Theoretical analysis of the saturation phase of the1/1 energetic-ion-driven resistive interchange mode

J. Varela

E-mail: jvrodrig@fis.uc3m.es

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, SpainNational Institute for Fusion Science, National Institute of Natural Science,Toki, 509-5292, JapanOak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA

D. A. Spong

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA

L. Garcia

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain

S. Ohdachi

National Institute for Fusion Science, National Institute of Natural Science,Toki, 509-5292, Japan

K. Y. Watanabe

National Institute for Fusion Science, National Institute of Natural Science,Toki, 509-5292, Japan

R. Seki

SOKENDAI, Department of Fusion Science, Toki/Gifu and National Institutefor Fusion Science, National Institute of Natural Science, Toki, 509-5292, Japan

Y. Ghai

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA

Abstract. The aim of the present study is to analyze the saturation regimeof the energetic-ion-driven resistive interchange mode (EIC) in the LHD plasma.A set of non linear simulations are performed by the FAR3d code that uses areduced MHD model for the thermal plasma coupled with a gyrofluid modelfor the energetic particles (EP) species. The hellically trapped EP componentis introduced through a modification of the averaged drift velocity operator toinclude their precessional drift. The non linear simulation results show similar 1/1EIC saturation phases with respect to the experimental observations, reproducingthe enhancement of the n/m = 1/1 resistive interchange modes (RIC) amplitudeand width as the EP β increases, the EP β threshold for the 1/1 EIC excitation,the further destabilization of the 1/1 EIC as the population of the helically trappedEP increases and the triggering of burst events. The frequency of the 1/1 EIC

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calculated during the burst event is 9.4 kHz and the 2/2 and 3/3 overtonesare destabilized, consistent with the frequency range and the complex modestructure measured in the experiment. In addition, the simulation shows theinward propagation of the 1/1 EIC due to the non linear destabilization of the3/4 and 2/3 EPMs, leading to the partial overlapping between resonances duringthe burst event. Finally, the analysis of the 1/1 EIC stabilization phase showsthe excitation of the 1/1 RIC as soon as the flattening induced by the 1/1 EIC inthe pressure profile vanishes, leading to the retrieval of the pressure gradient atthe plasma periphery and the overcoming of the RIC stability limit.

PACS numbers: 52.35.Py, 52.55.Hc, 52.55.Tn, 52.65.Kj

Keywords: Stellarator, LHD, EIC, MHD, AE, energetic particles

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1. Introduction

Low density and high ion temperature dischargesin Large Helical Device (LHD) show magneticfluctuations similar to the fishbone oscillations [1,2]. These perturbations are observed if the plasmais strongly heated by perpendicular neutral beaminjectors (NBI) [3] and the resistive interchange modes(RIC) are unstable at the plasma periphery [4, 5,6]. These perturbations are called 1/1 energetic-ion-driven resistive interchange mode (EIC), triggered bythe resonance between the precession motion of thehelically trapped EP generated by the perpendicularNBI (around the 10 kHz) and the RIC [7].

The 1/1 EIC can be included in the family of theenergetic particle modes (EPM) [8], modes destabilizedfor frequencies in the shear Alfven continua if thecontinuum damping is not strong enough to stabilizethem [9, 10]. The destabilization of the EPM shouldbe avoided because the heating efficiency of the devicedecreases due to the loss of EP before thermalization.

Different phases are observed during the 1/1 EICsaturation [7]. First, the RIC is unstable beforethe excitation of the 1/1 EIC once the density ofhelically trapped EP increases above a given threshold(EP β), identified in the experiment as a plasmaperturbation with a frequency around 4 kHz (PhaseI). Next, the 1/1 EIC is further destabilized as theEP β increases leading to a larger mode width andthe inward expansion of the perturbation (Phase II).Then, the frequency of the 1/1 EIC suddenly increasesto 9 kHz and the instability shows a bursting behaviorwhile a complex radial structure appears betweenr/a = [0.6, 0.85] as the perturbation keeps propagatinginward (Phase III). Finally, a steady magnetic islandappears at the radial location of the 1/1 rationalsurface, the 1/1 EIC stabilizes and the RIC is triggeredagain (Phase IV).

LHD is a helical device heated by three NBI linesparallel to the magnetic axis with an energy of 180keV and two NBI perpendicular to the magnetic axiswith an energy of 32 keV. Advanced LHD operationscenarios as high β discharges require the intensiveuse of the perpendicular NBIs, leading to a largeaccumulation of helically trapped particles at theplasma periphery and the destabilization of 1/1 EIC,limiting the maximum β of the experiment.

The aim of the present study is to analyze thesaturation of the 1/1 EIC reproducing the different

instability phases observed in the experiment, inparticular the burst event. The simulations areperformed using the FAR3D gyro-fluid code [11, 12, 13,14]. The numerical model solves the reduced non linearresistive MHD equations coupled with the momentequations of the energetic ion density and parallelvelocity [15, 16, 17]. The linear wave-particle resonanceeffects required for Landau damping/growth are addedfor the appropriate Landau closure relations, as wellas the parallel momentum response of the thermalplasma required for coupling to the geodesic acousticwaves [18]. Six field variables evolves starting from anequilibria calculated by the VMEC code [19].

This paper is organized as follows. Themodel equations, numerical scheme and equilibriumproperties are described in section 2. The analysis ofthe non linear simulation during the saturation phaseof the 1/1 EIC is performed in section 3. Next, theconclusions of this paper are presented in section 4.

2. Equations and numerical scheme

Following the method employed in Ref.[20], a reducedset of equations for high-aspect ratio configurationsand moderate β-values (of the order of the inverseaspect ratio) is derived retaining the toroidal anglevariation based upon an exact three-dimensionalequilibrium that assumes closed nested flux surfaces.The effect of the energetic particle population in theplasma stability is included through moments of thefast ion kinetic equation truncated with a closurerelation [21], describing the evolution of the energeticparticle density (nf ) and velocity moments parallel tothe magnetic field lines (v||f ). The coefficients of theclosure relation are selected to match analytic TAEgrowth rates based upon a two-pole approximation ofthe plasma dispersion function. A detail description ofthe model equations and numerical scheme is includedin the appendix.

