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Current Applied Physics 42 (2022) 60–70

Available online 5 August 2022

Progress in gyrokinetic validation studies using NBI heated L-mode discharge in KSTAR

D. Kim

^a

, J. Kang

^b

, M.W. Lee

^a

, J. Candy

^c

, E.S. Yoon

^d

, S. Yi

^b

, J.-M. Kwon

^b

, Y.-c. Ghim

^a

, W. Choe

^a

, C. Sung

^a^,^*

aKorea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, South Korea

bKorea Institute of Fusion Energy, 169-148 Gwahangno, Yuseong-gu, Daejeon, 34113, South Korea

cGeneral Atomics, San Diego, CA, 92121, USA

dDepartment of Nuclear Engineering, UNIST, 50, UNIST-gil, Eonyang-eup, Ulju-gun, Ulsan, 44919, South Korea

A R T I C L E I N F O Keywords:

Gyrokinetic Validation study CGYRO KSTAR Energy flux Zeff

A B S T R A C T

Progress in the first gyrokinetic validation study using KSTAR NBI heated L-mode discharge is reported in this paper. The energy flux levels simulated from gyrokinetic code, CGYRO[J. Candy et al., J. Comput. Phys. 324, 73–93 (2016)] were compared with experimental levels in this study for validation purposes. The linear stability analysis indicates that trapped electron modes (TEM) are the most unstable ion-scale modes at r/a =0.5. The simulated energy flux was under-predicted compared to the experimental energy flux level within their uncertainties. We also observed that simulated energy flux levels were sensitive to the input parameters related to impurity density profile such as effective charge, Zeff, and inverse gradient scale length of impurity and main ion, a/Lnc and a/Lni, respectively. For the conclusive future validation studies, we identified the Zeff profile, which can give constraints on not only impurity but also main ion profiles, as necessary input.

1. Introduction

High pressure (high density and temperature) plasmas should be confined long enough to have a sufficient number of fusion reactions to provide sufficient fusion energy for electricity generation. This condition is often referred to as triple product, which is the product of density, energy confinement time, and temperature (nτT). This condition can be expressed differently such that the transport level in high-pressure fusion plasmas should be minimized. Since transport phenomena are directly related to plasma performance, having a validated transport model based on the understanding of fusion plasma transport is essential not only to improve fusion plasma performance but also to obtain the predictive capability required to design future fusion devices.

The experimental transport level observed in fusion devices is much higher than that of neoclassical collisional transport, which indicates collisional transport is not the main transport mechanism in fusion plasmas. Turbulent transport is believed to be mainly responsible for this discrepancy. Nonlinear gyrokinetic theory [1,2] is the most advanced framework used to describe the turbulent transport processes in fusion plasmas. To utilize the gyrokinetic model to predict the performance of

future fusion plasmas, we need to validate the current gyrokinetic model through a quantitative comparison between the simulated physical quantities from the gyrokinetic model and the measurements. Gyroki- netic validation studies have been conducted extensively in many tokamaks including DIII-D [3–6], Alcator C-Mod [7–11], AUG [12,13], and JT-60U [14]. However, gyrokinetic validation using KSTAR plasmas has not been reported yet. The recent progress of fluctuation diagnostics and analysis tools in KSTAR motivated us carry out a gyrokinetic analysis of KSTAR plasmas.

In this paper, we report progress on gyrokinetic validation studies in KSTAR. One KSTAR L-mode discharge was chosen for this, and the simulated energy flux levels are compared with the experimental levels as an initial step. Since this is the first-round of validation study, we made efforts to identify the additional constraints required for the next round of validation study whether to conclude agreement or disagree- ment between the model and experiment. We also study characteristics of turbulence identified by gyrokinetic simulation in the KSTAR L-mode discharge. The remainder of this paper is as follows. Section 2 provides the experimental analysis required for this work. Linear and nonlinear gyrokinetic analysis is given in Section 3. We discuss what we need to

* Corresponding author.

E-mail address: [email protected] (C. Sung).

Contents lists available at ScienceDirect

Current Applied Physics

journal homepage: www.elsevier.com/locate/cap

https://doi.org/10.1016/j.cap.2022.07.015

Received 9 June 2022; Received in revised form 15 July 2022; Accepted 21 July 2022

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prepare in the next round of gyrokinetic analysis in Section 4. A summary and conclusions are given in Section 5.

