Toroidal Field Ripple Induced Excursion of Banana Orbit

Transcription

1 Toroidal Field Ripple Induced Excursion of Banana Orbi in Tokamak Plasmas Gao Qingdi Key words TF ripple Banana orbi Local magneic well Thermonuclear plasma ions wihin a okamak mus be achieved wih a finie number of oroidal field (TF coils. This resul in a rippled oroidal field srucure, and consequen disorions in fas ion orbi wih poenially rapid loss of he affeced ions. The ripple loss is an imporan issue for he design of fuure okamak reacors such as ITER because i resul in reduced alpha heaing as well as poenially severe localized wall damage in fusion reacors. In addiion, TF ripple also causes loss of neural beam injeced fas paricles deerioraing he plasma performance. Ripple losses of fas ions have been shown o cause significan power losses in JET, JT-60U, TFTR and Tore Supra okamaks [-4]. The TF ripple affec he ranspor of energeic paricles in differen ways depending on he magneic field srucure. Paricles can be locally rapped in he magneic well formed by ripple and subsequenly los via grad-b drif. Anoher way o cause ripple loss is ha ripple modifies he banana orbi of oroidally rapped paricles leading o sochasic ripple diffusion. Under conservaion of he magneic momen he rippled field produces variaions in he velociy componen parallel o he magneic field. These variaions cancel over mos of he banana orbi, bu are significan in modifying he ime spen near he banana ip leading o a verical jump of he ip for successive banana bounces. If he sep-size of he banana excursion is large enough, energeic paricles will loss via sochasic banana diffusion [5].. Ripple induced excursion of banana ip The oroidal magneic field in a okamak is aken o be he form B R0 B = 0 [ δ ( r, θ sin N φ ] ( R

2 B max min δ ( r, θ = (B max and B min are he maximum and minimum field magniude B max B + B min a consan major radius and elevaion is he oroidal field ripple (TF ripple, N he number of oroidal field coil, R and r he major and minor radius, θ and φ he poloidal and oroidal angle in a okamak configuraion. For Nq >>, he /R variaion of he oroidal field along a magneic field line is slow compared o he ripple modulaion. Near he urning poin of a rapped paricle banana orbi, we use he major radius of he banana ip, R, and i poloidal angle, θ o parameerize he magneic field along field lines. As one moves a oroidal angle φ along field line, he poloidal angle moves a small value φ/q, and R can be approximaed as φ R = R ( + ε sinθ ( q ε = r / R. Subsiuing R ino Equaion (, he magneic field around he banana ip can be wrien as B R0 B = 0 [ η( r, θ δ ( r, θ sin ], (3 R = Nφ is oroidal ripple coordinae, and η ( r, θ = ε sinθ / Nq is a parameer much smaller han uniy in he plasma cross-secion. To calculae a number of imporan quaniies for a ripple field, he normalized model magneic field is defined as ~ B B /( B R / R = η δ sin (4 0 0 The difference in he normalized field srengh beween successive ripple maxima is B = πη (5 which is independen on he TF ripple δ. As he locaions of he firs maximum and minimum are cos = π + ( η / δ, = π cos ( η / δ, he deph of ripple well is found, max min ~ ~ ( ( cos w B B = δ η η ( η / δ (6 max min From Eq. (6, i is found ha w = 0 when η = δ, and w has no physical value when η > δ, which means ha for

3 δ ( r, θ ε sinθ /( Nq (7 local magneic wells are eliminaed. The crierion [Equaion (7] divides he plasma crosssecion ino wo regions, he ripple-well region oroidal wells exis, and he res of cross-secion he oroidal wells do no exis. In he ripple well region energeic paricles are los dominanly hrough B drif in he verical direcion due o paricles becoming locally rapped in he ripple well. In he ripple well free region, he ripple modifies he banana orbi causing banana drif diffusion. The loss of perfec axisymmery of he magneic field due o he TF ripple modifies he banana orbi. To illusrae he physics picure of he banana modificaion, we consider he oroidal componen of he magneic field variaion δ B, which causes an oscillaion in he poloidal velociy,θ &. Following a field line, for which φ = φ 0 + qθ, δ & θ = δv /( R & (8 φ As paricle moves abou along he banana orbi i conserves energy, magneic momen, E = mv + µ B and µ = mv / B ( v and v are he velociy componen parallel and perpendicular o he magneic field respecively, he variaion of he paricle velociy is δv v δb v δb =, δ & θ = & θ (9 v B v B This velociy variaion diverges a he ip v = 0, and i modifies he ime spen near he banana ip when paricle moves along banana orbi. Consequenly he variaions of θ & coupled wih he usual B drif cause ne excursions across he magneic surfaces. In order o ge he ne excursion of a paricular banana rajecory, we have o evaluae he random variaion of he bounce ime and muliply i by he radial drif. The drif velociy of energeic paricle is v d = ( v + v Ωi v B B B (0 Ω i is gyro-frequency. According o he analysis above dominan par of he excursion arises from he region near he banana ip. By using he verical drif a he banana ip, he ne 3

4 excursion of a paricular banana rajecory is δz = ερ v ( r ρ is gyro-radius of he energeic paricle a he banana ip, v he oal velociy, and he variaion of bounce ime caused by he TF ripple. To find, he posiion of he banana urning poin in he limi δ 0 (namely he ripple free case, is used o parameerize he ripple-phase of a banana orbi. In he ripple free case, he ime requesed by a paricle raveling around he urning poin is ( s = v L dl v / v ( s ( L is he inegraion roue along he banana orbi, v he paricle velociy a v = 0. By using he model magneic field near he banana ip, (s can be wrien as ( s = R Nv ( η d ( η ( η (3 Here, we consider for he fac ha he bounce ime variaion of a paricle is due o he velociy variaion arising dominanly near he urning poin. Similarly, in he rippled field he ime requesed by a paricle raveling around he urning poin is ( a = R Nv a ( η d ( η ( η δ sin (4 a, he lower limi of inegraion, is he posiion of urning poin in he rippled field case, which is deermined by solving he ranscendenal equaion, η + δ sin = η (5 a a Wih = ( s ( a, he verical excursion, normalized o he gyro-radius ρ, is aken o be δ q z = D( η, εn sinθ (6 D η, is a dimensionless displacemen. For small η, ( D( η, = d d (7 ( a ( η η + δ sin / η a 4

