Isotope effects on energy transport in the core of ASDEX-Upgrade tokamak plasmas: Turbulence measurements and model validation

The new generation of tokamak fusion reactors aiming to achieve and exceed break-even conditions will operate with a 50:50 mixture of deuterium and tritium fuel to maximize the fusion reaction rate and arrive at reactor-relevant conditions more easily. Successful design and operation of these reactors requires accurate predictive modeling of plasma confinement in mixed ion species plasmas. There is a long history of research on the effects of ion mass on tokamak confinement properties dating back to the early 1990s.^1,2 A recent review article by Weisen³ addresses the multiple observations over the last three decades. The general experimental observation is that global energy confinement time increases with ion mass $τ E ∝ m i 0.09 − 0.47$ (where m_i refers to the ion mass)³ across a wide range of experimental conditions. This positive dependency is found in contrast to the popular “gyro-Bohm” scaling of heat diffusivity $χ g B ∝ T e B ρ i *$ (where $ρ i * = ρ i / a$⁠, ρ_i is the ion Larmor radius and a is the device minor radius, with $ρ i = T i m i / ( Z eff e B )$ where T_i is the ion temperature, Z_eff is the effective charge, e is the electron charge, and B is the magnetic field). The gyro-Bohm scaling⁴ leads to a negative dependency of global energy confinement on ion mass $τ E , g B ∝ a 2 / χ g B ∝ m i − 1 / 2$⁠. The disparity between experiments and simple theory—beneficial to energy confinement on deuterium–tritium plasmas—has been termed the isotope effect.

Turbulence diagnostics are ideally suited to directly measure local transport properties and can provide unambiguous evidence of the transport response to a changing ion mass. McKee et al.⁵ studied the $ρ i *$ scaling of density fluctuation properties by scanning the magnetic field in non-dimensionally similar discharges. It was shown that turbulence radial correlation lengths, fluctuation amplitudes, and decorrelation times follow a gyro-Bohm scaling. Hennequin et al.⁶ found a similarly good match with a gyro-Bohm scaling in density fluctuations over a wider range of scales spanning $k ⊥ = 6 − 30$ cm⁻¹ (where k_⊥ refers to fluctuations perpendicular to the tokamak's magnetic field). Both of these experiments used the magnetic field to vary $ρ i *$⁠; transport scaling experiments, which vary the ion mass to change $ρ i *$⁠, are less common. Holzhauer and Dodel⁷ reported density fluctuation measurements in deuterium and hydrogen over $k ⊥ = 3 − 15 cm − 1$ using collective laser-light scattering in the ASDEX tokamak. Density fluctuation amplitudes in deuterium were found to be larger than hydrogen in both the core and near-edge regions of the plasma.

All of these experiments have studied density fluctuation properties, yet electrostatic fluctuation-driven radial electron heat flux⁸ depends not only on fluctuations in electron density but also temperature, potential, as well as their coherency and cross-phase angles. Correlation electron cyclotron emission (CECE) can be used to measure long-wavelength temperature fluctuations,^9,10 frequency spectra, and correlation lengths.¹¹ Deng et al.¹² reported an isotopic dependence of electron temperature fluctuations at mid-radius (⁠ $r / a = 0.5$⁠) in the Rijnhuizen tokamak project using a 3D ECE-imaging diagnostic.¹³ In direct contrast to the observations in density fluctuations and a gyro-Bohm scaling, Deng et al. found higher T_e fluctuation levels in hydrogen over deuterium with a ratio close to the inverse of the ion mass ratio.

This publication aims at unraveling these contrasting observations. It presents recent experiments from the ASDEX Upgrade (AUG)¹⁴ tokamak to study the differences in turbulent electron temperature fluctuations and their effect in energy transport between plasmas with varying ion masses: hydrogen and deuterium. Local measurements of ion-scale (⁠ $k ⊥ ρ s < 0.2$⁠, where $ρ s = T e / T i ρ i$ is the sound Larmor radius using the deuterium ion mass in the definition of ρ_i. N.B. The deuterium ion mass is used in all further mentions of ρ_s unless otherwise stated) electron temperature fluctuations and density-temperature fluctuation cross-phase have been performed with a correlation electron cyclotron emission (CECE) diagnostic^15,16 and a perpendicular incidence X-mode reflectometer¹⁷ in the outer core (⁠ $ρ Tor = 0.65 − 0.8$⁠, where $ρ Tor$ is a radial coordinate defining contours of constant toroidal flux¹⁸) low-field side midplane, of electron cyclotron resonance heating (ECRH)-heated L-mode plasmas.

The present study makes use of these turbulence measurements as the basis of a validation study^19,20 of the quasi-linear trapped gyro-Landau fluid (TGLF-SAT2)²¹ and nonlinear gyrokinetic GENE²² codes. Accurate quantification of the agreement between experiments and simulations requires a careful consideration of the sources of uncertainty on both fronts as well as the place of each observable in the primacy hierarchy.²⁰ Since there have been many documented cases of fortuitous good agreement between experiment and simulation when using few observables,²⁰ validation efforts should ideally use as many observables as possible across the primacy hierarchy. This study makes use of up to four different observables, two of which consist of fundamental turbulence properties. Finally, most previous validation efforts have focused exclusively on deuterium plasmas.^23,24 This is the first validation study across different ion-mass plasmas.

The present article is structured as follows. It begins with a description of the plasma conditions and kinetic profile fits in Sec. III. Fluctuation measurements are presented in Sec. IV A. Modeling results follow starting with power balance in Sec. V A. Quasi-linear and non-linear gyrokinetic simulations are then addressed in Secs. V B and V C. The validation study is then finally presented in Sec. VI followed by the conclusions.

The ion mass affects several plasma parameters such as the ion thermal velocity, ion cyclotron frequency, Alfvèn velocity, ion drift velocities, and electron-ion momentum and energy exchange terms.^3,25,26 Experimentally, it has been observed to affect the period of magneto-hydrodynamic (MHD) events such as sawtooth and edge-localized modes.¹ However, a key aspect dominating particle, energy, and momentum transport is how the small-amplitude, high-frequency fluctuations in plasma density, temperature, and potential from drift-wave instabilities²⁷ respond to changes in ion mass. These fluctuations have been broadly identified as the major contributor to the anomalously large amount of cross-field energy and particle transport in tokamaks.^8,28

The ion-mass scaling of the energy confinement time can be studied through dimensionless parameter scaling.^29,30 In this approach, the thermal diffusivity can be expressed in the form $χ = T e e B F ( ρ i * , β , ν * , q , T e / T i , L X , plasma shape , … )$⁠, where $F$ is a function depending on the normalized ion Larmor radius (⁠ $ρ i *$⁠), the ratio of kinetic to magnetic pressures (β), the collisionality (⁠ $ν *$⁠), safety factor (q), ratio of electron to ion temperatures, gradient scale lengths of density and temperature (LX), and plasma shape—among others.^30,31

Diffusive transport arguments suggest a scaling $χ ∼ ( Δ l ) 2 / Δ t$⁠, where $Δ l$ stands for the fluctuation's characteristic length scale and $Δ t$ for the correlation time. If both $Δ l$ and $Δ t$ scale with ρ_i,³² then $F ∝ ρ i *$ and a gyro-Bohm scaling is obtained. The gyro-Bohm scaling results from fine-scale, collisionless, electrostatic, micro-instability linear drift-wave modes, and it is captured by quasi-linear and non-linear gyrokinetics in the local, electrostatic, collisionless, flowless limit with adiabatic electrons.³ If $Δ l$ scaled instead with $ρ i 1 / 2$—as can be the case in linear ion-temperature gradient (ITG) theory in a toroidal geometry³²—then $F$ is independent of ρ_i and a Bohm scaling is obtained. The name comes from the empirical Bohm diffusion coefficient, highly successful in describing fully ionized magnetized laboratory plasmas $χ B = 1 16 T e e B$(Ref. 33, Chap. 5). Whether heat transport scales Bohm or gyro-Bohm can have important consequences in the amount of external power required by future burning plasma devices.^29,30

Tokamak experiments feature several ion-mass-dependent effects^3,26 that can lead to a reversal of the gyro-Bohm mass scaling. E × B shear stabilization (either via bulk rotation³⁴ or zonal flows^35,36), collisional trapped electron mode (TEM) stabilization,³⁷ and electromagnetic effects at high beta³⁸ are a few examples. Additionally, global full-f gyrokinetic simulations of ITG-dominated plasmas have shown the pervasiveness of a Bohm-like $ρ i *$ scaling in energy confinement of both electron and ion-heated L-mode plasmas.³⁹ Recent publications have also highlighted that tokamak edge-specific effects (dominated by electron drift-wave turbulence) are also an important part of the puzzle. The non-adiabatic response of fast electron parallel motion,⁴⁰ as well as edge-specific electromagnetic and collisional effects,⁴¹ can exhibit counter gyro-Bohm behavior.

