Validation of quasi-linear turbulent transport models against plasmas with dominant electron heating for the prediction of ITER PFPO-1 plasmas

For ASDEX Upgrade plasmas, a set of ASTRA-TGLF simulations with evolving density, evolving temperature and evolving poloidal flux profiles was performed with the boundary at ρ_B = 0.95. Cold gas puff neutrals were used as the major particle source in these simulations. The source of cold gas puff neutrals was adjusted at ρ ⩾ 0.61 such that the total number of particles in experiment and simulation deviate from each other by less than 4% (see figure 6). We recall that ρ = 0.95 is the radial location where the ion heat flux analysis at the L–H transition was performed in [10, 11].

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Comparing the predicted density profile from TGLF-SAT1geo with experimental profiles from Thomson scattering and IDA, one observes reasonable agreement between experiment and simulation. As it will be shown in section 5.6, TGLF is able to reproduce the isotope effect on the density profile and provides good density profile predictions for hydrogen as well as for deuterium.

For JET-ILW plasmas, all simulations were performed with the boundary at ρ_B = 0.82. The particle source in the simulation domain [0; 0.82] is negligibly small such that the density profiles evolve to meet the zero flux condition (Γ_e ≈ 0, where Γ_e is the particle flux for electrons). One observes (see representative example in figure 7) that in JET simulations with evolving density, temperature and poloidal flux profiles, predicted density profiles are very close to the measured profiles from HRTS. There is only a small tendency to slightly overpredict the turbulent inward pinch, which leads to an overestimation of the normalized logarithmic gradient of density R/L_ne around mid-radius. As it will be shown in section 5.3, TGLF also overestimates R/L_Te in the JET simulations. This observation together with the results presented in [44] can explain why TGLF can predict a too large stationary R/L_ne in the considered JET-ILW plasmas. It was shown in [44] that the stationary R/L_ne satisfying Γ_e = 0 increases with increasing R/L_Te due to the increasing thermodiffusive pinch. In JET simulations with prescribed temperature profiles, i.e. simulations maintaining the correct normalized logarithmic gradients of T_e and T_i (within the experimental uncertainty), the predicted n_e-profiles match extremely well the HRTS profiles. Moreover, by performing a set of simulations with different prescribed current density and safety factor profiles, or with the inclusion of sawtooth relaxations of the current density profile, we have also observed a sensitivity of the predicted density profile peaking on the shape of the safety factor profile. The simulations presented in figure 7 have been obtained with evolving current density profiles, without the inclusion of any sawtooth relaxation in the centre, leading to a stationary safety factor profile in the ASTRA-SPIDER calculation which is in very good agreement with the one obtained by the EFIT magnetic equilibrium reconstruction.

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There is a large interest among the scientific community to push the boundary as close as possible to the separatrix [16, 39]. In previous versions of TGLF, several problems limited the possibilities of performing simulations with a boundary position ρ_B close to 1 (e.g. [40]).

Like its predecessor GLF23, TGLF cannot be expected to reproduce the edge transport barrier (ETB) that is present in H-modes [39]. One can see in the left plot of figure 8 that the TGLF-SAT1geo version with incorporated geometric factor still fails to reproduce the pedestal in the T_e-profile. The same observation is made in the ion temperature profile. In simulations which have the boundary condition in the ETB region at ρ_B = 0.95, electron and ion temperatures are systematically underpredicted not only close to the boundary, but over the full radius ρ ⩽ ρ_B in H-mode conditions. This is a result of the stiffness of turbulent transport that is reproduced by gyro-Landau fluid models such as TGLF, as shown in [41]. As the blue curve in figure 8 shows, significantly better predictions are obtained when the boundary conditions is placed on the pedestal top at ρ_B = 0.82. For the presented study, the missing turbulent transport suppression in the ETB region in TGLF-SAT1geo is not a major problem since the focus is on the prediction of Q_i,edge in L-mode plasmas. Recent simulations with an integrated modelling workflow, which applies TGLF in the core plasma and also includes a pedestal transport model and MHD pedestal stability calculations, have allowed the modelling of H-mode plasmas over the entire cross-section [42].

