Catto, P.J.

Fulop, T.

Helander, P.

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PSFC/JA-98- 11Neutral Diffusion and AnomalousEffects on Ion Flow ShearT. Fuilkp, Peter J. Catto, P. HelanderApril 1998Plasma Science and Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USAThis work was supported by the US Department of Energy, Grant No. DE-FG02-91ER54109. Reproduction, translation, publication, use, and disposal, in whole or inpart, by or for the US Government is permitted.Submitted for publication in: Physics of PlasmasNeutral diffusion and anomalous effects on ionflow shearT. Fiil6p(a), Peter J. Catto(b) and P. Helander()(a) Department of Electromagnetics, Chalmers University of Technology,S-412 96 G6teborg, Sweden(b) Plasma Science and Fusion Center, Massachusetts Institute of Technologyand Lodestar Research Corporation, Cambridgc, MA 02139(c) UKAEA Fusion, Culham Science Centre (UKAEA/Euratom Association),Abingdon, Oxon, OX14 3DB, United KingdomAbstractBulk ion flow can be modified by ion-ncutral interactions in the edge regionof tokamaks where neutral atoms are abundant. In this region, the standardneoclassical expression for the ion flow is not consistent with the experimentalobservations. Previous work in the plateau regime [Valanju, et al., Phys FluidsB 4, 2675 (1992)] suggests that the discrepancy is caused mainly by momen-tum loss due to ion-neutral charge exchange. The present work shows that inthe absence of neutral diffusion, charge exchange cannot affect the neoclassicaltransport of the ions significantly if the neutral mean-free path is short and theneutral density is low compared to the plasma density. However, the inclusionof neutral diffusion and anomalous processes in the plateau regime can sub-stantially modify parallel ion flow inside the separatrix as determined by theflux surface averaged momentum constraint. Moreover, the local shear in theflow may be determined by a balance between neutral diffusion and anomalouseffects which may then modify the turbulent transport.1Email: tunde@elmagn.chalmers.seI IntroductionThe influence of the neutral atoms on the ion parallel flow has been of interest recently,since the discrepancy between experimental measurements, [1], and neoclassical pre-dictions of the ion flow are in some cases correlated with neutral concentration.Earlier work, [2, 3], suggested that charge exchange (CX) processes have an impor-tant role in modifying the ion flow in the edge plasma, where the plasma temperatureand density are lower than in the core and the neutral concentration is high. How-ever, the analysis of Ref.[2] is not consistent with its neutral short mean-free pathassumption, since it assumes stationary neutrals. Reference [3] used the neoclassicalmoment approach including the effect of CX processes with stationary neutrals andmade quantitative comparisons with experiment to show that neutrals have a dra-matic effect on the main ion poloidal rotation. However, the assumption that theparallel neutral flow Vjl, ~ 0 is inappropriate for edge plasma where the neutral pen-etration depth and CX mean-free path for a neutral are on the order of a centimeter.The assumption of stationary neutrals made in Refs.[2, 3] cannot be made withoutalso including anomalous diffusion. The neutrals will only have vanishing mean par-allel velocity if they have not yet collided with any ions. In that case they are movinginto the plasma, which in steady state requires a balancing efflux of ions. Moreover,we will show that without the neutral diffusion, CX processes would not affect thetransport of ions significantly.The purpose of this work is to derive an ion drift kinetic equation that includesthe effect of neutral diffusion due to CX. Using the resulting drift kinetic equation we2demonstrate how the CX process and anomalous transport modify the parallel andpoloidal ion flow, assuming short mean-free path neutrals and plateau ions. The abil-ity of