Beidler, C.

Helander, P.

Kasilov, S.

Kernbichler, W.

Maassberg, H.

Marushchenko, N.

Turkin, Y.

English

MPG.PuRe

Calculations of ECCD with parallel momentum conservation and finite collisionality

Calculations of ECCD with parallel momentum conservationand finite collisionalityN.B. Marushchenko1, C.D. Beidler1, P. Helander1, S.V. Kasilov2,3,W. Kernbichler3, H. Maassberg1, Y. Turkin11 Max-Planck-Institut für Plasmaphysik, IPP-EURATOM Association, Greifswald, Germany2 Institute of Plasma Physics, NSC KIPT, Kharkov, Ukraine3 Institute für Theoretische Physik, Association EURATOM-ÖAW, TU-Graz, AustriaIntroduction . The adjoint approach [1] is an advanced and convenient method for calculationof the ECCD in plasmas which is applicable for both tokamaks andstellarators. A key pointof the adjoint approach is the choice of model for the corresponding Spitzer function, whichshould conserve parallel momentum in like-particle collision and must be valid for desiredrange of collisionality. The highly collisional (classical) limit, ν∗e → ∞ (here,ν∗e = νeR/ιv isthe monoenergetic collisionality,νe(v) = νee(v) + νei(v) is the collision frequency,R andι arethe major radius and the rotational transform, respectively), where trapped particles do not playany role, gives the upper limit for CD efficiency, while the oppsite collisionless limit,ν∗e → 0,which accounts for trapped particles, tends to underestimate it. The latter, for which rapid andaccurate solvers have been developed recently (see [2] and references therein), is appropriatefor calculations of ECCD in hot plasmas. The intermediate collisi nal regime, with small butfinite collisionality, ν⋆e ≪ 1, where the contribution of barely trapped electrons can also benon-negligible, requires special attention [3].In this work, we consider the generalized Spitzer problem inview of tools for calculation ofECCD efficiency suitable for implementation in ray-tracing codes.Description of the model.For calculations of ECCD efficiency, the (adjoint) drift kineticequation must be solved [1],v‖∇‖(χfeM) + C lin(χfeM) = −νe0v‖vthb feM . (1)the general solution of which describes all necessary physics including the finite collisionality.Here,χ(r;v) is the local response function andC lin is the linearized collision operator withconservation of parallel momentum. This problem is 4D (3D for tokamaks) and to date there areno well-tested solvers which are generally applicable. TheNEO-2 code (see [4] and referencestherein) solves this problem by the field-line-tracing technique for all collisionalities, but thepresent version is applicable only for 3D problem and is not yet suitable for stellarators.As an alternative approximation, the momentum correction technique (mct) [5] can be ap-plied. This model is based on the concept of a collisionality-dependent “effective trapped parti-cle fraction”,f efftr (ν⋆e (x)) with the limitsfefftr (ν⋆e → ∞) = 0 andf efftr (ν⋆e → 0) = ftr, which de-scribes the integral effect of finite collisionality (here,ftr the “geometrical” fraction of trapped38th EPS Conference on Plasma Physics (2011) P2.102particles). This function can be constructed from the neoclassical monoenergetic transport co-efficients, calculated by DKES (for details, see [5]). In this model, the solution of Eq.