The present model was already used to study thelinear stability of LHD plasma, particularly the AEdestabilization by EP [22], AE stability of plasmawith multiple EP populations [23], the effect ofthe tangential NBI current drive on the stability ofpressure and EP driven MHD modes [24] as well asEIC stabilization strategies [25], showing a reasonableagreement with the observations. In addition,non linear studies were performed reproducing thesawtooth-like and internal collapse events observed in

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Ti (keV) ni (1020 m−3) βth (%) VA (107 m/s)2 0.25 0.32 1.1

Table 1. Thermal plasma properties in the reference model(values at the magnetic axis). The first column is the thermalion temperature, the second column is the thermal ion density,the third column is the thermal β and the fourth column is theAlfven velocity.

LHD plasma [26, 27, 28, 29].

2.1. Equilibrium properties

A fixed boundary equilibrium from the VMEC code[19] was calculated during the LHD shot 116190after the destabilization of an EIC event includingthe EP component of the pressure. The electrondensity and temperature profiles were reconstructedby Thomson scattering data and electron cyclotronemission. Table 1 shows the main parameters of thethermal plasma. The cyclotron frequency is ωcy =2.41 · 108 s−1.

The magnetic field at the magnetic axis is 2.5 Tand the averaged inverse aspect ratio is ε = 0.16. Theenergy of the injected particles by the perpendicularNBI is Tf,⊥(0) = 40 keV, but we take the nominalenergy Tf,⊥(0) = 28 keV (vth,f = 1.64 · 106 m/s)resulting in an averaged Maxwellian energy equal tothe average energy of a slowing-down distribution.Figure 1 (a) shows the iota profile, (b) the normalizedpressure profile, (c) the thermal plasma density (blackline) and temperature (red line) profiles and (d) theEP density profile. It should be noted that the effectof the equilibrium toroidal rotation is not included inthe model for simplicity. The effect of the Dopplershift on the instability frequency caused by the toroidalrotation is small, particularly for a mode located in theplasma periphery, as it is observed in the experiments.

Figure 1. (a) Iota profile, (b) normalized pressure (thermal+ EP pressure) profile, (c) thermal plasma density andtemperature profiles and (d) density profile of the helicallytrapped EP.

2.2. Simulations parameters

The dynamic and equilibrium toroidal (n) and poloidal(m) modes included in the study are summarizedin table 2. The mode selection covers the resonantrational surfaces between the middle plasma and theplasma periphery, thus the inner plasma region (r/a <0.5) is not included in the simulations in order to reducethe computational time and avoid the destabilization ofundesired modes in the plasma core. The simulationsare performed with a uniform radial grid of 400 points.In the following, the mode number notation is n/m,which is consistent with the ι = n/m definition for theassociated resonance.

n m0 [0, 6]1 [1, 2]2 [2, 4]3 [2, 6]

Table 2. Dynamic and equilibrium toroidal (n) and poloidal(m) modes in the simulations.

The closure of the kinetic moment equations (6)and (7) breaks the MHD parities so both paritiesmust be included for all the dynamic variables.The convention of the code with respect to theFourier decomposition is, in the case of the pressureeigenfunction, that n > 0 corresponds to cos(mθ+nζ)and n < 0 corresponds to sin(−mθ−nζ). For example,the Fourier component for mode 3/4 is cos(3θ+4ζ) andfor the mode −3/ − 4 is sin(3θ + 4ζ). The magneticLundquist number is assumed S = 5 · 106, consistentwith the S value at the plasma periphery of LHDplasma, reproducing the stability properties regardingthe plasma resistivity for the 1/1 resistive interchangemode (RIC) and 1/1 EIC.

The density and temperature of the population ofhelically trapped EP generated by the perpendicularbeam are calculated by the code MORH [30, 31]. Forsimplicity, no radial dependency of the EP energyis considered and the EP density profile given bythe MORH code is fitted to the following analyticexpression:

nf,⊥(r) =(0.5(1+tanh(δr·(rpeak−r))+0.02)(0.5(1+tanh(δr·rpeak))+0.02) (1)

with the location of the EP density gradient profiledefined by the variable rpeak = 0.85 and the flatnessby δr = 10. A previous study indicated the passingparticles generated by the tangential NBIs could havea stabilizing effect of the EIC [25]. Nevertheless, theEIC saturation phase can be studied independently ofthe passing particle effects, reason why the passingparticles are not included in the simulations forsimplicity.

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The Landau closure in the model is based ontwo moment equations for the energetic particles,which is equivalent to a two-pole approximation of theplasma dispersion relation. The closure coefficientsare adjusted by fitting analytic AE growth rates.Such assumptions are consistent with a Lorentzianenergy distribution function for the energetic particles.The lowest order Lorentzian can be matched eitherto a Maxwellian or to a slowing-down distributionby choosing an equivalent average energy. For theresults given in this paper, we have matched the EPtemperature to the mean energy of a slowing-downdistribution function.

The simulations begin with a thermal β 50%smaller with respect to the original equilibria,increasing to 60% after t = 104τA0. The RIC isunstable for 60% of the original thermal β thus thevalue of the thermal β is no further increased duringthe simulation, avoiding numerical instabilities andthe destabilization of the 1/2 interchange mode in themiddle plasma region. Once the non linear simulationfor the thermal plasma reaches t = 2 · 104τA0, the EPdestabilizing effect is turned on by linearly increasingthe EP β. Figure 2 indicates the EP β values alongthe simulation. The EP β in the experiment increasesalong the discharge because the perpendicular NBIgenerates a larger population of trapped EP, thus theEP β may increase along the simulation to reproducethe different phases of the 1/1 EIC observed in theexperiment, linked to different EP β thresholds. TheEP β increments in the simulation are selected toavoid large variations of the AE/EPM stability inshort simulation times, that is to say, the simulationcan capture the effect of the EP β on the AE/EPMstability, identifying the EP β threshold linked to thedifferent phases of the EIC. Between t = 2.0 − 2.8 ·104τA0 the EP β increases from 0.0 to 0.01 by ∆β =0.002 each 2000τA0, reproducing the Phase I and theearly phase II of the 1/1 EIC. Next, the EP β is fixedto 0.01 until t = 3.6 · 104τA0, reproducing the phaseII. Finally,the EP β increases by ∆β = 0.0025 betweent = 3.7−4.0·104τA0 up to 0.02 leading to the triggeringof a burst event (Phase III). The stabilization phaseof the 1/1 EIC is analyzed separately (Phase IV),performing by a new simulation including only then = 1 toroidal family for simplicity.

