Kasilov, S.

Kernbichler, W.

Maassberg, H.

Martitsch, A.

MPG.PuRe

Evaluation of non-ambipolar particle fluxes driven by externalnon-resonant magnetic perturbations in a tokamak ∗A. F. Martitsch1, S. V. Kasilov1,2, W. Kernbichler1, H. Maassberg31 Fusion@ÖAW, Institut für Theoretische Physik - Computational Physics,TU Graz, Petersgasse 16, A–8010 Graz, Austria2 Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics andTechnology”, Akademicheskaya Str. 1, 61108 Kharkov, Ukraine3 Max-Planck Institut für Plasmaphysik, D-17491 Greifswald, GermanyIntroductionThe toroidal torque driven by external non-resonant magnetic perturbations (neoclassicaltoroidal viscosity) is an important momentum source affecting the toroidal plasma rotation intokamaks. The well-known force-flux relation directly links this torque to the non-ambipolarneoclassical particle fluxes arising due to the violation of the toroidal symmetry of the mag-netic field. A universal approach for this problem is usually provided by δ f Monte Carlomethods [1, 2] where the linearized drift-kinetic equation (LDKE) is solved directly (withoutbounce-averaging). Here, a more efficient universal approach for the numerical computationof these fluxes (without model simplifications of the LDKE) is described where the problemdimension is reduced by one using the quasilinear approximation. The only limiting conditionis that the non-axisymmetric perturbation field is small enough such that the effect of the per-turbation field on particle motion within the flux surface is negligible. Therefore, in addition tomost of the transport regimes described by the banana (bounce averaged) kinetic equation [3, 4]also such regimes as, e.g., ripple-plateau and resonant diffusion regimes are naturally includedin this approach. The approach has been realized in the code NEO-2 [5] for general tokamakgeometry using the full linearized collision operator. The quasilinear approach and the resultsof benchmarking the code NEO-2 with various existing models are the topic of this report.DefinitionsIn Boozer coordinates (r,ϑ ,ϕ) with re-defined flux surface label, 〈|∇r|〉 = 1, the poloidal andtoroidal components of the ion fluid velocity in the covariant notation areV ϑ =ck Bϕei√g〈B2〉dTidr, V ϕ =c√gBϑ(Er−1einid(niTi)dr)+qV ϑ , (1)where c, ei, ni, Ti, Er,√g, q and 〈. . .〉 are speed of light, ion charge, ion density, ion tem-perature, radial electric field, metric determinant, safety factor and the “flux surface” average,respectively. Here the coefficient k = 5/2−D32/D31 is determined by the neoclassical transportcoefficients D jk, which link the thermodynamic forces Ak and fluxes I j by the relationsI j =−nα3∑k=1D jkAk. (2)The thermodynamic fluxes are expressed through flux surface averaged particle (I1 =Γα ) andheat flux density (I2 =Qα/Tα ) and through the parallel flow density (I3 =nα〈V‖αB〉), whereas∗This work, supported in part by the European Commission under the contract of Associations between EU-RATOM and the Austrian Academy of Sciences, was carried out within the framework of the European FusionDevelopment Agreement. The views and opinions expressed herein do not necessarily reflect those of the EuropeanCommission. Andreas Martitsch is a Fellow of the Friedrich Schiedel Foundation for Energy Technology and theirsupport is gratefully acknowledged. The authors also gratefully acknowledge support from NAWI Graz.41st EPS Conference on Plasma Physics P1.049the thermodynamic forces are specified byA1 =1nαdnαdr− eαErTα− 32TαdTαdr, A2 =1TαdTαdr, A3 =eαTα〈E‖B〉〈B2〉 , (3)where α denotes the species and E‖ is the inductive electric field. The evolution of the radialelectric field is described by a simplified toroidal rotation equation neglecting contributions forneutral beam injection or other external