Gyrokinetic studies of core turbulence features in ASDEX Upgrade H-mode plasmas
A. Banon Navarro, T. Happel, T. Goerler, F. Jenko, J. Abiteboul, A. Bustos, H. Doerk, D. Told, ASDEX Upgrade Team
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n Gyrokinetic studies of core turbulence features in ASDEX UpgradeH-mode plasmas
A. Ba ˜n ´on Navarro, a) T. Happel, T. G ¨orler, F. Jenko,
1, 2, 3
J. Abiteboul, A. Bustos, H. Doerk, D. Told, and the ASDEX Upgrade Team Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstrase 2, 85748 Garching, Germany Max-Planck/Princeton Center for Plasma Physics Department of Physics and Astronomy, University of California, Los Angeles, California 90095 (Dated: 14 August 2018)
Gyrokinetic validation studies are crucial in developing confidence in the model incorporated in numerical simula-tions and thus improving their predictive capabilities. As one step in this direction, we simulate an ASDEX Upgradedischarge with the GENE code, and analyze various fluctuating quantities and compare them to experimental measure-ments. The approach taken is the following. First, linear simulations are performed in order to determine the turbulenceregime. Second, the heat fluxes in nonlinear simulations are matched to experimental fluxes by varying the logarithmicion temperature gradient within the expected experimental error bars. Finally, the dependence of various quantitieswith respect to the ion temperature gradient is analyzed in detail. It is found that density and temperature fluctuationscan vary significantly with small changes in this parameter, thus making comparisons with experiments very sensitiveto uncertainties in the experimental profiles. However, cross-phases are more robust, indicating that they are betterobservables for comparisons between gyrokinetic simulations and experimental measurements.
I. INTRODUCTION
It has been known for several decades that energy and par-ticle confinement in tokamak plasmas are mainly degradedby turbulence driven by steep temperature and density gradi-ents. For this reason, the characterization and understandingof these turbulent processes is a very important task in orderto improve the performance of present experiments as well asfuture fusion reactors.Due to the strong background magnetic field and the lowcollisionality in tokamak plasmas, gyrokinetic theory has beenestablished as the most appropriate theoretical framework forthe study of turbulent transport in the plasma core. In orderto improve confidence in the numerical results obtained withgyrokinetics and to establish a solid understanding of turbu-lent transport across the whole range of plasma parameters,it is very important to perform direct comparisons betweensimulations and experimental measurements. In this respect,with the recent development and improvements in fluctuationsdiagnostic, it is now possible to measure turbulence featureswith high precision, allowing for quantitative comparisons be-tween experimental data and results of nonlinear gyrokineticsimulations. These validation studies are crucial in developingconfidence in the models and improving the predictive capa-bilities of the numerical simulations.In a recent paper , we have compared density fluctuationlevels measured with a new Doppler reflectometer installedin ASDEX Upgrade and simulation results obtained with thegyrokinetic GENE code. We extend the previous work by ad-ditionally presenting simulation results of density wavenum-ber spectra, electron temperature fluctuation levels as well ascross-phases between different quantities. One of the rea-sons for analyzing electron temperature fluctuation levels and a) Electronic mail: [email protected] cross-phases is that a new Correlation Electron CyclotronEmission (CECE) system is expected to be installed and to bein operation in 2015 in ASDEX Upgrade. Therefore, the gy-rokinetic results presented in this paper can provide guidancefor the on-going development of the diagnostic. From a morefundamental point of view, we will also investigate the varia-tion of these quantities with respect to various physical inputparameters. This information can be used to characterize coreturbulence features in ASDEX Upgrade plasmas.The paper is organized as follows. In Section II, anoverview of the chosen plasma discharge analyzed is given.A description of the gyrokinetic simulation method used isdescribed in Section III. Micro-instability studies from lin-ear gyrokinetic simulations are outlined in Section IV. Themain results of the paper are shown in Section V. Core turbu-lence features such as heat fluxes, density fluctuation ampli-tudes and spectra, temperature fluctuation amplitudes as wellas cross-phases between these quantities will be presented indetail, followed by a discussion in Section VI. Finally, con-clusions and future work will be discussed in Section VII.
