Biancalani, A.

Bottino, A.

Carlevaro, N.

Di Siena, A.

Görler, T.

Montani, G.

Novikau, I.

Zarzoso, D.

MPG.PuRe

Nonlinear gyrokinetic investigation of energetic particle driven geodesic acoustic modes

Nonlinear gyrokinetic investigation of energetic particle driven geodesicacoustic modesA. Biancalani1, A. Bottino1, N. Carlevaro2, A. Di Siena1, T. Görler1, G. Montani2, I. Novikau1,and D. Zarzoso31 Max-Planck-Institut für Plasmaphysik, 85748 Garching, Germany2 ENEA C.R. Frascati - ENEA for EUROfusion, C.P. 65 - 00044 Frascati, Italy3 Aix-Marseille Université, PIIM, UMR 7345, 13397 Marseille, FranceIntroductionZonal (i.e. axisymmetric) radial electric fields, associated to zonal poloidal flows, are knownto develop in tokamak plasmas in the presence of turbulence.These zonal structures (ZS), areobserved as zero frequency zonal flows (ZFZF), nearly station ry, or finite-frequency geodesicacoustic modes (GAM) [1, 2]. In the presence of energetic particles (EP), ZS can also be excitedby inverse Landau damping, as EP-driven GAMs (EGAM) [3, 4, 5,6, 7].Here, we investigate the nonlinear dynamics of EGAMs, in theabsence and in the presenceof turbulence (see also Ref. [8]). The global gyrokinetic (GK) particle-in-cell code ORB5 isused. ORB5 was originally developed for ITG turbulence studies [9] and has more recentlybeen extended to an electromagnetic multispecies version [10]. Here, we focus on linear andnonlinear electrostatic collisionless simulations, where kinetic electron effects are neglected.EquilibriumWe use the same equil. as in Ref. [11]. The major radius isR0 = 1.0 m, the minor radiusis a = 0.3125 m, and the toroidal magnetic field at the axis isB0 = 1.9 T. In Ref. [11], flatq,n, T profiles were considered. Here, the safety factorq is given byq(ρ) = 1.74+1.3ρ2, withq(ρr = 0.45) = 2 at the reference radial position, corresponding tosr =√ψr/ψedge= 0.5 (withψ being the poloidal magnetic flux).The ion and electron temperatures are equal, withρ∗ = ρs/a= 0.00571 andρs= cs/Ωi beingthe sound Larmor radius (and withcs =√Te/mi being the sound velocity). Times are normal-ized to the ion cyclotron frequencyΩ−1i . The temperature and density gradients normalized tothe minor radiusa areκT = 3.7 andκn = 0.8 at s=0.5.The EP temperature profile is flat. The EP density will be considered either flat or with agaussian shape centered at the reference radiussr , with width δ/a= 0.1, wherea is the minorradius. In velocity space, the EP has a double bump-on-tail distribution in parallel velocity, as inRef. [5, 11] with the same normalized temperatureT̂h = Th/Ti = 1 and normalized mean parallel45th EPS Conference on Plasma Physics P2.1003velocity ζ̄ = v‖/vti = 4. Neumann and Dirichlet boundary conditions are imposed tothe scalarpotential, respectively at the inner and outer boundaries,s= 0 ands= 1.Linear dynamicsFor the equilibrium described in the previous section, and for a EP concentration ofnEP/ne=0.08, linear EGAMs simulations are performed. The linear frequency and growth rate of theEGAM for this EP concentration isωEGAM= 2.36·10−3Ωi = 0.94ωs= 4.3·105 rad/s,γEGAM=0.31·10−3Ωi = 0.12ωs= 0.56·105 rad/s.A study of the radial structure is also performed with linearEGAM simulations in the ab-sence of turbulence. Firstly, flat equilibrium radial profiles are considered for all species, likein the previous papers where EGAMs were studied with ORB5. The localized initial EGAMperturbation is observed to evolve in time, with decreasingvalue ofkr . On the contrary, when aradially localized EP density profile is introduced, the EGAM radial structure forms with char-acteristic width given by the EP profile (see Fig. 1). This behavior reflects the nature of EGAMsas forced modes, analogously to EPM for Alfvén-type modes.When we introduce non-flat density and temperature profiles of the thermal ions and elec-trons, the EGAM remains localized near the mid-radius, withsome changes in the structure andsome oscillations of the position of the peak in time, between s=0.4 and s=0.6.0 0.2 0.4 0.6 0.8 1−1−0.500.51sgrad(phibar)Er  t=0t=9720t=19440t=308200 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81sgrad(phibar)Er  t=0t=9280t=20200t=29020t=37820Figure 1: Radial structure of the linear EGAM electric field where all gradients are zero. On theleft, a flat EP density profile is initialized. On the right, the localized EP density profile is used.Nonlinear dynamicsIn this section, we discuss the results of nonlinear simulations of EGAMs, where a radiallylocalized EP density profile is initialized. A Krook-like operator acting as a heat source, modi-fied to conserve particles, momentum and zonal flows [12], is used for the thermal ions, whosefunction is to maintain the ion temperature profile close to the original one. At the same time,45th EPS Conference on Plasma Physics P2.1003the Krook operator acts as noise control. No source is applied to the EP, because we want to in-vestigate the effect of their nonlinear relaxation. For thesel cted EP density, i.e.nEP/ne= 0.08,the Krook operator used is found