1 EX/P1-18

Global gyrokinetic modeling of geodesic acoustic modes and shear Alfvéninstabilities in ASDEX Upgrade.∗

A. Biancalani1, A. Bottino1, S. Briguglio2, G.D. Conway1, C. Di Troia2, R. Kleiber3, A. Könies3,Ph. Lauber1, A. Mischenko3, B. Scott1, P. Simon1, G. Vlad2, X. Wang1, D. Zarzoso1, A. Zocco3,F. Zonca2,4, and the ASDEX Upgrade Team.1Max-Planck-Institute für Plasmaphysik, 85748 Garching, Germany2ENEA C. R. Frascati, Via E. Fermi 45, CP 65-00044 Frascati, Italy3Max-Planck-Institute für Plasmaphysik, 17491 Greifswald, Germany4Inst. for Fusion Theory and Simulation, Zhejiang University, Hangzhou, P.R.Chinacontact of main author: www.ipp.mpg.de/∼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 gy-rokinetic particle-in-cell (PIC)), EUTERPE (nonlinear global gyrokinetic PIC), LIGKA (linear global eigenvaluegyrokinetic), and XHMGC (nonlinear global hybrid PIC). In the first part of this work, results of axisymmetricsimulations 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 partwe show results of electromagnetic simulations of ITG with NEMORB. Finally, in the fourth part, we show resultsof 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 macro-scopic kinetic energy. The energetic particle (EP) population produced in the process of heatingtogether 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 plasmainstabilities redistribute the EP population making the plasma heating less effective. For thisreasons, it is important to have a proper theoretical framework to predict their instability thresh-old and eventually understand their nonlinear dynamics in future fusion reactors.

In this work, we investigate the dynamics of global instabilities with equilibrium profiles ofASDEX Upgrade (AUG) by means of collisionless numerical simulations. We focus in partic-ular 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 sym-metric component, which have been found to play a role in the regulation of turbulence inASDEX Upgrade [3]. Shear Alfvén waves (SAW) are transverse electromagnetic perturbationswhich propagate parallel to the ambient magnetic field with the characteristic Alfvén groupvelocity vA. 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 re-alistic tokamak profiles, and in absence of energetic particles. As a second step, we add apopulation 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 turbu-lence in gyrokinetic simulations [9]. To better understand the interaction between global modesand electromagnetic turbulence, in view of its future modeling, we find it useful to explore thecapability of the code NEMORB and simulate electromagnetic ion-temperature-gradient modes(ITG) [10]. ITG instabilities are drift-waves, driven unstable in tokamaks by ion temperaturegradients.

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2. 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 thecode ORB5 [12], which was written for studies of turbulence in tokamak plasmas. RecentlyNEMORB has been used also to investigate collective instabilities and has been successfullybenchmarked against analytical theory and other codes [13]. In this work, we also compareresults of AE obtained with NEMORB with results obtained with the global gyrokinetic PICcode EUTERPE. EUTERPE is an extension of the GYGLES code [14, 15]. As global nonlineargyrokinetic PIC codes, they offer the unique capability of investigating self-consistently wave-wave and wave-particle interaction saturation mechanisms of SAW instabilities, especially inthose regimes where a very high resolution in phase space is required. Results of axisymmetricmodes obtained with NEMORB are compared with LIGKA, a linear gyrokinetic global eigen-value code [16]. Moreover, results of AE obtained with the MHD-gyrokinetic hybrid PIC codeXHMGC [17, 18] are also shown, both in the absence and in the presence of energetic particles.

3. Geodesic Acoustic Modes (GAM).

FIG. 1.: Equilibrium profiles of AUGshot 28881.

The result of an extensive set of numerical simulations ofGAMs performed with NEMORB has been recently shownfor equilibria with flat equilibrium profiles and verifiedagainst analytical theory in various regimes, as a verifi-cation/benchmark phase [13]. In this work, we show theapplication of NEMORB to the modeling of AUG real-istic equilibrium profiles. We choose AUG shot 28881as a reference case. This shot is considered particularlyinteresting because it shows coexistence of axisymmetricglobal instabilities with n = 0 (GAMs/EGAMs) and ofnon-axisymmetric instabilities (in this case n = 1, 2) dur-ing off-axis neutral-beam-injection heating, in the currentrump-up phase [19]. One peculiarity of the equilibriumprofiles is the non-monotonicity of the temperature profile, which shows a maximum near themiddle-radius r/a=0.5 at time t=0.93 s, due to the off-axis heating.

