A Fully-Neoclassical Finite-Orbit-Width Version. of the CQL3D Fokker-Planck code

Transcription

1 A Flly-Neoclassical Finite-Orbit-Width Version of the CQL3 Fokker-Planck code CompX eport: CompX-6- Jly, 6 Y. V. Petrov and. W. Harvey CompX, el Mar, CA 94, USA

2 A Flly-Neoclassical Finite-Orbit-Width Version of the CQL3 Fokker-Planck code Y. V. Petrov and. W. Harvey CompX, el Mar, CA 94, USA ABSTACT The time-dependent bonce-averaged CQL3 flx-conservative finite-difference Fokker-Planck eqation (FPE) solver has been pgraded to inclde Finite-Orbit- Width (FOW) capabilities which are necessary for an accrate description of neoclassical transport, losses to the walls, and transfer of particles, momentm, and heat to the scrape-off layer. The FOW modifications are implemented in the formlation of the netral beam sorce, collision operator, F qasilinear diffsion operator, and in synthetic particle diagnostics. The collisional neoclassical radial transport appears natrally in the FOW version de to the orbit-averaging of local collision coefficients copled with transformation coefficients from local (, Z) coordinates along each giding-center orbit to the corresponding midplane comptational coordinates, where the FPE is solved. In a similar way, the local qasilinear F diffsion terms give rise to additional radial transport of orbits. We note that the neoclassical reslts are obtained for fll orbits, not dependent on a common small orbit-width approximation. eslts of validation tests for the FOW version are also presented. PACS COES: 5.65.Ff, 5.65.Cc, 5.55.Fa

3 . Introdction Finite-difference Fokker-Planck (FP) codes are essential comptational tools for the theoretical interpretation of existing plasma F and NBI heated experiments and for projections to new experiments. All present-day FP codes se varios simplifications becase a blnt solve of 6 kinetic eqation (3 spatial + 3 velocity) is a formidable problem even for modern sper-compters. Usefl redction of dimensionality can be achieved for most of the plasma volme in high temperatre tokamaks and other toroidally symmetric fsion energy devices, by averaging over periodic particle trajectories. A bonce-averaging (BA) procedre for toroidally symmetric magnetic geometry is based on assmption that bonce time (more generally, poloidal transit time) b of charged particles is mch smaller than the collision time or a characteristic time associated with any other diffsive process. This is commonly the case for energetic particles and most of the blk electrons and ions in tokamaks. It is also assmed that time variations of plasma parameters and the distribtion fnction are mch slower than the bonce time and gyro-period. Then, the fndamental kinetic eqation can be averaged over all three periodic motions of particles, therefore redcing the dimensionality from 6 to 3 [-4]. The CQL3 (Collisional QasiLinear 3) conservative finite-difference bonce-averaged Fokker-Planck eqation (FPE) solver [5, 6] is bilt on the collision model in the -in-velocity, -in-radis, zero-orbit-width (ZOW) bonce-average CQL code [7, 8]. The ZOW CQL3 was motivated by the FPP code [9], and is similar to other BA FP codes [-6], bt is different with regard to: greater generality, being mltispecies, having a flly relativistic option and having larger nmber of synthetic diagnostic tools. It ses flx-conserving differencing, and ths conserves particles, has a flly nonlinear Colomb collision operator, and derives

4 qasilinear F diffsion terms from wave data that is sally exported from the general, all-freqencies GENAY ray-tracing code [7]. It also models a netral beam sorce of ions, and is copled with F fll wave solver [8] and transport codes as part of the SciAC Integrated Plasma Simlator project [9]. Until recently CQL3 was restricted by a ZOW approximation which neglects giding-center drift perpendiclar to magnetic flx srfaces; the present work describes the new version of the code with the giding-center Finite-Orbit-Width (FOW) capabilities. The otline of the paper is as follows. In section, a general view of the FOW-FPE is smmarized, first in terms of action-angle variables which make orbit averaging simple, then transforming to more convenient variables. In section 3, the derivation of transformation coefficients is given. Section 4 provides the Jacobian for the transformation. In section 5, the bonce-average coefficients are derived in terms of the local diffsion tensor components and transformation coefficients. In section 6, we show that the FOW-FPE converges to the ZOW form in the limit of thin banana orbits. The particle sorce operator and the F qasilinear operator are described in sections 7 and 8. Qestions of the initial setp of the distribtion fnction and its rescaling at later time steps are discssed in section 9. The bondary conditions are discssed in section. Test reslts, based on National Spherical Tors Experiment, NSTX [] eqilibria, are provided in section. Section is the conclsion. Implementation of the internal bondary conditions connecting trapped and passing orbits is discssed in Appendix A.. General formlation of CQL3-FOW In the ZOW approximation, all orbits that cross the midplane at a given major radis coordinate (ot-board point of a magnetic flx srface) are confined to the same flx 3

5 srface; therefore, the bonce-averaging is redced to averaging over poloidal angle [8]. Then, the bonce-average FPE can be expressed throgh the comptational coordinates (,, ) where is the flx srface label, is the particle speed (relativistically, momentm per rest mass) and is the pitch angle at the oter-board midplane point (minimm B point) for a given flx srface. Sbscript refers to the minimm-b point for each given srface. The details of BA procedre in the ZOW approximation can be fond in Killeen [8] and the CQL3 code manal [6]. In contrast to the ZOW limit, bonce-averaging of the FPE with finite-width (FOW) orbits cannot be redced to averaging over poloidal angle. A general method for derivation of the FPE in this case is based on two main steps [-3]: () Using a Lagrangian formlation, write the six-dimensional kinetic eqation in the canonical action-angle space (J,) (note that the Jacobian of sch transformation is one); () Perform integration over canonical angles, which correspond to the three periodic motions: gyration arond magnetic field, orbital transit or boncing in poloidal angle, and toroidal precession of orbits. The redced eqation is a fnction of the remaining three action variables J, t f J J ij i J t f J, t F f J, t S( J, ), t i j (.) where the smmation convention is sed. A particle sorce sch as from NBI is represented by S. The orbit-averaged (canonical-angle averaged, to be exact) diffsion coefficients ij and advection-like coefficients F i are expressed throgh the corresponding spatially local vales and F at points along orbit i j ij J J (.) 4

6 and J i i F F. (.3) The local coefficients may inclde the collisional pls F diffsion coefficients, while F coefficients are from the collisional drag (and/or toroidal dc electric field). In these eqations, is the particle local momentm-per-rest-mass, enabling a relativistic treatment. In the eqations, one of the canonical action variables J corresponds to the magnetic moment, another - to the canonical toroidal anglar momentm p, and the third represents the toroidal magnetic flx enclosed by an orbit in the poloidal crosssection. The last one is not particlarly sitable for nmerical coding, especially for setting a comptational grid. In general, another set of invariants can be selected that identifies each orbit and which is convenient for setting the comptational grids. A nmber of choices for sch a set of invariants had been considered in literatre. One straightforward choice (not sed here) is I = (, /, p ). The derivation of the Jacobian of transformation from the canonical action variables to these type of I variables is shown in []. One shold notice, however, that this set of variables does not contain an isolated configration space coordinate. Instead of the mixed velocityradial variable p, other athors se a selected vale of a radial-type coordinate along the giding center orbit, sch as the maximm vale of the poloidal or toroidal magnetic flx along orbit [, 5], or the orbit-average vale of poloidal flx [4], or the flx srface radis at the innermost point of a passing orbit and radis at the bonce point of trapped particle orbit [3]. 5

7 For the FOW version of the CQL3, or choice of convenient variables is dictated by the desire to keep comptational grids as close as possible to the original ZOW version, and to maintain direct visalization of the reslts in velocity and configration space. Ths, we adopt I = (,, ), where is the major radis coordinate along the eqatorial plane (the midplane of tokamak, in case of p-down symmetrical eqilibrim); for each given point, the vale of particle speed (momentm-per-mass) and vale of the pitch-angle at this point determine a niqe orbit. The same as in CQL3-ZOW notation, sbscript refers to points on the midplane. Sch choice of I also yields an intitively clear meaning of the soltion of FPE: the distribtion fnction f(,, ) is the local distribtion fnction of giding centers at point on the midplane, as shown in figre. For any other point away from the midplane, the distribtion fnction, which is constant along each orbit, can be readily reconstrcted from the set of compted distribtions f( l, l, l ) at radial-grid points l. It is seen that each orbit contribtes to two distribtion fnctions on the midplane: f( a, a, a ) and f( b, b, b ), where a and b refer to the two points where orbit crosses the midplane (two legs of an orbit), and, of corse, the vale of f mst be eqal at these points. The transformation from canonical action space J to the adopted comptational space I = (,, ) reslts in the general form of the bonce-averaged FPE for the particle distribtion fnction f (sing notations of [4]): t f I I ij i I t f I, t F f I, t S t,, i j I. (.4) The Jacobian for the transformation, J/I, is sch that f is the nmber of particles in the volme element d 3 I. istinctive from eqations (.-.3), the 6

8 bonce-averaged diffsion coefficients and advection terms are now expressed throgh the transformation coefficients I/, rather than J/, i j ij I I (.5) and I i i F F (.6) where tensor describes a local diffsion, sch as cased by collisions or resonant F heating, and vector F corresponds to the collisional friction. Note that Colomb collisions and F QL diffsion are spatially local phenomena, changing only particle velocity, not position. However, scattering locally in velocity changes the spatial trajectory, and throgh this change the bonce-average radial diffsion terms are formed. In general, the r.h.s. of eqation (.4) may inclde other terms. For instance, the presence of dc toroidal electric field at particle g.c. position adds the following term into the local FPE, f t E q m f E qe m eˆ cos eˆ sin f, where ˆ, ˆ are the nit vectors in the local (, ) velocity space. After the e e transformation and bonce-averaging, we obtain f I, t i F f, t I t i E I E, where 7

