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Transport equations in tokamak plasmas a

J. D. Callen, b C. C. Hegna, and A. J. ColeUniversity of Wisconsin, Madison, Wisconsin 53706-1609, USA

Received 20 November 2009; accepted 29 December 2009; published online 8 April 2010

Tokamak plasma transport equations are usually obtained by ux surface averaging the collisionalBraginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc termsare added for neoclassical effects on the parallel Ohms law, uctuation-induced transport, heating,current-drive and ow sources and sinks, small magnetic eld nonaxisymmetries, magnetic eldtransients, etc. A set of self-consistent second order in gyroradius uid-moment-based transportequations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-basedapproach. The derivation uses neoclassical-based parallel viscous force closures, and includes all theeffects noted above. Plasma processes on successive time scales and constraints they impose areconsidered sequentially: compressional Alfvn waves GradShafranov equilibrium, ion radial forcebalance , sound waves pressure constant along eld lines, incompressible ows within a uxsurface , and collisions electrons, parallel Ohms law; ions, damping of poloidal ow . Radialparticle uxes are driven by the many second order in gyroradius toroidal angular torques on aplasma species: seven ambipolar collision-based ones classical, neoclassical, etc. and eightnonambipolar ones uctuation-induced, polarization ows from toroidal rotation transients, etc. .The plasma toroidal rotation equation results from setting to zero the net radial current induced by

the nonambipolar uxes. The radial particle ux consists of the collision-based intrinsicallyambipolar uxes plus the nonambipolar uxes evaluated at the ambipolarity-enforcing toroidalplasma rotation radial electric eld . The energy transport equations do not involve an ambipolarconstraint and hence are more directly obtained. The mean eld effects of microturbulence on theparallel Ohms law, poloidal ion ow, particle uxes, and toroidal momentum and energy transportare all included self-consistently. The nal comprehensive equations describe radial transport of plasma toroidal rotation, and poloidal and toroidal magnetic uxes, as well as the usual particle andenergy transport. 2010 American Institute of Physics . doi :10.1063/1.3335486

I. INTRODUCTION

Transport equations for describing tokamak plasmas

were initially advanced in the late 1970s. They were devel-oped by adapting the collisional Braginskii 1 transport equa-tions for plasma density and energy transport to the tokamak toroidal geometry. In addition, many other terms have beenadded over the years in an ad hoc manner to include otherprocesses important in tokamak plasmas: neoclassical effectson the parallel Ohms law 2,3 trapped particle effects on theresistivity, bootstrap bs current , anomalous plasmatransport induced by microturbulence, auxiliary heating,current-drive CD and ow sources and sinks, effects of small magnetic eld departures from axisymmetry, etc. Theresultant transport equations form the basis for ONETWO ,4

TRANSP ,5

and other transport modeling codes that are used tomodel present and future tokamak plasma experiments.

However, most tokamak plasmas are not in a collisionalregimecollision lengths are typically much longer than thetoroidal circumference of the tokamak. More rigorous andself-consistent plasma transport equations need to be devel-oped from a kinetic-based approach. Also all the effects

indicated in the preceding paragraph need to be includednaturally and self-consistentl y.

In this paper we develop 6,7 self-consistent radial plasma

transport equations from velocity-space uid moments of akinetic-based approach that includes all these effects fornearly axisymmetric tokamak plasmas using neoclassical-based parallel viscous force closures. 3 This new approachaccounts for lower order radial force balance constraints andsolves for the ows within ux surfaces before determiningthe net radial transport equations. This procedure is analo-gous to that used for determining radial neoclassical plasmatransport equations in nonaxisymmetric stellarator plasmassee for example Ref. 8 and references cited therein. This newapproach was originally developed 6,7 for determining the to-roidal rotation in tokamak plasmas. This paper extends that

analysis to include heat transport equations and takes intoaccount the poloidal heat ows that were neglected there.For additional details see Refs. 6 and 7.

This paper is organized as follows. Section II discussesthe fundamental kinetic equation from which the analysisbegins and the resultant uid moment equations that areused. Also, the key assumptions, small gyroradius expansionused for ordering various terms and lowest order radial forcebalance considerations are presented there. Section III de-scribes the determination of the rst order plasma owswithin ux surfaces. The transient evolution of poloidal and

aPaper NI3 1, Bull. Am. Phys. Soc. 54 , 180 2009 .

b Invited speaker. Electronic mail: [email protected]. http:// www.cae.wisc.edu/~callen .

