PhysPlasmas_16_082504

Toroidal flow and radial particle flux in tokamak plasmasJ. D. Callen,a� A. J. Cole, and C. C. HegnaUniversity of Wisconsin, Madison, Wisconsin 53706-1609, USA

�Received 16 April 2009; accepted 29 July 2009; published online 17 August 2009�

Many effects influence toroidal flow evolution in tokamak plasmas. Momentum sources and radialplasma transport due to collisional processes and microturbulence-induced anomalous transport areusually considered. In addition, toroidal flow can be affected by nonaxisymmetric magnetic fields;resonant components cause localized electromagnetic toroidal torques near rational surfaces inflowing plasmas and nonresonant components induce “global” toroidal flow damping torquethroughout the plasma. Also, poloidal magnetic field transients on the magnetic field diffusion timescale can influence plasma transport. Many of these processes can also produce momentum pinchand intrinsic flow effects. This paper presents a comprehensive and self-consistent description of allthese effects within a fluid moment context. Plasma processes on successive time scales �andconstraints they impose� are considered sequentially: compressional Alfvén waves �Grad–Shafranovequilibrium and ion radial force balance�, sound waves �pressure constant along a field line andincompressible flows within a flux surface�, and ion collisions �damping of poloidal flow�. Finally,plasma transport across magnetic flux surfaces is induced by the many second order �in the smallgyroradius expansion� toroidal torque effects indicated above. Nonambipolar components of theinduced particle transport fluxes produce radial plasma currents. Setting the flux surface average ofthe net radial current induced by all these effects to zero yields the transport-time-scale equation forevolution of the plasma toroidal flow. It includes a combination of global toroidal flow damping andresonant torques induced by nonaxisymmetric magnetic field components, poloidal magnetic fieldtransients, and momentum source effects, as well as the usual collision- andmicroturbulence-induced transport. On the transport time scale, the plasma toroidal rotationdetermines the radial electric field for net ambipolar particle transport. The ultimate radial particletransport is composed of intrinsically ambipolar fluxes plus nonambipolar fluxes evaluated at thistoroidal-rotation-determined radial electric field. © 2009 American Institute of Physics.�DOI: 10.1063/1.3206976�

I. INTRODUCTION

Determining the magnitude, radial profile, and evolutionof toroidal flow in tokamak plasmas is an important issue forboth present tokamak plasmas and ITER1—for E�B flowshear control of anomalous transport, prevention of lockedmodes, control of edge localized modes via resonantmagnetic perturbations, etc. Many effects2 influence theevolution of toroidal flow in tokamak plasmas. Radialtransport of toroidal flow due to collision-induced3–7 andmicroturbulence-induced “anomalous” processes obtainedusing “mean-field” theory8–13 is usually considered withinthe context of an equilibrium axisymmetric magnetic fieldmodel. In addition, the toroidal flow is constrained by fastertime scale processes and can be affected by three-dimensional �3D� nonaxisymmetric �NA� magnetic fieldcomponents,14–21 magnetic field transients,22–25 and exter-nally supplied toroidal momentum sources �e.g., from neutralbeam injection26 �NBI��.

Heretofore these diverse effects on the plasma toroidalflow have been mostly considered separately and in an adhoc fashion. In this paper a detailed framework is developedfor describing toroidal flow evolution in tokamaks that

accounts for all these physical effects simultaneously andself-consistently. A “neoclassical equilibrium” is obtained fortime scales longer than the collision time scales. It is brieflyrevisited to add self-consistently possible effects due to mi-croturbulence and momentum sources �e.g., noninductivecurrent drives �CDs�� on the lowest order plasma flows andparallel Ohm’s law in tokamak experiments. The overall to-roidal flow equation is developed using an approach origi-nally developed27,28 for a neoclassical description of flows inNA �e.g., stellarator� toroidal magnetic confinement systems,but here extended to include microturbulence, magnetic fieldtransients, and momentum source and sink effects.

NA magnetic field components in tokamaks arise fromcoil irregularities, active control coils, and plasma collectivemagnetic distortions �e.g., neoclassical tearing modes and re-sistive wall modes�. The NA components are typicallyvery small compared to the equilibrium magnetic field:

B /B0�10−3. Thus, they will be considered to be of first or-der or smaller in the small gyroradius expansion, which willgreatly facilitate analysis of the 3D effects.

This paper uses a fluid moment approach to develop acomprehensive description of the various effects on the par-ticle fluxes, toroidal flow, radial electric field, and resultantnet radial particle flux in a tokamak plasma. An earlier andmore limited and abridged version of this work has been

a�Electronic mail: [email protected]. Website: http://www.cae.wisc.edu/~callen.

PHYSICS OF PLASMAS 16, 082504 �2009�

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http://dx.doi.org/10.1063/1.3206976

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presented elsewhere.2 The magnetic field B, electric potential�, and plasma parameters will be expanded in terms of theiraxisymmetric equilibrium plus gyroradius-small NA pertur-bations. Both externally induced perturbations and collectiveplasma fluctuations are taken into account. Effects on succes-sively longer time scales will be considered sequentially: onthe compressional Alfvén wave time scale, the radial ionforce balance yields a relation between the poloidal and tor-oidal flows within a flux surface and the radial electric field;on Coulomb collision time scales, the density, temperature,and pressure of each plasma species become constant along amagnetic field line and flows within a magnetic flux surfacebecome incompressible; and then, for times longer than theion collision time, the poloidal ion flow is usually damped toa diamagnetic-type flow dependent on the ion temperaturegradient.

Finally, there are many second order �in a small gyrora-dius expansion and the 3D NA magnetic field magnitudes�radial particle transport fluxes induced by toroidal torques ona plasma species, both collision-induced intrinsically ambi-polar ones and possibly nonambipolar ones. The collision-induced ones include classical,3 Pfirsch–Schlüter �PS�,banana-plateau,5,6 and paleoclassical29 particle fluxes plusones induced by parallel noninductive and dynamo CDs, andthe EA�B /B2 pinch. Possibly nonambipolar fluxes arecaused by NA neoclassical toroidal viscosity �NTV� flowdamping effects,14–17,27,28 perpendicular viscosities,3,4,7 toroi-dal torques on resonant surfaces due to nonideal effects �e.g.,due to resistivity18 and two-fluid diamagnetic flows19� on NAresonant magnetic field components combined with NTVeffects,20 polarization flows, microturbulence,8–13 magneticfield transients,22–25 and momentum sources. Nonambipolarcomponents of the particle fluxes cause radial plasma cur-rents. Setting the flux surface average of the total radial cur-rent to zero �so, from charge conservation and the time de-rivative of Gauss’s law, the radial electric field does notincrease monotonically in time� yields2,15,27,28 a transport-time-scale toroidal flow evolution equation. The radial elec-tric field is determined from the toroidal flow. The net radialparticle flux is the sum of the collision-induced intrinsicallyambipolar particle fluxes and the nonambipolar particlefluxes evaluated at the toroidal-flow-determined radial elec-tric field.

We make a number of assumptions to facilitate deter-mining comprehensive equations for the particle fluxes andevolution of toroidal rotation in tokamak plasmas.

�1� Small gyroradius expansion, which to zeroth orderyields magnetohydrodynamic �MHD� force balanceequilibrium, flows within flux surfaces at first order, andsecond order “radial” transport fluxes

�2� Axisymmetric lowest order magnetic field structure withnested toroidal flux surfaces �i.e., no magnetic islands inregion of interest�

�3� Gyroradius-small magnetic field nonaxisymmetriessuch that toroidal nonaxisymmetries �NAs� in the mag-netic field B are first order or smaller in the gyroradiusexpansion

�4� Banana-plateau collisionality regime, where electron

and ion collision lengths are long compared to the po-loidal periodicity length 2�Rq and hence plasma prop-erties are constant on magnetic flux surfaces

�5� Gyroradius-small plasma fluctuations, which lead tosecond order microturbulence-induced anomalous radialplasma transport

�6� Slow poloidal magnetic field transients, which at theirfastest occur on the transport time scale

�7� Poloidal damping of electron heat flow is neglected inobtaining the parallel neoclassical Ohm’s law

No explicit inverse aspect ratio ��r /R0� expansion willbe made, except in estimating the magnitude of some termsand in neglect of some minor O��2� terms. While the analy-sis will be valid over most of a tokamak plasma, because ofassumptions �4� and �5�, it might not apply to its very edge.In particular, very close to the magnetic separatrix in di-verted tokamak plasmas, assumption �4� can be violated, inwhich case a two-dimensional �radial and poloidal� descrip-tion of plasma transport is needed. Also, in the edge ofL-mode tokamak plasmas fluctuations can become so largethat assumption �5� is violated, in which case a fluid-typeturbulent plasma description may be needed there. Becauseof assumption �7�, while scalings of all the collisional radialparticle fluxes will be correct, some of their numerical coef-ficients will only be approximate. Finally, for simplicity werestrict our analysis to a two species plasma—electrons plushydrogenic ions.

This paper is organized as follows. Section II specifiesthe plasma and magnetic field models and perturbation pro-cedure. Section III discusses the successive time scales andthe constraints they impose. The radial particle fluxes in-duced by the various effects indicated above and the plasmatoroidal flow evolution equation that results from setting tozero the net radial currents from their nonambipolar compo-nents are developed in Sec. IV. �Appendix A describes therelation of fluid flow velocities to guiding center velocities,and Appendix B describes the gyroviscosity and its effects.�Some interesting properties of the various effects in the evo-lution equation for the plasma toroidal rotation are discussedin Sec. V. Finally, Sec. VI summarizes the main results ob-tained in this paper.

II. PLASMA AND MAGNETIC FIELD MODELS

We use a plasma description based on fluid moments ofa general plasma kinetic equation for each species, whichincludes the Vlasov operator, the Coulomb collision operator,and sources. Some of the possible plasma particle and mo-mentum sources and sinks are those due to collisions withneutrals, fast ions injected via energetic neutral beams,26 cur-rent sources, and radio-frequency waves interacting with theplasma. Such a description is sometimes called a two-fluiddescription. However because we are considering thebanana-plateau collisionality regime, we will use the kineti-cally derived closure relations for the �typically first order inthe small gyroradius expansion� neoclassical parallel viscousforces �B0 ·� ·�.6 The magnetic field description will bebased on a lowest order axisymmetric magnetic field, whichis composed of nested magnetic flux surfaces. That is, any

082504-2 Callen, Cole, and Hegna Phys. Plasmas 16, 082504 �2009�


NA magnetic field perturbations will be assumed to be smallenough that they do not cause magnetic islands in the radialregions being considered. The dimensionless radial coordi-nate � will be based on the toroidal magnetic flux, as iscustomary for tokamak plasma transport modeling codes.The first order perturbations will include both the “average”�axisymmetric� poloidal variations in the equilibrium �PS ef-fects� and effects due to “zero-toroidal-average,” gyroradius-small perturbations.

