Neoclassical Transport Modeling Compatible with a Two-Fluid Transport Equation System

Transcription

1 Neoclaical Tranport Modeling Compatible with a Two-Fluid Tranport Equation Sytem Mituru HONDA, Atuhi FUKUYAMA 1) and Noriyohi NAKAJIMA 2) Japan Atomic Energy Agency, Naka , Japan 1) Graduate School of Engineering, Kyoto Univerity, Kyoto , Japan 2) National Intitute for Fuion Science, Toki , Japan (Received 1 November 2010 / Accepted 17 January 2011) A neoclaical tranport model compatible with a ytem of two-fluid equation i propoed, with an emphai on the heat flux contribution. The model i baed on the moment approach and i capable of accurately reproducing important neoclaical propertie through a imple expreion of the neoclaical vicoity tenor, with the aid of the NCLASS module. Applying neoclaical tranport theory in a fluid context, we confirm the reproducibility of firt-order flow, poloidal flow, neoclaical reitivity, boottrap current and particle flux. c 2011 The Japan Society of Plama Science and Nuclear Fuion Reearch Keyword: neoclaical tranport, neoclaical vicoity, heat flux, two-fluid equation, tokamak DOI: /pfr Introduction A one-dimenional tranport code TASK/TX coniting of two-fluid equation coupled with Maxwell equation ha been developed to invetigate the evolution of a plama elf-conitently, including a radial electric field, poloidal and toroidal rotation [1] a well a neutral tranport [2]. The code ha been applied, for example, in the phyic reearch on toroidal rotation induced by the lo of fat ion due to toroidal field ripple [3] and induced by the charge eparation of fat neutral due to the nearperpendicular NI intalled in JT-60U [4]. TASK/TX typically olve the continuity equation, the equation of motion in the radial, poloidal and toroidal direction, and the thermal tranport equation for electron and ion repectively, together with Maxwell equation. The et of equation which TASK/TX olve i eentially different from what a conventional tranport code olve, i.e., flux urface-averaged, one-dimenional diffuive tranport equation. The chief difference between them are ummarized in the following. (1) A conventional code typically calculate particle tranport only for ion pecie due to quai-neutrality and (2) require an explicit flux-gradient relationhip for a particle flux. (3) It olve a magnetic diffuion equation conitent with Faraday law, Ampère law without the diplacement current and Ohm law. (4) It determine poloidal rotation olely a an output uing an external neoclaical module and poibly olve a toroidal momentum equation analogou to a thermal tranport equation. In contrat, (1) TASK/TX olve the continuity equation for both electron and ion without explicit quai-neutrality condition and (2) alo olve an equation of motion in the radial direction, inauthor honda.mituru@jaea.go.jp dependently. Thee two difference are alo dicued in Ref. [5] in detail. (3) Maxwell equation are olved to decribe the behavior of magnetic field and electric field and the evolution of current denitie i calculated by olving equation of motion for electron. (4) Poloidal and toroidal rotation are calculated by olving equation of motion for ion. In that ene, the uual way for incorporating a tranport model/module into a conventional tranport code cannot be directly adopted with TASK/TX [5]. A imilar cae hold for neoclaical tranport. Neoclaical tranport play a very important role in toroidal plama, and, therefore, all chief neoclaical phenomena have to be reproduced in each tranport code for toroidal plama. Even when we focu only on an ideal axiymmetric plama, there are many important neoclaical characteritic that mut be conidered: neoclaical particle and thermal tranport, the neoclaical Ware pinch [6], neoclaical reitivity, boottrap current, neoclaically-driven poloidal flow, the radial electric field and o forth. In conventional tranport code, an external module uch a the NCLASS module [7] or the Matrix-Inverion method [8] etimate every quantity correponding to each neoclaical effect individually, and thee quantitie are then ubtituted into an appropriate term in the tranport equation: for example, the Ware pinch and particle diffuivity for a particle tranport equation; the neoclaical heat pinch and thermal diffuivity for a thermal tranport equation; and neoclaical reitivity and boottrap current for a magnetic diffuion equation. We, therefore, have to invent a method for reproducing all neoclaical effect in a twofluid TASK/TX ytem. Thee neoclaical tranport module build on the neoclaical moment approach of Hirhman and Sigmar [9]. The moment approach provide a method c 2011 The Japan Society of Plama Science and Nuclear Fuion Reearch