3. Saturation of the 1/1 EIC

The 1/1 EIC phases I to III are reproduced by asingle non linear simulation as the EP β increases.Figure 3, panels a and b, show the time evolution of thepoloidal component of the magnetic field perturbation(Bθ) at different plasma regions. A fast oscillatingperturbation appears between r/a = 0.8−1.0 (pink and

Figure 2. EP β along the simulation.

cyan lines in panel a), indicating the destabilization ofthe 1/1 EIC in the plasma periphery at t = 2.6 ·104τA0

and the transition from the Phase I to the Phase II .The perturbation amplitude increases during the PhaseII and reaches its maximum amplitude during the burstevent triggered in the Phase III (t = 3.9 · 104τA0). Inaddition, the perturbation extends from the plasmaperiphery to the middle plasma region (blue line inpanel b). Regarding the kinetic and magnetic energy(KE and ME, respectively) of the modes (including then = 0 toroidal family contribution), there are two localmaximum observed after the 1/1 EIC is excited andduring the burst event, although the KE/ME energypeak is 3 times larger in the burst event.

Figure 4 indicates the time evolution of thenormalized energy for the dominant modes among thephases II and III of the simulation, excluding the mode0/0. The consecutive local maxima / minima of the1/1 mode energy (red line) corresponds to cycles offurther destabilization / saturation of the 1/1 EIC,correlated with an enhancement / weakening of Bθoscillations at the plasma periphery. It should be notedthat the local maxima of the 1/1 mode are linked tolocal maxima of the overtones 2/2 (green line) and3/3 (pink line) energy, particularly large during thebursting event. From t = 3 · 104τA0 there are localmaxima of the 3/4 mode energy indicating the modedestabilization. Once the 3/4 mode is unstable, 3/4mode kinetic (panel a) and magnetic (panel b) energyshow a large enhancement at the end of the phaseII, larger regarding 1/1 mode kinetic and magneticenergy during the phase III. Consequently, the largestperturbation of the surrounding plasma flows as wellas the strongest magnetic perturbation is caused bythe 3/4 mode during the phase III, although thedominant perturbations during the phases I and IIare generated by the mode 1/1. Likewise, the energyassociated to the EP destabilizing effect (panel c) is

Non linear EIC 6

Figure 3. Time evolution of the poloidal component of themagnetic field perturbation (a) at r/a = 1.0 (pink line) and 0.8(cyan line), (b) inner-middle plasma at r/a = 0.6 (blue line),0.4 (red line) and 0.2 (black line), (c) normalized kinetic energyand (d) normalized magnetic energy. The dashed orange lineindicates the beginning of the EIC Phase II and the dashed darkred line the beginning of the Phase III. The long dashed greenline shows the simulation time when the EP β linearly increases.The yellow long dashed line indicates the simulation time whenthe burst even is triggered.

largely dominated by the 1/1 mode and overtonesduring the phases I and II, indicating that there isan energy transfer from the 1/1 EIC towards the 1/1mode overtones as well as the 3/4 mode. On the otherhand, the leading destabilizing effect is caused by the3/4 mode after the burst event.

In the following, each 1/1 EIC phase is analyzedseparately.

3.1. Phase I: 1/1 EIC destabilization

Figure 5 indicates the evolution of the electrostaticpotential eigenfunction of the 1/1 mode during thePhase I and early Phase II of the non linear simulation.In addtion, the growth rate and frequency is calculatedperforming linear simulations using the profiles of thenon linear simulation. If the EP β ≤ 0.004 the modeis a RIC (panels a and b) because the eigenfunction issymmetric with respect to the mode parities and thefrequency is lower than 1 kHz. The RIC is triggered ifthe pressure profile gradient is large enough to exceedthe stability limit of the interchange modes at theplasma periphery, although the RIC amplitude, widthand growth rate is also affected by the driving of the

Figure 4. Time evolution of the dominant modes normalized(a) kinetic (b) magnetic and (c) EP energy. The mode 1/1 isindicated by the red line, the 2/2 by the green line, the 3/3 bythe pink line, the 2/3 by the brown line and the 3/4 by the cyanline.

EP, increasing as the EP β grows. On the other hand,if the EP β = 0.006 the eigenfunction of the 1/1 modeparities are not symmetric, the mode frequency is 4kHz and the growth rate increases almost one orderof magnitude, indicating the destabilization of the 1/1EIC. The 1/1 EIC amplitude, width and growth ratealso increases with the EP β although this dependencyis stronger regarding the RIC.

Figure 5. 1/1 mode electrostatic potential eigenfunctioncalculated in the non linear simulation if (a) EP β = 0.002,(b) 0.004, (c) 0.006 and (d) 0.008. The mode growth rate andfrequency is included.

It should be noted that the EP β threshold todestabilize the 1/1 EIC is 0.006 in the simulation,similar to the critical β measured in the experiment.In addition, the increase of the amplitude and width of

Non linear EIC 7

the RIC and 1/1 EIC is also observed in the experimentas the EP population grows.

3.2. Phase II: inward propagation of the instability

The saturation of the 1/1 EIC begins in the Phase II assoon as the non linear effects dominate the instabilityevolution, as can be observed in the eigenfunction ofthe electrostatic potential of the n = 0 to 3 toroidalmode families (fig. 6). At t = 3.0 · 104τA, panel a,the 1/1 EIC amplitude increases and the overtones2/2 and 3/3 are unstable. In addition, the mode 0/0is destabilized indicating that the 1/1 EIC modifiesthe nearby plasma flows generating the so-called shearflows, that is to say, part of the 1/1 EIC energy istransferred toward the thermal plasma generating areadjustment of the equilibrium flows in the plasmaperiphery. A detailed analysis of the generation ofshear flows by the 1/1 EIC is out of the scope of thisarticle and will be the topic of a future study. Atpresent, the analysis of the simulation data presumesthe stability of the 1/1 EIC is affected by the presenceof shear flows, particularly during the saturation phaseof the instability once a feedback effect between themode and the shear flows exit. At t = 3.2 · 104τA,panel b, the mode 3/4 is triggered at r/a = 0.76 dueto the non-linear coupling with the mode 1/1, so that afraction of the 1/1 EIC energy is also channeled towardmodes of different toroidal families. The same way, att = 3.4 · 104τA, panel c, the mode 2/3 is triggeredaround r/a = 0.7. Later, at t = 3.6 · 104τA, paneld, the amplitude of the 3/4 mode keeps increasingindicating that the energy channeling continues whilethe 1/1 EIC is active. Consequently, the enhancementof the electron temperature perturbation measuredat different radial location during the Phase II [7]could be caused by the combined effect of the furtherdestabilization of the 1/1 EIC (increase of the 1/1mode width), the excitation of the 2/2 and 3/3overtones as well as the non linear destabilization of the3/4 and 2/3 modes, leading to the inward propagationof the perturbation.