II. OVERVIEW OF THE PLASMA DISCHARGE
The ASDEX Upgrade discharge analyzed in thispaper was operated in the high-confinement regime (H-mode).It was planned to study the turbulence characteristics in boththe ion temperature gradient (ITG) and trapped electron mode(TEM) regimes, through a transition from one regime to an-other. This transition can be achieved by modifying locally theelectron temperature gradient, which affects the TEM instabil-ity. This is obtained by changing the electron cyclotron reso-nance heating power (ECRH) by steps during the discharge.An overview of several relevant time traces is given inFig. 1. The ECRH power P ECRH is deposited at ρ pol = 0 . from . − . seconds. Here, ρ pol is the normalized poloidalflux radius. At this heating location, P ECRH is varied sub-
FIG. 1. (color online) Time traces of ASDEX Upgrade discharge . (a) ECRH heating power, (b) electron temperature and(c) logarithmic electron temperature gradient at different radial posi-tions. The data is analyzed in the time intervals shaded in grey. sequently between . , . , . and . (a). The influ-ence of the stepped heating power can be clearly observedin the electron temperature T e (b). Finally, the strongest in-crease of the logarithmic temperature gradient ω T e is observedat ρ pol ≈ . when an ECRH power of . MW is applied(c). For this reason, this case and the one without ECRH willbe analyzed in detail, as examples of the two extreme cases.They correspond to the time windows shaded in grey in Fig. 1.Within these time windows, we have simulated three differentradial locations: ρ pol = { . , . , . } , making a total of sixdifferent scenarios to be studied with gyrokinetic simulations.The physical parameters for each of these six cases aregiven in Table I. There, the reference length is defined as L ref = p Ψ tor , sep /πB ref , where Ψ tor , sep is the toroidal fluxat the separatrix and B ref is the magnetic field on axis. Typ-ically, this reference length is comparable but not identical tothe tokamak minor radius. The logarithmic gradients are de-fined as in Ref. 2: ω X = − X d X d ρ tor with X ∈ { T i , T e , n e } ,and ρ tor the normalized toroidal flux radius. The magneticshear is given as ˆ s = ρ tor q d q d ρ tor , where q is the safety factor,and the electron beta is defined as β e = 2 µ n e T e /B . Here, T i and T e are the ion and electron temperature, respectively,and n e is the electron density. Time [s] 2.65-2.95 2.65-2.95 2.65-2.95 3.65-3.95 3.65-3.95 3.65-3.95 ρ pol ρ tor ˆ s q ω T i ω T e ω n e β e [%] T i [keV] T e [keV] n e [10 m − ] R axis /L ref B ref [T] L ref [m] ρ s [cm] TABLE I. Physical parameters for the six simulated ASDEX Up-grade cases. R axis is the major radius at the magnetic axis and ρ s = c s / Ω i is a reference gyroradius defined with the ion soundspeed c s = p T e /m i and the ion gyrofrequency Ω i . III. OVERVIEW OF THE GYROKINETIC SIMULATIONMETHOD
The turbulence data obtained in this paper are pro-duced with the gyrokinetic code GENE, which solves self-consistently the gyrokinetic-Maxwell system of equations ona fixed grid in five dimensional phase space (plus time): twovelocity coordinates ( v k , µ ) and three field-aligned coordi-nates ( x, y, z ) . Here, z is the coordinate along the magneticfield line, while the radial coordinate x and the binormal coor-dinate y are orthogonal to the equilibrium magnetic field. Thevelocity coordinates are, respectively, the velocity parallel tothe magnetic field and the magnetic moment. GENE has thepossibility to simulate either a flux-tube (local simulations) ora full torus (global simulations). In the former option, it isassumed that the relevant turbulent structures are small withrespect to the radial variation of the background profiles andgradients. This allows to use periodic boundary conditionsand thus the coordinates perpendicular to the magnetic fieldare Fourier transformed ( x, y ) → ( k x , k y ) . For this work,only the local version of the code has been employed.The GENE code is physically quite comprehensive and in-cludes many features (see Ref. 3 for more details). For theASDEX Upgrade scenario studied in this paper, the followingfeatures of the GENE code were used: two particle species(deuterons and electrons), electromagnetic effects by solvingthe parallel component of Amp`ere’s law, external E × B shear,parallel flow shear and a linearized Landau-Boltzmann colli-sion operator with energy and momentum conserving terms .Unless stated otherwise, the magnetic equilibrium geometryis taken from the TRACER-EFIT interface . Additionally,GyroLES techniques have been used to reduce the accumula-tion of energy at the smallest scales (see Refs. [6–8]). Furthersimulation details, such as resolution grid, box sizes, etc., aregiven in the following sections. IV. MICRO-INSTABILITY STUDIES: LINEARGYROKINETIC SIMULATIONS
In order to calculate turbulent transport fluxes, densityand temperature fluctuation amplitudes, etc., nonlinear gy-rokinetic simulations are necessary. Nevertheless, linear gy-rokinetic simulations can provide useful insights. For in-stance, they may allow us to identify the underlying micro-instabilities which drive the turbulence present in the exper-iments. They can also be used for convergence studies and,since they are usually computationally cheap, they can also beused to do scans in different physical parameters.
