Transport processes and entropy production in toroidally rotating plasmas with electrostatic turbulence

A new gyrokinetic equation is derived for rotating plasmas with large ﬂow velocities on the order of the ion thermal speed. Neoclassical and anomalous transport of particles, energy, and toroidal momentum are systematically formulated from the ensemble-averaged kinetic equation with the gyrokinetic equation. As a conjugate pair of the thermodynamic force and the transport ﬂux, the shear of the toroidal ﬂow, which is caused by the radial electric ﬁeld shear, and the toroidal viscosity enter both the neoclassical and anomalous entropy production. The interaction between the ﬂuctuations and the sheared toroidal ﬂow is self-consistently described by the gyrokinetic equation containing the ﬂow shear as the thermodynamic force and by the toroidal momentum balance equation including the anomalous viscosity. Effects of the toroidal ﬂow shear on the toroidal ion temperature gradient driven modes are investigated. Linear and quasilinear analyses of the modes show that the toroidal ﬂow shear decreases the growth rates and reduces the anomalous toroidal viscosity. © 1997 American Institute of Physics. (cid:64) S1070-664X (cid:126) 97 (cid:33) 01302-5 (cid:35)


I. INTRODUCTION
In most magnetically confined toroidal plasmas, the observed particle and heat fluxes across the magnetic flux surfaces are dominated by the turbulent or anomalous transport, 1 which greatly exceed the predictions of the neoclassical transport theory. [2][3][4] However, in some operational regions of tokamak plasmas such as high-confinement modes ͑H-modes͒ 5 and reversed shear configurations, 6 there have been observed transport barriers with a significant reduction of anomalous transport. Generally, large radial electric field shear ͑or sheared flow͒ is considered as a cause of such reduction of the transport level. Determination of profiles of the radial electric fields or the sheared flows requires relevant analysis of the momentum balance equations. Since the viscosity involved in the neoclassical transport gives a significant contribution in the momentum balance, we should consider both the neoclassical and anomalous transport processes simultaneously in order to investigate the interaction of the sheared flow and the turbulent fluctuations.
In our previous works, 7,8 the neoclassical and anomalous transport are formulated in the synthesized framework, although that theory treats the E؋B flow velocity as on the order of the diamagnetic drift velocity. In the present work, our theory is extended to that for the rotating plasma with large flow velocities on the order of the ion thermal speed. Then, it is not valid to use conventional drift-kinetic and gyrokinetic equations, [9][10][11][12][13][14][15] in which the flow velocities are assumed to be O (␦v Ti ). Here v Ti ϵ(2T i /m i ) 1/2 denotes the ion thermal velocity, ␦ϵ i /L the drift ordering parameter, i ϵv Ti /⍀ i the ion thermal gyroradius, and L the equilibrium scale length. Hazeltine and Ware derived the driftkinetic equation for the plasma with large flows on the order of the ion thermal speed. 16 That equation has a complicated structure including the gyroviscosity term although it reduces to a simplified form in the case of the toroidally rotating axisymmetric plasma, for which the ion neoclassical transport coefficients were obtained by Hinton and Wong 17 and by Catto et al. 18 Artun and Tang 19,20 derived the gyrokinetic equation in the case where the large equilibrium flows exist. In the present work, we derive the new form of the gyrokinetic equation for the large flow case. Our gyrokinetic equation contains the term responsible for the perpendicular anomalous viscosity ͑or Reynolds stress͒, which is not included in the equation by Artun and Tang. This gyrokinetic equation, which is written in a compact form for the toroidally rotating axisymmetric system, is useful to express the anomalous transport and the resultant anomalous entropy production. It is emphasized that, in the rotating plasma, the shear of the toroidal flow or the radial electric field shear enters both the neoclassical and anomalous transport equations as an additional thermodynamic force, and that the products of the toroidal flow shear and the conjugate neoclassical and anomalous toroidal viscous fluxes make significant contributions to the total entropy production. These contributions of the toroidal flow shear and viscosities are considered as higher-order small quantities in ␦ by conventional treatments. The turbulent fluctuations and the resultant anomalous transport are influenced by the flow shear or the radial electric field shear contained as the additional thermodynamic force in the gyrokinetic equation.
Taking account of all the neoclassical and anomalous transport processes, we obtain balance equations for the particles, energy, toroidal momentum, entropy, and the fluctuation amplitude. Since the toroidal momentum is directly related to the radial electric field, the toroidal momentum balance equation describes the temporal evolution of the radial electric field. Through the anomalous viscosity term in the toroidal momentum balance equation, the fluctuations affect the flow and the associated radial electric field as a reaction to the sheared flow effect on the fluctuations. Thus, our extended theory gives a self-consistent description of the interaction between the fluctuations and the sheared flow based on the rigorous statistical kinetic foundation.