The conservative forms of the density and momentum�force balance� equations for each plasma species obtainedfrom the d3v�1,mv� fluid moments of a full plasma kineticequation including sources are �in laboratory coordinates x�

�n/�t�x + � · nV = Sn, �1�

�m�

�t�

x�nV� + � · �mnVV�

= nq�E + V� B� − �p − � · � + R + Sm. �2�

Here, V is the species flow velocity, −� ·� is the viscousforce density, R �Re nee�J� is the Coulomb collision dy-namical friction force density, and the other notation isstandard.3 The sources of density Sn and momentum Sm willbe assumed to be small and to mostly contribute at the trans-port level �i.e., at second order in the gyroradius expansion�.Also, the species label s has been suppressed for simplicity;it will be added as needed.

The lowest order axisymmetric equilibrium magneticfield B0 is composed of toroidal �Bt� and poloidal �Bp� com-ponents. It will be represented by

B0 = Bt + Bp � I � � + �� p

= �p� ��q − �� = �� t � − p � �� . �3�

Here, I�p��RBt is the poloidal current function, 2�p�� isthe “equilibrium” poloidal magnetic flux, which will be al-lowed to evolve on the slow transport time scale, � is thetoroidal �axisymmetry� angle, q�p��B0 ·�� /B0 ·�=dt /dp is the inverse of the rotational transform of themagnetic field B0, and is the “straight-field-line” �on amagnetic flux surface� poloidal angle. The Jacobian �g of thetransformation from laboratory �x� to the �nonorthogonal�toroidal magnetic flux coordinates �, , and � is

�g �1

�� · � � ��=

dp/d�B0 · �

=p�

I/qR2 , �4�

in which p��dp /d� aRBp. Here, ��t /t�a� is the di-mensionless radial �flux surface label� coordinate based onthe toroidal flux 2�t�dS�� ·B0=d3x B0 ·��. It rangesfrom zero �at the magnetic axis� to unity �at the plasma edgeor divertor magnetic separatrix of average minor radius a�.Also, R�x�=R�� ,� is the major radius to the laboratory po-sition x.

The flux surface average �FSA� of a scalar functionf�� ,� is defined by

�f �0

2�d�gf��,�

02�d�g

=0

2�df��,�/B0 · �

02�d/B0 · �

. �5�

Flux-surface-averaging is an annihilation operator for theparallel derivative along a magnetic field line,

�B0 · �f = 0. �6�

In addition, the FSA of the divergence of a vector functionC�� ,� is

�� · C =d

dV�C · �V =

1

V�

d

d��V��C · �� , �7�

in which V��0�d3x=2�0

�d�02��gd is the volume of the

� flux surface and V��dV /d�=2�02��gd. Thus, the FSA

is �f=2��d�gf /V�.The contravariant base vectors �ei��ui� for the

ui=� , ,� coordinate system are e��, e��, ande��. The covariant base vectors �ei��x /�ui� aree�=�g��, e=�g��, and e�=�g��.These nonorthogonal base vectors satisfy Kronecker deltarelations: ei ·e

j =�ij. Because of toroidal axisymmetry,

e��= e� /R and e�=R2��=Re� in which e�� / �� isthe unit vector in the � direction.

To lowest order in the gyroradius expansion the densityn, temperature T, and pressure p�nT of both plasma specieswill be constant5 on the poloidal magnetic flux surfacesp��. In first order we allow for zero-toroidal-average NA�3D� perturbations ��-dependent, denoted by tilde, due toinstability-induced fluctuations plus those induced by NA ex-ternally imposed magnetic field components� and poloidalvariations in the �-average �denoted by an overbar� plasmaparameters. Thus, we expand spatial dependences of n, T,and p as

p�x� = p0�� + � �p1��,� + p1��,,�� + O��2� , �8�

in which ��s /a�1 is the small gyroradius expansion pa-rameter. Here �s=vTs / cs is the most probable gyroradiusfor a species s with thermal speed vTs��2Ts /ms and gyro-frequency cs�qsB0 /ms.

The electric potential � will be expanded similarly. Inaddition, we define E=−��+EA in which EA�−�A /�t withA�t�−p�� for the magnetic field in Eq. �3�. The�-average inductive electric field is

EA = ��/�t + p� � � − t � � O��2� . �9�

Here, 2�� /�t is the �vacuum� loop voltage Vloop� �t� in-

duced by the rate of change in magnetic flux in the Ohmictransformer �central solenoid�. As indicated, it is spatiallyconstant but a function of time t; it represents a gauge trans-

formation choice. Further, 2�p represents the inductive�Lenz’s law� voltage induced within the plasma by poloidalmagnetic flux transients �e.g., from turning on a noninductive

CD�. The t term represents toroidal magnetic flux transientsinduced by radial motion and shaping of the plasma. Allthese slow, transport-time-scale transient effects are dis-cussed in Sec. IV A below.

082504-3 Toroidal flow and radial particle flux… Phys. Plasmas 16, 082504 �2009�


Because of the toroidal axisymmetry in the equilibrium,the zero-toroidal-average perturbations can be expanded in aFourier series. For example,

p1 = �n

pne−in�, pn �1

2��

0

2�

d�ein�p1. �10�

The toroidal average of a function F�x� is defined by

F�x� �1

2��

0

2�

d�F�x� , �11�

which is a function of � ,. Note that since p1=0,

p�x� = p0�� + � p1��,� + O��2� . �12�

The toroidal average of NA perturbations times functionsthat do not depend on � vanishes. However, the toroidal av-erage of products of perturbations does not in general vanish;

for example, p1�=�np−n�n usually does not vanish.The total �equilibrium plus perturbed� magnetic field

will be expanded as

B = B0��,� + � �B� + B�� + O��2� . �13�

The parallel � and perpendicular � components are definedrelative to the equilibrium magnetic field B0,

A� �B0 · A

B02 B0, A� �

− B0� �B0� A�B0

2 . �14�

The magnetic field magnitude can be approximated by

B � �B� = �B02 + 2� B0B� + �2B�

2 + �2B�2�1/2

B0��,� + � B��,,�� + O��2� . �15�

The magnetic field perturbations B� and B� are expanded inFourier series analogous to that for p1 in Eq. �10�.

Perpendicular gradients of instability-induced fluctua-tions will be assumed to scale as 1 /� to reflect the shortradial scale length of drift-wave-type perturbations. Thus, forexample, ��p1��1 /��0�1. In contrast, parallel gradi-ents of fluctuations will be assumed to scale with the overalltokamak plasma dimensions and hence as �0; thus, ��p1

��0��1. Gradients of average plasma properties willalso be assumed to scale as �0. Hence, while ��p1��0�1,we scale ��p1��1.

III. SUCCESSIVE TIME SCALES AND PROCESSES

The plasma toroidal flow evolves on the long transporttime scale ��seconds�. Its evolution arises from effects thatare formally second order or smaller in the gyroradius ex-pansion. To obtain an equation for evolution of the plasmatoroidal flow on this long time scale, we need to take accountof faster processes and constraints they impose on plasmabehavior. To analyze the various time scale processes, wewill analyze sequentially three independent �but not orthogo-nal� components of the momentum �force balance� equation:radial �O��0�, enforces radial force balance on fast compres-sional Alfvén time scale�, parallel �O��, determines first or-der flows on a flux surface on intermediate ion collision time

scale�, and toroidal �O��2�, determines radial particle trans-port fluxes and the desired toroidal plasma flow equation�.

Summing the density and momentum equations in Eqs.�1� and �2� over species, we readily obtain to zeroth order theideal MHD plasma equations ��m /�t+� ·�mV=0 and�mdV /dt=J�B−�P. In obtaining these equations we haveneglected viscous force effects, which are first order in thegyroradius expansion and source effects, which are assumedto be of first or higher order. Neglecting electron inertia, thelowest order electron momentum equation yields the colli-sionless two-fluid Ohm’s law E+V�B= �J�B−�pe� /nee.In these equations �m is the mass density, V Vi is theplasma � ion� flow velocity, J�niqiVi−neeVe is the currentdensity, and P� pe+ pi is the total plasma pressure. Addingthe nonrelativistic Maxwell equations � ·B=0, ��B=�0J,and �B /�t=−��E and an isentropic equation of state�d /dt��ln P /�m

��=O��2�→0, which is derivable from energybalance equations, completes the zeroth order plasma de-scription.

The fastest time scale MHD processes are compressionalAlfvén waves, which propagate perpendicular to magneticfield lines. On time scales longer than their natural waveperiods ��A�a /cA��s�, together with the conditionB0 ·�P0=0⇒ P0= P0�p�, they cause tokamak plasmas tocome into an MHD force balance equilibrium with J0�B0

=�P0= �dP0 /dp��p. This relation together with the equi-librium Maxwell equations yields the Grad–Shafranov equa-tion for determining the equilibrium poloidal flux functionp��.

Replacing J0�B0 by �P0=�pe0+�pi0 in the �-averageideal �collisionless� two-fluid Ohm’s law yields

0 = ni0qi�E0 + Vi� B0� − �pi0, �16�

in which the lowest order average electric field is electro-static: E0�−��0�p�. This MHD equilibrium ion force bal-ance equation can also be obtained directly from the equilib-rium limit �d /dt→0� of the ion force balance equation inEq. �2�—by neglecting the frictional �R�� and viscous�� ·�� forces and momentum sources �Sm�� ,�2�, whichare all higher order in the gyroradius expansion.

Taking the radial �e�� projection of Eq. �16� yields

�t � Vi · �� = − �d�0

dp+

1

ni0qi

dpi0

dp� + qVi · � . �17�

This equation provides a relation between the average

�designated by overbar� toroidal flow Vi ·�� Vt /R, lowest

order radial electric field E��−e� ·��0��=−��d�0 /d��=−��d�0 /dp�p�, toroidal ion diamagnetic flow

−�1 /ni0qi��dpi0 /dp�, and poloidal ion flow Vi ·�� Vp /r.However, it does not specify any of these quantities. Since inobtaining Eq. �17� we divided through by qiB0, the poloidaland toroidal ion flows are first order in the gyroradius��s�1 /qsB0�. Thus, these flows within tokamak flux surfacesare first order in the small gyroradius expansion; however,for simplicity of notation we will not always indicate this byadding a subscript 1 to these flows. Transport flows in theradial direction �i.e., V ·�� will be of second or higher orderin the gyroradius expansion.