PHYSICS OF PLASMAS 17 , 056113 2010

1070-664X/2010/17 5 /056113/9/$30.00 2010 American Institute of Physics17 , 056113-1

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toroidal magnetic uxes and their effects on plasma transportare described in Sec. IV. The radial parti cle uxes induced bythe many toroidal torques on the plasma 7 are summarized inSec. V. Also, the toroidal plasma rotation equation for atokamak 6,7 obtained by setting the net radial current inducedby the nonambipolar uxes to zero to enforce plasmaquasineutrality is summarized there. This key equation alsodetermines the radial electric eld and the resultant ambipo-

lar radial particle ux. Section VI develops the electron andion energy transport equations. Finally, the results of thispaper are summarized in Sec. VII which also discusses somekey new features of this comprehensive transport model.

II. BASIC EQUATIONS AND ASSUMPTIONS

We begin with a plasma kinetic equation PKE that in-cludes the FokkerPlanck Coulomb collision operator C f and kinetic sources S f for each plasma species,

f

t x+ v f +

q

mE + v B

f

v= C f + S f . 1

Fluid moment equations for the species density n, owvelocity V, temperature T , and conductive heat ow qare obtained by taking velocity-space moments

d 3v 1 , mv , mv 2 / 2 , v mv 2 / 2 of this PKE,

n t x

+ nV = S n , 2

t xmnV + mn VV

= nq E + V B p + R V + SV , 3

32

p t x

+ q + 52 pV = Q + V p : V + S E ,4

q t x

=q

mq B

T

m

52

n T + R q

Vq + qV + 23 V qI+

52

pV V + Sq + . 5

As indicated, in this conservative form of these equations theuid moment properties are evaluated at a laboratory posi-tion x. Here, the species pressure is p nT , and RV and Rqare the collisional friction and heat friction forces. Thedensity, momentum, net energy, and net heat ow sources areS n, SV , S E

S E V SV mV 2 / 2 S n, and Sq Sq 5T / 2m

SV mVS n . The at the end of Eq. 5 indicates higherorder terms that will be neglected here. Th e remaining nota-tion and denitions are relatively standard. 13 The uid mo-ments required to close these equations are the stress tensor and heat stress tensor ; the needed parallel viscousforces they induce are given in Sec. III.

A number of assumptions are made to facilitate theanalysis. 1 Particle gyroradii are small which to zeroth or-der yields magnetohydrodynamic MHD force balance equi-librium and the radial ion force balance, ow on ux sur-faces at rst order and second order radial transport uxes.2 Magnetic ux surfaces are nested and toroidally axisym-

metric to lowest order i.e., there are no magnetic islands inthe region of interest . 3 Nonaxisymmetries NA in themagnetic eld are rst order in the gyroradius, which causesthe toroidal ow damping rate to be one order smaller thanthe poloidal ow damping rate. 4 Collision lengths are longcompared to the plasma toroidal circumference so plasmaproperties are constant on magnetic ux surfaces, which isvalid for most tokamaks except possibly in a cold plasmaedge. 5 Electron and ion distribution functions areMaxwellian to lowest order in the gyroradius expansione.g., there are no radio-frequency-wave-induced zeroth or-

der distribution distortions . 6 Microinstability-inducedplasma uctuations are gyroradius small and lead mostly tosecond order anomalous plasma transport across ux sur-faces. 7 Impurity ow velocities are approximately equal tohydrogenic ion velocities, which simplies inclusion of im-purity effects. Finally, 8 magnetic eld transients are slowenough that they only occur on the plasma transport orlonger time scale.

A small gyroradius expansion is used to order the vari-ous terms in the uid moment equations: / a 1 inwhich is the particle gyroradius in the magnetic eld and ais the minor radius of the plasma. For example, the plasmapressure is expanded as

p x = p0 + p 1 , + p 1 , , + O 2 . 6The lowest order species pressure p0 is only a function of the

magnetic ux surface label . The rst order toroidal-angle-averaged pressure p 1 , represents the PrschSchlterpoloidal pressure variation on a ux surface. Finally, p 1 , , represents the gyroradius-small nonaxisymmetricperturbations induced by externally imposed magnetic eldsor collective plasma instabilities.

Because of lowest order axisymmetry in the toroidalangle , perturbations will be expanded in a Fourier series: p 1 = n p nein , p n 1 / 2 02 d e in p 1. The n=0 term de-nes the average over toroidal angle,

p x 1

2 0

2

d p x = p0 + p 1 , + O 2 . 7

Perpendicular scale lengths of uctuations will be presumedto be on the gyroradius scale. Hence, perpendicularderivatives of uctuations will be assumed to be large:

p 1 1 / 0. However, parallel derivatives of uc-tuations will be assumed to be on the macroscopic ascale and hence small: p 1 0 . Derivatives of equi-librium components p0, p 1 will be O 0 , .

