Neoclassical electron and ion transport in toroidally rotating plasmas

Neoclassical transport processes of electrons and ions are investigated in detail for toroidally rotating axisymmetric plasmas with large ﬂow velocities on the order of the ion thermal speed. The Onsager relations for the ﬂow-dependent neoclassical transport coefﬁcients are derived from the symmetry properties of the drift kinetic equation with the self-adjoint collision operator. The complete neoclassical transport matrix with the Onsager symmetry is obtained for the rotating plasma consisting of electrons and single-species ions in the Pﬁrsch–Schlu¨ter and banana regimes. It is found that the inward banana ﬂuxes of particles and toroidal momentum are driven by the parallel electric ﬁeld, which are phenomena coupled through the Onsager symmetric off-diagonal coefﬁcients to the parallel currents caused by the radial thermodynamic forces conjugate to the inward ﬂuxes, respectively. © 1997 American Institute of Physics. S1070-664X 97 00806-9


I. INTRODUCTION
Improved confinement modes of tokamak plasmas such as high-confinement modes ͑H-modes͒ 1 and reversed shear configurations 2 are attracting considerable attention as promising means for achieving controlled fusion. Such a reduction of the transport level is generally considered as caused by the large radial electric field shear ͑or sheared flow͒. In the Japan Atomic Energy Research Institute Tokamak-60 Upgrade ͑JT-60U͒, 3 the internal transport barrier ͑ITB͒ with the steep ion temperature gradient is formed in the region where the gradient of the toroidal flow is steepest. 4 In rotating plasmas with the large flow velocities on the order of the ion thermal speed v Ti , the toroidal flow shear influences the transport of particles, heat, and momentum as an additional thermodynamic force, although, in conventional neoclassical theories, [5][6][7] the flow velocities are assumed to be on the order of ␦v Ti and the direct effects of the flow shear on the transport do not appear in the lowest order. Here ␦ϵ i /L is the drift ordering parameter, i the ion thermal gyroradius, and L the equilibrium scale length. It is important to derive the transport equations including the flow shear effects at the same order as particle and thermal transport for understanding the ITB physics. Neoclassical ion transport equations for rotating plasmas were obtained by Hinton and Wong 8 and by Catto et al. 9 However, neoclassical electron fluxes are also required for a comprehensive description of transport processes. For example, the neoclassical parallel ͑bootstrap͒ current is associated with the parallel electron viscosity and is necessary for determining the equilibrium configuration selfconsistently. In the present work, we derive full transport equations for neoclassical electron and ion fluxes in the rotating plasma with the toroidal flow velocity on the order of the ion thermal speed.
Hereafter we consider only axisymmetric systems, for which the magnetic field is given by where is the toroidal angle, ⌿ represents the poloidal flux, and I(⌿)ϭRB T . In the axisymmetric systems, the poloidal flow decays in a few transit or collision times and the lowestorder flow velocity V 0 is in the toroidal direction: 8 where ⌽ 0 (⌿) denotes the lowest-order electrostatic potential in ␦ ͑which corresponds to ⌽ Ϫ1 in the paper by Hinton and Wong 8 ͒ and E 0 ϵϪٌ⌽ 0 ϵϪ⌽ 0 Јٌ⌿. The toroidal angular velocity V ϭϪc⌽ 0 Ј is directly given by the radial electric field and is a flux-surface quantity. For particle species a with the mass m a and the charge e a , the phase space variables (xЈ,,,) are defined in terms of the spatial coordinates x in the laboratory frame and the velocity vЈϵvϪV 0 in the moving frame as 8,10 xЈϭx, ⑀ϭ Here (e 1 ,e 2 ,bϵB/B) are unit vectors which form a righthanded orthogonal system at each point, vЈϭv ʈ where ⌽ 1 ϵ⌽ 1 Ϫ͗⌽ 1 ͓͘ϭO (␦)͔ is the poloidal-angledependent part of the electrostatic potential. The magnetic flux surface average is denoted by ͗•͘. The lowest-order distribution function is the Maxwellian which is written as

͑5͒
where the temperature T a ϭT a (⌿) and N a ϭN a (⌿) are fluxsurface functions although generally the density n a depends on the poloidal angle through ⌶ a and is given by 8 n a ϭN a exp ͩ Ϫ ⌶ a T a ͪ .