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In L-mode plasmas, previous versions of TGLF and GLF23 are known to underestimate turbulent heat transport close to the edge, especially at low plasma current and high q₉₅ [16, 43]. It was shown in [16] that early versions of TGLF-SAT1 still underpredict turbulent transport fluxes close to the edge (although less strong than TGLF-SAT0). The right plot in figure 8 visualizes this shortage of edge electron heat transport and the resulting pedestal-like structure in the T_e-profile for the original version of TGLF-SAT1. An analogous observation is made for the predicted ion heat transport and the ion temperature profile. One observes that with TGLF-SAT1geo with improved geometry factor, significantly better predictions of the heat conductivities χ_e and χ_i close to the edge are obtained than with the original TGLF-SAT1. As a result, electron and ion temperature profiles predicted with the more recent TGLF-SAT1geo smoothly follow the experimental profiles.

Electron temperature predictions close to the edge were considered in the previous section. This section will focus on electron temperatures closer to the centre of the plasma for which accurate estimates are needed to ensure good predictions of third harmonic ECRH absorption in PFPO-1.

The left plot in figure 9 shows a comparison between experimental and predicted electron temperatures at ρ = 0.32. This radial position was chosen since it is close to the sawtooth inversion radius of many plasmas in the database which makes the analysis less dependent on the sawtooth model and more specific to the transport model. Furthermore, at the highest ECRH powers, indications for significant amounts of superthermal particles are found at positions closer to the magnetic axis. The figure contains all 105 data points in the database. To avoid systematic underestimations of H-mode T_e-profiles as described in section 5.2, simulations for H-mode plasmas in AUG were preformed with boundary position ρ_B = 0.82 whereas simulations for L-mode plasmas in AUG were performed with ρ_B = 0.95. Simulations for the JET-ILW plasmas were all performed with ρ_B = 0.82.

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One observes that low electron temperatures are well-predicted whereas at high T_e, there is a tendency to underestimate T_e(ρ = 0.32). A possible explanation is provided by the right plot of figure 9 that shows that TGLF could not correctly take into account the temperature ratio T_i/T_e. Here the temperature deviation in the central region (ρ = 0.32) is plotted as a function of the ion to electron temperature ratio T_i/T_e at ρ = 0.6, that is, at a radial location which is in the middle of the confinement region, and thereby in the middle between the central location and the radial location of the boundary condition in the simulations. The main idea behind this comparison is connected with the consideration that in the transport modelling the predicted values in the central region of the plasma are determined by the level of transport coefficients in the confinement region (that is, around mid-radius), which also determine the temperature gradients in this region. Therefore the correlation between the temperature deviation in the centre has been examined as a function of various parameters around ρ = 0.6. We have found that there is a tendency to underestimate central electron temperatures at low T_i/T_e, whereas, at high T_i/T_e, one observes a slight overprediction of central T_e. There is a clear correlation between the relative deviation (T_e(TGLF) − T_e(Exp))/T_e(Exp) at ρ = 0.32 and T_i/T_e with a Pearson correlation coefficient of 0.63.

This dependence on T_i/T_e observed in electron temperature predictions has several consequences: since T_i/T_e correlates strongly with the collisionality ${\nu }^{{\ast}}\propto {n}_{\mathrm{e}}{Z}_{\text{eff}}{T}_{\mathrm{e}}^{-2}$ in the experiment, there is also an analogous ν*-dependence in TGLF-SAT1geo with underestimations of T_e(ρ = 0.32) at low collisionality and a tendency to overestimate T_e(ρ = 0.32) at high collisionality (Pearson correlation between (T_e(TGLF) − T_e(Exp))/T_e(Exp) at ρ = 0.32 and ν*(ρ = 0.6) is 0.41). An analogous observation was already made for TGLF with saturation rule SAT0 and GLF23 in [39]. It was also shown in [39] that this collisionality dependence is already weaker in TGLF-SAT0 than in GLF23 due to the more sophisticated model of trapped particles in TGLF. Nevertheless, the present study shows that a dependence on the collisionality of the deviation of the TGLF-SAT1geo predictions for the experimental electron temperature is still present, with a slight overestimation of the central electron temperature at high collisionality and an underestimation at low collisionality.

