Global gyrokinetic modeling of geodesic acoustic modes and shear Alfvén instabilities in ASDEX Upgrade.

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1 1 EX/P1-18 Global gyrokinetic modeling of geodesic acoustic modes and shear Alfvén instabilities in ASDEX Upgrade. A. Biancalani 1, A. Bottino 1, S. Briguglio 2, G.D. Conway 1, C. Di Troia 2, R. Kleiber 3, A. Könies 3, Ph. Lauber 1, A. Mischenko 3, B. Scott 1, P. Simon 1, G. Vlad 2, X. Wang 1, D. Zarzoso 1, A. Zocco 3, F. Zonca 2,4, and the ASDEX Upgrade Team. 1 Max-Planck-Institute für Plasmaphysik, Garching, Germany 2 ENEA C. R. Frascati, Via E. Fermi 45, CP Frascati, Italy 3 Max-Planck-Institute für Plasmaphysik, Greifswald, Germany 4 Inst. for Fusion Theory and Simulation, Zhejiang University, Hangzhou, P.R.China contact of main author: biancala Abstract. In this work, we investigate the dynamics of global instabilities observed in ASDEX Upgrade (AUG) by means of collisionless linear numerical simulations. We focus in particular on geodesic acoustic modes (GAM) and shear Alfvén eigenmodes (AE). The numerical tools we use are the codes NEMORB (nonlinear global gyrokinetic particle-in-cell (PIC)), EUTERPE (nonlinear global gyrokinetic PIC), LIGKA (linear global eigenvalue gyrokinetic), and XHMGC (nonlinear global hybrid PIC). In the first part of this work, results of axisymmetric simulations of GAMs with gyrokinetic codes NEMORB and LIGKA with AUG equilibrium profiles are shown. In the second part, we show results of energetic particle driven GAMs (EGAMs) with NEMORB. In the third part we show results of electromagnetic simulations of ITG with NEMORB. Finally, in the fourth part, we show results of single-n (with n the toroidal mode number) numerical simulations of shear Alfvén eigenmodes with NEMORB, EUTERPE and XHMGC. 1. Introduction. Plasma heating is essential for reaching appropriate fusion temperatures in tokamak plasmas, but as a side effect global modes can become unstable by converting thermal energy into macroscopic kinetic energy. The energetic particle (EP) population produced in the process of heating together with the alpha particles produced in fusion reactions are important actors in this chain, driving plasma oscillations unstable via resonant interactions. On the other hand, the plasma instabilities redistribute the EP population making the plasma heating less effective. For this reasons, it is important to have a proper theoretical framework to predict their instability threshold and eventually understand their nonlinear dynamics in future fusion reactors. In this work, we investigate the dynamics of global instabilities with equilibrium profiles of ASDEX Upgrade (AUG) by means of collisionless numerical simulations. We focus in particular on geodesic acoustic modes (GAM) [1, 2, 3] and shear Alfvén eigenmodes (AE) [4, 5, 6]. GAMs are axisymmetric plasma oscillations with mainly electrostatic and nearly poloidal symmetric component, which have been found to play a role in the regulation of turbulence in ASDEX Upgrade [3]. Shear Alfvén waves (SAW) are transverse electromagnetic perturbations which propagate parallel to the ambient magnetic field with the characteristic Alfvén group velocity v A. As a first step of theoretical modeling of energetic particle driven instabilities, the frequency of GAM and AE is calculated here by means of numerical simulations for realistic tokamak profiles, and in absence of energetic particles. As a second step, we add a population of energetic ions and describe results of simulations of energetic ion driven GAMs (EGAMs) [7, 8]. EGAMs have been observed to significantly modify and modulate the turbulence in gyrokinetic simulations [9]. To better understand the interaction between global modes and electromagnetic turbulence, in view of its future modeling, we find it useful to explore the capability of the code NEMORB and simulate electromagnetic ion-temperature-gradient modes (ITG) [1]. ITG instabilities are drift-waves, driven unstable in tokamaks by ion temperature gradients.