9 F i E i i I I q m E cos E sin. (.7) In addition, eqation (.4) also incldes the particle sorce term corresponding to NBI. It mst be expressed throgh the same set of variables as f(,, ). The derivation of the sorce term is given in section Transformation coefficients The physical meaning of the transformation coefficients in eqations (.5-.6), for or choice of I = (,, ), is to describe how an infinitesimal change in local -space at an arbitrary point (, Z) of particle orbit modifies the vales of (,, ) at the corresponding orbit position at the eqatorial plane. The derivation of transformation coefficients is based on npertrbed orbits specified by the conservation of energy, magnetic moment and toroidal anglar momentm, which we write in normalized form as (assming no radial electric field potential) = sin /B = sin /B (q/m) pol (,Z) + b cos = (q/m) pol ( ) + b cos, (3.a) (3.b) (3.c) where m is the mass, q is the charge, is the major radis, b = B /B is the ratio of the toroidal component of the magnetic field to the magnetic field strength, and pol is the poloidal magnetic flx normalized by. For brevity, eqation (3.c) for p conservation is written for the case of positive plasma crrent I and positive toroidal field B (conter-clockwise, if viewed from top). In the general case for I and B directions, the pitch-angle between particle velocity and the direction of the 8

10 9 magnetic field is defined in sch a way that cos >, if the particle travels in cocrrent direction. In the matrix form, the components of I/ are I. (3.) The lower row is composed of zeros becase the gyro-phase is an ignorable coordinate in the local space. The elements in the top row are fond by applying / to each side of eqations (3.), = / (3.3a) = (sin cos /B ) ( /) (sin /B ) (B / ) ( /) (3.3b) b cos = (q/m)( / ) ( /) + ((b )/ ) cos ( /) + + b ( /)cos b sin ( /), (3.3c) where pol ( ). This leads to cos sin cos ) ( cos cos B B b b m q b b, (3.4) and / is expressed throgh / as B B cos sin. (3.5) Similarly, by applying / to each side of eqations (3.), we obtain

11 b sin b B B sin cos cos q ( b ) sin B cos b m B cos, (3.6) and / is expressed throgh / B B sin cos sin sin cos cos B B. (3.7) Also, in the absence of the radial electric field, / =. 4. The Jacobian for the transformation It is convenient to consider a transformation from the canonical action variables to an intermediate set of invariants, and then a sbseqent transformation to or set I = (,, ). In relativistic form, the npertrbed Hamiltonian can be written as H c ( ) c c, (4.) and the action variables J and J 3 are J m, (4.) q q B q J3 p / m pol b. (4.3) m In the above, m is the rest mass and q is the particle charge, with sign. For the J, we only need to know that H/J = / b, the bonce (or transit) freqency. Also, H/J = qb/(m) = c / is the gyro-freqency; H/J 3 is the anglar freqency of orbital toroidal precession the exact expression is not needed.

12 Consider now the intermediate set of invariants K = (H, qj /mh, mj 3 ), and calclate the Jacobian of the transformation from this set to the canonical action space J. We have H/J = c /, H/J = / b, and also c H J H m q H J m q J, b H J m q H J m q J, (mj 3 )/J = (mj 3 )/J =, and (mj 3 )/J 3 = m. Other terms are not important. Then, the Jacobian is. b H q J K (4.4) Next, we calclate the Jacobian of transformation from the K set to or target set I. After writing the Hamiltonian, J and J variables in eqations (4.)-(4.3) in terms of (,, ), we obtain H/ = /, H/ = H/ =, and also H B H J m q cos sin, (4.5) B H B B H J m q sin, (4.6) 3 sin mj mb, (4.7) 3 cos b m q mj, (4.8) and the other terms are not needed. Then, the Jacobian is. sin cos cos sin 3 B B b b m q B H m I K (4.9)

13 Finally, the reslting Jacobian for the transformation to I is fond from eqations (4.4) and (4.9) as J/I = K/J - K/I, that is. sin cos cos / sin 3 b B B b b m q B m q I J (4.) Note that the expression nder the absolte vale sign, if divided by cos, is the same as the denominator in the transformation coefficients, as in eqation (3.4). This expression becomes zero for the stagnation orbits the g.c. drift orbits that collapse to a point in the (, Z) plane bt still move toroidally becase is not zero [4]. On the other hand, the nmerator in eqation (3.4) or (3.6) also becomes zero for sch orbits. To avoid singlarity in the code, the transformation coefficients like / or / are set to zero in the proximity of the stagnation orbits srface in (,, ). 5. The components of the diffsion tensor We now retrn to the eqation (.5-.6) for the bonce-average diffsion tensor components. Adopting notations sed in [8, Chapter ] or CQL3 manal [6], the components of the local diffsion tensor are written as / sin / / / F C C B. (5.) Also, for the collisional drag term, we have ) sin /( / A F. (5.)

14 For example, in the case of diffsion by Colomb collisions, the coefficients A, B, C,, F are expressed throgh derivatives of the osenblth potentials [5]. A backgrond local distribtion fnction f b (, Z,, ) at a given point of particle orbit is expanded in Legendre polynomials as b fb (, ) Vl ( ) P l (cos ). (5.3) l The expansion coefficients V b ( l ) are frther sed to define integrals like M l b b (l ) V V ( )( / ) d l l, (5.4) where c. These integrals are incorporated into the definition of the osenblth potentials and their derivatives. etails on the definition of the local collision coefficients A throgh F can be fond in [8, Chapter ] or the CQL3 manal [6]. The backgrond species in eqation (5.3) may inclde not only Maxwellian ions and electrons, bt also the species for which the FPE is being solved (referred to as general species in the code manal). In this case the collision operator becomes non-linear. The complexity of its evalation in the code comes from the fact that for the orbit averaging, the local distribtion f b (, Z,, ) mst be known at many discretized (, Z) points along each orbit in the plasma cross-section, while the soltion f f(,, ) is only obtained at the eqatorial plane. To accelerate this calclation, the reconstrction of the local f from the eqatorial f is accomplished by a fast mapping procedre withot the need of tracing an orbit from the local point back to the midplane. Essentially, the procedre tilizes a set of arrays defined over niform grids in (,, p ); these arrays store information on radial coordinates of orbit legs at the midplane ( and ) and corresponding pitch-angles ( and ). For a given local (, Z,, ) point, the vales of (, p ) are readily 3

15 4 evalated, and the nearest indices in the (,, p ) grids are identified, pointing to the vales of (,, ), whence the local f b (, Z,, ) is reconstrcted. For the F-indced qasilinear diffsion, the local coefficients B, C and F are given in section 8. Here, we focs on the general form of the bonce-average coefficients in terms of the local components and transformation coefficients. Based on I/ form given by eqation (3.) and tensor in eqation (5.), the components of bonce-average tensor / / j i ij I I are fond to be sin F C B (5.5a) sin F C B (5.5b) sin 3 3 F C B (5.5c) sin F C B (5.5d) sin 3 3 F C B (5.5e) sin 33 F C B (5.5f) The components of bonce-average collisional friction term F ) / ( i i I F are sin A F (5.6a) sin A F (5.6b)

16 A F3 (5.6c) sin The first two coefficients, F and F, affect the shape of the distribtion fnction in velocity space (, ), while F 3 coefficient can modify the radial profile of f (,, ), so it is an advection/pinch term. For the collision coefficients, the bonce-averaging is performed nmerically by evalating the vales inside brackets at a set of points along the npertrbed orbits. Typically, arond points are selected along each orbit. In contrast to particle simlation codes, these orbits are only sed as a framework for calclating the bonce-average terms, finding the loss cone in (,, ) space, etc. Therefore, the orbits are only needed to be traced once, and the nmber of orbits is relatively small determined by the grid sizes in (,, ). For the F diffsion operator, if being constrcted from ray-tracing data, the bonce-averaging procedre does not reqire the data on the whole orbit. The local (, ) space at the intersection of a given orbit with a ray element can be mapped directly onto the midplane, withot the need of other points along the orbit. More details are given below. In the absence of a radial electric field, the expressions for the ij and F i can be simplified by recalling that / = and / =. The other transformation coefficients in this case are given by eqations ( ). Finally, we rewrite the FPE from eqation (.4) as 5

17 S f f f f F f f f f F f f f f F t f (5.7) This form for the bonce-average FPE, together with eqations ( ), provides a general prescription for adding any type of local velocity-space diffsion process into the eqation, nder the condition that it can be defined similar to the form of eqation (5.), with local coefficients being a fnction of (, Z,, ). The above expressions exhibit the basic physics of FOW neoclassical transport in or constants of motion (COM) coordinates: the local collision/f velocity diffsion leads to scattering in (, ) space at each point along the orbit, weighted by the dependence of the COM coordinates on local velocity, I i /, and mltiplied by the local collision and F diffsion coefficients. We emphasize that this formlation of neoclassical transport applies to the fll g.c. particle orbits, with no assmption, as is the sal neoclassical theory, that g.c. orbits have a narrow radial extent compared to the plasma radis. Moreover, the radial electric field is amenable to this treatment, bt is reserved for ftre work. We expect that examination of the individal particle flows in or COM space will provide additional insight into neoclassical transport, beyond what is obscred by the distribtion fnction averaging in moment theory; for example, net radial particle, toroidal momentm, and energy transport may be comprised of low and high speed particle flows in opposite radial directions as discssed in (,) space [6].