͑6͒
This dependence of the density n a on the poloidal angle is one of the causes which complicates the derivation of the classical and neoclassical transport coefficients for the rotating plasma. For plasmas consisting of electrons and singlespecies ions with charge e i ϵZ i e, we have 8 where the charge neutrality condition ͚ a e a n a ϭ0 is used and m e /m i (Ӷ1) is neglected. In toroidally rotating axisymmetric systems, the linearized drift-kinetic equation is written as 8 where C a L denotes the linearized collision operator ͓see Eq. ͑8͒ in Ref. 11͔ and ḡ a is defined in terms of the first-order gyrophase-averaged distribution function f a1 as Here ͐ l dl denotes the integral along the magnetic field line, and E ʈ (2) ϵb•(Ϫٌ⌽ (2) Ϫc Ϫ1 ‫ץ‬A/‫ץ‬t) is the second-order parallel electric field. The thermodynamic forces (X a1 ,X a2 ,X V ,X E ) are flux-surface quantities defined by The neoclassical entropy production 10,11 is kinetically defined in terms of f a1 and C a L and is rewritten in the thermodynamic form by using Eq. ͑8͒. The surface-averaged total neoclassical entropy production is given by where the neoclassical transport fluxes (⌫ a ncl ,q a ncl /T a ,⌸ a ncl , J E ) conjugate to the forces (X a1 ,X a2 ,X V ,X E ) are defined by Here ⌫ a ncl , q a ncl , and ⌸ a ncl denote the surface-averaged radial fluxes of particles, heat, and toroidal ͑angular͒ momentum, respectively, and J E represents the surface-averaged parallel current. The neoclassical transport equations connecting the conjugate pairs of the fluxes and forces are written as where the transport coefficients are dependent on the radial electric field through the toroidal angular velocity The remaining parts of this work are organized as follows. In Sec. II, using the formal solution of the linearized drift kinetic equation ͑8͒, we prove the Onsager symmetry of the neoclassical transport matrix for the rotating plasma consisting of electrons and multi-species ions with arbitrary collision frequencies. In Sec. III, we describe the transport fluxes other than the neoclassical fluxes to give the total transport of the particles, heat, and toroidal momentum. In the cases of single-species ions, the explicit forms of the neoclassical transport matrices for the Pfirsch-Schlüter and banana regimes are given in Sec. IV and Sec. V, respectively. Conclusions and discussion are given in Sec. VII. Appendix A shows the classical transport coefficients for the rotating plasma. In Appendix B, the first-order parallel flows and the parallel momentum equations, which are useful to derive the neoclassical transport equations, are obtained from the drift kinetic equation. The parallel viscosity coefficients for the plateau regime, from which all the plateau transport coefficients except for L VV can be derived, are shown in Appendix C.

II. ONSAGER SYMMETRY OF NEOCLASSICAL TRANSPORT EQUATIONS FOR ROTATING PLASMAS
In order to prove the Onsager symmetry of the neoclassical transport equations, it is useful to note that the solution of the linearized drift kinetic equation ͑8͒ is written as where G abm (mϭ1,2) and G aM (M ϭV,E) are defined as the solutions of the following equations: Substituting Eq. ͑15͒ into Eq. ͑13͒, and comparing it with Eq. ͑14͒, we find that the neoclassical transport coefficients are given by Similarly, the second equation in Eqs. ͑16͒ is divided into parts with these symmetries, which is straightforward and not shown here. Next, we regard the transport coefficients as functions of V and write them as the sum of even and odd parts with respect to V : for example, L mn ab (m,nϭ1,2) given in Eq. ͑17͒ are written as gives the Onsager symmetry for the coefficients L mn ab (m,nϭ1,2) which is rewritten as L mn ab (V )ϭL nm ba (ϪV ). In the same way as above, we can derive the Onsager symmetry for the other coefficients. To summarize, the Onsager relations for all the neoclassical transport coefficients are given by L mn ab ͑V ͒ϭL nm ba ͑ϪV ͒ ͑m,nϭ1,2͒, L mM a ͑V ͒ϭϪL Mm a ͑ϪV ͒ ͑mϭ1,2;M ϭV,E ͒.
We can see from the derivation here that the Onsager relations in Eq. ͑24͒ are robustly valid even for the cases of multi-species ions and arbitrary collision frequencies.
Here, let us consider the case in which the system has up-down symmetry B()ϭB(Ϫ) (: a poloidal angle defined such that ϭ0 on the plane of reflection symmetry͒. In this case, it is convenient to use the transformation T :(v ʈ Ј ,,V )→(Ϫv ʈ Ј ,Ϫ,ϪV ). We note that, under the transformation T , W aV is invariant while W am (mϭ1,2) and W aE change their signs. Also, the operators v ʈ Јb•ٌ and C a L commute with T . Therefore, we see that G aV is symmetric and G abm (mϭ1,2) and G aE are antisymmetric with respect to T . Then, it is found from Eq. ͑17͒ that L mn ab , L mE a , L VV , and L EE are even while L mV a and L VE are odd in V . Thus, for the system with up-down symmetry, we obtain the restricted forms of the Onsager relations,