The magnetic eld will be represented by an averageB0 B t + B p= I + p= p q = t p plus small O perturbations B

due to three-dimensional 3D NA and collective responses: B = B0 , + B + B + O 2 , B B0 , + B + O 2 . Here,

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B0 eB0 e

mene

eeMe

B02 U e

B02 Q e

. 18

The collisional friction and heat friction forces are 3

B0 R eVB0 R eq = mene ee Ne

1nee

B0 J

2

5neT eB

0 q

e

. 19

The reference electron-electron collision frequency is 3

1

ee

4

3 4 nee

4 ln

4 02me

2v Te

3 5 1011 ne m

3

T e eV3/ 2 . 20

In the asymptotic e 1 banana collisionality regime thesymmet ric ma trix of dimensionless electron viscosity coef-cients is 3,10,11

Me e00 e01 e01 e11

= f t f c

0.53 + Z eff 0.62 + 32 Z eff

0.62 + 32 Z eff 1.39 + 13

4 Z eff .

21Here, f t 1 f c is the fraction of trapped particles

3 inwhich f c 3 / 4 B0

201 / Bmax d / 1 B0 11.46

+0.46 Ref. 10 is the fraction of circulating particles.For an impure plasma Z eff = in i Z i

2 / ne is the effective ioncharge. Multicollisionality forms of Me are given in Refs. 3,10, and 11. The symmetric matrix of electron friction forcecoefcients is 3,10,11

Ne e00 e01 e01 e11

= Z eff

32 Z eff

32 Z eff 2 + 134 Z eff

. 22

Finally, the FSA parallel external forces and heat forces onthe electrons are

B0 F eVB0 F eq = ne0e B0 E

A + B0 SeVtot

me/ T e0 B0 Seqtot . 23

The sources of electron momentum and heat ow are com-posed of both direct and uctuation-induced sources:B0 SeV

tot B0 S eV + B0 F eV

fl and B0 S eqtot B0 Seq+ B0 F eq

fl . The net electron noninductive momentum source

is B0 S eV = B0 SeV meV eS en and B0 Seq is the net

noninductive source of electron heat ow. The uc-tuation-induced electron momentum source is B0 F eVfl

B0

men

e0V

e V

e+

e n

e0e B

0 V

eB and

the analogous heat ow source is

B0 F eqfl B0 V

eq e + q eV e + 2 / 3 V e q eI

+ 5 pe0/ 2 B0 V e V

e ,

which involves many apparently small, rarely calculated ormeasured quantities.

The uctuating electron ow V e and heat ow q e hereare uctuating uid ows in the laboratory frame of refer-ence. As discussed in Appendix A of Ref. 7, the species uidow V is related to the guiding center ow Vg d 3v vd f / ninduced by the guiding center drift velocity vd that is usually

calcula ted in drift-kinetic and gyrokinetic calculationsthrough 1 V = V g + 1 / nq M . Here, 1 / nq M V= B0 p / nqB 2 is the diamagnetic ow induced by theplasma magnetization M pB / B2 produced by the mag-netic moments of the charged particles in the plasma. Theuid heat ow q is related to the guiding center heat owsimilarly.

The electron parallel viscous force can be written in

terms of the parallel current using the electron poloidal owfunction in the form

nee B02 U e = B0 J + nee B02 U i I

dP 0d p

. 24

Using this result and the denitions in the preceding para-graphs, the matrix equation in Eq. 17 becomes

mee ee

Me + Ne B0 J2e

5T eB0 q e

=

B0 F eVB0 F eq

+1

neeMe

IdP 0/ d p + nee B02 U i

InedT e0/ d p. 25

This matrix equation can be solved for the parallel currentand electron heat ow induced by the parallel external andviscous forces by inverting the Me + Ne matrix.

The parallel current solution of Eq. 25 can be writtenin the form of an extended neoclassical parallel Ohms law,

B0 E A = nc B0 J B0 J drives . 26

Here, the neoclassical parallel resistivity is

nc menee

2 ee

1

Ne + Me 001

menee

2 ee

1

Ne 001

1

Sp. 27

As indicated, when viscosity effects are negligible, this is just the reciprocal of the Spitzer electrical conductivity:1 / Sp = me e / ne0e2 2 + Z eff / 2 + 13 / 4 Z eff in which e Z eff / ee . This formula for the Spitzer conductivity istypically accurate to within about 1% for Z eff 14, but in-correct by about 5% for Z eff . Greater accuracy can beobtained by including energy-weighted heat ow effectsand inverting the resultant 3 3 matrix equation. However,such effort is not warranted because the intrinsic uncertainty

in the Coulomb collision operator is 1/ln 1

/17 6%,which is larger than errors in Eq. 27 . Neglecting electron

heat ow effects, the Ne and Me matrices reduce to their 00elements. Then, the neoclassical parallel resistivity becomessimply nc 1+ e00 / e00 , in which me e / ne0e2 isthe perpendicular resistivity and for 1 in the bananacollisionality regime e00 / e00 1.46 0.53+ Z eff / Z eff .