2. Experimental transport analysis 2.1. Experimental setup

We selected one NBI heated L-mode discharge (shot 21631) in KSTAR for the validation study. This discharge was operated with magnetic field, BT =2.5T on the axis, plasma current, IP =0.6 MA on the flat top, total NBI power, PNBI =2.9 MW, line integrated density, ne ~2 .0×10¹⁹m⁻³, safety factor at ψ=0.95 where ψ is a normalized poloidal flux, q₉₅~5, and normalized beta,β_N~1.33. Fig. 1 (a)-(c) show the time series data of the plasma parameters in this discharge. We chose this discharge since diagnostics required for basic transport analysis such as Thomson scattering [15], charge exchange spectroscopy (CES) [16], and motional stark effect (MSE) diagnostic [17] were available in this discharge and MHD activities were also very weak (~0.3G) as shown in Fig. 1 (d). The time slice at 2050 ms is selected for detailed transport and gyrokinetic analysis. At 2050 ms, peak amplitudes of MHD are almost negligible. Thus, it is possible to do validation work without considering the MHD effects at 2050 ms.

2.2. Experimental analysis

Kinetic EFIT [18] analysis using OMFIT [19] was performed for experimental profile and transport analysis. This kinetic equilibrium analysis is an iterative process among profile analysis, power balance analysis, and equilibrium reconstruction. In the profile analysis, experimental profiles are obtained from the measurements. Electron density (ne) and temperature (Te) are measured by Thomson scattering [14], which consists of 14 core channels and 17 edge channels with 20 ms time resolution. Charge exchange spectroscopy (CES) [15] is used to give ion temperature (Ti) and toroidal velocity (Vt) with toroidally 32

channels and poloidally 16 channels. TRANSP [20,21] is used as power balance analysis with NUBEAM [22]. NUBEAM can provide fast ion information using the Monte Carlo technique. Equilibrium reconstruction is performed using EFIT [23] with constraints of the internal magnetic field measured by the Motional Stark Effect (MSE) diagnostic [16] and kinetic pressure profile including the measured thermal pressure and fast ion pressure obtained from NUBEAM. Since profile analysis results will affect power balance analysis and equilibrium reconstruction and the results of equilibrium reconstruction also affect the other two analysis results, we iteratively repeat these three analyses until the results are converged. During this iterative analysis, we kept the consistency among analyses and also among measurements. Fig. 2 (a) shows the line averaged density measured by the Millimeter-wave interferometer (blue line) and calculated using density profile from Thomson scattering data at 2050 ms (red dot). Electron density profile was scaled to be consistent between line averaged density level derived from Thomson scattering data and measured from single channel of interferometer. Fig. 2 (b) demonstrates the consistency of stored energy between power balance analysis using TRANSP and equilibrium reconstruction by EFIT, shown as the green and blue lines, respectively.

They are well matched within 1%.

2.3. Uncertainty quantification

A quantitative comparison is possible with uncertainties of simulated and experimental quantities. Therefore, uncertainty quantification is one of the necessary processes for the validation study. The uncertainties of ne,Te,Ti,their inverse gradient scale lengths normalized by minor radius, a, a/Lx =a∇X/X, and toroidal velocity (Vt)profile, are quantified to calculate the uncertainties of simulated and experimental energy flux.

The random data are extracted from the normal distributions whose mean and standard deviation are taken from the measured value and the uncertainty respectively, in each diagnostic channel. By fitting these random data sets with weights adjusted by the measurement uncertainties, a profile sample can be generated. Repeating this process several times, profile samples of more than 3000 are obtained. Here, the number of repetitions is chosen by considering the time, cost and accuracy. By applying the χ²test, inappropriate profile samples that fitting was not successful are eliminated. Then, the uncertainty of profiles is obtained by calculating the standard deviation of the generated profile samples. Through the error propagation process with a numerical method, the uncertainty of the inverse gradient length scale is calculated. Here it is possible to consider the covariance term using the profile samples in the error propagation formula. Fig. 3 (a)-(g) show the results of uncertainty quantification of ne, Te, Ti, their inverse gradient scale lengths, and Vt. At r/a =0.5, where r/a is the normalized radius by minor radius a, the relative uncertainty of ne is ~11.9%, Te ~13.4% ,

Ti–0.6%, Vt ~0.5%. The uncertainty of a/Lne is calculated as 24.3%, a/ LTe ~20.5%, and a/LTi ~3.8%. Here, the relative uncertainty levels of ion temperature and toroidal velocity are small compared to other quantities. This is because the estimated measurement error of CES diagnostic obtained from Gaussian fitting in the data analysis is very small (~1%). Other error sources which can affect the CES measurements should be identified in the future.