III. TRANSPORT FLUXES OTHER THAN NEOCLASSICAL FLUXES
The particle, heat, and toroidal momentum fluxes for species a consist of the neoclassical and other transport parts, and are written as 10 Here the classical fluxes ⌫ a cl , q a cl , and ⌸ a cl are caused by particles' gyromotion with collisions. Their definitions and the classical transport coefficients for the rotating plasma are shown in Ref. 10. The momentum conservation in collisions assures the intrinsic ambipolarity of the classical particle fluxes, which implies that ͚ a e a ⌫ a cl ϭ0 is valid for arbitrary values of the radial electric field. In Appendix A, the classical transport coefficients for the case of single-species ions are given. The fluxes ⌫ a (E) , q a (E) , and ⌸ a (E) ͓see Eq. ͑20͒ in Ref. 10 for their definitions͔ are given from the inductive electric field E (A) ϵϪc Ϫ1 ‫ץ‬A/‫ץ‬t and do not contribute to the entropy production. The anomalous transport fluxes ⌫ a anom , q a anom , and ⌸ a anom are driven by turbulent fluctuations and are defined in terms of the fluctuation-particle interaction operator ͓see Eq. ͑26͒ in Ref. 10͔. Then, the intrinsic ambipolarity of the particle fluxes ⌫ a anom and ⌫ a (E) are separately derived from the charge neutrality: ͚ a e a ⌫ a anom ϭ ͚ a e a ⌫ a (E) ϭ0. The fluxes ⌫ a H , q a H , and ⌸ a H are defined in Ref. 10, and are related to the thermodynamic forces X a1 , X a2 , and X V through the nondissipative antisymmetric transport coefficients which satisfy the Onsager symmetry Eq. ͑24͒ ͓see Eqs. ͑23͒ and ͑24͒ in Ref. 10͔ and vanish if the system has updown symmetry.
It can be shown that the sum of the neoclassical fluxes , the symmetry is also valid for the transport coefficients for their total fluxes in Eq. ͑27͒. From the charge neutrality and the momentum conservation in collisions with Eq. ͑27͒, we find that the particle fluxes (⌫ a ncl ϩ⌫ a H ) are intrinsically ambipolar: ͚ a e a ͑ ⌫ a ncl ϩ⌫ a H ͒ϭ0. ͑28͒ Using the ambipolarity condition in Eq. ͑28͒, the number of the pairs of the particle fluxes (⌫ a ncl ϩ⌫ a H ) and the conjugate thermodynamic forces appearing in the transport equations can be reduced by one without breaking the Onsager symmetry of the transport matrix. 11 In the following sections, we consider the case of single-species ions (i) with charge e i ϭZ i e per particle, and derive the transport equations combining the five transport fluxes ͓(⌫ e ncl ϩ⌫ e H ),(q e where the Onsager symmetry for the 5ϫ5 transport matrix is already guaranteed. Here, we have neglected ⌸ e ncl ϩ⌸ e H which is O (m e /m i ) smaller than ⌸ i ncl ϩ⌸ i H . The first thermodynamic force X e1 * is defined by With this reduction, we see that the O (␦) radial electric field Ϫ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿ disappears from the thermodynamic forces.
Thus, in the axisymmetric toroidally rotating system, the O (␦) radial electric field neither affects the transport nor is determined by the ambipolar condition. Recall that the transport coefficients in Eq. ͑29͒ depend on the radial electric field of O (␦ 0 ) ͓not O (␦)͔ through the toroidal angular velocity V ϭϪc‫ץ‬⌽ 0 /‫ץ‬⌿ and that the thermodynamic force X V is proportional to the radial gradient of the O (␦ 0 ) radial electric field ͓see Eq. ͑10͔͒.