2 2 EX/P The theoretical models. The main numerical tool we use is the nonlinear gyrokinetic code NEMORB. NEMORB [11] is a global gyrokinetic PIC code, derived as the multispecies, electromagnetic version of the code ORB5 [12], which was written for studies of turbulence in tokamak plasmas. Recently NEMORB has been used also to investigate collective instabilities and has been successfully benchmarked against analytical theory and other codes [13]. In this work, we also compare results of AE obtained with NEMORB with results obtained with the global gyrokinetic PIC code EUTERPE. EUTERPE is an extension of the GYGLES code [14, 15]. As global nonlinear gyrokinetic PIC codes, they offer the unique capability of investigating self-consistently wavewave and wave-particle interaction saturation mechanisms of SAW instabilities, especially in those regimes where a very high resolution in phase space is required. Results of axisymmetric modes obtained with NEMORB are compared with LIGKA, a linear gyrokinetic global eigenvalue code [16]. Moreover, results of AE obtained with the MHD-gyrokinetic hybrid PIC code XHMGC [17, 18] are also shown, both in the absence and in the presence of energetic particles. 3. Geodesic Acoustic Modes (GAM). The result of an extensive set of numerical simulations of GAMs performed with NEMORB has been recently shown for equilibria with flat equilibrium profiles and verified against analytical theory in various regimes, as a verification/benchmark phase [13]. In this work, we show the application of NEMORB to the modeling of AUG realistic equilibrium profiles. We choose AUG shot as a reference case. This shot is considered particularly interesting because it shows coexistence of axisymmetric global instabilities with n = (GAMs/EGAMs) and of non-axisymmetric instabilities (in this case n = 1, 2) during off-axis neutral-beam-injection heating, in the current FIG. 1.: Equilibrium profiles of AUG shot rump-up phase [19]. One peculiarity of the equilibrium profiles is the non-monotonicity of the temperature profile, which shows a maximum near the middle-radius r/a=.5 at time t=.93 s, due to the off-axis heating. NEMORB AUG f [khz] rho_pol FIG. 2.: Frequency spectrum of GAMs for AUG shot 28881, obtained with NEMORB. FIG. 3.: Same plot obtained with LIGKA, for two different times of interest.

3 3 EX/P1-18 We consider the equilibrium profiles at this particular time for a first modeling of this shot, and perform numerical simulations of axisymmetric perturbations in the absence of EP with NEMORB and LIGKA. Electrostatic simulations with adiabatic electrons are performed with NEMORB. The result of the two codes is in good agreement, and shows that the GAM continuum reflects consistently the inversed-slope behavior of the temperature profile, with a peak of the frequency near r/a = Energetic-particle induced GAM (EGAM). Electrostatic simulations of EGAM with NEMORB, with flat equilibrium profiles. Ions are gyrokinetic, whereas electrons are adiabatic and drift-kinetic. Different concentrations of energetic particles have been considered, modeled following a shifted Maxwellian in parallel velocity, as in Ref. [2, 21]. The very good quantitative agreement between analytic predictions and electrostatic simulations with adiabatic electrons has been recently published in Ref. [21]. Passing electrons are treated adiabatically, since they are not expected to contribute to the damping/excitation of modes within the acoustic range of frequencies. However, trapped electrons might have an impact on the excitation of EGAMs, due to the resonance between the bounce motion and the mode, as occurs with the standard GAM [22]. Nevertheless, as we can see from our results, trapped kinetic electrons have little impact on the mode. Though they seem to damp the EGAM, their effect is not significant enough and the results are close to the ones with adiabatic electrons. This means in particular that the predicted frequencies and growth rates in the electrostatic limit should be recovered in electromagnetic simulations. 3 Comparison ω for q=2 Comparison γ for q=2 Frequency [c s /R ] NEMORB adiabatic NEMORB trapped Growth rate [c s /R ] NEMORB adiabatic NEMORB trapped n EP /n th n EP /n th FIG. 4.: EGAM frequencies (left) and growth rates (right) for electrostatic simulations with adiabatic electrons (solid black line) and trapped kinetic electrons (blue crosses) for a safety factor q=2. The growth rate is decreased likely due to the damping provided by the bounce motion of trapped electrons. 5. Ion temperature gradient modes (ITG). In this section, we show results of electromagnetic simulations of ITG instabilities performed with NEMORB, in the view of a more general modeling with coexistence of global modes and turbulence, to be done in the near future. The equilibrium profiles of the CYCLONE case were chosen to model shot #81499 of DIII-D described in Ref. [23]. The profiles away from the rational surfaces are chosen here according to the itm standard cyclone case, as in Ref. [24].