Coulomb collisions cause electrons to thermalize �be-come Maxwellian� on the electron collision time scale��1 /�e�10 �s for Te�1 keV, ne�3�1019 m−3, andZeff�3�. Similarly, ions thermalize on the ion collision timescale ��1 /�i�ms for Ti�1 keV�. In doing so they causetheir corresponding species temperatures to equilibrate alongmagnetic field lines over distances of order the collisionlength ��vT /��102 m. From assumption �4� above thiscauses5 the species density and temperature to become con-stant on poloidal flux surfaces on the collision time scale ofthe species: n0=n0�p� and T0=T0�p� for t�1 /�. On thissame time scale the species flow velocity becomes a defin-able and physically meaningful quantity.

Next, consider the form of the density equation �1� ontime scales longer than the species collision time. Since thelowest �first� order equilibrium flows lie within a flux sur-face, they can be written in terms of their poloidal �V ·��and toroidal �V ·�� components as

V1 � eV · � + e�V · �� . �18�

This representation applies to either electrons or ions. Alter-natively, the equilibrium flows within a flux surface can berepresented in terms of their components parallel to � � � andcross �∧, perpendicular to B0 but within a flux surface� theequilibrium magnetic field B0,

V1 = V�B0/B0 + V∧. �19�

The equilibrium cross flow V∧ in each species is obtained bytaking the cross product of B0 with the lowest order equilib-rium �� /�t→0� momentum equation �2�,

Vs∧ =B0� �p

B02 �d�0

dp+

1

ns0qs

dps0

dp� . �20�

The two terms in parentheses represent the usual equilibriumE0�B0 �VE� and diamagnetic �V�� flows.

The comparable first order perturbed cross flow for eachspecies is obtained by taking the cross product of the lowestorder perturbed momentum equation from Eq. �2� with B0,

Vs∧ =1

B02B0� ��1 +

1

ns0qs� ps1� � VE + V�. �21�

The terms in parentheses represent the perturbed E�B0 anddiamagnetic flows. Since the first order perturbed flows havea nonzero radial component, they lead to the anomalous ra-

dial particle flux �n1V∧ ·��. The relation of the fluctuating

fluid flow velocity V∧ to the guiding center flow velocityusually calculated in gyrokinetics �see, for example, Ref. 12�is discussed in Appendix A.

Because the equilibrium density n0 only changes on thetransport time scale �� /�t��2� and the radial transport fluxesand density sources are second order, the lowest order aver-age density equation �1� reduces to

� · V1 = 0 + O��2� . �22�

The divergence of a vector C is defined by � ·C= �1 /�g��i�� /�ui��gC ·ei�. Thus, using the facts that V1 has

no radial component and the equilibrium B0 is axisymmetric�� /��→0�, this reduces to

�B0 · ��

��V1 · �

B0 · �� = 0. �23�

The solution6 of this partial differential equation is that apoloidal flow function U is constant on a flux surface,

U�p� �V1 · �

B0 · �=

V�

B0+

V∧ · �

B0 · �. �24�

Using the poloidal component of first order E0�B0 and dia-magnetic flows within a flux surface from Eq. �20� yields

V∧ · �

B0 · �=

I

B02�d�0

dp+

1

ns0qs

dps0

dp� � −

I

B02�∧, �25�

in which �∧�p�=�∧�� is constant on a flux surface.The currents flowing within the flux surface can be simi-

larly determined. Using Eqs. �19� and �20�, we obtain

J � �s

ns0qsVs1 � J� + J∧ = J�

B0

B0+

B0� �P0

B02 , �26�

which is the usual sum of the parallel and diamagnetic cur-rent densities. Summing nq times the poloidal flow compo-nents in Eqs. �24� and �25�, or using the fact that for aquasineutral plasma the current density is also incompress-ible �� ·J=0�, yields

KJ�p� �J · �

B0 · �=

J�

B0+

I

B02

dP0

dp. �27�

The constant KJ is obtained from the FSA of B02 times this

equation:

KJ =�B0J��B0

2+

I

�B02

dP0

dp. �28�

Using this form in Eq. �27� yields

B0J� =�B0J�B0

2

�B02

− IdP0

dp�1 −

B02

�B02� . �29�

Here, �B0J� is the FSA parallel current, which will be deter-mined from a parallel Ohm’s law below and the last termrepresents the PS current whose FSA vanishes.

Similar considerations and orderings of the species en-ergy balance equation and the heat flux fluid moment equa-tion yield analogous formulas6 for the first order equilibriumheat flows within a tokamak flux surface,

Q�p� �q1 · �

B0 · �=

q�

B0+

q∧ · �

B0 · �, �30�

qs∧ · �

B0 · �=

5

2

I

qsB02

dTs0

dp. �31�

Finally, the FSA of the parallel �B0·� component of the low-est order parallel ion heat flux equation6 is �B0 ·Rqi

=0+O��2� in which Rqi

�−�iqi is the ion heat friction force.Solving Eq. �30� for qi� and substituting it in this lowestorder relation yield6



Qi =1

�B02�B0

2qi∧ · �

B0 · �� =

5

2

ni0Ti0I

qi�B02

dTi0

dp. �32�

The �first order� average parallel force balance for eachspecies is obtained from the parallel �B0·� component of themomentum equation. Its most convenient form here is ob-tained by using the density equation �1� to write

m��/�t��nV� + � · �mnVV + ��

= mn��V/�t + V · �V� + � · � + mVSn. �33�

The viscous stress tensor � has components along B ��,across magnetic field lines but within flux surfaces ��∧�, andperpendicular to flux surfaces ��,3

� = �0�� + ��∧ + �2��. �34�

For the parallel force balance the most important viscousforce will be the parallel component of −� ·��, which wewill discuss below. The gyroviscous stress �∧ indicates adiamagnetic-type B��V effect. The gyroviscous force−� ·�∧ it induces is higher order for equilibrium flows.However, it is significant for fluctuating flows �see AppendixB�. The forces due to perpendicular viscous stresses �� arehigher order and will not contribute to the first order parallelmomentum equation.

Thus, taking the FSA of the B0 �parallel� component ofthe � average of the momentum equation in Eq. �2� with allquantities expanded as in Eq. �8�, employing Eq. �33�, andeliminating �1 and p1 terms using Eq. �6� yield the O��parallel force balance equation for each species,

mn0��B0V�

�t= n0q�B0 · EA − �B0 · � · �� + �B0 · R

+ �B0 · �Sm − mVSn�

− �B0 · �mn0V1 · �V1 + � · �∧�

+ n0q�B0 · V∧� B� . �35�

The mn0V1 ·�V1 term represents the force caused by theReynolds stress introduced by the fluctuations. To lowestorder the �-average gyroviscous force � ·�∧ produces�see, for example, Ref. 30� the “gyroviscous cancellation”

of the V� ·�V contribution to the Reynolds stress force�see Appendix B�. The Maxwell-stress-type term

n0q�B0 · V∧� B�=−n0q�B� · ��1+ �1 /n0q�� p1� will yielddynamo-type contributions to the parallel Ohm’s law below.Both of these contributions are typically small for drift-wave-type fluctuations in the hot core of tokamakplasmas.12,31

The electron Coulomb collision frictional force den-sity is given by3 Re ne0e�J� /�� +J� /��. The perpendicu-lar and parallel �Spitzer� electrical conductivities are��ne0e2 /me�e and �� 1.96 �� for Z=1. Thus, the equi-librium limit �� /�t��e� of the electron momentum equation�35� yields an equation for the FSA parallel current5,6 in theplasma,

�B0J� = ��B0 · EA + ��/ne0e��B0 · � · �e� + �B0 · JCD

+ �B0 · Jdyn . �36�

The FSA parallel electron viscous force can be writtenas6 �B0 ·� ·�e� mene0�e00�B0

2Ue in which the neglectedpoloidal electron heat flow damping effects induced byQe only modify some numerical coefficients. Here,�e00 2.24��e is the viscous damping frequency6 �forZi=1 and ��1� of the poloidal electron flow Ue inwhich � is the inverse aspect ratio, which here representsthe variation in �B0�=B0�� ,� on a flux surface: ��Bmax

−Bmin� / �Bmax+Bmin� r /R0. Using the definitionJ� =ne0e�Vi� −Ve�� and Eqs. �24� and �25� for both electronsand ions, the parallel viscous force contribution to the paral-lel Ohm’s law in Eq. �36� can be written as

��

ne0e�B0 · � · �e� −

��

��

�e00

�e�B0J� + �B0 · Jbs . �37�

Here, the bootstrap current is defined by

�B0 · Jbs −��

��

�e00

�e� I

p�

dP0

d�− ne0eUi�B0

2� . �38�

Thus, in Eq. �36� the parallel electron viscous force in Eq.�37� produces6,29 both the trapped particle modification��e00��e� of the Spitzer parallel electrical conductivity�� and the bootstrap current.

The “noninductive” CD in Eq. �36� is induced by elec-tron momentum and particle sources,

�B0 · JCD � − ��/ne0e��B0 · �Sem − meVeSen� . �39�

Most noninductive CDs �e.g., due to electron cyclotron CD�result from the average electron parallel momentum�B0 ·Sem they impart to the plasma.

Fluctuation-induced electron Reynolds- and Maxwell-type stresses cause a dynamo-type parallel current,

�B0 · Jdyn =��

ne0e�B0 · �mene0Ve · �Ve + � · �e∧�

+ ��B0 · Ve∧� B� . �40�

In tokamak plasmas the � electron Reynolds stress anddynamo-type terms are usually negligible.31,32

Taking account of the bootstrap current and trapped par-ticle effects from the parallel electron viscous force in Eq.�37�, Eq. �36� becomes a neoclassical parallel Ohm’s law thatincludes noninductive and dynamo CDs,

�B0 · EA = ��nc�B0J� − �1/��B0 · Jbs + �B0 · JCD

+ �B0 · Jdyn� . �41�

Here, the neoclassical parallel electrical resistivity is

��nc

1

��1 +

��

��

�e00

�e� . �42�

Summing the electron and ion FSA parallel momentumequations �35� and using the quasineutrality constraint



�sns0qs=0 and the momentum conservation relation forCoulomb collisions between species of �sRs=0 yield theoverall plasma parallel force balance

�m��B0V�

�t − �s

�B0 · � · �i�

− �s�B0 · �msns0Vs · �Vs + � · �s∧�

+ �B0 · J∧� B�+ �B0 · �s

�Ssm − msVsSsn� . �43�

The sums over species are dominated by the ion contribu-tions since the corresponding electron momentum contribu-tions are typically a factor of �me /mi�1 /60�1 smaller.The FSA of the equilibrium parallel ion viscous force, whichis O��, is given by6

�B0 · � · �i� mini0�B02��i00Ui + �i01

− 2

5niTi0Qi� .