IV. PFIRSCH-SCHLÜ TER REGIME
In the Pfirsch-Schlüter regime, the ratio of the particles' mean-free path to the equilibrium scale length ϵv Ta aa /L is used as a small ordering parameter, where v Ta ϵͱ2T a /m a is the thermal velocity and aa the collision time defined in Ref. 6. In this section, we derive the full neoclassical transport equations for the rotating plasma consisting of electrons and single-species ions in the Pfirsch-Schlüter regime. Here we retain terms up to O ͓(m e /m i ) 1/2 ͔ in order to obtain the electron transport coefficients, which are not considered in Refs. 8 and 9. In Ref. 8, the rotation speed divided by the ion thermal velocity V 0 /v Ti is assumed to be as small as 1/2 although, without this assumption, our results here are valid even for V 0 ϳv Ti .
According to the conventional analytical technique for the Pfirsch-Schlüter transport, let us expand the O (␦) distribution function ḡ a in terms of the expansion parameter as In the lowest-order with respect to , the linearized drift kinetic equation ͑8͒ reduces to C a L (ḡ a (Ϫ1) )ϭ0, from which we have , and T a1 (Ϫ1) are independent of the velocity variables (,). In Appendix B, some relations on the parallel flows and the parallel momentum balance equations are derived from the drift kinetic equation. The quantity u ʈ1 (Ϫ1) represents the O (␦ Ϫ1 ) average parallel flow velocity, and is related by by noting from Eqs. ͑4͒ and ͑7͒ that ⌶ a is dependent on the poloidal angle . In the next order with respect to , the linearized drift kinetic equation ͑8͒ is written as where X E ϵ͗BE ʈ (A) ͘/͗B 2 ͘ 1/2 is considered to be on the order of (Ϫ1) since it is balanced with the parallel current multiplied by the resistivity (ϰ (Ϫ1) ) although the other thermodynamic forces X a1 , X a2 , and X V are regarded as O ( 0 ) quantities.
where terms up to O ͓(m e /m i ) 1/2 ͔ are retained, and the 2ϫ2 matrices ⌳ e , U e , and V e are defined by Eq. ͑37͒ and In Eqs. ͑43͒-͑52͒, M T denotes the transpose of M and ͗M͘ is defined by ͗M͘ϵ͓͗M jk ͔͘ for an arbitrary matrix Mϭ͓ M jk ͔. It is found that, even if there is no up-down symmetry, the Pfirsch-Schlüter transport coefficients in Eqs. ͑43͒-͑52͒ satisfy the restricted version of the Onsager symmetry given in Eq. ͑25͒ since the inhomogeneity of the magnetic field is ignored within the mean free path in the Pfirsch-Schlüter regime ͓see also Eq. ͑A3͒ in Appendix A showing that the restricted version of the Onsager symmetry is also valid for the classical transport coefficients͔.

V. BANANA REGIME
In order to analytically obtain the neoclassical transport coefficients for the banana regime, we hereafter consider the large aspect ratio toroidal system and use the toroidal coordinates (r,,) where the minor radius r is a label for magnetic surfaces. The major radius is given by RϭR 0 ϩrcos (R 0 : the distance between the major axis and the magnetic axis͒ and r/R 0 Ӷ1 is assumed. The banana regime is represented by Ta aa ӷ(R 0 /r) 3/2 where Ta ϵv Ta /(qR 0 ) is the transit frequency and qϵrB T /(R 0 B P ) is the safety factor. When Ta aa ӷ1, the dominant parts of the radial transport fluxes in Eq. ͑27͒ are given by where the distribution functions h a (aϭe,i) are defined by Eq. ͑B2͒.
It is shown from Eqs. ͑4͒, ͑6͒, and ͑7͒ that, in the lowestorder with respect to the inverse aspect ratio r/R 0 Ӷ1, the electron and ion densities are also regarded as surface functions: n e ϭZ i n i ӍN e where O (m e /m i ) terms are neglected. Then, the surface-averaged parallel momentum balance equations for Ta aa ӷ1 are obtained from Eqs. ͑B4͒, ͑B5͒, and ͑B6͒ as