0 0.2 0.4 0.6 0.8 120

30

40

50

60

rho_pol

f [k

Hz]

NEMORB−AUG−28881

FIG. 2.: Frequency spectrum of GAMs forAUG shot 28881, obtained with NEMORB.

FIG. 3.: Same plot obtained with LIGKA, fortwo different times of interest.

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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 withNEMORB and LIGKA. Electrostatic simulations with adiabatic electrons are performed withNEMORB. The result of the two codes is in good agreement, and shows that the GAM contin-uum reflects consistently the inversed-slope behavior of the temperature profile, with a peak ofthe frequency near r/a = 0.5.

4. Energetic-particle induced GAM (EGAM).

Electrostatic simulations of EGAM with NEMORB, with flat equilibrium profiles. Ions aregyrokinetic, whereas electrons are adiabatic and drift-kinetic. Different concentrations of en-ergetic particles have been considered, modeled following a shifted Maxwellian in parallel ve-locity, as in Ref. [20, 21]. The very good quantitative agreement between analytic predictionsand 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 damp-ing/excitation of modes within the acoustic range of frequencies. However, trapped electronsmight have an impact on the excitation of EGAMs, due to the resonance between the bouncemotion and the mode, as occurs with the standard GAM [22]. Nevertheless, as we can see fromour results, trapped kinetic electrons have little impact on the mode. Though they seem to dampthe EGAM, their effect is not significant enough and the results are close to the ones with adi-abatic electrons. This means in particular that the predicted frequencies and growth rates in theelectrostatic limit should be recovered in electromagnetic simulations.

0 0.1 0.2 0.3 0.40.5

1

1.5

2

2.5

3

nEP

/nth

Fre

qu

en

cy [

cs/R

0]

Comparison ω for q=2

NEMORB adiabatic

NEMORB trapped

0 0.1 0.2 0.3 0.4

−0.1

−0.05

0

0.05

0.1

0.15

nEP

/nth

Gro

wth

ra

te [

cs/R

0]

Comparison γ for q=2

NEMORB adiabatic

NEMORB trapped

FIG. 4.: EGAM frequencies (left) and growth rates (right) for electrostatic simulations with adiabaticelectrons (solid black line) and trapped kinetic electrons (blue crosses) for a safety factor q=2. Thegrowth 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 performedwith NEMORB, in the view of a more general modeling with coexistence of global modes andturbulence, to be done in the near future. The equilibrium profiles of the CYCLONE case werechosen to model shot #81499 of DIII-D described in Ref. [23]. The profiles away from therational surfaces are chosen here according to the itm standard cyclone case, as in Ref. [24].

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0 200 400 600 800 1000 1200 1400 1600 1800−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

t [Ωi

−1]

γ [Ω

i]

Contributions to γ ITG deuterium

tot

//

grad−B

curv

grad−P

ExB

EbxB

FIG. 5.: Power channels of electromagneticITG simulations with NEMORB.

The electron temperature at the rational surface isset by ρ∗ = ρs/a = 1/185, where the sound gy-roradius is ρs = cs/Ωi, with cs =

√Te/mi being

the sound velocity and Ωi being the ion gyrofre-quency. The ion temperature is Ti = Te. The tem-perature gradient is R0/LT i = 6.91, and the den-sity gradient is given by ηi = Ln/LT i = 3.15. Thetemperature profile away from the rational surfacewas not relevant in Ref. [23], because only local(i.e. flux-tube) codes were used. The value ofβe = 8πneTe/B

2 chosen for this simulation isβe = 10

−4. No energetic particle population ispresent in these simulations.

