An extensive validation effort performed for a modest-beta NSTX NBI-heated H-mode discharge predicts that electron thermal transport can be entirely explained by electron-scale turbulence fluctuations driven by the electron temperature gradient mode (ETG), both in conditions of strong and weak ETG turbulence drive. Thermal power-balance estimates computed by TRANSP as well as the shape of the high-k density fluctuation wavenumber spectrum and the fluctuation level ratio between strongly driven and weakly driven ETG-turbulence conditions can be matched by nonlinear gyrokinetic simulations and a synthetic diagnostic for high-k scattering. Linear gyrokinetic simulations suggest that the ion-scale instability in the weak ETG condition is close to the critical threshold for the kinetic ballooning mode instability, and nonlinear ion-scale gyrokinetic simulations show that turbulence might be in a state reminiscent of a Dimits' shift regime, opening speculation on the role that ion-scale turbulence might play for the weak ETG condition. A simulation that matched all experimental constraints is chosen to project high-k turbulence spectra in NSTX-U, revealing that the new high-k system [R. Barchfeld et al., Rev. Sci. Instrum. 89, 10C114 (2018)] should be sensitive to density fluctuations from radially elongated streamer structures. Two schemes are designed to characterize the radial and poloidal wavenumber dependence of the density fluctuation wavenumber power spectrum around the streamer peak, suggesting future high-k fluctuation measurements could be sensitive to an asymmetry in the kr spectrum introduced due to the presence of strong background flow shear.

Micro-turbulence is believed to be responsible for the anomalous transport of particles, energy, and momentum across the confining magnetic field in fusion plasma experiments. In the core of conventional tokamaks, micro-instabilities can be driven at ion gyro-radius scales (kρs<1), such as the ion temperature gradient mode (ITG1) and trapped electron mode (TEM2), and at electron scales (kρs>1) such as the electron temperature gradient (ETG) mode.3–6 These micro-instabilities can ultimately develop into turbulence and drive so-called anomalous transport that can largely exceed predictions from neoclassical collisional theory.7 Here k is the wavenumber of the turbulence perpendicular to the confining magnetic field, and ρs=cs/Ωc,i is the ion-sound gyro radius evaluated at the ion sound speed cs=Te/mi, and Ωc,i=eB/mi is the ion gyro-frequency.

Analysis of micro-stabilities in the core of spherical tokamak (ST) plasmas, mainly NSTX8 and MAST,9 has shown that a number of additional instabilities to the well-known ITG/TEM/ETG can become unstable and drive anomalous electron thermal transport. These can be represented in regime plots showing the dominant linear instability depending on the parametric regime of operation. In Figs. 1(a)–1(c), the regime plots schematically show the dominant linear instability present in various NSTX H-modes at r/a0.60.7 using the gyrokinetic code GYRO,10 vs local parameters (Fig. 1 is courtesy of Guttenfelder11). The traditional electrostatic ITG/TEM/ETG instabilities are most unstable at rather low values of electron beta βe. In conditions when the ion-scale instability is suppressed, ETG can now be the main driver of anomalous electron thermal transport at low βe and normalized electron temperature a/LTe exceeding the critical temperature gradient a/LTe,crit,12 as will be supported in this article. The micro-tearing mode (MT13–15) can become unstable at high collisionality νe/i and high βe, while the kinetic ballooning mode (KBM16) is predicted to be unstable at lower collisionality and high βe and high αMHD (MHD α parameter). The reader is referred to Ref. 11 for precise definitions and normalizations.

FIG. 1.

Regime plots showing dominant instability from local GYRO linear simulations corresponding to various NSTX H-modes at r/a0.60.7, vs local parameters (courtesy of Guttenfelder8). (a) βe vs νe/i. (b) βe·a/LTe vs αMHD,unit. (c) a/LTea/LTe,crit vs γE. Figure 1 from Ref. 11 has been updated with results from NSTX H-mode 141 767 studied in this article, indicated by the blue and green stars (conditions of strong and weak ETG turbulence drive, respectively). The definition of βe uses the vacuum BT0 value at the magnetic axis while and αMHD,unit use the convention Bunit in the GYRO code.10,24 Adapted with permission from Guttenfelder et al., Nucl. Fusion 53, 093022 (2013). Copyright 2013 IOP Publishing.

FIG. 1.

Regime plots showing dominant instability from local GYRO linear simulations corresponding to various NSTX H-modes at r/a0.60.7, vs local parameters (courtesy of Guttenfelder8). (a) βe vs νe/i. (b) βe·a/LTe vs αMHD,unit. (c) a/LTea/LTe,crit vs γE. Figure 1 from Ref. 11 has been updated with results from NSTX H-mode 141 767 studied in this article, indicated by the blue and green stars (conditions of strong and weak ETG turbulence drive, respectively). The definition of βe uses the vacuum BT0 value at the magnetic axis while and αMHD,unit use the convention Bunit in the GYRO code.10,24 Adapted with permission from Guttenfelder et al., Nucl. Fusion 53, 093022 (2013). Copyright 2013 IOP Publishing.

Close modal

The present article focuses on NSTX H-mode 141767, which has been extensively analyzed in Refs. 17–20. A total of 2 MW of NBI heating was used to heat the plasma in the time of interest (no RF heating), producing toroidal rotation levels of characteristic Mach number M0.2. Line-integrated density ne6×1015 cm−2 and toroidal magnetic field BT0.5 T on the magnetic axis, while MHD activity was quite low (total toroidal beta βTTOT5%6%). The plasma current was purposely ramped down from Ip1.10.9 MA, separating strongly driven and weakly driven ETG turbulence regimes at r/a0.7 as measured via high-k scattering.21 The experimental profile values at r/a0.7 are mapped into regime plots in Fig. 1 for both the strong and weak ETG conditions (blue and green stars), suggesting that the KBM and the micro-tearing mode are expected to be linearly stable for the strong ETG condition (blue), while the KBM could be close to marginality in the weak ETG condition (green). A discussion on the potential role of the ion-scale instability and turbulence is given in III. Importantly, the electron temperature gradient mode (ETG) is expected to be unstable in both conditions, motivating a detailed study of ETG-driven transport.

The rest of the article proceeds as follows: Sec. II describes the experimental evidence, gyrokinetic simulations, and a succinct review of the comparisons in Ref. 18 between the experimental values of electron thermal transport and measurements of density fluctuations vs the predicted ETG-driven transport and synthetic density fluctuation spectra. Section III discusses the role that the ion-scale instability and turbulence might play in the weak ETG condition. Section IV contains new projections for future high-k measurements in NSTX-U using nonlinear gyrokinetic simulation and a synthetic diagnostic for the new high-k scattering system. A brief summary and conclusions are stated in Sec. V.

In this section, we review the experimental measurements, nonlinear gyrokinetic simulations, and the experiment-model comparisons of ETG fluctuations and transport from NSTX NBI heated H-mode 141767 at r/a0.7, focusing on a strongly driven and a weakly driven ETG turbulence condition. Additional details can be found in Refs. 17, 18, and 20.

Direct experimental evidence of ETG-driven turbulence was provided via density fluctuation measurements from the 280 GHz high-k scattering system on NSTX.21 The high-k scattering system had three operational channels at each time of interest, sensitive to turbulence wavenumbers in the range (kR,kZ)(1119,2.43.5) cm−1, with normalized values (kRρsexp,kZρsexp)(813,1.52.5) for the strong ETG condition. For the weak ETG condition, (kR,kZ)(1526,1.5–3) cm−1, or equivalently (kRρsexp,kZρsexp)(1018,12). The quoted values for ρsexp0.7 cm use the TRANSP22 local values of Te0.4 keV and magnetic field BT0.4 T. In what follows, we will use the normalization of the ion-sound gyro radius by Bunit to perform quantitative comparisons via synthetic diagnostic for high-k scattering, which we will note ρs (here ρs0.2 cm using Bunit1.44 T). In this work, we denote kR the wavenumber component along the major radius R, and kZ is the component in the vertical direction. The difference in notation with respect to Refs. 18 and 19 is clarified on  Appendix A. These wavenumber components are computed independently for each measurement channel of the diagnostic via ray tracing calculations, following the propagation of a single ray modeling the microwave beam propagation in the plasma. The wavenumber resolution of the observed electron density fluctuations is Δk±0.7 cm−1 and a corresponding radial resolution ΔR±3 cm. A view from the top of the ray-tracing solution for the probe and scattered beams is shown in Fig. 9(a).