4 4 EX/P1-18 The electron temperature at the rational surface is set by ρ = ρ s /a = 1/185, where the sound gyroradius is ρ s = c s /Ω i, with c s = T e /m i being the sound velocity and Ω i being the ion gyrofrequency. The ion temperature is T i = T e. The temperature gradient is R /L T i = 6.91, and the density gradient is given by η i = L n /L T i = The temperature profile away from the rational surface was not relevant in Ref. [23], because only local (i.e. flux-tube) codes were used. The value of β e = 8πn e T e /B 2 chosen for this simulation is β e = 1 4. No energetic particle population is present in these simulations. γ [Ω i ] 1.5 x Contributions to γ ITG deuterium tot // grad B curv grad P ExB E b xb t [Ω i ] FIG. 5.: Power channels of electromagnetic ITG simulations with NEMORB. The code runs in electromagnetic mode, without collisions. The evolution of the mode in time can be divided in a first transient phase, up to t 6, followed by a linear phase. The value of the linear growth rate measured with J E is γ = Ω i =.23 c s /L n. The structure of the mode at a given toroidal angle φ, and at time t = 1 Ω 1 i, can be seen in Fig. 6.. We can see that the mode is radially localized around the rational surface at r =.5a, with radial half-width dρ.2 given by the the radial half-width of the temperature gradient profile. We can also see that the mode is formed by the coupling of several modes. At the low-field side of the tokamak (θ = ), the mode has a well defined poloidal structure with k directed along θ, whereas at the high-field side (θ = π), the characteristic ballooning behavior can be observed, where the mode loses the well defined poloidal structure and k has component both in the θ and in the s directions (with s = r/a). FIG. 6.: Poloidal snapshots of the scalar potential (left) and vector potential (right) at t=1 Ω 1 i, for electromagnetic ITG. The spatial structure of the amplitude of the scalar potential shows a clearly stronger signal on the low field side, and weaker at the high field side. The vector potential, on the other hand, shows a sinusoidal m=1 behavior, with very small intensity at θ = and θ = π. The ratio of vector potential over scalar potential at t=1 is about 1 3. Quantitative comparisons to recent analytical theories are left for future work [25].

5 5 EX/P Shear Alfvén Eigenmodes (AE). A) Toroidicity induced Alfvén eigenmode (TAE) in the ITPA equilibrium. In this section we present results of electromagnetic simulations performed with NEMORB, EUTERPE and XHMGC for a particular equilibrium with large aspect ratio, which is used as a test case of benchmark on AE frequency, between these three codes. The equilibrium is chosen consistently with the framework of the Energetic particle Topical Group of the International Tokamak Physics Activity (ITPA) [26], except for the value of beta, which has been chosen as β e = A circular magnetic surface tokamak equilibrium is considered, characterized by major radius R = 1m, minor radius a = 1m, safety factor q = q + (q a q )(r/a) 2, q = 1.71, q a = 1.87 and on-axis magnetic field B = 3T. The bulk plasma is characterized by flat ion (Hydrogen) and electron temperatures, T i = T e = 1keV, and densities. A single toroidal mode number, n = 6 is considered, and no EP particle population is initialized. ω / ω ci m = 11 m = 1 EUTERPE NEMORB XHMGC ideal MHD TAE m = 1 m = s FIG. 7.: Frequency spectrum of TAE obtained with NEMORB, for the ITPA equilibrium, and β e = A m = 7 m = 8 m = 9 m = 1 m = 11 m = 12 m = r / a FIG. 8.: Radial structure of the several poloidal components, obtained with EU- TERPE. We initialize a perturbation with poloidal mode numbers m = 1 and m = 11, and let it evolve in time, with Dirichlet boundary conditions imposed at the radial boundaries r = and r = a. We measure the time evolution of the parallel component of the vector potential at one poloidal section calculate the Fourier transform in time, which yields the mode frequency. The mode frequency measured with NEMORB, EUTERPE and XHMGC is then plotted in a graph together with the analytical SAW continuous spectrum (see Fig. 7.). The continuous spectrum is the frequency spectrum of the natural local oscillation of the equilibrium, and describe where the mode energy is going to be converted due to continuum damping [27]. For the two poloidal components of interest, the finite value of the aspect ratio opens a gap in the continuous spectrum due to toroidal curvature [28, 29], near the center of the radial grid, s = r/a.5. This gap corresponds to a window in frequency where the mode is not affected by continuum damping. In this gap, a discrete frequency eigenmode exists, which takes the name of toroidicity induced Alfvén eigenmode (TAE). The measurements of NEMORB, EUTERPE and XHMGC are found to be within the toroidicity induced gap, even though small differences between the 3 different values are found. The reason of the discrepancy is still to be investigated. The spatial structure of the TAE is also investigated radially and in the poloidal plane. We decompose it in Fourier components in the poloidal angle θ, for each radial position, and find that the radial structure is dominant for the modes m = 1 and m = 11, and their sum is peaked at the radial position of the continuum gap (see Fig. 8.). We show also poloidal snapshot of the