2 to compute all important neoclaical effect by olving teady-tate fluid momentum balance equation in a direction parallel to the magnetic field, with ome kinetic treatment, for particle and heat flow. ecaue thee momentum equation are roughly but eentially the ame a the equation of motion olved in TASK/TX, we conceived the idea of adding neoclaical vicoity term to the equation of motion to reproduce neoclaical effect. Hopefully, we will be able to decribe elf-conitently the neoclaical effect olely by olving the et of two-fluid equation with neoclaical vicoity term, once we know the neoclaical vicoitie. The poloidal heat flow can drive the poloidal (particle) flow in proportion to the temperature gradient, a fact which will be validated in Sec. 4.3, and thi effect may play a ignificant role in driving the poloidal flow in a plama with an internal tranport barrier. A will be een in the next ection, however, TASK/TX doe not include the moment equation for heat flux, implying that it doe not reproduce the heat flux-driven poloidal flow within the original framework [1]. A noted in the previou reearch [3,4], thi driving mechanim did not play a key role becaue plama with a moderate temperature gradient were conidered. We have to include the heat flux contribution in TASK/TX to fulfil the condition of completene in neoclaical tranport theory and to extend the applicability of the code, by coupling TASK/TX with NCLASS. We note that neoclaical theory a it relate to the moment approach i found in Ref. [10, 11] in addition to Ref. [9] and it i reviewed particularly well in Ref. [12]. The ret of thi paper i organized a follow. Apect of the baic equation of TASK/TX in aociation with thi tudy are decribed in Sec. 2. Analytical invetigation demontrate the exitence of firt-order flow parallel and perpendicular to the magnetic field in the TASK/TX ytem, a hown in Sec. 3. Detail related to neoclaical tranport modeling in TASK/TX are given in Sec. 4, a well a a demontration of the reproducibility of poloidal flow. Section 5 i devoted to a comparion of neoclaical reitivity and boottrap current calculated by TASK/TX, NCLASS and the Sauter model [13, 14]. It i demontrated in Sec. 6 that the radial particle flux can be reproduced accurately. Finally, a ummary and dicuion are given in Sec. 7. denity n, the radial flow velocity u r, the poloidal and toroidal flow velocitie u θ, u φ. The equation are [1] n = 1 r r (rn u r ) + S, (1) r (ru rm n u r ) + 1 r m n u 2 θ t t (m n u r ) = 1 r +e n (E r + u θ φ u φ θ ) r (n T ), (2) t (m n u θ ) = 1 r 2 r (r2 u r m n u θ ) + 1 [ r 3 ( uθ )] m r 2 n μ + e n (E θ u r φ ) r r r +F NC θ + F θ, (3) t (m n u φ ) = 1 r r (ru rm n u φ ) + 1 ( ) u φ rm n μ + e n (E φ + u r θ ) + F φ, (4) r r r where m and e are the ma and charge, repectively. The perpendicular vicoity repreent anomalou tranport contribution due to turbulent fluctuation. Here, S repreent the ource and ink term for particle; F NC, the force related to neoclaical effect, which will be dicued in detail in the following ection; and F θ and F φ, other force. In thi paper, we do not olve the thermal tranport equation, even though they are included in the code. Rather, we fix the temperature profile intead to focu our attention on the neoclaical effect on particle. Maxwell equation are imultaneouly olved uing thee equation for the evolution of the electromagnetic field; that i, the radial, poloidal and toroidal electric field E r, E θ, and E φ, and the poloidal and toroidal magnetic field θ and φ, repectively. Further detail of the code are decribed in Ref. [1 3]. 3. Firt-Order Flow in TASK/TX In thi ection, we recall the characteritic of firtorder flow in an axiymmetric configuration and then analytically confirm that thee characteritic are alo inherently preent in the TASK/TX ytem. We intend to demontrate that it i poible to achieve the reult provided olely by uing the radial force balance equation, thereby indicating the validity of thee reult in all colliionality regime. 2. Multi-Fluid Tranport Modeling, TASK/TX We briefly decribe the main equation, which become the bai for the following dicuion. Currently, the bai equation of TASK/TX eentially build on a concentric circular equilibrium. In thi ene, the haping effect of an equilibrium and the Shafranov hift have not yet been reflected in the code. The following flux urface-averaged quantitie for pecie are elf-conitently olved in the code a the initial boundary value problem: the particle Tranport ordering Neoclaical tranport theory relie on an ordering ued to categorize tranport phenomena baed on their magnitude. The baic ordering aumption i that Larmor radiu ρ i much maller than the macrocopic cale length L: δ ρ L 1. The macrocopic cale length L i typically defined by patial change in macrocopic parameter, uch a preure.

