Kapper, G.

Kasilov, S.

Kernbichler, W.

Marushchenko, N.

MPG.PuRe

Modeling of electron cyclotron current drivefor finite collisionality plasmas in Wendelstein 7-XG. Kapper1, W. Kernbichler1, S.V. Kasilov1,2, N.B. Marushchenko31 Fusion@ÖAW, Institute of Theoretical and Computational Physics, TU Graz, Austria2 Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics andTechnology”, Ukraine3 Max Planck Institute for Plasma Physics, EURATOM Association, GermanyIntroductionThe low-shear concept of Wendelstein 7-X avoids low-order rational flux surfaces in theplasma core since large magnetic islands can be formed there. In turn, such a resonant surfaceis placed at the plasma edge, thus producing an island divertor configuration. In order to pre-serve these features, which can be destroyed even by the arise of a relatively small bootstrapcurrent, this current should be compensated by electron cyclotron current drive (ECCD) [1]. Inmodern fusion devices the evaluation of the generalized Spitzer function (GSF), which servesas an ECCD efficiency, is usually performed in the low collisionality limit typical for these de-vices, where one can effectively use the bounce averaged kinetic equation. At the same time, athigh densities and mild electron temperatures, which can be expected at the initial stage of thedevice operation and at the start up of plasma discharge, finite collisionality effects should betaken into account. Such an approach requires the solution of the high dimensional (4D) drift ki-netic equation, which is a more difficult task as compared to the 2D bounce averaged equation.In this work, the impact of finite collisionality effects on ECCD in a high-mirror configurationof Wendelstein 7-X is studied using a combination of the codes NEO-2 [2], a 4D drift kineticequation solver for plasmas with finite collisionality using the full linearized Coulomb collisionoperator and realistic device geometry, and TRAVIS [3], a ray-tracing code developed for stud-ies of electron cyclotron current drive, heating and emission. Recently, this code combinationwas used to study the influence of finite collisionality on ECCD in tokamaks, where the gener-alized Spitzer problem at a given magnetic flux surface is essentially 3D [2]. The 4D problem ofstellarators led to the development of a new interface between the codes based on precomputeddata of NEO-2 for a realistic magnetic model from an equilibrium code and plasma parameterprofiles from a transport code. As in Ref. [2], the non-relativistic limit is used here for the GSF,which is compared to asymptotic limits of low and high plasma collisionality.42nd EPS Conference on Plasma Physics P1.164Code combination within the adjoint approachA commonly used technique for current drive modeling is the adjoint approach [4], wherethe flux surface averaged parallel current density,〈 j‖B〉= e lc〈∫d3 p∂ ḡ∂p·ΓRF〉, (1)is expressed by an adjoint GSF ḡ computed by NEO-2 and a microwave induced quasilineardiffusion flux in phase space ΓRF computed by TRAVIS, where 〈. . .〉 denotes flux surface av-erage, e is the electron charge, B is the magnetic field module and lc is the mean free path [2].The adjoint function is related by ḡ(v‖) = −g(−v‖) to the usual GSF g, which is a solution tothe conductivity problem written in terms of integrals of motion in velocity space as follows,v‖h ·∇ fMg− L̂CL fMg =Blcv‖ fM. (2)Here, v‖ is the parallel velocity, h is a unit vector along the magnetic field, fM is a Maxwellianand L̂CL is a linearized collision integral. Cross-field rotation has been ignored in Eq. (2) becauseit has small effect on the GSF in finite collisionality regimes as well as at low collisionalitieswhere the electrons are mostly in the 1/ν regime. Therefore, a 4D problem (2) can be reducedto a 3D problem for a single field line (position on field line is given by a line parameter ϕs)which is long enough to cover the given flux surface (labeled by the normalized toroidal fluxvalue s) densely. Representing the energy dependence of the GSF g(r,p) by a series expansionover normalized Laguerre polynomials of the order 3/2 (Sonine polynomials) L(3/2)m (z), thesolution to Eq. (2) isg(r,p) =M∑m=0gm(ϕs,η)L(3/2)m (z) , (3)where η = p2⊥/(p2B) and z = p2/(2mT ) = mv2/(2T ) are the normalized perpendicular adi-abatic invariant and energy, respectively. The GSF expansion coefficients gm(ϕs,η) are com-puted by NEO-2 using a conservative finite difference scheme on a 2D grid over ϕs and η withan adaptively refined η-grid in order to resolve the trapped-passing boundary layer. The spatialdependence of g on Boozer coordinates (s,ϑ ,ϕ) required by TRAVIS is reconstructed by aninterface with the help of linear interpolation on a non-uniform 3D grid formed by (discretizedover ϕs) segments of (long enough) field lines representing a prescribed set of flux surfaces.ResultsIn this work, two ECCD scenarii in W7-X, namely the X2- and the O2-scenario, are investi-gated for three different collision models, i.e., long mean free path (LMFP) limit by the