͑57͒
where the dimensionless friction coefficients are given by In Eqs. ͑55͒ and ͑56͒, we have used the notation q a ϵq a Ϫ͗⌶ a ͘⌫ a (aϭe,i) and the 13-moment ͑13M͒ approximation to express the friction forces in terms of the flows. Now, let us use the banana regime parameter (R 0 /r) 3/2 ( Ta aa ) Ϫ1 Ӷ1 to expand the distribution functions as The lowest order of the linearized drift kinetic equation is written as v ʈ

Јb•ٌh a
(0) ϭ0 which shows that h a (0) is independent of the poloidal angle : Thus h a (0) (aϭe,i) make no direct contribution to the neoclassical fluxes as shown by substituting Eq. ͑60͒ into Eq. ͑54͒. In the next order, the linearized drift kinetic equation gives Then, we have the solvability conditions 6 for Eq. ͑61͒ as where h a (0) is divided into the even (ϩ) and odd (Ϫ) parts in v ʈ Ј : We need to calculate the lowest-order parallel viscosities 2T a ]Ϫ 5 2 ͖ B•ٌh a (1) ͘ (aϭe,i) in order to obtain the radial fluxes in Eq. ͑54͒. It is found from Eq. ͑61͒ that only the odd part h a (Ϫ) is necessary for calculation of those lowest-order parallel viscosities. Following the standard procedure by Hirshman and Sigmar, 6 we can obtain the solution ĥ a (Ϫ) of Eq. ͑63͒ and derive the parallel viscosities, which are written in the linear forms of the poloidal flows: The dimensionless coefficients a j (aϭe,i; jϭ1,2,3) are defined by where the velocity-dependent collision frequency D a ϭ D a (x) (xϵvЈ/v Ta ) is defined in Ref. 6, and ⌼ denotes the square of the toroidal flow velocity normalized by the sound wave velocity: where the ion-electron collision contributions of O (m e /m i ) are neglected. Appendix C shows the parallel viscosity coefficients for the plateau regime, from which all the plateau transport coefficients except for L VV can be derived. Now, by using Eqs. ͑55͒-͑57͒, ͑65͒, and ͑B4͒, we can •ٌh a ͘ (aϭe,i) and the parallel current J E ϭe͗n e (u ʈi Ϫu ʈe )͘/͗B 2 ͘ 1/2 in the linear forms of the thermodynamic forces (X e1 * ,X e2 ,X i2 ,X V ,X E ).
We find that the effects of the toroidal flow velocity ͑not its shear͒ on the electron and ion parallel viscosities for the banana regime are included only through ⌼ in Eq. ͑66͒. As in Ref. 8, we have used here the approximate expression for the parallel velocity v ʈ Ј of the trapped particle in the toroidally rotating plasma with the large aspect ratio