18 7 6. ZOW limit It is instrctive to obtain a ZOW form of FPE from the FOW form, to compare with the original version in CQL3 [6]. Starting from eqation (3.c) for the conservation of p, we can simply set the terms with b or b to zero, reslting in conservation of pol along particle orbit. In this case, the nmerator in eqation (3.4) and (3.6) becomes zero, and then cos sin cos sin B B. (6.) Also, / =, as before. All other transformation coefficients are eqal to zero. Then, / B (6.a) C (6.b) sin F (6.c) / A F (6.d) sin F (6.e) with other terms being zero. Then the FPE casts into (sorce term is omitted) f f f F f f f F t f (6.3) where the Jacobian becomes

19 8 p 3 b sin cos sin B B B I J, (6.4) with b /. It is clear that in the ZOW limit, the FPE is redced to a - dimensional differential eqation, to be solved in (, ) for each radial coordinate. In the Jacobian, the factors that are not dependent on or can be canceled ot. After sbstitting coefficients from eqations (6.) and sing =, the ZOW-FPE takes the form of sin sin sin sin f F f C f f C f B f A t f (6.5) This eqation can be recognized as the one sed in the CQL3-ZOW version [6]. From comparison of the FOW form given by eqation (5.7) with the ZOW form in eqation (6.3), one can see what kind of new physics has been added nder the FOW formlation. We will refer to the new terms F 3, 3 = 3, 3 = 3 and 33 as the radial transport terms, or -transport terms, for brevity. These terms give rise to the neoclassical bootstrap crrent, radial diffsion, and also pinches (term F 3 ), cased by orbit modifications by collisions, F heating, and also by toroidal electric field. ring tests, we can easily disable those -transport terms to stdy their role. This is another advantage of sing the explicit radial coordinate in or set of I = (,, ).

20 7. The bonce-averaged NBI sorce operator The particle sorce operator mst be formed in terms of the eqatorial coordinates I = (,, ) in which the FPE is being solved. Sppose a particle is born at a point (, Z) in plasma cross-section, with velocity. In modeling of a netral beam sorce of fast ions, the birth points for each of many particles, for each of several beam energies, are weighted by a particle sorce rate which reflects the geometry and physics considerations in the netral beam injection/deposition modle. In CQL3, the netral beam modle is based on the Monte Carlo beam ionization code NFEYA [7], generalized in beam geometry and with pdated ionization cross-section data. A single ion birth point sorce can be written in Cartesian coordinates as Sk 3 r r 3 (7.) k k 3 3 with normalization S k d r d. After transforming to the canonical action-angle variables (J,) (with ( r, ) ( J, Θ) ) and performing averaging over angles, we have an averaged sorce s k, sk 3 3 J J ( ) (7.) k 3 3 with normalization ( ) d J. The physical meaning of eqations (7.) and s k (7.) is not the same becase the former is conting individal particles, while the latter is conting orbits as a whole. It may happen that two particles born at different (, Z) points have exactly same J k vector, i.e., same set of invariants, and ths the same giding center orbit. In the code, each of these particles contribtes to the bonce-averaged operator, effectively accmlating the weight factor mentioned above. 9

21 The next step is a transformation to or set I = (,, ), bt before we proceed, a discssion on or technical realization of a delta-fnction sch as sed in eqation (7.) is sefl. Nmerically, the delta-fnction can be represented by a linear hat fnction (considering one dimension for clarity): J J k (7.3) k J J max, J J where J is the half-width of the hat fnction, and /J is its height. The vale of J shold be at least eqal to the grid size J grid. For example, if J = J grid, we can attribte the peak vale of the hat fnction, /J, to the nearest grid point, so that J Jk dj J grid. J In another example, with twice larger J = J grid, the hat fnction contribtes to the three adjacent grid points, and we have J Jk dj Jgrid J grid J grid. J J J Switching now from J space to I = (,, ), the sorce fnction is transformed into 3 I Ik s k (7.4) 3 ( ) J I 3 3 with normalization ( ) J I d I. s k In contrast to eqation (7.), where an orbit is detected at a single point J k, the sorce fnction in eqation (7.4) contribtes to the two points that correspond to the two legs of an orbit at the eqatorial plane. We can write

22 s k, a k, a 3 ( ) k, a J I a k, a (7.5) for leg a on the midplane, and similarly for leg b. In eqation (7.5) all qantities are evalated at the midplane (sbscript is omitted here, for clarity). Nmerically, the sorce strength assigned to each leg of an orbit mst be same becase in the bonce-averaged FPE the sorce is a fnction of COM, and ths it takes one vale along an orbit with given J k. This condition can be satisfied by the following consideration. The representation of delta-fnctions by hat fnctions is not independent at the two legs of orbit. Nmerically, an orbit is represented by an orbit having finite extent J. At the eqatorial plane, the footprint of sch orbit is I,a = J / J/I a at leg a, and I,b = J / J/I b at leg b. Since the vale of J is the same at two legs, while the vales of Jacobian J/I are different, the footprints I,a and I,b are also different. Based on this consideration, the following procedre is sed for the definition of hat fnctions. First, the vales of grid J/I are compared at the two legs of an orbit. Sppose that grid,a J/I a > grid,b J/I b. Then, at leg a we set,a = grid,a so that the sorce vale is assigned to the nearest radial grid point l = l a, s( l la, j ja, i ia ) 3 grid, a grid, a grid, a ( ) J I a. (7.6) In the above, we also assmed that the half-width of the hat fnction in and is,a = grid,a and,a = grid,a. At leg b, however, the half-width of the radial hat fnction shold be consistent with the reqirement,a J/I a =,b J/I b (and we keep,b = grid and,a = grid where the grid sizes in and are the

23 same for all radial points). Ths,,b = grid,a J/I a / J/I b. Then, the sorce vale assigned to the nearest radial grid point l = l b is s( l lb, j jb, i ib ) 3, b grid grid ( ) J I b. (7.7) It can be seen that it is eqal to the sorce at l = l a in eqation (7.6). However, since,b > grid,b, which follows from the assmption grid,a J/I a > grid,b J/I b, there are also contribtions from the same orbit (of width J ) to several other radial points next to l = l b, given by s( l grid, b, b lb L, j jb, i ib ) 3, b grid grid ( ) max L, J I b. (7.8) The largest vale of L in the above is determined by L <,b / grid,b. With this procedre, the condition of having the same sorce vale assigned to 3 3 each leg is satisfied, and also we recover the normalization ( ) J I d I at leg a and at leg b, if the integral (in nmerical sense) is taken over the footprint of the orbit, i.e. over the grid points [l b L; l b +L]. This approach allows solving the FPE at each leg independently. As an alternative approach, the sorce operator (also the diffsion and advection terms) cold be defined at one leg of an orbit only, and the soltion fond at sch leg cold be simply copied to the other leg. However, this means that at many radial grid points the distribtion wold be assembled ot of nmeros pieces taken from different radial and pitch-angle grid points. Not only does this approach introdce a technical difficlty in determining which part of the distribtion fnction at a particlar radial coordinate shold be fond by solving the FPE and which shold be copied from elsewhere, bt also it can add an excessive nmerical noise from discontinities in sch patched distribtion fnction. s k

24 Finally we note that in the limit of thin orbits, legs a and b of an orbit fall into the same radial bin, and we have (see eqation (6.4)) s l, j, i grid grid grid 3 ( ) J I grid grid grid B sin Bp. (7.9) where / b. This expression can be recognized as the sorce operator in CQL3-ZOW [6]. 8. Qasilinear operator The general form of the bonce-averaged diffsion coefficients for the F operator is the same as that for the collision operator, as specified by eqation (5.5). The difference is, of corse, in the particlar expressions for the local diffsion coefficients in eqation (5.). Here, we consider how to obtain the qasilinear diffsion coefficients based on data passed to CQL3 from a ray-tracing code, sch as GENAY [7]. The details of sch procedre in the ZOW case can be fond in the CQL3 manal [6], and here we generalize to the FOW form. We assme that each ray trajectory has an effective perpendiclar width w in the poloidal plane; w is associated with the spectral distance between rays, and can be fond from the power P = P(k )k flowing in a ray channel and the energy flx S pol in the poloidal plane, S w P. (8.) pol The factor comes from the assmption that the ray channel is toroidally niform. The effective width w is measred perpendiclar to S pol. Then, 3

25 w P P ~ Spol S E k pol k, (8.) where ~ pol S is the energy flx [8] per nit component E k of the discretized k wave spectrm. The vales of each ray are obtained from ray-tracing data. ~ S pol, P, k and F electric field components along In addition, each ray is discretized into spatially local segments ( ray elements ) in the poloidal plane. The length l of a given element is generally chosen so that ray radial extent is less than the smallest radial distance between flx srfaces that go throgh the comptational radial grid points. For each ray element, the local qasilinear coefficients to be sed in eqations (5.) and (5.5) can be expressed throgh the parallel/perpendiclar components as B cos cos sin sin (8.3a) C ( )cos sin (cos sin ) (8.3b) F sin sin cos sin cos (8.3c) where is the local pitch angle. In accord with the random phase approximation for each wave-particle interaction, smmation of the diffsion coefficients from all the elements gives the global F diffsion coefficient to be sed in eqation (5.7). For the ion species, in the non-relativistic limit, the parallel/perpendiclar components can be written as [9, 3] q δ k nc n ( k ) m n q ) δ k nc n ( k )( n c m n (8.4a) (8.4b) 4