With respect to transport, estimates of the electron thermal power are computed from power balance calculations in TRANSP.22 A total electron thermal power of Pe1.48±0.33 MW was obtained for the strong ETG condition, and Pe1.02±0.23 MW for the weak ETG condition (the ± sign denotes the 1σ error bar computed via error propagation analysis). As discussed in detail in Refs. 17 and 18, the reduction of ETG fluctuations at the weak ETG time was correlated with decreased electron thermal power levels. For reference, ion thermal power Pi0.50.6 MW for both conditions, lying within the uncertainty range of neoclassical levels predicted by NEO.23 

The predictions taken from Fig. 1, along with the experimental electron-scale fluctuation measurements17,18 and the anomalous electron thermal power levels from TRANSP, all motivate a detailed study of electron-scale turbulence and transport. In what follows, we will use gyrokinetic simulation to predict turbulent electron thermal transport levels and to model electron-scale density fluctuations measured via high-k scattering. The latter will be enabled via a synthetic diagnostic developed as part of this work.

Local nonlinear gyrokinetic simulations were performed with the gyrokinetic code GYRO,10,24 only accurately resolving electron-scale fluctuations characteristic of ETG turbulence kρs1. Owing to the neoclassical ion thermal transport levels, ion-scale turbulence existing at kρs1 is assumed to play a negligible role. This is also consistent with linear and nonlinear ion-scale gyrokinetic simulations carried out in Ref. 18 for the present conditions. Some caveats to these assumptions are discussed in Sec. III.

Simulations were performed at the scattering location r/a0.7, resolving three gyrokinetic species (e,D,C), including electron collisions (νei1 a/cs, but not ion collisions: cf. Sec. III B), background flow and flow shear (M0.2,γE0.10.2 a/c s,γp1 a/cs), and fully electromagnetic fluctuations (δϕ,δA||,δB||). Linear background profiles were simulated employing nonperiodic boundary conditions in the radial direction with typical buffer widths Δb2–3ρs. An unusually large electron-scale simulation domain was employed [(Lr,Lθ)=(20,20.6)ρs], translating to a wavenumber grid resolving minimum nonzero wavenumbers dkθρs0.3 and dkrρs0.3. Here kθ and kr are, respectively, the poloidal and radial wavenumbers of the turbulence that are internally defined in GYRO10 and CGYRO.25 The particular choices for the radial and poloidal wavenumber resolutions are krρsϵ[0.3,40] and kθρsϵ[0.3,65–88], depending on the plasma condition [see Refs. 18 and 19 for additional details and discussion on the (kr,kθ) grid]. Parallel resolution employed 14 poloidal grid points (×2 signs of parallel velocity), 12 energies, and 12 pitch-angles (6 passing + 6 trapped). This choice of numerical grids was made according to previous convergence and accuracy tests for the GYRO code simulating micro-instabilities in the core NSTX11 and was also tested for convergence in the present conditions.18 The flux surface geometry is implemented by a Miller equilibrium.26 The numerical resolution parameters result in simulations that consume 12 M CPU hours per simulation, carried out in the leadership supercomputer Edison at NERSC.

In this subsection, we compare results from the nonlinear electron-scale gyrokinetic simulations to experimental values of electron thermal transport as well as to electron-scale turbulence measurements from high-k scattering using a synthetic diagnostic.19 Using experimental profile values as input shows that electron-scale gyrokinetic simulation corresponding to the strong ETG condition can only provide 30% of the experimental electron thermal power estimate from TRANSP (recall Peexp1.48±0.33 MW) and motivates a sensitivity study of the electron thermal power to the ETG turbulence drives. With respect to the weak ETG condition, the experimental electron thermal transport level can be matched within experimental uncertainty of the simulation drive inputs a/LTe,a/Lne (normalized electron temperature and density gradients18).

We present sensitivity scans corresponding to the strong ETG condition for the main ETG turbulence drive mechanisms (a/LTe,a/Lne,q,ŝ). Here a/LTeaTe/Te and a/Lneane/ne, q is the magnetic safety factor, and ŝ is the magnetic shear (we note a/LTe,a/Lne interchangeably by T,n in the rest of this paper). The background electron temperature and density gradients are scanned within +1σ experimental uncertainty, while q is scaled by 10% and ŝ is scanned by +20%, all in the direction of increasing the ETG drive (specific values are summarized on Table V in  Appendix C). The uncertainties in a/LTe,a/Lne were computed from uncertainties in the background electron density and temperature profiles followed by a Monte Carlo analysis approach, and the values of 10%q and +20%ŝ were taken as reasonable estimates of the equilibrium reconstruction uncertainty.

A total of five simulations corresponding to the strong ETG condition are used to compare to experiment for the following observables: electron thermal power Pe (MW), the shape of the wavenumber spectrum S(k), and the fluctuation level ratio between the strong and the weak ETG conditions SstrongETG/SweakETG. As shown in Refs. 18 and 19, these result in powerful metrics that allow a quantitative assessment of the fidelity of each simulation in the comparison to the experimental quantities. The rationale behind this work is to find simulation conditions for the strong ETG case where local nonlinear gyrokinetic simulation can simultaneously explain all the mentioned observables.

1. Electron thermal transport

Figure 2(a) summarizes the electron thermal power comparisons. The experimental level lies in the gray horizontal band at Peexp1.48±0.33 MW, computed via TRANSP. The uncertainties in Pe are computed via error propagation of the electron power balance equation implemented in TRANSP. The base simulation uses the experimental input profile parameters in the GYRO inputs and shows clear under-prediction of the experimental power level. Scaling (+a/LTe,a/Lne) by 1 σ (noted as (T,n) in black) increases the predicted Pe with respect to the base case, but still under-predicts the experimental Pe. However, scaling q and ŝ in addition to a combination of a/LTe or a/Lne in the GYRO inputs is shown to reproduce the experimental electron thermal power levels, as shown by the purple and light green bars. Scanning all four parameters (a/LTe,a/Lne,q,ŝ) is shown to drive ETG too strongly (noted (T,n,q,ŝ) in the dark green), leading to an over-prediction of the experimental Pe.

FIG. 2.

(a) Comparisons between experimental (TRANSP) and simulated electron thermal power from local electron-scale simulation corresponding to the strong ETG condition. Scans in the equilibrium inputs Te,ne are made within experimental uncertainty, safety factor q is varied by 10% and magnetic shear ŝ by 20%. (b) Comparisons between experiment and simulation of the fluctuation level ratio between the strong and weak ETG conditions, quantified by the metric log10[SstrongETG/SweakETG], where . denotes the average power between channels 1–3. (c) Comparisons between the measured high-k density fluctuation wavenumber spectrum and simulated wavenumber spectrum by local electron-scale GYRO simulation corresponding to the strong ETG condition. Adapted with permission from Ruiz et al., Plasma Phys. Controlled Fusion 61, 115015 (2019). Copyright 2019 IOP Publishing.

FIG. 2.

(a) Comparisons between experimental (TRANSP) and simulated electron thermal power from local electron-scale simulation corresponding to the strong ETG condition. Scans in the equilibrium inputs Te,ne are made within experimental uncertainty, safety factor q is varied by 10% and magnetic shear ŝ by 20%. (b) Comparisons between experiment and simulation of the fluctuation level ratio between the strong and weak ETG conditions, quantified by the metric log10[SstrongETG/SweakETG], where . denotes the average power between channels 1–3. (c) Comparisons between the measured high-k density fluctuation wavenumber spectrum and simulated wavenumber spectrum by local electron-scale GYRO simulation corresponding to the strong ETG condition. Adapted with permission from Ruiz et al., Plasma Phys. Controlled Fusion 61, 115015 (2019). Copyright 2019 IOP Publishing.

Close modal

Figure 2(a) shows that local, nonlinear, electron-scale gyrokinetic simulation is able to both under- and over-predict the experimental electron thermal transport levels Pe (MW) within a sensitivity scan of the most pertinent ETG drive terms. Additionally, Fig. 2(a) shows that simultaneously scanning a/LTe and a/Lne by 1σ was not sufficient to match Pe, but modifying q and ŝ was necessary to reproduce the experimental thermal power levels. This highlights the significant impact of q and ŝ in the present conditions, which will be further highlighted Sec. II C 2–II C 4.

2. Fluctuation level ratio

Owing to the absence of absolute calibration in the high-k scattering power level, an assessment of the measured fluctuation power compared with simulated fluctuation power is not possible. Here we choose to quantify the fluctuation level ratio between the strong and weak ETG conditions. The metric log10[SstrongETG/SweakETG] is chosen, where . denotes the arithmetic average power between channels 1–3. The use of the log10 is motivated by the fact that the quantity SstrongETG/SweakETG can vary by 23 orders of magnitude in the sensitivity scans performed. In the comparisons shown, only the quantity SstrongETG is varied, while SweakETG is kept constant, and calculated from a simulation for the weak ETG condition which matched the electron thermal power Pe (scaling a/LTe and a/Lne by 1σ uncertainty, not shown here).