Experiments were performed on ASDEX Upgrade,¹⁴ a moderate field (⁠ $B T = 2.5$ T), medium-sized (major radius R = 1.65 m, minor radius a = 0.5 m), tungsten walled, diverted tokamak. The plasma presented in this study is a relatively low density (line average $n e = 2.5 × 10 19$ m⁻³), electron-heated L-mode featuring a plasma current of 1 MA and an on-axis toroidal magnetic field of $B T =$ −2.37 T (where positive is defined as counterclockwise around the torus looking from the top). An external power of 600 kW is applied using second-harmonic electron cyclotron resonance heating (ECRH) at 140 GHz. This ECRH power is deposited in the plasma core $ρ Tor < 0.2$⁠. This L-mode reference was first studied in Ref. 24 and has been repeated in both hydrogen (H-36 770) and deuterium (D-36 974).

It is not possible to perform identical discharges in which the ion mass is changed without affecting many key plasma quantities³ such as the radial profiles and gradients. In order to attempt relatively similar conditions between isotopes with a limited number of discharges, the strategy was to keep the plasma current, shape, and external power constant and control the core density via active feedback. Electron and ion-temperature profiles were allowed to evolve freely. In order to measure temperature fluctuations with low statistical error, the discharges featured a stationary flat-top phase lasting 2.5 s. Figure 1 shows the time traces of key parameters of the two discharges under study. It shows that the core line-averaged density, external ECRH power, and plasma current are well matched. The peak electron temperature is larger in deuterium. Neutral-beam injection (NBI) blips are required to measure kinetic profiles of ion temperature and toroidal rotation velocity. The energy confinement time τ_E is obtained via TRANSP modeling⁴² and is calculated by adding the thermal energy continued in electrons and ions and dividing by the power sources (Ohmic and ECRH) minus power sinks (convection, radiation, and charge exchange).⁴³  τ_E is larger in deuterium than hydrogen, in agreement with the anti-Gyro-Bohm global experimental trends (⁠ $τ E ∝ m 0.09 − 0.47$⁠, Ref. 3).

Figure 2 shows average plasma radial profiles during the stationary phase. Electron density is measured using a Thomson-scattering (TS) diagnostic,⁴⁵ a 5-channel deuterium cyanide (DCN) laser interferometer, and a lithium-beam emission spectroscopy diagnostic.⁴⁶ Electron temperature profiles are measured with TS and a calibrated multi-channel electron cyclotron emission radiometer.⁴⁷ Ion temperature and toroidal rotation are measured with the charge–exchange recombination spectroscopy (CXRS) diagnostic.⁴⁸ This diagnostic relies on the charge–exchange reaction between fast neutral particles with carbon or boron impurities in the plasma, and hence, it requires either neutral beam injection (NBI)-heated plasmas or NBI beam blips. As seen in Fig. 1, beam blips of 20 ms every 300 ms are used in these experiments. These short NBI blips do not significantly affect ion temperatures but could induce a finite amount of toroidal torque (i.e., see Fig. 5 in Ref. 49). Toroidal rotation profile fits in Fig. 2(d) take into consideration all data points inside the beam blip and are hence an indicative overestimate of the true intrinsic rotation. The scatter observed in the rotation data did not allow accurate backward extrapolation to obtain true intrinsic rotation as has been done in previous studies.⁴⁹ The equilibrium used to map the diagnostics and to perform transport modeling is obtained from the inverse Grad–Shafranov equilibrium solver integrated data analysis equilibrium (IDE).⁵⁰ It uses pressure profiles and current diffusion modeling to constrain the equilibrium reconstruction.⁵¹ Comparing the IDE equilibrium with the non-kinetic standard Complete Interpretive Suite for Tokamak Equilibria (CLISTE) equilibrium reconstruction⁵² leads to differences in mapping of CECE channels on the order of 0.02 in ρ_Tor.

Figures 2 and 3 show radial profile fits (solid lines) and their logarithmic gradients, respectively. Logarithmic gradient scale lengths are defined as $1 / L X = − d / d ρ Tor ( ln ( X ) )$⁠, where X stands for the specific radial profile. Quantitative sensitivity studies,²⁰ as presented in Sec. V B, rely on a robust estimate of the fit, its derivative, and their respective uncertainties. Care has been taken to fit profiles with the latest, most sophisticated, profile-fitting techniques available.

Profile fits for electron density and temperature were obtained using the integrated data analysis (IDA) method.⁵³ IDA combines several diagnostics within a Bayesian probability framework taking into account the physics and calibration uncertainties of each diagnostic using forward modeling. Since turbulent transport modeling relies critically on accurate profile fits and their uncertainties, the latest IDA algorithm has been employed in this study.⁵¹ It uses a Markov chain Monte Carlo (MCMC) sampling technique, which explores the probability distribution function of a Bayesian data analysis approach, which allows a profile estimate from uncertain data input using non-linear forward models. The resulting fit is the mean of the distribution of samples with their standard deviation being the error.

Because of the specific aim of these experiments: validation of turbulence and transport properties, smooth gradients are required. Thus, the number of spline points was decreased to eight spline knots, roughly half of the usual setting: 19 knots for T_e and 14 knots for n_e. A direct consequence of the reduced spline knots is less variation between samples leading to a smaller estimated error. However, the resulting gradient profiles are within the margin of error of the standard IDA-MCMC and match values obtained with functional fits [i.e., modified hyperbolic tangent function (mtanh⁵⁴) and polynomial spline fits²⁵]. Ion temperature and toroidal rotation are obtained from a single diagnostic: CXRS. Thus, Gaussian process regression (GPR) has been used to estimate profile fit and gradients.^51,55 GPR intrinsically evaluates the fit, its derivative, and their uncertainty and allows for rigorous uncertainty quantification.

Figure 2 shows that density profiles are well matched, inside error bars, for both isotopes. Electron temperature is larger in deuterium within the range $ρ Tor = 0.2 − 0.8$⁠. Higher T_e in deuterium vs hydrogen was similarly observed for density matched Ohmic L-mode ASDEX experiments.¹ Ion-temperature fits show the opposite pattern: a larger ion temperature in hydrogen within the range $ρ Tor = 0 − 0.8$⁠. Toroidal rotation is larger in deuterium than hydrogen and are both positive. Note however that in the region where fluctuation measurements are available, $ρ Tor = 0.65 − 0.8$ (defined as ROI for region of interest henceforth), all profiles can be found within $2 σ$ errors of each other. Logarithmic gradient scale lengths inside the ROI shown in Fig. 3 are well matched within uncertainty in the electron density, electron temperature, and toroidal rotation. The only clear difference can be seen in the ion-temperature gradient, which is larger in H than D, although the difference is just outside of error bars [Fig. 3(c)].

Doppler backscattering^56,57 (DBS, also known as Doppler reflectometry) has been used to measure the perpendicular rotation velocity of fluctuations (v_⊥) in deuterium and hydrogen discharges. Perpendicular $⊥$ here refers to normal to the magnetic field. v_⊥ is composed of both a plasma background E × B velocity (v_E×B) and a phase velocity of turbulent fluctuations (v_ph) components.⁵⁸  Figure 4 shows both the scattered signal spectrograms and inferred v_⊥ profiles over plasma radius. These measurements reveal that the momentum input of NBI beam-blips (required for CXRS measurements) leads to important changes in v_⊥: both in absolute value and profile for both isotopes. The spectrograms in Fig. 4 show that immediately after the beam blip and for the next ∼100 ms, the scattered signal's spectrum shows larger Doppler shift values than ∼100 ms before the next beam blip. Figures 4(b) and 4(d) thus show the v_⊥ profiles over plasma radius 100 ms before and after the beam-blip. It appears that the increase in v_⊥ is more significant in the deeper in the core $ρ Tor < 0.7$⁠. Regardless of the beam-blip dynamics, Fig. 4 shows that v_⊥ profiles are significantly larger in deuterium (⁠ $v ⊥ ∼$ 1.3 km/s) than hydrogen (⁠ $v ⊥ ∼$ 0.3 km/s) inside the ROI. This stands in contrast to v_Tor measurements in Fig. 2 and is further discussed in Sec. V C.