͑68͒
In the right-hand side of Eq. ͑68͒, the term proportional to ⌼ is derived from the poloidal dependent part of the potential function ⌶ a which consists of the electrostatic potential and the effective gravity potential due to the centrifugal force ͓see Eqs. ͑2͒, ͑4͒, and ͑7͔͒. The sum of this poloidal variation ⌶ a and the poloidal magnetic variation forms the well for trapped particles, which is expressed by (r/R 0 ) ϫ(1ϩ⌼/x 2 )cos in Eq. ͑68͒. Thus, the toroidal rotation increases the trapped particles' population and accordingly the parallel viscosity coefficients as shown by the enhancement factor (1ϩ⌼/x 2 ) 1/2 in the right-hand side of Eq. ͑66͒.
In order to obtain the full transport equations, we need to also derive the linear thermodynamic expression of ͗͐d 3 v͓m i I 2 (v ʈ Ј) 2 /B 2 ϩR 2 B P 2 /B͔C i L (ḡ i )͘ which is necessary for the radial flux of the toroidal momentum in Eq. ͑54͒. This requires the solution ĥ i (ϩ) of Eq. ͑62͒, which can be given as in Ref. 8 by minimizing the positive definite functional: where C ii L denotes the linearized ion-ion collision operator with the ion-electron collisions neglected. Then we have the approximate solution Thus, the final banana transport formulas form a 5ϫ5 system of the coefficients as follows: where S ϵ(e 2 n e ee /m e )l 22 e /͓ l 11 e l 22 e Ϫ( l 12 e ) 2 ͔ denotes the Spitzer resistivity and B P ϵٌ͉⌿͉/R represents the poloidal magnetic field. The dimensionless friction coefficients l jk e ( j,kϭ1,2) are written in Eq. ͑58͒ and the dimensionless viscosity coefficients a j (aϭe,i; jϭ1,2,3) are given by Eqs. ͑66͒ and ͑67͒. Recall that ⌿ is used to define the radial transport fluxes and the radial thermodynamic forces in such a way that q a ϵ͗q a •ٌ⌿͘ and X a2 ϵϪ‫ץ‬T a /‫ץ‬⌿. When we use r instead of ⌿ to define the radial fluxes and forces, the resultant transport coefficients are immediately given by replacing ٌ͉⌿͉ in Eqs. ͑71͒-͑80͒ with the unity.
We find that, in the large aspect ratio system, the banana transport coefficients in Eqs. ͑71͒-͑80͒ are much larger than the classical transport coefficients in Eq. ͑A2͒ by a factor of O ͓q 2 (R 0 /r) 3/2 ͔ except for the diagonal banana coefficient for the toroidal momentum transport All the coefficients in Eqs. ͑71͒-͑80͒ are functions of V 0 as seen from the explicit appearance of V 0 and from the flow-dependent viscosity coefficients ͓see Eq. ͑66͔͒. From Eq. ͑75͒ with the small electron mass terms neglected, the toroidal flow dependence of the ion thermal diffusivity L 22 ii appears through where the enhancement factor F(⌼) for the ion thermal diffusivity is fitted for 0р⌼р1 as This enhancement factor is in good agreement with that given by Catto et al.,9 F(⌼)ϭ1ϩ0.75⌼Ϫ0.60⌼ 2 ϩ0.26⌼ 3 ͓see Eq. ͑98͒ in Ref. 9 and note that ⌼ is written as X in their notation͔, in spite of the difference between the solution methods: our calculation is based on the moment expansion method with the 13M approximation while they use the variational technique. We find from Eqs. ͑67͒, ͑71͒, and ͑81͒ that the banana particle diffusivity and the banana electron and ion thermal diffusivities are monotonically increasing functions of ⌼. This is because the potential well due to the toroidal rotation increases the number of the trapped particles as mentioned after Eq. ͑68͒. The transport coefficients in Eqs. ͑71͒-͑80͒ satisfy the restricted version of the Onsager symmetry given in Eq. ͑25͒ since we have used the large aspect ratio approximation where the magnetic surfaces (rϭ const) have circular cross sections. A well-known pair of Onsager symmetric neoclassical transport coefficients is that of L 1E e and L E1 e (ϭϪL 1E e ) ͓see Eq. ͑74͔͒. The off-diagonal coefficient L 1E e Ͻ0 indicates that the parallel electric field X E gives the inward particle flux ⌫ e ϭL 1E e X E Ͻ0 due to trapped particles, which is known as the Ware pinch effect. 12 The counterpart L E1 e (ϭϪL 1E e ) represents that the thermodynamic force X e1 * produces the parallel current ͑the bootstrap current͒ J E ϭL E1 e X e1 * . Since the toroidal momentum transport ⌸ i and the flow shear X V enter the transport equations for the toroidally rotating plasma as a new conjugate flux-force pair, there appears a new physically important pair of Onsager symmetric neoclassical transport coefficients L VE and L EV (ϭϪL VE ). The e ͓see Eq. ͑79͔͒ shows that the parallel electric field X E gives the inward toroidal momentum flux ⌸ i ϭL VE X E ͑which has the opposite sign to V 0 ) due to the pinched trapped ions with the mean toroidal velocity V 0 . From its partner L EV (ϭϪL VE ), we find that the flow shear X V drives the parallel current J E ϭL EV X V .
It is shown from Eqs. ͑4͒, ͑6͒, ͑30͒, and ͑79͒ that the sum of the currents driven by X e1 * and X V is rewritten e /n e ٌ͉⌿͉)͓Ϫ(‫ץ‬P/‫ץ‬r) ϩn i m i (V 0 2 /R)(‫ץ‬R/‫ץ‬r)Ϫ(n i m i V 0 2 /2T i )(‫ץ‬T i /‫ץ‬r)͔ with the flow shear term canceled. Here the total pressure gradient term ‫ץ‬ P/‫ץ‬rϵ‫(ץ‬n e T e ϩn i T i )/‫ץ‬r and the centrifugal force term n i m i (V 0 2 /R)(‫ץ‬R/‫ץ‬r)ϭϪn i m i V 0 •ٌV 0 •ٌR appear since they give the perpendicular current J Ќ through the equilibrium equation J؋B/cϭٌ PϪn i m i V 0 •ٌV 0 and accordingly drive the electron poloidal flow, from which the electron parallel viscosity ͓see Eq. ͑65͔͒ and therefore the neoclassical parallel current are produced.
Using the current coefficients in Eqs. ͑74͒, ͑77͒, ͑79͒, and ͑80͒ with the viscosity coefficients ͑67͒, the full expression of the parallel current for the toroidally rotating plasma is written as where Z i ϭ1 and 0р⌼р1 are assumed, and the centrifugal force term of O ͓⌼(r/R 0 )(dP/dr)͔ is neglected.