internal42nd EPS Conference on Plasma Physics P1.1640 0.2 0.4 0.6 0.8 1024681012sne[1013cm−3],Te[keV],Zeffplot 1sdfplot 2plot 30.60.70.80.91ι-Density neTemperature TeEffective charge ZeffRotational transform ι-1Figure 1: Temperature, density, effectivecharge and rotational transform profilesover the normalized toroidal flux.TRAVIS model, finite collisionality by NEO-2 and thehigh-collisional limit using the classical Spitzer func-tion [5]. Profiles of plasma parameters shown in Fig. 1correspond to the onset of the LMFP regime (colli-sionality is roughly three times smaller than needed forthe plateau regime). The microwave beam is launchedoff axis (Z = −0.1 m) in the horizontal plane with thepoloidal launch angle α = 5.5 deg (angle between thisplane and the beam). Dependencies of the total currentgenerated in the plasma on the toroidal launch angle β(angle between the beam and the meridian plane containing the launcher) are shown in Fig. 2.It can be seen that in the standard X2-scenario the current is by 25% different from the LMFPlimit, as it can be expected, while for the advanced O2-scenario the current significantly differsfrom both, LMFP and high collisionality limits. In the latter case, characterized by weak absorp-tion and respectively smaller current, a significant part of this current is produced by particlesfrom the trapped region. This is better seen from Fig. 3 where the distributions of the absorbedpower and generated current along the microwave beam are shown for β = 18.5 deg. Due to thestrong absorption in the X2 case, most power is absorbed by strongly passing particles, whosedistribution function is not significantly different from the GSF in the LMFP limit, before thebeam reaches the cold resonance. In the O2-scenario weak absorption results in power absorp-tion and respective counter-current by passing particles on the other side of cold resonance, aswell as in a significant amount of wave energy absorbed by trapped particles whose current ismainly determined by the position of the absorption region but not significantly by the parallelwave number [6, 2]. This can be used to increase the total toroidal current.−30 −20 −10 0 10 20 30−4−3−2−101234·10−3β [deg]I tor/Pinj[A/W]Low collisionalFinite collisionalHigh collisional1−30 −20 −10 0 10 20 30−8−6−4−202468·10−4β [deg]I tor/Pinj[A/W]Low collisionalFinite collisionalHigh collisional00.20.40.60.81Pabs/PinjPabs/Pinj1Figure 2: Total toroidal current by passing (dashed), trapped (dotted) and all (solid) particles as functionof the toroidal launch angle β for X2-mode (left) and O2-mode (right). The fraction of the absorbedpower Pabs/Pinj is shown for the O2-mode only, since for the X2-mode there is total absorption.42nd EPS Conference on Plasma Physics P1.1640.66 0.68 0.7 0.72051015202530354045l [m]dP/dl[MW/m]Power absorption (X2)10.65 0.7 0.75 0.8 0.8501234567l [m]dP/dl[MW/m]Power absorption (O2)10.66 0.68 0.7 0.72−150−100−50050100l [m]dI/dl[kA/m]Low collisionalFinite collisionalHigh collisional10.65 0.7 0.75 0.8 0.85−14−12−10−8−6−4−20246l [m]dI/dl[kA/m]Low collisionalFinite collisionalHigh collisional1Figure 3: Local power absorption (upper plots) and local current generation (lower plots) for X2-mode(left) and O2-mode (right) by passing (dashed), trapped (dotted) and all (solid) particles along the raytrajectory. Cold resonance position at l = 0.73 m is marked by a vertical (cyan) line.ConclusionThis report extends the ECCD studies for tokamaks [2] to stellarator geometry. The studiesof the X2- and O2-scenarii in a high-mirror configuration of W7-X at the onset of the LMFPregime show that in the O2-scenario finite collisionality effects play a significant role on ECCDcurrent, which cannot be approximated by asymptotical collisionality limits. This effect is sig-nificant also in the X2-scenario, where the current is increased by 25%. Properties of ECCD ontrapped particles can be used to improve the efficiency in advanced (O2, X3) scenarii, whichare only available at high densities, where the cut-off prohibits the standard (O1, X2) scenarii.AcknowledgmentsThis work has been carried out within the framework of the EUROfusion Consortium andhas received funding from the Euratom research and training programme 2014-2018 under grantagreement No 633053. The views and opinions expressed herein do not necessarily reflect thoseof the European Commission. The authors gratefully acknowledge support from NAWI Graz.References[1] M. Romé, et al., Plasma Phys. Control. Fusion 40(4), 511 (1998)[2] W. Kernbichler, et al., EPJ Web of Conferences 87, 01006 (2015)[3] N. B. Marushchenko, et al., Comput. Phys. Commun. 185(1), 165 (2014)[4] T. M. Antonsen and K. R. Chu, Phys. Fluids 25(8), 1295 (1982)[5] L. Spitzer, Jr. and R. Härm, Phys. Rev. 89(5), 977 (1953)[6] P. Helander and P. J. Catto, Phys. Plasmas 8(5), 1988 (2001)42nd EPS Conference on Plasma Physics P1.164

Modeling of electron cyclotron current drive for finite collisionality plasmas in Wendelstein 7-X

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