.(2.31) and (2.32) are the generalized version of
equations of motion obtained by Balescu [9].
In the large aspect ratio circular configuration, that is, the
inversed aspect ratio,   1 , from Eq.(2.2.28) we get
dr
  d sin 
dt
(2.2.33)

d
1

   d cos
dt qR0
r
(2.2.34)
  (  2 sin
2
0
2
0

2
)
1
2
(2.2.35)
are the values at   0 point. When
and  0
where   0
2
 0  2  0 , the particle will be trapped mainly in the outer part of
the cross section forming banana a orbit. If particle is near the
magnetic axis the orbit is like a potato [4] and called potato orbit.
With conservation of the canonical momentum in toroidal
direction
e
P  m0 u - 
c
, the equation of motion in the toroidal
direction is
E
du
  R R0 d   0 R0
dt
B
,
(2.2.36)
The electric field can be transformed away in Eq. (2.2.32) by
moving to a frame having a velocity,  f 
du f
dt
where
uf

E
Bp
. Eq. (2.2.36) is then
E
R0 dE
  R R0 d   0 R0
B p dt
B
(2.2.37)
represents toroidal velocity in the moving frame. If the
radial electric field is time-dependent,
dE
dt
term will lush a toroidal
flow. The radial electric field may play an important role in the L-H
mode transition [10]. The second term is the  // B acceleration [1].
The third term is the parallel electric field acceleration.
Figure 1 Normalized precession velocity of the trapped electrons
versus k1 , s is the magnetic shear. For s  0 and k1  0.8 the
particles precession is reversed.
Figure 2 Normalized precession velocity of the circulating
electrons versus k, s is the magnetic shear. For s  0 the precession
of all the particles is reversed.
References
[1] John Wesson, Tokamaks, The Oxford Engineering Science Series 48, Second edition 1997.
[2] Robert G. Littlejohn, J. Plasma physics 29, 111(1983).
[3] Lichtenberg A J and Lieberman M A, Regular and Stochastic Motion, Applied Sciences 38,
(Springer-Verlag New York Inc. 1983).
[4] Wang Z T 1999 Plasma Phys. Control. Fusion 41 A679.
[5] Hazeltine R D, Mahajan S M and Hitchcock D A 1972 Phys. Fluids 24 1164.
[6] Wang Z T, Long Y X, Dong J Q, Wang L and Zonca F 2006 Chin. Phys. Lett. 23 158.
[7] Connor J W, Hastie R J, and Martin T J, 1983 Nucl. Fusion 23 1702.
[8] Wang Z T, Long Y X, Wang A K, Dong J Q, Wang L and Zonca F 2007 Nucl. Fusion 47
accepted.
[9] Balescu R 1988 in Transport Processes in Plasma, (North-Holland, Amsterdam. Oxford. New
York. Tokyo), Vol.2, P.393.
[10] Wang Z.T. and Le Clarir G., 1992 Nucl. Fusion 32 2036.