Figure 2(b) shows the result of the fluctuation level ratio comparisons between the five simulations for the strong ETG condition and the experimental range in gray. The experimental level is shown by the gray horizontal band, where the uncertainty is calculated from uncertainties in the fluctuation levels by channels 1–3 of the high-k scattering system. The base simulation is shown to clearly under-predict the experimental fluctuation level ratio, shown by the magenta bar. This means that the strong ETG simulation with the base nominal experimental parameters (magenta) is too weakly driven to reproduce the ratio of fluctuation levels, which is also consistent with Fig. 2(a). Simulations scanning (a/Lne,q,ŝ)-light green and (a/LTe,a/Lne,q,ŝ)-dark green exhibit an over-prediction in the fluctuation level ratio with respect to the levels measured by experiment, suggesting that for these parameter scans, ETG was too strongly driven in the strong ETG condition. However, simulations scanning (a/LTe,a/Lne)-black and (a/LTe,q,ŝ)-purple both lie within the experimental range. As in Fig. 2(a), Fig. 2(b) shows how the fluctuation level ratio between the strong and weak ETG conditions can be both under- and over-predicted by gyrokinetic simulation when sensitivity scans in the main ETG drive mechanisms are performed. This shows that local nonlinear electron-scale gyrokinetic simulation can explain the fluctuation level ratio between the strong and weak ETG conditions.

3. Wavenumber spectra shape

Comparisons of the shape of the wavenumber spectrum are shown in Fig. 2(c), corresponding to the same five gyrokinetic simulation scans for the strong ETG condition. The experimental wavenumber spectrum is depicted by blue circles and is scaled in amplitude in order to yield comparisons of the shape of the wavenumber spectrum.

Figure 2(c) shows that neither the base simulation (magenta) nor the simulation scaling (a/LTe,a/Lne) (black) is able to reproduce the shape of the experimental wavenumber spectrum, computed from the three high-k scattering channels. Figure 2(c) also qualitatively shows that the best agreement in the shape of the wavenumber spectrum is shown for simulations which scaled q and ŝ in the inputs, in addition to combinations of a/LTe and a/Lne, as shown in purple, light green, and dark green. This highlights once more the significant sensitivity that q and ŝ have on the present ETG conditions, now showing how q and ŝ can modify the shape of the ETG spectrum. Particularly, the best agreement is shown for the simulation run with scaled (a/Lne,q,ŝ), depicted in light green. This shows that electron-scale simulation can also explain the shape of the measured density fluctuation wavenumber power spectrum.

4. Experiment-model comparison of the frequency and wavenumber spectrum

As detailed in Ref. 18, the fidelity of the previous comparisons can be quantitatively assessed via the use of validation metrics. These showed that simulations scanning (a/LTe,q,ŝ) and (a/Lne,q,ŝ) exhibit the best quantitative agreement. Remarkably, both simulations were shown to reproduce the experimental level of the electron thermal power [Fig. 2(a)]. Simulation scanning (a/Lne,q,ŝ) will be used in Sec. IV to make projections of frequency and wavenumber spectra with the new high-k system to be installed in NSTX-U.27 

In Fig. 3 are shown the frequency and wavenumber spectra comparisons for the simulation scanning (a/Lne,q,ŝ). Although the frequency spectrum was strongly influenced by Doppler shift,18,19 a successful validation work should still match the measured frequency spectrum as a consistency check of the simulated turbulence fluctuations and the synthetic diagnostic. Figure 3(a) shows that the synthetic frequency spectrum agrees well with the measured spectrum. As expected, the f =0 peak and the background noise level are different between experiment and simulation (due to stray radiation and possibly electronic noise setting the experimental background), but only the broadband fluctuations at f1 MHz should be visually compared. For reference, all five simulations presented here exhibited reasonably good agreement in the peak and the width of the frequency spectrum. As shown in Fig. 3(b), the agreement in the wavenumber spectrum is also satisfactory.

FIG. 3.

(a) Frequency spectra and (b) wavenumber spectra comparisons between experiment and simulation, corresponding to the best matched GYRO simulation for the strong ETG condition [simulation scanning (a/Lne,q,ŝ) in the inputs], as quantified by the validation metrics in Ref. 18. Frequency spectra corresponds to channel 1, where channels 2 and 3 exhibit similar agreement. Reproduced with permission from Ruiz et al., Plasma Phys. Controlled Fusion 61, 115015 (2019). Copyright 2019 IOP Publishing.

FIG. 3.

(a) Frequency spectra and (b) wavenumber spectra comparisons between experiment and simulation, corresponding to the best matched GYRO simulation for the strong ETG condition [simulation scanning (a/Lne,q,ŝ) in the inputs], as quantified by the validation metrics in Ref. 18. Frequency spectra corresponds to channel 1, where channels 2 and 3 exhibit similar agreement. Reproduced with permission from Ruiz et al., Plasma Phys. Controlled Fusion 61, 115015 (2019). Copyright 2019 IOP Publishing.

Close modal

Following successful validation of the electron-scale gyrokinetic simulations for the strong ETG condition in Sec. II, in this section we discuss the potential role that the ion-scale instability and turbulence might play in the weak ETG condition. Linear gyrokinetic simulations in subsection III A suggest that the ion-scale instability is expected to play a negligible role in the strong ETG condition, while the experimental profiles are shown to sit close to the critical KBM threshold in the weak ETG condition. Subsection III B shows how ion–ion collisions can strongly affect the ion-scale turbulent fluctuation levels via zonal flow damping in these near-marginal ion-scale turbulent conditions for the weak ETG case.

The linear effect of electron beta βe on the ion-scale instability is shown to be different between the strong and weak ETG conditions. This might be specifically significant for the weak ETG condition since the experimental value of βe is shown to be very close to the instability threshold of the KBM instability.

Figures 4(a) and 4(b) show the real frequency ωr and linear growth rate γ from scans in the electron beta βe carried out for a particularly unstable ion-scale wavenumber (kθρs=0.48) for the strong ETG condition. At low βe, an ion-directed, ballooning parity instability is shown to be the fastest growing mode, possibly identified to a trapped electron mode or ubiquitous mode (noted TEM). The linear growth rate in (b) is shown to be weaker than the background E × B shearing rate. At increasing value of βe, the linear growth rate is shown to decrease with βe up to values of βe0.25%, in which the dominant instability is shown to transition to an electron-directed, tearing parity micro-tearing mode (MT). The experimental βe value is shown to sit right at the critical beta for the transition between the TEM and MT mode. For beta values >0.25%, the linear growth rate exhibits a weak, destabilizing dependence with βe, characteristic of the MT mode. However, the MT linear growth rate is very close to the background E × B shearing rate. For even greater βe values, the MT mode becomes subdominant to an ion-directed kinetic ballooning mode, exhibiting high growth rate values that clearly surpass the E × B shearing rate value. This βe scan clearly shows three different instabilities can co-exist for experimentally relevant NSTX plasma parameters. However, we point out that the TEM and MT modes are the most experimentally relevant for the present case, which are predicted to be suppressed by strong E × B shear in this condition and consistent with negligible transport produced by nonlinear ion-scale gyrokinetic simulation (not shown here). Note βe here uses the Bunit convention in GYRO, and the uncertainty of βe is on the order of 10%. The beta scans presented also preserved the value of beta-prime β fixed. The αMHD,unit value is shown to be 0.5 in this condition, suggesting the KBM may not be experimentally relevant in this condition as suggested by Fig. 1 and previous work.11 

FIG. 4.

(a) Real frequency from a linear beta scan for kθρs=0.48 for the strong ETG condition. (b) Corresponding linear growth rate from a linear beta scan for kθρs=0.48.

FIG. 4.

(a) Real frequency from a linear beta scan for kθρs=0.48 for the strong ETG condition. (b) Corresponding linear growth rate from a linear beta scan for kθρs=0.48.

Close modal

The black curve in Fig. 4 shows linear scans computed including the full electromagnetic field perturbations (δϕ,δA||,δB||). The blue linear scans only included (δϕ,δA||) perturbations. The TEM and MT modes are shown to be unaffected by the inclusion of δB|| perturbations; however, the KBM is shown to be stabilized without δB|| perturbations.

Figure 5 shows the corresponding βe scans for the weak ETG condition. In this case, an electron-directed, ballooning parity TEM mode is most unstable at low beta. The growth rate is shown to be higher than the background E × B shearing rate. When full perturbations of the electromagnetic field are included (black line), the TEM mode transitions to an ion-directed, ballooning parity KBM mode at βe0.35%, with linear growth rates greatly exceeding the background E × B shearing rate. The transition from TEM to KBM happens within 20% of the experimental βe value. This suggests the KBM might have an important role to play in the weak ETG condition. If only (δϕ,δA||) perturbations are included in the simulation, the KBM is stabilized and the TEM transitions to an electron-directed, tearing parity MT mode that ultimately transitions to the KBM for βe0.9% (greatly exceeding the experimental βe). It is worth pointing out how the TEM at low beta is shown to be stabilized when δB|| perturbations were not included in the simulation. The uncertainty value of βe is about 10%, showing the transition to KBM is outside uncertainty. However the αMHD,unit value is shown to be 0.7–0.8 in this condition, suggesting KBM may be experimentally relevant as suggested by Fig. 1 and by previous work.11,28 It may also be that additional effects such as the inclusion of a fast-ion species could lower the KBM limit within the uncertainty range of the experimental beta value. Additional analysis should be performed to address this speculation.