anomalous transport to influence parallel dynamics has been noted previously,[4]. The sharp radial (and perhaps poloidal) variation of the neutrals and anomaloustransport cause them to introduce a strong shear into the poloidal flow and therebythe electrostatic potential. Consequently, the neutrals and anomalous processes mayinfluence the sheared flow stabilization observed in reduced transport regimes, [5].By working in the plateau regime, we are relying on a large aspect ratio to makeexpansions valid. However, this somewhat artificial limit allows us to illustrate themechanisms at work more simply.The rest of the paper is organized as follows. In Section II we derive the kineticequation of the ions, including the effect of neutral and anomalous ion diffusion. InSection III we solve a simplified version of the kinetic equation in the plateau limit andinvestigate the modification of the parallel ion flow due to the diffusive terms. SectionIV is devoted to a discussion of the results. Finally, in Section V we summarize theconclusions.II Derivation of the kinetic equationIn the neutral kinetic equation we retain charge exchange and ionization so thatv -Vf7, = X(fi, f.) - vf" , (1)3where X denotes the charge exchange operator and v, = n,(o-v)z is the ionizationfrequency. We then letS=h (2)nito obtainv n,, fi + h) = (av). (njfi - njf, ) - vzf, = -(v. + v.4)h - i,,2 " fj, (3)\fnl niwhere v, = n (o-v), is the CX frequency for neutrals. Note that this definition of theCX collision frequency differs by a factor n,/ni from the one used in Ref.[1-3] wherev, referred to the ions. The short mean-free path solution (Vt < WV.) to Eq.(1)valid for v, > v, ish- 1 (1" - z nf+... (4)V. +v \. n } v. + vz niwhere vt = 2T/mj and W is the local equilibrium scale length associated with theedge in the presence of neutrals. Inserting h into the sum of the ion and neutralkinetic equationsv V(fi + f,)+ a . Vjf - C(f)= (5)where a = (Ze/M)JE + (v x B)/c] and C is the ion-ion collision operator, givesV-V 1 + Vx nn" f] + a -Vjf-C(fi)= V- [v V - (" vfi] , (6)v. + , nj V. + vl niwhere the right hand side represents neutral diffusion. Ionization appears diffusivebecause it is occurring in the presence of CX which causes the neutrals to responddiffusively, but it is important to remember that Eq.(6) is only valid for v, Z v.Writing+ (7)4where fi =< f; >, is the gyroaveraged part of the ion distribution function and f, =fi - fi is the gyrophase dependent part, and assuming Q > vii, with Q = ZeB/Mc,and IEJ < 1(v x B)/c, the gyrophase dependent term is given by; = x b -V [(1 +,q) (8)where b = B/B and we have introduced y = v~n,/n;(v + vs). Inserting Eq.(8) andgyroaveraging Eq.(6) yields the sum equationviivil [(1 + 7) A] + (1 + 7) va V (1+ ?) + (1 + 7) ((6OVE ' V i , fludC =v [ V: ((v)'fi)] UJi, (9)V. +v \-niwhere U is the neutral diffusion operator, (vv)1 v 2/2(1 - bb) + vAbb, and Vd =[v 2/(2Q)]b x V In B+(v /)b x (b -Vb). We retain anomalous transport to balanceneutral diffusion by introducing the fluctuating E x B drift 6vE and the fluctuatingportion of the ion distribution function 6fi with (...) denoting an average overturbulent fluctuations. If we note that bfi ~ r-6VE -V6' with r, the correlation timefor the fluctuations, thenDanfi -= -(( vE ' bf)i),)fuct (D , (10)where D is the anomalous diffusivity and we assume electrostatic fluctuations.Changing to v and '= vjj/v variables givesVil = - a2 sin 0 + , (11)where e = r/R, 0 is the poloidal angle, r and R the minor and major radii, and M isthe magnetic mirror term. In the new variables, the gyroaveraged sum equation with5the r7 ~ n,/n; < 1 terms neglected becomeso + Mg - Cg = -vd -Vf - Mfd + Uf + DAfi, (12)where w = v i/qR, C is the linearized collision operator, and we have letfi = fM + fd + g = fM + f + g, (13)with V/i; a lowest order flux function,mi M. /2 p(_ EfM = n( exp(-(E - e1)/T) (14)and E = Mv2/ 2 + e1.III Solution and flows in the plateau regimeTo lowest order we may solve for g by neglecting Mg and (U+Dai)fi. In the plateauregime, E < 1 allows us to neglect Mg. Then using Vji/vu ~ qpi/eW < 1 in Mfd,gives g/fM ~ qpi/W < 1. Note that the assumption V1ii/vu ~ qpi/EW can beverified a posteriori. The neglect of neutral and anomalous diffusion to lowest ordercorresponds to Vd - VfM > (U + Da,)fv which for D, ~ U requires n,/ni <Wi/,/RG. To next order we keep all the terms in Eq.