Jenko, F.

Peeters, A.

Scott, B.

Teo, C.

English

Peeters, A. G.

Teo, C.-Y.

Max Planck Society eDoc Server

Self-Consitent Computation of Transport Barrier Formation by Fluid Drift Turbulence in Tokamak Geometry

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IAEA-F1-CN-69 TH 1/05Self-Consistent Computation of Transport Ba rrier Formation b yFluid Drift T urbulence in T okamak GeometryB. Scott, F. Jenko, A. G. P eeters, A. C-Y. T eoMax Planc k Institut fur Plasmaph ysikEuratom AssociationD-85748 Garc hing bei Munchen, German y1IAEA-F1-CN-69 TH 1/05AbstractSELF-CONSISTENT COMPUTATION OF TRANSPORT BARRIER F ORMA TIONBY FLUID DRIFT TURBULENCE IN TOKAMAK GEOMETRY(1) Computations of turbulence from the electromagnetic gyrouid model are per-formed in a ux surface geometry represen ting the actual MHD equilibrium of theASDEX Upgrade edge ux surfaces. The transition to ideal ballooning seen in simplegeometries as the plasma beta rises is suppressed, lea ving the transport at quantita-tively realistic levels. Computations for core parameters at half-radius geometry sho wsignicant contribution due to the nite beta electron dynamics, possibly remo vingthe standard ITG threshold. (2) Strong inward vorticity transport in edge turbulence,resulting from ion diamagnetic o ws, ma y lead to a build up of mean ExB v orticityfast enough to cause an H-mode transition. (3) F riction of mean ion ows againstneutrals involves both toroidal and poloidal ow componen ts, leading to a nite radialcurrent due to a given ExB prole even with zero poloidal rotation.1. Electromagnetic Gyrouid Model, T okamak GeometryW e report on numerical studies of tokamak edge turbulence, transport, and bi-furcation phenomena whic h form part of the dev elopment towards a comprehensiv e,self-consistent description of the transport barrier formation process from rst princi-ples.A necessary ingredient of any such model is a model for lo w frequency magnetisedplasma drift dynamics whic h is as complete as possible. The curren t state of the art is thegyrouid model, whic h in principle can incorporate all three of the principal ingredientsof drift turbulence: ion temperature gradien t (ITG) dynamics [1,2], drift Alfv en (DALF)dynamics of the passing electrons [3], and trapped electrons [4]. No model as y et hasall three; we report the rst results from a gyrouid model con taining DALF dynamics,showing its importance not only in the plasma edge but also to predominan tly ITGturbulence in the core.The form ulation of an electromagnetic gyrouid model is a straigh tforward exten-sion of the standard one, dropping the requiremen t that the electrons behave adiabat-ically and allowing the parallel electric eld to have an inductive part [5]. This addskinetic Alfven eects to the parallel dynamics of the electrons, following their evolutionself consistently. The underlying dynamics is the same as in the uid model [3,6], withthe additions that dissipation is more properly handled and that nite Larmor radiuseects (notably, polarisation) are treated to arbitrary order.Flux surface geometry ma y be accurately treated with a ux tube model in an yaxisymmetric system pro vided separatrices are avoided [7]; here we add global consis-tency with the procedure and denitions in [8]. An equilibrium is computed using theHELENA code [9], the desired radius is pic ked out, the local metric is computed; andthen the metric, curvature operator, magnetic eld strength, and unit v ector divergenceare stored as functions of the parallel coordinate for use by the turbulence code. TheASDEX Upgrade model and parameter regime are shown in Fig. 1.2The results of using an accurate geometry can be striking. The more traditionalballooning coordinate model [7], basically a sheared slab model with curvature termsadded, yields the ideal ballooning instability at values of the plasma beta below theexperimentally observed L-to-H transition. One proposal to remedy this is to invokediamagnetic stabilisation, but in the absence of a pedestal in the pre-transition L-mode this is not available. We nd that the use of the ASDEX Upgrade equilibriumleads to a partial suppression of ideal ballooning, due to the increased shear but alsoto the shaping eects, a strong function of ellipticity. The transport is at realisticlevels { total power through the s95 surface slightly above 3MW for