6 6 EX/P1-18 FIG. 9.: Poloidal snapshots of the vector potential of the TAE simulation in the ITPA equilibrium, obtained with NEMORB (left) and EUTERPE (right). parallel component of the vector potential for simulations with NEMORB and EUTERPE (see Fig. 9.). We can see that in both cases two main coronas are present, an inner one and an outer one, relatives to the two poloidal dominant modes. As next steps, simulations of TAE in the presence of EP will be performed, with realistic tokamak profiles. B) Beta scan. A scan with beta is performed with NEMORB for axisymmetric SAW modes, to measure the code response when we consider SAW: B=2.4*1 4 G, R = cm, ε =.1, q=2, ρ*/2 = 1/1, (m,n)=(1,) pressures which have more and more re-.16 v.14 A / qr alistic values for tokamak profiles. In order to NEMORB m /m =2 i e simplify the geometry and concentrate on the.12 beta scan, we show here results for an equilibrium.1 with very large aspect ratio ɛ =.1 and evolving only axisymmetric SAW perturbations..8 We choose ρ = 1/5, and heavy.6 electrons (m i /m e = 2). The result, is that.4 in this regime the code gives very good agreement.2 with analytical MHD theory up to values 1 4 characteristic of nowadays tokamak plasmas β e = 8π n e T e /B This proves the feasibility of gyrokinetic-ions FIG. 1.: Beta scan for axisymmetric perturbations / drift-kinetic electrons simulations of tokamak with NEMORB. plasmas with realistic values of beta, at least in this range of ρ and for these electron masses. Stability for long simulations has still to be investigated. C) Beta induced Alfvén eigenmodes. Numerical simulations with XHMGC have been performed in the presence of a population of EP. The equilibrium profiles have been chosen in order to investigate the opening of a continuum gap at low frequencies, due to the finite pressure coupling with toroidal curvature. In this gap, another discrete mode exists, which is named beta induced Alfvén eigenmode (BAE) [3, 31]. As a result, we find that the BAE observed with XHMGC has a frequency which lies just below the continuum accumulation point (see Fig. 11.). We have performed a scan with fast ion concentration, finding that the frequency does not change sensibly, whereas the growth rate ω [Ω i ]

7 7 EX/P1-18 scales linearly [32]. Later these cases will be analyzed with NEMORB and benchmarked with XHMGC. Meanwhile, related nonlinear saturation mechanisms will be investigated by using Hamiltonian mapping techniques [33]. ω/ω A φ(r,ω) 2 tω A = 3x18., tω A = Sum over m,n l-min,l-max= 1, 8 log-scale x 6.749E n-continuum-wres= 2 r/a HMGC code - ENEA - Frascati.126 ω r τ A ω r n H /n i γ.8 γτ A FIG. 11.: Frequency spectrum of BAE obtained with XHMGC, compared with incompressible MHD continuum (dotted line) and compressible continuum (continuous line). FIG. 12.: Scan of BAE frequency and growth rate with fast ion concentration. 7. Conclusions. In fusion plasmas, fast ions in the MeV energy range have velocities comparable with the typical Alfvén speed and can therefore resonantly interact with SAW. In addition, SAW group velocity is directed along the magnetic field line and, therefore, fast ions can stay in resonance and effectively exchange energy with the wave [5, 6]. For typical plasma core parameters, SAW instabilities are the most efficient mechanism of redistribution of EP. Transport due to multi-mode interaction, strongly nonlinear transport due to bursting avalanche events, and their relative importance in burning plasma conditions, are phenomena which are still not fully understood. For these reasons, a comprehensive understanding of the EP transport requires a self consistent model treating collective instabilities and particle transport on the same footing. We have shown numerical simulations of global instabilities obtained with gyrokinetic and hybrid codes. Both simplified and realistic equilibrium profiles have been chosen, depending on the case of interest. In the case of geodesic acoustic modes, ASDEX Upgrade equilibria have been considered and numerical modeling of frequency has been performed with eigenvalue code LIGKA and initial value code NEMORB. In the presence of energetic particles, EGAM simulations have been performed and effect of kinetic electrons has been shown. Ion temperature gradient modes have also been investigated by means of electromagnetic simulations with NEMORB. Finally, we have shown results of simulations of toroidicity induced Alfvén eigenmodes and beta induced Alfvén eigenmodes with NEMORB, EUTERPE and XHMGC.