(12) to evaluate the flux surfaceaveraged parallel momentum constraint which gives the parallel ion flow as modifiedby neutral and anomalous diffusion.Making the usual plateau replacement Cf = -vf, the lowest order version ofEq.(12) becomesW + g = 21 2 sin0 + s -(_2) , (15)6where(V sin 9 - W cos 9)Qg=fM (16)andV2 + 2 'E~ e<> F+ V 2 + T2) 2Qr V1 17S= 22 ++ +  T + -2 q R 712(17)We use a prime to denote a radial derivative. Equations (15)-(17) are the same as inRef.[2].To obtain the parallel ion flow, in the plateau regime, we return to Eq.(12), takethe vil moment, and perform a flux surface average. Most of the terms in Eq.(12) willvanish, leaving only terms assumed to be higher order, namelyv Mgdov) - (J v (U + Da,)fddV , (18)where (...) denotes flux surface averaging. This equation determines the parallel ionflow, V1jj, and represents a balance between the damping of the parallel flow on theleft and the accumulation of momentum due to diffusion on the right. Evaluating thedifferent terms in Eq.(18) givesfr niv, Tp' e(>' T 2QE V _ = 19 1 ((n")OVj )]4 QR pi 2,92r v, Or ++(( V f V VI1 , ' l". (19)where (...), denotes a poloidal average, i.e. (n,)o = (2r)-' j dOn,. In the absence ofneutrals and anomalous processes Eq.(19) reduces to the usual plateau result.The modification due to neutrals and anomalous processes can be estimated bywriting the parallel ion flow schematically asVi = GU,,, (20)7where the usual plateau result is_T qIp' e?' T! 'U -1 = T + --- + - . (21)miE E pi T 2T]Using Eq.(10) to crudely estimate the anomalous term as(((bvE ' V(T6VE - VniV1i)),)J'1.a)o ~ D (22)givesG-1 ~ 1+ O[ R + . (23),ue ni W2 W2In the expression for G, A = vti/(v±,+v) is the mean-free path and (n-,g)"/(nJii) ~W~ 2 ~(nnVj1 vi)"/(n,,V6). Using the preceding, the poloidal flow may be writtenasVi = (c/eBni)[(1 - G)(p's + eniJV) - GniT/2]. (24)Consequently, the neutral diffusion and anomalous processes may introduce strongshear in the poloidal flow.To see whether substantial changes are possible we make comparisons between thevarious terms in the flux surface averaged momentum constraint, using the followingset of parameters: safety factor q = 3, inverse aspect ratio E = 1/3, the neutral to iondensity ratio n,,/ni = 10-3, the ion thermal velocity vi = 107 cm/s, the characteristicradial scale length W = 2 cm, major radius R = 1 m, mean-free path A = 1 cm andD = 1 m2/s. Comparing the magnitude of the mirror term with the neutral diffusionterm(f v Ufdd v) _-2nE 2 qRA\ (25)(f viMgd3v) ni W 28so that these terms are comparable. Thus, within the limitations of a plateau ordering,taking the neutral diffusion term the same order as the mirror term is justified at theedge, just inside the separatrix. Comparing this anomalous diffusion term with theneutral diffusion term we obtain(f v Dfd 3 v) ni D (26)(f v Ufdd3V) n,,,viAhTherefore, the two terms are comparable and the contributions from both should beincluded in the ion parallel flow. That both appear should not be surprising since inthe steady state the anomalous outward diffusion of ions is balanced by an influx ofrecycled neutrals.To verify that our orderings are consistent we note that Eq.(18) implies thatMg ~ Ufd while the neglect of U in our lowest order equation (15) requires wag/> >UfM. As a result, we need Vji/evi ~ qp/(e2W) > 1 which allows us to satisfy ourassumptionn, R ,0 qRA E2W (27)n > - ~ - -- . (27ni 1W fni e2 y 2  q,We need to ensure the magnetic field satisfies qpi/W < 1. Thus for the parameterslisted B ~ 10 kG is required. The reason such a low B is needed can be traced backto the limitations of the plateau model which requires E < 1 while we have takenE = 1/3.IV DiscussionIn Ref.