A spatially constant effective ion charge (Z_eff) is estimated using bremsstrahlung radiation in the background CXRS spectra.⁵¹ The deuterium plasma featured an average $Z eff = 1.8 ± 0.7$ while the hydrogen plasma $Z eff = 1.5 ± 0.5$⁠. It can be inferred thus that the hydrogen plasma contains a smaller impurity density, which is consistent with bolometer measurements.^25,59 Total radiation power inside the separatrix is $P rad − D = ( 405 ± 40 )$ kW in deuterium and $P rad − H = ( 353 ± 25 )$ kW in hydrogen. A neutral particle analyzer was used to measure the ratio between hydrogen and the total amount of hydrogen neutral particles—both hydrogen and deuterium—arriving at the plasma periphery. The ratio is less than (5 ± 2)% during the deuterium discharge and (90.6 ± 0.7)% during the hydrogen discharge. These measurements provide confidence that fueling was sufficient and that wall-recycling did not lead to a mixed-species plasmas in either discharge.

Sawtooth frequencies are determined via spectral analysis of core ECE channels to be $f D S T ≈$ 0.56 kHz and $f H S T ≈$ 1.2 kHz. Beam blips are observed to strongly decrease the sawtooth frequency; these estimates were thus taken in-between beam blips when the sawtooth frequency was steady. The smaller deuterium sawtooth frequency is in agreement with an trend reported in ASDEX experiments,¹ which observed that the sawtooth frequency scaled as $f S T ∝ A i − 0.54$⁠, where A_i is the atomic mass number.

A CECE diagnostic^15,16 was used to measure broadband (<1 MHz), low-amplitude (⁠ $< 2 %$⁠) electron temperature fluctuations perpendicular to the magnetic field during both hydrogen and deuterium discharges inside the outer core region $ρ Tor = 0.65 − 0.8$⁠. These small fluctuations have been associated with drift-wave turbulence in tokamaks, believed to be a major contributor to cross-field transport.^8,60 The CECE diagnostic is a heterodyne radiometer operating in F-band between 110 and 125 GHz. The fluctuations observed in the single ECE radiometer signal cannot be interpreted as turbulent temperature fluctuations⁶¹ because of the incoherent nature of black-body electron cyclotron emission in the core of a tokamak and the radiometer detection process.⁶² AUG's CECE radiometer uses frequency-domain cross-correlation^62,63 between two radially close-by ECE channels to circumvent the large natural amount of black-body noise in the signal and access small turbulent temperature fluctuations.

AUG's CECE diagnostic features 24 intermediate-frequency (IF) channels, which allow the measurement of radial temperature fluctuation frequency spectra dT_e(f) up to 1 MHz bandwidth (anti-aliasing low-pass filter), fluctuation levels (⁠ $d T e / T e$⁠), and radial correlation lengths [ $L r , c$(T_e)] simultaneously. The frequency spacing between filter center frequencies is 250 MHz, and each channel features a filter with a (200 ± 50) MHz bandwidth. This spacing leads to a separation of about 4–5 mm in real space and a radial resolution of $Δ ρ Tor ≈ 0.01$ in a typical L-mode plasma. Using a forward model for ECE radiation transport (the ECRad code⁶⁴), the radial width of the 1/e amplitude of the birth-place distribution of observed intensity (Ref. 65, p. 21) in the experiments was found to be 5.5 mm at $ρ Tor = 0.75$⁠. Thus, each channel samples a different plasma region radially. In contrast, the $4 − 5$ mm radial separation in space is generally smaller than the measured radial correlation length of temperature fluctuations, found in previous studies to be ∼10 mm.²⁴ Hence, two adjacent channels are believed to observe statistically the same fluctuations.

Figure 5 shows the magnitude of the cross-coherence function $| γ x y |$ (Ref. 63, p. 32) of three selected pairs of channels in both deuterium and hydrogen discharges. The deuterium spectra show a broad peak in between $20 and 40$ kH, which increases in amplitude with radius. The hydrogen discharge shows, in contrast with deuterium, a similar peak but at much lower frequency. This high-frequency broadband feature is often observed in CECE spectra, and it has been associated with drift-wave turbulence.^67, Figure 5 shows an increased level of fluctuations with radius. This is an expected result consistent with previous CECE measurements⁶⁶ as well as density fluctuation levels measured with other diagnostics (see, for example, Refs. 5 and 68). From the spectra in Fig. 5, it could be ascertained that, while the limited sampling volume of the CECE diagnostic distorts the real fluctuation spectra and phase velocity, the poloidal rotation velocity of fluctuations is larger in deuterium than hydrogen. It is a well-known fact that drift-wave turbulence propagates perpendicular to the magnetic field,⁶⁹ nearly normal to the CECE diagnostic line of sight. A larger perpendicular velocity of fluctuations would result in a higher frequency modulation of ECE radiation, which in turn would result in what appears like a larger “Doppler shift” of the CECE spectrum. The perpendicular rotation velocity of fluctuations, just as is the case in the DBS diagnostic,⁵⁸ is combination of the background plasma rotation and the phase velocity of fluctuations. Figure 4 shows that deuterium plasmas do rotate significantly faster (⁠ $v ⊥ ≃$ 1.5 km s⁻¹) inside the ROI than their hydrogen counterparts (⁠ $v ⊥ ≃$ 0.3 km s⁻¹), a factor of $× 5 − 6$⁠. It is important to mention that AUG's CECE line-of-sight featured a 2 deg angle with respect to the normal to the flux surface during this experiments. Thus, the changes in the location of the ECE drift-wave feature are believed to be due to modulation of the ECE radiation and not a Doppler shift. GENE simulations presented in Sec. V C further support this assertion.

Figure 6 shows that fluctuation amplitudes in deuterium are larger than their hydrogen counterparts, well outside statistical error bars and the statistical limit at ∼0.1%. Figure 6(b) shows the ratio between the rms fluctuation levels in Fig. 6(a), $d T e / T e$(H) and $d T e / T e$(D). Figure 6(b) shows that the ratio between the amplitudes in H/D follows an ion Larmor radius H/D ratio inside $2 σ$ error bars.

L_r,cs in both deuterium and hydrogen seem to broadly decrease with increasing plasma radius. Correlation lengths in deuterium are also clearly larger than those in hydrogen, well outside error bars in over 70% of channels. The ion Larmor radius multiplied by an arbitrary constant of 5.5 is plotted with a dotted line in Fig. 7(a). This Larmor radius scaling seems able to roughly explain both the radial decrease over radius and the difference between the isotopes. Motivated by discussions in Refs. 5 and 69, the correlation lengths of both ions were divided by their respective ion Larmor radius as shown in Fig. 7(b). The result is that both ions seem to obey a 5–7ρ_i scaling, roughly independent of ion mass, specially for channels outside $ρ Tor >$ 0.75. Figures 6 and 7 demonstrate that a local gyro-Bohm scaling of temperature fluctuations holds in the outer core of these L-mode plasmas in AUG.

The CECE diagnostic currently uses the mm-wave quasi-optics of AUG's profile ECE system as described by Classen et al.⁷⁰ It uses a 25 dB gain D-band rectangular horn antenna, which couples to three high-density polyethylene lenses and measures in the low-field side midplane of AUG plasmas. The antenna and RF hardware sit on a rail that can be moved by 60 mm radially and allows the probing beam poloidal waist location to be adjusted. A wire grid selects a vertical polarization in the laboratory frame, which leads to $≈$ 2% signal losses due to a finite edge B-field pitch of $≈ 11 °$ during the L-modes studied here. The electrodynamics reciprocity theorem is used here to assume that the incoherent ECE power coupled into the antenna is well described by the antenna radiation pattern.⁷¹ 

The finite antenna pattern in the poloidal plane and the limited radial extent of the measured ECE radiation result in an unavoidable limited sampling volume, which leads to biased measurements of turbulence properties (Ref. 72, Chap. 5). The limited poloidal sampling volume leads to reduced measured fluctuation amplitudes and increased correlation lengths;⁷³ it acts as a sort of spatial low-pass filter for ion-scale fluctuations in wavenumber space.

It is important to note that the diagnostic beam radius in the poloidal plane is not constant throughout the entire sampling range in Figs. 6 and 7 shown above. Vacuum complex-beam ray-tracing, also known as Arnaud's method,⁷⁴ has been used to model the beam properties through the optics system.⁷⁰ Since the CECE antenna was placed slightly off-axis due to practical reasons and the beam's size was of the order of the lens' dimensions, traditional Gaussian beam q-parameter calculations (Ref. 75, Chap. 2) are not applicable. Assuming vacuum propagation through the vessel (no plasma), the position of the rail was optimized such that the beam waist was found at $ρ Tor =$ 0.75—close to the middle of the sampling range. At $ρ Tor = 0.75$⁠, the CECE diagnostic filters fluctuations perpendicular to the magnetic field with $k ⊥ < 2$ cm⁻¹ (⁠ $k ⊥ ρ s < 0.2$⁠) due to its antenna E-field beam radius of $≈$ 7.6 mm. The beam radius varied between 7.8 mm at $ρ Tor = 0.67$ and 14 mm at $ρ Tor = 0.84$⁠. However, as is well-known from experience modeling ECRH beam propagation, the geometric-optics approach used in ray-tracing may not be appropriate inside fusion-relevant plasmas where large gradients can give rise to significant diffraction effects. The TORBEAM code⁷⁶ has been used to better quantify diffraction effects, and it has been found that the beam's E-field radius can significantly increase by roughly a factor of $∼ 2$⁠. TORBEAM predicts the beam radius varied between 27 mm at $ρ Tor = 0.67$ and 19 mm at $ρ Tor = 0.84$⁠. The plasma is sufficiently far from a cutoff that refractive effects have been tested to be in the mm-range. Nonetheless, both ECRad and TORBEAM consider refraction to estimate an accurate ECE emission position.