Currents driven by the total radial pressure gradient bscurrent and electron momentum sources are

B0 J drives B0 J bs + J CD + J dyn . 28

The bs, non inductive CD, and dynamo dyn currents aredened to be 7



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B0 J bs = b00 I dP 0d p

+ nee B02 U i + b01 ne0 I

dT e0d p

,

29

B0 J CD = B0 SeV

+ c01 me/ T e0 B0 Seq

ne0e nc , 30

B0 J dyn = B0 F eV

fl + c01 me/ T e0 B0 F eqfl

ne0e nc . 31

Here, the coefcients are b00 Ne + Me 1 Me 00,b01 Ne + Me 1 Me 01, and c01 Ne + Me 01

1 / Ne + Me 001

= e01 + e01 / e11 + e11 . Neglecting electron heat oweffects and U i , we obtain b01 0 and the Ne and Mematrices reduce to their 00 elements; then, the bs currentis given approximately by B0 J bs e00 / e00 + e00

I dP 0 / d 0 B0 / B p dP 0 / d in the banana re-gime. The heat ow source effects in Eq. 30 are illustrativeof the extra electron cyclotron current drive effects discussed

in Ref. 12.Next we determine U i and Q i . The reference ion colli-

sion frequency is analogous to Eq. 20 with e i : 1 / ii= 4 / 3 4 n i Z i4e4 ln / 4 0 2mi2v Ti3 . The ion frictionand viscosity coefcient matrices N i and Mi are the same asthe electron ones in Eqs. 22 and 21 with 10 Z eff Z inwhich for a combination of hydrogenic ions subscript i, Z i =1 and a small admixture of impurity ions subscript I Z I n I Z I

2 / n i = ne / n i Z eff 1. Assuming impurities have thesame ow velocities as hydrogenic ions, there is no colli-sional friction between them: B0 R iV =0. Expanding Eq.3 as in Eq. 6 , averaging over , FSA and summing over

species, the plasma equilibrium t 1 / i parallel force bal-ance becomes 6,7

0 B0 i + B0 SVtot . 32

Here, the summed parallel viscous force has been approxi-mated by its ion component because it is larger thanthe electron one by mi / me 1. Also, the externally imposedand uctuation-induced due to Reynolds and Maxwellstresses FSA parallel forces on the plasma are

B0 SVtot = B0 s SsV msV sS

sn

s B0 msns0V s V

s + s

+ B0 J B .

As explained in the paragraph preceding Eq. 24 , the uc-tuating ows here are uid ows that result from a combi-nation of guiding center and diamagnetic ows.

To rst order in the gyroradius expansion the FSA equi-librium parallel heat ow equation is

0 = B0 i + B0 R iq + mi/ T i0 B0 Siqtot . 33

Here, we have B0 S iqtot B0 S iq + 5 / 2 pi0V i V

i

B0 V iq i + q iV i + 2 / 3 V i q iI . Since the heat friction

B0 R iq = min i0 / ii 2 / 5n i0T i0 B0 q i , using Eq. 13and the ion version of Eq. 18 this equation can be solved toyield

Q i = cU U i cT I

q i B02

dT i0d p

+ ii B0 Siq

tot / B02

min i0 i11 + i11. 34

Here, cU i01 / i11 + i11 and cT i11 / i11 + i11 . Substi-tuting this result into Eq. 32 yields

U i = k i I

q i B02

dT i0d p

+ U i drives . 35

The neoclassical offset poloidal ion ow coefcient is

k i i01

i00

1

1 + i11 i012 / i00 / i11. 36

For a pure electron-ion plasma i.e., Z 0 in theasymptotic ion banana collisionality regime i 1, 1 ,this yields the usual results: i01 / i00 =1.17 and k i=1.17 / 1+0.67 . Possible poloidal ows driven by exter-nal sources and uctuations are

U i drives =

cdrmin i0 B0

2

B0 SVtot

i00/ ii+

mi/ T i0 B0 Siqtot

i11 + i11 / ii.

37

Here, cdr = i11 + i11 / i11 + i11 i012 / i00 .

Since V = U i B0 = I / qR 2 U i , substituting thispoloidal ion ow into Eq. 8 yields the relation

t = d d p +1

n i0q i

dp i0d p

+ p, p I

R2U i . 38

This relation does not determine either t or d 0 / d p;it only yields a relation between the radial electric eld

d 0 / d p and the toroidal rotation frequency t . The to-roidal rotation or radial electric eld is determined from thenonambipolar particle uxes on the longer plasma transporttime scale. 7

IV. MAGNETIC FLUX TRANSIENT EFFECTS

The poloidal and toroidal magnetic uxes p and t evolve during plasma start-up, addition of CDs, and ap-proach to steady state on magnetic eld diffusion time scales.These slow O 2 effects have been negligible in the pre-ceding O 0 , 1 analyses, but need to be included in thecomprehensive transport equations. Using B = Awith A = t p in Faradays law in the form