3 Thi ordering i called the mall gyroradiu ordering, drift ordering or tranport ordering. aed on thi ordering, we know that the time derivative i coniderably maller: t O(δ2 ), and the flow velocity V i to firt order in δ: V δv th. 3.2 Firt-order flow perpendicular to Theory In an axiymmetric ytem, the magnetic field i defined a = φ ψ + I φ, where I(ψ) = R φ i a poloidal current and a flux function, a function of the flux urface ψ alone, a well. A fluid moment equation for flow velocity i given by m n du dt = p π +e n (E+u )+R, (5) where π denote the vicoity tenor and R, the exchange of momentum. The convective derivative d/dt i defined in a frame moving at the fluid velocity u. Taking the vector product with from the left, we have to leading order in δ u E 2 + b p m n Ω, (6) coniting of the E drift and the diamagnetic drift. Here b /. The electric field i given by E = Φ A/t, but we now conider only the electrotatic term becaue to thi order the electromagnetic term i negligible. With the aid of the relationhip ψ = I 2 b R ˆφ, (7) Equation (6) i rewritten a ( u = ω R ˆφ I ) b, (8) ω Φ ψ 1 p e n ψ. (9) Here, ˆφ = R φ i the unit vector in the toroidal direction. We call u a firt-order diamagnetic velocity, in accordance with Ref. [10] TASK/TX In the TASK/TX ytem under the aumption of a concentric circular equilibrium, the radial direction i orthogonal to the poloidal direction unlike the traight field-line coordinate. The poloidal direction i geometrically determined, and the magnetic field i expreed a where ˆθ i the unit vector in the poloidal direction. From the definition of perpendicular direction, the fluid velocity perpendicular to i written a u = b (u b) = u r ˆr + φ ( φu 2 θ θ u φ )ˆθ θ ( φu 2 θ θ u φ ) ˆφ, (10) where ˆr i the unit vector in the radial direction. The velocity u r croing a flux urface i one order maller than u θ and u φ and, thu, negligible when we conider the flow to leading order in δ. aed on the tranport ordering, the inertia and centrifugal term in Eq. (2) are negligibly mall and then we obtain φ u θ θ u φ 1 p e n r + Φ r. (11) Subtituting thi force balance relation into Eq. (10) yield u ( φ u θ θ u φ ) φ ˆθ θ ˆφ 2 = 1 ( 1 p R θ e n r Φ ) R θ φ r 2 ( =ω R ˆφ I ) b, ˆθ + R2 θ 2 which i identical to Eq. (8), implying that the TASK/TX equation ytem incorporate the flow perpendicular to. We have ued /ψ (1/R θ )/r in the final tep of thi derivation. 3.3 Firt-order flow parallel to Theory To leading order in δ, the particle flow i incompreible on a flux urface, while it parallel and perpendicular component are not individually divergence-free due to the inhomogeneity of the trength of the magnetic field along a field line. Thi characteritic yield a parallel return flow expreed a (ee e.g. Sec. 8.5 in Ref. [12]) Iω n u = n b + K, (12) where K i an integral contant and a flux function a well. Taking the calar product of Eq. (12) with θ, we find K : u θ K (ψ) n û θ = n θ. (13) In fact, K /n i identical to the contravariant component of the flow, û θ. We note that û θ doe not have a velocity dimenion and i alo a flux function. Finally, combining Eq. (8) with Eq. (12), we obtain the firt-order particle flow expreed a ˆφ = θ ˆθ + φ ˆφ, u = ω R ˆφ + û θ. (14)

4 When we take the calar product of u = u + u with p = φ ψ, we ee the relationhip: u p 2 p = u p 2 p + u p. (15) 2 p Obviouly, the factor of u on the left-hand ide (LHS) of the equation i unity and that of in the econd term on the RHS i equivalent to û θ, a een in Eq. (13). Comparing Eq. (15) to Eq. (12), we have u = V 1 + û θ, (16) V 1 u p 2 p = Iω. (17) In thi ene, V 1 i the flow along the magnetic field which would cancel the poloidal component of the diamagnetic velocity, u. A imilar cae hold for the firt-order heat flux parallel to the magnetic field q, expreed a q = 5 2 p V 2 + ˆq θ, (18) V 2 = I T m Ω ψ. (19) ˆq θ i defined in the ame fahion a in Eq. (13) TASK/TX The velocity parallel to the magnetic field i obtained by u = (b u)b = θu θ + φ u φ b. With the help of the radial force balance relation in Eq. (11), thi expreion can be deformed in the following: u = 1 φ ( θ u φ φ u θ ) + θ u θ + 2 φ u θ θ θ 1 [ ( 1 p R φ e n ψ + Φ ) + u ] θ 2 ψ θ = 1 ( Iω + u ) θ 2, (20) θ where we have ued the definition of ω, a given in Eq. (9). Recalling K = n û θ = n u θ / θ in a circular concentric equilibrium, we finally have 4.1 Neoclaical vicoity tenor A een in the derivation of the preceding ection, the perpendicular flow in Eq. (8) can be determined uniquely in term of the thermodynamic force, while the parallel flow in Eq. (12) cannot becaue the poloidal flow û θ i unknown, which i generally a function of the colliion frequency. Thi i alo the cae for heat flux. In etimating the poloidal flow, we hould take into account the neoclaical vicoity term dependent on the colliionality regime, a term which regulate the behavior of poloidal flow. aed on the moment approach, uing u and q a given in Eq. (16) and (18), bearing in mind that there i no firt-order parallel vicoity force ariing from the ψ derivative of the flow [10], yield the flux urfaceaveraged vicoity tre a follow: π = 3 ( ) 2 ( 2ˆq θ μ 1 û θ + μ 2 5p θ = 3 ( ) 2 ( 2ˆq θ μ 2 û θ + μ 3 5p ), (21) ), (22) where the contant 3 ( ) 2 ha been choen o that μ 1 reduce to raginkii claical vicoity coefficient [15] in the colliional limit. Here, θ denote the heat flux tenor analogou to π. The unknown coefficient μ j ( j = 1, 2, 3) are called neoclaical parallel vicoity coefficient and are expreed a integral of the velocity dependent vicoitie over the velocity pace [16]. We adopt in a practical manner the expreion of Eq. (1) in Ref. [7] and Eq. (20) in Ref. [16] a the vicoitie in the banana and Pfirch-Schlüter regime, repectively. The chief advantage of thi approach i that there i no need to numerically olve any complicated kinetic equation to etimate the neoclaical vicoitie. 