Due to the finite ECE radial emission width, estimated at ∼5 mm, radial fluctuations are filtered under $k r < 5$ cm⁻¹ (⁠ $k r ρ s < 0.5$⁠). Since the toroidal viewing angle is only about 1°, the ECE emission radial size is dominated by relativistic mass-shift broadening (Ref. 72, p. 165).

Finally, note that CECE measures predominantly fluctuations perpendicular to the main magnetic field. Due to the much larger conductivity along the magnetic field lines, turbulent structures are highly elongated in the parallel direction (⁠ $k ⊥ ≫$ k_∥).⁸ The antenna pattern in the parallel direction (⁠ $w ∥$⁠) is thus very small in comparison with $k ∥ w ∥ ≪ 1$⁠.⁷³ The contribution to the experimentally measured fluctuation levels from fluctuations parallel to the field is thus negligible.

Coupling a fluctuation reflectometer⁷⁷ (in perpendicular incidence to the cutoff layer, not in the Doppler backscattering⁵⁵ configuration) and a ECE radiometer through the same antenna and aligning the reflectometer cutoff and ECE resonance has been used as an attempt to measure the cross-phase angle between density and temperature fluctuations, α_nT. This approach was pioneered by Häse et al.⁷⁸ in W7-AS and later demonstrated in DIII-D.^23,79 While α_nT is not directly responsible for cross-field energy transport,²⁷ it has been shown to respond to changes in experimental gradients and dominant micro-instabilities.⁸⁰ In the experiments reported here, a W-band heterodyne system¹⁷ operating in X-mode polarization was used as a fluctuation reflectometer by aligning the antenna in perpendicular incidence to the cutoff and sharing the line-of-sight with the F-band correlation radiometer described above.

Although the reflectometer phase has been most often associated with analytic treatments of the reflectometer signal as density fluctuations,^77,81 the reflectometer amplitude has been shown to give a stronger dn_e-dT_e cross-coherence signal with ECE signals experimentally.⁷⁹ Furthermore, cross-phase angles between the reflectometer's amplitude and ECE signals were found to be identical to those between reflectometer's phase and ECE signals.⁷⁹ 

Previous experimental results have also shown that the homodyne signal, $A cos ϕ$ (where A and $ϕ$ stand for the reflectometer amplitude and phase, respectively)—which responds linearly with the amplitude A—can measure density fluctuation correlation lengths in agreement with probe diagnostics.^82,83 Two-dimensional full-wave⁸⁴ as well as physical-optics⁸⁵ simulations attest to the increased reliability of the reflectometer amplitude over the phase signal to access radial correlation lengths. However, most studies used cross-correlations between reflectometer channels and do not address directly under which conditions the fluctuation reflectometer's amplitude and/or phase signal by itself is a valid kernel of density fluctuations.

There can be adverse experimental conditions, termed the non-linear regime, which lead to large localization uncertainty.⁸⁶ Using heat flux-matched gyrokinetic simulations presented in Sec. V C, density fluctuation level $d n e / n e$ and radial correlation lengths can be approximated as 0.5% and 10 mm. These can be used as input to the criteria for the non-linear regime presented by Gusakov and Popov⁸⁶ as $γ = ( ω / c ) 2 l c x x c ( d n e , rms / n c ) 2 ln ( x c / l c x ) ≥ 1$⁠, where l_cx is the radial correlation length of turbulence, x_c is the cutoff distance from the plasma edge (∼8 cm here), n_c is the density at the cutoff layer, $d n e , rms$ is the rms value of density fluctuations, and ω is the sampling frequency. The regime in this experiments is thus linear since $γ ≈ 0.1$⁠. This implies that the reflected signal's amplitude and phase are mostly coming from the cutoff region.⁸⁶ 

Recent simulation results have cast doubt as to whether the reflectometer-amplitude signal is an unbiased kernel of density fluctuations. It has been shown that there may be a cross-phase angle between density fluctuations and the reflectometer-amplitude signal depending on linearity, the radial turbulent wavelength spectrum, and the sampling beam radius of curvature.⁸⁷ While these results are from a conference proceeding and have not been peer-reviewed, they are a clear warning that the accurate interpretation of the experimental cross-phase measurements between reflectometer-amplitude and ECE signals requires careful modeling. Full-wave modeling is beyond the scope of this work. Yet, it is deemed interesting to present experimental cross-phase measurements nonetheless.

Figure 8 shows both the magnitude of the complex cross-coherence and cross-phase spectra between the W-band reflectometer-amplitude signal and a single CECE channel signal during deuterium (34 626) and hydrogen (39 726) discharges. These discharges are repeats of 36 974 and 36 770 presented above. The reflectometer frequency and CECE channel positioning were chosen to find a cross-phase angle around $ρ T = 0.75$ to agree with the location of simulations presented below. In discharge 34 626, the magnitude of the cross-coherence was largest at precisely $ρ T = 0.75$ as shown in Fig. 8. For 39 726, the peak cross-coherence was found in a deeper channel (⁠ $ρ T =$ 0.7), but the statistically significant cross-phase angle found on this channel was identical to that observed in $ρ T = 0.74$ as reported in Fig. 8. The reflectometer amplitude signal has been chosen because cross-correlation with the reflectometer phase led to significantly lower cross-coherence: below the 2.5σ level, and thus not statistically significant.

The cross-phase angle is found to change with isotope mass, outside of error bars. On average, inside the region of significant coherence (⁠ $| γ c | > 2.5 σ$⁠), $α RefA − ECE = 100 ± 16 °$ in deuterium and 153  $± 20 °$ in hydrogen. Experimental error bars come from the root sum square of the statistical variance of the cross-power phase spectrum⁶⁶ and the standard deviation of the cross-phase inside the region of significant coherence. In both discharges, the cross-phase angle is positive: the reflectometer-amplitude signal leads the ECE signal.

Previous studies have highlighted that cross-phase magnitudes moving toward zero are consistent with an increased trapped-electron mode (TEM) drive in the non-linear turbulence mix.⁸⁰ Since $| α RefA − ECE$(H) $| > | α RefA − ECE$(D) $|$⁠, it can be similarly hypothesized, experimentally, that there is a stronger electron turbulence drive in the deuterium discharge turbulence mix. This hypothesis is consistent with the increased ion-temperature gradient (LT_i) in Fig. 3(c), which drives a stronger ion turbulence mix in the hydrogen discharge.

The TRANSP code⁴² is used to empirically interpret transport features of both H and D discharges. TRANSP is a 1.5D equilibrium and transport solver.⁸⁸ It uses a fluid description of the plasma to solve magnetic field diffusion as well as particle, momentum, and energy conservation equations. Figure 9 shows the inferred power balance electron and ion heat fluxes. These definitions follow transport equations in Ref. 89.

The input density and temperature profiles are shown above in Sec. III. The kinetic IDE equilibrium is used to define the simulation domain and boundaries. The Kadomtsev model⁹⁰ is used to ensure that the effects of sawteeth in the current density—and hence in the Ohmic power—are taken into consideration. Measured Z_eff and bolometry radiation radial profiles are additional inputs. Figure 9 shows error bars in heat fluxes. These were obtained by running 25 separate TRANSP simulations varying input uncertainties within $± 2 σ$ based on the specific uncertainties of all input profiles: n_e, T_e, T_i, P_rad, and Z_eff. The input uncertainties were Gaussian-distributed and featured a mean around zero and a standard deviation of $1 σ$⁠. At $ρ Tor = 0.73$⁠, the standard deviation of the resulting heat fluxes was 18% and 23% for the electron and ion heat flux, respectively.

The mean electron heat flux is larger in deuterium than hydrogen by about ∼20%, but it lies well inside error bars in the ROI (⁠ $ρ Tor = 0.65 − 0.8$⁠). The opposite is seen in the ion heat flux where Q_i in H is larger than D by about ∼60%. This difference is larger than the difference in electron heat flux, but it is still inside 1σ error bars in the ROI. However, for most of the deep core $ρ Tor < 0.5$⁠, the ion heat flux is larger in H than D outside of error bars.