(1) isrepresented (similar to the collisionless solution) as a product of functions only of the globalvariables,x = v/vth andλ = v2⊥/(v2b) with b = B/Bmax,χeff(x, λ) = sign(v‖)Heff(x; λ) Keff(x). (2)In contrast to the collisionless solution,Heff(x; λ) contains also the monoenergetic collisionality(i.e. velocity) as a parameter,Heff(x; λ) = 11− f efftr (ν⋆e (x))((1− ftr)H(λ) + h(1− λ)δf efftr (x))(3)(here,δf efftr (x) = fefftr (ν⋆e (x)) − ftr, H(λ) is the collisionless pitch-dependence (see [2]), andh(y) is the Heaviside function). WhereasH(1) = 0, at finite but lowν⋆e a trapped-passing-particle boundary layer of width∝ √ν⋆e appears allowing for an additional nearly constantcontribution only in the passing domain. This simple “off-set” modeling is very reasonableoutside of this boundary layer, but fails within the layer. The contribution within the layer scalesasν⋆e and is negligible compared to the “off-set” modeling withδfefftr ∝√ν⋆e . The generalizedSpitzer function,Keff(x), which takes into account also the monoenergetic viscositydue to thefinite collisionality, can be caclulated from a 1D integro-diffential equation (for details, see [5]).This approximate model can be easily applied for ray-tracing alculations. Despite its limitedapplicability (contrary to NEO-2, the model is applicable only for low collisionality,ν⋆e ≪ 1),this model is very attractive, being applicable for arbitrary configurations. This approximatemodel can be easily applied for ray-tracing calculations. Despite its limited applicability (con-trary to NEO-2, the model is applicable only for low collisionality, ν⋆e ≪ 1), this model is veryattractive, being applicable for arbitrary configurations.Qualitatively, the impact of finite collisionality in ECCD arises in two ways. The first is areduction of drag due to the barely trapped electrons and thesecond is a direct contribution ofbarely trapped electrons (if allowed by the cyclotron resonance condition). The leading orderfinite collisionality correction in ECCD scales as [3]δj‖ ∝√ν⋆e · jcoll‖ (here,jcoll‖ is the currentin collisional limit). Both these effects are described by the local solution of Eq.(1) and partlyrepresented in the approximate solution Eq.(2). Below, we apply the “off-set” mct-model forcircular tokamaks and benchmark it against the NEO-2 code.Numerical results. In Fig. 1, the ECCD efficiency for X2-mode with fixed obliqueness asfunction of (normalized) magnetic field,nY = nωce/ω with n = 2 is shown. Two points oflocation of the RF-source withN‖ = 0.256 (β = 15◦) are chosen, which correspond to theminimum (left) and maximum (right) ofB. The plasma parameters chosen guarantee a ratherlarge collisionality effects. Apart from the models with finite collisionality effects (labeled as38th EPS Conference on Plasma Physics (2011) P2.1020.97 0.98 0.99 1.00-0.20.00.20.40.6 = <j ||>/<p abs>nY = n ce/minB: hsl pmc mct neo-20.98 0.99 1.00 1.01 1.02 1.03-0.20.00.20.40.6 = <j ||>/<p abs>nY = n ce/maxB: hsl pmc mct neo2Figure 1:The local ECCD efficiency,η = 〈j‖〉/〈pabs〉, calculated by the different models for X2-mode withN‖ = 0.256 for a tokamak withǫ = r/R = 0.2 in Bmin andBmax points (left and right,respectively). Plasma parameters:ne = 1020 m−3, Te = 1 keV andZeff = 1.“neo-2” and “mct”), also the collisionless model with parallel momentum conservation (“pmc”)and its high-speed-limit (“hsl”) were applied as reference.When only passing electrons are in resonance, both the “off-set” mct-model and NEO-2give a similar deviation from the collisionless solution. If, however, the resonance line touchesthe passing/trapped boundary, current diffusion from passing to barely trapped domain shouldappear. In Fig.1, this point corresponds to a sharp change ofthe slope in collisionless and mct-solutions. At the same time, the local solution from NEO-2 correctly describes the current dif-fusion and this is well seen from the smoothed transition. For the maximum ofB, the “off-set”mct-model somewhat overestimates the ECCD efficiency in comparison with the NEO-2 re-sults. Nevertheless, the qualitative