a typical pre-transition L-Mode situation. Finally, the dependence on collisionality is weak at ASDEXUpgrade parameters, with the local  at the boundary below which Braginskii thermalconduction breaks down and Landau damping takes over the dissipation. These resultsare summarised in Fig. 2. The parameter ̂ = (csqR=2vALp)2 reects the strengthof drift wave dynamics relative to Alfven transients (it is important to note that thesecouple to the electron pressure [3]), where Lp is the pressure scale length. In these termsthe MHD parameter M =  q2Rr is 1 at ̂  8 and the experimental transition isat ̂  20 The sharp transition at ̂  5 is identied as the onset of ideal ballooningby means of mode structure diagnostics, especially the probability distribution of paseshifts of the component waves [5]. This is shown in Fig. 3.Core parameters dier from the edge situation principally through the much smallervalue of the scale ratio qR=Lp. Due to the higher beta, we still have ̂  1 althoughthe electron thermal speed is much faster than the drift wave phase velocities. Thismakes rn rather than rpe important to the electron dynamics. We nd that thecombination of nite  1i = d logn=d log Ti and nite ̂  1 brings very strong Alfvendynamics into the ITG mode system, due to the ability of the inductive electric eld tooppose the static force imbalance given by the parallel gradients of the electron densityand electrostatic potential. The passing electrons are therefore nonadiabatic, possiblyable to completely remove the ITG threshold. Exploratory computations are shown inFig. 4. This situation is almost completely collisionless. Interpretations concerning thetransition from edge to core turbulence [10] will be strongly aected.2. Vorticity Transport and Rotation DampingAs Fig 1 shows, we do not nd a parameter transition one could apply to the L-to-H transition observed in tokamak edges, in contrast to other work [11]. The reasonsfor the discrepancy are yet unknown, but many possible causes (periodic boundaryconditions, Braginskii thermal conduction, ux tube domain aspect ratio, and globalconsistency) have been ruled out. The present results have been obtained with severaldierent numerical treatments of the Alfven dynamics (upwinding, a predictor/correctorscheme), and with both the uid [6] and gyrouid [5] models. The likelihood that theyare accurate is large.We turn therefore to prole eects, particularly the self consistent interaction be-tween turbulence, ExB rotation, and the density and temperature proles. There is3much circumstantial evidence that the L-to-H transition involves all of these in con-cert [12]. One very promising result from the above computations is the existence ofa signicant inward transport of total ion vorticity. This has an obvious origin: gy-roviscosity. The total vorticity is made up of ExB and diamagnetic components (in agyrouid model, ne ni, where ni is the gyrouid ion density, not the total ion densitywhich is equal to ne), but due to gyroviscosity it is advected by the ExB velocity alone.This breaks the symmetry which keeps the purely ExB Reynolds stress small on averageat small scales. It is a necessary result that if epi is isotropic at small scales, then theturbulent ExB transport of pi radially outward implies the inward transport of r2?pi.Then, since the prole of pi responds according to its own equation, if these diamag-netic vorticity uctuations build up a mean vorticity prole the latter will result in theform of an ExB rotation prole, which due to the accumulating mean vorticity will besheared. This is basically a modication of the standard predator/prey scenario withinthe same paradigm [13]. Investigation of the self-consistent prole reaction to this vor-ticity transport by means of a nonlocal extension to the electromagnetic gyrouid codeis in progress.Consideration of the ExB rotation prole will have to include the presence of neutralfriction. Since this acts on both the toroidal and poloidal rotation components, onecannot consider solely poloidal rotation (the only component damped by neoclassicalviscosity). We have attempted a detailed calculation of these eects in ASDEX Upgrade,taking the sheared rotation prole as given and applying Monte Carlo techniques tocompute the net radial current induced by both neoclassical and neutral dissipation[14]. For solely poloidal ow the