Inside the ROI, heat diffusivities are very close between both isotopes. Electron heat diffusivities are about $χ e ≈ 1.5 − 2$ m² s⁻¹, very similar for both ions. Ion heat diffusivities are larger in hydrogen than deuterium but well inside error bars between $χ i = 2 − 3$ m² s⁻¹. Smaller profile error bars are required to arrive at relevant empirical evidence of any isotope effects in heat diffusivities in the ROI. It appears, nonetheless, that heat diffusivity hints at a Bohm-like scaling of transport, which could be due to $ρ i *$ effects as suggested by Idomura.³⁹ Simulations⁹¹ showed that $ρ i *$ effects were important from $1 / ρ i *$ 150 to 600, which include the $1 / ρ i * ∼ 550$ range of the present experiments.

The trapped gyro-Landau fluid (TGLF) code has been used to model and describe the turbulent energy transport of the discharges above. TGLF uses the trapped gyro-Landau-fluid equations⁹² to provide an accurate approximation to the linear eigenmodes of tokamak drift-wave instabilities. The resulting eigenmodes are then used to compute quasi-linear energy fluxes. The quasi-linear approach uses a saturated turbulence intensity model based on a large set of non-linear gyrokinetic turbulence simulations.⁹³ The latest and most accurate non-linear saturation rule, SAT2,⁹⁴ has been employed in this study. This new saturation rule includes the full 3D dependence (poloidal angle, radial, and poloidal wavenumber) of the saturated electrostatic potential fluctuation intensity. It has been verified and calibrated against a database of CGYRO turbulence simulations.⁹⁵ 

TGLF uses the spectral shift method⁹⁶ to account for E × B shear suppression of turbulence. It calculates the E × B shear rate⁹⁷ from TRANSP estimates of the radial electric field E_r invoking radial force balance, neoclassical estimates of poloidal rotation, and experimental toroidal rotation input.

Figure 12 shows the dominant mode linear growth rates and real frequency spectra obtained from TGLF using nominal profile inputs. Nominal refers to simulations that use the original profile fit gradients and Z_eff values from experiment. These plots are normalized by $c s / a$⁠, where c_s is the electron sound speed: $c s = T e / m D$ and a is the plasma minor radius. Discharge-specific temperature profile and a constant deuterium ion mass are used in this normalization to meaningfully compare growth rates. Figure 12 shows that both shots feature unstable ion-temperature gradient (ITG) modes propagating in the ion-diamagnetic direction at long wavelengths $k ⊥ ρ s < 1$⁠. Growth rates are larger at $ρ Tor = 0.76$ than at 0.66 in both isotopes as can be expected from mixing length arguments given the increased profile gradients with increasing radius. Furthermore, dominant-mode linear growth rates under $k ⊥ ρ s < 0.2$ appear larger in hydrogen. Since these plasmas are ITG dominated, this increase in H over D could be due to the higher ion-temperature gradient in the ROI $ρ Tor = 0.65 − 0.8$ as shown in Fig. 3(c).

Figure 13(a) shows TGLF's raw temperature fluctuating intensity and weight functions for the case of the hydrogen discharge at $ρ Tor = 0.75$ using nominal input gradients. Linear interpolation is used to increase the k_⊥ resolution of the wavenumber integral. There are cases, as shown in Fig. 13(a), where the fluctuation amplitude at the first k_⊥ is non-zero. Since TGLF does not provide a solution at k_⊥ = 0, the first k_⊥'s A² value is assigned for $A 2 ($ $k ⊥ = 0 )$⁠. This choice is based on GENE simulations where the temperature fluctuation k-spectra showed a non-zero amplitude at $k ⊥ = 0$ due to zonal (radial) components excited via nonlinear coupling.

Before comparing TGLF temperature fluctuation measurements with experimental CECE measurements, two additional corrections were required. First, the magnetic reconstruction was used to find the local magnetic field pitch angle at the measurement locations, which corresponded to $∼ 11 °$⁠. The cosine of this angle was taken to account for the projection of the vertical antenna pattern in the k_⊥ space defined by TGLF. Second, as shown inRef. 99, temperature fluctuations may have quite anisotropic components parallel and perpendicular to the main field. The TGLF fluctuation intensity $A tot 2 ( k y )$ accounts for the total fluctuation level, while CECE measures predominantly the perpendicular component⁷³ (see Sec. IV A). Since exclusively perpendicular temperature fluctuations could not be extracted from TGLF, a factor of 30% scaling factor has been applied in this study $d T e , ⊥ ≈ 1.3 d T e , Total$ based on GENE simulations⁹⁹ at $ρ Tor =$ 0.75 as shown in Sec. V C.

TGLF also calculates the quasi-linear cross-phase angle between density and temperature fluctuations in wavenumber space, which can be compared with experimental measurements. The Gaussian weight function developed above can be used in a simple weighted average to extract a single-valued synthetic TGLF α_nT estimate. This approach ignores the spatial properties of the reflectometer antenna pattern and takes exclusively the sampling volume of the ECE radiometer. This is justified on the basis that the spatial volume common to both diagnostics is limited by the smallest footprint between the two.¹⁰⁰ Since both use the same line of sight, the relevant volume is limited by the ECE radiometer antenna pattern, which is the smallest of the two because of its higher microwave receiver frequencies. Figure 13(b) shows a sample nT k-spectrum and the weight function. Inside the relevant wavenumber range covered by the Gaussian weight function, the nT cross-phase is found negative. It disagrees with the positive sign of the experimental cross-phase measurements between the reflectometer amplitude and the ECE radiometer as presented in Sec. IV A. However, the absolute value of the nT cross-phase angle seems to be close to experiments $∼ 100 ° − 150 °$⁠.

Prior to comparing turbulence simulations with experiment, a “base” case is sought in order to ensure that the heat fluxes are in agreement with experiment and the turbulence state represents experimental conditions. Thanks to the speed of TGLF simulations and the availability of local fluctuation data, the Validation via Iterative Training of Active Learning Surrogates (VITALS)¹⁰¹ framework was used. VITALS exploits surrogate-based strategies using Gaussian processes and sequential parameter updates to achieve the combination of plasma parameters that best matches experimental transport measurements within diagnostic error bars. In this paper, the VITALS framework is used to vary the input uncertainties of logarithmic gradients (⁠ $L n e , L T e , L T i$⁠), E × B shear rate (⁠ $ω E × B$⁠), and Z_eff in an attempt to find the best match to experimental output electron and ion heat fluxes as well as turbulence properties: CECE temperature fluctuation levels and nT cross-phases.

Figure 14 shows both nominal and VITALS results and how much they disagree in percentage with the experimental values. Nominal results provide a look into how discrepancies in heat fluxes are linked to discrepancies in turbulence properties and serve as a useful contrast with VITALS “base-case” results.

The first striking feature of Fig. 14 is that ion heat fluxes (Q_i) are strongly overestimated; factors of $× 4 − 7$ are observed in nominal simulations, while factors of $× 2 − 4$ are seen in VITALS simulations. While VITALS clearly brings Q_i toward agreement, the discrepancies are maintained well outside $2 σ$ error bars. In contrast, electron heat flux (Q_e) is much better estimated. Nominal simulations also overestimate Q_e but only by factors between $× 1.5 − 2$⁠. VITALS successfully brings Q_e to within $1 σ$ error-bar agreement, with the only exception of hydrogen at $ρ Tor = 0.75$⁠. Temperature fluctuation levels (⁠ $d T e / T e$ where the subscript $⊥$ denoting perpendicular to the field has been dropped for convenience) as well as the absolute values of density-temperature cross-phase angles (⁠ $| α n T |$⁠) are found in relative good agreement in nominal simulations already. VITALS brings both quantities toward closer agreement with experiment and agreement within $1 σ$ is found except for $| α n T |$ at $ρ Tor = 0.75$ in deuterium. TGLF does recover successfully the experimental pattern observed in Fig. 6, namely, larger $d T e / T e$ in deuterium than hydrogen. $| α n T |$ is found in agreement within 2.5  $σ$ error in most cases, but does not change significantly between isotopes and thus does not recover the pattern observed in Fig. 8.

In summary, varying input gradients, v_E×B shear, and Z_eff within 1σ errors allows for a good match of electron heat flux Q_e and $d T e / T e$⁠. While the α_nT cross-phase angle is negative in all simulations, its absolute value seems to be in the same ball-park as experimental measurements. However, $| α n T |$ in the simulations fails to change significantly with isotope mass. Ion heat fluxes are found to be consistently overpredicted in both isotopes, particularly strongly in the case of hydrogen. These results encourage a revision of the quasi-linear saturation rules in different isotopes and/or a reconsideration of experimental T_i radial profiles and LT_i gradients in both isotopes. Section VI B below attempts to quantify the agreement between experiments and TGLF simulations.

