托卡马克磁场位形中带电粒子的运动解读

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,

B e dt d m ?=υυ (1.1.1)

where m is the mass, e is the charge, υ is the velocity B is the

magnetic field. If the magnetic field is uniform in z-direction the components of the equation are

y c x dt

d υωυ= , x c y dt d υωυ-= (1.1.2) 0=dt

d z υ (1.1.3) wher

e m eB =ω is the cyclotron frequency. The solutions are

t c x ωυυsin ⊥=, t c y ωυυcos ⊥= (1.1.4)

.const z =υ (1.1.5)

The equation (1.14) can be integrated,

t x c ωρcos -= t y c s i n ρ= (1.1.6) where eB

m c ⊥⊥==υωυρ is the Larmor radius. Thus the particle has a 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.

B E ? 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 B E ? drift.

The equation of motion is

)(B E e dt d m ?+=υυ (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

B e dt d m y x υυ=, )(B E e dt d m x y

υυ-=

The solution of the equations can be written

B E t c x +=⊥ωυυsin , t c y ωυυcos ⊥= (1.2.2)

B E

is the B E ? 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

B e dt d m x y

υυ-= (1.2.3)

where y B B B '+=0 d x x υυυ+=0 (1.2.4)

The unperturbed motion of the particle is written by

t c x ωυυsin 0⊥=, t y c ωρsin = (1.2.5)

We assume that both the gradient

y B ' and drift d υ are small, then,

we have

0000B y B B dt d e m d x x y υυυυ-'--= (1.2.6)

Taking the time average gives the drift

⊥??=υρυ221B B B d

(1.2.7) where eB m ⊥

=υρ, 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//B e i R m dt d m c ?+=⊥⊥υυυ （1.2.8）

where c i is the unit vector outward along the radius of the curvature.

It is similar to Eq.(1.2.1) for the case of the

B E ? drift with the force

eE replaced R m /2//υ. By analogy the curvature drift is given by R

c d ωυυ2//= (1.2.9) Since c ω takes the sign of the charge, the electron and ion have

opposite drift, the drift direction is

B i e c ?. 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

])([B t E e dt d m ?+=υυ (1.2.10)

The electric field can be transformed away by moving to an accelerated frame having a velocity

2B B E f ?=υ (1.2.11) The equation of motion is then

B dt E d B m B e dt d m ?-?=2υυ (1.2.12)

This equation is similar to Eq.(1.2.1) with

E e being replaced by

B dt E d B m ?-2. The polarization drift is therefore dt E d B B B dt E d eB m c d ωυ1)(2=??-= (1.2.13) The polarization current, which play an important role in the neoclassical tearing modes, is

dt E d B j m p 2ρ= (1.2.14)

where m ρ 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 q U q T t q q L -= (2.1.1)

where the q and q

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, i q

0=??-??i i q L q L dt d (2.1.2) which is derived from a variation principle (?=0Ldt δ).

If we define the Hamiltonian by

∑-=i i i t q q L p q

t q p H ),,(),,( (2.1.3) where i i q L

p ??= . According to Eq.(2.1.2), we get a form of equations of motion by Hamiltonian,

i i q H

p ??-= (2.1.4)

i i p H

q ??= (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 i p and i q said to be conjugate variables.

2. Canonical transformation

In tokamak configuration, the relativistic Hamiltonian of a charged particle can be expressed as

Φ++-+-+-=e c m c R RA c e P A c e P A c e P H Z Z R R 42022222]/)()()[(φφ (2.2.1) where A R , A Z , and A φ are the vector potential components of the magnetic field, Φ is the electrical potential, 0m is the rest mass, and

e is the charge. P R , P φ, P Z, are the canonical momenta conjugate to R, φ, and Z respectively,

0R R R A c e u m P += (2.2.2) R c

e +0φφφA u Rm P = (2.2.3) 0Z Z Z A c e u m P += (2.2.4)

where γυ=u and 2/122)/1(c u +=γ is the relativistic factor. The magnetic field can be expressed as

φφ?+ψ???=I B (2.2.5)

where ψ 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

, ln , 00R A R R I A A Z R ψ-=-==φ (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,

))(ln exp(22000000020001ZX tg R m X R R R m X R m F -Ω-ΩΩ-=α (2.2.7)

where

ln 0000R R R m X C Ω= (2.2.8)

and Ωc is the toroidal gyro-frequency, ρ the Larmor radius, α the gyro-phase, subscripts o and c refer to the values at the magnetic axis and the guiding center respectively. X and α are the new coordinates conjugate to the momenta 24 2αραρsin R sin Z P C X +

+= (2.2.9) 2120ραC m P Ω= (2.2.10)