VI. CONCLUSIONS AND DISCUSSION
In this work, we have studied neoclassical transport for the axisymmetric system with the large toroidal flow velocity (ϳv Ti ). In the toroidally rotating plasma, the transport equations involve a new pair of the transport flux and the thermodynamic force: the radial flux of the toroidal momentum and the toroidal flow shear which is proportional to the radial electric field shear. For general rotating plasmas consisting of multi-species particles in arbitrary collisional regimes, the Onsager symmetry of the neoclassical transport matrix is proved by using the formal solution of the linearized drift kinetic equation with the self-adjoint collision operator, and its restricted form for the system with up-down symmetry is also shown. The complete neoclassical electron and ion transport equations are derived for the Pfirsch-Schlüter and banana regimes in the case of single-species ions and the Onsager symmetry is directly confirmed by them.
We have found that the toroidal rotation causes the centrifugal force and the poloidal variation of the electrostatic potential, which result in the increase of the trapped particles and therefore the enhancement of the parallel viscosities, the particle and thermal diffusivities, and the transport coefficients concerned with the pressure gradient driven ͑boot-strap͒ current and the Ware pinch. It is also shown that the parallel inductive electric field drives the inward banana flux of the toroidal momentum in addition to the Ware pinch of the particles. These inward particle and momentum fluxes driven by the parallel electric field are related through two pairs of the Onsager symmetric off-diagonal coefficients to the parallel currents driven by the radial thermodynamic forces conjugate to the inward fluxes, respectively. For large toroidal flows such as observed in JT-60U, 4,13 toroidal flow effects on the parallel current coefficients ͓see Eqs. ͑82͒ and ͑83͔͒ are roughly estimated to reach the order of ϳ10% of those without taking account of the flow effects, and thus should not be neglected for accurate calculation of the current profile and the magnetic configuration.
Nagashima et al. 13 obtained from the JT-60U experiment the toroidal momentum diffusivity and the inward velocity v inward for the toroidal momentum transport. These are related to the transport coefficients given in the present paper by L VV ϭn i m i R 0 2 ٌ͉⌿͉ 2 and L VE e X E ϭϪ n i m i R 0 V 0 v inward ٌ͉⌿͉, and are written from the results in Sec. V as ϭ(1/ͱ2)( 1 10 q 2 ϩ 3 5 ) i 2 / ii and v inward ϭ 2.411Z i Ϫ1 F E1 e (⌼)(r/R 0 ) 1/2 cE ʈ /B P for the banana regime. Their experimental results give typically ϳ1 m 2 /s and v inward ϳ1 m/s, which are much larger than the predictions by the above neoclassical model ϳ10 Ϫ4 m 2 /s and v inward р0.1 m/s. Thus, the radial transport of the toroidal momentum is considered to be dominated by the anomalous processes. In our previous paper, 10 the anomalous transport fluxes for the rotating plasma are formulated based on the gyrokinetic equations, and the simple expression for the anomalous toroidal momentum diffusivity is given for the mixing length level of the ion temperature gradient ͑ITG͒ driven turbulence ͓see Eq. ͑66͒ in Ref. 10 where i A corresponds to ͔. This mixing length type estimation can give a larger momentum diffusivity on the order of the experimentally observed one. However, in order to describe the anomalous pinch of the toroidal momentum and explain the significant reduction of the transport at the ITB, a more elaborate investigation on the anomalous transport fluxes in the rotating plasma is required as a future task.