The code runs in electromagnetic mode, without collisions. The evolution of the mode in timecan be divided in a first transient phase, up to t ' 600, followed by a linear phase. The value ofthe linear growth rate measured with J · E is γ = 1.01 · 10−3Ωi = 0.23 cs/Ln. The structureof the mode at a given toroidal angle φ, and at time t = 1000 Ω−1i , can be seen in Fig. 6.. Wecan see that the mode is radially localized around the rational surface at r = 0.5a, with radialhalf-width dρ ' 0.2 given by the the radial half-width of the temperature gradient profile. Wecan also see that the mode is formed by the coupling of several modes. At the low-field side ofthe tokamak (θ = 0), 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 andin the s directions (with s = r/a).

FIG. 6.: Poloidal snapshots of the scalar potential (left) and vector potential (right) at t=1000 Ω−1i , forelectromagnetic ITG.

The spatial structure of the amplitude of the scalar potential shows a clearly stronger signal onthe 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 = 0 and θ = π. The ratioof vector potential over scalar potential at t=1000 is about 10−3. Quantitative comparisons torecent analytical theories are left for future work [25].

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6. 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 atest case of benchmark on AE frequency, between these three codes. The equilibrium is chosenconsistently with the framework of the Energetic particle Topical Group of the InternationalTokamak Physics Activity (ITPA) [26], except for the value of beta, which has been chosen asβe = 2 · 10−4. A circular magnetic surface tokamak equilibrium is considered, characterizedby major radius R0 = 10m, minor radius a = 1m, safety factor q = q0 + (qa − q0)(r/a)2, q0 =1.71, qa = 1.87 and on-axis magnetic field B0 = 3T. The bulk plasma is characterized by flat ion(Hydrogen) and electron temperatures, Ti = Te = 1keV, and densities. A single toroidal modenumber, n = 6 is considered, and no EP particle population is initialized.

0 0.2 0.4 0.6 0.8 1s

0

0.002

0.004

0.006

ω /

ωci

EUTERPE

NEMORB

XHMGC

ideal MHD TAE

m = 10

m = 11 m =

10

m =

11

0 0.2 0.4 0.6 0.8 1r / a

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

A||

m = 7m = 8m = 9m = 10m = 11m = 12m = 13

FIG. 7.: Frequency spectrum of TAE obtainedwith NEMORB, for the ITPA equilibrium, andβe = 2 · 10−4.

FIG. 8.: Radial structure of the severalpoloidal components, obtained with EU-TERPE.

We initialize a perturbation with poloidal mode numbers m = 10 and m = 11, and let it evolvein time, with Dirichlet boundary conditions imposed at the radial boundaries r = 0 and r = a.We measure the time evolution of the parallel component of the vector potential at one poloidalsection calculate the Fourier transform in time, which yields the mode frequency. The modefrequency measured with NEMORB, EUTERPE and XHMGC is then plotted in a graph to-gether with the analytical SAW continuous spectrum (see Fig. 7.). The continuous spectrum isthe frequency spectrum of the natural local oscillation of the equilibrium, and describe wherethe mode energy is going to be converted due to continuum damping [27]. For the two poloidalcomponents of interest, the finite value of the aspect ratio opens a gap in the continuous spec-trum due to toroidal curvature [28, 29], near the center of the radial grid, s = r/a ' 0.5. Thisgap corresponds to a window in frequency where the mode is not affected by continuum damp-ing. In this gap, a discrete frequency eigenmode exists, which takes the name of toroidicityinduced Alfvén eigenmode (TAE). The measurements of NEMORB, EUTERPE and XHMGCare found to be within the toroidicity induced gap, even though small differences between the 3different 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. Wedecompose it in Fourier components in the poloidal angle θ, for each radial position, and findthat the radial structure is dominant for the modes m = 10 and m = 11, and their sum is peakedat the radial position of the continuum gap (see Fig. 8.). We show also poloidal snapshot of the

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FIG. 9.: Poloidal snapshots of the vector potential of the TAE simulation in the ITPA equilibrium, ob-tained with NEMORB (left) and EUTERPE (right).

parallel component of the vector potential for simulations with NEMORB and EUTERPE (seeFig. 9.). We can see that in both cases two main coronas are present, an inner one and an outerone, relatives to the two poloidal dominant modes. As next steps, simulations of TAE in thepresence of EP will be performed, with realistic tokamak profiles.