FIG. 5.

(a) Real frequency from a linear beta scan for kθρs=0.48 for the weak ETG condition. (b) Corresponding linear growth rate from a linear beta scan for kθρs=0.48.

FIG. 5.

(a) Real frequency from a linear beta scan for kθρs=0.48 for the weak ETG condition. (b) Corresponding linear growth rate from a linear beta scan for kθρs=0.48.

Close modal

Although very preliminary, these linear beta scans for ion-scale modes have shown how the KBM might be experimentally relevant in the weak ETG condition, but not at the strong ETG condition, which as shown in this article is dominated by ETG.

All the available experimental evidence shown in this article has pointed to ETG-driven turbulence as the main contributor to electron thermal transport both in the strong and weak ETG conditions. However, the state and the role that ion-scale turbulence might play in the weak ETG condition is still not fully understood and requires more careful analysis which we partly present here.

We collect the available evidence for the weak ETG condition: (1) the measured high-k fluctuation power is lower than for the strong ETG condition; (2) the ion-scale linear growth rate can exceed the background E × B shearing rate, and lies close to the linear βe threshold for the KBM; (3) ion thermal transport is very close to the neoclassical levels predicted by NEO;23 (4) nonlinear electron-scale gyrokinetic simulation can reproduce the electron thermal power within uncertainty in T and n;18 and (5) ion-scale nonlinear simulation showed that the background E × B shear can suppresses ion-scale turbulence.18 Although the ion-scale instability might be linearly unstable, nonlinear simulation shows that ion-scale turbulence is suppressed, suggesting a negligible role for the weak ETG condition. In this section, we show ion-scale turbulence is nonlinearly suppressed by large zonal flow amplitudes, in a state reminiscent of a Dimits' shift regime,29 and possibly linked to the recently observed subcritical fluctuations.30 However, a strong TEM/KBM branch is destabilized within experimental uncertainty, yielding unphysical electron thermal power levels up to 10 times the experimental value.

Nonlinear ion-scale gyrokinetic simulations carried out for the weak ETG condition show the important effect of ion–ion collisions on zonal flow. For two decades, it has been known that ion–ion collisions can act as a damping mechanism on the zonal flow.31–33 These were not modeled in the simulations presented in Sec. II. Ion–ion collisions are expected to be negligible for finite turbulence amplitudes dominated by turbulence interactions with the zonal flow. In situations of fully developed turbulence commonly encountered in the tokamak core, ion–ion collisions tend to play a negligible role in damping the zonal flow. However, ion–ion collisions can become important in conditions where the turbulence is very close to the nonlinear threshold. This effect can be enhanced at high collisionality values νei1 cs/a in the present conditions. These are precisely the conditions found for the weak ETG case and motivate this preliminary analysis on the effect of ion–ion collisions on ion-scale turbulence.

The ion-scale simulations model radial and poloidal wavenumbers necessary to resolve ion-scale turbulence, here krρsϵ[0.08,4] and kθρsϵ[0.08,1.1]. We model drift-kinetic electrons, gyrokinetic main deuterium ions, and a gyrokinetic carbon impurity. Simulations were performed at the local scattering location r/a0.7, including electron-ion collisions νei=4πnee4logΛ/(2Te)3/2me1/21 cs/a (via a pitch-angle scattering Lorentz operator), ion–ion collisions νii=νeimemi(TeTi)3/2nineZi2, fully electromagnetic fluctuations (δϕ,δA||,δB||), background toroidal flow, and flow shear (M0.2, perpendicular flow shear γE0.10.2 and parallel flow shear γp1). Linear background profiles were simulated, employing nonperiodic boundary conditions in the radial direction, implemented in GYRO via buffer damping regions of radial width Δb8ρs. Parallel resolution employed 14 poloidal grid points (×2 signs of parallel velocity), 12 energies, and 12 pitch-angles (6 passing + 6 trapped). This choice of numerical grids was made according to previous convergence and accuracy tests for the GYRO code, simulating micro-instabilities in the core of NSTX,11 and was also tested for convergence for the present conditions.18 

Figure 6 shows the total, saturated values of the electron and ion thermal power Pe, Pi (MW) computed via ion-scale gyrokinetic simulation for different values of the driving parameter a/LTe. All simulations were run using a constant, scaled value of a/Lne of 1 − σ uncertainty. In red are shown simulations run without the inclusion of ion–ion collisions. Looking at the red curve, ion-scale simulation predicts negligible Pe and Pi for a/LTe<5 values, but a highly unstable TEM/KBM branch for a/LTe>5. The letters A–C denote simulations run with input values of a/LTe=4.5,4.8,5.4, respectively. The magenta curves are the result of three simulations that were restarted from A–C by including ion–ion collisions on the zonal flow n =0. The saturated values of Pe and Pi from such simulations are shown by the letters D–F. Figure 6 shows that ion–ion collisions can have a substantial effect on the turbulence saturated state for A and B, leading to the highly unstable turbulence regimes D and E. However, C and F predict essentially the same turbulence state and transport. These results suggest that ion–ion collisions are mostly important in close to threshold conditions (close to marginality as in A and B), but not far from threshold (as in C).

FIG. 6.

(a) Electron thermal power from ion-scale simulation run with scaled density gradient value within 1σ uncertainty, for different values of electron temperature gradient a/LTe. In red is shown simulation output when the value of ion–ion collision frequency νii was set to 0. In magenta is shown simulation output when the value of ion–ion collision frequency was set to the experimental value νii=νiiexp was set to 0. (b) Same as (a) but showing the ion thermal power Pi instead of the electron thermal power Pe.

FIG. 6.

(a) Electron thermal power from ion-scale simulation run with scaled density gradient value within 1σ uncertainty, for different values of electron temperature gradient a/LTe. In red is shown simulation output when the value of ion–ion collision frequency νii was set to 0. In magenta is shown simulation output when the value of ion–ion collision frequency was set to the experimental value νii=νiiexp was set to 0. (b) Same as (a) but showing the ion thermal power Pi instead of the electron thermal power Pe.

Close modal

The role of ion–ion collisions on A–C are more clearly shown in Figs. 7(a)–7(c). These show time traces of the electromagnetic field fluctuation amplitudes (δϕ,δA||,δB||) for the turbulence (n>0) and zonal flow (n =0), corresponding to simulations A–F in Fig. 6. Simulations in Fig. 7 are run with the same value of electron density gradient (scaled by 1 − σ) but different electron temperature gradients as input. All three simulations in Fig. 7 are divided in three time phases. In the first time phase, simulations are started with a zero value of E × B shearing rate γE and zero value of ion–ion collision frequency νii,ZF=0. In this phase, turbulence fluctuations follow the linear exponential growth phase for the first 50–100 a/cs. After the linear growth phase, fluctuations saturate at high amplitudes without the suppressing effect of γE. In the second phase of the simulations, the experimental value of E × B shear is included and corresponds to phases A–C, respectively, in Figs. 7(a)–7(c). In this phase, the finite n turbulence fluctuations are suppressed by the effect of γE. In Figs. 7(a) and 7(b), the state is dominated by strong potential fluctuations of the zonal flow (indicated by ϕn=0, black dashed line) and the turbulence is shown to be fully suppressed. Negligible transport is predicted in this time phase, as can also be seen from A and B in Fig. 6. In Fig. 7(c), the state is dominated by strong finite-n, turbulence fluctuations, producing high values of transport (corresponding to C in Fig. 6). The unrealistic transport values greatly exceed the experimental transport levels ∼ tenfold.

FIG. 7.

Time traces of the electromagnetic field root mean square fluctuation amplitudes (ϕ,A||,B||) for the turbulence (n>0) and zonal flow (n =0), corresponding to simulations A–F in Fig. 6. All simulations were run with a scaled value of a/Lne within 1 − σ uncertainty. (a) Simulation was run using a/LTe=4.5. (b) Simulation was run using a/LTe=4.8. (c) Simulation was using a/LTe=5.4.

FIG. 7.

Time traces of the electromagnetic field root mean square fluctuation amplitudes (ϕ,A||,B||) for the turbulence (n>0) and zonal flow (n =0), corresponding to simulations A–F in Fig. 6. All simulations were run with a scaled value of a/Lne within 1 − σ uncertainty. (a) Simulation was run using a/LTe=4.5. (b) Simulation was run using a/LTe=4.8. (c) Simulation was using a/LTe=5.4.