4.2 Moment equation of neoclaical fluxe Taking the calar product of Eq. (5) with yield to firt order in δ π = (R + e n E (A) ), (23) and imilarly for the heat flux we have θ = H. (24) Iω n u = n b + n û θ, which i abolutely equivalent to Eq. (12). 4. Neoclaical Tranport Modelling aed on the Moment Approach In order to explain our neoclaical modeling, we briefly recall the moment approach [9] and then demontrate a mechanim to realize neoclaical effect in the TASK/TX ytem Here E (A) i the inductive element in the parallel electric field and H i a parallel heat friction. Thee equation contitute the balance equation for momentum and heat. The kinetic treatment of the friction force R and H tipulate the relationhip between the friction force and the parallel flow. Therefore, R and H are expreed by the claical friction coefficient lij [9] and the parallel flow, and they are valid in all neoclaical regime. Subtituting the neoclaical vicoity tenor given in Eq. (21) and (22) and the expreion of the parallel friction force and the parallel flow given in Eq. (16) and

5 (18) into the balance equation given in Eq. (23) and (24) yield the imultaneou equation in term of the poloidal flow, expreed a 3 ( ) 2 ( ) μ 1 μ 2 μ 2 μ ûθ 2ˆq θ 3 5p ( ) l = 11 l 12 l 12 l V 1 + û θ 2 22 V 2 + 2ˆq θ 5p 2 + e n E (A). (25) 0 Once the inductive parallel electric field E (A) i known, olving the imultaneou equation yield the poloidal flow and ubequently the particle and heat flow in the parallel direction. Roughly peaking, Eq. (25) include the equation that are practically olved in NCLASS. 4.3 Qualitative interpretation of neoclaical flow It i meaningful to focu on the relationhip between neoclaical vicoitie and flow by conidering the limit to a large apect ratio, ɛ 1. In thi limit, the vicoitie μ j are generally much maller than the friction force lij, and ubequently the ratio of the former to the latter can be ued a a mall expanion parameter [9,10]. Neglecting the E (A) contribution, to lowet (zeroth) order in the ratio, we have R = 0 and H = 0. Momentum conervation in Coulomb colliion guarantee that the friction force depend olely on relative parallel particle flow. Therefore, the olution to Eq. (16), (18) and (25) mut have the form: u V and q 0. The former reult indicate that all pecie have a common flow equal to V to lowet order in the ratio. aed on R = 0 and quai-neutrality, umming Eq. (23) over all pecie including electron yield π = 0. Subtituting the parallel flow given in Eq. (16) and (18) and the neoclaical vicoity term given in Eq. (21) into thi equation, we have after averaging it over the flux urface and umming it over all pecie V = (μ 1 V 1 + μ 2 V 2 ). μ 1 ecaue electron neoclaical vicoity i in general maller by a factor of (m e /m i ) 1/2 than ion vicoity, the common flow V i almot excluively governed by ion motion. For intance, conidering a pure plama coniting only of electron and ion without impuritie, the ion poloidal flow i expreed a û iθ = 1 2 ( V V 1) = 1 2 μ i2 V 2i μ i1 = μ i2 I μ i1 Z i e T 2 i ψ, (26) applying Eq. (19). Thi form provide an important reult: poloidal flow i proportional to the temperature gradient via the neoclaical vicoitie. Uually there i a finite and negative temperature gradient in a plama, uch that the poloidal flow i alway driven by the temperature gradient, owing to the heat flux contribution, μ i2. If the contribution from the heat flux to the momentum were ignored, i.e., μ i2 = 0, the poloidal flow would never be neoclaically driven and would jut be damped due to the neoclaical vicoity μ i1 at ome point, a clearly een in Eq. (21). Phyically, the damping of the poloidal flow in connection with the firt term in parenthee in Eq. (21) i due to colliion between trapped particle and paing particle. 4.4 Implementation of the neoclaical vicoity term in TASK/TX In a perfect axiymmetric tokamak, there are no neoclaical vicoitie working in the toroidal direction, baed on tandard neoclaical theory. Roughly peaking, toroidal flow may be balanced by an external torque input and tranport of momentum, free from neoclaical tranport: thi fact will be dicued in detail in another paper. In fact, we found that the neoclaical moment approach build on the equation of motion in a direction parallel to, a een in the preceding ection, and the equation of motion in the toroidal direction ha nothing to do with the moment approach. A parallel equation of motion i developed in the TASK/TX ytem by umming the poloidal equation of motion given in Eq. (3), multiplied by θ and the toroidal equation of motion given in Eq. (4), multiplied by φ. Conidering the above tatement, we find that we hould include the neoclaical vicoity tenor only in the