Table 3 shows the growth rate and frequency ofthe n = 1 to 3 dominant modes (the modes withthe largest growth rate of each toroidal family) duringthe Phase II, calculated by linear simulations usingthe profiles obtained from the non linear simulations.The mode 3/4 has a frequency around 12 kHz andthe eigenfunction of the mode parities are asymmetric,indicating that the 3/4 mode is an EPM. On the otherhand, the mode 2/3 has a frequency below 1 kHz, asymmetric eigenfuncion with respect to mode paritiesand a growth rate 5 times smaller than the 1/1 EIC,thus the 2/3 is an interchange mode.

Now, the effect of the 1/1 EIC on the equilibriumprofiles is analyzed. Figure 7 shows the evolution of

Figure 6. Electrostatic potential eigenfunction of the n = 1to n = 3 toroidal mode families calculated in the non linearsimulation at (a) t = 3.0 · 104τA, (b) t = 3.2 · 104τA, (c)t = 3.4 · 104τA and (d) t = 3.6 · 104τA.

n/m Time Growth rate f(104τA0) (γτA0) (kHz)

1/1 2.9 0.0067 7.161/1 3.0 0.0057 6.151/1 3.2 0.0059 6.393/4 3.2 0.0079 12.211/1 3.4 0.0058 6.322/3 3.4 0.0010 < 13/4 3.4 0.0078 12.231/1 3.6 0.0058 6.262/3 3.6 0.0010 < 13/4 3.6 0.0082 12.59

Table 3. Growth rate and frequency of the n = 1 to n = 3modes during the Phase II.

the EP density and pressure profiles during the PhaseII. There is a flattening of the EP density and pressureprofiles around r/a = 0.88, the radial location of the1/1 rational surface. The largest deviation from theequilibrium profiles is observed at t = 2.9 · 104τA0 oncethe 1/1 EIC growth rate reaches a local maximum(also identified by the local maxima of the KE andME, see fig 4). The 1/1 EIC saturates because thegradient of the EP density profile around the 1/1rational surface is weaker, leading to a decrease of thefree energy required to sustain the 1/1 EIC unstable.For this reason the mode growth rate decreases. Itshould be noted that the EP β of the simulationis not modified after the 1/1 EIC is triggered, eventhough the experimental observations indicate thatpart of the helically trapped EP population is lost afterthe plasma relaxation. The EP β of the simulationdoes not decrease because the destabilizing effect ofthe EP is maximized to reduce the simulation timerequired to reproduce the different 1/1 EIC phases.

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With regard to the effect of the 1/1 EIC on thepressure, the pressure is also flattened around the 1/1rational surface, leading to a decrease of the pressuregradient at the plasma periphery. Thus, the stabilityof the EPM/AE and pressure gradient driven modesare non linearly connected, in a manner resulting inthe stabilization of the RIC during the saturation ofthe 1/1 EIC. The displacement of ∆(r/a) = 0.0025between the flattening induced in the pressure and EPdensity profiles is caused by the model resolution.

Figure 7. Evolution of the (a) EP density and (b) pressureprofiles towards the Phase II (Equilibrium + 0/0 perturbationcomponent). The black broad line indicates the equilibriumprofiles. The orange arrow indicates the radial location of thelargest deviation from the equilibrium.

Figure 8 shows an isosurface and the evolution ofthe poloidal contour of the EP density perturbation.The 1/1 EIC perturbation is located at the plasmaperiphery (panel b). There are two other perturbationslinked to the 3/4 EPM (t = 3.2 · 104τA, panel c) andthe 2/3 interchange mode (t = 3.4 · 104τA, panel d),although the strongest perturbation is caused by the1/1 EIC along the Phase II.

In summary, the inward propagation of theinstability observed in the experiment during thePhase II of the EIC saturation could be caused bythe non linear destabilization of modes from differenttoroidal families, particularly the 3/4 EPM and the 2/3

Figure 8. (a) Isosurface and poloidal contours of the EPdensity. The yellow rectangle indicates the location of thepoloidal contour plotted. Evolution of the poloidal contour of theEP density perturbation at (b) t = 2.9 ·104τA, (c) t = 3.2 ·104τAand (d) t = 3.4 · 104τA. The yellow arrows indicate the locationof the instabilities.

interchange mode. These modes are excited due to theenergy transferred from the 1/1 EIC, channeled towardthe thermal plasma for the case of the 2/3 interchangemode and through non linear couplings with the 3/4EPM.

3.3. Phase III: burst event

The burst event during the Phase III is triggered oncethe EP β of the simulation increases to 0.02, betweent = 3.9 − 4.0 · 104τA (see fig 3 and 4). The 2/2 and3/3 overtones are further destabilized showing a modeamplitude similar to the mode 1/1 during the burstevent (fig 9, panels a, c, e and g). In addition, theamplitude of the 0/0 mode is almost 2 times largerwith respect to the 1/1 mode (t = 4.0 ·104τA), pointingout a strong perturbation of the equilibrium and thetriggering of a large plasma relaxation, consistent withthe local maxima of the KE and ME (see fig 3c and d).The energy channeled from the 1/1 EIC toward thethermal plasma and by non linear coupling with othertoroidal families leads to the further destabilizationof the middle-outer plasma region (please comparethe poloidal contours of the EP density perturbationbetween the panels b, d, f and h). Table 4 showsthe growth rate and frequency of the n = 1 to 3dominant modes during the burst event. At t =3.9 ·104τA the growth rate of the mode 1/1 is two times

Non linear EIC 9

larger with respect to the Phase II and the frequencyincreases to 9.36 kHz. In addition, the dominantmodes of the n = 2 and 3 toroidal families are theovertones 2/2 and 3/3 with frequencies of 12.8 and 22kHz, respectively. The overtones frequency and theasymmetry of the eigenfunction of the mode paritiesindicate that these modes are EPMs. Consequently,the simulation reproduces the increase of the 1/1 EICfrequency to 9.4 kHz measured in the experiment,as well as the instability enhancement at the plasmaperiphery (0.88 < r/a < 0.91) and the complex modestructure between 0.6 < r/a < 0.85 during the burstevent. It should be noted that at t = 4.0 · 104τA thedominant mode of the n = 3 toroidal family is the3/4, consistent with the inward displacement of theinstability.

n/m Time Growth rate f(104τA0) (γτA0) (kHz)

1/1 3.9 0.0119 9.362/2 3.9 0.0137 12.823/3 3.9 0.0221 22.061/1 4.0 0.0135 9.532/2 4.0 0.0170 13.303/4 4.0 0.0267 18.15

Table 4. Growth rate and frequency of the n = 1 to n = 3dominant modes during the burst event.

