Variations of the Magnetic Moment of Fast Ions in Spherical Tori

Transcription

1 TH/P3- Variaions of he Magneic Momen of Fas Ions in Spherical Tori V.A.Yavorskij), D.Darrow), V.Ya.Goloborod ko), N.Gorelenkov), U. Holzmueller- Seinacker3), M.Isobe4), S.M. Kaye), S.N.Reznik), K.Schoepf3) )Insiue for Nuclear Research, Ukrainian Academy of Sciences, Kiev, Ukraine )Princeon Plasma Physics Laboraory, New Jersey, U.S.A. 3)Insiue for Theoreical Physics, Universiy of Innsbruck, Associaion EURATOM- OEAW, Ausria 4)Naional Insiue for Fusion Science, Toki-shi, Japan conac of main auhor: Absrac. Due o he srongly inhomogeneous magneic field in low-aspec-raio spherical ori he magneic momen of fas ions confined herein appears o be non-conserved. We examine he non-adiabaic fas ion moion in oroidal configuraions as effeced by he flux surface shape and by ripples of he magneic field. Specific aenion is paid o poloidal ripples of axisymmeric spherical ori, which arise from he combined effecs of plasma paramagneism and a large Shafranov shif of noncircular flux surfaces in he case of a highbea plasma, causing o vary non-monoonically on he ouboard of he plasma column in he viciniy of he mid-plane. The srong sensiiviy of non-adiabaic variaions of he magneic momen o weak small-scale poloidal perurbaions of flux surfaces and of is demonsraed. The invesigaion presened helps o undersand he naure of he enhanced non-adiabaiciy observed in he numerical EFIT field of NSTX, and i enables he evaluaion of he accuracy of analyical esimaes of non-adiabaic changes of he magneic momen. The effec considered is expeced o subsanially impac fas ion ranspor in spherical okamaks as well as in sellaraors.. Inroducion The magneic momen µ of fas ions confined in axisymmeric low-aspec-raio spherical orus (ST) plasmas is no well conserved due o he pronounced inhomogeneiy of he magneic field [,]. In his paper we evaluae, boh analyically and numerically, he magniudes of non-adiabaic [3] variaions of he magneic momen of fas ions in STs as induced by perurbaions of he flux surface (FS) shape and/or by small-scale (in poloidal angle χ) modulaion of along he field line. Our consideraion is based on a model magneic field (χ) wih non-circular bu poloidally smooh FSs [4], which has been exended o accoun for small-scale poloidal modulaions (χ) ypical for high bea ST equilibria. These perurbaions arise from he combined effecs of plasma paramagneism and a large Shafranov shif of noncircular flux surfaces, causing = o vary non-monoonically as a field line is followed from is inner o he ouer mid-plane crossing poin. The resuling smallscale modulaion of he magneic field we will call poloidal field (PF) ripples in analogy o he oroidal field ripples.. Effec of poloidal field ripples on non-adiabaic jumps of µ. Theory Evaluaing non-adiabaic jumps of he magneic momen µ, which occur when paricles cross he ouer mid-plane (χ=) of he axisymmeric oroidal magneic field = + p, we follow Refs. [,3] and derive for he magniude of relaive non-adiabaic jumps he expression V Mˆ /8 = Re exp iϑ /8 + iζ s + + O( ) () µ V