neutrals produce a small addition to the radial frictioncurrent [15]. With both toroidal and poloidal ow considered, the end state is zeropoloidal ow and a balance between neutral friction and toroidal ow for the given ExBprole. However, the corresponding damping rate of the ExB vorticity is comparableto or smaller than the ion collision frequency, smaller than what one nds in the morecollisional uid regime in which one can use standard bulk viscosity [17]. It is thereforeimportant to use expressions for damping that are well behaved in the weakly collisionalregime [2], in which the edge as well as core regions of modern tokamaks nd themselves.3. Scrape O Layer and Kinetic EectsEarlier investigations of coupling between closed and open eld line regions showedthe transition between the two to be gradual, with a radial width of about 2 to 3 cm [18].On open eld lines in the scrape o layer (SOL) region, the turbulence character is muchmore interchange like due to the lack of eld line connection { the driven range is atlonger wavelength and the coupling between density and potential is much weaker dueto the sheath conditions at the end of the eld lines. This is not yet fully incorporatedinto the gyrouid codes (but see [19], but our earlier experience with simplied models(zero uctuations or zero parallel gradient at eld line ends) shows much the samecharacter as in [18]; without closed eld line connection there is a rk = 0 componentwhich behaves largely electrostatically.4A companion model to the uid suite is our phase space drift kinetic electron code[20]. It has also been extended to include the kinetic Alfven parallel dynamics that lieat the heart of the DALF system. Gyrokinetic ions for this code lie in the near future,but in the meantime several benchmarks with the Landau uid DALF code have beencarried out in the simple ballooning geometry. Transport at zero collisionality as afunction of ̂ agrees well with the uid code, with the largest discrepancy at 0.4 versus0.25 for the range 1 < ̂ < 10. The kinetic model also produces the ideal ballooningtransition The validity of the uid models incorporating the simple Landau dampingclosures of [1,2] is helped by the fact that magnetic shear reduces the role played bywave/particle trapping of electrons moving along magnetic eld lines [20]. Investigationof the validity of the ion Landau closure in the presence of the strong DALF dynamicsat core parameters discussed above is urgently needed; this will be treated in the nearfuture.4. Further DirectionsAmong the most critical things to do next is to build the gyrokinetic phase spacecode mentioned above. Techniques developed in gyrokinetic particle codes in currentuse will be applied to the gyrokinetic ion distribution function, which is tied to theelectron function through the gyrokinetic polarisation equation [21]. Besides buttressingthe gyrouid model in the electromagnetic regime, this will provide a useful check ondiscreteness eects in the particle codes, especially in the representation of the higher-order moments appearing in the gyrouid model. Electron trapping will be incorporatedself consistently as well.Another need is a nonlocal version of the gyrouid model to treat self consistentpedestal physics, an electromagnetic gyrouid version of [22]. Radial variations of theux surface geometry in the edge are expected to have a signicant eect on the spatialvariation of the turbulence drive, transfer, and dissipation.Interaction of the turbulent transport with the proles and vice versa is anotheravenue gaining increasing attention. At the moment the DALF model results are beingused as one of several components in a numerical function to give the transport interms of background quantities, in the Multi Mode Model transport code [23]. It isalso importat to consider the self consistent dynamical evolution of the turbulence andtransport, especially if the proles change rapidly; it is also interesting to use coupledmodels to arrive at a global description, without paying penalties for machine size orthe separation of time scales [24]. We plan to do this with the B2/EIRENE edge code[25] in the near term; self-consistent mean eld treatment of the transport, as in [14]for the ow dynamics, will be necessary to include several anomalous thermal transfereects as well as the basic transport of the prolesReferences[1] DORLAND, W., HAMMETT, G.W., Phys. Fluids B 5 (1993) 812.