1. Nominal parameter set
In linear simulations we calculate the growth rate and fre-quency of the most unstable mode present in the system for agiven binormal wave vector k y and k x = 0 . In this paper wechoose to present k y in cm − instead of the more common k y ρ s units. This has been done in order to compare to experi-mental results. For reference, ρ s values are given in Table I.In Fig 2, we display the growth rates ( γ ) and frequencies ( ω ) for the cases at ρ pol = 0 . , because similar conclusionsare obtained for the other cases. In the figure, the negative fre-quencies are represented by dashed lines. The grid resolutionwas { x, z, v k , µ } = { , , , } . Convergence tests wereperformed at higher resolutions and confirm the validity ofthe results. Several observations can be made. First, for lowwavenumbers, ITG is the dominant instability. This is indi-cated by a positive frequency, which with the present normal-ization represents a frequency in the ion diamagnetic direc-tion. Second, for all the cases analyzed, the TEM mode is sta-ble (studied with an eigenvalue solver). In fact, as was alreadyshown in Ref. 1, only with the combination of a much higherelectron temperature gradient and a lower ion temperature gra-dient than the ones measured experimentally, do TEM modesbecome unstable . Third, for higher wavenumbers, ETG is thedominant instability (indicated by a negative frequency). Fi-nally, there is practically no effect of the ECRH on the ITGgrowth rates. The main effect of ECRH is to increase thegrowth rates of ETG modes, possibly leading to a subsequentincrease of the electron heat flux for these cases. However, inboth cases, we expect their contribution to be small with re-spect to the ITG contribution. Moreover, ETG modes are notexpected to influence density and temperature fluctuations atlow wavenumbers, which are dominated by ITG. Since theseare the scales measured by the diagnostics we are consideringhere, in this work we will limit ourselves to wavenumbers upto k y = 10 cm − , thus excluding ETG modes, but allowingfor a significant reduction in computational resources.
2. Sensitivity studies with respect to the main physicalparameters
Physical parameters such as temperature, density, magneticequilibrium profiles, etc., are measured with experimental un- k y [cm −1 ]10 k H z Growth rate [γ] no ECRH1.8 MW ECRH k y [cm −1 ] Frequency [ω]ρ pol =0.7ITG ETG ω >0 ω <0
FIG. 2. (color online) Linear growth rates (left) and frequencies(right) for the cases at ρ pol=0 . versus the binormal wavenumber k y . The dashed lines in the figure indicate negative frequency values.Positive frequencies refer to modes drifting in the ion diamagnetic di-rection and negative in the electron diamagnetic direction. For thesecases, they correspond to ITG and ETG modes, respectively. certainties. Since these values are used as input in the gyroki-netic codes, the uncertainty in these quantities could affect thesimulation results. Therefore, sensitivity studies are carriedout, with the aim of studying the effect of the different uncer-tainties on the simulation results. Ideally, these studies shouldbe done in nonlinear gyrokinetic simulations. However, due tothe expensive computational effort associated, this is in prac-tice unfeasible. For this reason, this sensitivity study is donewithin linear gyrokinetic simulations.We have studied the sensitivity of the linear growth ratewith respect to a variation of ±
20 % in the nominal valuefor different physical parameters, such as: logarithmic iontemperature gradient ( ω T i ), logarithmic electron temperaturegradient ( ω T e ), logarithmic electron density gradient ( ω n e ),electron to ion temperature ratio ( T e /T i ), collisionality ( ν col ),safety factor ( q ) and magnetic shear ( ˆ s ). The main results aresummarized in Table II. For simplicity, we show only casesat ρ pol = 0 . , although similar conclusions were obtained tothe other radial positions. The sensitivity studies for q andfor ˆ s have been done using a Miller-type magnetic equilib-rium . As is shown in Table II, the peak of the growth ratesare practically insensitive with respect to changes of ± in ω n , ω T e , ν col , q and ˆ s . Changes in the peak of the growth rateup to
20 % are found for T e /T i variations. The most criticalparameter is ω T i , whose ± variation modifies the growthrate by up to (see Fig. 3). Based on these results, onecould expect that the uncertainties in ω T i will have the largestinfluence in nonlinear gyrokinetic simulations. In the follow-ing, we will mainly focus on the influence of this parameteron nonlinear simulations. The influence of the E × B shearis in general also expected to have a relevant impact on thetransport in nonlinear simulations. However, the low value ofthe E × B shear for this particular discharge is such that itsinfluence can safely be neglected. No ECRH ∆ γ . MW ECRH ∆ γω Ti × . -38 % ω Ti × . -30% ω Ti × . +25 % ω Ti × . +26% ω T e × . -4 % ω T e × . -5% ω T e × . +5 % ω T e × . +5% ω n e × . +0 % ω n e × . -5% ω n e × . +0 % ω n e × . +6 % T e /T i × . +14 % T e /T i × . +11 % T e /T i × . -18 % T e /T i × . -9% q × . -14 % q × . -9 % q × . +5 % q × . +6% ˆ s × . +0 % ˆ s × . -5% ˆ s × . -4 % ˆ s × . +5 % ν col × . +5 % ν col × . +6% ν col × . -4 % ν col × . -5 %TABLE II. Percentage difference in maximum growth rate with re-spect to the nominal values for cases at ρ pol = 0 . for various pa-rameters ( ω Ti , ω Te , ω n , T i /T e , q , ˆ s , and ν col ). y [cm −1 ]020406080 γ [ k H z ] no ECRH ω T i × 0.8ω T i ω T i × 1.2 y [cm −1 ] ω T i × 0.8ω T i ω T i × 1.2 ρ pol =0.7 FIG. 3. (color online) Linear growth rates versus the binormalwavenumber k y for the cases at ρ pol = 0 . with respect to the varia-tion of the logarithmic ion temperature gradient. V. CORE TURBULENCE FEATURES: NONLINEARGYROKINETIC SIMULATIONS
In order to predict and compare with experimental results,nonlinear simulations are required. For the selected discharge,the grid resolution needed is { × × × × } points in { x, y, z, v k , µ } coordinates. A convergence test ofthe results with that resolution has been performed by com-parison with nonlinear simulations with a double resolution inthe perpendicular directions for a few cases. Moreover, per-pendicular box sizes have been chosen in such a way that sev-eral correlation lengths fit in the box and convergence checkshave also been done on this respect to ensure the validity ofthis choice. Results of the simulations are time-averaged overa range well exceeding the correlation time of the underlyingturbulence. A. Turbulence ion and electron heat fluxes
In this section, we compare the experimental ion and elec-tron heat fluxes obtained through power balance analysis withthe ASTRA code and the results from nonlinear GENE simu-lations. As was done previously, we have grouped in the sameplot the cases taken at the same radial position. The resultsare shown in Fig. 4.
FIG. 4. (color online) Comparison of experimental heat fluxes(dashed-lines) with those obtained from gyrokinetic nonlinear GENEsimulations (markers) using the nominal parameters and the variationwith respect to logarithmic ion temperature gradient for all cases. Ionheat fluxes (left) and electron heat fluxes (right) are depicted. Therows represent the radial positions ρ pol = 0 . , . , . . The experi-mental values are obtained through power balance analysis with theASTRA code and the shaded regions are used to indicate the uncer-tainty of the ASTRA values. In this case, a error is assumed forall cases. In blue are the discharge parameters without ECRH heat-ing and in red with ECRH heating. The nominal parameter in eachcase is colored differently to distinguish it from the rest. Ion (electron) heat fluxes are shown in the left (right)columns in Fig. 4. The rows represent radial positions ρ pol =0 . , . , . . The dashed lines indicate the ASTRA results andtheir shaded regions are used to indicate the uncertainty of theASTRA values, where a
20 % uncertainty is assumed . Basedon the linear sensitivity studies, for each case, several simula-tions were performed varying the logarithmic ion temperaturegradient in steps of ± with respect to the nominal values,up to a maximum of ± variation. GENE simulation re-sults are represented by the markers in the figure. The caseswithout ECRH are colored in blue and with ECRH in red andfor each set, the simulations with the nominal parameters arecolored differently (in green for the cases without ECRH andin yellow for the cases with ECRH). The statistical error bar isan estimation of the standard deviation of the set of means ofconsecutive temporal sub-domains of the saturated state. Sev-eral conclusions can be obtained from Fig. 4. For the nominalparameters, the cases with ECRH produce more ion and elec-tron heat flux than the ones without ECRH for all positions.Moreover, at ρ pol = 0 . , the ion heat fluxes match the experi-mental values without having to vary the gradient with respectto the nominal value. At this position, only the electron tem-perature must be increased by . However, for the otherpositions, the values of the heat fluxes obtained with the nom-inal parameters clearly overestimate the heat fluxes obtainedwith ASTRA by a factor of − . We need to decrease ω T i by in order to match the experimental results for the caseswithout ECRH. Whilst, for the cases with ECRH heating, ithas to be decreased by a maximum of . In Fig. 5, onlythe flux-matched results are compared with the experimentalheat fluxes. We can conclude that agreement of the transportlevels within the errors bars can be achieved, since even smalluncertainties in the temperature profile itself may translate torelatively large ones (up to − ) in the logarithmic gra-dients . pol Q [ M W ] Ions no ECRH1.8 MW ECRH pol
Electrons
FIG. 5. (color online) Comparison of experimental heat fluxes(dashed-lines) with those obtained from gyrokinetic nonlinear GENEsimulations (markers) using the flux-matched simulations.