Close modal

In the third phase of the simulations, the experimental value of the ion–ion collision frequency is included, preserving the previous parameters constant (γE, etc.). They are phases D–F, respectively. In Figs. 7(a) and 7(b), the large value of the zonal flow amplitude is shown to be strongly damped by the inclusion of finite νii. This has the effect of destabilizing finite-n perturbations since now the weaker zonal flows are not able to efficiently regulate the turbulence. In D and E, finite-n perturbations are shown to saturate at very high amplitudes. As can be seen from Figure 6, phases D and E produce some 6–8 MW of electron thermal power, clearly over-predicting the experimental value of Peexp1 MW. In Fig. 7(c), turbulence was already shown to be strongly driven in phase C, and the inclusion of ion–ion collisions in F is shown to have a negligible effect on the dynamics of the zonal flow, barely affecting the saturated levels and transport. In summary, Figs. 6 and 7 suggest that ion–ion collisions can have a very important effect on the ion-scale turbulence amplitude via the damping of zonal flows in close to marginal conditions and at high values of collisionality νei1 cs/a. However, the effect is shown to be negligible in far from threshold conditions, exhibiting finite amplitude turbulence fluctuations.

Figures 8(a) and 8(b) show the time traces of electromagnetic field fluctuation amplitudes (δϕ,δA||,δB||), similar to Fig. 7, but this time corresponding to simulations carried out for the nominal, experimental profile values (in particular, experimental a/Lne and a/LTe). As in Fig. 7, simulations are started with a zero value of E × B shearing rate (γE=0). In Fig. 8(a), the ion–ion collision frequency is initially set to 0 (νii,ZF=0); in Fig. 8(b), the ion–ion collision frequency is set to the experimental value (νii,ZF=νiiexp). In the second phase, the experimental value of γE is included, which shows to suppress the finite-n modes. However, in the absence of ion–ion collisions in Fig. 8(a), the zonal flows are not saturated and exhibit n = 0 potential perturbations that have a linear dependence in time (|δϕn=0|2t). This behavior is not observed in the presence of ion–ion collisions in Fig. 8(b) (after t 120 a/cs). This suggests the undamped zonal flow amplitudes in Fig. 8(a) are linked to the absence of ion–ion collisional damping on the zonal flows. In Fig. 8(b), one can observe sudden bursts of finite-n perturbations at t450,650,1100 a/cs (bold black line). These are in fact lone, large-scale eddies, occupying most of the simulation domain and producing non-negligible levels of instantaneous transport; however, they are eventually damped by the strong zonal flows present, producing negligible time-averaged transport levels. The observation of lone, large-scale eddies, occupying most of the simulation domain, is reminiscent of the subcritical transition to ITG turbulence observed in MAST gyrokinetic simulations.30 

FIG. 8.

Time traces of electromagnetic field root mean square fluctuation amplitudes (ϕ,A||,B||), similar to Fig. 7, but this time carried out for the nominal, experimental profile values (in particular, experimental a/Lne and a/LTe). (a) The simulation is started at t =0 a/cs using ion–ion collision frequency set to 0 (νii,ZF=0). (b) The simulation is started at t =0 a/cs using ion–ion collision frequency set to the experimental value (νii,ZF=νiiexp).

FIG. 8.

Time traces of electromagnetic field root mean square fluctuation amplitudes (ϕ,A||,B||), similar to Fig. 7, but this time carried out for the nominal, experimental profile values (in particular, experimental a/Lne and a/LTe). (a) The simulation is started at t =0 a/cs using ion–ion collision frequency set to 0 (νii,ZF=0). (b) The simulation is started at t =0 a/cs using ion–ion collision frequency set to the experimental value (νii,ZF=νiiexp).

Close modal

The behavior of undamped, linear dependence of the zonal flow in time observed in Fig. 8(a) is consistent with seminal work by Rosenbluth and Hinton.31,32 When subject to a random and statistically stationary noise source,31,32 showed that the zonal mean square potential can increase linearly in time. This behavior appears to be qualitatively similar to that observed in the outer core of this NSTX H-mode, in which the strongly suppressed finite-n fluctuations could act as the random and statistically stationary noise source in Refs. 31 and 32 (this might not be the case in the case of finite fluctuation amplitudes). One last time phase is shown in Fig. 8(a) by including ion–ion collisions (after t1130 a/cs). In this regime, the main effect of finite νii is on the zonal flow amplitude, exhibiting a nearly exponential decay. This behavior was also observed in Refs. 31 and 32. Unfortunately, the simulation in Fig. 8(a) was not run for long enough in order to observe the final saturated state of the zonal flow due to the large computational time needed for completion.

These ion-scale gyrokinetic simulations have shown that ion–ion collisions can have an important effect in damping zonal flows in close to threshold conditions, but not in far from threshold conditions. Strong zonal flows are shown to develop in the absence of ion–ion collisional damping on the zonal flow, and ion-scale turbulence fluctuations are strongly suppressed. When ion–ion collisions are included, zonal flows are strongly damped and ion-scale turbulence grows to unphysically large amplitudes. This suggests zonal flows, in addition to background E × B shear, can strongly suppress ion-scale turbulence in the weak ETG condition. Additionally, we have shown that a TEM/KBM branch can become unstable and lead to unphysically large ion and electron thermal powers. Previous work performed for an NSTX H-mode11,28 has in fact highlighted that a hybrid TEM/KBM mode can produce experimentally relevant Pe. Further analysis would need to be performed to shed further light on this issue.

There are important caveats worth pointing out in the analysis of this subsection. We have used the “simple” Lorentz collision operator, which only models pitch-angle scattering collisions and no energy-diffusion. The high collisionality of these plasmas (νei1 cs/a) suggests the Lorentz model might not be valid in these regimes, and advanced collisional models such as implemented in CGYRO25 might be needed. In fact, recent work using the CGYRO code has shown that the residual zonal flow potential calculated using the advanced Sugama collision operator can vary by close to an order of magnitude at high levels of electron collisionality (similar to those used here) with respect to the simpler Lorentz model implemented in GYRO. This work was carried out simulating ITG turbulence for the GA standard case.34 The results presented in this subsection motivate further work to understand the role of ion–ion collisions on the zonal flow, and the possible implications for the turbulence saturated levels and for cross-scale coupling interactions in realistic plasma conditions for the spherical tokamak.

As a clarifying note, ion–ion collisions were shown to play a negligible role in the electron-scale simulations from Sec. II.

Having successfully validated electron-scale gyrokinetic simulations in the current NSTX discharge, in this section we make use of the predicted turbulence fluctuations via gyrokinetic simulation to make preliminary projections for the new high-k scattering system that is currently under development for NSTX-U.27 The new high-k scattering system will employ a 693 GHz microwave beam that will replace the 280 GHz beam in NSTX. The new scattering geometry and smaller beam wavelength is projected to enable measurements of up to k>40 cm−1 in a spatial range from r/a=0.1 out to r/a=0.99. A detailed description of the Gaussian probe beam, launch optics, scattering and receiver optics, receiver hardware, and electronics is presented in Ref. 27.

The projections that follow represent a reduced subset of potential high-k measurements in NSTX-U. We restrict ourselves to r/a 0.7 in the hypothetical scenario of NSTX discharge 141767 which was the object of the validation work presented in Sec. II. Although limited in scope, the projections show the enhanced capability for measuring the most relevant parts of the high-k density fluctuation wavenumber spectrum (so-called streamers5,6), which were not possible using the high-k system in NSTX.

The new high-k scattering geometry was designed to enable measurements of higher kθ and smaller kr wavenumber components of the turbulence than those accessed by the NSTX high-k system. Figure 9(a) shows a view from the top of the NSTX high-k scattering system, where the trajectories of the probe and scattered beams from 141767 are computed via ray-tracing. The near-tangential scattering geometry is suited for high kr and smaller kθ components. The scattering geometry for the NSTX-U high-k system is shown in Figs. 9(b) and 9(c). The probe beam in the new high-k scattering system propagates near the midplane but more perpendicularly to the magnetic field than the NSTX system, allowing lower kr and higher kθ measurements.

FIG. 9.

(a) View from top of the NSTX high-k scattering geometry for 141767 [Reproduced from Phys. Plasmas 22, 122501 (2015). Copyright 2015 AIP Publishing LLC]. The tangential geometry of this system was suited for performing high-kr and lower kθ wavenumber components of high-k turbulence. (b) View from top of the new NSTX-U high-k scattering geometry for the hypothetical scenario of shot 141767 (upward scattering). (c) Poloidal cross section of the new high-k scattering geometry for 141767. The new geometry of the probe and scattered beams allows for higher kθ measurements than in the NSTX high-k system.

FIG. 9.