ACKNOWLEDGMENTS
The author ͑H. S.͒ thanks Professor M. Okamoto for his encouragement of this work.
This work is supported in part by the Grant-in-Aid from the Japanese Ministry of Education, Science, and Culture, and in part by the U.S. Department of Energy, Grant No. DE-FG05-80ET-53088.

APPENDIX A: CLASSICAL TRANSPORT FOR TOROIDALLY ROTATING PLASMAS CONSISTING OF ELECTRONS AND SINGLE-SPECIES IONS
The classical transport equations for the rotating plasma consisting of electrons and multi-species ions are derived in Appendix A of Ref. 10

͑A1͒
where the classical transport coefficients are given by ͫ ͑ L cl ͒ 11 ee ͑L cl ͒ 12 ee Here, the poloidal magnetic field is given by B P ϭٌ͉⌿͉/R and the collision times aa (aϭe,i) are defined in Ref. 6. The dimensionless friction coefficients are given by l 11 e ϭZ i , l 12 e ϭ 3 2 Z i , l 22 e ϭͱ2ϩ 13 4 Z i , l 22 i ϭͱ2, and l i V ϭ 3 5 ͱ2.
The classical ion particle flux is given from the electron particle flux through the intrinsic ambipolarity condition ⌫ i cl ϭ⌫ e cl /Z i . With respect to the small mass ratio m e /m i , we have retained the terms up to O ͓(m e /m i ) 1/2 ͔ in Eqs. ͑A1͒ and ͑A2͒ although the terms of O (m e /m i ) such as ⌸ e cl have been neglected. We see from Eqs. ͑A2͒ that the classical transport coefficients satisfy the Onsager symmetry ͑ L cl ͒ mn ab ͑V ͒ϭ͑L cl ͒ mn ab ͑ϪV ͒ϭ͑L cl ͒ nm ba ͑V ͒ ͑a,bϭe,i;m,nϭ1,2͒, ͑ L cl ͒ mV a ͑V ͒ϭϪ͑L cl ͒ mV a ͑ϪV ͒ϭ͑L cl ͒ Vm a ͑V ͒ ͑aϭe,i;mϭ1,2͒, ͑A3͒ ͑ L cl ͒ VV ͑V ͒ϭ͑L cl ͒ VV ͑ϪV ͒, which has the same form as Eq. ͑25͒ and is valid even without up-down symmetry since the classical transport is a spatially local process.

APPENDIX B: THE FIRST-ORDER PARALLEL FLOWS AND PARALLEL MOMENTUM BALANCE EQUATIONS
From Eqs. ͑8͒ and ͑11͒, the linearized drift kinetic equation is rewritten as v ʈ Јb•ٌh a ϪC a L ͑ ḡ a ͒ϭ 1 T a f a0 W aE X E , ͑B1͒ where h a ϵḡ a Ϫ 1 T a f a0 ͑ U a1 X a1 ϩU a2 X a2 ϩU aV X V ͒. ͑B2͒ Multiplying Eq. ͑B1͒ by the unity and (/T a Ϫ5/2), and integrating them in the velocity space give the continuity equation and the energy balance equation of O (␦) as Integrating Eq. ͑B3͒ along the magnetic field line, we have the O (␦) parallel flows: n a u ʈa1 ϵ ͵ d 3 v f a1 v ʈ Ј ϭB⌫ a ͑ ⌿ ͒ϩ n a cI e a B ͑X a1 ϩ⌬ a X a2 ϩm a R 2 V X V ͒, ͑B4͒