In addition, the uncertainties of radial electric field, Er and ExB shearing rate, ω_EXBare calculated using a similar way where Er=_Z^∇p^C

CenC+ VtorBpol− VpolBtor and ω_EXB = − ^r_q_dr^d

(c REr

Bp

). The Bpol, Btor, q, and Vpol

denote poloidal, toroidal magnetic field, safety factor, and poloidal velocity. Here, the uncertainties of carbon temperature and toroidal velocity are used from CES data. Carbon density measurements were not available in this discharge. In this analysis, carbon density is obtained from TRANSP with the assumed flat effective charge, Zeff

(

=∑

jZ²_jnj/ne, wherejisionspecies )

profile for the calcula- Fig. 1. The time traces of (a)plasma current, (b) total NBI power, (c) line

averaged density, and (d) peak amplitude of MHD activities of KSTAR discharge 21631. Dash line denotes the target time for the present validation work.

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tion of the first term of Er. The toroidal and poloidal velocities of Car- bons were used for the second and third term of Er. The poloidal velocity is calculated by NEO, a kinetic neoclassical code [24]. For uncertainty quantification of the neoclassical poloidal velocity, the randomly selected inputs from profile samples are used as NEO inputs, then Vpol

profile samples are obtained. The uncertainty of Vpol can be estimated by calculating the standard deviation of the Vpol profile samples. The

uncertainty of radial electric field and shear rate are depicted in Fig. 3 (h) and (i). The VtorBpol term is dominant in Er (Vpol/Vtor ~O(1/100) at core region in this discharge), and since the uncertainty of Vtor is small, the uncertainty of Er is also calculated to be small. The uncertainty of equilibrium data Bpol, Btor, and q are neglected. The quantified uncertainties of parameters are used for gyrokinetic validation study here.

The uncertainty of the experimental energy flux is quantified using Fig. 2. (a) Time trace of line averaged density from interferometer (blue curve) and calculated from Thomson scattering data at t =2050msec (red dot) (b) Time histories of stored energies calculated from equilibrium reconstruction by EFIT (blue) and power balance analysis by TRANSP (green). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3. Radial profiles of (a) ne (b) Te (c) Ti (d) a/Lne (e) a/LTe (f) a/LTi (g) Vt (h) Er (i) ωEXB. Shaded bands denote uncertainties of each profile.

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generated profile samples. The profile samples of ne,Te,Ti,and Vt are utilized in power balance code, TRANSP as inputs. The standard deviation of TRANSP results is estimated as the uncertainty of the experimental energy flux. The experimental ion and electron energy fluxes with their uncertainties are shown in Fig. 4. At r/a =0.5, the uncertainty of the experimental ion energy flux, Q_i, is ~8.5% while the experimental electron energy flux, Qe,is ~10.7%. It is noteworthy that flat Zeff profile, assumed frequently when Zeff profile measurements were not available as can be found in previous studies [3,7,11,25], with the value of 2, i.e., Zeff=2 was assumed in this analysis. Previous study show that the Zeff

value of KSTAR NBI heated L-mode plasma is between 1.2 and 2.5 [26].

The energy fluxes and uncertainties obtained TRANSP results with different Zeff values (Zeff = 1.2 and 2.5) are shown in Fig. 5. The experimental energy flux was set as the possible range of energy flux with the Zeff =[1.2,2.5].