Figure 10 shows the evolution of the EP densityand pressure profiles in the Phase III. The flatteningof the profiles around the 1/1 rational surface isbroader with respect to the Phase II (panels a andb), particularly once the burst event is triggered at t =3.9·104τA. There is further flattening in the EP densityand pressure profiles caused by the rational surface 3/4around r/a = 0.76 (panels c and d), showing the largestexclusion with respect to the equilibrium profiles att = 4.0 · 104τA. There is a third flattening induced bythe 2/3 rational surface around r/a = 0.69, showinga smaller deviation from the equilibrium profiles withrespect to the 1/1 and 3/4 rational surfaces, mainlyobserved in the EP density profile, indicating that the2/3 interchange mode destabilized during the phase IIevolved to a 2/3 EPM due to a the non linear couplingwith the 1/1 EIC. Figure 11 shows the electrostaticpotential eigenfunction of the 2/3 and 3/4 EPMs,growth rate and frequency. The destabilization of themode 0/0 by the 2/3 and 3/4 EPMs indicates thatthese modes also generate a local modification of thesurrounding plasma flows, although the effect is weakerthan for the 1/1 EIC because the 0/0 mode amplitudeis smaller. Consequently, the energy channeled fromthe 1/1 EIC by non-linear couplings towards thethermal plasma and the 2/3 interchange mode as wellas the 3/4 EPM modifies the plasma flows, pressure

Figure 9. Evolution of the electrostatic potential eigenfunctionand poloidal contour of the EP density perturbation in the nonlinear simulation at (a) t = 3.8 · 104τA, (c) t = 3.9 · 104τA, (e)t = 4.0 · 104τA, (g) t = 4.1 · 104τA and (i) t = 4.2 · 104τA. Theyellow box indicates the modes eigenfunction and the poloidalcontour of the EP density perturbation during the burstingevent.

and EP density profiles between r/a ≈ 0.65−0.92, thatis to say, the middle-outer plasma region is destabilizedas the experiment observations indicate.

In short, the burst event triggered in the non linearsimulation shows an increase of the 1/1 EIC frequencyto 9.35 kHz, similar to the experimental measurement.Also, the complex mode structure near the 1/1 rationalsurface could be caused by the destabilization of the2/2 and 3/3 overtones. The inward propagation ofthe instability could be explained by the non lineardestabilization of the 2/3 and 3/4 EPMs due to the

Non linear EIC 10

Figure 10. Evolution of the (a) EP density and (b) pressureprofiles of the 1/1 EIC, Evolution of the (c) EP density and(d) pressure profiles of the 3/4 EPM. Evolution of the (e) EPdensity and (f) pressure profiles of the 2/3 EPM (Equilibrium+ 0/0 perturbation component). The black broad line indicatesthe equilibrium profiles. The orange arrow indicates the radiallocation of the strongest deviation from the equilibrium.

Figure 11. Electrostatic potential eigenfunction in the nonlinear simulation of the (a) 2/3 EPM and (b) 3/4 EPM att = 4.0 · 104τA.

energy channeled from the 1/1 EIC. Therefore, themultiple resonances triggered at nearby radial locationsand frequency ranges could lead to an enhancementof the EP losses, a resonance overlapping alreadyproposed and analyzed by other authors [32].

3.4. Phase IV: stabilization

Now, the stabilization phase (Phase IV) of the 1/1EIC is analyzed. For simplicity, a new simulations isperformed including only the n = 1 toroidal familyin the non linear simulation, excluding the n = 2

and 3 toroidal mode families. The analysis targetis reproducing the plasma conditions that lead tothe transition from an unstable 1/1 EIC to the RICtriggering observed in the experiment. Figure 12shows the EP β along the simulation. EP β = 0.01between t = 1.0 − 2.3 · 104τA, increasing to 0.025 att = 2.8 · 104τA. From t = 2.9 · 104τA the EP β isset to zero reproducing the EP losses caused by the1/1 EIC. It should be noted that in LHD dischargesnot all the EP population is lost after the 1/1 EIC istriggered, only a fraction of the EP population, thus achain of instabilities are observed in the experiment asthe EP β exceeds or decreases below the destabilizationthreshold of the 1/1 EIC. To avoid the destabilizationof another 1/1 EIC along the simulation the EP β isset to zero, so that the study can be focused in thetransition from the 1/1 EIC to the RIC. In addition,the EP β = 0.025 before setting the EP β to zero inorder to maximize the destabilizing effect of the EP onthe transition between the EIC phases III to IV. Thatway, it is easier to analyze the effect of the EP on theRIC stability before and after the transition, leading toa more didactic evaluation of the experimental data.

Figure 12. EP β along the simulation.

Figure 13a shows the time evolution of Bθ atdifferent radial locations in the Phase IV. From t =1.3·104τA the plasma periphery is unstable once the EPβ threshold of the 1/1 EIC is exceed, further enhancedas the EP β increases. There is a transition fromhigh frequency to low frequency Bθ oscillations afterthe EP β is set to zero, indicating a change of theperturbation source. Figure 13b shows an increase ofthe KE and ME after the 1/1 EIC is triggered reachingits maximum once the EP β = 0.025. The KE and MEdecrease after the EP β is set to zero, almost an orderof magnitude smaller with respect to the 1/1 EIC,pointing out that the low frequency mode is potentiallyless hazardous for the device performance.

Figure 14, panels a and b, shows the eigenfunction

Non linear EIC 11

Figure 13. (a) Time evolution of the poloidal component of themagnetic field perturbation at r/a = 1.0 (pink line), 0.8 (cyanline), 0.6 (blue line), 0.4 (red line) and 0.2 (black line). (b)Normalized kinetic (black line) and magnetic (red) energy. Thehorizontal dashed green lines indicates the simulation time whenthe EP β increases linearly from 0.01 to 0.025. The horizontalorange dashed line indicates the simulation time when the EP βis set to zero.

of the electrostatic potential during the Phase IV. Att = 2.8·104τA the 1/1 EIC is unstable with f = 3.4 kHzand a growth rate of γτA0 = 0.012. At t = 3.4 · 104τAthe frequency of the mode decreases below 1 kHz andthe eigenfunction shows symmetric 1/1 mode parities,thus the mode is a RIC with a growth rate almosttwo orders of magnitude smaller than the 1/1 EIC.It should be noted that the amplitude of the mode0/0 decreases by a 60%, indicating that the effectof the RIC on the equilibrium plasma flows is alsosmaller. The evolution of the pressure profiles shows aflattening at t = 3.4 · 104τA, panel d, consistent withthe saturation of a pressure gradient driven mode. Onthe other hand, the EP density profile shows only asmall deviation from the equilibrium profile (panel c).