2 wih = i s s ll dω, = i dvll ( ) V, () where V and V ll are he paricle velociies perpendicular and, respecively, parallel o he magneic field, is he adiabaiciy parameer (raio of full ion gyro-radius, ρ L =V/ω, o he curvaure radius of a magneic field line, R c ) and V is he oal paricle velociy; furher, he ime momens = s correspond o he saionary gyro-phase poins, ω ( s )=, τ =/, = r r ζ = arcg denoes he phase shif, and ˆM is a, τ =[τ τ ], ( ) facor of order uniy, which is deermined by he FS shape parameers. To inroduce a magneic field perurbaion o (χ) in accordance wih he FS represenaion {R (r,χ), Z (r,χ)} of [4] we use R=R and Z= Z +Z = Z [- (r)(-m(r)cos χ ) cosχ], (3) where r denoes he FS radius and he funcions (r)<< and M(r)> define he magniude and he poloidal scale of FS perurbaion, respecively. Taking (r)= (r/a) and M(r) =.7 wih (a)=.4.6 and a being he plasma radius, he chosen perurbaion Z conforms qualiaively wih he axisymmeric perurbaion. (χ) cos(6χ) ha fis he equilibrium FSs and (χ) in NSTX (wih β = 4%) beer han (χ). This is demonsraed in NSTX, β = 4%, r =.58m = + = EFIT.6.8 a).6 NSTX (4%), model = +, r=.445m Z (m) (T) (χ)/ (χ)..8.4 M=.5 M=.7 χ χ=6.3 χ= R (m) Poloidal angle (deg) poloidal angle, χ, [deg] Fig. FSs and poloidal dependences of model and EFIT magneic fields in NSTX (β=4%). FS radius is r=.58 m. Fig. Poloidal dependences of of model magneic filed (Eq.(3)) for =.6 and M=.7 and M =.5. FS radius is r=.455 m. Fig. where he FSs and (χ) are displayed for a given flux surface radius r =.58 m for he unperurbed ( = ), he perurbed ( =.3) and he EFIT magneic field, respecively. The main discrepancy beween and he EFIT field is observed a he ouboard par of he plasma column. I is seen ha he above induces an addiional modulaion of (χ) such ha he modeled is congruous wih he EFIT field. The ypical poloidal dependence of (χ) a r =.445 m ) is illusraed in Fig. for M =.7 and.5, and for =.6 ( =.6). As seen, reducing M from M =.7 o M =.5 resuls in an increase of he ypical poloidal scale χ from 3 o 79. Roughly, his corresponds o a decrease of he poloidal mode number of PF ripples from m 6 o m. Nex we examine he effec of poloidal field ripples on he value of / by using he Taylor expansion of () = () + () abou he mid-

3 3 plane, i.e. ( ) = + ω ( ) + Ο( ) wih ( ) he value / is given by 3 ω =. Following Ref. [3], 3 =, (4) ω V χ ll + where = (=). Using he FS represenaion, Eq. (3), for he poloidal ripple induced enhancemen of he non-adiabaic change of he magneic momen, we hus obain ( ) ( ) exp = exp ( M ) + ( ) ( ) p M + d / dr, (5) where (r) represens he Shafranov shif. Taking ino accoun ha, in NSTX, he raio p / a he ouer mid-plane, i is apparen from Eq. (5) ha poloidal ripples wih of he order of several percens may enhance he non-adiabaic variaions of he magneic momen of fas ions more han -3 imes (if M 3 and />5). This conclusion is confirmed by numerical calculaions presened in he following secion.. Resuls of numerical calculaions Z (m) a) NSTX (β=4%), E=3keV = = + <<3 <<45 = = R (m) R (m) R (m) R (m) d) crucial effec of small-scale poloidal perurbaions of he magneic field on he paricle moion. While in he unperurbed magneic field he non-adiabaic variaion of µ during 7 bounce periods is of order % (Fig.4a), he perurbaion. (χ) cos(6χ) effecs a non-adiabaic change of µ of order % (Fig.4. Doubling of he perurbaion yields non-adiabaic of order 3% (Fig.4, which is comparable o he 45% variaion of he magneic momen observed in he numerical field (Fig. 4d). However, we noe ha he raher srong non-adiabaiciy observed in he EFIT field may be possibly caused also by small-scale numerical errors. From Fig.b we see small-scale poloidal modulaions of remaining in discrepancy wih he perurbed EFIT Fig. 3 Full gyro-orbis of 3 kev comoving deuerons in NSTX during 7 bounce periods. Figures 3 and 4 display he full gyro-orbis and he ime variaions of he magneic momen of 3 kev co-moving deuerons in he unperurbed (Figs. 3a, 4a), he perurbed (Figs. 3b, 3c, 4b, 4 and he numerical EFIT magneic field (Figs. 3d, 4d) of NSTX wih β = 4%. All orbis sar a R =.433 m in he equaorial plane wih an iniial velociy saisfying he resonance l ω /ω b = 5, where ω and ω b are he values of he averaged gyro- and he a).5 bounce frequency, respecively. Fig. 4.5 =, NSTX(β=4%), E =3keV demonsraes he d µ()/µ bounce averaged () d) = +, NSTX(β=4%), E d =3keV = +, NSTX(β=4%), E d =3keV µ µ bounce averaged.5 EFIT, NSTX(β=4%), E =3keV d Time (au) Fig.4 Variaions of µ of 3 kev co-going deuerons in NSTX during 7 bounce periods