5 A numerical calculaion of D η, shows ha when η > δ, for which ripple-wells ( are eliminaed, D is sinusoidal wih respec o leading o an oscillaory excursion. The difference in he excursion ampliude beween consecuive urning poin causes non-closure of banana orbi, which, for small values of he excursion, resul in a complicaed verical moion of he banana ip on consecuive bounce. This verical moion is deermined by wo periodiciies, namely he oroidal periodiciy of ripple field and he periodiciy of he banana procession moion, and i is essenially periodic, and does no lead o losses. If he excursion ampliude becomes large, however, i will lead o decorrelaion of he banana orbi causing sochasic banana diffusion [5]. To show he characerisics of he banana excursion, we calculae he rippled field induced banana jump in HL-A. The sysem of oroidal magneic field in HL-A consis of 6 D-shaped coils. Compared wih oher okamaks, HL-A is of less oroidal field coils. Though he smaller number of coils is beneficial o accessibiliy for auxiliary heaing insrumen and diagnosics, hey generae higher TF ripple. The conours of he calculaed TF ripple δ(r,θ are of an elongaed D-shape as shown in Fig. he dashed line circle indicaes he plasma boundary. Since he separaion beween he plasma and he big D-coils is raher large, he ripple ampliude is less han % a he ouer plasma edge, and his is saisfacory for avoiding excessive ripple well loss. In he cenral region of he plasma, however, here is an area wih ripple in excess of Fig. TF ripple conours of he HL-A okamak. The circles wih dashed and doed line indicae he plasma boundary and shear reversal flux surface respecively 0.0%, which would induce sochasic banana diffusion. In he HL-A okamak (R=.64m, a=0.4m, B T =.8T, I p =0.48MA, TRANSP modeling demonsraes ha a quasi-saionary reversed magneic shear (RS discharge can be esablished by profile conrol [6]. The sudies on he opimizaion of he curren densiy profile sugges ha reversed magneic shear is desirable for confinemen, sabiliy and boorap alignmen. In many okamaks, RS plasmas develop an inernal ranspor barrier 5 Fig. Safey facor q on he plasma mid-plane versus major radius.

6 (ITB ha produces improved cenral confinemen. Neverheless, he fac ha q (safey facor value inside RS plasmas is usually higher han ha in convenional (non-rs okamak plasmas resul in sronger sochasic banana diffusion [7]. For he quasi-saionary RS configuraion in HL-A, he minimum safey facor q min =.78 locaing a x min ( r min /a = 0.63, and he q profile on he plasma mid-plane is shown in Fig.. The shear reversal flux surface is indicaed by a doed line circle in Fig.. We choose wo ypical poin on he shear reversal flux surface (P and S as shown in Fig. o calculae he ripple induced banana ip excursion, δz. These wo poin are locaed in he cenral plasma region wih ripple greaer han Accouning for he Shafranov shif, δ =.3 0-4, η = a poin P (θ = π/, and δ = , η = a poin S (θ = π/4. I is shown ha δz is approximaely sinusoidal wih respec o (Fig. 3, and i ampliude is less han 0.ρ a poin S, bu is larger han ρ a poin P due o he larger ripple value (δ ~0-3 a his poin. In order o show he effec of ripple on he banana excursion we evaluae he ampliude of he oscillaory excursion a poin S for differen ripple values. I is found ha he ampliude increases almos linearly as he TF ripple δ increasing.0 δz/ρ S Fig. 3 Banana ip excursion δz versus, full line corresponding o δz a poin P, doed line corresponding o 0 δz a poin S. P Fig. 4 Ampliude of he sinusoidal excursion versus he TF ripple δ. (Fig. 4. The maximum excursion a poin S can be several gyro-radius when δ increases by Conclusions Under conservaion of he magneic momen, he rippled field produces variaions in he velociy of rapped paricles. These variaions cancel over mos of he banana orbi, bu are 6

7 significan in modifying he ime spen near he banana ip. This leads o excursion of he ip posiion for successive banana bounces. When he excursion is large enough, he rapped energeic paricles loss rapidly via sochasic banana diffusion. We calculae he ripple field induced excursion of he banana ip in an RS configuraion in HL-A. I is shown ha he excursion is sinusoidal wih respec o, and i ampliude can be larger han one gyro-radius in he cenral plasma region wih he TF ripple δ ~ 0-3. The TF ripple affec he excursion significanly. I is found ha he ampliude of he sinusoidal excursion increases almos linearly as he TF ripple δ increasing. References. Puvinskij S.V., e al., Nucl. Fusion 34 ( Tobia K., e al., Nucl. Fusion 34 ( Redi M.H., e al., Nucl. Fusion 35 ( Basiuk V., e al., Proc. Inernaional Conf. on Plasma Phys. (Innsbruck, 99 Vol 6C, Par I (European Physical Sociey Goldsone R.J., e al., Phys. Rev. Le. 47 ( Gao Qingdi, e al., Nucl. Fusion 40 ( Gao Qingdi, Sochasic ripple diffusion of energeic paricles in reversed magneic shear okamak, o be published 7