A / t x + E A = 0 and the FSA of R2 times theseequations yield 7 for the toroidal and poloidal ux evolutionequations,

t t x

= u G t

t , 39

p t x = B0 E A I R2 t u G p . 40

Here, u G u G = B p E A / p I R2 is the t gridspeed 13 and 2 / t V loop

t is the toroidal loop voltage

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induced by the Ohmic heating solenoid. Using the parallelOhms law in Eq. 26 for B0 E A and Eqs. 15 and 16 forthe parallel current 0 B0 J yields a diffusion equation forpoloidal ux p on a toroidal ux surface t ,

p p

t t = D + p S , 41 D

nc

0, S =

t +

nc

I R2B0 J drives . 42

Tokamak plasma properties are determined in terms of the poloidal magnetic ux p: 1 the GradShafranov equa-tion determines p x given P p and I p ; 2 classicaland neoclassical transport are determined 14 across poloidalux surfaces p; and 3 the drift-kinetic and gyrokineticequations use poloidal ux variables and f 0 = f Max p sothe canonical toroidal angular momentum emerge s as a natu-ral constant of motion. Thus, we need to transform 14 the uidmoment equations from determining the species density, mo-mentum, and energy at a laboratory position x to determiningthem on a poloidal ux surface pi.e., n / t x n / t p,etc. However, for low collisionali ty tokamak plasmas thistransformation should rst be made 14 in the drif t-k inetic orgyrokinetic equation. This transformation adds 15 a paleo-classical diffusion-type operator D f D , f / a 2 2 f tothe right side of Eq. 1 .

The effects of the paleoclassical transport operator Dwill be illustrated through their inuence on the perturbed, -avera ged and FSA density equation 2 . First, we notethat 15,7

D n0 p n0

+ n0u G +1

V

2

2V D n0 .

43

Here, p p / p with p d p / d represents p surfacemotion relative to the t -based radial coordinate , D

D / a 2, 1 / a 2 2/ R2 / R2 1 / a 2 and u G= 1 / V V / t . Including the se transformation effects,the FSA density equation becomes 7

1V

t pV n0 + p

n0

+1

V

V = S n . 44

The quantity V n0 is the number of particles between the and + d surfaces; it is an adiabatic plasma property. Theradial coordinate t / Bt 0 in transport codes 4,5 is basedon the relatively immobile 2,3,13 toroidal magnetic ux t ;however, the uid moments n , T , V are determined on poloi-dal ux surfaces p. The p n0 / term takes account of the slow magnetic eld diffusion time scale motion of the p surfaces relative to the t surfaces. The total O 2 par-ticle ux for each species is 7

= a + na + pca . 45

Here, a n0 V2 u G is the intrinsically ambipolarparticle ux driven by axisymmetric collisional processes,

na n 1V 1 is the possibly nonambipolar uctuation-

induced particle ux, and pca / V D

n0 is the ambi-

polar paleoclassical particle ux that results from the coordi-nate transformation. 7

V. RADIAL PARTICLE FLUXES, PLASMA TOROIDALROTATION, AND ELECTRIC FIELD

In Secs. II and III the O 0 radial and O 1 parallel

components of the species force balance equation 3 wereanalyzed. The O 2 particle uxes will be determined fromthe toroidal angular component of the force balance equa-tions. A vector identity for determining the second order ra-dial uxes is e R2 = Re ,

e nV B0 = nV e B0 = nV p. 46

Thus, the e component of the force balance shows the par-ticle ux is induced by j toroidal torques T j e F j on theplasma species by the various uid forces F j,

0 = e nq V B0 + j

F j q p = j

T j. 47

Taking the toroidal angular e component of the spe-cies force balance, transforming from x to p, and averagingover and a ux surface yields a particle ux with 15components. 7 There are six collision-induced intrinsicallyambipolar uxes due to toroidal torques from cross frictionclassical , parallel friction PrschSchlter , parallel vis-

cosity banana-plateau , noninductive current drives, dyna-mos, and the E A B p pinch, plus the paleoclassical one dueto the coordinate transformation. In addition, there areeight possibly nonambipolar uxes due to toroidal torquesexerted on plasma species by polarization ows induced by t / t 0, neoclassical toroidal viscosity NTV from small3D NA magnetic elds, perpendicular viscosities

classical,

neoclassical, and paleoclassical , Reynolds and Maxwellstresses induced by uctuations, J B forces in the vicinityof low order rational surfaces induced by resonant eld er-rors, poloidal ux transients, and external momentumsources. 7