A further consequence of this dependence on T_i/T_e is the dependence of the predictive capability of T_e(ρ = 0.32) on the heating method. Since in the JET-ILW plasmas with ICRH, the temperature profiles are closer together than in AUG plasmas with ECRH, one observes slightly overpredicted T_e-profiles in JET simulations. Moreover, as the colour coding in figure 9 shows, there is a strong tendency to underestimate T_e-profiles especially at low line-averaged density which is the result of the weaker coupling between electron and ion heat channel at low density (${P}_{\mathrm{e}\mathrm{i}}\propto {n}_{\mathrm{e}}^{2}$). Figure 10 shows the example of such a plasma at low line-averaged density and thus low T_i/T_e in this data set. This plasma phase, corresponding to the time window from 5.25 s to 5.55 s in the AUG discharge ♯36982, is characterized by an ECRH power of 2.7 MW, with a temperature ratio T_e/T_i = 3.8 at ρ = 0.5 and T_e/T_i ≃ 6.5 close to the magnetic axis.

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Restricting the analysis to more ITER-PFPO-1 specific heating conditions only for AUG plasmas with 0.8 MW ⩽ P_ECRH ⩽ 1.8 MW (1.3 MW in AUG lead to the same heating power per surface element as 20 MW in ITER), one observes a tendency to underestimate T_e(ρ = 0.32) with a relative deviation of (−10 ± 6)%. Therefore, one can expect a slight underestimation of the absorbed X3 ECRH power in PFPO-1 simulations. However, it is worth to note that in pure Ohmic plasmas (P_ECRH = 0 MW), predictions of T_e(ρ = 0.32) are very good with a relative deviation of only (−3 ± 4)% since in these plasmas, electron and ion temperatures are closer to each other than at high ECRH power. This is a valuable observation since especially at low ECRH powers/low electron temperatures, third harmonic ECRH absorption is very sensitive to central electron temperatures [2]. Accurate central electron temperature predictions ensure precise estimates of X3 central ECRH absorption in Ohmic plasmas and plasmas at very low P_ECRH.

Since the cross section for fusion reactions depends strongly on the temperature of the ions in the plasma, accurate predictions of T_i are necessary especially for simulations of the fusion power operation phase in ITER. It was already shown in [39] that with TGLF-SAT0, ion temperature profiles, and therefore also neutron yields, are better predicted than with GLF23. As described in section 5.2, TGLF-SAT1geo with the sophisticated geometric factor further improves the predictive capability close to the edge and predicts ion temperature profiles following very well the experimental profiles close to the edge. This section will therefore focus on the ion temperature predictions around mid-radius and in the centre of the plasma.

The left plot in figure 11 compares the experimental ion heat conductivities to the effective ion heat conductivities determined from TGLF-SAT1geo being applied to the measured experimental profiles. One observes that in the experiment, χ_i shows almost no dependence on T_e/T_i whereas TGLF predicts significantly increasing ion heat transport with increasing T_e/T_i. This results in significant overestimations of ion heat fluxes at high T_e/T_i. Therefore, especially in plasmas at low density and high ECRH power (and thus, high T_e/T_i), central ion temperatures are considerably underestimated. The JET-ILW plasmas in the database, in contrast, feature extremely good predictions of the central ion temperatures, with only very small overpredictions (see example in the right plot of figure 11). This is a consequence of the considerable amount of auxiliary heating power directly deposited to the ion heat channel which results in a comparably low T_e/T_i, closer to 1.

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As outlined in the previous section, a dependence on T_e/T_i in the predictive capability leads to an analogous dependence on the collisionality ${\nu }^{{\ast}}\propto {n}_{\mathrm{e}}{Z}_{\text{eff}}{T}_{\mathrm{e}}^{-2}$. One observes a strong tendency in TGLF-SAT1geo to overestimate ion heat transport at low collisionality and underestimate ion heat transport at high collisionality. In the examination of the correlations between the deviation of the ion temperature in the central region (ρ = 0.32) and the plasma parameters in the confinement region around ρ = 0.6, as motivated and already described for the electron temperature simulations in the previous subsection, a Pearson correlation of 0.48 between (T_i(TGLF) − T_i(Exp))/T_i(Exp) at ρ = 0.32 and ν*(ρ = 0.6) has been found. Because of the strong dependence of the electron–ion energy exchange on the density (${P}_{\mathrm{e}\mathrm{i}}\propto {n}_{\mathrm{e}}^{2}$), the T_e/T_i-dependence translates also into a ${\bar{n}}_{\mathrm{e}}$-dependence in ECRH plasmas with underestimations of T_i especially at low line-averaged density and overestimations of T_i especially at high line-averaged density (see figure 12).