The Eulerian nonlinear gyrokinetic code GENE²² has been used to further understand the turbulent transport in this pair of discharges and to compare turbulence measurements with state-of-the-art gyrokinetic modeling.^99,102

These simulations include electromagnetic effects and collisions modeled with a linearized Landau operator and employ a field-line tracing algorithm to obtain the flux surface shape and metric from the kinetic IDE equilibrium reconstruction. Furthermore, external E × B shear flow and parallel flow shear effects are retained in nonlinear simulations. By default, the main ions and electrons are considered as active gyrokinetic species with the actual mass ratios. The $Z eff$ value is either considered in two-species simulations as a corresponding rescaling of the ion-electron collisionality or used to derive the density of a third gyrokinetic species by assuming fully ionized boron impurities with density gradients, temperature, and its gradient taken from the main ion species (with density of the latter accordingly reduced). A summary of the physics inputs is shown in Table VIII in  Appendix B.

First, the GENE linear solver was used to hint at the most important micro-instability dominating turbulent transport in these discharges. Figure 15 shows the linear growth rate of nominal-input case scenarios in both hydrogen and deuterium. Scans of input gradients and Z_eff (inside 1σ) are included to hint at the possibility of mixed turbulence drives inside experimental input error bars. In the case of hydrogen, regardless of input changes to gradients and/or Z_eff, all linear growth rates at low-k propagate in the ion-diamagnetic direction. In the case of deuterium, however, there are cases where an increased electron temperature gradient and/or density gradient can turn the lowest part of the spectrum into a dominant electron mode, very likely trapped electron mode (TEM) at such low-k. Additionally, contrasting hydrogen and deuterium growth rates in Fig. 15 shows that, inside ion-scales, the linear growth rates in hydrogen are larger than in deuterium. This is similar to what was observed in TGLF growth rates in Fig. 12(a). Again, the larger ion-temperature gradient in the hydrogen discharge could be contributing to stronger ITG growth rates.

Non-linear gyrokinetic simulations have also been performed at $ρ Tor = 0.75$⁠. These are ion-scale, local (flux-tube) simulations. A synthetic diagnostic has been employed to quantitatively compare non-linear GENE simulation outputs with experimental data. GENE's field-aligned fluctuating quantities can be mapped into a 2D space in lab-coordinates where experimental sampling volumes can be directly applied to mimic experimental measurements with great fidelity. Since the CECE diagnostic would predominantly measure the perpendicular component of temperature fluctuations due to its antenna pattern, the perpendicular component of temperature fluctuations is extracted from GENE simulations⁹⁹ to compare with experiment. A “point spread function” (PSF) after Holland et al.¹⁰² is used to model the effect of finite diagnostic sampling volume on the measured turbulence properties.⁷³ The PSF consists of a 2D Gaussian with anisotropic radial and vertical dimensions. Using a forward model for ECE radiation transport (the ECRad code⁶⁴), the radial 1/e amplitude of the birth-place distribution of observed intensity (Ref. 65, p. 21) was found to be 5.5 mm at $ρ Tor = 0.75$⁠. The vertical 1/e-amplitude of the electric field beam radius (w) was obtained using complex-beam ray-tracing through the optics and TORBEAM inside the plasmas as outlined in Sec. IV B. Just as done in the TGLF synthetic diagnostic above, since the CECE measures ECE power, the power beam radius $w Pow = w / 2$ is used in the GENE PSF. The GENE synthetic diagnostic has been implemented in Python²⁴ and has been numerically verified to provide identical results to previous implementations in IDL.^99,102

Figure 16 compares perpendicular temperature fluctuation frequency spectra from GENE and experiment in both hydrogen and deuterium plasmas. Both figures feature “nominal” and “best” synthetic GENE traces, which stand for simulations performed with the original input gradients and those where the best overall match of heat flux and turbulence quantities has been found. The best simulations are the result of an empirical 15-simulation scan of input gradients and Z_eff within $1 σ$ experimental errors. These best match simulations were also found when using a third gyrokinetic species to model Z_eff in both hydrogen and deuterium simulations. The large computation time required for convergence of these non-linear simulations did not allow a thorough scan of input parameters and/or the use of a VITALS-like tool as in the case of TGLF (see Sec. V B). However, the TGLF-VITALS results informed the gradient quantities to be modified toward a better agreement with experiment.

Note that the y-axis of the plots in Fig. 16 are not in arbitrary units but in physical relative fluctuation amplitude squared $( d T e / T e ) 2$ per kHz; neither the experimental data nor the simulation's synthetic diagnostic output have been scaled relative to each other. The physically correct scaling is obtained using the cross-power spectral density (CPSD) algorithm found (Ref. 63, pp. 301–327). Hann windows are used for spectral leakage and set to 50% overlap for frequency-domain averaging. In order to reproduce the experimental conditions as accurately as possible, time traces from two separate channels featuring the diagnostic line-of-sight and inter-channel separation are used in the cross-power calculation. The synthetic CPSD was initially found to feature many oscillations due to the short time periods of 2D turbulence data from the GENE simulation, lasting only about $1 − 2$ ms. In order to reduce this scatter, PSF time-traces from 5 pairs of volumes placed before and after the simulation point were used to arrive at the mean cross-power trace in Fig. 16. Since flux-tube non-linear GENE simulations were run using local profile information at $ρ T = 0.75$⁠, the turbulence conditions in the neighboring channels are believed to be identical. The error bars in the synthetic spectra are the standard deviation of the mean.

The experimental $d T e / T e$ frequency spectrum from CECE data is taken from the integrand in Eq. (1) (see Sec. IV A). This is because the CECE radiometer is not absolutely calibrated and because crosstalk can occur between filters, requiring the active subtraction of the background coherence. Nonetheless, the same CPSD algorithm is used to compute experimental and synthetic GENE frequency spectra. Equation (2) is used for the experimental error bars, which appear relatively small in Fig. 16 due to the large number of data points acquired during a 2.5-s flat-top phase at 4 MHz sampling.

Figure 16(a) shows GENE synthetic $d T e / T e$ spectra in comparison with experimental CECE results in deuterium. Figure 16(a) contrasts the effects of using different models to estimate the vertical antenna pattern. Using complex-beam ray-tracing leads to significantly smaller (⁠ $× 2 − 3$⁠) vertical beam radii when compared with TORBEAM results (see Sec. IV B). Using the same GENE simulation output, the vertical sampling dimension was changed between vacuum complex ray-tracing and TORBEAM estimates. A narrower beam radius selects a larger portion of the k-space and hence leads to a larger $d T e / T e$ fluctuation level. Figure 16(a) shows that the vacuum estimate contains fluctuations in agreement with experiment inside $5 − 50$ kHz, but it also features significantly more power in higher frequencies, which is not seen in experiment. The integrated $d T e / T e$ spectrum over all frequencies leads to an overestimate of the total fluctuation level when compared with CECE data. Using the TORBEAM w_Z estimates results in a slight underestimate when compared with the experimental observations, but the integrated $d T e / T e ( f )$ can be found in better agreement with experiment, inside $1 σ$ error bars. The better agreement between TORBEAM-GENE and CECE data provides confidence that the TORBEAM beam radius is likely a better estimate of the beam radius in experiments. In all future observables presented in this paper, the TORBEAM beam-radius estimate will be used.

AUG's CXRS diagnostic measures light emitted from boron impurities; thus, E_r can be found directly from measured boron density gradients, temperatures, and toroidal rotation together with the IDE magnetic reconstruction. The poloidal rotation can be either inferred from asymmetries in toroidal rotation¹⁰⁷ or calculated from neo-classical estimates. In the case of the discharges studied here, poloidal rotation estimates were only available inside $ρ Tor <$ 0.5; thus, the code NCLASS(CAR48)¹⁰⁸ was used to estimate $v Pol ( D ) = 0.3$ and $v Pol ( H ) = 0.25$ km s⁻¹ at $ρ Tor =$ 0.75. Lebschy¹⁰⁶ showed that neoclassical poloidal rotation estimates were in agreement within error bars with poloidal rotation measurements in typical AUG L-mode plasmas inside $ρ Tor = 0.7 − 0.8$⁠. Table I shows the estimates of E_r from CXRS. The uncertainty in these estimates was calculated via uncertainty propagation taking: profile errors as shown in Figs. 2 and 3, a 20% error on the boron density profile gradient, a poloidal rotation uncertainty of σv_Pol = 0.2 km s⁻¹,¹⁰⁶ and no error in the magnetic reconstruction.