Figure 14. Evolution of the electrostatic potential eigenfunc-tion at (a) t = 2.8·104τA and (b) t = 3.4·104τA. Evolution of the(c) EP density and (d) pressure profiles. The black broad line in-dicates the equilibrium profiles. The orange arrow indicates theradial location of the strongest deviation from the equilibrium.

In summary, the simulation reproduces the 1/1EIC stabilization due to the decrease of the helicallytrapped EP population and the RIC destabilizationonce the stabilizing effect of the 1/1 EIC vanishes andthe pressure profile gradient builds up again at theplasma periphery.

4. Conclusions and discussion

Two non linear simulations are performed by theFAR3d code analyzing the saturation of the 1/1 EIC.The simulation results show a reasonable agreementwith the experimental data [7].

The non linear simulations calculate an EP βthreshold of 0.006 for the destabilization of the 1/1EIC, similar to the experiment. In addition, theenhancement of the RIC and 1/1 EIC width andamplitude as the EP β increases is also reproduced.

The inward propagation of the instability from theplasma periphery to the middle plasma region is ob-served in the simulations. The inward propagation iscaused by the non linear destabilization of the modes2/3 and 3/4. The 1/1 EIC energy is channeled towardsthe thermal plasma, leading to the destabilization ofthe 2/3 interchange mode in the phase II, as well as bynon linear coupling with the 3/4 and 2/3 EPMs in thephases II and III, respectively. Figure 15 shows the evo-lution of the pressure perturbation between the middleand outer plasma region. The evolution of the pressureperturbation has similarities with the evolution of theelectron temperature perturbation measured in the ex-periment (please see the fig. 7b of the reference [7]).If the 1/1 EIC phases of the experiment and the sim-ulation are compared, the numerical model reproducesthe enhancement of the perturbation at r/a = 0.88,0.75 and 0.68 caused by the destabilization of the 1/1,3/4 and 2/3 modes during the Phase II. In addition,once the 3/4 mode is destabilized, the perturbation ex-pands outward from r/a = 0.75 to 0.85 (white dottedarrow) followed by the inward expansion of the 1/1perturbation from r/a = 0.9 to 0.78 (gray dotted ar-row). Such consecutive counter-propagating perturba-tions are also observed in the experiment [33]. Afterthe burst event is triggered during the Phase III (solidyellow line) the 1/1, 3/4 and 2/3 perturbations overlapand the instability propagates further inward (dottedyellow line), consistent with the inward propagationof the instability observed in the experiment betweenr/a = 0.88 and 0.65. It should be noted that certaindifferences are observed between the experimental dataand the simulation results, particularly with respect tothe radial transport. The simulations show predom-inant horizontal contourns and some vertical mixing,although the experimental contourns of the electrontemperature are mainly vertical, that is to say, the ra-

Non linear EIC 12

dial transport in the experiment is stronger. Presentdeveloping efforts of the numerical model are dedicatedto improve the transport description reducing the dis-crepancy between simulation and experimental obser-vations. Consequently, the transport analysis is out ofthe scope of this study. In addition, FAR3d code is asingle fluid model thus electrons and ions cannot be an-alyzed separately, reason why the comparison betweenthe measured electron temperature perturbation andthe pressure perturbation provided by the simulationshould be understood as a first order approximation.Such information is useful to analyze the inward prop-agation of the instability observed in the experiment,although a two fluid model is required to provide amore complete description of the burst event.

The linear simulations performed using theprofiles obtained in the non linear simulation show thatduring the Phase II the 3/4 perturbation is an EPMwith a frequency of 12.2 kHz and the 2/3 perturbationan interchange mode, although the mode 2/3 evolves toan EPM with a frequency of 5.25 kHz during the PhaseIII. The 3/4 and 2/3 EPMs are destabilized becausepart of the 1/1 EIC energy is channeled by non linearcouplings toward these modes. On the other hand, the2/3 interchange mode is triggered because the 1/1 EICenergy is also transferred toward the thermal plasma.

The non linear simulations also indicate the effectof the 1/1 EIC on the equilibrium profiles, particularlyby the destabilization of the 0/0 mode. The saturationof the 1/1 EIC begins as soon as the EP densityprofile flattens around the 1/1 rational surface, leadingto a decrease of the EP density profile gradient atthe plasma periphery as well as the available freeenergy required to destabilize the mode. It shouldbe noted that the destabilization of the 1/1 EIC leadsto a loss of the EP population and a decrease of theEP β in the experiment, although this effect is onlyincluded in the simulation dedicated to the analysis ofthe stabilization phase of the 1/1 EIC (Phase IV) forsimplicity and computational efficiency. In addition,once the 1/1 EIC is triggered the pressure profileshows a flattening around the 1/1 rational surfaceand the pressure gradient at the plasma peripheryis weaker. This result indicates that the stabilityof the EPM and pressure gradient driven modes areconnected non linearly. Consequently, the RIC isstable after the 1/1 EIC is triggered because theRIC saturates due to the flattening of the pressureprofile induced via the 1/1 EIC. On top of that, the1/1 EIC and the 3/4 EPM modify the surroundingequilibrium plasma flows because the 0/0 componentof the electrostatic potential is destabilized, consistentwith the experiment observations [7]. On the otherhand, the effect of the 1/1 EIC on the -ι profile issmall because there are only small changes on the

radial variation of the 0/0 component of the poloidalflow, particularly at the plasma periphery due to theproximity of the device coils (data not shown).