sources∂∂ t ∑αmα〈gϕϕnαVϕα〉+1S∂∂ rSΠrϕ = TNAϕ , (4)where mα is α species mass, Πrϕ is the momentum flux density due to neoclassical and anoma-lous transport and T NAϕ is the density of the toroidal torque driven by external non-axisymmetricmagnetic field perturbations. This toroidal torque density is linked with non-ambipolar particlefluxes ΓNAα via the flux-force relation,T NAϕ =−1c√gBϑ ∑αeαΓNAα = −νtmini〈gϕϕ(V ϕ −V ϕin)〉, (5)whereby the rotation relaxation rate νt (toroidal viscosity frequency) and the “intrinsic” (“off-set”) rotation velocity V ϕin take a particular simple form if the electron particle flux is negligible,νt =e2i g(Bϑ)2 DNA11c2miTi〈gϕϕ〉 , V ϕin = ckNAei√gBϑ dTidr with kNA = DNA12DNA11− 52+B2ϕ k〈B2〉〈gϕϕ〉 . (6)Quasilinear ansatzFor the evaluation of non-ambipolar particle fluxes within the standard neoclassical approachthe solution of the LDKE (which is 4D) is required. Within quasilinear approach, which reducesthe problem dimension by one, the perturbation field is split into an axisymmetric and smallnon-axisymmetric part, B=B0 + δB. It is sufficient to find the solution to LDKE up to linearorder in the perturbation field amplitude, f1= f10 + f11. Expanding the non-axisymmetric partsof the magnetic field and of the solution in a Fourier series over a field aligned toroidal angle,ϕ0 = ϕ−qϑ ,δB(ϑ ,ϕ0) = Re∞∑n=1Bn(ϑ)einϕ0, f11(ϑ ,ϕ0) = Re∞∑n=1fn(ϑ)einϕ0, (7)the set of equations for the axisymmetric and the non-axisymmetric parts isL̂(0) f10 =mαcBϕv2 (2−ηB0)2eα√g0B30∂B0∂ϑ∂ fM∂ r, (8)L̂(n) fn =mαcBϕv2 (2−ηB0)2eα√g0B30(∂Bn∂ϑ− inB20BnBϑ0 Bϕ− 2BnB0∂B0∂ϑ)∂ fM∂ r+ v‖Bϑ0B20(BnB0∂B0∂ϑ− ∂Bn∂ϑ)η∂ f10∂η− BnB0L̂cL f10. (9)Here, the velocity space variables are (v,η) where η = v2⊥/(v2B0), fM is a local Maxwellian,L̂(n) ≡ v‖Bϑ0B0∂∂ϑ− L̂cL + in(ΩtE +ΩtB), (10)41st EPS Conference on Plasma Physics P1.049L̂cL is the full linearized collision operator and ΩtE and ΩtB are toroidal rotation frequencies dueto the electric and magnetic drift, respectively. The non-ambipolar particle flux density, whichdetermines the coefficients DNA11 and DNA12 , is expressed through Fourier amplitudes asΓNAα =∞∑n=1πmαc4eα√g0Bϑ0 2π∫0dϑB20−1 2π∫0dϑB20∞∫0dvv41/B0∫0dη2−ηB0√1−ηB0 ∑σ=±1n Im fnB∗n. (11)In NEO-2, the dependence of f10 and fn on the velocity module v is discretized by a seriesexpansion over associated Laguerre polynomials of the order 3/2 and the resulting set of coupled2D equations is solved using a finite-difference scheme on an adaptive (ϑ ,η) grid.Benchmarking resultsFor benchmarking, a tokamak configuration with circular cross-section and aspect ratio A = 3.8is used. The results for the full linearized collision model correspond here to the ion component.The perturbation field is taken in the form of a single harmonic, δB= εMB0(r,ϑ)cos(mϑ +nϕ).Since transport coefficients have a simple, quadratic dependence on εM, this quantity has beenset to 1 in all plots below. For the non-linear models, such as the DKES code [6], results areobtained for εM = 10−3 and are then rescaled. Results of benchmarking with the DKES codeare presented in Fig. 1. There scans over the collisionality parameter ν∗ = 2νqRv−1T of thediffusion coefficient D11 normalized to the mono-energetic plateau diffusion coefficient Dp =πqvT ρ2L/16R are shown for various perturbation modes and for various radial electric fieldsgiven in terms of toroidal Mach numbers (normalized toroidal rotation velocity values) Mt =ΩtERv−1T . Besides bounce-averaged regimes the ripple plateau and resonant diffusion regimesare clearly reproduced by both codes.10−8 10−6 10−4 10−2 100108106104102ν∗D11/DpA=10, m=0, n=1NEO-2DKES10−8 10−6 10−4 10−2 100108106104102100ν∗D11/DpA=10, m=0, n=3NEO-2DKES10−8 10−6 10−4 10−2 100108106104102100ν∗D11/DpA=10, m=0, n=18NEO-2DKES10−8 10−6 10−4 10−2 1001010108106104102100ν∗D11/DpA=10, m=−3, n=3NEO-2DKESFig. 