As in our previous works, 7,8 let us start from an ensemble-averaged kinetic equation for species a: where C a is a collision term, I a is a term representing the effects of external sources such as neutral beam injection, and D a is a fluctuation-particle interaction term defined by Here ͗•͘ ens denotes the ensemble average and we divided the distribution function ͑the electric field͒ into the ensembleaveraged part f a (E,⌽) and the fluctuating part f a (Ê , ). Throughout this paper, the magnetic fluctuations B are not considered although generalization to the case with the magnetic fluctuations is straightforward. The source term I a is assumed to be a quantity of O(␦ 2 ). 17 Then we should note that the linearized drift-kinetic equation and the gyrokinetic equation are not affected by I a . Thus the neoclassical and anomalous transport equations are not changed by the source term, although the balance equations of particles, energy, and momentum derived from Eq. ͑1͒, which are O (␦ 2 ), involve source terms caused by I a .
In deriving the drift-kinetic and gyrokinetic equations, the perturbative expansion in the drift-ordering parameter ␦ is utilized. When we apply this expansion procedure to the system with the large flow on the order of the ion thermal velocity, it is useful to observe particles' gyromotion from the moving frame with that flow velocity V 0 . 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 . Hinton and Wong 17 showed that, in the axisymmetric systems, the poloidal flow decays in a few transit or collision times and that the lowest-order flow velocity V 0 is in the toroidal direction and is derived from where ⌽ 0 (⌿) denotes the lowest-order electrostatic potential in ␦ and E 0 ϵϪٌ⌽ 0 ϵϪ⌽ 0 Јٌ⌿. We should note that the toroidal angular velocity V ϭϪc⌽ 0 Ј is directly given by the radial electric field and is a flux-surface quantity. The lowest-order electrostatic potential is written as ⌽ Ϫ1 in the paper by Hinton and Wong 17 although it is denoted by ⌽ 0 in the present work since we follow the Littlejohn's drift ordering rule 21 to regard the electric charge e ͑instead of ⌽) as the parameter of O (␦ Ϫ1 ): eϭe Ϫ1 . As in the work by Hinton and Wong, 17 let us introduce the phase space variables (xЈ,,,) which are defined in terms of the spatial coordinates x in the laboratory frame and the velocity vЈϵvϪV 0 in the moving frame as where (e 1 ,e 2 ,bϵB/B) are unit vectors which forms a righthanded orthogonal system at each point, and vЈϭv ʈ In the definition of the energy variable , ⌶ a is given by where ⌽ 1 ϵ⌽ 1 Ϫ͗⌽ 1 ͓͘ϭO (␦)͔ is the poloidal-angledependent part of the electrostatic potential. The magnetic flux surface average is denoted by ͗•͘. It is shown that and are conserved along the lowest-order guiding center orbit: (d/dt) 0 ϭ(d/dt) 0 ϭ0 where •ϵ͛d/2 represents the gyrophase average. The guiding center velocity is defined by where we should note that the centrifugal force Ϫm a V 0 •ٌV 0 ϭm a R(V ) 2 ٌR and the Coriolis force tribute to the guiding center drift velocity. Applying the same procedure as in Ref. 8 to Eq. ͑1͒, we obtain the double-averaged kinetic equation over the statistical ensemble and the gyrophase angle which is valid up to O(␦ 2 ) and is written as where • and •˜denote the average and oscillating parts with respect to the gyrophase angle , respectively, and the differential operator L is defined by Here and hereafter, the spatial gradient operator ٌ is taken with (,,) fixed. The gyrophase-dependent parts of the first and second-order distribution functions are given by where C a L denotes the linearized collision operator ͓see Eq. ͑8͒ in Ref. 22͔ and the integration constants related to ͐ d are uniquely determined by the conditions f a (1) ϭ f a (2) ϭ0. Here the second-order gyrophase-dependent distribution functions f a C and f a A are associated with collisional ͑classical͒ and turbulent ͑or anomalous͒ dissipation processes, respectively, while f a H is related to the higher-order small corrections to the drift orbit with no dissipation. 8 The lowest-order solution of Eq. ͑8͒ is the Maxwellian distribution function which is written as

͑11͒
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

͑12͒
The charge neutrality ͚ a e a n a ϭ0 imposes the constraints on ⌽ 1 and N a . For plasmas consisting of electrons and a single species of ions with charge e i ϵZ i e, we have 17 where m e /m i (Ӷ1) is neglected. It is emphasized that the density n a and the temperature T a as well as the toroidal ͑angular͒ velocity V should be specified for the lowest-order description of the rotating plasma. In the first-order in ␦, Eq.
͑8͒ reduces to the linearized drift-kinetic equation as shown in Sec. III, the solution of which gives the neoclassical transport. The second-order part of Eq. ͑8͒ describes behavior of the distribution function in the transport time scale ϳ␦ Ϫ2 L/v Ta in which the density, the temperature, and the toroidal velocity vary due to the transport processes. The anomalous transport fluxes are defined by the solution of the new gyrokinetic equation in Sec. IV, and the shear flow effects on the anomalous transport due to the ion temperature gradient driven modes are investigated as an example in Sec. V. In the next section, we find balance equations of the particles, energy, toroidal momentum, and entropy for the rotating plasma.