B) Beta scan.A scan with beta is performed with NEMORB for axisymmetric SAW modes,

10−4

10−3

10−2

10−1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

βe = 8π n

e T

e /B

2

ω [

Ωi]

SAW: B=2.4*104 G, R = 166.7 cm, ε

0 = 0.01, q=2, ρ*/2 = 1/100, (m,n)=(1,0)

vA / qR

NEMORB mi/m

e=200

FIG. 10.: Beta scan for axisymmetric perturbationswith NEMORB.

to measure the code response when we con-sider pressures which have more and more re-alistic values for tokamak profiles. In order tosimplify the geometry and concentrate on thebeta scan, we show here results for an equi-librium with very large aspect ratio � = 0.01and evolving only axisymmetric SAW pertur-bations. We choose ρ∗ = 1/50, and heavyelectrons (mi/me = 200). The result, is thatin this regime the code gives very good agree-ment with analytical MHD theory up to valuescharacteristic of nowadays tokamak plasmas.This proves the feasibility of gyrokinetic-ions/ drift-kinetic electrons simulations of tokamakplasmas 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 ofEP. The equilibrium profiles have been chosen in order to investigate the opening of a continuumgap 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) [30, 31].As a result, we find that the BAE observed with XHMGC has a frequency which lies justbelow the continuum accumulation point (see Fig. 11.). We have performed a scan with fastion concentration, finding that the frequency does not change sensibly, whereas the growth rate

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scales linearly [32]. Later these cases will be analyzed with NEMORB and benchmarked withXHMGC. Meanwhile, related nonlinear saturation mechanisms will be investigated by usingHamiltonian mapping techniques [33].

10-3

10-2

10-1

1x 6.749E-10

|φ(r,ω)|2 ∆tωA0= 3x1008.0, tωA0=1799.40Sum over m,n l-min,l-max= 1, 8log-scale

r/a

|ω/ωA0|

HM

GC

cod

e -

EN

EA

- F

rasc

ati

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

n-continuum-wres= 2

0.105

0.112

0.119

0.126

0.002

0.004

0.006

0.008

0.001 0.002 0.003 0.004 0.005n

H /n

i0

ωrτ

A γτA

ωr

γ

FIG. 11.: Frequency spectrum of BAE obtainedwith XHMGC, compared with incompressibleMHD continuum (dotted line) and compress-ible continuum (continuous line).

FIG. 12.: Scan of BAE frequency and growthrate with fast ion concentration.

7. Conclusions.

In fusion plasmas, fast ions in the MeV energy range have velocities comparable with the typicalAlfvén speed and can therefore resonantly interact with SAW. In addition, SAW group veloc-ity is directed along the magnetic field line and, therefore, fast ions can stay in resonance andeffectively exchange energy with the wave [5, 6]. For typical plasma core parameters, SAW in-stabilities are the most efficient mechanism of redistribution of EP. Transport due to multi-modeinteraction, strongly nonlinear transport due to bursting avalanche events, and their relative im-portance 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 consistentmodel treating collective instabilities and particle transport on the same footing.

We have shown numerical simulations of global instabilities obtained with gyrokinetic and hy-brid codes. Both simplified and realistic equilibrium profiles have been chosen, depending onthe case of interest. In the case of geodesic acoustic modes, ASDEX Upgrade equilibria havebeen considered and numerical modeling of frequency has been performed with eigenvaluecode LIGKA and initial value code NEMORB. In the presence of energetic particles, EGAMsimulations have been performed and effect of kinetic electrons has been shown. Ion tem-perature gradient modes have also been investigated by means of electromagnetic simulationswith NEMORB. Finally, we have shown results of simulations of toroidicity induced Alfvéneigenmodes and beta induced Alfvén eigenmodes with NEMORB, EUTERPE and XHMGC.

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8. *AcknowledgmentsThis work is the result of a collaboration carried out within the framework of the Nonlinearenergetic particle dynamics (NLED) European Enabling Research Project, and has receivedfunding from the European Unions Horizon 2020 research and innovation program under grantagreement number 633053. The views and opinions expressed herein do not necessarily reflectthose of the European Commission. Simulations were performed on the IFERC-CSC Heliossupercomputer within the framework of the ORBFAST project.

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