3. Gyrokinetic simulation 3.1. Simulation setup

Gyrokinetic analysis was performed at r/a =0.5 using CGYRO [27], a multiscale Eulerian gyrokinetic solver. CGYRO uses the rigorous ρ^*_s-ordering of sugama [2], which is required for a consistent formulation of transport theory. Here, ρ^*_s=ρ_s/a,where ion Larmor radius, ρ_s=cs/ωi

with ion sound speed, cs, and ion cyclotron frequency, ωi. In this analysis, the ordering is justified because 1/ ρ^*_s~290. Miller geometry parameterization [28] is used to consider geometric effects of non-circular shaped magnetic flux surfaces, and the effects of collision are calculated through Sugama operator [29]. Electromagnetic effects are considered by including fluctuations of vector potential parallel to the magnetic field, δA‖, as well as electrostatic potential fluctuations, δϕ. Parallel magnetic field fluctuations, δB‖are not included in the analysis since we observed that its effect was negligible from the linear stability analysis. We mainly considered the ion scale turbulence (kθρ_s≤1 and krρ_s≤4.9 where kθ and kr are poloidal and radial wave number, respectively). In addition, fast ions generated by NBI are treated as another species in this analysis.

The experimental profiles shown in Fig. 3 were used as input parameters of the gyrokinetic analysis. The main parameters for the gyrokinetic analysis at r/a =0.5 are as follows: ni/ne = 0.6836, nc/ ne =0.03333, nfi/ne =0.1164, a/Lni =1.534, a/Lnc =1.833, a/Lnfi = 3.590, a/Lne =1.833, a/LTi =1.436, a/LTc =1.564, a/LTfi = 0.39683, a/LTe =2.3, safety factor, q =1.9, magnetic shear, s =0.44, and Ti/Te

=1.1. Here, carbon is used as the impurity species since carbon is the dominant impurity species in most KSTAR discharges with carbon wall and divertor. Since Zeff profile measurements were not available in this discharge, a flat profile with the value of 2 was assumed in most simulation runs.

3.2. Linear stability analysis

The linear stability analysis was performed to understand the characteristics of the ion scale turbulence (kθρ_s≤1) at r/a =0.5. In this analysis, the positive and negative signs of the real frequency indicate the electron and ion diamagnetic directions, respectively. Fig. 6 shows the changes in real frequency and linear growth rate of the most unstable mode when fast ions are included. Here, experimental input parameters are utilized as input. We can see that the real frequency of the most unstable mode is in the electron diamagnetic direction in both cases without fast ion and with fast ion. However, when fast ions are included, the linear growth rate decreased up to 30%. This decrease can be due to the effects of fast ion including dilution of the main thermal ion and the increased β. In the validation study perspective, this result indicates that fast ion should be included in this study since energy flux levels predicted from nonlinear simulations will be also affected significantly by fast ions.

Inverse gradient scale lengths of density and temperature (a/Ln, a/LTe, and a/LTi) and collision frequency (υ) were varied to understand the characteristics of the most unstable mode. In the scan of a/Ln, inverse density gradient scale lengths of all species are varied together to keep quasi-neutrality conditions shown in Appendix. The linear gyrokinetic simulation results with ±15% changes of a/LX are shown in Figs. 7 and 8. The upper and bottom rows denote the growth rate and the real frequency, individually. Each column plots show the scan results of a/Ln,a/LTe, υ, and a/LTi. The blue dots exhibit the results of reference case with the experimental input parameters. The green and red dots correspond to the case with 15% increased and decreased values of the reference input parameters, respectively. Fig. 7 (a) and (b) show that the real frequency of the dominant mode stays in the electron diamagnetic direction while we change a/Ln and a/LTe by ± 15%. However, increased a/Ln and a/LTe destabilize the most unstable modes and make the real frequency moves to the electron diamagnetic direction as can be seen in Fig. 7 (d) and (e). These results are consistent with the characteristics of trapped electron mode (TEM) [30], suggesting that the most unstable mode for kθρ_s≤1 at r/a =0.5 are TEMs. We can also see that reduced collision frequency moves the real frequency of the most unstable mode to a deeper electron diamagnetic direction and increases its growth rate from Fig. 7 (c) and (f). This suggests that this TEM is likely to be collisionless TEM rather than dissipative TEM. In addition, Fig. 8 (a) and (b) show that the most unstable mode is sensitive to a/LTi as well, implying that this mode is not pure TEM and other instabilities are coexisted or coupled.