8 8 EX/P *Acknowledgments This work is the result of a collaboration carried out within the framework of the Nonlinear energetic particle dynamics (NLED) European Enabling Research Project, and has received funding from the European Unions Horizon 22 research and innovation program under grant agreement number The views and opinions expressed herein do not necessarily reflect those of the European Commission. Simulations were performed on the IFERC-CSC Helios supercomputer within the framework of the ORBFAST project. References [1] WINSOR, N., et al., Phys. Fluids 11 (1968) 2448 [2] SUGAMA, H., WATANABE, T. H., J. Plasma Physics 74 (27) 139 [3] CONWAY, G. D., et al., Phys. Rev. Letters 16 (211) 651 [4] CHENG, C. Z., et al., Ann. Phys 161 (1985) 21 [5] CHEN, L., ZONCA, F., Nucl. Fusion 47 (27) S727 [6] LAUBER, Ph., Phys. Reports. 533 (213) [7] FU, G., et al., Phys. Rev. Lett. 11 (28) 1852 [8] ZARZOSO, D., et al., Phys. Plasmas 19 (212) [9] ZARZOSO, D., et al., Phys. Rev. Lett. 11 (213) 1252 [1] RUDAKOV, L. I., SAGDEEV, R. Z., Sov. Phys. Dokl. 6 (1965) 498 [11] BOTTINO, A., et al. Plasma Phys. Controlled Fusion 53 (211) [12] JOLLIET, S., et al., Comput. Phys 177 (27) 49 [13] BIANCALANI, A., et al, Numerical validation of the electromagnetic gyrokinetic code NEMORB on global axisymmetric modes accepted for public. in Nucl. Fusion (213) [14] KORNILOV, V., et al., Phys. Plasmas 11 (24) 3196 [15] MISCHENKO, A., et al., Phys. Plasmas 15 (28) [16] LAUBER, Ph., et al. Journal of Comp. Phys. 226 (27) 447 [17] BRIGUGLIO, S., et al., Phys. Plasmas 2 (1995) 3711 [18] WANG, X., et al., Phys. Plasmas 18 (211) 5254 [19] LAUBER, Ph., et. al, Off-axis NBI-driven modes at ASDEX Upgrade, IAEA technical meeting on energetic particles, Beijing (213) [2] GIRARDO, J. B., et al., Phys. Plasmas 21 (214) 9257 [21] ZARZOSO, D., et al., Nucl. Fusion 54 (214) 136 [22] ZHANG, H.S., LIN, Z., Phys. Plasmas 17 (21) 7252 [23] DIMITS, A.M., et al., Phys. of Plasmas 7 (2) 969 [24] FALCHETTO, G., et al., Phys. of Plasmas 5 (28) [25] CONNOR, J.W., et al., Plasma Phys. Controlled Fusion 54 3 (212) 353 [26] KÖNIES, A., et al., Benchmark of gyrokinetic, kinetic MHD and gyrofluid codes for the linear calculation of fast particle driven TAE dynamics, IAEA-FEC, ITR/P1-34 (212). [27] CHEN, L., HASEGAWA, A., Phys. Fluids 17 (1974) [28] KIERAS, C.E., TATARONIS, J.A., J. Plasma Physics 28 (1982) 395. [29] CHENG, C.Z., et al., Ann Phys 161 (1985) 21. [3] CHU, M.S., et al., Phys. Fluids B 4 (1992) [31] ZONCA, F., et al., Plasma Phys. Control. Fusion 38 (1996) 211. [32] WANG, X. et al., Studies of nonlinear dynamics of wave-particle interactions in Tokamak plasmas based on Hamiltonian mapping techniques, Joint Varenna-Lausanne international workshop, Varenna, Italy (214) [33] BRIGUGLIO, S., et al., Analysis of the nonlinear behaviour of shear-alfvén modes in Tokamaks based on Hamiltonian mapping techniques, submit. to Phys. Plasmas (214)

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