�44�

Here, �i00 1.1��i and �i01 1.17�i00 for Z=1 and��1 in the asymptotic banana collisionality regime6 �i.e.,for ��i��iRq /�3/2vTi�1�. They are damping frequencies forthe poloidal ion flow and heat flow.

Physically, the ion parallel viscous force �B0 ·� ·�i�is caused by ion collisional drag by trapped particles�f t�1.46�� on the untrapped �circulating, fc�1− f t� ionsthat carry the parallel �poloidal� flow. This viscous forcedamps the poloidal �ion� flow to an ion temperature gradientdiamagnetic-type flow because hotter ions are more colli-sionless than bulk ions and hotter regions at smaller radiidamp less than colder ones at larger radii.

Thus, Eq. �43� is an evolution equation for the average

parallel ion flow Vi� or the average poloidal ion flow functionUi. The parallel and poloidal ion flows come into equilib-rium �see Ref. 33 and references cited therein� on the ioncollision time scale t�1 /�i�ms. In the absence of the usu-ally small parallel momentum sinks and sources on the sec-ond through fourth lines of Eq. �43�, the equilibrium poloidalion flow can be obtained by setting the parallel ion viscousforce in Eq. �44� to zero,

Ui0 −

�i01

�i00

− 2

5niTi0Qi = cp

I

qi�B02

dTi0

dp. �45�

The poloidal flow coefficient k=cp��i01 /�i00 is 1.17 for��i�1 and ��1. However, cp depends on the ion collision-ality regime and �;6 with impurities it also depends on gra-dients of impurity densities and temperatures.34 It is oftenevaluated using the NCLASS code.35

When sinks and sources on the second and third lines ofEq. �43� are significant, they cause a poloidal ion flow of

Ui�p� = Ui0 −

�B0 · �mini0Vi · �Vi + � · �i∧�mini0�i00�B0

2

+�B0 · J∧� B� + �B0 · �s�Ssm − msVsSsn�

mini0�i00�B02

.

�46�

The additions to the poloidal ion flow Ui0 in Eq. �45� are

induced by Reynolds stresses, dynamo-type effects, and ion“flow drive” effects due to �B0 ·Sim, respectively. They areusually thought to be small in tokamak plasmas �see Eqs.�22� and �25� and discussion in Ref. 12�.

Having determined the average poloidal ion flowVi ·�=UiB0 ·�, we substitute it into the �t expressionobtained in Eq. �17� to obtain the more specific toroidal ro-tation frequency relation �for t�1 /�i�ms�,

�t = − �d�0

dp+

1

ni0qi

dpi0

dp� +��p =�i∧ +��p. �47�

Here, the poloidal flow contribution to �t is

��p��,� �I

R2Ui�� cpI2

qiR2�B0

2dTi0

dp, �48�

in which the last approximate form neglects the usually smallparallel momentum sink and source terms. Since for rigidbody rotation �t must be a flux function, tokamak plasmasrotate toroidally as a rigid body only when the poloidal flowis negligible so �t �i∧�p�.

The neoclassical literature contains a number of formsfor the average ion flow velocity V1 in addition to thosegiven in Eqs. �18� and �19�. In particular, using the definitionof �∧ at the end of Eq. �25� in Eq. �47�, one can write thefirst order average ion flow within a flux surface as6

Vi =�i∧e� + UiB0 =�i∧�p�R2 � � + Ui�p�B0. �49�

The �� projection of this representation of Vi readily yieldsthe toroidal rotation frequency given in Eq. �47� above. Also,the plasma parallel flow speed can be written as

V� � B0 · Vi/B0 =�i∧�p�I�p�/B0 + Ui�p�B0

− R�d�0

dp+

1

ni0qi

dpi0

dp� + k

R02

qiR

dTi0

dp. �50�

In Eqs. �48�–�50�, R�R�� ,� R0�1+� cos �.For given radial ion pressure and temperature gradients,

Eq. �47� provides a consistency relation between the plasma

toroidal rotation �t�V ·� Vt /R and the lowest order ra-

dial electric field E��−��p��d�0 /dp�. However, it doesnot specify either of these quantities. To proceed further weneed to determine either the radial electric field or the toroi-dal flow—from their own evolution equations. We proceedby calculating the radial particle transport fluxes and hencenet radial current in the spirit of seeking to calculate theradial electric field.



IV. TRANSPORT TIME SCALE

The collision-induced radial particle fluxes of electronsand ions are mostly determined from the second order fluxn0V2 ·�p. To determine them we first note that

e� · V� B0 = − V · e� � B0 = V · �p = p�V · �� . �51�

Thus, the radial particle flux can be determined from thetoroidal angular component �e� ·� of the force balance Eq.�2�, i.e., from the toroidal torques on the plasma species.

Next, consider the e� projection of the � ·mnVV term inEq. �2�. Employing a vector, tensor identity for a generaltensor T, it can be written in the form

e� · � · T = � · �e� · T� − �e�:T . �52�

Using a cylindrical coordinate system R ,� ,Z with R=0being the � symmetry axis, since �R= eR and �e�= e�� e� /��=−e��1 /R�eR, one can show that

�e� = ��Re�� = eRe� − e�eR. �53�

Since this is an antisymmetric tensor, its contraction with anysymmetric tensor TS vanishes: �e� :TS=0. Hence, for anysymmetric tensor TS, Eq. �52� simplifies to

e� · � · TS = � · �e� · TS� . �54�

Therefore, since mnVV is a symmetric tensor, the toroi-dal component of the terms on the left side of the momentumequation �2� can be written as

e� · ��/�t�x�mnV� + � · �mnVV��

= �/�t�x�mn�e� · V�� + � · �mn�e� · V�V� . �55�

Thus, the � average of the toroidal �e� ·� component of themomentum equation �2� for each species with all quantitiesexpanded as in Eq. �8� can be written �in laboratory coordi-nates x� to lowest order as

�/�t�x�mn0�e� · V1�� + � · mn�e� · V1�V1

= n0q�e� · EA + V2 · �p� + qn1V1 · �p

+ n0qe� · V1� B − e� · � · � + e� · R + e� · Sm.

�56�

After taking into account the requirement for ambipolarity ofnet radial particle fluxes, the FSA of Eq. �56� will produce anequation for the toroidal angular momentum densitymn0�e� ·V1 and hence �t.

A. Transient magnetic field effects

The poloidal and toroidal magnetic fields in tokamaksevolve during plasma startup and the approach to steady stateon the slow magnetic “skin” diffusion time scale. Also, non-inductive CD and other poloidal flux transient effects, par-ticularly radially localized ones such as electron cyclotronheating �ECH� and CD, can cause transient poloidal fieldeffects on transport time scales of the plasma thermodynamicparameters n, T, and e� ·V1. Thus, we need to allow for tran-sient magnetic flux effects in analyzing Eq. �56�.

To do so, we need to determine our density and toroidalflow equations on poloidal flux surfaces p, upon which driftkinetics and gyrokinetics are based—because Grad–Shafranov equilibria determine the poloidal flux surfaces,and neoclassical transport22 and microturbulence-inducedanomalous transport are determined relative to them. Mag-netic field transient effects were not considered above be-cause they are O��2�; hence they were negligible in obtain-ing the O��0� radial force balance and O�� parallel forcebalance and flows within flux surfaces.

Equations for the slow transport-time-scale evolution ofthe poloidal p and toroidal t magnetic fluxes can be ob-tained from Faraday’s law. Representing the magnetic fieldby B=��A, Faraday’s law can be written as �� A /�t �x−��+EA�=0. Using the vector potential A=t�−p��for the B0 field given in Eq. �3�, the temporal evolution of thetoroidal and poloidal magnetic fluxes at a given laboratoryposition x are

� �t

�t�

x= − e · EA = −

qR2Bp · EA

I, �57�

� �p

�t�

x= + e� · EA −

��

�t= R2 � � · EA −

��

�t. �58�

Here, �� /�t�Vloop� �t� /2� is a spatial constant �gauge vari-

able� that is usually positive �see Eqs. �66�–�68� below� todrive current in the plasma. It represents the toroidal loopvoltage induced in vacuum by the rate of change in magneticflux in the central solenoid �Ohmic transformer� of a toka-mak. In tokamak modeling codes that include flux surfaceevolution,23–25 it is usually taken to be zero; then thetransformer-induced flux change is imposed as a boundarycondition on the plasma domain and causes the poloidal fluxto increase linearly with time for an Ohmically heated toko-mak plasma in steady state. The �� /�t�0 term is intro-duced here so that, in near steady-state conditions, the poloi-dal flux function p�� , t� represents poloidal flux changeswithin the plasma and does not increase approximately lin-early with time t.

Multiplying these equations by 1 /R2, taking their � av-erages and flux-surface-averaging them yields

� �t

�t�

x= −

q�Bp · EAI�R−2

� − uG�t

�� t, �59�

� �p

�t�

x=

�� · EA�R−2

−��

�t. �60�

The small radial “grid” speed uG of toroidal flux surfacesrelative to laboratory coordinates �x� defined in Eq. �59� is23

uG � �uG · �� =�Bp · EAp�I�R

−2. �61�

Since our radial coordinate � is based on the toroidal fluxsurfaces, this grid speed represents the radial speed of ourcoordinate system relative to fixed laboratory coordinates�x�. It is caused by inductive poloidal electric fields producedby the poloidal magnetic field system as it moves the toka-



mak plasma radially or changes its cross-section �e.g., ellip-ticity and triangularity� in the �=constant plane. Note alsothat on a flux surface the time rate of change in the toroidalflux surface differential volume V��dV /d� is given by23,25

the compressibility of the toroidal flux surfaces,

� 1

V�

�V�

�t��

= �� · uG =1

V�

�

��V��uG · �� . �62�

Using the relation Bp��=B0− I��, Eq. �61� canbe rearranged to yield for the toroidal electric field

�� · EA/�R−2 = �B0 · EA/�I�R−2� − uGp� . �63�

The parallel electric field �B0 ·EA in the plasma is obtainedfrom the parallel Ohm’s law in Eq. �41�. The parallel plasmacurrent can be written in terms of p as5,29

�0�B0J� = I�R−2�+p, �64�

in which the second order cylindrical-type operator is

�+p �I

�R−2V�

�

�� 2

R2 �V�

I

�p

��

1

�

�

��

�p

��.