II. BALANCE EQUATIONS OF PARTICLES, ENERGY, TOROIDAL MOMENTUM, AND ENTROPY
Here, equations describing temporal evolutions of the particle density, energy, toroidal momentum, and entropy are given in the magnetic-surface-averaged forms. The collision operator C a as well as the fluctuation-particle interaction operator D a in the ensemble-averaged kinetic equation ͑1͒ con-serves the particle number. Then, taking the zeroth moment and the magnetic surface average of the kinetic equation, we obtain the particle density equation:
The surfaced-averaged energy balance equation is written as ‫ץ‬ ‫ץ‬t ͳ 3 2 p a ϩn a e a ⌽ 1 ʹ ϩ 1 where p a ϵn a T a is the pressure, q a the surface-averaged radial heat flux, ⌸ a the surface-averaged toroidal viscosity ͑or the radial flux of the toroidal angular momentum͒, and E (A) ϵϪc Ϫ1 ‫ץ‬A/‫ץ‬t the inductive electric field. The firstorder flow velocity u a1 ϭu ʈa1 bϩu Ќa1 is incompressible (ٌ•u a1 ϭ0) and its perpendicular component is driven by the pressure gradient, the first-order radial electric field, and the centrifugal force as In the right-hand side of Eq. ͑15͒, the O (␦) radial electric field Ϫ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿ is contained in the second term Ϫe a ⌫ a ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿ as well as in the third and sixth terms through u Ќa1 ͓see also Eq. ͑18͒ in Ref. 8͔. These terms associated with the O (␦) radial electric field give Ϫe a ⌫ a A ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿. Then, we find that the right-hand side of Eq. ͑15͒ contains the anomalous energy exchange due to the fluctuations, which is given by Ϫe a ⌫ a A ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿ and Q a A ͓see Eqs. ͑55͒ and ͑56͔͒. It is shown in the same way as in Eq. ͑64͒ of Ref. 8 that the sum of these anomalous energy exchange terms does not depend on the magnitude of the O (␦) radial electric field. Thus, in the axisymmetric toroidally rotating system, the energy balance described by Eq. ͑15͒ is independent of the O (␦) radial electric field although it is affected by the O (␦ 0 ) radial electric field Ϫ‫ץ‬⌽ 0 /‫ץ‬⌿(ϭV /c). When V 0 Ӷv Ti or r/RӶ1 (r denotes the minor radius͒, the terms associated with ⌽ 1 in Eq. ͑15͒ are ignorable since we have e a ⌽ 1 /T a ϳO ͓(r/R)(V 0 / v Ti ) 2 ͔ from Eq. ͑13͒, assuming T a ϳT i .
Taking the flux surface average and species summation of toroidal velocity moment of the kinetic equation, the equation, which determines the temporal evolution of the toroidal angular velocity V , is obtained as ‫ץ‬ ‫ץ‬t ͳ͚ͩ a m a n a ͪ R 2 V ʹ ϩ 1

͑17͒
where the first and second terms in the left-hand side represent the time derivative of the total toroidal angular momentum and the divergence of the total toroidal angular momentum flux, respectively. In the right-hand side of Eq. ͑17͒, the torque by the external sources such as neutral beam injection is given, and v is the covariant toroidal component of the particle velocity in the laboratory frame:

͑18͒
The transport fluxes ⌫ a , q a , and ⌸ a are written as Here the fluxes with the superscript (E) represent the inductive-electric-field-driven parts given by These inductive-field-driven fluxes are not dissipative in that they are not involved in the entropy production processes. The fluxes with the superscript ''H'', ''cl'', and ''anom'' are given by the second-order gyrophase-dependent distribution functions f a H f a C and f a A , respectively, as The neoclassical fluxes ⌫ a ncl q a ncl and ⌸ a ncl are given by the first-order gyrophase-averaged distribution function, which is obtained by solving the linearized drift-kinetic equation. The definitions of the neoclassical fluxes and their properties related to the linearized drift-kinetic equation are shown in the next section.
Without knowledge of the solution of the linearized drift-kinetic equation, we can obtain the transport equations relating the fluxes (⌫ a H ,q a H /T a ,⌸ a H ) and (⌫ a cl ,q a cl /T a ,⌸ a cl ) to the radial gradient thermodynamic forces (X a1 ,X a2 ,X V ) which are flux-surface quantities defined by The transport equations for (⌫ a H ,q a H /T a ,⌸ a H ) are given by where the transport coefficients (L H ) 1V a and (L H ) 2V a are given by which are independent of the collision frequency. Here R 2 B P 2 ϭٌ͉⌿͉ 2 . The fluxes ⌫ a H and q a H /T a are shown to be rewritten in terms of the parallel gyroviscosity as )͘, respectively. It is wellknown that the gyroviscosity is nondissipative. In fact, the antisymmetry of the transport matrix in Eq. ͑23͒ shows that the transport fluxes (⌫ a H ,q a H /T a ,⌸ a H ) are nondissipative, or that they give no entropy production: The fluxes (⌫ a H ,q a H /T a ,⌸ a H ) are contained in the neoclassical fluxes defined by Hinton and Wong. 17 In the present work, we regard the neoclassical fluxes as caused by the guiding center motion with collisions, which are defined in terms of the gyrophase-averaged distribution function determined by the drift kinetic equation in Sec. III ) also contain the nondiagonal collision-independent parts, i.e., the flow shear driven particle and heat fluxes, and the pressure gradient driven toroidal momentum flux.
The classical fluxes (⌫ a cl ,q a cl /T a ,⌸ a cl ) and the anomalous fluxes (⌫ a anom ,q a anom /T a ,⌸ a anom ) are rewritten in terms of the gyrophase-dependent part of the linearized collision operator C a L ( f a1 ) and that of the fluctuation-particle interaction operator D a , respectively, as

͑27͒
In Appendix A, by rewriting the collisional frictions in Eq. ͑26͒ in terms of the perpendicular flows and the gyroviscosity, the classical transport equations relating (⌫ a cl ,q a cl /T a ,⌸ a cl ) to (X a1 ,X a2 ,X V ) is derived and the Onsager relations for the classical transport coefficients are shown. The classical entropy production defined kinetically in terms of f a1 and C a L is written in the thermodynamic form as the inner product of the fluxes and the forces: The properties of the anomalous transport and the anomalous entropy production are shown in Sec. IV with the help of the gyrokinetic equation for the fluctuating part of the distribution function. The entropy per unit volume for species a is defined by of which the lowest-order expression is given by temporal variation of the surface-averaged entropy density ͗S a ͘ is described by represents the external entropy source. Here the surfaceaveraged radial entropy flux J S a tot consists of the convection and conduction parts as where (S a /n a Ϫ⌶ a /T a ) is a flux surface quantity in the lowest order in ␦. We have redefined the anomalous fluxes by ⌫ a A ϵ⌫ a anom , ⌸ a A ϵ⌸ a anom , and where the second term in the right-hand side represents the turbulent transport of the fluctuation-particle interaction energy and ͗͗•͘͘ denotes a double average over the magnetic surface and the ensemble. The surface-averaged entropy production rate ͗ a tot ͘ is given by

͑34͒
It is shown that the classical, neoclassical, and neoclassical contributions in Eq. ͑34͒ are separately positive definite.

III. NEOCLASSICAL TRANSPORT IN TOROIDALLY ROTATING PLASMAS
In toroidally rotating axisymmetric systems, the driftkinetic equation derived by Hazeltine and Ware 16 where ḡ a is defined in terms of the first-order gyrophaseaveraged 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 defined by Eq. ͑22͒ and