26 q m n δ k n c n ( n c ) (8.4c) with ( / ) J E J ( E ie ) J ( E ie ) (8.4d) n n n x y n x y where the argment of the Bessel fnctions J n is k /c, c is the cyclotron freqency, is the wave anglar freqency, k and k are the wavenmber components perpendiclar and parallel to the ambient magnetic field, and F field components E x and E y are in the direction perpendiclar to the ambient magnetic field. The smmation is meant to cover all negative and positive nmbers, bt the resonance condition sets practical limits for the harmonic nmber n to be within a narrow range. elativistic expressions (sed in CQL3 for electron heating applications) are given in [3], and analysis of the affected momentm space regions in the CQL3 manal [6]. For bonce-averaging, we consider a crossing of a given ray element with a drift orbit. The local orientation of the orbit path is described by the giding center drift velocity V gc,pol (the projection onto the (, Z) plane), while the direction of the ray element is characterized by the direction of the energy flx in the poloidal plane S pol which is proportional to the ray grop velocity projected onto the poloidal plane. The contribtion of each ray element to the bonce-averaged coefficient is determined by the portion of orbit path overlapped by the ray element. For instance, if the ray is crossing a particle orbit at right angle, the length of the overlapping path is w. The integral over the orbit is redced to orbit dl V pol gc,pol loc b b w V gc,pol loc (8.5) 5

27 where loc is a combination of the local coefficients B, C, F from eqations (8.3)-(8.5) mltiplied by proper transformation coefficients, i.e. one of expressions inside the angle brackets in eqations (5.5). Since the length of the ray element is smaller than the distance between flx srfaces, several ray elements may contribte to the same radial bin, each with a weight factor proportional to l/ l where l is the radial grid size of the radial bin l at the midplane that contains the leg of the orbit that goes throgh the ray element. Therefore, the contribtion from a given ray element mst inclde the l/ l factor as well as the w factor; so, the contribtion is proportional to the area of ray element, lw. In fact, the orientation of the ray element with respect to the orbit path is not important. If the angle between S pol and V gc,pol is less than 9, the overlapping path becomes larger than w bt the projection of the ray element onto the midplane (following the orbit) becomes smaller. The overall contribtion is still proportional to the ray element area lw. Then, for frther derivations, we assme that the crossings are at right angles. For a proper mapping of a ray element length onto the midplane, attention shold be paid that the volme in configration space (and also in phase space) associated with an orbit changes from a local (, Z) point to the corresponding eqatorial point. For a given ray element, we consider two orbits passing throgh the two end points of the element, so that they are separated by distance l. At the midplane, the separation between the two orbits is l. By expanding the toroidal anglar momentm at the midplane and also at the ray element position, we find l q( e norm q( pol ) mb ( e ) mb norm e ) l (8.6) 6

28 where the nmerator is evalated at the ray element position, e norm being the nit vector normal to the orbit path (sch that e norm V gc,pol = ) and e being the nit vector in the -direction; the denominator is evalated at the midplane. Note that in the ZOW limit, the terms with b are absent, and the nit vector e norm is parallel to the gradient of pol. Then, eqation (8.6) is simplified to l /l = (B p ) ray-el /(B p ), which is incidentally a description of how the distance between the flx srfaces is changed with the poloidal distance. Going back to eqations (8.4), we notice that the physical vales, inclding components of wavenmber, F field components and cyclotron freqency are specific to a given ray element, and they are obtained from a ray-tracing code. On the other hand, the delta-fnction in these eqations defines a resonant line in the local velocity space, which may span over a large range in energies and pitch angles. We can write k n δ ( n )/ k δ (8.7) c c δ k k res where res ( nc) / k, a fixed vale for a given ray element. Nmerically, the delta fnction is replaced by the linear hat fnction δ (8.8) res max, res where is the half-width of the hat fnction, and / is its height. The vale of is set to be large enogh to cover the biggest grid cell in comptational velocity space. The hat fnction defines a strip in the local (to ray element) velocity space. This strip region can be poplated with, ) points or, alternatively, with (, ) ( points that are mapped onto the midplane by following the particle that has (, ) 7

29 velocity components at the ray element position. Ths, in contrast to the ZOW case, a ray element may contribte to different radial bins at the midplane, depending on the particlar vale of (, ) within the local resonance strip. In a smmary, the contribtion from a given ray element, from a single point (, ) of the local resonance strip is given by loc j, i, l w V b gc,pol l l q( e norm q( pol ) mb ( e ) mb norm e ) ( ), j, i, l loc (, ). (8.9) The indices (j,i,l) correspond to ( j, i, l ) grid point at the midplane to where an orbit is traced, for a particle with (,) at the ray element position. The l grid point is associated with the radial bin width l. The additional factor ( /) corresponds to mapping of the local -grid bin to the midplane. We assme that the local -grid is over-dense, so that several -points from the resonance strip can contribte to the same ( j, i, l ) point. Then, the contribtion from one -point is proportional to the ratio of the footprint, which is ( /), to,j,i,l, which is the accmlated vale of ( /), contribted by other -points. There is no similar factor for the mapping becase here we assme that =, and the -grid is the same at the ray element and at the midplane. Note that the loc (,) coefficient incldes the weight factor from eqation (8.8). To calclate the total contribtion from one ray element, the eqation (8.9), withot,j,i,l, is evalated for each point in the resonance strip, with vales accmlated in corresponding ( j, i, l ) grid points at the midplane; afterwards, the reslt is divided by,j,i,l. The procedre is repeated for all ray elements and rays, with vales of loc added p at 8

30 the midplane grid points, to form the qasilinear operator for the FPE eqation. The described procedre is qite different and more complicated than that for the ZOW form of QL operator [6]. For instance, in the ZOW version, there is no need to form a local -grid within the resonance strip all points are mapped into one radial bin on the midplane, so that the mapping procedre can be easily done in reverse, i.e., from the midplane -grid points back to the local -space. In spite of the technically different approach, tests have shown, as consistency reqires, that in the ZOW-limit the FOW-QL diffsion coefficients converge to the original ZOW-QL coefficients with good accracy. In a typical application of CQL3 to F ion heating, the distribtion fnction of ions (the general species) may develop a strong non-maxwellian tail, and therefore the power deposited to ions may well exceed the vale calclated in the ray tracing code via linear damping mechanism (with all species being Maxwellian). The power deposited to ions at a given radial bin l is determined by the power flowing in ray channels, which makes p the QL diffsion coefficients, and also by the shape of the ion distribtion fnction, i.e., the soltion of FPE fond at each time step. As the distribtion fnction evolves, more power is absorbed by ions and conseqently less power in ray channels becomes available for the remaining portion of ray elements. The code iterates between calclation of the QL coefficients, the consistent distorted distribtion fnctions, and the damping of the ray energy along the ray channels, to achieve self-consistent non-maxwellian distribtions and damping of the F. In the final note to this section, we wold like to nderline the main difference in formation of the collision and qasilinear operators. The BA collision operator tilizes a set of npertrbed sampling orbits traced for each grid point ( j, i, l ), 9

31 pls the COM-table that maps a local (, Z,, ) point back to the midplane point(s) for the reconstrction of the local distribtion fnction. In contrast, the formation of the qasilinear operator does not reqire the data on a set of points along each orbit. The QL operator is formed by a direct COM-mapping from a local (, ) space at each ray element position back to the midplane points where the operator is formed. Ths, every ray element is acconted for, as long as the resonance condition is satisfied. 9. Initial distribtion fnction and rescaling The initial distribtion fnction over the (,, ) comptational space mst be set in sch a way that it has the same vale at both legs of each particle orbit. This reqirement is dictated by the fact that in general (at the bonce time scale) the soltion of FPE mst be a fnction of COM, and as sch, it has the same vale at any point along orbit. If we want to set the initial f (, ) at each given radial point as a Maxwellian type distribtion, the density and temperatre in sch fnction can only be a fnction of COM, rather than the local coordinate. Technically, this is achieved by the following procedre. For each grid point ( j, i ) at a given radial point l, the vale of bonce-average poloidal flx pol is fond dring initial orbit tracing and stored in an array over grid indices (j,i,l). We also assme that the initial profiles of density n and temperatre T are set as a fnction of the generalized radial coordinate, which can be niqely determined from pol. Then, we se the vale of pol = pol j,i,l to determine n( pol ) and T( pol ), and to define the vale of the Maxwellian-like distribtion fnction for the particlar orbit. This vale is attribted to both legs of the orbit. It shold be noted that with sch procedre, the local 3

32 distribtion fnction f(, ) at each given radial point l is not isotropic in becase orbits that start at l have different vales of pol (see figre ). In fact, this deviation from the isotropic distribtion acconts, in a typical case, for abot 5% of the total bootstrap crrent, as discssed in section, and can be interpreted as a finite orbit width magnetization crrent; bt it is not the actal bootstrap crrent ntil the collisional transport (particlarly the radial terms) re-adjsts the distribtion. The reqirement of having the same vale at both legs of each orbit mst be also maintained when a rescaling of the soltion is done dring time iterations. In the ZOW version, sch rescaling is applied to the distribtion fnctions at each radial coordinate independently, so that the radial density profile remains nchanged dring simlations. In the FOW variant, the vales of the distribtion fnction at different radial coordinates that are copled by the same COM mst be scaled by the same factor. The easiest way to satisfy this condition is to rescale the distribtion fnction at every radial coordinate by one and the same factor based on total nmber of particles in plasma volme. Sch rescaling keeps the total nmber of particles in plasma volme nchanged; however, the profile of density calclated from f is allowed to evolve. For a given soltion of FPE f (,,, t), the total nmber of particles in plasma can be calclated from 3 f J / I ddd N( t) (9.) where the integral over incldes only one leg of each orbit. The integrand in eqation (9.) times (d d d ) gives the nmber of particles in a particle tbe defined by d, d and d at the (,, ) point and the particle trajectories. The initial vale of total nmber of particles, N(t=), and the vale obtained at a given 3