Table I shows that the estimates from CXRS and DBS (post-beam-blip) agree inside 1  $− σ$ errors in deuterium but only inside 1.5  $− σ$ in hydrogen. It also shows that taking DBS data well away (200 ms) from the NBI beam-blip, the E_r estimate can change significantly—well outside error bars. Since NBI beam-blips are known to input momentum to the plasma⁴⁹ and since CXRS measures the toroidal rotation during the beam-blip exclusively, it is safe to argue that CXRS E_r estimates are biased and likely overestimate the E × B rotation. Furthermore, since CXRS E_r estimates feature the highest amount of uncertainty, the DBS pre-beam-blip E_r estimates are believed to be the most reliable to represent the background E × B rotation.

Figures 16(b) and 16(c) show the effect of using either the CXRS or DBS pre-beam-blip E_r estimates on the shape of the synthetic CECE spectrum and how these compare with experiments. The most salient feature on both isotopes is that a larger E_r leads to a synthetic CECE spectra peaking at higher frequencies. Furthermore, these figures thus support the hypothesis proposed in Sec. IV A that differences in plasma rotation perpendicular to the CECE line-of-sight lead to changes in the $d T e / T e$ frequency spectra shape as shown in Fig. 5.

Figure 16(b) shows that the simulation using CXRS's E_r estimate leads to a broad spectrum featuring power in frequencies above 50 kHz, which is not found in experiments. On the other hand, the “GENE-best” simulations in Fig. 16(b) featuring the DBS pre-blip E_r features fluctuations peaking in a frequency range about 20 kHz, underneath that of experiments. In the case of hydrogen, shown in Fig. 16(c), the CXRS E_r simulations show a much broader spectrum than experiments featuring significant amounts of power in frequencies above 30 kHz. The DBS pre-blip E_r estimates bring the synthetic $d T e / T e$ spectrum into much better agreement with experiments.

It must be noted that only the local value of E_r was changed in these flux-tube simulations at $ρ Tor =$ 0.75. Changing only the E_r amplitude had no measurable impact (beyond random error) on heat fluxes nor turbulence observable quantities—such as $d T e / T e$⁠. The E × B shearing rate is calculated from Eq. (7) using the CXRS input v_Tor, ignoring the first two terms in Eq. (7) as is done by default in GENE core simulations. CXRS and DBS E_r simulations in Figs. 16(b) and 16(c) only explore the changes of E_r amplitude on the synthetic CECE diagnostic spectrum. Future work could attempt to calculate radial profiles of fluctuation phase velocity and measure radial profiles of poloidal rotation across the ROI. These would allow to consistently explore the effects of both E_r amplitude and its shear on turbulent energy transport.

Temperature fluctuation-level estimates from GENE and experiment are also shown in Fig. 16. The GENE-best estimates reproduce the experimental observation that $d T e / T e$ fluctuation levels are smaller in hydrogen than deuterium. The ratio $d T e / T e$(H) $/ d T e / T e$(D) from the GENE-best estimates is 0.77 ± 0.16 compared to the CECE estimate of 0.76 ± 0.14. GENE-estimated $d T e / T e$ fluctuation levels, thus, agree with a gyro-Bohm scaling picture suggested in Sec. IV A.

Figure 17 shows the synthetic GENE estimates of the radial correlation length compared with experiment. The synthetic radial correlation length from GENE is determined by calculating the normalized cross-correlation coefficient $ρ x y , Δ r$ between one fixed channel and neighbor channels with increasing separation.

Since the synthetic GENE PSF time traces do not contain any background radiometer black-body noise and provide $d T e / T e$ traces, $ρ x y , Δ r$ can be computed directly using cross-correlation functions at zero time-delay, namely, $ρ x y , Δ r = R x y ( τ = 0 ) / R x x ( τ = 0 ) R y y ( τ = 0 )$⁠. The roundabout FFT approach is used to compute these, following (Ref. 63, pp. 328–333). Given the limited duration of the time traces, averaging is performed to minimize the effect of numerical uncertainty on the final fit. This averaging consists of calculating $ρ x y , Δ r$ for six different inter-channel separations both increasing and decreasing in major radius across the line-of-sight. This process is performed for a central channel aligned with the measurement location at $ρ Tor = 0.75$ and 3 central points both above and below in major radius. The uncertainty is found from the standard deviation of the mean at each inter-channel separation. Experimental CECE $ρ x y , Δ r$ s are found using Eq. (3) as derived in  Appendix A. This equation allows an uncalibrated radiometer to access physically equivalent $ρ x y , Δ r$ s. Experimental error bars come from applying the error propagation formula on Eq. (3).

The fitting technique and the estimate of fit errors (confidence intervals) are central to a proper quantitative comparison between experiment and synthetic GENE results. CECE experimental $ρ x y , Δ r$ can be fit by a Gaussian function. The reduced chi-square statistic (⁠ $χ 2$⁠) (Ref. 109, pp. 63–65) is used to quantify the goodness of fit. CECE Gaussian fits in Fig. 17 feature $χ 2 = 0.96$ and $χ 2 = 0.4$ for D and H, respectively. Since both of these $χ 2$ s are close to the ideal 1.0, it can be ascertained that the Gaussian function matches the experimental data in accordance with the experimental uncertainty. On the other hand, the synthetic GENE $ρ x y , Δ r$ s are found to decrease less sharply over inter-channel distance and a Gaussian function leads to $χ 2 ≫ 1$⁠. Instead, a Lorentzian function [ $F L ( x ) = γ A o / ( x 2 + γ 2 )$] is used in these cases, which leads to $χ 2 ∼ 0.7 − 1.7$⁠. Evaluating the fit confidence interval is a subtle matter prone to underprediction when using least-square optimization. To ensure numerically accurate errors on the fit parameters are obtained, the Bootstrap method was used.¹¹⁰ It uses a Monte Carlo algorithm, which prepares a large number of different input data sets (∼400) where each data point features normal-distributed random amounts of error inside experimental uncertainty. The least-square fit algorithm is then run independently on these data sets. The sample mean of the bootstrapped parameter estimates is taken as the final fit, and the standard deviation of the output data set as the error bars—plotted in Fig. 17.

One salient, puzzling, feature of Fig. 17 is that the synthetic GENE best heat-flux match $ρ x y , Δ r$ decays just like the CECE experimental data with a Gaussian function, but the hydrogen GENE best heat-flux match $ρ x y , Δ r$ features “tails” best fit by Lorentzian instead of a Gaussian function. The reason behind this discrepancy is not clear. Furthermore, half width at half maximum (HWHM) estimates in deuterium seem to agree between GENE and experiment. However, hydrogen GENE $L r , c$ overestimates experimental findings. Moreover, Fig. 17 also shows that the synthetic GENE correlation length is larger in D than H, reproducing the experimental finding that $L r , c$(D) $> L r , c$(H). The CECE ratio $L r , c$(H) $/ L r , c$(D) $= 0.68 ± 0.09$ agrees roughly inside 2σ error against the GENE ratio $L r , c$(H) $/ L r , c$(D) $= 0.81 ± 0.09$⁠. A gyro-Bohm scaling of $L r , c ∝ 1 / m$ is thus reproduced by GENE, within error.

Finally, Fig. 18 shows a comparison between the experimental and synthetic-GENE nT cross-phase angles. The GENE cross-phase angle is obtained via Eq. (4) where the kernels for density and temperature fluctuations are the fluctuating density and perpendicular temperature traces from the 2D GENE synthetic diagnostic. A single PSF is applied to both density and temperature fluctuation maps: identical physical locations and PSF functions are used for both dn_e and dT_e maps. This is justified in the basis that the reflectometer airy width and vertical beam radius are both expected to be larger than their respective ECE emission sizes, and thus, the smaller volume of the two is taken.⁷⁹ In order to reduce random noise error due to the short duration of synthetic GENE time-traces, the synthetic cross-phase corresponds to the mean from 5 volumes placed before and after the measurement location at $ρ Tor = 0.75$ along the experimental line-of-sight. The error bars in the synthetic spectra are the standard deviation of the mean. Figure 18 shows that GENE, in agreement with TGLF results presented above, also predicts a negative nT cross-phase angle. This is found in contrast with the experimental cross-phase angle between the reflectometer's amplitude and ECE signals, which was found positive (see Sec. IV A).

Figure 18 shows that—in the case of both isotopes—if the experimental α_nT undergoes a sign reversal, the cross-phase angle is found relatively close with GENE synthetic estimates. In the case of hydrogen, the negative of the experimental cross-phase angle agrees within error bars. In deuterium, the angle differs by $∼ 30 °$⁠, yet agrees inside $2 σ$ error bars. In short, while the nT cross-phase angle of GENE synthetic diagnostics and experimental results does not agree in sign, their amplitude $| α n T |$ is found to be in agreement. Another interesting result found in Fig. 18 is that the synthetic GENE cross-phase angle is higher in amplitude in H than D, recovering the experimental trend observed. The higher amplitude and more out-of-phase cross-phase angle in hydrogen is consistent with the stronger, unambiguous, ITG drive observed in the linear GENE simulations in Fig. 15.

