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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. EB 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 EB 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 EB 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 . EB 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 EB 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 PPZ, 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 ddXd 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 u2 )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 2qR0 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 u20 c2 m0 u20 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 2qR0 K (k ) (2.2.25) (2.2.26) u20 u20 s E (k ) E (k ) k2 c q bc [ (1 )] 2 p rR0 K (k ) 2 p R02 K (k ) q bc u20 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 k12 . 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 e0 ( 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 EB 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.