For quasineutrality the transport time scale charge con-tinuity equation requires J =0. Setting the J in-duced by the nonambipolar uxes to zero yields 6,7 a compre-hensive toroidal torque balance equation for the toroidalangular momentum density Lt min i0 R2 t neglectingsmall electron terms ,

1V

t pV Lt e i

NA e i

1

V

V i + e J

B

p Lt

+ e s

S sV . 48

Here, V Lt is the plasma toroidal angular momentum be-tween the and + d ux surfaces, which is an adiabaticquantity. The terms on the right represent NTV effects,collision-induced perpendicular viscosity, Reynolds andMaxwell stresses induced by uctuations, poloidal ux tran-



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sients, and externally supplied momentum sources.Once this equation is solved for Lt the toroidal rotation is

given by t Lt / min i0 R2 . Hence from Eq. 38 the ra-dial electric eld E d 0 / d is

7

E = t R2 p

R2 p + 1n i0q idp i0d . 49

This E or t causes the electron and ion nonambipolarradial particle uxes to become equal i.e., ambipolar :

ena E = Z i i

na E to satisfy J =0, which was usedto obtain the t or E equation. Hence, the net ambipolarparticle ux is the sum of the intrinsically ambipolar uxes

a + pca and the potentially nonambipolar uxes evaluated at

the electric eld E , na E . It is easiest to evaluate forelectrons since the dominant nearly canceling terms in theJ that is set to zero to obtain Eq. 48 result from

nona mbipolar ion particle uxes this is the so-called ionroot 16 ,

enet

ea + epc

a + ena E = i

net . 50

This is the net ambipolar particle ux

to be used in thedensity continuity equation on a p ux surface given in Eq.44 . See Eqs. 87 93 in Ref. 7 for the seven intrinsically

ambipolar particle uxes and Eqs. 106 113 there for theeight potentially nonambipolar particle uxes.

VI. ENERGY TRANSPORT EQUATIONS

The energy transport equations on the transport timescale / t 2 are obtained by expanding all quantities inEq. 4 using perturbation expansions like that in Eq. 6 .Then, we take the toroidal angle average of the equations.Finally, we ux surface average the resultant equations andtransform them from laboratory x to poloidal ux p co-ordinates to obtain for each species

32

p0

t pln p0V

5/ 3 +32

p p0 t

+1

V

V = Qnet .

51

Here, ln ps0V 5/ 3 is the collisional entropy between the

and + d ux surfaces; without dissipation or energy

sources it is conserved. The p term accounts for motion of p surfaces relative to the t surfaces.

The total FSA radial heat uxes are

q 2 +5

2 p0 V 2 uG + p 1V 1 + pc

= q2 +

5

2n

0T

1V

1 +

pc+

5

2T

0, 52

which is comprised of conductive plus convective heatuxes. The particle ux in the convective heat ux5 / 2 T 0 is the net ambipolar one specied in Eq. 50 . The

second order conductive heat ux q 2 is obtain ed similarly tohow the radial particle uxes were obtained 7 via Eq. 46from the toroidal angular component of the perturbed, -averaged, FSA and transformed heat ow equation 5 ,

q 2 = cl + PS + bp + fl + Sq . 53

The lowest order classical

cross heat friction , PrschSchlter parallel heat friction , banana-plateau parallel heatviscosity and paleoclassical transform to p collision-induced heat uxes, and uctuation- and source-inducedconductive radial heat uxes obtained from the e R2 component of Eq. 5 for each species are

cl T s0B0

qs B02 R sq , 54

PS IT s0qs p

1

B02

1

B02 B0 R sq , 55

bp IT s0

qs B02 p

B0 s , 56

pc 1

V

V D

3

2ns0T s0 , 57

fl ms

q s p

5

2

e p s1 T s1ms

+ ps0 e V s1 V

s1 +

1

V

V e V

s1q s1 + q s1V s1 + 2 / 3 V s1 q s1I

1

pe q s1 B , 58

Sq ms

qs pe Ssq

. 59

Here, the perpendicular viscosity effects 7 have been ne-glected since they give O 2 smaller contributions to theheat uxes. The rather complicated uctuation-driven contri-

butions in fl given by Eq. 58 are usually much smallerthan the typically dominant 5 / 2 n0 T

1V

1 uctuation-

driven contribution to Eq. 52 because they mostly in-volve the smaller ow uctuation magnitudes in the toroidaldirection compared to the typically larger ones in the radialdirection. As explained in the paragraph preceding Eq. 24 ,



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the uctuating ows and heat ows here are uid ows thatresult from a combination of guiding center and diamagneticows. However, as with the uctuation-induced particle uxsee the discussion in Appendix A of Ref. 7 , the uctuating

diamagnetic ow does not contribute to the uctuation-induced heat transport; thus the lowest order uctuation-induced contribution to is just due to guiding center ow,i.e., 5 / 2 n0 T

1V

1 5 / 2 n0 T

1V

g .