�65�

Substituting Eq. �63� into Eq. �60� and using the neoclassicalparallel Ohm’s law in Eq. �41� and the definition of �B0J� inEq. �64� yield a diffusion equation for the poloidal flux,29

� �p

�t�

x= D� �

+p − S − uGp� . �66�

Here, the magnetic field diffusivity D� and parallel “CDsources” S of poloidal flux are29,36,37

D� ��

nc

�0, �67�

S =��

�t+

1/��

I�R−2��B0 · �Jbs + JCD + Jdyn�� . �68�

As indicated in Eq. �59�, the toroidal flux moves onlyslightly relative to laboratory coordinates due to the usuallysmall grid speed uG. In contrast, the poloidal flux is muchmore mobile; it moves radially in response to nonideal MHDeffects from the parallel electric field �B0 ·EA caused bychanges in ��

nc and �B0J�, the bootstrap current, and nonin-ductive and dynamo CDs. Thus, tokamak modelingcodes23,25 usually use a magnetic flux coordinate systembased on the toroidal flux in the plasma region, as we have inthis paper. Taking into account Eq. �59�, the temporal changein the poloidal flux on a toroidal flux surface is given by

p � � �p

�t�t

= D� �+p − S. �69�

The density and momentum equations in Eqs. �1� and �2�are written in terms of laboratory coordinates �x�. However,drift-kinetic and gyrokinetic analyses used for obtainingcollision- and fluctuation-induced transport fluxes areperformed22 in terms of poloidal magnetic flux coordinatesp , ,�. In particular, they are performed on poloidal mag-

netic flux surfaces, so the guiding center canonical toroidalangular momentum p�g�mv�I /B−qp naturally emerges asa constant of motion.

Transforming the drift-kinetic equation and hence itsfluid moments in Eqs. �1� and �2� from laboratory to p , ,�coordinates requires taking account of the fact that the equa-tion for the poloidal magnetic flux p in Eq. �66� includespossible transient and grid motion effects and poloidal fluxdiffusion. A transformation procedure based on the math-ematical characteristics of the drift-kinetic equation on poloi-dal flux surfaces has been developed;36 also, a simplifiedmodel illustrating its key physics has been presented.37 Inresponse to a Comment38 on the derivation in Ref. 36, amore systematic derivation based on an analysis of the sec-ond order radial guiding center motion was published.39 Thenet result36,37,39 of this coordinate transformation is that a“paleoclassical” diffusion-type operator D�f� is added to theright side of the drift-kinetic equation, which then becomes

� � f

�t�p

+ �v� + vd� · �f + E� f

�E= C�f� + D�f� . �70�

Here, in general we have f = f�p , ,� ,� ,E , t�. For a lowestorder distribution function f that is constant on a flux surface

�i.e., f�� f�p ,� ,E , t��, the FSA of the paleoclassical op-erator can be written using Eq. �7� as39

�D� f�� − �p

� f

��+ �� · fuG +

1

V�

�2

��2 �V�D� f� . �71�

Here, �pindicates motion in the toroidal-flux-based � coor-

dinate as the radial position of a given p surface changesduring poloidal flux transients within the tokamak plasma.Since remaining on a given p surface during transients re-quires dp=0=d��p /��+dt�p /�t, the � motion requiredto remain on the same t toroidal flux surface is36,37,39

�p� p/p� . �72�

The second term in Eq. �71� indicates grid �coordinate� speedeffects as indicated in Eq. �62�. The final term representspaleoclassical transport effects due to the guiding centers ofcharged particles being carried along with the diffusing po-loidal magnetic flux.36,37,39 The normalized magnetic diffu-sivity in Eq. �71� has units of s−1,29

D� �D�a2 ,

1

a2 �1

�R−2� ��2

R2 � . �73�

Here, a is an effective minor radius of the � flux surface.The scaling of the operator D�f� in Eq. �71� is deter-

mined from the scaling of its diffusion coefficient, which isthe magnetic field diffusivity from Eq. �67�: D��

nc /�0

��e�e2 in which �e�c / p is the electromagnetic skin depth.

Since �e is of order the ion gyroradius, the terms D�f� intro-duced in the drift-kinetic equation are O��2�; they have beennegligible up to now because they were not needed in theO��0 ,�1� analyses in Sec. III.



B. Density conservation equation

Before solving the �-average toroidal momentum equa-tion �56� for the collisional radial particle flux n0V2 ·�p, wediscuss the net particle flux that arises in the density conser-vation equation. The relevant density conservation �continu-ity� equation can be obtained from the FSA of the d3v,moment of the drift-kinetic equation in Eq. �70�. First, wenote that25 for a lowest order Maxwellian distribution func-

tion fM�p� that is constant on a flux surface,

�d3v� fM /�t �p= ��n /�t �p

= �1 /V�� /�t �p�V�n0�. Also, the

FSA of the D�f� contribution to the right side of the density

equation is �d3vD� fM�= �D�n0�. Thus, taking the d3vmoment of the drift-kinetic equation in Eq. �70�, averagingover toroidal angle � and taking its FSA using Eq. �7�, wefind that in terms of the toroidal-flux-surface-based coordi-nate �, the O��2� density conservation equation on a poloidalflux surface p is

� 1

V�

�

�t�p

�V�n0� + �p

�n0

��+

1

V�

�

��V�� = �Sn . �74�

This density equation can also be obtained by taking the FSAof the � average of Eq. �1� with all quantities expanded as inEq. �8� and adding �D�n0� to its right side to take account ofits transformation from laboratory �x� to poloidal flux coor-dinates. The quantity dN��V�n0�d� is the number of par-ticles contained between the � and �+d� flux surfaces. Thequantity V�n0 is an “adiabatic” plasma property5,6,23–25 that isconserved in the absence of particle fluxes, density sources,and effects caused by poloidal flux changes induced by �p

�p�0.For each plasma species the particle flux in Eq. �74� is

�� a + �an

na + �pca . �75�

It includes the “direct” second order ambipolar �superscripta� collision-induced flux,

��a � n0��V2 − uG� · ��; �76�

possibly nonambipolar �superscript “na”� anomalous �sub-script “an”� fluxes induced by correlations of first order den-sity fluctuations and fluid flows,

�anna � �n1V∧ · ��; �77�

and the ambipolar paleoclassical particle flux given by

�pca = −

1

V�

�

��V�D�n0� = − D�

�n0

��− n0Vpc. �78�

Here, the paleoclassical particle pinch speed �s−1� is

Vpc =1

V�

�

��V�D�� . �79�

The grid velocity uG has been incorporated into the defini-tion of the collisional part of ��

a in Eq. �76� via �V2 ·��→ ��V2−uG� ·�� because while5,6,22,23 collisional �classical

and neoclassical� transport theory determines particle fluxesacross poloidal flux surfaces, it uses a toroidal flux coordi-nate system ��t� that moves with grid velocity uG.

All the contributions to the particle flux in Eq. �75� areO��2� or smaller in the gyroradius expansion. The units of �are m−3 s−1 �because � is dimensionless and ��1 /a�m−1�. Note also that the particle fluxes are defined relativeto �� here rather than to �p as was used in Ref. 2; thus, the�� ·�p�� ·��p�=�p� in Ref. 2 is p� times the �defined here.

The toroidal force balance equation �56� also needs to betransformed from laboratory to poloidal flux coordinates.Analogous to the effects in the density equation �see discus-sion between Eqs. �69� and �74� above�, the transformationadds a term e� ·D�d3vmvf�=e� ·D�mn0V1� to the right sideof Eq. �56�. Then, since the unit vector e�� / ��=R��does not change with � and hence �e� /��=0, we find that theFSA of this contribution becomes

�e� · D�mn0V1� − �p

�

��mn0�e� · V1�

+ �� · �mn0�e� · V1�uG�

+1

V�

�2

��2 �V�D�mn0�e� · V1� . �80�

Here, the approximate equality indicates that O��2� inverseaspect ratio terms have been neglected in bringing the Rfactor in e��Re� through the � derivatives.

Thus, transforming Eq. �56� to poloidal flux coordinates,taking its FSA, and solving it for the second order radialparticle fluxes for each species yield

p��a + �an

na� � �n0�V2 − uG� · �p + �n1V∧ · �p

= −1

q��e� · R + n0q�e� · EA − �e� · � · ��

−1

q�e� · D�mn0V1�

+� 1

qV�

�

�t�p

�V�mn0�e� · V1�

+1

qV�

�

��V�mn0��e� · V1��V1 · ��

− �e� · n0V1� B −1

q�e� · Sm . �81�

Here, Eq. �7� has been used to obtain the fluctuation-inducedcontribution to the particle flux on the fourth line. Successivelines in Eq. �81� indicate that the radial particle flux hasmany O��2� components: collision-induced fluxes, collision-induced paleoclassical processes, polarization flux due totime-dependent toroidal flows, microturbulence-induced

Reynolds stresses, effects due to the Maxwell stress J� Binduced by fluctuations and NA magnetic field components,and toroidal momentum sources.