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Considering all ITER-like heated AUG plasmas with P_ECRH ∈ [0.8 MW; 1.8 MW] in the database, predictions of T_i(ρ = 0.32) are very good with a relative deviation of (−1 ± 9)% on average (±standard deviation).

The present results can also be considered in combination with recent applications of TGLF-SAT1 in conditions with larger ion heating fraction than those modelled in this study. With dominant ion heating, the transport model has been found to predict a stiffness in the relation between the ion heat flux and the logarithmic ITG which is weaker than the experiment and than companion gyrokinetic simulations [45]. Thereby the TGLF-SAT1 model appears to provide increasingly good predictions of the ion temperature in conditions where the ion to electron heating fractions are comparable and where the dominant unstable modes are provided by a combination of ITG and TEM, like in the present study in conditions of T_e ⩾ T_i. In contrast, conditions with dominant ion heating usually feature an overprediction of the ion temperature by TGLF [45] and conditions with largely dominant electron heating and large electron to ion temperature ratio show an overprediction of the ion heat transport. The present picture appears to be consistent also with the good performance of TGLF-SAT1geo in the prediction of AUG H-mode plasmas with comparable ion and electron heating fractions, recently reported in [46].

Correctly predicting the transition from L-mode to H-mode requires a reliable prediction of the edge ion heat flux Q_i,edge, which is evaluated here at ρ = 0.95, consistently with [10]. The edge electron heat flux will not be examined in detail here since it was shown to not play a relevant role in the occurrence of L–H transitions [10].

Figure 13 compares the experimental edge ion heat fluxes, obtained from power balance simulations, to the predicted edge ion heat fluxes from ASTRA-TGLF simulations. One observes very good predictions of Q_i,edge (except for one outlier that can be explained by its relatively high density of ${\bar{n}}_{\mathrm{e}}=4.5{\times}1{0}^{19}\enspace {\mathrm{m}}^{-3}$ leading to a high sensitivity of P_ei on the temperature profiles) with a slight tendency to underestimate edge ion heat fluxes. On average (±standard deviation), the relative deviation between experiment and simulation is (Q_i,edge(TGLF) − Q_i,edge(Exp))/Q_i,edge(Exp) = (−8 ± 13)% in simulations with evolving density and temperature profiles. Prescribing the density profile does not significantly improve the prediction quality (see left plot in figure 13).

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As discussed in figure 4, the edge ion heat flux is more prone to errors in the temperature profiles at low ECRH power. Nevertheless, very precise predictions with only small underestimations are observed also at comparably low ECRH powers of P_ECRH ∈ [0.8 MW; 1.8 MW] with a relative deviation of (−9 ± 10)%. Rather underestimating than overestimating Q_i,edge in this power range makes predictions of L–H transitions in PFPO-1 a bit conservative, i.e. if ASTRA-TGLF predicts ${Q}_{\mathrm{i},\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}{ >}{Q}_{\mathrm{i},\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}^{\mathrm{L}-\mathrm{H}}$, one can assume with an increased level of confidence that this condition will also be satisfied in reality.

Restricting the analysis to all ECRH L-mode plasmas in the database, one finds a relative deviation of (−10 ± 14)%. This shows again the high accuracy (with small underpredictions) of estimations of Q_i,edge to be applied to the prediction of L–H transitions.