͑37͒
The parallel inductive electric field E ʈ (A) is not included in the work by Hinton and Wong 17 but it is retained by Catto et al. 18 The functions (W a1 ,W a2 ,W aV ,W aE ) are defined by In the same way as in Ref. 22, the neoclassical entropy production is kinetically defined in terms of f a1 and C a L and is rewritten in the thermodynamic form by using Eq. ͑35͒. The surface-averaged total neoclassical entropy production is given by where the fluxes (⌫ a ncl ,q a ncl /T a ,⌸ a ncl ,J E ) are defined by The neoclassical transport equations are written as where the transport coefficients are dependent on the radial electric field through the toroidal angular velocity V ϭϪc⌽ 0 Ј . By using the self-adjointness of the linearized collision operator and the formal solution of Eq. ͑35͒, we can prove that the neoclassical transport coefficients satisfy the Onsager symmetry which is given by L mn ab ͑V ͒ϭL nm ba ͑ϪV ͒ ͑m,nϭ1,2͒, It is noted that the transport coefficients given in Eqs. ͑23͒ and ͑24͒ satisfy the same Onsager symmetry as Eq. ͑42͒. If the system has up-down symmetry, the neoclassical transport coefficients are shown to be more restricted by the relations: where the charge neutrality ͚ a e a n a ϭ0 and the momentum conservation by collisions ͚ a F a1 ϭ0 are used. The ambipolarity condition given by Eq. ͑45͒ is intrinsically valid for an arbitrary value of the radial electric field although neither ͚ a e a ⌫ a ncl ϭ0 nor ͚ a e a ⌫ a H ϭ0 is separately valid without updown symmetry. The O (␦ 0 ) radial electric field Ϫ‫ץ‬⌽ 0 /‫ץ‬⌿(ϭV /c) is determined not by the ambipolar condition Eq. ͑45͒, which is O (␦), but by the toroidal momentum balance equation ͑17͒, which is O (␦ 2 ). On the other hand, the O (␦) radial electric field Ϫ‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿ neither affects the transport nor is determined by the ambipolar con-dition. Using the ambipolarity condition, the number of the thermodynamic forces in the transport equations for ͓⌫ a ncl ϩ⌫ a H ,(q a ncl ϩq a H )/T a ,⌸ a ncl ϩ⌸ a H ,J E ͔ is reduced by one, although the transport coefficients in the reduced transport equations retain the Onsager symmetry as shown in Ref. 22.
Hinton and Wong, 17 and Catto et al. 18 derived the detailed expressions of the ion neoclassical transport coefficients. We have derived the detailed expressions of the full transport matrix including the electron-ion cross coefficients, which will be reported elsewhere. We find that the coefficient L VE describes the inward flux of the toroidal momentum caused by the parallel electric field, which is associated with the Ware pinch effect 23 corresponding to the coefficient L 1E e .

IV. GYROKINETIC EQUATION AND ANOMALOUS TRANSPORT IN TOROIDALLY ROTATING PLASMAS
We assume that any fluctuating field F is written in the WKB ͑or eikonal͒ form: where the eikonal ͐ xЈ k Ќ •dxЈ represents the rapid variation in the directions perpendicular to the magnetic field lines with characteristic scale lengths k Ќ Ϫ1 ϳ i . The gyrokinetic ordering employed here for the turbulent fluctuations is written in terms of ␦ϵ i /L as where the characteristic parallel wavelength is given by k ʈ ϳL Ϫ1 . When a frequency of the fluctuation with the perpendicular wave number k Ќ ϳ i Ϫ1 is observed in the laboratory frame, it contains the rapid component due to the high plasma rotation, which is written as Ϫ1 ϵk Ќ •V 0 ϳ␦ Ϫ1 v Ti /L. The frequency 0 in Eq. ͑47͒ is defined by 0 ϵ(Ϫ Ϫ1 )ϳv Ti /L. The fluctuating part of the distribution function is divided into the adiabatic and nonadiabatic parts as where L a (k Ќ )ϵk Ќ •(vЈ؋b)/⍀ a . Appendix B shows that the nonadiabatic part of the distribution function satisfies the following nonlinear gyrokinetic equation: where ‫‪t‬ץ/ץ‬ 0 ϵe Ϫi Ϫ1 t (‫ץ/ץ‬t)e i Ϫ1 t ϭ‫ץ/ץ‬tϩi Ϫ1 , and J 0 (␥Ј) is the zeroth-order Bessel function of ␥Јϵk Ќ Ј v Ќ Ј /⍀ a . In the right-hand side of Eq. ͑49͒, we have defined the forces (X a1 A ,X a2 A ,X aV A ,X aT A ) as and the fluctuating functions (ŵ a1 ,ŵ a2 ,ŵ aV ,ŵ aT ) as Here, the last term in the definition of ŵ aV is not contained in the gyrokinetic equation by Artun and Tang 19,20 but needed to derive the anomalous viscosity which reduces to the Reynolds stress in the fluid limit ͓see Eqs. ͑58͒ and ͑59͔͒.
In the same way as in Ref. 8, the contribution from the turbulent fluctuations to the entropy balance is represented by where the surface-averaged radial anomalous entropy flux is given by and the surface-averaged anomalous entropy production rate is written in the thermodynamic form as The anomalous fluxes (⌫ a A ,q a A /T a ,⌸ a A ,Q a A ) conjugate to the forces (X a1 A ,X a2 A ,X aV A ,X aT A ) are given by the correlations between ĥ a and (ŵ a1 ,ŵ a2 ,ŵ aV ,ŵ aT ) as The heating term due to the fluctuation-particle interaction operator D a in Eq. ͑15͒ is rewritten in the stationary turbulent states by We find from Eqs. ͑15͒, ͑19͒, ͑32͒, and ͑56͒ that not q a anom but q a A should be used as a radial anomalous heat flux in the energy balance equation, and that the fluctuations transfer the energy to the particles of species a through the two terms Ϫe a ⌫ a anom (‫͗ץ‬⌽ 1 ͘/‫ץ‬⌿) and Q a A Using the charge neutrality condition ͚ a e a n a ϭ0 (n a ϵ͐d 3 vf a ) or the Poisson's equation ٌ•Ê ϭ4 ͚ a e a n a for the self-consistent fluctuations, we have the ambipolarity of the anomalous particle fluxes ͚ a e a ⌫ a A ϭ0 and the cancellation of the total anomalous heat transfer ͚ a Q a A ϭ0, which shows that the self-consistent fluctuations cause no net heating of the total particles but result in the anomalous heat exchange between different species of particles. We obtain the balance equation for the fluctuation amplitude as