poloidal equation of motion a follow: F NC θ 1 π θ 2 3 ( ) 2 ( ) = θ 2 u θ 2ˆq θ μ 1 + μ 2, (27) θ 5p where we have ued û θ = u θ / θ a valid in the TASK/TX ytem. In a practical ene, TASK/TX ue the NCLASS module [7] to calculate the neoclaical vicoitie, given a 3 ( ) 2 μ j / 2, where j = 1, 2, and the poloidal heat fluxe, given a 2ˆq θ /5p. TASK/TX olve the momentum equation for the particle given in Eq. (2), (3) and (4) but never olve the momentum equation for heat. In thi ene, u θ, appearing in the firt term in the parenthee of Eq. (27), can be calculated a one of the dependent variable by olving the et of equation in TASK/TX, while ˆq θ in the econd term, temming from the momentum equation for heat, cannot be determined olely within the current framework of TASK/TX. Therefore, at preent, the poloidal heat flux ˆq θ i etimated by NCLASS: we would like to emphaize that thi i eentially the only variable for which NCLASS i ued for ubequent calculation in TASK/TX. We finally note that we may regard our model a propoed here to be a combination of the ba-

6 Fig. 1 ackground radial profile of (a) the electron denity and (b) the electron and ion temperature. Fig. 2 Comparion of the poloidal flow for (a) electron and (b) ion. The hort broken line in red (NCLASS) i the um of the long broken line in blue (NCLASS, thermo) and the chain line in green (NCLASS, E ). i equation of TASK/TX and Eq. (25), becaue ˆq θ i etimated in NCLASS by olving Eq. (25), in practice. 4.5 Comparion of the poloidal flow A expected from the dicuion in Sec. 4.2, NCLASS implemented in TASK/TX alo provide û θ = u θ / θ,a well a ˆq θ. Accordingly, we can directly compare the poloidal flow u θ a a dependent variable in TASK/TX to û θ a calculated directly in NCLASS. In other word, we compare the olution of Eq. (3) to that of Eq. (25), in term of the poloidal flow. Plama parameter comparable to the JT-60U are ued: the major radiu R = 3.2 m, the minor radiu a = 0.8 m, the toroidal field T0 = 3.2 T, and the plama current I p = 0.6 MA. The profile of n e, T e and T i are hown in Fig. 1. Temperature profile are fixed throughout the imulation. We conider a pure plama coniting of electron and ion only: the effective charge Z eff = 1.0. Focuing on neoclaically-driven poloidal flow, we et the anomalou particle diffuivity to zero. Thi choice tem from the fact that in TASK/TX modeling, an anomalou particle flux i achieved through a change in the poloidal force due to turbulence [5]. A comparion of the poloidal flow for electron and ion calculated by TASK/TX and NCLASS, repectively, i hown in Fig. 2 and indicate very good agreement for both cae. Thi finding mean that the neoclaicallydriven poloidal flow clearly can be reproduced in the TASK/TX ytem, following the introduction of Eq. (27) NCLASS alo provide each component of the total poloidal flow. Figure 2 (a) how that the electron poloidal flow i dominated by the contribution from the parallel electric field E, while Fig. 2 (b) how that the ion poloidal flow i mainly governed by the thermodynamic force proportional to the gradient of temperature and preure, temming from the heat flux contribution. In thi ene, the heat flux contribution to the particle flux i eential for accurately reproducing the neoclaical effect in the two-fluid modeling. Even if many term, uch a anomalou perpendicular diffuion, claical colliion, colliion with neutral, and charge exchange lo, are included in Eq. (3), the reultant poloidal flow i identical to the one etimated by NCLASS. In other word, the neoclaical vicoity tenor play a dominant role in determining poloidal flow. In contrat, a force model producing a quailinear particle flux due to turbulence [5], implemented in TASK/TX, could alter poloidal flow from a purely neoclaical poloidal flow, if it were activated. It i natural, however, that turbulence influence neoclaical characteritic a analytically elucidated by Shaing [17]. The neoclaical modeling of TASK/TX given in Eq. (27) ha many advantage. Firt of all, thi modeling never intervene in the two-fluid equation ytem; it imply require the neoclaical vicoity term to be added to the poloidal equation of motion. Secondly, an additional effect can be readily included in thi framework. For example, we have a new equilibrium tate of the poloidal

7 flow that tem from both the neoclaical and additional effect when we add a torque ource other than neoclaical tranport. Thi feature will be ueful to invetigate the behavior of experimentally oberved poloidal flow that deviate from neoclaical prediction [18]. Thirdly, thi modeling i compatible with elf-conitent modeling in term of the evolution of the polarization current. TASK/TX ha been applied to the tudy of E r and toroidal rotation driven by an ambipolar radial current created by NI [3, 4]. Thee tudie can be conducted uing the feature of TASK/TX which olve the poloidal equation of motion with Maxwell equation: the evolution of E r i neoclaically linked to that of the radial (polarization) current through that of the poloidal flow. Thi neoclaical modeling can reproduce the above important neoclaical characteritic of a plama without violating the current framework of TASK/TX. The relationhip between E r and the radial current will be dicued in detail in another paper. Two important