Gyrokinetic calculations of the neoclassical radial electric field in stellarator plasmas

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1 PHYSICS OF PLASMAS VOLUME 8, NUMBER 6 JUNE 2001 Gyrokineti alulations of the neolassial radial eletri field in stellarator plasmas J. L. V. Lewandowski Plasma Physis Laboratory, Prineton University, P.O. Box 451, Prineton, New Jersey J. Williams and A. H. Boozer Department of Applied Physis, Columbia University, New York, New York Z. Lin Plasma Physis Laboratory, Prineton University, P.O. Box 451, Prineton, New Jersey Reeived 12 September 2000; aepted 15 Marh 2001 A novel method to alulate the neolassial radial eletri field in stellarator plasmas is desribed. The method, whih does not have the inonveniene of large statistial flutuations noise of the standard Monte Carlo tehnique, is based on the variation of the ombined parallel and perpendiular pressures on a magneti surfae. Using a three-dimensional gyro-kineti f ode, the alulation of the radial eletri field (E r ) in the National Compat Stellarator Experiment G. H. Neilson et al., Phys. Plasmas 7, has been arried out. It is shown that a diret evaluation of E r based on a diret alulation of the radial partile flux is not tratable due to the onsiderable noise Amerian Institute of Physis. DOI: / I. INTRODUCTION The lak of toroidal symmetry of stellarators requires a fully three-dimensional desription of the plasma. The departure from axi-symmetry in stellarator plasmas leads to enhaned neolassial losses in the low-ollisionality regime. Another related feature of the effet of the non-axisymmetry of the plasma is to strongly modify the drift orbits of the partiles see the reviews by Sadgdeev and Galeev 1 and by Kovrizhnykh 2. Over the past few years, there has been a renewed interest in the so-alled quasi-axisymmetri QA stellarator onept; that is the equilibrium magneti field strength is approximately symmetri in the magneti toroidal angle, after transformation to Boozer oordinates. 3 A stellarator experiment based on the QA onepts is urrently being designed in the United States; 4 the National Compat Stellarator Experiment NCSX 5 is a three-field period, low-aspetratio onfiguration whih has good transport and stability properties. 5 7 One important feature of the QA onept is that the plasma an rotate in the diretion of quasiaxisymmetry, and it may be possible to exploit and ontrol the formation of transport barriers, as in advaned tokamak plasmas. 8 Sine the radial eletri field is a major ontender 9 in the formation of transport barriers, an aurate alulation of the radial eletri field is an important aspet of QA plasmas. The reader who is not familiar with urrent trends in stellarator design an onsult Ref. 10. In this artile we desribe a novel method to alulate the neolassial radial eletri field in asymmetri toroidal plasmas. The low-noise method, whih exploits the advantages of the f algorithm, is based on the variation of the ombined parallel and perpendiular pressures (P and P ) on a magneti surfae. As an example, we have alulated the radial eletri field E r for the C82 onfiguration of the NCSX plasma. 5 It is shown that a diret Monte Carlo alulation of the radial eletri field based on the radial partile flux is not tratable due to the large statistial flutuations; interestingly, the gyrokineti alulation using the variation of P and P on a given magneti surfae does not exhibit large statistial deviations, whih allows for a determination of E r. The artile is organized as follows; in Se. II, we desribe the method used to alulate the radial partile flux. The numerial method, the omputational details and the results are given in Se. III. Conluding remarks and future appliation of the method are presented in Se. IV. II. THE METHOD In this setion, we desribe the method used to determine the radial eletri field based on the loal variation of the parallel and perpendiular pressures (P and P, respetively. The quantities P and P are evaluated by taking appropriate veloity moments of f f f 0, where f is the total distribution funtion whereas f 0 is its equilibrium part usually f 0 is taken to be a Maxwellian distribution. The perturbed part of the distribution funtion, f, evolves due to the ombined effets of magneti drifts and spatial inhomogeneity. Another subtle point regarding the numerial alulation is that the veloity moments for P and P are arried out in small annulus i.e., finite volume around a magneti surfae of referene; this point is disussed in more detail in Se. III. In stellarator geometry, it is onvenient, both for analytial and omputational purposes, to use magneti oordinates. The onfining magneti field B is written in Boozer oordinates 3 as X/2001/8(6)/2849/6/$ Amerian Institute of Physis