[2], the ion kinetic equation is employed with ionization neglected. Thereforev-Vfi+a-Vf 1 = C(fi)--X(fi, f,,) is considered. In the gyrophase averaged equation9for g, the replacementX(fi, f,) -> X(fM + fd, -fM) = (28)ni niis employed, which corresponds to setting V1,i = 0. However, from parallel neutralmomentum conservation Mn,1 v(Vli; - Vi1) = V11p,. so a consistent treatment wouldhave givenn n 7fV J Ji -V1X(fA, f,) - V. "(fdi - fd7 ) = v.I'fi ( 1  - Vi 1 ), (29)fni ni Twhich would have only yielded terms small in n,,/ni < 1 in the short mean-free pathlimit. In other words, since V'1i ~ Vi in the short mean-free path limit, the Galileaninvariance of the CX operator ensures negligible corrections unless diffusive terms areretained.Analogously, if Igi ~ VI/, the parallel ion flow will be unaffected also by theelastic ion-neutral collisions, to lowest order. This follows from the Galilean invarianceproperty of the collision operator giving Ci,,(fvi, fda) + C,,(fdi, fum) ~ 0, whichmeans that these terms will not contribute to the flux surface averaged momentumconstraint. This argument is independent of the form of the collision operator andcan be used for both the CX and elastic collision operators. In addition, the elasticion-neutral collision operator will give a diffusive contribution to the parallel ionflow, just as the CX operator does. However, the analysis of the elastic operatoris considerably more difficult, since it is a complicated Boltzmann operator. Termsqualitatively similar to those obtained from CX interactions between the neutrals andions are anticipated.10V Concluding remarksThe main conclusion of this work is that the neutral diffusion and anomalous pro-cesses affect the parallel and poloidal ion flows significantly. This observation is ofimportance for understanding the discrepancies between experimental observationsand neoclassical predictions of the flows. It also means that the lowest order ex-pression (Zeni/M)(E + c-'v x B) = V(pi + pn) is not automatically valid for edgeplasmas since neutral diffusion and anomalous processes can be substantial.The neutral density variation on the scale length of the penetration depth ( ~vu/l/IE~) introduces a strong shear into the parallel ion flow. Anomalous processesmay also cause (((bvE -Vf dvvtoi)<)fluc,)o to have strong radial variation. Asa result, the amount of shear in the parallel and poloidal ion flow just inside theseparatrix may be determined by a delicate balance between the effects of neutraldiffusion and anomalous processes which may thereby play a role in the suppressionof turbulent transport and the L to H mode transition.AcknowledgmentsWe would like to thank Sergei Krasheninnikov for enlightening discussions and T.F.is indebted to the Plasma Science and Fusion Center at Massachusetts Institute ofTechnology for its hospitality. This work was supported jointly by U.S. Departmentof Energy grant DE-FG02-91ER-54109, by the European Community under an asso-ciation contract between Euratom and Sweden and by the U.K. Department of Tradeand Industry and Euratom.11References[1] W. L. Rowan, A. G. Meigs, E. R. Solano, P. M. Valanju, M. D. Calvin, and R.D. Hazeltine, Phys. Fluids B 5, 2485, (1993)[2] P. M. Valanju, M. D. Calvin, R. D. Hazeltine, and E. R. Solano, Phys. Fluids B4, 2675, (1992)[3] P. Monier-Garbet, K. H. Burrell, F. L. Hinton, J. Kim, X. Garbet, and R. J.Groebner, Nucl. Fusion, 37, 403, (1997)[4] P. Helander, R. D. Hazeltine, and P. J. Catto, Phys. Rev. Lett. 77, 2479, (1996)[5] K. H. Burrell, Phys. Plasmas, 4, 1499, (1997)12

Neutral diffusion and anomalous effects of ion flow shear

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Neutral diffusion and anomalous effects of ion flow shear

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