1. Turbulence electron density amplitudes
Since turbulent fluxes are caused by plasma fluctuationson microscopic scales, it is necessary to validate gyrokineticcodes on a microscopic level. In this regard, a new Dopplerreflectometer has been recently installed in ASDEX Upgrade(see Ref 1 for more details on the diagnostic), which is able tomeasure electron density fluctuation amplitudes ( ˜ n e ). In orderto compare experimental and numerical results, a synthetic di-agnostic must be implemented in GENE to reproduce the mea-surement process of the reflectometer. Two kinds of syntheticdiagnostics can be employed. A first approach consists in sim-ply filtering the data in order to taking into account only thelocation and the wavenumbers that the diagnostic measures.This translates to take into account only fluctuations at the out-board mid-plane ( z = 0 ), averaged over a finite radial lengthand then to select the range of perpendicular wavenumber thatare measured in each case ( k measured y, min ≤ k y ≤ k measured y, max in GENE). A more sophisticated method uses a full-wave codeto simulate also the incidence and reflection of the wave intothe gyrokinetic turbulent data . This work is in progress andonly preliminary results are available with this synthetic diag-nostic (see Ref. 15). For this reason, we have only used thefiltering method for this work.Comparison of the experimental and simulated ˜ n e areshown in Fig. 6. The simulations that match the experi-mental ion heat fluxes are shown with a different marker todistinguish them from the rest. The fluctuation data is ana-lyzed considering only perpendicular wavenumbers between ≤ k y ≤ − ] . The experimental values are scaled by acommon factor since the measurements are in arbitrary units.For this reason, only the shape of radial turbulence level pro-files and the effect of ECRH can be used for comparison. Wedecided to scale the experimental values to try to match thecase without ECRH. As is shown in the left plot of Fig. 6,we obtain a remarkable agreement between experimental andsimulations results in the radial trend. For the case of . MWECRH, there is also a good agreement in the turbulence levelprofile. However, with the scale used, the fluctuation levelsare clearly underestimated with respect to the ones measuredexperimentally. In particular, the flux-matched results presentthe biggest discrepancy with respect to the experimental mea-surements. Finally, from this figure we can also observe howsensitive the density fluctuations are with respect to variationsin the ion logarithmic temperature gradient. For instance, a reduction in the logarithmic gradient can reduce densityfluctuation levels by more than a factor of . FIG. 6. (color online) Electron density fluctuation amplitudes at dif-ferent radial positions. Blue-plus markers represent the data withoutECRH heating, red-cross markers with ECRH and experimental re-sults are in full circles. The flux-matched (GENE f.m.) simulationsare marked differently. Fluctuation data is analyzed at the outboardmid-plane, averaged over a finite radial length and with perpendicu-lar wavenumbers between ≤ k y ≤ − ] .