33 time step, N(t), are sed to rescale the distribtion fnction f (,,, t) by the N(t=)/N(t) factor same at each radial coordinate. It shold be noted that in ZOW limit, the Jacobian in eqation (9.) is redced to the form given by eqation (6.4), and then we have N ( t). (9.) ( Bp / B ) d f / bd sind The integral f / bd sind can be interpreted as a field-line density for a flx srface with ot-board radial coordinate, that is, the total nmber of particles in a flx tbe defined by the perpendiclar-to-b area at.. Bondary conditions In general, the differencing of the FPE is formlated throgh the particle flx in the (,, ) space. The expressions in the sqare brackets of eqation (5.7) are the corresponding three components of the flx. For an internal point (i, j, l) of the (,, ) grid, we consider a trapezoidal cell with faces at i, j, l half-mesh points where the advection and diffsion coefficients F- from eqation (5.7) mst be defined. For the cells at the edge of each grid, we impose the peripheral bondary conditions. They are same as in the ZOW version [8, 3], except the flxes in and direction now inclde radial terms, and also there is an additional radial flx. As a rle, the flxes throgh the oter face of the edge cell are assmed to be zero. For example, for the last point in grid, j = j max, the coefficients F,, and 3 at j = j max +/ point are set to zero, which implies a zero flx leaving or entering the pper bondary in. An exception from this rle is F coefficient in case when it incldes a dc toroidal electric field in addition to the collisional friction, and the effect of particle acceleration is larger than the collisional drag in this case a free 3

34 streaming is allowed off the velocity grid [3]. For the = and = edge points, an additional condition is imposed in form of f. This is achieved by setting, and 3 to zero at i = and i = i max grid points. For the = bondary (j = grid point), the condition f is added. For the borders in the radial grid, we impose a zero radial flx condition at the otboard edge. At the first radial grid point, which is sally selected to be close to the magnetic axis, we do not add the radial transport terms F 3, 3, 3 and 33. This can be jstified by assming f at the plasma center, and also assming that the radial derivatives of the F 3, 3, 3 and 33 terms are also close to zero in the plasma core. In ftre work, we plan to extend the radial grid to the inner edge of the plasma. This is dictated by necessity to inclde the conter-passing orbits that reside at < axis ; those are the orbits close to O-type stagnation orbits [4]. With sch extension of the radial grid, the bondary will be at the plasma edge only. Here, we only gave a brief smmary on the peripheral bondary conditions, as the approach to their formlation is not so different from the ZOW case. The main complication in the FOW formlation is rather the Internal Bondary Conditions (IBC) which mst be applied to the regions in (,, ) space that are connected by the pinch orbits [4]. A set of orbits converging to the pinch orbit is shown in figre. In the figre, the barely trapped kidney -shaped orbit with legs at = 45 cm and 68 cm can scatter by collisions into a pinch orbit, and then frther into a conter-passing orbit with legs at cm and 7 cm. In this example the IBC connects flxes of particles at the ( a = 45cm, a 45) point with flxes at the ( b cm, b 9) point. The IBCs are formlated in sch a way that the total flx of particles across the two bondaries is conserved, i.e., the flx of particles leaving the trapped region mst eqal the flx entering the 33

35 passing region. Also, the symmetry of flxes at the trapped side of each bondary is sed these flxes are eqal in magnitde bt opposite in, as they describe the same trapped particle being scattered towards (or away from) each bondary at eqal rate. An important simplification in the ZOW case is that the trapped-passing borders are straight lines in (, ) space, and that each IBC connects two bondaries at the same given radial coordinate. In the Fll-FOW case, the two legs of the largest banana orbit cross the midplane at different radii; hence, the IBCs shold connect the flxes at different radial grid points. Before discssing the IBC in more detail, it is sefl to describe the other important bondaries in (, ) space for each given, and their correspondence to the bondaries in a more commonly sed COM space (E,, p ) of energy, magnetic moment and toroidal anglar momentm [33-36]. Let s start with the (p, ) E map and specify the bondaries in it. The sbscript E signifies that this map is a slice of (E,, p ) space at a given energy level. Based on eqations (3.), we can eliminate cos, and express as a fnction of p, sin [( p q ) ( mb )] (.) m / B B In this form, p and are the variables, and (and ) can be considered as parameters. Parameters, pol and B-field are sbscripted by as a reminder that they are evalated at the midplane (althogh in general, they cold be evalated at any other (, Z) point). In the (p, ) E space, this eqation defines an inverted parabola; the position of its vertex, which corresponds to a cos = point, depends only on. The parabola for orbits started at = 3 cm is plotted with the dashed crve in figre 3. The left arm of the parabola corresponds to conter-going orbits 34

36 (inclding conter-passing and trapped orbits started with cos < at = 3 cm); the lowest point at this arm corresponds to cos =. Similarly, the right arm of the parabola corresponds to co-going orbits (co-passing and trapped orbits started with cos > at = 3 cm); the lowest point at this arm corresponds to cos = +. In configration space, the orbits corresponding to the dashed line are shown in figre 4. Also, in figre 3, two other important parabolas are plotted: a smaller one (centered at abscissa vale.93) for orbits started at the smallest plasma radis, = min, and the large one for orbits started at the largest plasma radis, = max. The major radii points min and max mark the eqatorial positions of the vacm chamber wall. In the smaller parabola, the right arm represents co-going orbits that are passing ot of plasma, and therefore are lost, while in the larger parabola, the left-arm/conter-going orbits are lost. The stagnation point-like orbits are marked with o and o lines. We follow the designations sed in efs. [4, 37] and in particlar [38]. Line o corresponds to the point-like orbits that exist at > axis region; these orbits become = orbits in the ZOW limit. Line o corresponds to the point-like orbits that are located in part of region at < axis. The x line corresponds to the pinch orbits. In the (, Z) plane, each pinch orbit can be viewed as being composed of two orbits passing throgh the one and same X-point on the midplane: a smaller conter-passing orbit, and a larger orbit that looks like a co-passing bt actally a kidney -shaped trapped orbit (the sign of alters not far from the midplane, as shown in figre ). These two orbits have the same vales of (E,, p ); therefore, they are represented by the same line in figre 3 marked x (x, x3), meaning that the complete pinch orbit x is the combination of the two orbits x and x3 described above. However, one of the orbits (x3) can be lost to the chamber wall, and then the pinch orbit becomes incomplete, 35

37 having only the x part. Therefore the length of x and x3 lines in figre 3 is not the same. In fact, the only portion of the line where orbits x3 are confined is between points A and S in the figre. We also add one more line that corresponds to orbits that have = at the eqatorial plane. By setting cos to zero, the eqations for sch line become p = q pol ( ), (.a) m B( /. (.b) ) In figre 3, this line is designated as tp (for trapped-passing ); it is plotted by varying the vale of in the above eqations. It consists of two arms the lower arm corresponds to the range < axis, and the pper arm to > axis. It is seen that the pper arm closely approaches the o stagnation line. Therefore, the stagnation orbits in this region have the vales of pitch angle close to / ( ); the distinction of their pitch angle from / grows with energy. Also, the lower arm of the tp line follows closely parallel to the x pinch-orbit line. The separation between these two lines reflects the fact that the point where becomes zero along the kidney part of the pinch orbit is not exactly at the eqatorial plane bt rather at some finite distance off the midplane. The points where the dashed parabola crosses other crves are of particlar importance. For example, point a in the figre separates trapped from co-passing orbits. Points b and d are two other crossing points of the dashed line with the tp line. The narrow region between b and d corresponds to a grop of tiny co-passing orbits (in, Z space) that are nested on both sides from the point-shaped stagnation orbit c. Orbit d starts with = at = 3 cm, and encloses the grop of tiny 36

38 co-passing orbits jst to the right of orbit c. Orbit e is the last confined trapped orbit. All orbits between d and e are trapped orbits with the first leg being at = 3 cm, and the other leg somewhere at > 3 cm. For the orbit e, the other leg is at = max. Note that, on the dashed crve, point e is the trapped orbit started as a conter-going particle, while on the parabola for = max orbits, it is the same orbit bt it is started as a co-going particle. Point f is the intersection of the dashed line with the pinch-stagnation line x. Points below f on the dashed line are the conterpassing orbits (in figre 4, these orbits are labeled as ff ). Points between e and f are lost banana orbits, with starting leg at = 3 cm, and the other leg being otside of plasma ( > max ). Point f itself corresponds to the pinch orbit and two other associated orbits, as discssed above: the pinch orbit lanched from = 3 cm with the other leg being otside plasma; the conter-passing orbit with starting leg at = 3 cm and the other leg at = 34 cm (the location of the X-pinch point); the kidney orbit that has one leg at = 34 cm and the other otside of plasma. The intersection of the dashed line with tp line jst below the point f is not important in present context; it marks an orbit that has = at the midplane at 34 cm; however, the other leg of this orbit is not at = 3 cm, bt rather somewhere otside of plasma; it is close in size to the kidney orbit mentioned above. The very same point also represents a conter-passing orbit with legs at = 3 cm and = 35 cm (no = point along orbit), i.e., one of ff orbits in figre 4. Most of the complication in describing of the (p, ) E space comes from the fact that a single point may represent two orbits, which might have no common points in configration space. Also, the (p, ) E map contains no information on the orbit leg positions at the midplane. 37