The high-k simulation results of electron scale linear stability analysis at kθρ_s≤52 and r/a =0.5 are depicted in Fig. 9 (a) and (b). These figures show the sensitivity of electron scale turbulence to a/LTe. The most unstable mode shown in Fig. 9 is destabilized and its real frequency moves to electron scale as a/LTe increases, consistent with the characteristics of electron temperature gradient mode (ETG) at r/a =0.5. Next, we need to estimate the importance of multi-scale effects including

Fig. 4.Experimental energy fluxes of (a) ion, (b) electron energy flux are plotted as function of r/a. Shaded bands denote the uncertainty of the experimental energy flux. At r/a =0.5, relative errors of ion and electron energy flux are 8.5% and 10.7%.

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transport directly induced by electron scale turbulence and multi-scale interactions between ion scale and electron scale turbulence. The previous studies [10,31] found that the multi-scale simulations were important when γ_high_/_low= (γ_high₋_k/khigh)/(γ_low₋_k/klow)was larger than 1 or γ_high−_k/γ_low−_kwas larger than 40 where γ_high−_kand γ_low−_kare the

peak linear growth rate of electron and ion scale, respectively. Here, γ_high−_k(γ_low−_k) denotes the peak higk-k (low-k) linear growth rate and khigh (klow) is the corresponding wave number. In the case studied here, γ_high/lowis 0.33 and γ_high−_k/γ_low−_kis ~15, suggesting multi-scale effects are not significant in this case. Considering computing resource and the Fig. 5. Experimental energy fluxes of (a) ion and (b) electron energy flux are plotted as function of r/a with different Zeff. Shaded bands denote the uncertainty of the experimental energy flux.

Fig. 6. kθρ_sspectrum of (a) real frequency and (b) linear growth rate for the cases with and without fast ion at r/a =0.5. The (+) and (−) sign of real frequency indicate the electron and ion diamagnetic direction, respectively.

Fig. 7. (top) kθρ_sspectrum of real frequency; (bottom) linear growth rate with scan of the a/Ln (left), a/LTe (middle), collision frequency, ν (right) at r/a =0.5. The (+) and (−) sign of real frequency indicate the electron and ion diamagnetic direction, respectively.

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above analysis results, we first focused on the single scale (ion scale) simulation and, non-linear multi-scale simulation was not performed.

However, it is noteworthy that the significance of multi-scale effects is not confirmed in this study since the criteria used here were based on the linear growth rates. Non-linear multi-scale simulations should be performed to confirm the importance of multi-scale effects in this case, which is remained as future work.

3.3. Nonlinear gyrokinetic simulation

To calculate the simulated energy flux, nonlinear gyrokinetic simulations were performed for ion scale turbulence (kθρ_s≤1). The numerical parameters in the nonlinear simulation including resolution parameters (nradial, nξ, nenergy, nθ,nφ) and box size (Lx,Ly) were determined by convergence tests. Here, nradial, nξ, nenergy, nθ,nφ, Lx,and Ly are the number of radial, pitch angle, energy, poloidal, and toroidal grid points, and radial and poloidal box size, respectively. Experimental input parameters used in the linear simulation were used in this convergence test but fast ions were not included because of the limited computational resources. The numerical parameters used in the reference case in this convergence test are the following: [nradial, nξ, nenergy, nθ,nφ] =[8,16,16,28,162], and [Lx,Ly] =[101.63 ρ_s,93.78 ρ_s]. The convergence test results are shown in Table 1. The changes in energy flux levels were less than 16% by a 66% increase of nradial and less than

8% by 25% increase of other resolution parameters and box size. The simulated energy fluxes are obtained by time-averaging after saturation, and its uncertainty was estimated from the standard deviation of the time-averaging of the simulated energy flux levels, as shown in Fig. 10.

Fig. 8. kθρ_sspectrum of (a) real frequency and (b) linear growth rate for a/LTi at r/a =0.5. The (+) and (−) sign of real frequency indicate the electron and ion diamagnetic direction, respectively.

Fig. 9. kθρ_sspectrum for kθρ_s≤52 of (a) real frequency and (b) linear growth rate for a/LTe at r/a =0.5.

Table 1

Convergence test results with different resolution parameters and box sizes. QD,QC, and Qe indicate the gyroBohm normalized main ion, carbon, and electron energy flux. The numerical parameters of the reference case are [ nradial, nξ, nenergy, nθ,nφ] =[8,16,16,28,162] and box size is [Lx,Ly] =[101.63 ρ_s,93.78 ρ_s].