The simulation calculates an increase of the 1/1EIC frequency to 9.4 kHz during the burst event,similar to the experimental observations. In addition,the overtones 2/2 and 3/3 are destabilized showingthe largest growth rate of the respective toroidalfamilies. The overtone destabilization as well as theperturbation overlapping with the 3/4 and 2/3 EPMscould explain the complex mode structure measuredin the experiment. The inward propagation of theinstability can be also tracked by the transition of thedominant mode of the n = 3 toroidal family betweenthe 3/3 to the 3/4 (mode with the largest growth rate)during the burst event, as well as the EP and pressureprofiles flattening that appear around the 3/4 and 2/3rational surfaces. In addition, the large decrease ofthe EP population measured during the burst eventcould be caused by the partial overlapping between theresonances of the 1/1 EIC, 3/4 EPM and 2/3 EPM,leading to the enhancement of the outward transportof the helically trapped EP population [32].

The non linear simulation that analyzes thestabilization of the 1/1 EIC caused by loss of the EPpopulation shows the destabilization of a RIC as soonas the stabilizing effect of the 1/1 EIC vanishes. Thisis because the flattening induced in the pressure profiledisappears and the profile gradient recovers, exceedingthe stability limit of the interchange modes at theplasma periphery. It should be noted that the energyof the RIC is almost one order of magnitude smallerwith respect to the 1/1 EIC, thus the RIC is a lessdangerous instability for device performance.

Present simulations include a reduced set oftoroidal families to minimize the simulation timealthough retaining the essential physics requiredto analyze the saturation phase of the 1/1 EIC,particularly the non linear coupling between then = 1 and the n = 0, 2 and 3 toroidal families.Nevertheless, future studies will include an extendednumber of toroidal families as well as the effect ofthe helical couplings; this is expected to have ameaningful contribution during the bursting event as aprevious study indicated [8]. In addition, the collectivetransport driven by the EIC will be evaluated by thederivation of an effective transport coefficient for theEP.

An improved LHD performance requires theattenuation or avoidance of the 1/1 EIC. The linearstability and optimization strategies of the 1/1 EICare already discussed in other publications from theexperimental and theoretical point of view [8, 25, 34,35, 36, 37], although a detailed study of the nonlinear evolution is mandatory to analyze the saturation

Non linear EIC 13

Figure 15. Evolution of the pressure perturbation between r/a = 0.65 − 0.9. The orange dashed lines indicate the transitionbetween the 1/1 EIC phases. The dotted white/gray lines indicate the counter-propagating perturbations during Phase II. The solidyellow arrow indicates the burst even and the dotted yellow line the inward propagation of the 1/1 EIC perturbation.

phase of the instability. Some LHD operation scenariosrequire strong plasma heating by the perpendicularNBIs thus a large population of helically trappedEP is generated, exceeding the stability limit of the1/1 EIC. The non linear simulation results indicatethat the most injurious effect of the 1/1 EIC onthe device performance takes place during the burstevent. The resonance overlap caused by the nonlinear destabilization of the 3/4 and 2/3 EPMs lead tosignificant losses in the helically trapped EP populationbefore thermalization, reducing the heating efficiencyof the perpendicular NBIs and limiting the maximumβ reached during the discharge. Consequently, thetransition from the Phase II to III of the EIC 1/1should be avoided. A possible indicator of the EPβ threshold for the phase II to III transition couldbe the electron temperature fluctuations nearby therational surfaces 3/4 and 2/3, indicating a large energychanneling from the 1/1 EIC and the destabilization ofthe 3/4 and 2/3 EPMs as precursors of the burst event.Dedicated experiments in LHD will be suggested toconfirm the present study results.

Appendix

The model equations and numerical scheme areindicated in this appendix. The model formulationassumes high aspect ratio, medium β (of the order ofthe inverse aspect ratio ε = a/R0), small variation ofthe fields and small resistivity. The plasma velocityand perturbation of the magnetic field are defined as

v =√gR0∇ζ ×∇Φ, B = R0∇ζ ×∇ψ, (2)

where ζ is the toroidal angle, Φ is a stream functionproportional to the electrostatic potential, and ψ is theperturbation of the poloidal flux.

The equations, in dimensionless form, are

∂ψ

∂t=√gB∇‖Φ+ηε2JJζ−

(1

ρ

∂ψ

∂θ

∂

∂ρ+∂ψ

∂ρ

1

ρ

∂

∂θ

)Φ(3)

∂U

∂t= −vζ,eq

∂U

∂ζ

+√gB∇||Jζ −

1

ρ

(∂Jeq∂ρ

∂ψ

∂θ− ∂Jeq

∂θ

∂ψ

∂ρ

)

− β02ε2√g (∇√g ×∇p)ζ − βf

2ε2√g (∇√g ×∇nf )

ζ

−v · ∇U −

(1

ρ

∂ψ

∂θ

∂

∂ρ+∂ψ

∂ρ

1

ρ

∂

∂θ

)Jζ +DU∇2

⊥U (4)

∂p

∂t= −vζ,eq

∂p

∂ζ+dpeqdρ

1

ρ

∂Φ

∂θ

+Γpeq

[(∇√g ×∇Φ

)ζ−∇‖v‖th

]+∂p

∂ρ

1

ρ

∂Φ

∂θ− 1

ρ

∂p

∂θ

∂Φ

∂ρ− Γp

(∇√g ×∇Φ

)ζ− β0B

2n0(J − -ιI)

(∂

∂θ− -ι

∂

∂ζ

)pv||,th +Dp∇2

⊥p (5)

∂v‖th

∂t= −vζ,eq

∂v||th

∂ζ− β0

2n0,th∇‖p

+∂v||,th

∂ρ

1

ρ

∂Φ

∂θ− 1

ρ

∂v||,th

∂θ

∂Φ

∂ρ

+β0B

2n0(J − -ιI)

(∂ψ

∂ρ

1

ρ

∂p

∂θ− 1

ρ∂ψ∂θ

∂p

∂ρ

)

+B

(J − -ιI)

(∂

∂θ− -ι

∂

∂ζ

)v2||,th +Dvth∇2

⊥v||,th (6)

Non linear EIC 14

∂nf∂t

= −vζ,eq∂nf∂ζ−v2th,fε2ωcy

Ωd(nf )− nf0∇‖v‖f

−nf0 Ωd(Φ) + nf0 Ω∗(Φ)

+∂nf∂ρ

1

ρ

∂Φ

∂θ− 1

ρ

∂nf∂θ

∂Φ

∂ρ

+B

(J − -ιI)

(∂

∂θ− -ι

∂

∂ζ

)nf v||,f +Dnf∇2

⊥nf (7)

∂v‖f

∂t= −vζ,eq

∂v||f

∂ζ−v2th,fε2ωcy

Ωd(v‖f )