Finally, the net rate of energy input to the species Qnet inEq. 51 is given in general by

Qnet Q + V2 uG p0 + V 1 p 1 + V 1 p 1

: V1 + S

E . 60

Successive terms on the right will be specied in greaterdetail in the following paragraphs.

The total collisional energy exchange between speciesis1 Q d 3v m v V 2 / 2 C f = d 3v mv 2 / 2 mv V C f =sign qs Q V R V. Thus, the FSA of the -average of Qhas three parts,

Q sign qs Q V 1 R V V 1 R V . 61The rst term is the collisional energy transfer rate fromelectrons to ions due to their differing temperatures. For ionsit is Q d 3v mev 2 / 2 C ei f e =3 me / mi ne / ee T e T i ; itis Q in the electron energy equation because of collisionalenergy conservation. As discussed in the next paragraph, thesecond term represents primarily Joule heating. The nalterm in Eq. 61 represents uctuation-induced parallel andperpendicular Joule heating; it has been found to be negli-gible for some typical tokamak plasma conditions. 17

The net collisional Joule heating rate can be obtained byneglecting the last term in Eq. 61 and summing over spe-cies: Q J s Q s = s V s1 R sV . The rst order ows andcurrents can be rep rese nted in terms of their parallel andcross components by 2,3,7

V s1 = U s p B0 + s p e , 62

s d 0d p +1

n s0q s

dp s0d p

, 63

J s

ns0qsV s1 = K J p B0 dP 0 p

d pe . 64

As noted before Eq. 14 , B02 K J = B0 J + IdP 0 / d p. Also,we nd from Amperes law 0J B0 = I + p , that K J J / B0 = 1 / 0 dI / d p. Thus,the toroidal and poloidal components of the current are

B0 = I R2 ,

J B0

=B0 J

B02

dP 0/ d pB0

1 I 2 R2

B02 , 65

J B0

K J p = 1

0

dI

d p. 66

Note that the toroidal current J has both a FSA com-ponent and a small PrshSchlter type component. The

poloidal current J is typically smaller than the toroi-dal current by a factor of B p / Bt / q.

Using these ow and current representations, the rstrow of Eq. 17 for the FSA parallel electron friction forceB0 R eV , the relation B0 E A = B0 e E A

+ u G p , and from collisional momentum conservationR iV = R eV, we nd for the Joule heating

Q

J s U s B0 R sV + s e R sV

= K J B0 R eV

ne0e e

a

ne0

dP 0d

B0 E

A

B0 dP 0d p

= B0 E A J B0

ea

ne0

dP 0d

+J

B0

B0 e B0 SeVtot

ne0e . 67The P0dV work by the collision-induced classical, PrschSchlter and banana-plateau particle ux e

a is usually neg-

ligible. For Ohmically heated tokamak plasmas where theJoule heating from the average toroidal current induced bythe toroidal inductive electric eld usually dominates overthat induced by the poloidal current for B p / Bt / q 1 ,we obtain simply Q J B0 E A B0 J / B02 . The FSA in-ductive parallel electric eld B0 E A is provided by the ex-tended parallel neoclassical Ohms law given in Eq. 26 .

Implicitly, the Joule heating result in Eq. 67 is obtainedin the ion rest frame. Thus, for ions we have

Q i Q V i1 R iV . 68

The corresponding collisional energy exchange rate for elec-

trons is

Q e Q

J Q

V e1 R eV . 69

The second Qnet term in Eq. 60 represents the pdV work done by an outowing species of particles. For eitherspecies it is

V s2 uG ps0 = ea / ns0 dp s0/ d . 70

This contribution cancels part of the small P0dV work termin Eq. 67 . The third Qnet term in Eq. 60 represents pdV work by uctuations as they induce radial particle transport.Using the vector identity V p= pV p V , it can beseen that when uctuating ows are incompressible the re-maining p 1V 1 = 1 / V / V p 1V 1 term justchanges the 5/2 factor in the convective heat ow in Eq. 52to 3/2, as is well known. The fourth and fth Qnet terms inEq. 60 represent viscous heating. Using the vector identitythat : V = V V , the ow velocity represen-tation in Eq. 62 and the fact that 2,3,7 e A =0, theseviscous heating terms reduce for each species to 3

Q visc V 1 p 1 : V 1 = U B0 , 71

which is mn U 2 B0

2 . However, it just cancels part of thetypically small poloidal current contribution to Eq. 67 .