The database contains AUG plasmas in hydrogen as well as in deuterium which allows one to examine the predicted mass dependence in ASTRA-TGLF simulations. Experimentally, isotope effects in ASDEX Upgrade were extensively studied in [47] where several isotope mass dependent mechanisms were identified which can influence the energy confinement. Considering the presence of such a mass dependence in turbulent transport, it is considered to be problematic for ITER PFPO-1 simulations that TGLF is exclusively calibrated on deuterium plasmas although all discharges in PFPO-1 will be performed either in hydrogen or in helium [25]. This section will therefore examine the mass dependence predicted by TGLF-SAT1geo that is calibrated against ten CGYRO simulations of deuterium plasmas [17].

Gyrokinetic simulations and analytic theory presented in [48] suggest decreasing stationary R/L_ne with increasing main ion mass m_i in the core plasma. It is explained by the reduced turbulent inward pinch with increasing m_i, caused by the increasing collisional contribution of trapped electrons to outwards-directed convective particle transport and the increased reduction of the curvature pinch with increasing mass m_i, produced by the increased effective collisionality with increasing ion mass. As the plot at the top of figure 14 shows, this behaviour is indeed found in the experiment with lower R/L_ne for deuterium than for hydrogen. The plot also shows that TGLF is capable of reproducing this mass dependence in the particle transport. Note that TGLF also reproduces the flattening of the density profile with increasing ECRH power that was investigated in [49].

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For the heat conductivity χ, the gyroBohm scaling predicts χ/T^3/2 to be a constant increasing with the main ion mass (${\chi }_{\mathrm{g}\mathrm{B}}/{T}^{3/2}\propto \sqrt{{m}_{\mathrm{i}}}$). Considering the experimental results at the bottom of figure 14, one indeed finds ${\chi }_{\mathrm{e}}/{T}_{\mathrm{e}}^{3/2}$ to be approximately constant over all ECRH power levels. TGLF approximately reproduces the constant behaviour of ${\chi }_{\mathrm{e}}/{T}_{\mathrm{e}}^{3/2}$ (with slight overestimations of ${\chi }_{\mathrm{e}}/{T}_{\mathrm{e}}^{3/2}$ at ${\bar{n}}_{\mathrm{e}}\approx 2{\times}1{0}^{19}\enspace {\mathrm{m}}^{-3}$ as discussed in section 5.3). Moreover, ${\chi }_{\mathrm{e}}/{T}_{\mathrm{e}}^{3/2}$ from the power balance decreases significantly from hydrogen to deuterium discharges, and this variation is reasonably well reproduced by TGLF in the corresponding ASTRA simulations, as shown in figure 14. Such an anti-gyroBohm behaviour is also observed for plasmas at higher density. In companion ASTRA calculations in which the main ion mass is doubled from the hydrogen mass to the deuterium mass in the TGLF input, the resulting electron heat conductivities also decrease, although by a more limited amount, between 5% and 10%. This isotopic effect can be explained by the collisional stabilization of TEM leading to decreasing electron heat transport with increasing main ion mass, since the stabilizing impact of collisionality is stronger with heavier ion mass, that is, at smaller ion thermal velocity [50].

As shown in figure 15, in the predictions of the ion temperature, the replacement of hydrogen mass with deuterium mass leads to a small increase of the TGLF ion heat conductivity in the confinement region, by about 10% on average in the present AUG dataset. This weak positive dependence can be still considered to be acceptable in reproducing the AUG experimental observations in these conditions of dominant electron heating, where the experimental power balance ion heat conductivities are observed not to show a strong dependence on mass, consistent with previous studies [47, 51]. This behaviour of the TGLF model is also satisfactory in H-mode conditions with comparable ion and electron heating, where the TGLF modelling of H plasmas with H mass only leads to a small overprediction of the ion temperature [46]. Here the dominant effect in the core is consistently observed to be produced by the presence of different fast ion populations in H and D plasmas in both JET [45] and AUG [46]. Therefore, it can be also expected that in conditions of dominant ion heating in H plasmas, the use of H mass in TGLF for the modelling of the ion temperature can lead to a more significant overprediction of the ion temperature, as a consequence of the quasi-linear properties of the TGLF model which, in conditions of largely dominant ITG modes, more closely follow the gyro-Bohm scaling [45].