(a) View from top of the NSTX high-k scattering geometry for 141767 [Reproduced from Phys. Plasmas 22, 122501 (2015). Copyright 2015 AIP Publishing LLC]. The tangential geometry of this system was suited for performing high-kr and lower kθ wavenumber components of high-k turbulence. (b) View from top of the new NSTX-U high-k scattering geometry for the hypothetical scenario of shot 141767 (upward scattering). (c) Poloidal cross section of the new high-k scattering geometry for 141767. The new geometry of the probe and scattered beams allows for higher kθ measurements than in the NSTX high-k system.

Close modal

The new high-k scattering system is also designed with the capability for upward and downward scattering. Here we present projections for upward scattering in which the probe beam is launched directed slightly downwards from the midplane, while the scattered beams are directed upwards in most conditions [Fig. 9(c)]. This geometry will allow us to sample the most relevant parts of the high-k wavenumber spectrum. We adopt a Cartesian coordinate system (X, Y, Z) to describe the scattering geometry (Fig. 9), defining the corresponding wavenumber components (kx,ky,kz) (these Cartesian coordinates do not correspond to those defined in Ref. 19. Details in  Appendix A). We will also make use of the cylindrical coordinate system (R,Z,φ), where R=X2+Y2, Z = Z, and tanφ=Y/X. The corresponding cylindrical wavenumber components (kR,kZ,kφ) are defined in Sec. IV B and  Appendix A.

The probe beam is launched with azimuthal angle of 93° (with respect to the X-axis on the X-Y plane) and polar angle of 91° (inclination angle with respect to the Z-axis). The launching point is (X,Y,Z)=(0.4147,2.00809,0) m, and the collection points are assumed to be at Y =1.8 m [Figs. 9(b) and 9(c)]. The scattering location takes place at R135 cm, Z6 cm (corresponding to a poloidal angle θ4°). Modeling the probe and scattered ray trajectories with a ray-tracing code shows negligible ray refraction takes place in the current conditions, given the high frequency fi = 693 GHz of propagation with respect to the electron plasma and cyclotron frequencies (fpe70 GHz, fce20 GHz). Projections of the scattering location and wavenumber components differ by less than 1% with respect to modeling the probe and scattered beams by straight line trajectories. As a result, the projections presented here are based on straight line trajectories of the probe and scattered beams. Note the high-k scattering system in NSTX exhibited non-negligible refraction due to the propagation geometry and reduced frequency fi = 280 GHz [Fig. 9(a)], meaning ray-tracing was mandatory.

The selection of the turbulent wave-vectors k+ is dictated by the selection rules k+=kski,ω=ωsωi between the incident i, scattered s and selected turbulence wave-vectors and frequencies (ki,ωi),(ks,ωs), and (k+,ω).35–39 The anisotropic nature of turbulent fluctuations imposes the constraint k+·B=0 to be satisfied by scattering experiments and will be imposed in the k-selection schemes that follow. Additionally, the condition for coherent scattering |k+|λDe1 (Refs. 35–39) is largely satisfied in the present conditions.

With respect to spatial localization, the given scattering geometry suggests the vertical and toroidal dimensions of the scattering volume can be approximated by the beam waist a0, giving ΔZRΔφa0. Radial localization is achieved by taking into account the anisotropy of the magnetic field and the scattering geometry.40–42 Applied to the new high-k scattering geometry yields estimates ΔR5 cm for k=10 cm−1,43 which is greatly enhanced from simple estimates based on the linear overlap volume between the incident and scattered beams. Typically ΔR decreases for increasing k. This justifies the use of local gyrokinetic simulations to predict the turbulence spectra for the new high-k scattering system. The diagnostic wavenumber resolutions Δkθρs and Δkrρs can be computed from knowledge of ΔZ and ΔR (Sec. IV B). A value of ΔZa01.8 cm leads to a spectral resolution Δkθρs1.5. A reference value of ΔR=5 cm leads to Δkrρs0.06 for the new high-k system.

In this subsection, we briefly describe the implementation of synthetic high-k diagnostics using local gyrokinetic simulation and GYRO/CGYRO notation. This builds on previous development of synthetic diagnostics for beam emission spectroscopy and correlation electron cyclotron emission,44 Doppler backscattering,45 and high-k scattering46 using gyrokinetic simulation. The description presented is applied to the NSTX (documented in detail in Ref. 19) and NSTX-U high-k scattering diagnostics. Other high-k diagnostics could also be modeled using the synthetic description presented here, provided that the scattering volume ΔV could be written in separable form ΔVΔRΔZRΔφ. Specifics about the high-k scattering geometry are encapsulated in the selected wavenumber k+ and the spatial resolutions ΔR,ΔZ, and RΔφ, which will vary from system to system. The present description could also readily be adapted to other gyrokinetic codes using the corresponding internally defined wavenumber grid, ρs normalization, etc.

The synthetic diagnostic takes as input the raw simulated electron density fluctuating field δn(kr,kθ,θ,t), the selected k+ by high-k scattering, and the diagnostic resolutions in the radial ΔR, vertical ΔZ, and toroidal RΔφ directions. A posteriori filtering of the GYRO simulated raw density fluctuation field around k+ yields the synthetic density fluctuation signal δnU(k+,t), which is used for the computation of spectra (details in Refs. 18 and 19). Here we characterize the measured k+ and the diagnostic resolutions that are fed into the synthetic diagnostic.

Ray-tracing (or beam tracing or equivalent calculations) provides the measured wavenumber components in cylindrical coordinates (kR,kZ,kφ) ( Appendix A), whereas gyrokinetic codes employ field-aligned geometry with perpendicular coordinates (kr,kθ) in the case of GYRO and CGYRO. The wavenumber components (kr,kθ)+ of the selected wave-vector are computed from the cylindrical components (kR,kZ)+ via the following wavenumber mapping:19 

kr+rqαrkθ+=RrkR++ZrkZ+1qαθkθ+=1rRθkR++1rZθkZ+,
(1)

where the geometric quantities r and q (safety factor) are evaluated locally, as well as the r and θ derivatives of the field-line label α=α(r,θ) and the flux-surface shape functions R(r,θ) and Z(r,θ). The cylindrical components of the wave-vector k+=(kR,kZ,kφ) correspond to the projections of k+, respectively, along major-radius R, vertically Z, and along the toroidal direction φ. Formula 1 assumes that the cylindrical kR+ and kZ+ components only represent the central ray of the probing beam and do not vary with toroidal angle. Strictly speaking, a third equation corresponding to mapping the kφ+ component should be appended to Eq. (1), namely, rqRkθ+=kφ+. When the scattering condition k+·B0 is well satisfied, the kφ equation is redundant and the mapping is completely determined by Eq. (1). When k+·B0 is not well satisfied, misalignment occurs which can be described via different selected kθ+ from Eq. (1) and the kφ equation. As shown in Ref. 19, the toroidal direction introduces no additional effect on the filtering for essentially all Δφ for well-aligned NSTX high-k scattering measurements. Additionally, the condition k+·B=0 will be imposed from the outset for the new high-k scattering system in NSTX-U. As a result, we use a 2D synthetic diagnostic in the R and Z plane which neglects toroidal variation (equivalent to Δφ0) where the toroidal wavenumber component kφ is absent in the description. A different coordinate system would need to be used to correctly incorporate 3D geometry (Ref. 19 and  Appendix A). The full mapping including the kφ equation could be employed to assess the effect of misalignment on the scattering amplitude, but is not the object of the present study.

The measurement range of three high-k scattering channels from shot 141767 is represented in Fig. 12(a) by three white dots on top of the raw GYRO spectral density S(kr,kθ)|δn(kr,kθ)|2 at the outboard midplane (θ = 0) corresponding to the strong ETG condition. These correspond to a measurement of krρs23 and kθρs35 (recall ρs uses the Bunit convention12,26). The density fluctuation wavenumber spectrum is anisotropic in (kr,kθ), peaking at finite krρs1 and kθρs48. This corresponds to tilted, radially elongated eddies known as ETG streamers.5,6 The additional colored dots around the streamer peak in Fig. 12 correspond to potential high-k measurements in NSTX-U. The spectral density in Fig. 12(a) has been averaged during the turbulent saturated phase of the gyrokinetic simulation (during a time period 14 a/cs).

The selected k+ can be interpreted as the wave-vector providing the dominant contributions to the scattered signal. A range of wave-vectors centered around k+ also contribute to the measured fluctuation power depending on the wavenumber resolution of the diagnostic. The dimensions of the scattering volume in the radial ΔR and vertical ΔZ determine the diagnostic wavenumber resolutions Δkr and Δkθ (the toroidal dimension RΔφ is absent in the 2D description). These are well approximated by Δkrρs2|r|ΔRρs and Δkθρs2κqΔZ(α/θ)ρs for θ0 (Ref. 19) (here κ is elongation, while α/θ and |r| are functions characterizing the flux surface geometry. More details in  Appendix B).