39 Or comptational space (, ) is shown in figre 5, for the selected radial point = 3 cm at the midplane. Each node of the (, ) grid is depicted with a point (these may fse visally into radial lines), except in the loss region. Solid lines are calclated with the help of (p, ) E maps by nmerically finding the intersection points between the parabola representing orbits at = 3 cm with all relevant bondary lines sch as tp, o, x, and the two limiting parabolas (orbits lanched from min and max ). This procedre is repeated for a set of (p, ) E maps with different energy levels, i.e., different. The vales are saved and then plotted in (, ) map as a parametric fnction ( bndy ( ), bndry ( )), for each bondary line. The only exclsion from this procedre is the stagnation line o; it is fond analytically. In comparing to the (p, ) E map, figre 3, we shold follow the dashed line 5 kev in the figre. The points labeled a throgh f correspond to the same orbits in both maps. Orbits between points a and b are the trapped orbits with the other leg at < 3 cm, and orbits between points d and e are the trapped orbits with the other leg at > 3 cm. Orbits between b and d are the tiny passing particle orbits, inclding the stagnation orbit c. Orbits between e and f are the lost orbits (banana orbits with the other leg otside of plasma). There are no complete pinch orbits at energy level 5 kev, in the sense of line S-A in figre 3. Orbit f can be viewed as the half of pinch orbit the conter-passing branch that is confined, while the kidney -shaped branch is lost. The complete pinch orbits appear at lower energies in figre 5, where dark trianglar regions are present. The size of the right-hand-side triangle, in the co-passing cone, is determined by the length of x3 line in (p, ) E space, and this length depends on. Note that in the (p, ) E space, althogh all three borders x, x, x3 are shown with a single line, the length of x3 is 38

40 shorter than that of the other two; line x3 is only present over the dark triangle, between points A and S. The analysis of (, ) map is important for the Fll-FOW model. It gives an insight on how the IBC shold be defined. In figre 5, the trapped-passing bondaries (they start at the origin of coordinates and pass throgh points a, b, d, and f) are seen to be crved and limited in space. However, as discssed above, we are not generally interested in the bondaries between trapped and passing particles, bt only in those associated with pinch orbits. For example, point a in the figre corresponds to a smooth transition from co-passing elliptical orbits to -shaped orbits that have = point. These -shaped orbits shrink in size as the pitch angle approaches point b in the figre, where the tiny -shaped orbits smoothly transform into tiny elliptical orbits of passing particles. There is no pinch orbit involved at points a, b or d, so there is no need in IBC. In figre 3, the region where complete pinch orbits exist is shown with the short line S-A, bt, if the energy dimension is added, it becomes a srface that expands towards low energies and vanishes at high energies. The projection of this srface onto or (, ) space in figre 5 is shown with the line that starts at the origin and ends at point B; we designate this bondary as -B. It appears to be an initial part of the -a line, bt, to be exact, the two lines do not coincide; they merge together only at zero-energy limit. Any selected point within -B bondary corresponds to the co-going, kidney -shaped branch of a pinch orbit. By collisional pitch angle scattering a barely passing orbit on the cocrrent side can be transformed into a kidney orbit and then into the kidney branch of the pinch orbit. At the X-point on the midplane, the particle may jmp onto the conter-going branch of the pinch orbit, and then frther transform into the conter-passing orbit. Sch a conter-passing orbit has its starting leg at different 39

41 radis, smaller than the given = 3 cm. Therefore, the IBC shold connect a point at the -B bondary with the conter-passing side bondary, sch as -C in figre 5, bt at a different radial coordinate. The other-side radial coordinate depends on the energy of a particle, implying that the bondary -B at each given radis is linked to many other radial points, except in the low-energy ZOW limit when they are at the same radial coordinate. For the formlation of IBC, for a given energy ( vale) we consider the two pitch-angle grid nodes that are nearest to the -B physical bondary on two sides of it. Starting from one of these nodes at one side of the -B line, the vale of the distribtion fnction is extended into the virtal point on the other side of the -B line. The same procedre is carried ot for the opposite side of the -B line. This procedre allows formlating the pitch-angle flxes at each side of the bondary. As shown above, the bondary -B is connected to the bondary -C at another radis, where the same consideration of flxes across bondary is made. Ths, for each given energy, we have for nknown vales of virtal distribtion fnction. The for eqations that are needed to find these vales consist of the two continity conditions for distribtion fnction across each bondary (one at -B line and one at -C line), the symmetry of -flx on the trapped-particles side of -B line with that of -C line, and the conservation of total -flx throgh the -B and -C bondaries. More details are given in the Appendix.. Application of the FOW version for ion heating For illstration and verification of the fll-fow capabilities, we consider a typical NSTX scenario of plasma heating by means of NBI and High Harmonic Fast Waves (HHFW). Fast ions prodced by NBI can effectively interact with HHFW throgh 4

42 wave-particle resonant finite-larmor-radis effects, that is, throgh the k /c terms in the QL diffsion coefficients. With the magnetic field varying from approximately.4 to.4 T within the plasma cross-section, the waves at 3 MHz freqency can enconter a wide range of cyclotron harmonics for + ions / ci =, althogh from ray-tracing, they can reach the resonance layers with / ci = 4. For simplicity, the density and temperatre profiles of backgrond + ions and electrons are assmed to be parabolic in the coordinate (the sqare root of the normalized toroidal flx), with T()/T() =./. kev, n()/n() = 3e3/3e cm -3. The + fast ions are continosly generated by NBI at the main energy peak 65 kev and sbpeaks 33 kev and kev, at total power level of.3 MW. The total F power is. MW, althogh only a small portion,.3-.5 MW is delivered to the ions. ay data is from GENAY [7] with f = 3 MHz and n at the antenna location. The general reslts are presented in figre 6, showing average-energy profiles, which are obtained by integration of the distribtion fnction, i.e., the compted soltion of FPE, over velocity space at each midplane coordinate. Althogh not a flx-srface averaged (FSA) characteristic, sch diagnostic qantity is a good reflection of a particlar distribtion at a given coordinate; the srface averaging wold mix together the midplane distribtion fnctions from many radial points. As seen in the figre, after the NBI+HHFW heating is applied, the energy is increased in the plasma core from.5 kev (which is <energy> 3/ T i at t =, before heating) to abot 3 kev; the increase is from non-thermal tail ions. In comptations withot the radial transport terms F 3, 3, 3, 3, 3 and 33 ( no -transp in the figre), the energy profile exhibits strong variations of magnitde at different radial points. Similar energy profile variations are also observed in ZOW rns and are a 4

43 conseqence of localized ion heating by several resonances present in the plasma cross-section. However, when the -transport terms in the collision and QL operators are added, the isolated peaks in energy profile are spatially diffsed and the profile becomes very smooth (bold line in figre 6). In general, the effect of the -transport terms is a redction of energy content de to additional transport cased by 3, 3, 3, 3 and 33 terms in eqation (5.7); withot these terms, the particles are lost only throgh pitch-angle diffsion into the loss cone. More insight into the effects of radial transport can be gained from inspection of the distribtion fnction plots, sch as shown in figre 7 and figre 8. With the radial terms being enabled, there are certain changes in the distribtion fnction in the energy range - kev, as indicated in figre 8(d). For the radial coordinate =.5 ( = 38 cm) the effect from the -transport terms is a redction of the highenergy tail at specific pitch angles. However, in the plasma core, as seen in figre 8(a,b), the radial transport seems to enhance the tail; apparently, it is poplated by fast ions from neighboring radial points. It is also instrctive to look at the plots of the radial diffsion coefficients. In figre 9, the contor levels of the radial diffsion term 33 from eqation (5.7) are plotted. The collision diffsion coefficient, shown in part (a), is largest at near-thermal energies, where it reaches 4 cm /sec. At abot 3V th,i it drops to x 3 cm /sec for the trapped particles. For the passing particles, it is typically below cm /sec. In contrast, the F QL radial diffsion coefficient, shown in part (b), exhibits several peaks at energies arond 5 kev with vales p to 4x 4 cm /sec and vanishes at thermal energies. The isolated peaks are related to the presence of several IC resonances in plasma. In general, at energies above kev the radial diffsion by F heating of ions seems to prevail. It shold be noted that the shape and the magnitde of the radial transport terms 3, 3 and 33 in the QL 4

44 operator strongly depends on the particlar wave type and on the injected F power, however, in most types of F ion heating one wold expect elevated vales in the high-energy tail. Another important characteristic obtained in the FOW calclations is the plasma crrent. To facilitate the comparison with available models, we calclated the FSA profile of plasma crrent density <j > FSA. This is done by reconstrcting the local distribtion fnction at a set of poloidal points for each magnetic srface, from the compted soltion of the FPE at the midplane. It is clear that for a given magnetic srface, the orbits from different radial grid points at the midplane can contribte to the vale of <j > FSA, in contrast to the ZOW limit in which each srface confines orbits that have one and the same radial coordinate at the midplane. The profile of ion crrent density <j > FSA in the case of NBI+HHFW heating is shown in figre (a). Most of the crrent at <. is prodced by NBI, while at >.6 it is de to bootstrap crrent. In calclations presented in figre (b), the NBI and F heating were trned off, therefore the calclated <j > FSA is only the bootstrap crrent. This is compared to the profile of crrent based on the analytical fit model for the bootstrap crrent [39, 4], considering in or case the ion component only. For the analytic model, the profiles of FSA density and energy mst be provided; those are obtained from the FOW rn (the initially nearly parabolic profiles are slightly changed by the radial transport, especially at the plasma edge). emarkably, the two crrent profiles are qite similar. In the FOW calclations, all radial transport terms 3, 3, 3, 3 and 33 were enabled. Withot them, the profile of <j > FSA from the distribtion fnction is significantly lower, p to only 4-5 % at = A deviation from the analytical model profile can be attribted to wide ion orbits in the NSTX-level eqilibrim field. It is noteworthy that the crrent compted with CQL3 is lower 43