C. Ambipolar collisional particle fluxes

Consider first particle fluxes induced by the collisionalfriction force Re nee�J� /�� +J∧/��=−Ri �neglect of po-loidal heat flows changes some numerical coefficients butdoes not affect the qualitative results or scalings�,

p��a � −

1

qs�e� · Rs − n0�e� · EA . �82�

A vector identity relating the toroidal, parallel, and poloidaldirections is useful for analyzing �e� ·Rs,

e� � R2 � � =IB0

B02 −

B0� �p

B02 . �83�

The collisional-friction-induced toroidal torque is thus

1

qs�e� · R =

I

qs�B0 · R

B02 � −

1

qs�B0� �p

B02 · R�

= −ne0I

��B0J�

B02 � +

ne0

�� p�2

B02 � dP0

dp. �84�

The FSA J� term can be worked out using Eqs. �29� and �36�,

�B0J�

B02 � =

�B0J��B0

2+ �J�B0� 1

B02 −

1

�B02��

=��B0 · EA + ��/ne0e��B0 · � · �e�

�B02

+�B0 · JCD + �B0 · Jdyn

�B02

− IdP0

dp� 1

B02�1 −

B02

�B02�2� . �85�

Next, we use this result in Eq. �84� to obtain the totalcollision-induced particle flux from Eq. �75�, which hasseven contributions: classical �cl�, PS, banana-plateau6 �bp�collisionality regime, and paleoclassical �pc� fluxes plusthe radial particle fluxes due to the noninductive and dynamoparallel currents �B0 ·JCD and �B0 ·Jdyn, and theEA�B0 /B0

2 pinch effect,5

��a + �pc

a = �cl + �PS + �bp + �pc + �CD + �dyn + �EA. �86�

Here, for each species we have defined

�cl � −ne0

�� 2

B02 � dP0

d�, �87�

�PS � −ne0I2

��p�2� 1

B02�1 −

B02

�B02�2� dP0

d�, �88�

�bp �I

e�B02p�

�B0 · � · �e� , �89�

�pc � − D�dne0

d�− ne0Vpc, �90�

�CD �ne0I

��p��B02

�B0 · JCD , �91�

�dyn �ne0I

��p��B02

�B0 · Jdyn , �92�

�EA � −ne0

p��e� · EA�1 −

I2�R−2�B0

2 � −I�Bp · EA

�B02 � . �93�

Note that all seven of these particle fluxes are the same forelectrons and ions.5,6,40 Thus, they are ambipolar �superscripta� and cause no FSA radial plasma current,

�Ja · �� s

qs��s�a + �spc

a � = 0. �94�

For a Fick’s law particle flux form �=−�D�n ·��=−D��2�dn0 /d��, the collision-induced diffusion coeffi-cients are of the order Dcl��e�e

2, DPS��q2�� /��Dcl, Dbp

��e00�e2Bt

2 /Bp2 ��q2 /�3/2�Dcl, and Dpc��

nc /�0�Dcl /�e,which is the largest D for the usual situation where �e

� pe / �B02 /2�0��3/2 /q2�1. In addition there are nondiffu-

sive radial fluxes due to the Ware pinch5,6 ��B0 ·EA, from�bp via the parallel viscous force in Eq. �37� using Eq. �41��,paleoclassical pinch flow �79�, outward flows due to nonin-ductive co-CD and dynamo effects, and EA�B0 /B0

2 inwardpinch.5

D. Possibly nonambipolar particle fluxes

Next, consider the particle flux induced by the toroidal�e� ·� component of the viscous force density −� ·�,

p�� 1/qs��e� · � · � . �95�

The viscous stress tensor � was described in Eq. �34� above.The parallel stresses, which are in principle the largest, canbe written in the Chew–Goldberger–Low form

�� = �p� − p��BB/B2 − I/3� . �96�

They can be split into their axisymmetric �superscript A� andNA �superscript NA� parts, which when averaged over thefluctuations can be written as

�� = ��A + ��

NA. �97�

Using equilibrium average quantities in the definition of ��

in Eq. �96� and J0�B0= ��B0��B0 /�0=�P0 yields

� · ��A = B0�B0 · �� p� − p�

B02 � +

p� − p�

B02 � �B0

2

2+ �0P0�

−1

3� �p� − p�� . �98�

Taking the toroidal angular �e� ·� component of this viscousforce, the terms on the second line vanish because of axisym-metry �� /��→0� in the equilibrium. Thus, using Eq. �6�, the



FSA of the toroidal viscous torque in an axisymmetric mag-netic field vanishes,5,6

�e� · � · ��A = I��B0 · �� p� − p�

B02 �� = 0. �99�

Alternatively, since ��A is a symmetric tensor, one can use

the vector, tensor identity in Eq. �54�, and the FSA propertyin Eq. �7� to obtain the same result. Physically, this toroidalviscous force vanishes because there is no toroidal inhomo-geneity in the equilibrium magnetic field strength to impedeflow in the toroidal �axisymmetry� direction.

However, NA �3D� magnetic field components in Eq.�15� cause toroidal torques on toroidally flowing plasmas

that are typically second order in B� and hence in the gyro-radius expansion. They are usually calculated15–17 in the ab-sence of fluctuation and momentum source effects from therelation �e� ·� ·�s�

NA=qsp��s��NA. The net ion torques, which

are typically a factor of order �mi /me�1/2�60 larger than theelectron ones, can be written in the generic form2,14–17,41

�e� · � · �i�NA = mini0�it

Beff2

B02 ��R2�t − �R2�� . �100�

Here, �it �s−1� is an effective ion viscous toroidal dampingfrequency due to transit-time magnetic pumping,14,15

banana-drift,16,41 and ripple-trapping effects �see Ref. 17 and

references cited therein�. Moreover �Beff /B0�2 is a weighted

sum over all m ,n components of B�, which represents theeffective magnetic field NA through which the tokamakplasma flows toroidally.

The intrinsic �or “offset”42,43� rotation frequency

�� R2��/�R2 �cp + ct��1/qi��dTi0/dp� �101�

is a diamagnetic-type toroidal rotation frequency propor-tional to the radial ion temperature gradient. It is caused byions of different energy drifting radially at different speeds.In most NA transport processes, ions on the tail of the iondistribution drift radially more rapidly than thermal ions; thiscauses ��0 for dTi0 /dp�0. The poloidal coefficient cp

in Eq. �101� was discussed after Eq. �45�; the toroidal coef-ficient ct ranges from �0.67 to 2.4, depending on which NAprocess is involved.2,41

Next, we consider the cross and perpendicular stressesand their effects. The cross stress �∧�B0��V is a sym-metric diamagnetic-type stress tensor �see Ref. 30 and refer-ences cited therein� for which an approximate form is givenin Eq. �B1�. The gyroviscous force it induces is given in Eq.�B2�. The FSA of its toroidal angular component vanishes forthe equilibrium first order neoclassical flow velocity V1 forup-down symmetric plasmas,7 which we will assume. �Forup-down asymmetric plasmas it is very small unless there is“extreme asymmetry,” perhaps near a divertor separatrix, inthe PS collisionality regime.7� Also, for collisionless plasmasit vanishes.44 Thus, for equilibrium flows we will assume

�e� · � · �∧A = 0. �102�

However, for fluctuations it does not vanish. That is,�e� ·� ·�∧�0 �see Appendix B�.

The ion perpendicular stresses cause both classical3 andneoclassical4,7 radial transport of the plasma toroidal angularrotation �t through their respective perpendicular viscousforces �e� ·� ·�s�

cl and �e� ·� ·�s�nc . They cause diffusive ra-

dial transport of �t with diffusion coefficients �classical3��t

cl 0.3�i�i2 and �neoclassical4� �t

nc 0.1q2�i�i2. It is impor-

tant to note that the neoclassical toroidal angular momentumdiffusivity �t

nc is a factor of order 0.1�3/2 smaller than the ionneoclassical radial ion heat diffusivity5 �i��q2 /�3/2��i�i

2. Inaddition, neoclassical radial transport of toroidal momentumincludes a momentum pinch type effect4,7 that is comparablein magnitude to the diffusive radial transport of toroidal mo-mentum. In most tokamak plasmas of fusion interest, theclassical and neoclassical toroidal angular momentum diffu-sivities �t are less than 0.1 m2 /s; hence the radial transportof �t they induce is usually negligible. The combination ofall these cross and perpendicular classical and neoclassicalviscous stress effects is discussed in detail in Ref. 7.

Paleoclassical processes due to the �e� ·D�mn0V1� de-fined in Eq. �80� also produce radial transport of toroidalangular momentum. The D�-induced part of electron and ionpaleoclassical effects can be characterized as perpendicularviscous torques on the toroidal angular momentum density ofeach species. The ion paleoclassical viscous torque is largerthan that for electrons by mi /me�3672�1. It is caused bythe ion D�-induced term in Eq. �80�,

�e� · � · �i�pc −

1

V�

�2

��2 �V�D�Lt�

= −1

V�

�

��V��D�

�Lt

��+ LtVpc�� , �103�

in which the plasma toroidal angular momentum �torque�density is

Lt � mini0�e� · Vi = mini0�R2�t . �104�

The ion term −�p�� /��mini0�e� ·Vi in Eq. �80� leads to an

additional toroidal torque density on the right side of Eq.�56� of −�p

��Lt /��. It is caused by poloidal magnetic field

transients in the plasma that produce p�0 and is analogousto the �p

�n0 /�� term in Eq. �74�. The �� ·mini0�e� ·Vi�uGion term in Eq. �80� contributes, as in particle fluxes, toproperly determining the neoclassical perpendicular viscositydiscussed above when there is grid motion of the toroidalflux surfaces.

Paleoclassical radial momentum transport in Eq. �103�causes radial diffusion of Lt with a diffusivity �t

pc�D�, plusa momentum pinch Vpc that is the same as that for the den-sity, i.e., Eq. �79�. The paleoclassical momentum diffusivityis the same as the magnetic field diffusivity: D��

nc /�0

�1400Zeff /Te�eV�3/2, which becomes less than about0.1 m2 /s for Te�1 keV. While it can be significant in



Ohmic-level tokamak plasmas and toward the plasma edge,it is usually negligible in the hot plasma core unlessfluctuation-induced transport is suppressed.

Taking into account all of the classical, neoclassical, andpaleoclassical processes of radial transport of �t, the totalcollision-induced perpendicular viscous torque is

�e� · � · �s� � �e� · � · ��s�cl + �s�

nc + �s�pc �

�e� · � · �s�pc . �105�

Combining this with the parallel viscous force due to 3D NAmagnetic field components, the radial particle fluxes inducedby � and � viscous forces become

p�� 1/qs��e� · � · �s�NA , �106�

p�� 1/qs��e� · � · �s� . �107�

The other particle fluxes in Eq. �81� are induced by po-larization flows, perturbation-induced Reynolds and Max-well stresses, poloidal magnetic flux transients, and momen-tum sources,

p��pol �1

qsV��

�t�p

�V�msns0�e� · Vs1� , �108�

p��Rey �1

qsV�

�

��V� s�� , �109�

p��Max � − �e� · ns0V1� B , �110�

p��J�B � − �e� · ns0V�mnNA � B�mn

NA , �111�

p��p��p

qs

�

��mn0�e� · V1� , �112�

p��S � − �1/qs��e� · Ssm . �113�

The toroidal Reynolds stress s�� has been considered influid models9,10 but is now usually obtained fromgyrokinetic-based theories.11–13 The definition of it thatemerges from this fluid-based approach is

s�� msns0�� · Vs∧��Vs · e�� + �� · �∧ · e� . �114�

The gyroviscous stress �∧ in the last term is described inAppendix B and an approximate form of it is specified in Eq.�B4�; this term provides a partial gyroviscous cancellation�see discussion before Eq. �B2�� of the first term in Eq. �114�.Also, we have split the V� B contributions into afluctuation-induced Maxwell stress part �Max and a part �J�Binduced by externally imposed NA resonant �at q=m /n�magnetic perturbations, which will be discussed after Eq.�120� below.