Fourier analysis of the synthetic signal δnU(k+,t) yields the frequency power spectrum of fluctuations, represented by the spectral density S(k+,ω)=1TVs|δnU(k+,ω)|2n0. Here T is the collection time, n0 is the background density, and the brackets denote an ensemble average. This quantity can be directly compared to the measured high-k frequency spectrum [Fig. 3(a)]. The density fluctuation wavenumber power spectrum S(k+) can be computed from the frequency power spectrum S(k+,ω) via integration within a prescribed frequency band (as in experiment) and can be directly compared to the measured high-k wavenumber spectrum [Fig. 3(b)].

With views of sampling the high-k density fluctuation wavenumber power spectrum, we introduce two different schemes designed to predominantly measure the poloidal kθ and radial kr wavenumber dependence of the spectrum around the streamer peak. These projections assume that a full suite of eight scattering channels are available for a given experiment.

For each case, we use k+=kski to select a turbulence wave-vector k+ expressed in Cartesian coordinates. Following the procedure outlined in detail in Refs. 18 and 19, we map k+ to the components (kr,kθ) employed by GYRO/CGYRO using Eq. (1) (the Cartesian coordinates defined here do not correspond to those in Ref. 19. Details in  Appendix A). In the present 2D description, we use the dimensions of the scattering volume in the radial ΔR and vertical ΔZ directions to define the filter functions applied on the turbulence amplitude δn via the synthetic diagnostic. For each set of k+, we then deploy the synthetic diagnostic to the output of a nonlinear gyrokinetic simulation that scanned (n,q, and ŝ) from their experimental values, one of the two that best matched all the experimental observables in Sec. II. The frequency S(kr+,kθ+,ω) and wavenumber spectra S(kr+,kθ+) are computed following the details from Sec. IV B.

1. Measuring thekθdependence of the spectrum

In order to scan the kθ dependence of the turbulent wavenumber spectrum we perform scans in the vertical Cartesian coordinate kz. This is a good approximation since the projected measurements are made near the outboard equatorial plane (for θ = 0 the second equation in Eq. (1) reduces to kθ+=κq/(α/θ)kz+). In order to scan kz, we work in Cartesian coordinates to perform the following procedure for selecting k+: (1) choose a particular value of k (cm−1), (2) choose a particular ky component, and (3) use k·B=0 to solve for the corresponding kx and kz components (select kz>0 for upwards scattering). Once a particular kz is found, we iterate for different k, which will select different kz (and kx) keeping ky fixed. In addition to scanning kz, this procedure also introduces a different kx component for each iteration. This translates to a small but finite kr in the field-line following frame [Eq. (1) reduces to krkR/|r| for θ0, which contains a kx dependence]. Introducing a small but finite kr is desired in plasma conditions with strong flow shear since the peak of the spectrum is expected to take place at a finite kr, helping us probe the wavenumber spectrum as close as possible to the streamer peak. The streamer peak for the particular case study presented here takes place at krρs1. An example of selected k+ using this kθ-scanning procedure is schematically shown by the red dots overlaid on the simulated spectral density S(kr,kθ) in Fig. 12(a).

In Figs. 10(a) and 10(b) are shown views from the top and poloidal cross section of the probe and scattered beams issued by the previous procedure for k-selection. The selected k+ (colored arrows) are scaled by a constant for visualization purposes. The black arrow shows the magnetic field projection on each plane. The varying k+ are all perpendicular to B by design and primarily scan kz. Mapped to the (krρs,kθρs)+ components in GYRO/CGYRO, these correspond to kθ+ρs[230] while kr+ρs1 (the specific values are given on  Appendix B). In Fig. 10(c) are shown the corresponding frequency spectra issued from deployment of the synthetic high-k diagnostic for the different k+. Even a moderate Mach number (M0.18) produces a Doppler shift ωD that is 2 orders of magnitude larger than the plasma frame frequency of fluctuations. The peak of the frequency spectrum is shifted up to frequencies of f10 MHz for the higher kθ components, and is shown to roughly follow the Doppler shift frequency ωD=k+·v(r/qR)Mkθ+. Higher frequencies would be expected for measurements with higher kθ+ and higher Mach number M. The amplitude of the fluctuations spans three orders of magnitude in power, and is shown to decrease for increasing kθ+, while the width of the spectrum becomes wider for increasing kθ+.

FIG. 10.

(a) View from the top of the probe and scattered beam trajectories for measuring the dependence in kθ of the density fluctuation spectrum. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

FIG. 10.

(a) View from the top of the probe and scattered beam trajectories for measuring the dependence in kθ of the density fluctuation spectrum. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

Close modal

Figure 12(b) shows the power of the synthetic spectral density S(kr+,kθ+) for the eight selected wavenumbers k+ by the kθ-scanning procedure. These correspond to the red dots on Fig. 12(a). The amplitude of the synthetic spectral density peaks at wavenumbers kθρs56 (kθρe0.1), and decays with a power-law dependence for higher kθ characteristic of turbulence in the inertial range of the cascade. A power-law fit to the spectrum reveals |δn|2kθ3.84. Future measurements of the peak of the spectrum and the spectral exponent in kθ could prove critical for validating existing turbulent theories and could serve as potential observables in future validation studies.

2. Measuring the kr dependence of the spectrum

Here we design a scheme to probe the kr dependence of the density fluctuation power spectrum. We perform two sets of scans focusing separately on positive kr>0 vs negative kr<0. By separately covering (kθ>0,kr>0) and (kθ>0,kr<0), we ensure full coverage of the spectrum in kr and kθ around the streamer peak [the negative kθ can be computed by the reality condition δn(kr,kθ)=δn(kr,kθ)*].

We propose to sample the kr dependence of the spectrum using the following k-selection procedure: (1) choose a particular k (cm−1), (2) choose a particular kz>0 component (upwards scattering), and (3) use k·B=0 to solve for the corresponding kx and ky components. Once particular kx and ky are found, we can iterate the procedure for different k which will select different kx and ky while keeping kz fixed (negative kr<0 corresponds to ky<0, while kr>0 corresponds to ky>0). This procedure results in scanning primarily kR (and subsequently kr) while introducing small variations in kθ. A fixed kz = 4 cm−1 component is chosen to probe the spectrum around the streamer peak given by kθρs56.

Figures 11(a) and 11(b) show a view from the top and poloidal cross section of the probe and scattered beams and the corresponding selected k+ having kr+<0. The selected k+ are scaled by a constant for visualization purposes (colored arrows). Note how this procedure primarily scans the radial kR component. The mapped radial wavenumber components range from krρsϵ[6,0]. This k-selection procedure introduces variations in the poloidal wavenumber in the range kθρs[2,5]. This might appear counter-intuitive since the same kz is selected for each of the eight wave-vectors k+ [the second equation in Eq. (1) reduces to kθ+=κq/(α/θ)kz+ for θ = 0 implying kθ+ should be constant for fixed kz+]. The variation in kθ+ is in fact due to the off-midplane corrections to the previous formula, suggesting the mapping given by Eq. (1) should be employed even for small off-midplane locations as small as θ04° presently. This is due to the effect of the R/θ term in Eq. (1), which is enhanced for large kR and in strongly shaped plasmas such as in spherical tokamaks. A similar conclusion was reached in Ref. 19 when mapping to kr and kθ for the high-k scattering diagnostic in NSTX. Figure 11(c) shows the frequency spectra computed via synthetic diagnostic corresponding to each of the selected k+ in Figs. 11(a) and 11(b). Even though small variations are introduced in kθ, the frequency spectrum here does not exhibit as wide variation in the frequency range as in the previous scan in kθ and now peaks between f(12) MHz for all selected k+. The amplitude of the fluctuation spectrum span around three orders of magnitude, while no appreciable variation is observed in the width of the frequency spectrum for the different kr+.

FIG. 11.

(a) View from the top of the probe and scattered beam trajectories for measuring the dependence of the density fluctuation spectrum for kr<0. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

FIG. 11.

(a) View from the top of the probe and scattered beam trajectories for measuring the dependence of the density fluctuation spectrum for kr<0. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

Close modal

The selected (kr,kθ)+ components are overlaid on the 2D wavenumber spectrum in Fig. 13(a) and depicted by the blue dots. One can visually see the peak of the spectrum appears to have been sampled following this procedure to scan kr<0. In Fig. 12(c) is shown the amplitude variation of the synthetic spectral density S(kr+,kθ+) for the eight selected wavenumbers k+ by the kr-scanning procedure. The blue dots correspond to those in Fig. 12(a), and show a peak in the spectrum around kr+ρs1. For higher values of kr the spectrum is shown to exhibit a power-law dependence characteristic of the inertial range in the turbulent cascade, here given by |δn|2kr4.3 for |kr+ρs|>1. Note the span in kr+ is kr+ρsϵ[6,0.1] is a smaller range than in kθρsϵ[2,30]. The peak of the spectrum and the spectral index in kr could be relevant observables to validate existing turbulent theories against future high-k measurements.