͑57͒
Thus, in the stationary turbulent states, the anomalous entropy production driven by the turbulent transport equals the collisional dissipation of the fluctuating distribution function, which results in the positive definiteness of the total anomalous entropy production: sager symmetry for the quasilinear anomalous transport equations is described in Appendix C.
Compared to the conventional gyrokinetic equation, [9][10][11][12][13][14][15] our gyrokinetic equation ͑49͒ contains the flow shear X aV A as an additional thermodynamic force and the function ŵ aV related to the anomalous viscosity. It is instructive to derive the Hasegawa-Mima equation 24 from Eq. ͑49͒ and examine how the flow shear and the anomalous viscosity enter it. Considering that a collisionless plasma consists of adiabatic electrons and a single species of ions with charge e i ϵZ i e and low ion temperature T i ӶT e (k Ќ i Ͻ1), and assuming that the ion nonadiabatic distribution function has the form ĥ i Ӎ f i0 n i nad /n i , we obtain from Eq. ͑49͒ with the charge neutrality condition the generalized Hasegawa-Mima equation: where c s ϵ( is derived in the same way as in Frieman and Chen 15 and is equivalent to the Hasegawa-Mima equation derived by them except for the second and last terms in the left-hand side associated with the background flow V 0 . The fourth term in the left-hand side is related to the magnetic gradient and curvature drift. The last term in the left-hand side determines the exchange of energy between the fluctuating Ê ؋B flows and the background flow V 0 . To see this, we derive the energy balance equation from Eq. ͑58͒ as where De ϵT e /(4n e e 2 ) 1/2 is the electron Debye length, and v E (k Ќ )ϵϪi(c/B) (k Ќ )k Ќ ؋b is the Ê ؋B drift velocity due to the electrostatic fluctuations. Here we have used the relation (k Ќ k Ќ ):(R)(ٌ⌿)X V ϭ͓(k Ќ ؋b)(k Ќ ؋b)͔: (ٌV 0 ). The right-hand side of Eq. ͑59͒ represents the energy transfer from the background sheared flow to the fluctuations through the Reynolds stress multiplied by the flow shear. In this case, the anomalous ion viscosity ⌸ i A is rewritten in terms of the Reynolds stress as For pressure gradient driven fluctuations, the Reynolds stress can generate the shear flow, 25 which corresponds to the case where the energy transfer from the background flow to the fluctuations given in the right-hand side of Eq. ͑59͒ is negative:

V. EFFECTS OF SHEARED TOROIDAL FLOW ON ION TEMPERATURE GRADIENT DRIVEN MODES
Here we use the sheared slab geometry to consider the ion temperature gradient ͑ITG͒ driven modes localized in the bad curvature region of the large-aspect-ratio system. Let us assume that the plasma is collisionless and consists of adiabatic electrons and a single species of ions with charge e i ϵZ i e. Using the charge neutrality and the linearized version of the gyrokinetic equation ͑49͒ with the approxima-tions ͉k Ќ •v di / 0 ͉Ӷ1, ͉k ʈ v Ti / 0 ͉Ӷ1 ( 0 ϵϪ Ϫ1 ), and k Ќ i Ӷ1, we obtain the following linear eigenmode equation: where b s ϵk 2 s 2 (k : the poloidal wave number͒, ϵ 0 / * e , * e ϵk cT e /(eBL N ), and L N ϵϪ(dlnN e / dr) Ϫ1 . The radial distance from the mode rational surface normalized by s is denoted by xϵ(rϪr s )/ s (r s denotes the minor radius of the mode rational surface͒, and the magnetic shear length is defined by L s ϵRq/s with the safety factor q and the shear parameter sϵr͉dlnq/dr͉. Other parameters in Eq. ͑60͒ are defined by where ϵT i /(Z i T e ), i ϵL N /L Ti , and L Ti ϵϪ(dlnT i /dr) Ϫ1 . In Eq. ͑60͒, the parameters K and G destabilize the ITG modes. Especially, the parameter G, which contains the toroidal curvature term 2L N /R, causes the toroidal ITG modes. From Eq. ͑61͒, we find that G is modified by the terms resulting from the centrifugal force and the Coriolis force due to the toroidal rotation. The parameter ⌺ represents the effects of the sheared toroidal flow, and L E ϵϪ(dlnV /dr) Ϫ1 ϭϪ(dlnE r /dr) Ϫ1 denotes the gradient scale length for the toroidal flow ͑or for the radial electric field͒.
The linear eigenmode equation ͑60͒ is easily solved to give the dispersion relation: and the corresponding eigenfunction: where H n is the Hermite function of order n. If we put Gϭ␣ϭ0, Eq. ͑62͒ reduces to the dispersion relation obtained by Dong and Horton for the slab ITG modes in the sheared flow. 26 Now, let us consider the simple cases where b s , ␣, and (2nϩ1)L N /L s are negligibly small, and Kӷ1ӷG is satisfied. If there is no sheared flow ⌺ϭ0, we have from Eq. ͑62͒ the linear growth rate of the toroidal ITG mode: Im( )Ӎ(KG) 1/2 for KGӷ1. When the sheared flow is large enough to satisfy ⌺ϾK 2 /4, we obtain the linear growth rate of the sheared flow driven instability: Im( )Ӎ⌺ 1/2 . We find that the stability condition is approximately given by K͓1ϩ2(KG) 1/2 ͔Ͻ⌺ϽK 2 /4, which is rewritten by From this estimation, the stability window is expected to appear more clearly when K ӷ 1 is well satisfied. In order to have large K, hot ion temperature ϵT i /(Z i T e )ӷ1 or large ion temperature gradient i ӷ1 is required. Figure 1 shows the normalized linear growth rates Im( )ϭIm()/ * e obtained from Eq. ͑63͒ as functions of the flow shear parameter (L N /L E )(V 0 /c s ) for Kϭ5, 8,11 and Gϭ0.1 with no magnetic shear L N /L s ϭ0. Since the eigenmode equation ͑60͒ is derived for the long perpendicular wavelengths, we use the small poloidal wave number k s ϭ0.1 here although the dependence of the dispersion relation ͑62͒ on k s is weak for k s р0.3. The full kinetic treatment of the ITG mode dispersion relation, which is valid for arbitrary values of k s , was done by Dong and Horton. 26 From Eqs. ͑6͒, ͑13͒, and ͑61͒, we see that ␣ is proportional to (V 0 /v Ti ) 2 . We take ␣ϭ0 here by assuming that (V 0 /v Ti ) 2 is considerably smaller than unity ͓(V 0 /v Ti ) 2 р0.1 typically͔ although the results are almost the same as for the case of ␣ϳ0.1. It is found from Fig.1 that the increase of K broadens the width of the stability window but heightens the strength of the flow shear required for the stabilization, which is expected from the approximate stabil- ity condition ͑64͒. ͑Figure 1 recalls the effects of the radial electric field shear on the resistive interchange modes, 27 which have similar shear dependence of the linear growth rates with the stability window.͒ The real and imaginary parts of the normalized eigenfrequency ϵ 0 / * e for Kϭ8 are given as functions of (L N /L E )(V 0 /c s ) in Figs. 2͑a͒ and 2͑b͒, respectively, where the cases with different values of magnetic shear L N /L s ϭ0,0.01,0.05,0.1 are shown. As the flow shear increases, the real frequency has the sign of the ion diamagnetic frequency and its absolute value increases. The reduction of the growth rates by the flow shear becomes weaker for stronger magnetic shear.
Using the linearized gyrokinetic equation for the response of the distribution function to the electrostatic fluctuations, we can obtain quasilinearly the anomalous ion radial heat flux and the anomalous ion toroidal viscosity, which are given by where the contributions from the nondiagonal parts of the anomalous transport coefficients are neglected by using the approximations ͉k Ќ •v di / 0 ͉Ӷ1 and ͉k ʈ v Ti / 0 ͉Ӷ1. The anomalous ion thermal diffusivity i A and the anomalous ion toroidal momentum diffusivity i A in Eq. ͑65͒ are given by where k y and denote the representative wave number in the energy containing range of the fluctuation spectrum and the corresponding eigenfrequency, respectively. If we use the mixing length treatment to estimate the potential fluctuation amplitude as e͉ ͉/T e ϳ(k Ќ L) Ϫ1 with the characteristic wave number k Ќ ϳk y and the ion pressure gradient scale length LϳL N /(1ϩ i ), we have ͑In the case where the flow shear is the dominant destabilizing source, we should take account of V 0 /L E to estimate the fluctuation level as in Ref. 25.͒ The anomalous transport coefficients given by Eq. ͑67͒ depend on the flow shear through the linear response factor Im( )/͉ ͉ 2 , which is shown as a function of (L N /L E )(V 0 /c s ) in Fig. 3 using the same parameters as in Fig. 2. We see that the decrease of the growth rate Im( ) together with the increase of ͉ ͉ results in the abrupt cutoff of the anomalous transport coefficients for the flow shear ͑or the radial electric field shear͒ greater than a critical value. The dispersion relation obtained by Dong and Horton 26 using the kinetic integral equation predicts the critical value of the radial electric field shear for stabilization smaller than the results of our dispersion relation Eq. ͑62͒ obtained in the limit of low wave numbers and high phase velocities. Their kinetic dispersion relation also shows that the smaller magnetic shear is favorable for the stability window as shown here. The present transport analysis demonstrates the relationship between the energy and momentum transport in the presence of drift wave fluctuations in an axisymmetric toroidal plasma.