2 2850 Phys. Plasmas, Vol. 8, No. 6, June 2001 Lewandowski et al. B, Bg I, 1 S(E r,t) will provide a measure of the time variation of the harge Q. For one-speies simulation, one an then determine the radial eletri E r (0) by solving where and are the poloidal and toroidal angles, respetively; is proportional to the enlosed toroidal flux; is the rotational transform; and g() and I() are, within a multipliative onstant, the poloidal and toroidal urrents, respetively. In these oordinates the Jaobian of the transformation, J ( ) 1, satisfies JB 2 g() ()I()F(). Consider the momentum of the balane equation for the ions dv dt Pen E VB FR, 2 where V is the fluid veloity, is the mass density, R is the fore due to ollisions, F represents the possible applied fore, and PP bˆbˆ(ibˆbˆ)p is the pressure tensor; here bˆb/b is the unit vetor along B, P and P are, respetively, the parallel and perpendiular pressures. We note that Eq. 2 an be obtained by taking the first-order veloity moment of the kineti equation in the small gyro-radius limit. Taking the salar produt of e r/ where r is the position vetor with Eq. 2, and operating with......j(,,)dd, we obtain dq dt dl dt Pˆ T, where Pˆ (P P )/2, T (RF) e is the torque due to applied fores and ollisional drag; L is the toroidal omponent of the anonial momentum LVeA/, where A is the vetor potential and 2 is the poloidal flux. In deriving Eq. 3 we have assumed that the eletrostati potential yet to be determined is of the form (), and we have negleted the loop voltage (/t 0). Using Eq. 1, the first term on the left-hand side of Eq. 3 an be obtained from ene VB nv Be e Jd d e d n dq dt. Here Q is the total harge, nv is the partile flux, and d n J d d is an area element normal to the magneti surfae onst and pointing outwards. The first term on the right-hand side of Eq. 3, whih is related to the pressure tensor term in Eq. 2, is derived in the Appendix. The parallel and perpendiular pressures whih enter SPˆ / are alulated from the veloity moments of f whih, in turn, depends on the partile trajetories. The partile trajetories being affeted by the applied eletri field E (d/d), we may write SS(E r,t). After a few ollision times the distribution f will relax and S (E r ) 3 S E r (0) 0. Alternatively, one an measure the flux surfae-averaged radial partile flux r (E r ) and determine E (0) r suh that the partile flux vanishes. However, in quasi-axisymmetri toroidal onfiguration, the large statistial flutuations in r an be omparable to, or larger than, the time-averaged signal. As will be shown in the next setion, a diret measurement of the radial partile flux is too noisy to be of pratial use. Alternatively, a dynami alulation using the global gyrokineti toroidal ode GTC, 15 whih has been rigorously benhmarked against analytial tokamak neolassial transport theory, 16 shows that S(E r,t) reahes a low-noise asymptoti value after a few ion-ion ollision times. The general behavior of S is of the form S(E r,t)s 0 exp(ct/ ii )S 1exp(Ct/ ii ), where ii is the ion-ion ollision time, S 0 (E r )S(E r,t0), and C is a onstant of the order of unity. It is interesting to note that S provides information on the asymmetri part of the partile transport. As it turns out, S stritly vanishes in an axisymmetri onfiguration suh as an ideal, two-dimensional tokamak plasma. To show this, we use the definition of the fluxsurfae average and perform an integration by parts: SPˆ J d d J 1 Pˆ J. Noting that JB 2 is a flux surfae quantity it follows that J 1 J/2B 1 B/ and S P P B B, showing that S0 in an axisymmetri onfiguration. We note that the off-diagonal ontributions in the pressure tensor have been negleted in terms of the smallness parameter i /R 0 ). The inlusion of finite Larmor radius FLR effets, suh as in the paper of Rosenbluth et al., 17 an lead to nonambipolar transport. III. NUMERICAL METHOD AND RESULTS In this setion, we desribe the low-noise numerial method used to evaluate S(E r,t), whih provides a measure of the radial partile flux. The omputational domain and the ollision operators are also disussed. Finally we present speifi numerial results for the NCSX plasma. It is onvenient to write the parallel and perpendiular pressures in terms of their respetive Fourier omponents on a given magneti surfae; for example, the perpendiular pressure an be written as P m,n P m,n expimnn p, where N p is the number of field periods of the onfiguration and the Fourier oeffiients are alulated aording to 4 5 6