The effect of toroidal rotation on the synthetic nT cross-phase was found to be very important. Reversing the input rotation direction lead to a sign reversal in the nT cross-phase, while the magnitude was left unchanged. The negative angle reported here results from a careful transformation of CXRS toroidal rotation profiles (see Fig. 2) into GENE coordinates.

Figure 19 brings all these observations together and compares inputs and outputs of nominal and “best-match” simulations of both isotopes. These figures use percentages to highlight how much each quantity was varied in the input and how the non-linear GENE simulations responded to these inputs. $1 σ$ experimental uncertainty in both inputs and outputs is also shown. Two main observations can be made. First, nominal gradient inputs lead to a strong overestimation of the ion heat flux (Q_i) for both isotopes, specially for hydrogen with a $× 3$ overestimate. This can be expected from the hypersensitivity of non-linear simulations to input gradients.¹¹¹ The overestimation is not as strong as that seen in the TGLF results in Fig. 14. It is surprising to see that the turbulent properties do not react significantly to the strong Q_i overestimate, specially the nT cross-phase angle, which is supposed to address directly the turbulence mix. Varying input gradients and Z_eff within uncertainty, the best-heat flux match simulations succeed in bringing heat fluxes down to within 1σ error, yet a significant change toward agreement is not observed in the turbulence properties tracked in this study (see nT cross-phase in deuterium, for instance). Second, the electron heat flux is well matched by nominal deuterium simulations. Turbulence properties of the nominal deuterium simulation are then also found within 1.5σ error of experiment, which supports the fact that electron heat transport has been well modeled by nominal input parameters. This is not the case for the hydrogen nominal simulations, which show an overestimate of Q_e, $d T e / T e$⁠, and $L r , c$⁠. Best heat-flux match hydrogen simulations succeed in bringing the electron heat flux down and also bring $d T e / T e$ into 1σ agreement with experiment. However, the $L r , c$ is seen to increase further away from experiment, while the $| α n , T |$ changes by a negligible amount. This observation highlights the importance of having many experimental observables in experiment-modeling comparisons since agreement with one quantity can prove fortuitous. It also introduces the need for a composite validation metric, which quantifies the agreement between experiment and simulation across multiple observables, as discussed in Sec. VI below.

The hierarchy position of each observable is now addressed. The temperature fluctuation level $d T e / T e$ is considered to be first in the primacy hierarchy since it comes directly from the integration of the cross-coherence of two nearby radiometer channels and does not require any other diagnostic for evaluation. The radial correlation length $L r , c$ of temperature fluctuations is also considered first in the primacy hierarchy since it comes from a single diagnostic, and it is a fundamental property of the turbulence. The nT cross-phase angle α_nT is considered second in the hierarchy since it uses inputs from two different diagnostics and compares the properties of distinct fluctuating quantities. Finally, both ion and electron heat fluxes are considered third in the hierarchy because power-balance calculations take into account several experimental inputs as well as external models to arrive at the heat fluxes. In the case of TRANSP, density/temperature profiles require data from different diagnostics and a magnetic reconstruction, which also uses magnetic probe experimental data. Profiles are thus at least second in the hierarchy. Additionally, TRANSP makes use of external models for ECRH absorption, Ohmic power deposition, and sawtooth behavior, which rely on second-order profiles and push the power balance calculation further away from being a fundamental measurement.

The measured cross-phase angle between the reflectometer's amplitude and the ECE signal was found to feature an opposite sign when compared with the synthetic density-temperature cross-phase estimates from both TGLF and GENE. However, the magnitudes of these cross-phase angles (see Figs. 8, 13, and 18) seem to be in fair agreement between simulations and experiment. It has been suggested that there could be a phase difference between the reflectometer-amplitude signal and density fluctuations.⁸⁷ Without careful full-wave reflectometer modeling, which is beyond the scope of this publication, it is not clear whether the measured cross-phase angle between the reflectometer's amplitude and the ECE signal is truly an experimental observable for the nT cross-phase. Nevertheless, in order to perform an impartial validation effort, the Ricci parameter framework will be applied here to all observables first ignoring the nT cross-phase, second including the nT cross-phase regardless of the sign disparity, and finally using only the magnitude of the nT cross-phase as an observable.

Central to any rigorous validation effort is the careful evaluation of the sources of potential uncertainty in both experimental and simulation fronts. Experimental uncertainty is generally well described since it is intrinsic to rigorous empirical measurement efforts. Uncertainties for the different observables have been discussed in detail in the experimental results (Sec. IV A) and power-balance (Sec. V A) . On the other hand, simulation uncertainty is rarely addressed because of the difficulties quantifying it; however, it is just as important as experimental uncertainty to the quantification of experiment-simulation agreement. Beyond fundamental limitations inherent to any model, Ricci et al.¹¹³ mentioned potential sources of simulation uncertainty: limited numerical accuracy, finite time series (statistical uncertainty), and imprecise input parameters. The latter includes inputs to the simulation such as uncertainty in profiles, gradients, and Z_eff, as well as imprecise details on the synthetic diagnostic interpretation such as the sampling beam radius.

This section proceeds to use the Ricci parameter framework to compare experimental results with those obtained from TGLF as presented in Sec. V B. Four separate quantities will be used to validate TGLF: temperature fluctuation levels $d T e / T e$⁠, nT cross-phase angle α_nT, electron (Q_e), and ion (Q_i) heat fluxes.

Simulation uncertainty estimates of the output observables taken from TGLF simulations will be discussed now. Since TGLF calculates a unique solution to the turbulence properties and heat fluxes based on a given set of inputs, errors due to finite time series can be neglected. Errors due to limited numerical accuracy can affect the $d T e / T e$ and α_nT synthetic diagnostic estimate. The default TGLF k_⊥ resolution is 1.02 cm⁻¹ [see symbols in Fig. 13(a)]. In this study, the default k_⊥ grid was increased to 0.05 cm⁻¹ to define the diagnostic Gaussian function weight and linear interpolation was used to increase the k_⊥ resolution of the temperature fluctuation intensity function. If a stepwise interpolation was used instead, $d T e / T e$ estimates could change up to 10%. The interpolation method was tested to have negligible effects in the α_nT observable.

Another important source of uncertainty, overlooked by previous validation studies of this kind, is the sensitivity of TGLF to imprecise input parameters. Taking a similar approach as taken above to estimate TRANSP heat flux uncertainty (see Sec. V A), 30 independent simulations were performed where input gradients and Z_eff were varied to observe the sensitivity of TGLF outputs to input experimental uncertainty. Gradient and Z_eff input values were varied within the following 1σ experimental uncertainty (see Sec. III): $a / L n e$ by ±20%, $a / L T e$ by 30%, $a / L T i$ by 10%, and Z_eff by 20%. In order to get an average output error estimate from random input error, each of these input uncertainties was varied randomly (Gaussian-distributed) in 30 separate simulations with a mean variation of zero and a standard deviation of 1σ on each input. The results of this sensitivity scan yielded errors of $Δ d T e / T e / d T e / T e = 38 % , Δ α n T / α n T = 7 %$⁠, $Δ Q e / Q e = 33 % , Δ Q i / Q i = 44 %$⁠.

The synthetic diagnostic applied to TGLF outputs to extract $d T e / T e$ and α_nT is also affected by uncertainty in the sampling beam radius (effective antenna pattern at measurement location). Since the position of the launching antenna, quasi-optical lenses, and vacuum window is known with an accuracy of 5 mm, 40 different ray-tracing simulations were performed varying these positions arbitrary inside 2σ. Since the measurement locations were close to the beam waist, the standard deviation of changes to the beam radius was only 0.3 mm, while the average was observed to increase by 0.2 mm. Thus, a conservative error bar of 0.5 mm was applied to the beam radius in the synthetic diagnostic. This beam radius error leads to an average error of 4% in $d T e / T e$ and 3% in α_nT. This calculation ignores changes to the beam radius caused by inaccuracies or inhomogeneities in the refractive index of focusing elements, and it is left for future work to quantify this source of uncertainty. Another potential source of uncertainty in the estimation of the beam radius could be scattering of the beam due to turbulence in the scrape-off layer (SOL) and the confined plasma near the separatrix, see Ref. 114.

Tables II–IV show the Ricci validation tables for both isotopes at both locations studied here. These quantify the experiment-simulation agreement shown in Fig. 14. Both d_o and λ are equal to 1.0. These have been chosen to allow for a quick growth of R values toward 1.0 and observe a significant variation in χ_R between nominal and VITALS simulations; the fundamental results are not changed by this choice.¹¹³