45 than that from the model at <.8. This is in agreement with reslts obtained with the NEO code [4], particlarly at low collision rate. A more detailed comparison between CQL3-FOW and NEO sing the same plasma profiles and eqilibrim data is in consideration for a ftre work. Frther, figre (c) shows the reslts for the ion bootstrap crrent in a modified eqilibrim B: both the poloidal and toroidal magnetic field components are mltiplied by a factor of for. The larger B-field makes the orbits for times narrower, and now the match between the analytical model (assmes narrow orbits) and <j > FSA derived from soltion of FPE is almost perfect. The peak at the edge appears to be cased by the losses of conter-crrent going ions, which are not acconted by the analytical model. Interestingly, the compted profile of bootstrap crrent is not very sensitive to imposing of internal bondary conditions, the implementation of which was described in the previos section. Most of the crrent is generated by merely enabling the 3, 3, 3, 3 and 33 terms, while the IBC are disabled. This reslt appears to be consistent with the simplified model for bootstrap crrent as a correction to ZOWtype bonce-averaged FPE [4, 43]. Effectively, in terms of or fll-fow eqations, the simplified model only considers one radial term, namely 3 = 3 given by or eqation (5.5e), and only the last term in it, which is F / sin. Note that for the / starting leg of an orbit, eqation (3.6) gives / =, while for the other leg of a trapped orbit on the midplane, in the limit of thin banana orbits, we obtain / /(qb z /m). The other transformation coefficient, /, is approximately eqal to + and at the two legs of trapped orbit. Therefore, the change of the ( /)( /) factor from one leg of the trapped orbit to another is of order the 44

46 banana width, and we recover the simplified bootstrap-crrent generating term in [4], which is proportional to the width of the trapped orbit at trapped-passing bondary and bonce-average pitch-pitch diffsion coefficient. Ths, it appears that merely enabling the radial term 3 is sfficient to generate the bootstrap crrent, at least in the limit of narrow orbits. The weak dependence of the calclated crrent on the for extra eqations corresponding to the internal bondary conditions (see the end of section ) sggests that the presence of the 3, 3, 3, 3 and 33 terms largely enforces the conservation of flx and other conditions that were sed in the formlation of IBC.. iscssion and conclsions The finite-difference bonce-averaged Fokker-Planck code CQL3 has been pgraded to inclde the Finite-Orbit-Width (FOW) effects. This is achieved by transforming the FP eqation written in canonical action variables to another set of COM coordinates. The distinctive featre of or approach from that considered by other athors is the selection of the major radis at the eqatorial plane as one of the COM coordinates. erivations are provided for the transformation coefficients from a local point along orbit to the midplane COM coordinates, and also for the Jacobian of transformation from the canonical action variables to or coordinates. It is shown that they converge to a familiar form sed in the ZOW limit of bonce-average FPE. The details are given for the comptational implementation of the FOW forms of the collision operator, qasilinear F operator and the NBI particle sorce. Also we discss how to start the initial distribtion fnction and to rescale the distribtion fnction at each time step, in order to compensate for the particle losses throgh the radial and pitch-angle diffsion. The otlined rescaling procedre is consistent with 45

47 the reqirement of keeping the vale of distribtion fnction constant along the orbit, bt as a conseqence, the profile of plasma density mst be allowed to evolve; this is different from rescaling the distribtion fnction in the ZOW version when the profile cold be kept nchanged. A significant portion of this paper is dedicated to the identification of internal bondaries associated with the pinch orbits, which connect the flxes of nearly trapped co- and conter-passing particles at different radial coordinates. The analysis shows how the different types of particle orbits and bondaries in or (, ) space at a given radial coordinate are related to bondaries in a more commonly sed (p,, E) space. Effectively, this analysis helps to nderstand the forms of the loss cone, trapped particle regions and other featres of the local distribtion at a given radial coordinate. The implementation of the IBCs trned ot to be the most difficlt technical problem in development of the FOW version. Improvements are ongoing. The main challenges are related to the fact that the physical bondaries in (, ) space do not match the grid lines corresponding to constant pitch-angle, and also that the trapped orbit width sizes generally are not commensrate with the distance between the radial grid points. However, there are several properties of bondaries that lessen the impact of complications with the IBCs. First, we have shown that the nmber of links between the relevant (, ) bondaries at different coordinates is qickly redced with an increase of ion energy. For NSTX conditions, there are no links and therefore no IBC needed at energies above 57 kev. Second, we observed that the reslts of simlations are not very sensitive to the presence or absence of IBC. As a conseqence, a nmerical implementation of IBC may allow a certain leeway. 46

48 In contrast, the radial transport terms 3, 3, 3, 3 and 33 that appear in the FOW formlation are fond to be crcial in calclation of the bootstrap crrent and the radial diffsion. We emphasize that these radial terms are evalated sing realistic drift orbits rather than a first-order expansion of orbits arond flx srfaces. In general calclations, all for possible coefficients related to the radial coordinate mst be inclded, which are F 3 (the advection/pinch term), 3 = 3, 3 = 3 and 33. Similar radial terms (except F 3 ) appear in the bonce-average QL operator, bt their strctre in (, ) velocity space is very different from that in the collision operator. The flly-neoclassical FOW version of CQL3 has been applied to an NSTX ion heating scenario with NBI and HHFW sorces. The fast ions prodced by NBI are slowing down by collisions, bt on the other hand they are diffsed to higher energies by high harmonic fast waves. The simlation reslts demonstrate the code capability to describe the physics of transport phenomena in plasma with axiliary heating, in particlar, the enhancement of the radial transport of ions by F heating and the occrrence of the bootstrap crrent. Soltion of the fll 3 copled difference eqations, giving a large sparse set of order a million eqations, is accomplished iteratively sing the SPASKIT package [44]. Becase of the bonceaveraged type of the FPE, the reslts are obtained in a relatively short comptational time compared to higher dimensional soltions. A typical fll-fow rn time is hor sing 4 MPI cores. e to a time-implicit differencing, calclations with a large time step (tested p to dt =.5 sec) remain stable. 47

49 Acknowledgments We grateflly acknowledge spport by US oe Office of Fsion Energy awards E- FC-E54649, E-FG-4E54744, and E-SC664. This research sed resorces of the National Energy esearch Scientific Compting Center, a OE Office of Science User Facility spported by the Office of Science of the U.S. epartment of Energy nder Contract No. E-AC-5CH3. We thank r. Ken Kpfer for providing backgrond and motivation for this work. We appreciate the encoragement of r. Gary Taylor (PPPL), r. Pal Bonoli (MIT), and members of the SciAC Center for Simlation of Wave-Plasma Interactions. Appendix: Implementation of IBC Schematically the internal bondaries in (, ) space are shown in figre A. The dashed lines represent the physical bondaries, sch as lines -B and -C in figre 5. The solid lines designate the -grid nodes. We designate the physical bondary with </ as the lower trapped-passing bondary, and the nearest -grid points as ipl for the last passing particle and itl for the first trapped particle. Strictly speaking, the orbit corresponding to pitch angle (ipl) may not be a passing particle orbit bt rather a kidney shaped trapped particle orbit, as discssed in section, bt this is not important for the formlation of IBC. Similarly, we designate the physical bondary with >/ as the pper trapped-passing bondary, and the nearest grid points as it for the last trapped particle and ip for the first passing particle. Althogh the pper and lower bondaries are pictred together in one plot, in general those bondaries that are linked by IBC are at different radial coordinates, and the radial distance between sch bondary points becomes larger with particle 48

50 energy. This is illstrated in figre A, where the IBC links are shown for 3 kev deterim ions in NSTX eqilibrim conditions. The vertical dashed lines mark the position of radial grid points. The pitch angle grid cold be shown by a set of horizontal lines, bt these are omitted for clarity. It is seen that for 3 kev particles, there are only several links present between the pper and the lower trapped-passing bondaries. Each link connects the two legs of the largest trapped particle orbit. We designate the radial grid index at the lower bondary as l_a, and the radial grid index at the pper bondary as l_b. etrning to the diagram in figre A, we consider the pitch-angle flxes H at each bondary. This is the expression nder the / operator in eqation (5.7) that incorporates partial derivatives of the distribtion fnction, sch as f /, and therefore reqires the knowledge of the distribtion fnction at 7 adjacent grid points (three for each dimension) for the flx vale of H(,, ) at a given grid point. In particlar, for the ipl point, we need to se the adjacent ipl- and ipl+ points. The distribtion fnction at ipl+ point is not the vale of f at the itl point, bt rather it is a virtal ( ghost ) vale to be fond from the conservation of H flxes. We se a star symbol to distingish sch virtal points, as in f * ipl+. For each (, ) grid point we consider for virtal points: f * ipl+ is extended from the co-passing side (from the ipl point) into the trapped region, f * itl is extended from the trapped side back into the co-passing region, f * it+ is extended from the trapped side into the conter-passing region, and finally f * ip is extended from the conter-passing side (from the ip point) back into the trapped region. In a general form, the H flxes can be written as 49