neoclaical characteritic are neoclaical reitivity and boottrap current, the former of which regulate the penetration time of an ohmic current and the latter of which i the ole, elf-generating (boottrap) current in a plama without an external drive and i expected to play an indipenable role in a teady-tate, non-inductive plama. All tranport code for a plama in a tokamak mut take both effect into account. ecaue TASK/TX track the current evolution a electron motion by olving the equation of motion for electron, reitivity and boottrap current do not explicitly appear in the et of equation, unlike conventional diffuive tranport code. Thee effect are naturally embedded in the total current denity; hence, the only reult evident in the olution of thee equation i the total current denity in the radial, poloidal and toroidal direction, i.e., the radial electron flux and the parallel and toroidal current denitie. We anticipate that thee effect would appear olely by introducing the neoclaical vicoity tenor given in Eq. (27) into the et of equation. Therefore we hould confirm whether the vicoity term produce neoclaical reitivity and boottrap current in TASK/TX through ome other mean. In order to validate the TASK/TX reult, we extract the component aociated with reitivity and boottrap current from analytically-reduced formulae for the et of bai equation of TASK/TX. In thi repect, the comparion ha been already carried out in Ref. [1]. However, the neoclaical contribution from the heat flux to the particle flux wa not included at that point; hence, in ome ene the comparion wa incomplete. In thi ection, we approximately formulate a teady-tate analytical olution of the equation for electron, and after extracting correponding term from the teady-tate flux, we compare them with thoe from NCLASS and the Sauter model, briefly explained below, in a manner imilar to that in Ref. [1]. 5.1 Theoretical model NCLASS NCLASS, which calculate the neoclaical vicoitie and the heat flux ued in TASK/TX, i alo capable of determining neoclaical reitivity and boottrap current. Specifically, the former i figured directly by NCLASS; the latter i derived, a follow, from thermodynamic gradient and the NCLASS-upplied coefficient: js ( = CT, S T + C S p ) p,, (28) T where i the pecie and the prime denote the ψ- derivative. p 5. Reproducibility of Neoclaical Reitivity and oottrap Current Sauter model The CQLP code [19], olving the Fokker-Planck equation with a full colliion operator and including variation along the magnetic field line, coupled with adjoint function formalim, i ued to calculate coefficient for reitivity and boottrap current in arbitrary equilibrium and colliionality regime. The reult for a wide range of plama parameter are applied to model-fitted formulae, termed the Sauter model [13, 14]. The model ditinguihe itelf from NCLASS in term of it fat calculation peed due to the algebraic formulae and highly accurate etimate due to a full-colliion operator: the moment approach without the full-colliion operator may caue error up to 20% [7]. The Sauter model doe not include the effect of potato orbit particle. Alo, it cannot etimate anything neoclaical other than neoclaical reitivity and boottrap current, meaning that it alone doe not include all variable regarding neoclaical effect, unlike NCLASS. 5.2 Analytical etimation in TASK/TX A hown in Sec. 2.3 of Ref. [1], we may reduce the et of equation to obtain analytical expreion of electron flow by auming a teady tate and neglecting relatively mall factor and ion flow. ecaue neoclaical reitivity η i determined by the equation j = η 1 E, term proportional to E hould be extracted from electron teady-tate fluxe in the poloidal and toroidal direction. The electron parallel current can be written a φ u eφ + θ u eθ j = en e. (29) After analytically etimating the teady-tate poloidal and toroidal fluxe for electron, we ubtitute term proportional to E in the u eθ and u eφ equation into Eq. (29) to

8 Fig. 3 Comparion of TASK/TX uing Eq. (31) and (33), NCLASS and the Sauter model for etimate of (a) neoclaical reitivity and (b) boottrap current. obtain j = e2 n e θ F q eθ E E m e ν α 2, (30) en φ e where ν 3 i the contribution from the claical colliion and α i the non-dimenional parameter dependent on claical and neoclaical colliionalitie, both of which are defined in Ref. [1]. The heat flux contribution F q eθ i expreed a F q eθ = 3 ( ) 2 2ˆq eθ μ e2 θ 5p e, where the ubcript denote the component of the heat flux aociated with a quantity : E, in thi cae. Subtituting F q eθ E into Eq. (30) and dividing it by E yield η = m eˆν 3 (1 + α) e 2 n e 2 φ ( ) 2 E μ e2 2ˆq eθ en e 5p e E 1, (31) where E in the denominator in parenthee may be offet by E implicitly appearing in the heat flux contribution ˆq eθ, indicating that η doe not explicitly depend on E. In the claical limit, (1 + α) ν 3 ν ei 2 / 2 φ, and the heat flux contribution become nil. Subequently, Eq. (31) i reduced to η cl = ν m e m e ei N(Z e 2 eff )ν ei, (32) n e e 2 n e where ν ei i the electron-ion colliion frequency and N(Z eff ) i the correction function of the colliion frequency with repect to the effective charge Z eff, which take into account the electron-electron colliion effect [9]: In a pure plama with Z eff = 1, N(1) We, therefore, find