1. Normalized coeffi-cient D11D−1p from NEO-2 (Lorentz collision model)and DKES [6] as a func-tion of collisionality ν∗ forvarious perturbation modesand toroidal Mach numbersMt = 0 (♦), 2.8 · 10−7 (M),2.8 · 10−6 (), 2.8 · 10−5(◦), 2.8 · 10−4 (×), 2.8 · 10−3(+) and 2.8 · 10−2 (?). Thetoroidal rotation frequencydue to magnetic drift is setto zero for the four casesshown above. Aspect ratioand mode numbers are indi-cated in the titles.Results of the computation with the full linearized collision operator and the comparison tothe universal formula of Shaing et al [3] are shown in Fig. 2. A good agreement is seen in thedomain of validity of the bounce-averaged approach.41st EPS Conference on Plasma Physics P1.04910−7 10−5 10−3 10−1 10110−2100102104106108ν∗D11/DpA=10, m=0, n=3NEO-2Shaing10−7 10−5 10−3 10−1 10110810610410210010−2ν∗D11/DpA=10, m=0, n=18NEO-2Shaing10−7 10−5 10−3 10−1 101−0.500.511.522.533.5ν∗kNAA=10, m=0, n=3NEO-210−7 10−5 10−3 10−1 10100.511.522.533.5ν∗kNAA=10, m=0, n=18NEO-2Fig. 2. Normalized coeffi-cient D11D−1p from NEO-2(full collision model) andbounce-averaged model ofShaing [3] (upper panel) and“offset” rotation coefficientkNA from NEO-2 (lower panel)as functions of collisionalityν∗ for various perturbationmodes and toroidal Machnumbers Mt = 0 (♦), 2.8 ·10−7(M), 2.8 · 10−6 (), 2.8 · 10−5(◦), 2.8 · 10−4 (×), 2.8 · 10−3(+) and 2.8 · 10−2 (?). Theripple-plateau diffusion co-efficient [7] is shown with adashed line at the upper panel.The comparison of NEO-2 results with the full collision operator in the superbanana plateauregime to the respective asymptotic formula of Shaing et al [3] is shown in Fig. 3. The toroidalmagnetic drift is fixed by setting cTα (eαψa)−1 = |ΩtE |where ψa is the toroidal flux at the edge.10−9 10−7 10−5 10−3 10−1 101102104106108ν∗D11/DpA=10, m=0, n=3Mt=10−7Mt=10−6Mt=10−5NEO-2Shaing10−9 10−7 10−5 10−3 10−1 101102104106108ν∗D11/DpA=10, m=0, n=3Mt=−10−7Mt=−10−6Mt=−10−5NEO-2ShaingFig. 3. Normalized coefficientD11D−1p from NEO-2 (full col-lision model) and asymptoticalformula of Shaing [3] as a func-tion of collisionality ν∗ for var-ious toroidal Mach numbers.Aspect ratio and mode numberare indicated in the title.ConclusionsThe developed quasilinear version of the drift kinetic equation solver NEO-2 allows for an ef-ficient evaluation of the non-ambipolar particle flux, which determines the torque, in the wholecollisionality range. The only limiting assumption of sufficient smallness of the perturbationfield is valid in most cases of practical interest. The code has been benchmarked with a fewanalytical and semi-analytical models as well as with the DKES code and stays in good agree-ment in the validity domain of those models. As shown in the benchmarking, bounce averagedmodels can give significant errors already in case of mild toroidal Mach numbers (Mt ∼ 0.03 inthe examples here) not only for short scale perturbations typical for the toroidal field ripple butalso for medium scale perturbations such as perturbations produced by ELM mitigation coils.References[1] S. Satake, J.-K. Park, H. Sugama, and R. Kanno, Phys. Rev. Letters 107, 055001 (2011).[2] K. Kim, J.-K. Park, and A. H. Boozer, Phys. Rev. Letters 110, 185004 (2013).[3] K. C. Shaing, S. A. Sabbagh, and M. S. Chu, Nuclear Fusion 50, 025022 (2010).[4] Y. Sun et al., Phys. Rev. Letters 105, 145002 (2010).[5] W. Kernbichler et al, Plasma and Fusion Research 3, S1061 (2008).[6] S. P. Hirshman et al, Phys. Fluids 29, 2951 (1986).[7] A. H. Boozer, Phys. Fluids 23, 2283 (1980).41st EPS Conference on Plasma Physics P1.049

Evaluation of non-ambipolar particle fluxes driven by external non-resonant magnetic perturbations in a tokamak

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