The sum of the rest of the particle fluxes in Eq. �81�, inaddition to those included in Eq. �86�, for each species isthus

�anna = �� + �� + �pol + �Rey + �Max + �J�B + �p

+ �S.

�115�

These eight particle flux components are all, in principle,different for electrons and ions and hence nonambipolar �su-perscript na�.

E. Radial current and toroidal rotation equation

Next, we consider the FSA of the plasma current conti-nuity equation obtained by multiplying the FSA densityequation �74� by the charge qs and summing over plasmaspecies ��q��sns0qs�,

� 1

V�

�

�t�p

�V��q� + �p

��q��

+1

V�

�

��V��J · �� = 0.

�116�

Here, V��J ·��sqsV��s is the net FSA plasma current �inamperes� flowing radially across a magnetic flux surface. Forsimplicity, we have assumed that the particle sources are

ambipolar and hence that �sqs�Ssn=0, because inside toka-mak plasmas the electron and ion density sources Ssn areusually equal and hence ambipolar. Thus, using the FSA ofGauss’s law ��0� ·E0= ��q� and Eq. �7�, this last equationbecomes

1

V�

�

��

�t�p

V��0�E0 · �� + V��J · ��+ �p

�

�� 0

V�

�

��V��E0 · �� = 0. �117�

The radial electric field contribution here is �E0 ·��= ��2��0 /��. It is proportional to one of the key terms inthe toroidal rotation �t given in Eq. �47�. Hence, the radialion current due to the ion polarization particle flow �ipol inEq. �108� includes a term that is of the form �� /�t��mini0�R2��0 /�p� in Eq. �117�. This term is larger thanthe vacuum field term ��0E0 ·�� /�t in Eq. �117� by afactor of mini0�R2 / ��0p�

2��2� c2 /cAp2 �1, in which

cAp�Bp /��0mini0 is the Alfvén speed in the poloidal mag-netic field. The electric field term proportional to �p

is of asimilar magnitude relative to this inertial electric field termfrom �ipol. Thus, since typically c2 /cAp

2 �105�1, both of the�E0 ·�� electric field terms in Eq. �117� are negligible com-pared to the neoclassical MHD polarizability45 arising fromthe �� /�t��mini0�R2��0 /�p� term in �Lt /�t. This inertialterm produces the neoclassical perpendicular dielectric��=c2 /cAp

2 for this plasma when the poloidal flow is in equi-librium on the transport time scale, for which the equilibriumpoloidal flow is given in Eq. �46�.

Thus, as in stellarator neoclassical transporttheory,15,27,28 since the �E0 ·�� terms are negligible in Eq.�117�, this equation requires that there should be no net radialcurrent in the plasma on the transport time scale,



0 = �J · �� = �s

qs�s = �s

qs�sanna . �118�

As indicated, only nonambipolar particle fluxes cause netradial currents. Setting to zero the net radial plasma currentcaused by contributions produced by all eight nonambipolarparticle fluxes in Eq. �115� for both species yields

1

V�

�

�t�

�p

�V�Lt�

inertia

� − �e� · � · �i�NA�

non-res. NA B�

− �e� · � · �i��

cl,neo, paleo

−1

V�

�

��V��i��

ion Reynolds stress

+ �e� · J � B�

Maxwell stress

+ �e� · J�mnNA � B�mn

NA �

resonant NA Bmn

− ��p� Lt/��

�p�0 transient

+ �e� · �sSsm�

mom. sources

.

�119�

This comprehensive conservation equation for the toroidalangular momentum Lt�mini0�R2�t of a tokamak plasma isthe primary result of this paper. Here, the approximate equal-ity indicates that we have neglected the electron �, � vis-cous, and Reynolds stresses, which are usually smaller by afactor of the order of �me /mi�1 /60. Similar to the density,V�Lt represents the plasma toroidal angular momentum be-tween the � and �+d� flux surfaces; it is also an adiabaticplasma property that vanishes in the absence of dissipativeprocesses and momentum sources.

The Maxwell stress contributions in Eq. �119� are in-duced by microturbulence. They can be made more explicitby using Ampere’s law for the fluctuations and the vectoridentity C� ��C�=��C�2 /2�−C ·�C:

�e� · J� B = �e� ·1

�0�� B�� B�

= �e� ·1

�0B · �B� . �120�

However, for radially local fluctuations that do not connectto “external” magnetic fields, the average of this contributionover a narrow radial region of the plasma vanishes �see Eq.�28� and discussion in Ref. 12�.

The �e� · J� B toroidal torque vanishes for ideal MHDperturbations throughout the hot core of tokamak plasmas.18

However, in the vicinity of a low order rational surface �e.g.,q=m /n�, nonideal effects can allow an externally imposedNA magnetic field component to induce a finite parallel cur-

rent J�mn and nonzero B�mn within the thin nonideal bound-ary layer18,19 around the low order rational surface. This pro-

cess produces the toroidal torque �e� · J�mnNA� B�mn

NA in Eq.�119� in the vicinity of low order q=m /n rational surfaces ina toroidally rotating plasma.

F. Rotation, radial electric field, and net particle flux

The FSA toroidal plasma rotation frequency ��t is de-fined in terms of Lt by

��t �Lt

mini0�R2=

�R2�t�R2

=�∧ +�R2��p

�R2= − �d�0

dp+

1

ni0qi

dpi0

dp� +

I

�R2Ui

−1

p��d�0

d�+

1

ni0qi

dpi0

d�−

cp

qi

dTi0

d�� . �121�

Thus, determination of Lt and hence �R2�t from a solutionof Eq. �119� also determines the radial electric field,

E� ��

��· E0 = − ��

d�0

d�

= ��tp� +1

ni0qi

dpi0

d�−

I

�R2Uip��

��tp� +1

ni0qi

dpi0

d�−

cp

qi

dTi0

d�� . �122�

The �� factor accounts for the bunching together ��1 /a� of poloidal flux surfaces on the tokamak’s outboardside but stretched separations ��1 /a� on the inboardside due to the Shafranov shift, etc.

The resultant E� �or ��t� causes the electron and ionnonambipolar radial particle fluxes to become equal �i.e.,ambipolar� and produce no net radial current,

�eanna �E�� = Zi�ian

na �E�� ⇒ �Janna�E�� · �� = 0. �123�

Note that obtaining no net radial current in the plasma is thesame as determining ��t �or E�� from the evolution equationfor Lt given in Eq. �119�. Because Eq. �118� and hence Eq.�119� are determined predominantly by the sum of the non-ambipolar ion particle fluxes, the radial electric field E� ob-tained here is what is referred to as the “ion root” in thestellarator literature.46

Thus, from Eq. �75� the net ambipolar radial particle fluxis the sum of the intrinsically ambipolar fluxes ��

a+�pca and

the nonambipolar fluxes �anna evaluated at E�,

�enet = �e�

a + �epca + �ean

na �E�� = �inet. �124�

As indicated, since the nonambipolar ion particle fluxesmostly sum to zero to yield �J ·�� eZi�ian

na �E��0, it iseasiest to determine the net ambipolar particle flux using thenonambipolar electron fluxes.

Since electron perpendicular-viscosity-driven classicaland neoclassical particle fluxes are me /mi�1 /3672�1smaller than the corresponding already small ion ones, thelowest order electron particle flux is likely to be determinedby the paleoclassical particle flux ��pc� plus the electronReynolds and Maxwell stress particle fluxes,



�enet �pc

a + �eReyna �E�� + �eMax

na �E�� = − D�dne0

d�− ne0Vpc

−1

e

1

p�V�

�

��V� e�� −

1

p��e� · ne0Ve� B . �125�

However, electron particle fluxes due to parallel viscousforce effects from NA magnetic fields �via �e�� and momen-tum source effects �via �eS� may be important for particulartokamak plasma situations. Also, during poloidal magneticflux transients, the FSA density equation �74� has the addi-

tional contribution �p�ne0 /��= �p /p��ne0 /��. This

added ambipolar radial motion effect could lead, for ex-ample, to the “density pump-out” effect �see, for example,Ref. 47� as ECH is applied to a tokamak plasma, locallyrapidly increases Te, reduces D��

nc�1 /Te3/2, and causes

�p�p�0 and hence �ne0 /�t�−�p�ne /��0 there.

V. DISCUSSION

The seven contributions on the right of the toroidaltorque equation �119� have a wide variety of effects. In thefollowing we describe briefly for each contribution, in orderof its appearance in Eq. �119�, the parameter regime in whichit is likely to be significant, and if it were dominant whateffect it would have on the plasma toroidal rotation fre-quency ��t.

�1� NTV due to NA fields. When field errors are very large orlarge NA control fields are deliberately applied, the neo-classical toroidal viscous torque in Eq. �100� acts as adrag throughout the plasma. It relaxes the plasma toroi-dal rotation toward an “intrinsic” or offset42,43 rotationfrequency in the “counter” �to the current� directiongiven by Eq. �101�: ��t �� cp+ct��1 /qip��dTi0 /d��0.

�2� Collision-induced perpendicular viscosities. In plasmaswith approximately Ohmic-level heating and near theplasma edge where Te�1 keV, and when fluctuation-induced transport is suppressed, paleoclassical perpen-dicular viscosity in Eq. �103� may be dominant andcause ��t 0 through a combination of radial diffusionof Lt with diffusivity D� from Eq. �67� and an inwardpinch Vpc from Eq. �79�.

�3� Fluctuation-induced ion Reynolds stress. Various prop-erties of microturbulence can lead8–13 to radial diffusion,plus nondiffusive typically co-current intrinsic rotation�or “pinch velocity”�12,48–53 and “residual stress”13,54 ef-fects that do not depend on either ��t or its radial de-rivative.