51 H ipl+.5 = a ipl f ipl + a ipl+ f * ipl+ (from co-passing to trapped) H itl.5 = a itl f * itl + a itl f itl (from trapped to co-passing) H it+.5 = b it f it + b it+ f * it+ (from trapped to conter-passing) H ip.5 = b ip f * ip + b ip f ip (from conter-passing to trapped) Here, a and b coefficients are expressed throgh F,,,3 coefficients in eqation (5.7). In fact, each of the for flxes shold inclde a smmation of the terms (with corresponding virtal and actal f vales) over adjacent in (, ) grid points, bt these are omitted for clarity. In formlation of the flx conservation conditions, we shold keep in mind that the matching flxes occr at different radial points, sch as at a pair of dots connected by a link in figre A. The IBC conditions are:. Continity of the distribtion fnction at the lower trapped-passing bondary, at the l=l_a radial index: f * ipl+ f itl = f * itl f ipl.. Continity of f at the pper trapped-passing bondary, at the l=l_b radial index: f * ip f it = f * it+ f ip. 3. Symmetry of flx in the trapped region: H it+.5 l_b = H itl.5 l_a 4. Conservation of total flx: (H ip.5 H it+.5 ) l_b + (H itl.5 H ipl+.5 ) l_a =. 5

52 Ths, we have for nknown virtal points in the distribtion fnction that can be fond from the for eqations. In the nmerical coding, the for virtal points are added to the vector of soltion to be fond, while the matrix of coefficients contains for extra rows corresponding to the IBC above; these for extra nknowns and for extra rows are present in each (, ) grid section. eferences [] Bernstein I. B. and Molvig K. 983 Phys. Flids [] Cohen. H., Hizanidis K., Molvig K. and Bernstein I. B. 984 Phys. Flids [3] Kafman A.N. 97 Phys. Flids 5 63 [4] Kpfer K. 99 Fokker-Planck Formlation for F Crrent rive, Inclding Wave-riven adial Transport, Proc. IAEA Tech. Committee Mtg. on FWC in eactor Scale Tokamaks (Arles, 99) p 49 (available at compxco.com/cql3d.html) [5] Harvey.W. and McCoy M.G. 99 The CQL3 Fokker-Planck Code, Proc. IAEA Tech. Committee Mtg. on Advances in Simlation and Modeling of Thermonclear Plasmas (Montreal 99) p 489 (available throgh USOC/NTIS No 9396) [6] Harvey.W. and McCoy M.G. 99 The CQL3 Fokker-Planck Code (available as The CQL3 Manal at [7] Kerbel G.. and McCoy M.G. 985 Phys. Flids [8] Killeen J., Kerbel G.., McCoy M.G., and Mirin A.A. 986 Comptational Methods for Kinetic Models of Magnetically Confined Plasmas (New York: Springer- Verlag) [9] Goldston.J. 977 Ph.. thesis (Princeton University 977) 5

53 [] O Brien M.., Cox M., Warrick C.. and Zaitsev F.S. 99 Proc. IAEA Tech. Committee Mtg. on Advances in Simlation and Modeling of Thermonclear Plasmas (Montreal 99) p 57 [] McKenzie J., O Brien M., and Cox M. 99 Compt. Phys. Commn [] Girzzi G., Fidone I., and Garbet X. 99 Ncl. Fsion 3 [3] Hammett G. 986 Ph.. thesis (Princeton University 986) [4] Fkyama A. and Ueeda T. 99 Proc. 7 th Topical Conf. on Controlled Fsion and Plasma Heating (Amsterdam 99) vol 4B part III (Eropean Physical Society) p 5 [5] Goloborod ko V.Ya., eznik S.N., Yavorskij V.A., and Zweben S.J. 995 Ncl. Fsion [6] Yavorskij V.A., Andrshchenko Zh.N., Edenstrasser J.W., and Goloborod ko V.Ya. 999 Phys. Plasmas [7] Smirnov A.P. and Harvey.W. The GENAY ay Tracing Code (available at [8] Jaeger E.F., Harvey.W., Berry L.A. et al 6 Ncl. Fsion [9] Batchelor.B. 5 Phyics Today [] Ono M., Bell M.G., Bell.E. et al 5 Ncl. Fsion 4 45 [] Eriksson L-G. and Helander P. 994 Phys. Plasmas 38 [] Hively L.M., Miley G.H., and ome J.A. 98 Ncl. Fsion 43 [3] Zaitsev F.S., O Brien M.., and Cox M. 993 Phys. Flids B 5 59 [4] ome J. A. and Peng Y-K.M. 979 Ncl. Fsion 9 94 [5] osenblth M.N., Maconald W.M., and Jdd.L. 957 Phys. ev. 7 [6] Harvey.W., McCoy M.G., Hs J.Y., and Mirin A.A. 98 Phys. ev. Lett. 47 5

54 [7] Fowler., Holmes J., and ome J. 979 Oak ridge national laboratory report ONL/TM-6845 [8] Stix T.H. 96 The Theory of Plasma Waves (New York: McGraw-Hill) [9] Kennel C. and Engelmann F. 966 Phys. Flids [3] Lerche I. 968 Phys. Flids 7 [3] Stix T.H. 99 Waves in Plasmas 99 (New York: AIP) [3] Karney C.F.F. 986 Compter Physics eports 4 83 [33] Hs C.T. and Sigmar.J. 99 Phys. Flids B 4 49 [34] Herrmann M.C. 998 Ph.. thesis (Princeton University 998) [35] Borba. and Kerner W. 999 J. of Compt. Physics 53 [36] White.B. 6 The Theory of Toroidally Confined Plasmas (London: Imperial College Press) [37] oloc C.M. and Martin G. 995 Ncl. Fsion [38] Egedal J. Ncl. Fsion [39] Hirshman S.P. 988 Phys. Flids 3 35 [4] Sater O., Angioni C., and Lin-Li Y Phys. Plasmas [4] Belli E.A., Candy J., Meneghini O., and Osborne T.H. 4 Plasma Phys. Control. Fsion [4] Westerhof E. and Peeters A.G. 996 Compter Physics Commnications 95 3 [43] Harvey.W. and Taylor G. 5 Phys. Plasmas 559 [44] Saad Y. 3 Iterative Methods for Sparse Linear Systems nd ed. (Philadelphia: Society for Indstrial and Applied Mathematics) 53

55 Figre. Giding center orbits representing distribtion fnction at a given radial grid point = 35 cm, for one energy level E = 5 kev. Particles are lanched with different initial pitch angles eqi-spaced over [; ] range. Illstration is made for + ions in an NSTX eqilibrim field. The dashed line corresponds to the last closed flx srface, and the magnetic axis is marked with the +. Also shown is the reference flx srface that passes throgh the = 35 cm coordinate on the eqatorial plane. Orbits that reside inside the srface are started as co-crrent-going particles. 54

56 Figre. A set of orbits converging to the pinch orbit, for 3 kev + ions in NSTX. For orbits are started at =45 cm a nearly co-passing orbit with pitch angle = 45.6 (the kidney -shaped orbit with marked points where becomes zero), and three banana orbits with pitch angles 45.33, and One more orbit, conter-passing, is started at = cm with pitch angle =

57 Figre 3. Map of different orbit types and bondaries in space of normalized toroidal canonical momentm and magnetic moment, for 5 kev + ions in NSTX. Vales of min and max correspond to the innermost and otermost major radis coordinates of the vacm vessel at the eqatorial plane; q is the particle charge, axis is the vale of poloidal magnetic flx at the magnetic axis. ifferent types of orbits are shown with different colors, matching those in figres 4 and 5. 56

58 Figre 4. Orbits corresponding to the dashed line in figre 3. Orbits of conterpassing ions (a portion of dashed line below point f in figre 3) are marked here as ff. Orbits that reside inside the magnetic srface going throgh = 3 cm are on the right arm of the dashed line in figre 3, below point c. Orbits f-e from figre 3 are shown in yellow; some of them are traced otside of the vacm chamber allowing a completion of the poloidal trn, bt all of them are considered to be lost. The yellow orbits are followed by the red orbits e-d that are confined. 57

59 Figre 5. Map of different orbit types and bondaries in local velocity space at a given radial point = 3 cm on the midplane. ashed line 5 kev corresponds to the dashed parabola in figre 3 and orbits in figre 4. Line o corresponds to stagnation-orbits with different energies. The colors match those in figre 3, except the yellow color for lost orbits e-f is removed here. 58

60 Figre 6. Energy content of + ions as a fnction of the normalized radial coordinate (related to sqare root of toroidal magnetic flx). The lower dashed line shows the initial profile, the thin solid line the steady-state profile in case of NBI+F heating with disabled radial transport terms, and the bold line is for the NBI+F heating calclations with enabled radial transport. 59

61 Figre 7. Steady-state distribtion fnction of ions with NBI+F heating, shown for one radial point =.5 / = 38 cm at the midplane. The thin (black) lines are for the rn with disabled radial transport terms, and the bold (red) lines are for the calclations with enabled radial transport. Five grops of lines correspond to cts of the distribtion at five pitch angles. Largest energy on the grid is E norm = 3 kev. Noticeably, enabling the radial transport terms reslts in significant drop of the distribtion at / norm.-.6 velocity range. 6

62 Figre 8. Contor levels of distribtion fnction log (f) over local velocity space for two radial points at the midplane: (a) and (b) are for =.34 / = 4 cm, and (c) and (d) are for =.5 / = 38 cm. Parts (b) and (d) correspond to calclations with enabled radial transport. The distribtions are the steady-state soltions of FPE for + ions in NSTX conditions with NBI+HHFW heating. 6