2. Turbulence electron density spectra
The knowledge of the power-law spectra of a physicalquantity is important for the understanding of the underlyingphysics and useful for providing constraints for simple phys-ical models. Based on Kolmogorov-type arguments , turbu-lence is generally associated with universal power-law spec- -1 k ⟂ [cm −1 ]10 -6 -5 -4 -3 -2 -1 (cid:1) | ˜ n e | (cid:0) [ A . U .] no ECRH ω Ti ω Ti ×0.9ω Ti ×0.8 -1 k ⟂ [cm −1 ] ω Ti ω Ti ×0.9ω Ti ×0.8ω Ti ×0.7 ρ pol =0.8 5.3 5.9 5.2 6.0 FIG. 7. (color online) Electron density fluctuations spectra at ρ pol =0 . including the variation of the logarithmic ion temperature gradi-ent. The solid lines represent the wavenumber range where the fit toa power law was done. Spectral indices for the flattest and steepestspectra are also indicated in the figure for each case. T i Sp e c t r a l I n d e x no ECRH ρ pol =0.6ρ pol =0.7ρ pol =0.8 T i ρ pol =0.6ρ pol =0.7ρ pol =0.8 FIG. 8. (color online) Calculated spectral indices versus the logarith-mic ion temperature gradient for different radial positions and ECRHscenarios. A decrease of the spectral indices with respect to the iontemperature gradient is observed for most of the cases. pol Sp e c t r a l I n d e x no ECRH pol FIG. 9. (color online) Calculated spectral indices versus the radialposition. Blue-plus markers represent the data without ECRH heat-ing and red-cross markers with ECRH. The flux-match simulationsare marked differently. A decrease of the spectral indices with re-spect to the radial position is observed. tra. However, as shown in Refs 8, 17, and 18, this is gener-ally not the case in plasma turbulence, and different power-law spectra indices can be found depending on the type ofmechanism which drives or dissipates energy in the system.Furthermore, the knowledge of wavenumber spectra could beimportant for a clear identification of the turbulent regimesdriven by different microinstabilities and can be used to fur-ther validation of the gyrokinetic model. The results of the electron density fluctuation spectra at ρ pol = 0 . for different logarithmic ion temperature gradientsare shown in Fig. 7. The solid lines represent the wavenumberrange where a fit to a power law is shown: (cid:10) | ˜ n e | (cid:11) = a k − b ⊥ ,where b is the spectral index, and hi represents an average overall the coordinates except k ⊥ . Spectral indices for the flattestand steepest spectra are indicated in the figure. The calculatedspectral indices are also shown in Figs. 8 and 9 with respectto the ion temperature gradient and to the radial position, re-spectively. These figures present a clear qualitative behavior:the spectral index decreases with the increase of turbulencedrive. Additionally, the spectral indices also decrease whengoing from the inner to the outer core position. Moreover,the magnitude of the exponents cover a wide range of values,approximately from to . Although density fluctuation spec-tra were not measured with the Doppler reflectometer for thisdischarge, a similar trend has been also reported in Ref. 19 forvarious ASDEX Upgrade discharges.These results indicate that the turbulence driven by ITGmodes exhibits non-universal power laws, whose spectral in-dices could depend on several physical parameters. Futurework in this respect will be to study also if TEM modes ex-hibit similar properties. If this was the case, then it wouldbecome very difficult to distinguish a type of instability bymeasuring only its characteristic spectral index. B. Turbulence electron temperature amplitudes
At the time the discharge was performed, no temperaturefluctuation measurements were available. However, a Corre-lation Electron Cyclotron Emission (CECE) diagnostic is cur-rently installed on ASDEX Upgrade, and electron temperaturefluctuation profiles will be available in the future campaigns.The CECE diagnostic measures perpendicular electrontemperature fluctuations ( ˜ T ⊥ ,e ) in the long wavelength range(relevant for ITG and TEM modes) and is not sensitive shortwavelengths (ETG modes). This diagnostic presents an inher-ent limitation in the lowest fluctuation level that can be de-tected. This noise level depends on the physical parameters ofthe discharge, but typical values are between . − . .A detailed description of CECE modeling in DIII-D isgiven in Ref. 21. This synthetic diagnostic has already beenimplemented in GENE for DIII-D discharges and a simi-lar synthetic diagnostic will be implemented for ASDEX Up-grade discharges. However, this diagnostic could be not usedin this work since it requires the knowledge of the CECE con-figuration during the discharge. For this reason, we have con-sidered a simpler synthetic diagnostic, which consists in fil-tering the gyrokinetic data to the positions and wavenumbersthat are expected to be measured in ASDEX Upgrade. Con-sequently, the gyrokinetic data analysis results are restrictedto the outboard mid-plane position ( z = 0 ), averaged over thefinite radial length and summing all perpendicular wavenum-bers (since short wavelengths are not simulated). In order tobetter model the actual diagnostic, one should also apply afilter in the frequency space. However, this filter will also de-pend on the specific range of frequencies measured. Since wedo not have access to this information, we have considered allthe frequencies in the analysis. Therefore, the following re-sults should be only used as an approximated indication of thefluctuation amplitudes that could be detected with CECE forthis discharge. Nevertheless, we do not expect radial trendsto change with respect to a more sophisticated synthetic di-agnostic approach and only the amplitudes are likely to berescaled .The main results are shown in Fig. 10. The perpendicularelectron temperature fluctuation amplitudes go from a mini-mum of . at the inner position to a maximum of . atthe outer core position. Therefore, assuming a noise-level inthe order of . − . for the CECE diagnostic, this couldimply that only the fluctuations at the outer core positions(starting from ρ pol ≥ . ) could be detected. As for the caseof the density amplitudes, we also observe a large variation ofthe fluctuation amplitudes with respect to the changes in thelogarithmic ion temperature gradient. pol ˜ T ⟂ , e / T e [ % ] no ECRH pol FIG. 10. (color online) Percentages of electron perpendicular tem-perature fluctuation amplitudes at different radial positions. Blue-plus markers represent the data without ECRH heating and red-crossmarkers with ECRH. The flux-match simulations are marked differ-ently. The area shaded in grey indicates a typical noise level of theCECE diagnostic.