Reference nenergy =10 nθ =35 nξ =24 nradial =270 [Lx,Ly] =[135.51 ρ_s_{, 93.78}ρ_s_] _[L_x,Ly] =[136.19 ρ_s_{, 125.66}ρ_s_]

QD [QgB] 0.372 0.356 0.346 0.379 0.432 0.33 0.351

QC [QgB] 0.068 0.069 0.06 0.067 0.072 0.058 0.062

Qe [QgB] 1.093 1.083 1.004 1.085 1.212 0.955 0.99

Fig. 10.GyroBohm normalized energy flux of (black) main ion (magenta) carbon impurity (blue) electron without fast ion against normalized time to a/cs. Horizontal dash line is the interval for time-averaging to calculate the simulated energy fluxes. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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This uncertainty level is usually more than 16%, which is the largest variation observed in the convergence test. We, therefore, conclude that the numerical parameters used in the reference case are sufficient to simulate the experiment described in Section 2.

Fig. 11 shows the simulated ion and electron energy flux levels using the experimental input parameters with and without fast ions. As fast ions were included, both ion and electron energy fluxes decreased by 78% and 70%, respectively. This result is consistent with the linear stability analysis results shown in Fig. 6 and previous study results [32–34]. It also indicates that including fast ions are critical in this validation study. We should include fast ions since quantitative energy flux comparison results can be significantly different as the fast ion is included. We can also see that simulated energy flux levels are lower than the experimental energy flux levels at r/a =0.5 and the difference is larger than their uncertainties.

To see whether this result is valid within the uncertainties of input parameters, we observed changes in simulated energy flux levels by varying input parameters within their uncertainties. The linear stability analysis provided that linear growth rate of the most unstable mode is increased by increasing a/Ln and a/LTe and decreasing ν. In addition, ω_E×Bcould stabilize the turbulence. Therefore, to increase both ion and electron energy flux closer to the experimental level, a nonlinear simulation was run with an increase of a/Ln and a/LTe, and decrease of ν and ωE×B. As shown in Fig. 12, the simulated energy flux levels are still lower than the experimental level by increasing a/Ln and a/LTe ,or decreasing ν and ω_E×B within their uncertainties. Simulated energy flux of changed a/LTi is also under-predicted compared to the experimental level although it is not shown because its uncertainty is small (~4%).

Since the impurity information is missing in this analysis and a flat Zeff =2 profile was assumed in this analysis, we need to check whether the results shown in Figs. 11 and 12 can be changed or not due to input parameters related to impurity. We first vary the Zeff value itself between 1.2 and 2.5 in both linear and nonlinear runs as shown in Fig. 13. As Zeff

value varies from 1.2 to 2.5, real frequency of the most dominant mode moves electron diamagnetic direction and the linear growth rate in kθρ_s≥0.5 increases as shown in Fig. 13 (a) and (b). Nonlinear simulation run results show that ion energy fluxes are still underestimated compared to experimental energy flux while electron energy flux matches when Zeff =2.5. However, there is no case where both ion and electron energy fluxes match the experimental level. These results indicate that the simulated energy flux is sensitive on Zeff value so this value should be well known with its uncertainty for validation study. It is noteworthy that we have used Zeff profile and changing impurity density gradient scale length (a/Lnc) will affect the gradient scale length of the main ion (a/Lni). Since we do not know a/Lnc in this study, both a/ Lni and a/Lnc are not constrained. Local impurity gradient can be varied in large range including even sign change, i.e., peaked to hollow profile.

We therefore need to check the sensitivity of the simulated energy flux levels on these unconstrained parameters, i.e., a/Lnc and a/ Lni. As

shown in Appendix, inverse gradient scale lengths of density should satisfy ∑

jZjnja/Lnj=0 where j is ion species and electron, to maintain quasi-neutrality. In this sensitivity analysis, we varied a/Lni and a/Lnc

together to satisfy the above relation. The results are shown in Fig. 14.