4 field, which are apparenly responsible for he srong non-conservaion of µ even a moderae (E<3 kev) energies. The numerically calculaed non-adiabaic jumps of µ of 4 kev comoving deuerons in NSTX (β=4%, =.6) are depiced in Fig. 5 as a funcion of he gyro-phase θ when he paricle has crossed he mid-plane. The observed (θ) are seen o be in saisfacory agreemen wih analyical predicions, hus confirming he resonan naure of he non-adiabaic jumps of µ. Finally we consider he conribuion of poloidal field ripples o max. In Fig. 6 we display he numerically obained dependence of max on he magniude of he FS shape perurbaion,, for he parameers of Fig. 5. The dependence max () is essenially nonlinear. For.8 he conribuion of FP ripples is weak and numerical calculaions /µ cosθ µax. NSTX(β=4%), = +, r=.445m,.8 E d =4keV, λ =.7 /µ.4 max /µ max /µ. M=.5 M=.7 EFIT NSTX(β=4%), = +, =.6, M=.5, r=.445m, E =4keV, λ =.7 d Gyro-phase, [deg] Fig. 5: Non-adiabaic jumps of he magneic momen as a funcion he gyro-phase when he paricle has crossed he mid-plane Fig. 6: Magniude of non-adiabaic jumps of he magneic momen vs he magniude of FS shape perurbaion max () max (), whereas for >.8 we obained max ( ) ~. For =.7 corresponding o he EFIT poloidal modulaion of for a flux surface radius r =.445 m =.66a, he PF ripples enhance he non-adiabaic changes of he magneic momen of 4 kev deuerons more han imes. Fig. 6 demonsraes also he essenial reducion of nonadiabaic jumps of µ when he poloidal scale of field perurbaion χ is increased from 3 (M =.7) o 78 (M =.5). The red poin in Fig. 6 gives he value of max ( =.7, M =.5) ha is abou 35 imes less han max ( =.7, M =.7). Noe ha he value max () presened in Fig. 6 is in saisfacory agreemen wih formula of Eq. (5). 3. Discussion The poloidal ripple effec considered here can essenially impac on fas ion ranspor in spherical ori. The reason for he srong effec of small-scale poloidal -perurbaions on non-adiabaic jumps of he fas ion s magneic momen is heir significan influence on he local resonance ω = in he viciniy of he ST mid-plane. Noe ha his resonance can be significanly affeced also by small-scale oroidal perurbaions of non-axisymmeric magneic configuraions, e.g of a sellaraor field [5]. This is confirmed by Fig. 7 depicing he ime variaions of he magneic momen and of he full gyro-orbi of a 5 kev deueron in he poloidal cross secion of he Compac Helical Sysem (CHS, Japan) during more han 3 bounce periods τ b. The orbi sars a R =. m in he equaorial plane wih an iniial velociy V R =, V ϕ =.896V. I can be seen ha µ execues a non-adiabaic superbanana oscillaion wih a period abou ( )τ b feauring raher srong non-adiabaiciy, /µ

5 %, conrary o he expeced exponenially small exp(- /) for an adiabaiciy parameer ρ L /R c = (3 5)%. magneic momen, [au].8 τ a).6 b CHS, R =.m, E d =5keV, V /V=.896, V =, V > ϕ R Z Z, [m] CHS, R =.m, E =5keV, V /V=.896, V =, V > d ϕ R Z Y, [m] ime, [au] R, [m] X, [m] Fig. 7 Time variaions of µ and full gyro-orbi of a marginally confined 5 kev deueron in CHS during 4τ b. Evidenly, TF ripples in convenional okamaks may also resul in an enhancemen of non-adiabaic jumps of he magneic momen of fas ions, if TF Nq R ρ a L TF ( + d / dr)( Λ) N 5, where TF denoes he TF ripple magniude, N he TF coil number and Λ is he FS riangulariy. Acknowledgemen This work has been parially carried ou wihin he Associaion EURATOM-OEAW projec P4. The conen of he publicaion is he sole responsibiliy of is auhors and does no necessarily represen he views of he European Commission or is services. References [] V. A. Yavorskij, e al., Proc. 8h EPS Conf. on CFPP, Madeira,, P5.5; hp:// Nucl. Fusion 4 () [] J. Carlsson, Phys. Plasmas, 8 () 475 [3] R. J. Hasie, e al., Plasma Phys. Conr. Fusion, (IAEA, Vienna, 969) (969) 389 [4] V. A. Yavorskij, e al., Plasma Phys. Conr. Fusion, 43 () 49 [5] M. Isobe, e al., Nucl. Fusion, 4 () 73