[2] BEER, M.A., HAMMETT, G.W., Phys. Plasmas 3 (1996) 4046.5[3] SCOTT, B., Plasma Phys. Control. Fusion 39 (1997) 1635.[4] BEER, M.A., HAMMETT, G.W., Phys. Plasmas 3 (1996) 4018.[5] SCOTT, B., \Electromagnetic Gyrouid Turbulence in Tokamak Edge Geometries",Varenna Conference on Fusion Plasmas (1998).[6] SCOTT, B., Contrib. Plasma Phys. 38 (1998) 171.[7] BEER, M.A., COWLEY, S.C., HAMMETT, G.W., Phys. Plasmas 2 (1995) 2687.[8] SCOTT, B., Phys. Plasmas 5 (1998) 2334.[9] HUIJSMANS, G., \External resistive modes in tokamaks", Ph.D. thesis, Vrije Uni-versiteit Amsterdam (1991)[10] ZEILER, A., HALLATSCHEK, K., et al, these Proceedings, paper IAEA-F1-CN-69THP 2/02.[11] ROGERS, B., DRAKE, J.F., DORLAND, W., ZEILER, A., these Proceedings,paper IAEA-F1-CN-69 THP 2/01.[12] Plasma Phys. Control. Fusion 40 Special Issue: Papers from the 6th IAEA TechnicalCommittee Meeting on H-mode Physics, no 5, pp 531-861.[13] DIAMOND, P.H., LIANG, Y.-M., CARRERAS, B.A., TERRY, P.W., Phys. Rev.Lett. 72 (1994) 2565.[14] PEETERS, A.G., Phys. Plasmas 5 (1998) 763.[15] PEETERS, A.G., Plasma Phys. Control. Fusion 40 (1998) 703.[16] SHAING, K.C., YOKOHAMA, M., WAKATANI, M., HSU, C.T., Phys. Plasmas 3(1996) 965.[17] MCCARTHY, D.R., DRAKE, J.F., GUZDAR, P.N., HASSAM, A.B., Phys. FluidsB 5 (1993) 1188.[18] SCOTT, B., \Three Dimensional Computation of Collisional Drift Wave Turbulenceand Transport in Tokamak Geometry", Proceedings of the 23rd EPS Conference onControlled Fusion and Plasma Physics" (EPS, Kiev, 1996), a005, p 54.[19] XU, X., COHEN, R., et al, these Proceedings, paper IAEA-F1-CN-69 THP 2/03.[20] JENKO, F., SCOTT, B., Phys. Rev. Lett. 80 (1998) 4883.[21] PARKER, S.E., Lee, W.W., Phys. Plasmas 5 (1993) 77.[22] SCOTT, B., Plasma Phys. Control. Fusion 39 (1997) 471.[23] BATEMAN, G., KRITZ, A., et al, these Proceedings, paper IAEA-F1-CN-69 THP2/19.[24] LODESTRO, L.L., et al, these Proceedings, paper IAEA-F1-CN-69 THP 2/04.[25] SCHNEIDER, R., COSTER, D.P., et al, these Proceedings, paper IAEA-F1-CN-69THP 2/05.60 2•1019 4•1019 6•1019 8•1019 1•10200200400600n   (a - 2 cm)    [m   ] -3Ideal MHDunstableRadiation unstableL-modeH-modetype III ELMsELM-free H-mode(not stationary)Type I ELMsCompound ELMsType III ELMsL-ModeL Density LimitHASDEX UpgradeIdeal ballooning limitIp = 1 MAIp = 0.8 MAebT     (a - 2 cm)  [eV] eb ν = 0.5β̂ = 510 2030Fig. 1. (left) Flux surface geometry; the thick line denotes the s95 surface atwhich the edge uxtube metric is calculated. The reference scale is 50 cm.(right) Edge parameter diagram, showing some of the ̂ values used, and the = 0:5 line.Fig. 2. Edge transport results. (left) particle and (center) total heat trans-port in the ASDEX Upgrade ux surface model, for several ̂ as a function ofnormalised collision frequency ( is about 0:06  ). Heat transport in thesimple ballooning geometry (right) is strongly overestimated, especially by theappearance of ideal ballooning for ̂ > 5; diagnosis is in the next gure.7Fig. 3. Mode structure in the simple ballooning geometry, for (left to right)̂ = 1:0, 2.0, 5.0, and 10.0, respectively. Phase shifts () between the densityand potential uctuations for each component wave (ky), shown as probabilitydistribution functions. Contours 0.3, 0.5, and 0.7 are shown. Drift wave modestructure (left) yields to ideal ballooning (drive at lowest ky, followed by aninertial range) for ̂ > 5:0.Fig. 4. Core transport results, for particles and electron and ion energy, in theASDEX Upgrade ux surface model at the 0,6 normalised minor radius, for̂ = 1:0 (the actual value) and 0.1, showing the inuence of rn through theAlfven dynamics. The value of  here is 0.01.Fig. 5. Reynolds stress eects in the edge turbulence, preliminary result of thenonlocal model. (left) Inward diamagnetic vorticity transport, resulting fromthe outward ion heat transport. (center) The ExB Reynolds stress by itself issmall. (right) Small scale diamagnetic vorticity builds up at the inside edgeof a bounded domain, possibly leading to a strong ExB shear layer as at largescales the contribution of the diamagnetic to the total perpendicular ow issmall.8

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Self-Consitent Computation of Transport Barrier Formation by Fluid Drift Turbulence in Tokamak Geometry

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