C. Turbulence cross-phases
Doppler reflectometers can be coupled to the CECE diag-nostics to calculate cross-phases between electron density andtemperature fluctuations . This measurement is important forgyrokinetic validation studies since it represents a relationshipbetween different fluctuating quantities (density and tempera-ture in this case). In addition, this cross-phase could be alsorelated to the cross-phase that determines the turbulent heatfluxes (electrostatic potential and temperature fluctuations).For this reason, in addition to the cross-phase that can bemeasured experimentally, we will also show the cross-phasebetween electrostatic potential and electron density fluctua-tions, so we can relate the cross-phases measured experimen-tally to the turbulent heat fluxes. The cross-phases are heredefined as δ ˜ n, ˜ T ⊥ ,e = tan − ( ℑ (˜ n/ ˜ T ⊥ ,e ) / ℜ (˜ n/ ˜ T ⊥ ,e )) , and, δ ˜ n, ˜ φ = tan − ( ℑ (˜ n/ ˜ φ ) / ℜ (˜ n/ ˜ φ )) .Fig. 11 shows the variation of the cross-phase versus theion temperature gradient integrated over binormal wavenum-bers in the range of − ≤ k y ≤
10 cm − . For most of δ ˜ n , ˜ φ [ d e g .] no ECRH ρ pol =0.6ρ pol =0.7ρ pol =0.8 ρ pol =0.6ρ pol =0.7ρ pol =0.8 T i δ ˜ n , ˜ T ⟂ , e [ d e g .] T i a) b)c) d) FIG. 11. (color online) Calculated cross-phases for the differentcases versus the logarithmic ion temperature gradient. Cross-phasesbetween density and electrostatic potential fluctuations (a) and (b).Cross-phases between electron density and temperature fluctuations(c) and (d). The cross-phases seem to be rather insensitive with re-spect to changes in the logarithmic ion temperature gradients. the cases, the phases remain rather invariant with respect tothe variation of this parameter. Therefore, this result seems toindicate that the cross-phase is a better observable to identifythe type of instability which drives the turbulence in experi-ments, since TEM instability is expected to exhibit differentcross-phases .In Fig. 12, the cross-phase are displayed versus the radialpositions. Regarding the cross-phase between density andelectrostatic potential fluctuations (a) and (b), we see that theyare practically in phase, i.e. close to . In addition, an in-crease of the phase with the radial position is also observed,going from practically degrees at ρ pol = 0 . to approxi-mate by degrees at the outer core position. On the contrary,for the cross-phase between electron density and temperaturefluctuations (c) and (d), we see a decrease with the radial po-sition, going from around degrees in the inner positionto a value of degrees, which result in an increase of theelectron heat flux. Similar values of this cross-phase havealso been measured in DIII-D and calculated with GYRO inRef. 25. Furthermore, these values have also been found byGENE for these discharges, see Ref. 13. These observationscould be explained in the following way. In the inner position,the population of the trapped particles that contributes to theITG instability is small, so the electrons behave almost adi-abatically. Because of this, density and potential fluctuationsare in phase and density and perpendicular temperature fluctu-ations are almost out of phase (i.e., close to degrees), thusproducing negligible electron heat flux. With increasing theradial position, the population of trapped particles increaseand a deviation of the electron adiabaticity is observed. Forthis reasons, both cross-phases approach to degrees, withthe subsequent increase of electron heat flux.Finally, in Fig. 13, the colored contours display the cross-phases obtained from the nonlinear simulations, while the redsquares are used to display the cross-phases of the linear sim-ulations for the case without ECRH at ρ pol = 0 . . The agree-ment between linear and nonlinear cross-phases is remarkablygood. This is also observed for the other cases . This re-sult implies that linear simulations could be enough to com-pare with experimental results. δ ˜ n , ˜ φ [ d e g .] no ECRH 1.8 MW ECRH pol δ ˜ n , ˜ T ⟂ , e [ d e g .] pol a) b)c) d) FIG. 12. (color online) Calculated cross-phases versus the radialposition. Cross-phases between density and electrostatic potentialfluctuations (a) and (b). Cross-phases between electron density andtemperature fluctuations (c) and (d). The flux-match simulations aremarked differently.FIG. 13. (color online) Case without ECRH at ρ pol = 0 . : Com-parison of linear (markers) and nonlinear (contour) cross-phases asfunction of the binormal wavenumber where amplitudes increasefrom white to black. Left. Cross-phases between electron densityand electrostatic potential fluctuations. Right. Cross-phases betweendensity and temperature fluctuations. VI. DISCUSSION