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For electrons there is one additional term to be added tothe left side of Eq. 51 . It is the contributio n due to helicallyresonant paleoclassical transport processes, 18

q epc =

M

V

2

2V D

32

pe0 +3

2

pe0

. 72

Here, M 10 is a factor reecting the distance, relative to thepoloidal periodicity length, over which the electron tempera-ture is equilibrated as it diffuses radially with the diffusingpoloidal magnetic ux. 18 Also, = q p / / q p withq q / reects the motion required to stay on the sameq t / / p / surface in transient poloidal ux situ-ations where p , t p , 0 + p t . There is no paleo-classical M contribution to the ion energy balance because 18

the ion toroidal precessional drift frequency exceeds the ioncollision frequency in most tokamak plasmas.

VII. SUMMARY

Comprehensive plasma ow and transport equations for

describing tokamak plasmas have been derived with akinetic-based approach using a small gyroradius expansion.Relevant uid moment equations that include neoclassical-based kinetic closures and nonaxisymmeric perturbationsexternally imposed or from plasma uctuations have been

averaged over the toroidal angle and then FSA. The zerothorder Alfvn time scale radial force balance provides a con-straint relation for the toroidal plasma rotation t in Eqs. 8and 38 . The rst order parallel force balances yield theneoclassical-type parallel Ohms law in Eq. 26 and poloidalion ow in Eq. 35 for times longer than collision times.The radial uxes of particles in Eq. 45 and heat in Eqs.52 59 are obtained from toroidal angular components of

the momentum and heat ow equations, respectively. Theresultant tokamak plasma model includes transport timescale evolution equations for the toroidal magnetic ux t inEq. 39 , poloidal magnetic ux p in Eq. 41 , and toroidalangular momentum density Lt min i0 R2 t in Eq. 48 , inaddition to expanded versions of the FSA equations for thedensity n in Eq. 44 and species pressure p nT in Eq. 51 .The FSA plasma toroidal rotation frequency t

Lt / min i0 R2 determines the radial electric eld E in Eq.

49 that is required to obtain the net ambipolar radial par-ticle ux net in Eq. 50 .

Key attributes of and consequences from this new ap-proach for tokamak plasma transport equations are the fol-lowing. 1 The derivation of the radial particle ux andtoroidal ow are naturally joined. 2 The radial electric eldis determined self-consistently and enforces ambipolar radial

particle transport. 3 The mean-eld effects of microturbulence-induced uctuations on all plasma transportproperties are included self-consistentlyparallel Ohms lawEq. 31 , poloidal ion ow Eq. 37 , particle uxes Eq.45 , toroidal rotation Eq. 48 , heat uxes Eqs. 52 and58 , and heating Eqs. 60 and 61 . 4 Source effectse.g., energetic neutral beam momentum input and noninduc-

tive current drives are also included self-consistently. 5Paleoclassical n, t , and T diffusion and pinch effects plusthe electron heat transport in Eq. 72 are included naturally.Finally, 6 the radial motion of n, t , and T induced by p 0 poloidal ux transients are included for the rst time.These plasma transport equations follow naturally from ex-tended two-uid moment equations; hence they are consis-tent with extended MHD code frameworks and could pro-vide a basis for the Fusion Simulation Program.

ACKNOWLEDGMENTS

The authors are grateful to G. Bateman who stimulatedus to develop the simplied Joule heating formula in Eq.67 .

This research was supported by U.S. Department of En-ergy Grant No. DE-FG02-86ER53218.

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http://dx.doi.org/10.1103/RevModPhys.48.239http://w3.pppl.gov/transp/http://w3.pppl.gov/transp/http://dx.doi.org/10.1088/0029-5515/49/8/085021http://dx.doi.org/10.1063/1.3206976http://dx.doi.org/10.1063/1.864108http://dx.doi.org/10.1063/1.872465http://dx.doi.org/10.1063/1.872465http://dx.doi.org/10.1063/1.859671http://dx.doi.org/10.1063/1.3335486http://www.cptc.wisc.edu/http://dx.doi.org/10.1063/1.3258850http://dx.doi.org/10.1063/1.862654http://dx.doi.org/10.1063/1.1694191http://dx.doi.org/10.1063/1.2715564http://dx.doi.org/10.1063/1.2830823http://dx.doi.org/10.1063/1.2047227http://dx.doi.org/10.1063/1.2047227http://dx.doi.org/10.1063/1.2830823http://dx.doi.org/10.1063/1.2715564http://dx.doi.org/10.1063/1.1694191http://dx.doi.org/10.1063/1.862654http://dx.doi.org/10.1063/1.3258850http://www.cptc.wisc.edu/http://dx.doi.org/10.1063/1.3335486http://dx.doi.org/10.1063/1.859671http://dx.doi.org/10.1063/1.872465http://dx.doi.org/10.1063/1.872465http://dx.doi.org/10.1063/1.864108http://dx.doi.org/10.1063/1.3206976http://dx.doi.org/10.1088/0029-5515/49/8/085021http://w3.pppl.gov/transp/http://dx.doi.org/10.1103/RevModPhys.48.239

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