�4� Fluctuation-induced Maxwell stress. This tends to besmall for low � plasmas, but its electron componentcould be important in the radial particle flux.

�5� Resonant NA magnetic fields. They produce a toroidaltorque in the vicinity of low order resonant rational sur-faces where q=m /n. Toroidal flow inhibits penetrationof resonant field errors into the plasma by producing ashielding effect on the rational surfaces. Above a criticalfield error amplitude—termed the penetration

threshold—the plasma rotation can no longer suppressthe resonant torque, and plasma rotation at an m /n ra-tional surface rapidly �in a few milliseconds� locks to thewall �laboratory frame�.18,21 Then, the diffusive radialmomentum transport due to perpendicular collision-induced viscosity and the ion Reynolds stress slows to-roidal rotation throughout the plasma on a fraction of theglobal momentum confinement time scale.55 The net re-sult is then a toroidal rotation profile that is locked to thewall on low order rational surfaces with some residualtoroidal rotation between rational surfaces if the plasmatoroidal torque source �e� ·�sSsm is large.

�6� Poloidal magnetic flux transients. Poloidal flux tran-

sients that cause p�0 move Lt�mini0�R2�t radially

at a speed d� /dt= �p= p /p�. The rotation frequency

��t typically decreases with minor radius; thus, fromEq. �72� one usually has �Lt /��0. If, in addition,

p�0 �e.g., due to EC heating or turn-on of a non-inductive co-current source�, this effect dynamicallyreduces the plasma toroidal angular momentum onthe magnetic field diffusion time scale:

�1 /V�� V�Lt� /�t �p −�p

�Lt /�� D��Lt /��0.�7� Momentum sources. A net toroidal momentum source

�e� ·�sSsm applies toroidal torque to the plasma. For ex-ample, unidirectional tangential energetic NBI used toheat tokamak plasmas also causes a toroidal torque onthe plasma via the injected fast ions collisionallytransferring26 their momentum to the backgroundplasma—in the direction of the NBI. In steady state thistorque on the plasma is usually balanced by thefluctuation-induced ion Reynolds stress discussed in �3�above. Similar considerations apply to the radial currentinduced by “energetic” electrons involved in lower hy-brid CD �LHCD� experiments.56 In LHCD, the wavemomentum input transferred to the energetic electronsprovides a plasma momentum source via collisionaltransfer of the momentum in energetic electrons to thebackground plasma. This LHCD plasma momentumsource is in the direction opposite to the wave momen-tum input because the negatively charged energetic elec-trons cause a current in the opposite direction from thewave momentum input. Thus, co-current LHCD pro-duces a plasma torque in the counter-current direction.

In the preceding discussion, particularly point �7�, it wasassumed that the only effects of the sources and sinks arethrough their particle and momentum inputs. However, fortangential NBI the mechanical momentum source from thecollisional slowing down of the injected untrapped fast ionsonly comprises26 a fraction �1−� of the total momentuminput, because at birth the v� of the NBI-injected fast ions isless than their toroidal velocity. The remaining �� fractionresults from the change in the fast ion guiding center canoni-cal toroidal angular momentum p�g�mv�I /B−qip causedby the difference in their birth radius from the bounce aver-age of their radial guiding center position: �� f=p�� f. For a fast ion injection rate26 nf this causes aradial fast ion current �J f ·�� nfqf�� f. At the plasma



edge this radial current can also be induced by direct fast ionlosses to the surrounding vacuum vessel wall.

Equation �118� did not include this fast-ion-induced cur-rent. However, it should be added there,57 in which case thisequation becomes �J ·��+ �J f ·��=0. Multiplying Eq. �51�by nq, we see that J ·�p= �J ·��p�=e� ·J�B RJrBp.Thus, the toroidal torque the fast ion radial current inducescan be accommodated in the formalism of this paper by in-cluding an extra fast ion momentum source,57

�e� · S fm � − �J f · ��p� − RBpnfqf�� f. �126�

This is positive for tangentially co-injected ions that on av-erage drift inward of their birth radius and cause �� f�0.This extra toroidal angular momentum input due to a fast-ion-induced radial current causes a “return current” in theplasma:57 �J ·��=−�J f ·��.

For tangential NBI this momentum input is typically asmall fraction �� of the direct momentum input and hencecan usually be neglected; however, for near-perpendicularNBI, this fast-ion-induced radial current momentum sourceeffect can be dominant. Similar considerations apply to theradial current induced by the energetic electrons involved inLHCD �Ref. 56� due to the average radial guiding centermotion of the “extra species” of energetic electrons. In addi-tion, a probe inserted into a plasma58 that draws or emits acurrent Iprobe�V��Jprobe ·�� produces an analogous toroidalmomentum source �e� ·Spm=−�Iprobe /V��p� at the radial po-sition of the probe.

VI. SUMMARY

Three independent components of the plasma force bal-ance have been considered on sequential time scales: �1�plasma MHD force balance �16� on the fast Alfvén timescale, which leads to a fundamental relation �17� betweentoroidal and poloidal flows, and E�B and ion diamagneticflows; �2� plasma parallel force balance �43�, which on theintermediate ion collision time scale leads to specification�46� of the ion poloidal flow in terms mainly of the radial iontemperature gradient; and finally, �3� toroidal force balance�56� on the slow transport time scale, which leads to radialparticle fluxes.

The transport-time-scale density equation on a poloidalflux surface is specified in Eq. �74�. The net particle flux�� ·�� is composed of seven ambipolar collisionalfluxes �86� due to classical, PS, banana-plateau neoclassical,paleoclassical, CD, dynamo current, and EA�B /B2 pinchtransport processes plus eight nonambipolar fluxes �115� dueto parallel �NA� and perpendicular viscosities, polarizationflows, fluctuation-induced Reynolds and Maxwell stresses,externally imposed resonant NA magnetic fields, poloidalflux transients, and momentum source effects.

Finally, a new comprehensive evolution equation forthe plasma toroidal angular momentum density Lt

�mini0�R2�t in Eq. �119� has been obtained by requiringthe net radial current to vanish on the transport time scale.

The net radial current is determined by summing over spe-cies current contributions from all eight nonambipolar par-ticle fluxes. The toroidal rotation frequency ��t in Eq. �121�and hence radial electric field in Eq. �122� are obtained fromLt. For this radial electric field, the net electron and ion par-ticle fluxes become equal �ambipolar� �Eq. �124�� and aredetermined mainly by the electron particle fluxes indicated inEq. �125�.

This paper has also developed a general self-consistentframework for taking account of all the anomalous transporteffects of microturbulence on tokamak plasmas. These ef-fects are explicitly included in equations for the parallelOhm’s law �36� and �40�, poloidal ion flow �46�, particlefluxes �109� and �110�, momentum flux �114�, and plasmatoroidal angular momentum �119�.

ACKNOWLEDGMENTS

The authors are grateful to K. C. Shaing for many usefulcollaborations and discussions over the years and to theirDIII-D colleagues and S. A. Sabbagh �Columbia University/NSTX� whose experimental results stimulated this researchin large part. They are particularly grateful to A. M.Garofalo, K. H. Burrell, J. S. deGrassie, M. J. Schaffer, andE. J. Strait for many stimulating discussions about toroidalflows and their recent experimental tests �Refs. 42 and 43� ofone of our key predictions. They also gratefully acknowledgeuseful discussions of a preliminary manuscript draft with N.M. Ferraro, S. C. Jardin, J. E. Rice, and R. E. Waltz and ofECH density pump-out effects with R. Prater and F. Volpe.Finally, they thank R. J. Groebner for careful reading of thedraft manuscript.

This research was supported by U.S. DOE under GrantNos. DE-FG02-86ER53218 and DE-FG02-92ER54139.

APPENDIX A: FLUID AND GUIDING CENTER FLOWS,FLUXES

Since the first order perturbed flows have a nonzero ra-dial component, they lead to the anomalous radial particle

flux �n1V∧ ·��. The species fluid flow velocity V∧�VE

+V� is related to the guiding center flow velocity usuallycalculated in drift kinetics and gyrokinetics via3 nq�VE

+V��=nqVg+��M. Here, Vg�d3vvdfM /n is the flow ve-locity of particle guiding centers having drift velocities vd,and M�−pB /B2 is the magnetism produced by the mag-netic moments of the charged particles. Since to lowest order

in ��2�0p /B02�1 and the gyroradius parameter �, vd E

�B0 /B02, and ��M B0��p /B0

2, we obtain

Vg =� d3vvdfM/n0 VE, �A1�

VM = �1/n0q� � � M V�. �A2�

Thus, V∧= Vg+ V�+O�� ,�2�. Moreover in general, since



� ·��M=0, we have �1 /V�� /��V��n1V∧�= �� · nV∧= �� · nVg��1 /V�� /��V��g�, where �g is thefluctuation-induced particle flux evaluated in gyrokineticcodes �see, for example, Ref. 12�. Also, since E�B flowsare the same for electrons and ions, the net particle fluxes �g

determined from guiding center flows are ambipolar to low-est order.

APPENDIX B: GYROVISCOSITY AND ITS EFFECTS

The gyroviscous stress �∧ is a symmetric tensor that iscaused by diamagnetic-type B��V effects. It is describedin detail in Ref. 30. The gyroviscous stress tensor can bewritten to lowest order �in � and neglecting heat flux effects�for illustrative purposes as3

�∧ = �p/4 c��W∧ + W∧T� . �B1�

Here, W∧� b�W · �I+3bb� in which b�B /B, I is the iden-tity tensor, W=�V+ ��V�T− �2 /3�I�� ·V�, and the super-script T indicates the transpose of that tensor. For this formof the gyroviscous stress where the heat flux effects are ne-glected, to lowest order in � and � the gyroviscous force canbe written as30

− � · �∧ mnV� · �V = �m/q� � �M · �V . �B2�

This gyroviscous force cancels the V� ·�V part of V ·�V inEq. �33� and causes it to become approximately VE ·�V,which has been speculated59 to be in general Vg ·�V.

Since the �-average fluctuation-induced gyroviscousstress �∧ is a symmetric tensor, using Eq. �54� we find ingeneral that

�e� · � · �∧ = �� · �e� · �∧� =1

V�

�

��e� · �∧ · �� . �B3�

To lowest order in �, neglecting heat flux effects and gradi-ents of equilibrium quantities, the gyroviscous stress inducedby fluctuations is

�∧ �1/4 c0�p�W∧ + W∧T� . �B4�

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