that Eq. (32) recreate the claical reitivity. A imilar cae hold for boottrap current. oottrap current conit of term proportional to the preure and temperature gradient, p and T, typically rendered a in Eq. (28). The p -driven current originate from the diamagnetic drift, and the T -driven current originate from the heat flux. In thi ene, we have to include the T contribution calculated by NCLASS in order to etimate boottrap current. Subtituting term proportional to p and T in the u eθ and u eφ equation into Eq. (29) yield 1 j S = θ (1 + α) ν 3 φ ν p NCe r θ ef q eθ T φ m e, (33) where p i the total preure. The expreion of F q eθ T ha already been defined above. We hould emphaize that we have impoed a rough aumption in thi derivation in that we have ignored the contribution of ion flow. We will dicu the influence of neglecting the contribution of ion to boottrap current in the lat part of the next ection. 5.3 Comparion of neoclaical reitivity and boottrap current The imulation condition ued are identical to thoe given in Sec We compare neoclaical reitivity and boottrap current etimated by TASK/TX, NCLASS and the Sauter model. We ue Eq. (31) a the neoclaical reitivity model and Eq. (33) a the boottrap current model for TASK/TX. Good agreement on neoclaical reitivity η i obtained with thee three model, a hown in Fig. 3 (a). Epecially, reitivity calculated by NCLASS i nearly equivalent to that etimated by TASK/TX. Conidering that the neoclaical framework of TASK/TX build on the moment approach adopted in NCLASS, the fact that the reult from both are nearly identical demontrate that TASK/TX could incorporate the moment approach internally. The light difference in η between the Sauter model and the other may be acribed to the approach to modeling, i.e., a fitted formula baed on Fokker-Planck analye and the moment approach. Alo, agreement on boottrap current i obtained, a hown in Fig. 3 (b). A light deviation of the TASK/TX etimation from the other i oberved in the core region, but the maximum deviation fall within a factor of 1.2. When we compare the deviation in Fig. 3 (b) to a peak of the u iθ profile in Fig. 2 (b), we deduce that the deviation of boottrap current ha to do with the hape of the ion poloidal flow u iθ. Actually, the location of the maximum deviation may be linked to that of the maximum magnitude in u iθ. Thi finding lead to the deduction that neglecting the con-

9 Fig. 4 Comparion of TASK/TX uing Eq. (34), NCLASS and the Sauter model for etimate of boottrap current. The figure indicate that the contribution of the ion poloidal flow to the boottrap current a included in Eq. (34) i ignificant. tribution of ion flow produce the deviation, a deduction upported by the fact that the current i the um of the electron and ion flow. In thi cae, no ignificant ion toroidal flow exit becaue an external torque i not applied to the plama; only poloidal flow may exert an impact on boottrap current. We extend the approximate expreion of boottrap current given in Eq. (33) o a to include the contribution from u iθ, a follow: 1 j S = θ (1 + α) ν 3 φ ν p NCe r θ ef q eθ T φ m e ( ) ] θ φ ν NCe ν ei en e u iθ. (34) We ue a elf-conitent olution of u iθ calculated by TASK/TX, when etimating boottrap current uing Eq. (34). Figure 4 how a comparion of boottrap current among Eq. (34), NCLASS and the Sauter model: Eq. (34) reproduce quite accurately the boottrap current etimated by NCLASS a well a the Sauter model. Thi finding mean that the deviation in Fig. 3 (b) i in large meaure acribed to neglecting ion poloidal flow. Thi reult alo upport our contention that the neoclaical propertie of reitivity and boottrap current can be reproduced accurately in the TASK/TX ytem. Finally, we note that Eq. (34) i not quite appropriate a an approximate formula of boottrap current becaue the poloidal flow u iθ i explicitly included to decribe the current flow. Ideally, we hould have replaced u iθ in Eq. (34) with term not directly including the flow. However, it i quite difficult to analytically reduce the et of equation of TASK/TX to a et of approximate, linear formulae for all dependent variable. We, therefore, opt for the econd bet way to decribe boottrap current a in Eq. (34) by numerically calculating u iθ Reproducibility of Neoclaical Particle Flux aically, neoclaical tranport in term of particle and heat i much maller than turbulent tranport. Crofield tranport, with a tep ize typically imilar to a banana width, i on the order of O(δ 2 ), which i alo much maller than parallel, poloidal and toroidal flow. One of the important neoclaical tranport propertie i the Ware pinch [6], which i directed inward a long a E exit and may lead to a peaked denity profile toward the magnetic axi. A i clear in Eq. (1) and (2), there are no explicit neoclaical convection and diffuion term in the equation aociated with particle tranport. A uch, we hould clearly demontrate that neoclaical particle tranport doe occur in the TASK/TX ytem through the neoclaical vicoity tenor given in Eq. (27), a i the cae with the other neoclaical propertie mentioned above. In thi ection, we confirm the exitence of neoclaical particle tranport in our neoclaical modeling and then compare the particle flux with that directly etimated by NCLASS. 6.1 Radial particle flux Firt of all, let u recall how the cro-field radial particle flux Γ i i derived in neoclaical tranport theory. y derivation, we omit the ubcript i meaning charge tate, for the ake of implicity. Conidering the flux urfaceaveraged toroidal projection (R ˆφ) of the equation of motion given in Eq. (5), we have Γ ψ = 1 e R(Rφ + e n E (A) φ ), where we have neglected the inertia and vicoity term, which are much maller than the remaining term, due to the tranport ordering. Uing the relationhip given in Eq. (7), the radial particle flux may be decompoed a Γ ψ = 1 I e b (R + e n E) + 1 ψ (R e 2 + e n E) I e 2 (R + e n E (A) ) I R + e n E (A) e + Γ cl E (A) + n 2 I = e 2 I e R E (A) In π ) (1 2 2 ) (1 2 2 ) (1 2 2 ψ