As we increase a/Lni and decrease a/Lnc together while satisfying

∑

jZjnja/Lnj = 0, both simulated ion and electron energy fluxes increase remarkably. We found even the case that simulated ion and electron energy fluxes matched with experimental levels at the case with 1.5×a/Lni,ref and − 0.43×a/Lnc,ref. This result does not indicate that CGYRO can reproduce the experimental energy flux levels. This result indicates that this simulation is sensitive on the gradient scale lengths of impurity and main ion. Since both input parameters are not well constrained in this study, we cannot conclude that simulated energy flux levels are under-predicted compared to experimental levels. Instead, we found that impurity information missing in this study should be included for solid comparison between simulation and experiment for validation purpose.

4. Preparation for next round of gyrokinetic validation study using KSTAR plasmas

In the first gyrokinetic validation study using KSTAR plasmas, both electron and ion energy fluxes between experiment and gyrokinetic simulation were compared within their uncertainties. From this study, we identified what we need to improve for the next round of gyrokinetic validation study.

First is the Zeff profile information. The impurity density profile and ion density profile were obtained using flat Zeff =2 profile. Therefore, ni/ne, nc/ne, and the gradient of main ion and carbon impurity, used as the input parameters in the simulation, can be different from real experimental values. As shown in Figs. 13 and 14, simulated energy flux levels can be varied significantly by changing the impurity level and gradient of the main ion and impurity species. Since the simulated energy flux levels can be even matched with experimental levels in the scan of these input parameters. We cannot draw any conclusions about the validity of the gyrokinetic model used here from this first round of validation study results. For the next round of validation study, Zeff

profile should be constrained by experimental measurements.

In this study, we focused on the quantitative comparison between experimental and simulated energy flux levels. To conduct more reliable validation study, multi-level comparison is required. That is, the various quantities including fluctuations as well as energy flux from turbulence should be compared between experiment and simulation. All diagnostics have their own temporal and spatial resolutions and give the average values within their resolutions. Since simulation does not consider these measurement conditions, synthetic diagnostic [5], the model that transforms the output of simulation into diagnostic measurement data,

Fig. 11. The comparison results between experimental and simulated energy flux of (a) ion (b) electron at r/a =0.5. Error bar denotes uncertainty of the simulated energy flux. Red and blue are the with and without fast ion case, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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is required in the comparison of measurements and simulations.

Here we adopted and tested the synthetic diagnostic model devel- oped in the past [5] for the next steps of a gyrokinetic validation study using KSTAR plasmas. The synthetic diagnostic was applied to the

nonlinear simulation run with 1.5 a/Lni & − 0.43 a/Lnc, which predicted energy flux levels most closely compared to the experimental energy flux levels. Fig. 15 depicts the synthetic diagnostic results when the point spread function having gaussian shaped function with full width Fig. 12. Simulated energy flux for (a) ion, (b) electron of a/Ln (+24%), a/LTe (+20%), collision frequency ν (-24%), and ωE×B (-50%) within their uncertainty range at r/a =0.5. Shaded bands denote the experimental energy flux.

Fig. 13. (top) kθρ_sspectrum of (a) real frequency and (b) linear growth rate for Zeff =[1.2,2.5]; (bottom) Simulated energy flux of (c) ion and (d) electron for Zeff = [1.2,2.5] at r/a =0.5. Shaded bands denote the experimental energy flux.

Fig. 14.Simulated energy flux of (a) ion and (b) electron for varied a/Lni& a/Lnc with quasi-neutrality constraint at r/a =0.5. Shaded bands denote the experimental energy flux levels including their uncertainties.

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half maximum, FWHM =1.5 cm, is used. This resolution condition is similar to beam emission spectroscopy (BES) measurements condition in KSTAR. 5.5 Ms/s, which is enough to cover the fluctuation measured BES, is used as the sampling rate. The fluctuation level of simulated fluctuations is calculated by integrating the auto power spectrum over the frequency. The calculated fluctuation level of unfiltered and synthetic are 0.3% and 0.15%. This fluctuation level is relatively low for a fluctuation diagnostic to measure. This is somewhat consistent with the fact that there were no measurable turbulence fluctuations from BES in this discharge. By applying the synthetic diagnostic tool, we will compare the fluctuation levels between gyrokinetic simulations and diagnostic measurements in the next round of gyrokinetic validation study using KSTAR plasmas.