−(π

2

)1/2vth,f

∣∣∇‖v‖f ∣∣− v2th,fnf0∇‖nf + v2th,f Ω∗(ψ)

+∂v||,f

∂ρ

1

ρ

∂Φ

∂θ− 1

ρ

∂v||,f

∂θ

∂Φ

∂ρ

+B

(J − -ιI)

(∂

∂θ− -ι

∂

∂ζ

)v2||,f +Dvf∇2

⊥v||,f (8)

Equation (2) is derived from Ohm’s law coupledwith Faraday’s law, equation (3) is obtained fromthe toroidal component of the momentum balanceequation after applying the operator ∇∧√g, equation(4) is obtained from the thermal plasma continuityequation with compressibility effects and equation(5) is obtained from the parallel component of themomentum balance, the equations (6) and (7) aremoments of the kinetic equation for the energetic

particles. Here, U =√g[∇×

(ρm√gv)]ζ

is thetoroidal component of the vorticity, ρm the ion massdensity, ρ =

√φN the effective radius with φN the

normalized toroidal flux and θ the poloidal angle.The perturbation of the toroidal current density Jζ

is defined as:

Jζ =1

ρ

∂

∂ρ

(−gρθ√

g

∂ψ

∂θ+ ρ

gθθ√g

∂ψ

∂ρ

)

−1

ρ

∂

∂θ

(gρρ√g

1

ρ

∂ψ

∂θ+ ρ

gρθ√g

∂ψ

∂ρ

)(9)

v||th is the parallel velocity moment of the thermalplasma and vζ,eq is the equilibrium toroidal rotation.β0 is the equilibrium β at the magnetic axis, βfis the maximum value of the EP β (located at themagnetic axis in the on-axis cases but not in the off-axis cases) and nf0 is the EP radial density profilenormalized to the local maxima. Φ is normalized toa2B0/τA0 and ψ to a2B0 with τA0 the Alfven timeτA0 = R0(µ0ρm)1/2/B0. The radius ρ is normalizedto a minor radius a; the resistivity to η0 (its valueat the magnetic axis); the time to the Alfven time;the magnetic field to B0 (the averaged value at themagnetic axis); and the pressure to its equilibriumvalue at the magnetic axis. The Lundquist numberS parameter is the ratio of the resistive time τR =a2µ0/η0 to the Alfven time. -ι is the rotational

transform, vth,f =√Tf/mf/vA0 is the radial profile

of the energetic particle thermal velocity normalized tothe Alfven velocity at the magnetic axis vA0 and ωcythe energetic particle cyclotron frequency normalizedto τA0. qf is the charge, Tf is the radial profile of theeffective EP temperature and mf is the mass of theEP. The Ω operators are defined as:

Ωd =ε2πρ2ωbdb

[∂

∂θ

(1√g

)]−1·

1

2B4√g

[(I

ρ

∂B2

∂ζ− J 1

ρ

∂B2

∂θ

)∂

∂ρ

]− 1

2B4√g

[(ρβ∗

∂B2

∂ζ− J ∂B

2

∂ρ

)1

ρ

∂

∂θ

]+

1

2B4√g

[(ρβ∗

1

ρ

∂B2

∂θ− I

ρ

∂B2

∂ρ

)∂

∂ζ

](10)

Ω∗ =1

B2√g1

nf0

dnf0dρ

(I

ρ

∂

∂ζ− J 1

ρ

∂

∂θ

). (11)

Here the Ωd operator models the average drift velocityof the helically trapped particle and Ω∗ models thediamagnetic drift frequency. The parameter ωb = 100kHz indicates the bounce frequency and db = 0.01 mthe bounce length of the helically trapped EP guidingcenter. For more details regarding the derivation of theaverage drift velocity operator please see Ref.[8].

Each equation has a perpendicular dissipationterm with the characteristic coefficients Di = 2 · 10−8

normalized to a2/τA0 (i = U, p, vth, nf, vf is theperturbation index).

We also define the parallel gradient, perpendiculargradient squared (lowest order) and curvature opera-tors as

∇‖f =1

B√g

(∂f

∂ζ+ -ι

∂f

∂θ− ∂feq

∂ρ

1

ρ

∂ψ

∂θ+

1

ρ

∂feq∂θ

∂ψ

∂ρ

)(12)

∇2⊥ =

1

a21√g

1

ρ

∂

∂ρ

(ρgθθ

∂

∂ρ− gρθ

∂

∂θ

)

+1

ρ

∂

∂θ

(−gρθ

∂

∂ρ− gρρ

1

ρ

∂

∂θ

)(13)

√g(∇√g ×∇f

)ζ=∂√g

∂ρ

1

ρ

∂f

∂θ− 1

ρ

∂√g

∂θ

∂f

∂ρ(14)

with the Jacobian of the transformation,

1√g

=B2

ε2(J + -ιI). (15)

Equations 5 and 6 introduce the parallel momen-tum response of the thermal plasma. These are re-quired for coupling to the geodesic acoustic waves, ac-counting for the geodesic compressibility in the fre-quency range of the geodesic acoustic mode (GAM)

Non linear EIC 15

[38, 39]. The coupling between the equations of theEP and thermal plasma is done in the equation of theperturbation of the toroidal component of the vorticity(eq. 3), particularly through the fifth term on the rightside, introducing the EP destabilizing effect caused bythe gradient of the fluctuating EP density.

Equilibrium flux coordinates (ρ, θ, ζ) are used.Here, ρ is a generalized radial coordinate proportionalto the square root of the toroidal flux function, andnormalized to the unity at the edge. The fluxcoordinates used in the code are those described byBoozer [40], and

√g is the Jacobian of the coordinate

transformation. All functions have equilibrium andperturbation components represented as: A = Aeq+A.

The FAR3d code uses finite differences in theradial direction and Fourier expansions in the twoangular variables. The numerical scheme is semi-implicit in the linear terms and explicit in the non-linear terms using a two semi-steps method to ensure(∆t)3 accuracy. The finite Larmor radius and theelectron-ion Landau damping effects are excluded fromthe simulations for simplicity.

Acknowledgments

The authors would like to thank the LHD technicalstaff for their contributions in the operation and main-tenance of LHD. This work was supported by the Co-munidad de Madrid under the project 2019-T1/AMB-13648, Comunidad de Madrid - multiannual agreementwith UC3M (“Excelencia para el Profesorado Univer-sitario” - EPUC3M14 ) - Fifth regional research plan2016-2020 and NIFS07KLPH004

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