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We conclude that TGLF-SAT1geo is well suited to be also applied to other isotopes than just deuterium in conditions of dominant electron heating like those of AUG and JET considered in this work. In these conditions, the replacement of the hydrogen mass with the deuterium mass in the simulation of hydrogen plasma can lead to a slight overestimate of the central electron temperature but also a better prediction of the ion temperature, particularly with increasing ion heating fraction. A strong difference between predictions with hydrogen and with deuterium masses used as input of the TGLF model should be interpreted as the sign that the model is applied in conditions which go outside the domain over which it has been tested in the present work, and for which the present results cannot guarantee the reliability.

This section will analyse the growth rates γ and the real frequencies ω_r predicted by TGLF after convergence of the simulations. A short qualitative analysis is followed by a more quantitative comparison between TGLF spectra and linear gyrokinetic calculations performed with the GKW code [52].

Figure 16 shows the dominant micro-instability (the instability with the highest γ/(k_θ ρ_s), where k_θ is the poloidal wavenumber of the considered mode and ρ_s is the sound ion gyroradius) at two radial positions, r/a = 0.50 and r/a = 0.80 for all hydrogen plasmas in the database. Two type of space diagrams are considered, one in the engineering parameters ECRH power and density (figure 16, left and middle plots) and one in the physically dimensionless parameters provided by the electron to ion temperature ratio and the electron–ion collision frequency normalized to the sound speed divided by the connection length qR. For convenience of the reader, this normalized collisionality is still expressed with usual practical dimensions for densities and temperatures. At these radial positions, all H plasmas in the database are either ITG-dominated or TEM-dominated (at other radial positions, also ETG-dominated plasmas are found). TEM-dominance is predominantly observed at high ECRH power because the normalized logarithmic gradient of the electron temperature R/L_Te increases with increasing ECRH power which drives the TEM instabilities. ITG-dominated plasmas, on the other hand, are predominantly observed at high densities. This is explained by the collisionality (which increases with increasing density) that leads to a stabilization of TEM [53] whereas ITG modes are more weakly affected by collisions. Additionally, the temperature ratio T_e/T_i increases with increasing ECRH power and decreases with increasing density which supports the observed TEM- and ITG-regimes. However, results from linear gyrokinetic simulations in [53] suggest that the role of T_e/T_i is only secondary for the TEM ↔ ITG transitions compared to the impact of ν* and R/L_Te. With increasing radial position, both the trapped particle fraction and the collisionality increase. The latter effect prevails and this reduces the instability domain with TEM dominance and broadens that with ITG dominance when moving from mid-radius to the plasma edge. Interestingly, all of the points practically populate a single curve when plotted on the space diagram provided by the electron to ion temperature ratio and the collisionality (figure 16, right plot).

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For twelve hydrogen plasmas at q₉₅ ≈ 5, a set of linear gyrokinetic simulations with GKW was performed for the two radial positions r/a = 0.53 and r/a = 0.69. The left plot of figure 17 shows a comparison between TGLF spectra and GKW spectra for a purely Ohmic (P_ECRH = 0 MW) plasma. As this example shows, TGLF commonly predicts some dips on the ETG scale that are not reproduced by GKW. This results in narrower ETG spectra predicted by TGLF than computed by GKW. In saturation rule SAT1geo, the constructed effective growth rate γ_model does not depend on the ETG profile if γ < k_θ V_ZF (where V_ZF is the root mean squared E × B drift velocity of the fluctuating zonal flows). However, in ETG-dominated plasmas, these too narrow profiles affect γ_model and therefore also the predicted transport.

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Close to the edge (at r/a = 0.69), one finds a strong tendency in TGLF to underestimate electron scale growth rates whereas around mid-radius (r/a = 0.53), better predictions for γ at the ETG scale are found. This can be explained by the observation that TGLF tends to underpredict ETG growth rates at high collisionality, proportional to ${n}_{\mathrm{e}}R{Z}_{\text{eff}}/{T}_{\mathrm{e}}^{2}$ (see right plot in figure 17) or high T_i/T_e. The Pearson correlation between the relative deviation of max(γ/(k_θ ρ_s)) on the ETG scale and the collisionality ν* is − 0.52; the Pearson correlation between the relative deviation of max(γ/(k_θ ρ_s)) on the electron Larmor radius scale and T_i/T_e is similar with −0.48.