5. Summary and conclusion

In this paper, we present the first gyrokinetic validation study using KSTAR plasmas. For the gyrokinetic validation study, KSTAR discharge 21631 was selected. The experimental analysis including profile analysis and power balance analysis was performed first to generate input parameters for simulation and experimental energy flux profiles. In the experimental analysis, profile analysis, power balance analysis, and equilibrium reconstruction were conducted with iteration to keep the consistency among them. The uncertainty quantification was conducted

using profile samples from data weighted normal distribution and error propagation.

By conducting the linear stability analysis using CGYRO, the characteristics of ion scale turbulence at r/a =0.5 were analyzed. Since the most unstable mode is in the electron diamagnetic direction and a/Ln

and a/LTe destabilize this mode, it is very likely that the most unstable mode in this linear run is TEM. Nonlinear gyrokinetic analysis was performed once numerical input parameters were determined from the convergence test. The nonlinear gyrokinetic simulation with experimental input parameters underpredicted the energy flux compared to the experimental energy flux levels estimated from power balance analysis. We also found that the simulated energy flux levels were still under-predicted by changing input parameters within their uncertainties under the assumed flat Zeff profile.

However, when the impurity density gradient scale length was varied with the main ion density gradient scale length to satisfy quasi- neutrality constraint, the simulated energy flux was varied significantly. They are even matched with the experimental energy flux levels.

This does not indicate that the gyrokinetic simulation used here can reproduce the experimental energy flux levels, but indicate that impurity density gradient scale length should be constrained for the solid validation study. Thus, the next round of validation study requires Zeff

profile measurements, which can provide the constraint of ion and impurity density profile. In addition, multi-level comparison using syn- Fig. 15.(a) Overlays of point spread function on contour of the simulated (unfiltered) electron density fluctuations (b) auto-power spectrum (c) cross power spectrum of the simulated (black) and synthetic (red) electron density fluctuations in the lab frame in the most closely matched simulation case (1.5 a/Lni,ref & −0.43 a/Lnc,ref) to the experimental energy flux levels at r/a =0.5. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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thetic diagnostic will be performed in the next round of validation study.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to express our gratitude to the KSTAR team for their assistance in conducting the experiment for simulation and

validation studies. The authors also thank Dr. C. Holland in UCSD for allowing and providing synthetic diagnostic code for CGYRO output.

This study was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1F1A1076162), and the R&D Pro- gram of the KSTAR Experimental Collaboration and Fusion Plasma Research (EN2201-13). Computing resources were provided on the National Supercomputing Center with supercomputing resources including technical support (KSC-2020-CRE-0364) and on the KFE computer KAIROS, funded by the Ministry of Science and ICT of the Republic of Korea (KFE-EN2241). Part of the data analysis was performed using the OMFIT integrated modeling framework.

Appendix. Simultaneous changes in inverse density gradient scale lengths of plasma species

In this appendix, we document the quasi-neutrality condition used in this paper. To satisfy the quasi-neutrality, the sum of the charges of ion species and electrons should be almost neutral. That is, ΣjZjnj is zero where j is ion species and electron, and Zj is charge of j. As a reminder, our discharge has carbon as the only impurity species. Therefore, quasi-neutrality condition represents follow equation:

ne=ni+6nc+nFI (A.1)

where ne, ni, nc, and nFI denote the densities of electron, main ion, carbon impurity, and fast ions.

The relation obtained by differentiate the (A.1) is as follow:

∇ne= ∇ni+6∇nc+ ∇nFI (A.2)

From Σ_jZj∇nj =0, the below equation can be derived.

∑

j

Zjnja /

Lnj=0 (A.3)

In the gyrokinetic simulation with varied a/Lni& a/Lnc, scales of a/Lni and a/Lnc were adjusted while satisfying equation (A.3), without changing densities of the ion species and electron, and density gradient scale lengths of electron and fast ion. Therefore, nia/Lni+6nca/Lnc =constant here. We can calculate the scale of a/Lni and a/Lnc from (A.4):

nia/Lni+6nca/Lnc=niζ_ia/Lni+6ncζ_ca/Lnc (A.4)

where ζ_jdenotes the scale of a/Lnj. [ζ_i,ζ_c] could be calculated as follows: [1,1], [1.25, 0.2848], [1.5, − 0.4306], [1.75, − 1.1456] using information in Section 3.1.

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