10 + n E (A) 2 ψ + Γ cl, (35) where we have ued Eq. (23) in the final equality and implicitly aumed that n i the flux function. The firt term in Eq. (35) i the banana-plateau flux, which i driven by the urface-averaged preure aniotropie and i dominant in the long mean free path regime. The E component of the banana-plateau flux i aociated with the Ware pinch. The econd term i the Pfirch-Schlüter flux that reult from a poloidal variation of the friction force on a magnetic urface. The combination of the third and fourth term including the electric field decribe the motion of the magnetic flux urface [9]. When we conider the particle flux on a flux urface, thee term do not appear. The ixth term i defined a the claical flux. In NCLASS, the flux for particle with pecie and charge tate i can be decompoed into banana-plateau (P), Pfirch-Schlüter (PS) and claical (cl) component: Γ i Γ i ρ = Γ P i + Γ PS i + Γ cl i, where ρ denote the flux urface. Thi expreion i compatible with Eq. (35). 6.2 Comparion of neoclaical particle flux In thi comparion, although the amplitude of the claical flux i two order maller than that of the neoclaical flux, the claical flux i included in thi comparion, though negligible in practice. In a concentric circular equilibrium which TASK/TX adopt, 2 i identical to 2 and, thu, the Pfirch-Schlüter flux vanihe: the PS flux i of coure finite in an actual plama, but the magnitude i much maller than the banana-plateau flux. In the end, we virtually compare the P flux. A clearly een in Eq. (2), the radial particle flux Γ n u r i one of the dependent variable in TASK/TX. In thi ene, we do not need an analytically-reduced equation to extract the particle flux from the imulation reult, unlike reitivity and the boottrap current, a hown in Sec. 5. The neoclaical particle flux i readily obtained in TASK/TX directly by olving the et of equation with turbulent particle diffuivity et to zero. In doing o, we obtain the flux excluding the turbulence-particle flux in a imulation reult in TASK/TX. A comparion of the imulation reult for the electron particle flux Γ e, calculated by TASK/TX and NCLASS i hown in Fig. 5 in a manner indicating that the neoclaical particle flux i accurately reproduced over the profile in the TASK/TX ytem, a i the cae for other neoclaical propertie. 7. Summary and Dicuion We have developed neoclaical tranport modeling compatible with a ytem of two-fluid equation on which TASK/TX relie, with the aid of the NCLASS module. The baic idea wa borrowed from the moment approach ued in NCLASS a etablihed mainly by Hirhman [9]. Conidering that parallel friction acting on guiding center i eential for neoclaical tranport, we have imply introduced the parallel neoclaical vicoity tenor given in Eq. (27) into the poloidal equation of motion given in Eq. (3). Even though the neoclaical vicoitie and the poloidal heat flux are etimated by NCLASS, the poloidal particle flow u θ and any other neoclaical feature are elf-conitently calculated olely by olving the equation in TASK/TX. Through comparion of many important neoclaical propertie with NCLASS and ometime the Sauter model, the validity of the neoclaical modeling wa confirmed. A i uual with two-fluid equation, the chain of moment equation i truncated up to the econd order moment, the energy equation, to get a cloed et of equation for TASK/TX. In other word, the moment equation for heat flux are not olved in the code: intead, thee equation are olved uing Eq. (25) to obtain the poloidal heat flow ˆq θ. If thee equation are included in the et of equation, etimating the heat flux by olving the imultaneou equation given in Eq. (25), i.e., NCLASS, i not neceary, becaue ˆq θ become a dependent variable of the code. We imply have to calculate the vicoitie μ j ( j = 1, 2, 3) baed on their analytic formulae [7, 16] in term of neoclaical tranport in thi cae and then olve the et of equation, including the vicoity tenor given in Eq. (21) and (22). In doing o, we enable conitent etimation of the radial heat flux (tranport), imilar to the particle flux hown in Sec. 6. From the tandpoint of completene of the framework of neoclaical tranport theory, the code hould be improved o a to olve the moment equation for heat flux and alo to accommodate itelf to an arbitrary equilibrium. Fig. 5 Etimate from TASK/TX and NCLASS for the neoclaical particle flux. Acknowledgment Thi work wa upported by the Grant-in-Aid for Young Scientit () (No ) from The Minitry of Education, Culture, Sport and Technology (MEXT) and

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