3 Phys. Plasmas, Vol. 8, No. 6, June 2001 Gyrokineti alulations of the neolassial radial FIG. 1. Computational domain for f alulations of the radial eletri field. The magneti oordinates are, and. The partiles are initialized at the toroidal flux surfae. The width of the annulus is suh that / b 1, where b is the toroidal flux at the plasma boundary. P m,n 0 2 d 2 0 dmv 2 /2 f expimnn p d 3 v. 2 0 d 2 0 d 7 In pratie the partile trajetories are integrated in a volume enlosed between two neighboring toroidal flux surfaes /2 and /2 Fig. 1. Here is a magneti surfae of referene and is hosen so that / b, where b is the toroidal flux at the plasma boundary, is muh less than unity. However, due to the ombined effets of magneti drifts and bakground inhomogeneity, the partiles will drift outside the layer, the number of Lagrangian markers within the layer will derease in time, and the statistis assoiated with P and P will beome poorer. To bypass this diffiulty, partiles are uniformly loaded in the layer but the perpendiular and parallel pressures are monitored in the annulus Fig. 2. Introduing the radial oordinate r/b 0 where as before is the enlosed toroidal flux, one has (rr)(r)2b 0 rr from whih we get r/(2b 0 ). The radial omponent of the urvature drift veloity is of the order of V d v th ( th /R), where v th is the thermal veloity, th is the assoiated thermal gyroradius and R is the major radius; here we have used R B/B 1 R as the typial sale length of the magneti field inhomogeneity. Therefore the typial drift time within the layer is d r/v d, whih is hosen so that d r, where r is the relaxation time typially a few ion-ion ollision time. Alternatively, one an assume periodiity in the radial diretion, that is the partile that esapes the radial domain an be put bak in the same domain; in this ase, the number of Lagrangian markers remains onstant. Different shemes an be used and ompared to randomize the position (,,) and the pith of the partile. However, as in a real experiment, one is interested in the global radial profile of the eletri field; one important aspet of the method previously desribed is that it an easily be extended to the entire plasma volume, without additional assumptions regarding the boundary onditions. The guiding enter motion and the ollisions will spread the partiles toward equal density in pith and over the magneti surfae; therefore it is onvenient to make the replaement again using the partiular form of the Jaobian in Boozer oordinates and the definition of the volume element in magneti oordinates, d 3 xjdd d) d d J 1 1 d 3 x F 1 B 2 d 3 x. Therefore one an alulate the Fourier oeffiients for the perpendiular pressure aording to P m,n d3 xmv 2 /2 fb 2 expimnn p d 3 xb 2 d 3 v. The same method was used to evaluate the parallel pressure on the magneti surfae. As is well known in neolassial theory, the momentum and energy onservation properties of the ollision operator are important for aurate alulation of quantities suh as the radial partile flux. The gyrophased ollision operator for like-speies ollisions an be written as 18 C f f v 2 v f 2 2 v v f 2 v f v 2 2 f, where 0 v F, v 2 2FHG2v 2 G, 2 0 v 2 v HG, 0 v 2 H/2v 2 G/2, FIG. 2. The Lagrangian f markers are uniformly distributed in the layer of width ; the radial width is hosen suh that the typial drift time of an ion in the layer is larger than the relaxation time of the perturbed parallel and perpendiular alulated in the annulus entered at. 2 0 v 2 v 2 Hv 2 G. 10 In Eq. 10, 0 4n q 2 q 2 ln /(m 2 v 3 ) is the basi frequeny for ollisions of test partiles with bakground partile. F, G, and H are dimensionless funtions that an be written in terms of the Maxwell integral (x) 2 1/2 x 0 y 1/2 exp(y)dy, where xv 2 2 /v th and v th is the thermal veloity of the bakground partiles. The funtions

4 2852 Phys. Plasmas, Vol. 8, No. 6, June 2001 Lewandowski et al. FIG. 3. Magneti surfae of the three-field period National Compat Stellarator Experiment NCSX. Although the shape of the magneti surfae is largely different for that of a omparable tokamak, the equilibrium B field is approximately symmetri in magneti toroidal angle after transformation to Boozer oordinates. F, G, and H are F(x)(1m /m )(x), G(x)(x)1 1/(2x)d/dx and H(x)(x)/x. The test-partile drag and diffusion an be implemented by utilizing a Monte Carlo method due to Xu and Rosenbluth. 18 The partile weights are modified suh that the ollision operator annihilates a shifted Maxwellian. 16 For ion-eletron ollisions a Lorentz ollision operator is used, and its Monte Carlo implementation has been disussed elsewhere. 19 For the simulations presented in this artile, the trajetories of a set of Lagrangian markers have been integrated with a time step t/ ii Collisional effets are alulated every ten time steps. The onfining B field and the shape of the magneti surfaes (R,Z,) have been speified in terms of Fourier series Fig. 3; a set of 30 Fourier harmonis have been retained in the alulations. Other parameters are the on-axis magneti field B G, entral ion temperature T i (0)2.76 kev, entral eletron temperature T e (0)2.14 kev and entral plasma density n m 3 these parameters are the typial design parameters for NCSX. The magneti surfae of referene is loated at / b 0.7. At eah time step, the loal i.e., within ; see Fig. 2 perpendiular and parallel pressures from eah proessor element PE are olleted onto a single PE say PE0; the Fourier oeffiients for P and P are then alulated aording to Eq. 7. All numerial parallel omputations reported here have arried out with 16 PEs. The radial partile flux obtained