dependence is the samas with NEO-2.0.4 0.5 0.6 0.7 0.8 0.90123456dPabs/dl,  MW/mray length, m passing trapped totalFigure 2:dPRF/dl for 1 MW O2-mode,140 GHz,B0 = 2.5 T, in W7-X for ne =1020 m−3, Te = 2 keV, Zeff = 1.5.The role of finite-collisionality effects in W7-Xwas checked for both the X2- and O2-scenarios. Thelaunch-port is located near the “bean-shaped” planeand the RF-beam is directed to the inner-wall mirrorwith N‖ = 0.25. The calculations were performed bythe ray-tracing code TRAVIS (see [3])with the “off-set” mct-model implemented (the NEO-2 code is notyet applicable for stellarators).For the O2-mode, the plasma is optically “gray” andparticipation of trapped electrons in the cyclotron in-teraction is unavoidable. Despite launching near themaximum ofB, trapped electrons can absorb a significant part of the injected power. As ex-ample, the deposition along the ray is shown in Fig.2, where approximately 13% of the poweris absorbed by trapped electrons.38th EPS Conference on Plasma Physics (2011) P2.1020 2 4 6 8 100.00.51.01.52.02.5   O2, coll.less   O2, coll   O2, fin.collCD efficiency, *Te(0),  keV   X2, coll.less   X2, coll   X2, fin.coll0 2 4 6 8 100.00.10.20.30.40.50.60.7I fin.col. / I col.lessTe(0), keV X2 O2Figure 3:Left: Te-dependence of the total ECCD efficiency,ξ∗ = 32.7×10−20ne(0)R/Te(0)·Itor/PRF,calculated for both X2- and O2-modes withN‖ = 0.25 in W7-X with ne = 1020 m−3 andZeff = 1.5.Right: relative contribution to ECCD of the finite collisionality. Only single pass is considered.In order to check the importance of finite collisionality, the electron temperature is scannedwith all other parameters fixed. For comparison, both the X2-mode and the O2-mode were con-sidered. In Fig.3 (left), the total ECCD efficiency calculatedwith finite collisionality is plotted.For reference, both analytical collisional and collisionless limits are depicted. In Fig.3 (right),the impact of finite collisionality in ECCD,δIfc/Icl, is shown (here,δIfc = δIfc−Icl with Ifc andIcl are the toroidal current values calculated with and withoutfinite collisionality, respectively).The different electron temperatures correspond to different locations of the cyclotron res-onance line in velocity-space. Analysing the energy of electrons responsible for the domi-nant contribution to ECCD, one can estimate the (monoenergetic) collisionality for the reso-nance electrons. The effect of finite collisionality contributes to the ECCD practically up toTe = 10 keV. Nevertheless, since the applicability of the “off-set” mct-model is limited only bythe rangeν⋆e ≪ 1, the results forTe < 2 keV are rather doubtful.Summary. Finite collisionality effects in ECCD are important not onlyfor low but alsofor high electron temperature plasmas, especially in stellarators. Both the NEO-2 code (slow& precise) and the “off-set” mct-model (“quick & dirty”) produce qualitatively similar results.Large ECCD can be obtained only for low collisionality, and here the leading order collisionalitycorrection is calculated.References[1] T.M. Antonsen and K.R. Chu, Phys. Fluids25, 1295 (1982)[2] N.B. Marushchenko et al., Phys. Plasmas18, 032501 (2011)[3] Y.R. Lin-Liu et al., Proc. 26th EPS Conf. on Contr. Fusion andPlasma Physics, Maastricht,14 - 18 June 1999 ECA Vol.23J (1999) 1245 - 1248, P3.078[4] W. Kernbichler et al., Contrib. Plasma Phys.50, 761 (2010)[5] H. Maaßberg, C.D. Beidler and Y. Turkin, Phys. Plasmas16, 072504 (2009)38th EPS Conference on Plasma Physics (2011) P2.102

http://hdl.handle.net/21.11116/0000-0008-AEC5-6

Calculations of ECCD with parallel momentum conservation and finite collisionality

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