not to modify the EGAM lineargrowth rate and nonlinearsaturation level.Firstly, we investigate nonlinear EGAM simulations without t rbulence (see Fig. 2-left), byfiltering out all non-zonal perturbations at every time step. The EGAM is observed to grow lin-early and then saturate, then entering a deep nonlinear phase where the mode initially decays,then oscillations in amplitude occur. Here, we are not interested in the deep nonlinear phase, butonly in the level of the first saturation. Simulations with wave-particle + wave-wave nonlinear-ity are observed to saturate at a lower level with respect to simulations where the wave-particlenonlinearity only is switched on. Nevertheless, for the case of interest, the wave-particle non-linearity is shown not to be negligible, and provide at least50% of the saturation mechanism.Secondly, we study the evolution in time of the zonal radial electric field in simulationsof turbulence, without and with EP (see Fig. 2-right). In thesimulation without EP, the ZSis observed to grow due to the nonlinear drive of the ITG turbulence, then saturate. In thesimulation with EP, after the turbulence saturates, the EGAM starts growing on top of that, andsaturates at a higher level. The growth rate of the ZS due to the turbulence is also higher whenEP are present.Comparisons with the GK codes GENE and GYSELA are also performed in the absenceof turbulence. A good agreement of the linear and nonlinear dynamics is found, and a lower0 2 4 6 8 10x 10410−1100101102103104105max|∇φ 00(t)| [V/m]tMax of zonal Er in radial domain [0.25, 0.75]  EGAM, NL, WPEGAM, NL, WP+WW0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10410−1100101102103104105max|∇φ 00(t)| [V/m]tMax of Er in radial domain [0.25, 0.75]  ITG, NL, rel00, with krook4ITG+EGAM, NL, rel000, with krook4Figure 2: On the left, simulations of EGAMs without turbulenc , with wave-particle nonlinear-ity only (red line) and wave-particle + wave-wave nonlinearity (blue line). The maximum valueof the radial zonal electric field calculated in the radial domain 0.25< s< 0.75 is shown. Onthe right, simulations with turbulence, without EP (red line) and with EP (black line).45th EPS Conference on Plasma Physics P2.1003saturation level of EGAMs is confirmed when the wave-wave nonli earity is switched on.Conclusion and discussionThe linear and nonlinear dynamics of EGAMs is studied here, in the presence and absenceof turbulence. Several nonlinear mechanisms are studied separat ly, and in particular the wave-particle and wave-wave nonlinearities. Comparing the firstsaturation level of the simulationswith and without turbulence, we note that it does not depend on the presence of turbulence, forthe case of interest, beingδEr,max= 1.0 ·105 V/m (whereas it can change in the deep nonlinearphase). This means that the EGAM saturation mechanism for the case of interest is not thewave-wave nonlinearity between EGAM and ITG. We deduce thatt e EGAM saturates herefor a combined effect of wave-particle nonlinearity (i.e. EP redistribution in velocity space, andresonance detuning due to the EGAM frequency chirping) and wave-wave nonlinearity (ZFZFand second-harmonic generation [7]). A comparison of the saturated level with the reducedmodel provided in Ref. [11] is also done. We find a good agreement of the order of magnitudeof the saturated electric field.Acknowledgments: Valuable discussions are acknowledged with F. Zonca, Z. Qiu, L. Villard,S. Brunner, A. Mishchenko and P. Lauber. This work has been carried out within the frame-work of the EUROfusion Consortium and has received funding from the Euratom research andtraining programme 2014-2018 under grant agreement No 633053. The views and opinions ex-pressed herein do not necessarily reflect those of the European Commission. Simulations wereperformed on the Marconi supercomputer within the OrbZONE and ORBFAST projects.References[1] N. Winsoret al., Phys. Fluids11, 2448, (1968)[2] F. Zonca and L. Chen,Europhys. Lett.83, 35001 (2008)[3] G. Y. Fu,Phys. Rev. Letters101 (18), 185002 (2008)[4] Z. Qiu, F. Zonca, and L. ChenPlasma Phys. Control. Fusion52 (9), 095003 (2010)[5] D. Zarzoso et al.Phys. Plasmas19, 022102-1 (2012)[6] A. Di Siena, A. Biancalani, T. Görler, H. Doerk, I. Novikau, P. Lauber, A. Bottino, E. Poli,and the ASDEX Upgrade Team, submitted toNucl. Fusion(2018)[7] Z. Qiu, L. Chen and F. Zonca, “Kinetic theory of geodesic acoustic modes in toroidalplasmas: a brief review”, accepted for publication inPlasma Science and Tech.(2018)[8] D. Zarzoso, et al.Phys. Rev. Letters110 (12), 125002 (2013)[9] S. Jolliet, et al,Comput. Phys. Commun.177, 409 (2007).[10] A. Bottino, et alPlasma Phys. Controlled Fusion53, 124027 (2011)[11] A. Biancalani, et alJ. Plasma Phys.83 725830602 (2017)[12] B.F. McMillan, et al.,Phys Plasmas15, 052308 (2008)45th EPS Conference on Plasma Physics P2.1003

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Nonlinear gyrokinetic investigation of energetic particle driven geodesic acoustic modes

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