from a diret measurement of the radial partile flux arbitrary units is shown as the jagged urve in Fig. 4 in this ase the applied radial eletri field is 0 and the bakground distribution funtion f 0 has been loaded as a Maxwellian with V 0). It is noted that the noise is omparable to the signal; to determine the relation r r (E r ) from a diret measurement is not aurate. As a omparison, the thik line in Fig. 4 represents the FIG. 4. Radial partile fluxes arbitrary units at the magneti surfae / b 0.7 as obtained from a diret measurement thin, broken line and from the fluid moment approah thik line. The perpendiular and parallel pressures relax on a time sale of the order of a few ion-ion ollision times. The applied radial eletri field is zero and the time has been normalized to the ion-ion ollision time ii. radial partile flux alulated from the veloity moments P and P. The eletron urrent density alulated for large t as a funtion of the normalized radial eletri field E r ād/dr(e/t i (0)) is shown in Fig. 5. The eletron urrent density displays an almost linear dependene on E r. The ion urrent density, shown in Fig. 6, shows, however, a strong dependene on the E r parameter. The largest ion flux is obtained for E r 0.2. We note that the ion flux is typially two orders of magnitude larger than the eletron flux for small eletri field note the sale differene between Figs. 5 and 6. By inverting the relation i E r (0) e E r (0), we obtained E r (0) 0.87 statvolt/m, that is E r (0) 26.2 kv/m whih orresponds to the stable root. For illustrative pur- FIG. 5. Eletron urrent density as a funtion of the normalized radial eletri field E r ād/dr(e/t i (0)), where ā is the average minor radius of the last losed magneti surfae, and T i (0) is the ion temperature at the magneti axis.

5 Phys. Plasmas, Vol. 8, No. 6, June 2001 Gyrokineti alulations of the neolassial radial where bˆb/b is the unit vetor along B, I is the unit dyadi, and P and P are, respetively, the parallel and perpendiular pressures. We an also write Eq. A1 as PP BB P I, where P (P P )/B 2. The divergene of the pressure tensor then reads PBB P P BB P, A2 where the seond term on the right-hand side an be alulated using the relation B 2 /2B( B)(B )B so that BB 1 2 B 2 B B. A3 FIG. 6. Ion urrent density as a funtion of the normalized radial eletri field plasma parameters are the same as in Fig. 5. poses, the alulations have been arried out for a single magneti surfae more preisely, for a single annulus, but a global alulation would lead the variation of E r with the radial oordinate. IV. CONCLUDING REMARKS In this artile, we have presented a method to alulate the radial eletri field in stellarator plasmas; our method is partiularly useful for toroidal plasmas whih depart weakly from axisymmetry e.g., the quasi-asymmetri onept. It is has been shown that a diret measurement of the radial partile flux is very noisy, and not of muh pratial use for a diret alulation of E r. The moment approah, however, shows a relatively smooth behavior and reahes an asymptoti value after a few ion-ion ollision times. The method presented in this artile an be improved by inluding the off-diagonal terms whih are i /R 0 times smaller than the diagonal terms in the pressure tensor. However, sine the dominant symmetry breaking term in the B mn spetrum is muh larger than the smallness parameter i /R 0, one an, in first approximation, neglet the offdiagonal terms in the pressure tensor. An extension of the method presented in this artile an be used to determine the damping rates of toroidal and poloidal flows in stellarator plasmas; this is left for future work. ACKNOWLEDGMENTS One of us J.L.V.L. would like to thank W. W. Lee for enouragement. The method for arrying out redued-noise f Monte Carlo alulations in single-heliity magneti onfigurations is part of the dotoral thesis of one of us J.W.. Using Ampere s law, 4J B, and the radial fore balane, JB P 0, one gets B( B)4 P 0 and the divergene of the pressure tensor now reads PBB P P 1 2 B 2 4 P 0 P. A4 Taking the salar produt of Eq. A4 with e r/ where r is the position vetor and, one gets e PB B P P 2 B2 P, A5 where we used the fat that the equilibrium pressure is a flux surfae quantity, P 0 P 0 (). In Boozer oordinates, the produt of the Jaobian and the magneti field strength squared is a flux surfae quantity, that is, JB 2 F(), where J ( ) 1 denotes the Jaobian and F() isa linear ombination of the poloidal urrent, toroidal urrent and safety fator. We introdue the flux-surfae average operator as J dd. A6 It is easy to show that annihilates the B operator, that is, B G0 A7 for any funtion GG(,,). Noting that, in Boozer oordinates, B and B are flux-surfae quantities, the fluxsurfae average of Eq. A5 is APPENDIX: DIVERGENCE OF THE PRESSURE TENSOR The pressure tensor, P, an be written as e P 1 2 P P, A8 PP bˆbˆp Ibˆbˆ, A1 where we used Eq. A7 and

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