Focusing on the case at ρ pol = 0 . with . MW ECRH,the key results obtained in this paper can be illustrated inFig. 14. Here, we show the impact of the variation of the log-arithmic ion temperature gradient around the nominal valueon various observables. Ion and electron heat fluxes (a), elec-tron and temperature fluctuation amplitudes (b) and electrondensity spectral indices (c) are all very sensitive with respect A / A N o m . Q i Q e A / A N o m . ˜n e /n e ˜T ⟂,e /T e −40−30−20−10 0 10 20ω T i [%]0.00.20.40.60.81.01.21.4 A / A N o m . Spec ral Index −40−30−20−10 0 10 20ω T i [%]−150−100−50050100150 d e g . δ ˜n,˜φ δ ˜n,˜T ⟂,e ρ pol =0.7, 1.8 MW ECRHa) b)c) d) FIG. 14. (color online) Impact of the variation of the logarithmicion temperature gradient around the nominal value for the case at ρ pol=0 . with . MW ECRH on a) ion and electron heat fluxes b)electron density and temperature fluctuation amplitudes, c) electrondensity spectral index and d) cross phases. Figures a), b) and c) arenormalized with respect to the value at the nominal ion temperaturegradient. to small changes in the ion logarithmic temperature gradient.For instance, by decreasing ω T i by , the amplitudes canbe reduced by a factor of for the density and temperaturefluctuations and by a factor of for the heat fluxes. This resultimplies that the comparison of gyrokinetic simulation and ex-perimental measurements for these observables are very sensi-tive to uncertainties in the experimental input profiles. On thecontrary, cross-phases between density and temperature fluc-tuations and between density and electrostatic potential (d) arerather insensitive with respect to ω T i . This, together with thefact that linear and nonlinear cross-phases agreed also remark-ably well, indicates that cross-phases could be a good observ-able to compare (fast) linear gyrokinetic simulations with ex-perimental measurements. VII. CONCLUSIONS AND FUTURE WORK
We have analyzed, by means of gyrokinetic simulationswith GENE, core turbulence features of an H-mode dischargein ASDEX Upgrade. The main results of this paper canbe summarized as follows. Flux-matched simulations wereachieved by varying the nominal ion temperature gradient bya factor of − , which is within the uncertainty rangeof the experimental profiles. In addition, density fluctuationlevels show an agreement in the shape of the radial turbu-lence level profiles, although the effect of the ECRH on thefluctuation levels was not reproduced. Non-universal power-law spectra were found for turbulence driven by ITG modes.In particular, ITG instability exhibits spectral indices for thedensity fluctuation spectra which cover a broad range of val-ues. These values depend on the radial position and also onthe specific ion temperature gradient. Gyrokinetic simulationspredict a decrease of the exponents with respect to both theincrease of the ion temperature gradient and the increase ofthe radial position. These results could help validate futureanalytical theories and are useful for comparisons with othergyrokinetic codes and future measurements. Regarding theelectron temperature fluctuations, we observe for the inner po-sition ( ρ pol = 0 . ) fluctuation amplitudes which are close tothe sensitivity of the CECE diagnostic. Therefore, it is possi-ble that only measurements at positions larger than ρ pol = 0 . could be detected with this diagnostic in such discharges. Wehope that these results can provide guidance for the develop-ment of the CECE diagnostic that is currently being installedin ASDEX Upgrade. Finally, by analyzing cross-phases be-tween density and temperature fluctuations and between den-sity and electrostatic potential, we observed that linear andnonlinear cross-phases agree remarkably well, and that theyare rather insensitive with respect to the variation of the iontemperature gradient, indicating that cross-phases could be agood observable with experimental measurements for com-parisons.For future work, GENE and ASDEX Upgrade comparisonswill continue with the study of similar H-modes plasmas butwith higher ECRH power (up to . MW). These dischargesare expected to have peaked electron temperature profiles andallow TEM modes to be dominant. This will allow us tostudy fundamental differences between ITG and TEM modesfrom a microscopic level. Furthermore, additionally dedicateddischarges have already been conducted in which detailedwavenumber spectra have been measured with the Dopplerreflectometer. These comparisons, along with the inclusionof future CECE measurements, will help in further validatinggyrokinetic codes and the development of synthetic diagnos-tics.
ACKNOWLEDGEMENTS
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