That the moment is turned to be coordinate often occurs during area-conserved canonical transformation [3]. The other two canonical variables φP and φ do not change in the new coordinates. The old coordinates are connected with new ones through four identical equations,

αραρ

sin cos 0c R c R e m P Ω= (2.2.11)

X P Z -= (2.2.12)

)R cos exp(R R C C α

ρ-= (2.2.13)

24 2

αραρsin R sin P Z C X --= (2.2.14)

The Jacobian in the area-conserved transformation is unity [3], that is,

φατφαdXd

d dP dP JdP d x = （2.2.15） 1=J (2.2.16)

The exact Hamiltonian for the relativistic particles is

Φ++ψ+++Ω=e c m c e P R

R R P m H c c 4202222220}][1]cos sin )[(2{φααα (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;

Φ+++Ω=e c m c u m P m H c 42022200)2(φα (2.2.18) We form a new invariant [2],

?=∏dX P x

π21

(2.2.19) New momenta ζαP P ,,∏,which are the three invariants, are conjugate

to ζαη,, and b dt d ωη=, ζωζ=dt d , ?ψ=ψ-=0c e P ζ which is actually the position variable [5]. The bounce frequency and the precession frequency are obtained from Hamiltonian equations,

1)(-?∏?=∏??==

H

H dt d b ηω (2.2.20) )/()(H H dt d ?∏?ψ?∏?-=ψ??==??ζωζ (2.2.21)

For the trapped particles in the large aspect ratio configuration, that is, the inversed aspect ratio 1<<ε, we get

)]()1()([)/(812115

.00000k K k k E m P m qR t --Ω=∏πεα (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],

)

(2)/(105.000k K qR m P bt γεπωαΩ= (2.2.23) t p p p t G R m P k k K k E R m s P k K k E R m P 2000211120001120002)]1()

()([4]21)()([2ΩΩ=--ΩΩ+-ΩΩ=γγγωαααζ (2.2.24)

where p Ωis the poloidal gyro-frequency, 0/R r =ε, αφεP u m k 0200214Ω=, 212200)/21(c u m P C φαγ+Ω+

=, and s is the magnetic shear,)(1k E and )(1k K are complete elliptic functions, t G the normalized precession

velocity of the trapped particle seen in Fig.1,

For the circulating particles,

)(22000200k E u m qR r m c πσφ+Ω=∏ (2.2.25)

)(200k K qR u bc γσ

πωφ= (2.2.26)

02

020202020

2)()()]21()()([2ζφφφζωωγωγγωω+=Ω+=Ω+--Ω+=bc c p bc p p bc c q G rR u q k K k E R s u k k K k E rR u q (2.2.27)

where q is safety factor, c G is the normalized precession velocity of

the circulating particle seen in Fig.2, σ represents direction of the circulating particle and 212-=k k . 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

, )],([21200200Φ+ψ++Ω=e P X e P mR P H x C C φα (2.2.28) Equations of motion are

dR dt B B u R Ru R R E =-φ()02 , (2.2.29)

dZ dt B B u R Ru R Z E d =-+φυ()0

2 , (2.2.30)

where . )(1= , = 22

00d 0E 202

R u m P R B E R R R u c p E +ΩΩ=ψΦ-=αυω??The B E ? drift, B ? drift, Curvature drift are recovered. If 20R Ru R u E - is smaller than d υ, 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, φ, Z ) coordinates,

υυψ=B B R p d , (2.2.31)

υυφp p E z p

B B u R Ru R B B =-+()02d , (2.2.32) where φυR u =，υψ is in radial direction, while υp 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<<ε, from Eq.(2.2.28) we get

dr dt d =-υθsin (2.2.33)

d dt qR r

d θυυθφ=-10cos (2.2.34)

21

2202

0)2sin 2(θ

ευυυφφ⊥-= (2.2.35)

where 0φυ and 0⊥υ are the values at 0=θ point. When

002⊥<υευφ, 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 c e -0ψ=u m P φ, the equation of motion in the toroidal direction is

000φφυB E R R dt du d R Ω+Ω=, (2.2.36)

The electric field can be transformed away in Eq. (2.2.32) by moving to a frame having a velocity, p f B E =

υ. Eq. (2.2.36) is then 0000φ

φυB E R R dt dE B R dt du d R p f Ω+Ω+= (2.2.37) where f u represents toroidal velocity in the moving frame. If the radial electric field is time-dependent, dt dE

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 1k , s is the magnetic shear. For s

0≤ and 8.01≥k 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), V ol.2, P.393.

[10] Wang Z.T. and Le Clarir G., 1992 Nucl. Fusion 32 2036.