VI. CONCLUSIONS AND DISCUSSION
In this work, the synthesized theory of neoclassical and anomalous transport 7,8 is generalized to that for the rotating turbulent plasma with large flow velocities on the order of the ion thermal speed. Taking account of all transport processes, i.e., classical, neoclassical and anomalous transport processes, we have obtained balance equations for the particles, energy, toroidal momentum ͑or radial electric field͒, entropy, and the fluctuation amplitude, which are given by Eqs. ͑14͒, ͑15͒, ͑17͒, ͑30͒, and ͑57͒, respectively. We have also given the rigorous expressions which define the classical, neoclassical, and anomalous transport fluxes of the particles, energy, and toroidal momentum. Nonequilibrium thermodynamic properties such as the entropy production rate, the conjugate pairs of the fluxes and forces, and the transport equations with the Onsager symmetry are established from the basic kinetic equations as the first principle. The drift kinetic equation and the new gyrokinetic equation for the rotating plasma are used to formulate the neoclassical and anomalous fluxes which are connected to the conjugate thermodynamic forces by the corresponding transport equations.
In the presence of the high-speed toroidal flows, the shear of the toroidal flow ͑or the radial electric field shear͒ enters all the classical, neoclassical, and anomalous transport equations as an additional thermodynamic force, and accordingly influences the transport fluxes and the fluctuation level. On the other hand, through the anomalous viscosity term in the toroidal momentum balance equation, the fluctuations affect the flow or the radial electric field as a reaction to the sheared flow effect on the fluctuations. Thus, a selfconsistent description of the interaction between the fluctuations and the sheared flow is given.
Our gyrokinetic equation for the rotating plasma contains the new sheared flow driving term, from which the perpendicular anomalous viscosity is defined kinetically. We have derived the generalized Hasegawa-Mima equation from the gyrokinetic equation, and found that the kinetically defined anomalous viscosity reduces to the Reynolds stress in the fluid limit.
We have examined effects of the toroidal flow shear on the toroidal ITG modes by using the approximate dispersion relation derived from the gyrokinetic equation. It is found that there exists a stability window in the flow shear parameter space when the parameter K ͓see Eq. ͑61͔͒ is large ͑or when T i /T e ӷ1 or i ϵL N /L Ti ӷ1). As K increases, the width of the stability window is broadened and the threshold value of the flow shear is heightened. The flow shear stabilization is relatively stronger for the case of weak magnetic shear. We have given the quasilinear anomalous diffusivities for the heat and toroidal momentum transport using the linear eigenfrequencies and the mixing length argument. The anomalous toroidal momentum diffusivity is proportional to the anomalous thermal diffusivity and they decrease as the flow shear increases. In the improved confinement of the Japan Atomic Energy Research Institute Tokamak-60 Upgrade ͑JT-60U͒, 28 the internal transport barrier ͑ITB͒ with the steep ion temperature gradient is formed in the region where the gradient of the toroidal flow is steep, the magnetic shear is weak, and T i ϾT e . 29 These ITB formation conditions are in qualitative agreement with our results on the stabilization of the toroidal ITG modes by the sheared toroidal flow. In the stabilized region, there still remain the neoclassical fluxes which are presented in Sec. III.
where the coefficients l jk ab are the same ones as defined in Refs. 3  Ϫl 22 ͑ L cl ͒ VV ͑V ͒ϭ͑L cl ͒ VV ͑ϪV ͒, which has the same form as Eq. ͑43͒. Equation ͑A8͒ is valid even without up-down symmetry since the classical transport is a spatially local process. The momentum conservation by collisions assures the intrinsic ambipolarity of the classical particle fluxes ͚ a e a ⌫ a cl ϭ0, which reduces the number of the independent thermodynamic forces in the classical transport equations by one. As shown in Ref. 22, the reduced classical transport equations retain the Onsager symmetry.

APPENDIX B: DERIVATION OF THE GYROKINETIC EQUATION FOR TOROIDALLY ROTATING PLASMAS
The derivation of the gyrokinetic equation ͑49͒ for rotating plasmas with the toroidal flow velocity V 0 ϭO (v Ta )  where