FIG. 12.

(a) 2D wavenumber spectrum of the electron density S(kr,kθ)|δn(kr,kθ)|2 at the outboard midplane. White dots indicate the high-k measurement range for 141767 in NSTX, while red, black, and blue dots are projections for potential measurements with the new high-k system in NSTX-U. (b) Dependence of the synthetic spectrum S(kr+,kθ+) with kθ in a projected scan in kθ+. (c) Dependence of the synthetic spectrum S(kr+,kθ+) with kr in projected scans for kr+>0 and kr+<0.

FIG. 12.

(a) 2D wavenumber spectrum of the electron density S(kr,kθ)|δn(kr,kθ)|2 at the outboard midplane. White dots indicate the high-k measurement range for 141767 in NSTX, while red, black, and blue dots are projections for potential measurements with the new high-k system in NSTX-U. (b) Dependence of the synthetic spectrum S(kr+,kθ+) with kθ in a projected scan in kθ+. (c) Dependence of the synthetic spectrum S(kr+,kθ+) with kr in projected scans for kr+>0 and kr+<0.

Close modal

The peak of the spectrum at finite kr is evidence of an asymmetric spectrum. In fact, the density fluctuation wavenumber power spectrum S|δn(kr,kθ)|2 is expected to be asymmetric in kr in the presence of a symmetry breaking mechanism.47 Here the inclusion of perpendicular E × B and parallel flow shear in the nonlinear gyrokinetic simulations break an underlying symmetry in the gyrokinetic equation leading to an asymmetric spectrum in kr. The peak of the spectrum for a finite kr also introduces a tilt of the turbulent eddies in the perpendicular direction,48 and has previously been observed in nonlinear gyrokinetic simulations of ETG in NSTX49 and MAST.50 These projections suggest that high-k scattering could provide experimental evidence of such an asymmetry.

The same procedure is performed to probe the kr > 0 parts of the density fluctuation wavenumber power spectrum. Figures 13(a) and 13(b) show a view from the top and poloidal cross section of the probing, scattered and selected k+ characterized by kr+>0, while Fig. 13(c) shows the corresponding frequency spectra. Once more a small variation is introduced in kθ due to the high kR values and off-midplane corrections. This time kr is scanned from krρsϵ[0.1,6], while kθ is in the range kθρs[5,8]. The peak power of the frequency spectra spans two orders of magnitude and peaks at frequencies f(12) MHz. The frequency peak is again shown to be proportional to the selected kθ+, while no significant variation can be observed in the spectral width with kr+>0.

FIG. 13.

(a) View from the top of the probe and scattered beam trajectories for scanning the measuring the dependence of the density fluctuation spectrum for kr>0. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

FIG. 13.

(a) View from the top of the probe and scattered beam trajectories for scanning the measuring the dependence of the density fluctuation spectrum for kr>0. (b) Poloidal cross section on the (R, Z) plane corresponding to the geometry in (a). Colored arrows represent the measured k+ for each scan. (c) Frequency spectra computed via synthetic high-k diagnostic corresponding to the wave-vectors in (a) and (b).

Close modal

The selected k+ having kr+>0 are plotted by black dots over the 2D wavenumber spectrum in Fig. 13(a). The corresponding variation of the spectral power S(kr+,kθ+) with kr+>0 is given in Fig. 13(c) by the black dots. Note this time no peak is observed for positive kr>0, and the spectrum is shown to monotonically decrease. A power-law |δn|2kr2.06 can be fit to the spectrum for krρs1, a factor of two shallower than for negative kr<0 (note however the applicable range is |krρs|6). The asymmetry with respect to the peak of the spectrum and spectral exponent in kr could possibly be quantifiable in future scattering experiments.

We note that the values of the metrics presented here (peak of the spectrum and spectral indices in kr, kθ) are not affected by the synthetic diagnostic. The observable values issued from the synthetic diagnostic are in agreement with the corresponding values of the peak and spectral indices computed from raw density fluctuations taken as direct output from nonlinear gyrokinetic simulation.

In summary, in this article, we have reviewed previous validation work of nonlinear electron-scale gyrokinetic simulations via comparisons of density fluctuation spectra between experiment and simulation;18 we have discussed the potential role that the ion-scale instability and turbulence might play on the weakly driven ETG condition and used the strongly driven ETG condition to project future high-k measurements in NSTX-U.

Section II constitutes a succinct review of Refs. 18 and 19 and has given extensive experimental evidence supporting the thesis of ETG-driven turbulence as the main responsible driver of anomalous electron thermal transport in the outer core-gradient region of a modest-β NSTX NBI-heated H-mode. These conclusions are reached after performing detailed comparisons of electron thermal transport and measured high-k density fluctuation spectra between experiments and simulation. These have shown that the anomalous levels of electron thermal power can be reproduced by nonlinear electron-scale gyrokinetic simulation in conditions where high-k density fluctuations measured by high-k scattering were both low and high (respectively, alluded to weak and strong ETG drive times). Simulations were simultaneously able to reproduce the shape of the measured high-k wavenumber spectrum, as well as the ratio of fluctuation levels between strongly driven and weakly driven ETG turbulence conditions. As concluded in Ref. 18, this provides the strongest experimental evidence to date supporting ETG-driven turbulence as a main driver of anomalous electron thermal transport in the outer core of a modest-β ST plasma.

One important comment should be made with respect to the role of ion-scale turbulence, presented in Sec. III. For the strong ETG condition, it was shown in Ref. 18 that ion-scale turbulence cannot be destabilized within uncertainty in the turbulence drive terms. This is consistent with the maximum linear growth rate lying clearly below the background E × B shearing rate (Fig. 4). For the weak ETG condition, the situation is more subtle. We summarize the most important evidence for the weak ETG condition: (1) the measured high-k fluctuation power is lower than for the strong ETG condition, (2) ion thermal transport is very close to the neoclassical levels predicted by NEO,23 and (3) nonlinear electron-scale gyrokinetic simulation can reproduce electron thermal power within uncertainty in T and n (shown in Ref. 18). This evidence allowed us to conclude that ion-scale turbulence drives negligible electron thermal transport for the weak ETG condition. However, as shown in Sec. III A, the ion-scale instability lies very close to the linear KBM threshold (βcrit). Additionally, Sec. III B shows that ion-scale turbulence may be very close to the nonlinear threshold in a state possibly reminiscent of the Dimits-shift regime.29 Within experimental uncertainty in the turbulence drives (T,n) a TEM/KBM mode can be excited and lead to unphysically large ion and electron thermal powers.

These findings bring into question the role that the ion-scale instability and turbulence might play at the weak ETG condition. We argue, however, that it is unlikely that ion-scale turbulence might impact our conclusions since the constraints presented mainly concern the strong ETG condition for which we are confident that ion-scale turbulence is suppressed. It is possible, however, that the fluctuation level ratio might be affected by the presence of ion-scale turbulence in the weak ETG regime. This remains speculative owing to the neoclassical ion thermal transport levels: ion-scale turbulence would need to drive electron thermal transport but negligible ion thermal transport in order to comply with the neoclassical ion thermal transport constraint, but it remains a possibility. If this were true, the presence of ion-scale turbulence driving electron thermal transport for the weak ETG condition would imply a lower value of electron-scale electron thermal transport, which would increase the fluctuation level ratio presented with the current simulations (recall the fluctuation level ratio is between the strong and weak ETG drive). This might mean that the experimental value of the fluctuation level ratio could be matched at lower turbulence drives for the strong ETG condition. Additional simulations would need to be performed to confirm this speculation.

One additional possibility concerning the role of ion-scale turbulence is multi-scale interactions between ion-scales and electron-scales, which cannot be ruled out for the weak ETG condition. In fact, recent work51 has shown that the electron thermal power can be greatly enhanced via cross-scale interactions in conditions where ion-scale turbulence is close to the nonlinear threshold: a very similar situation to the weak ETG condition, as shown in Secs. III A and III B. This discussion also shows one weakness of the present work, namely, the lack of experimental constraint on ion-scale turbulence and further motivates the simultaneous use of multi-scale diagnostics in future validation studies.

Successfully validated nonlinear gyrokinetic simulations of ETG-driven turbulence are used to project high-k turbulence spectra by the new high-k scattering system in NSTX-U. Here no assessment is made on the practicality of the measurements, and it is possible that future measurements inspired by the high-k projections made will be challenging or even unfeasible. If experimentally feasible, they could guide future high-k turbulence studies in NSTX-U. Hypothetical measurements of the peak of the high-k fluctuation wavenumber spectrum and the spectral indices in kr and kθ could reveal experimental evidence of the anisotropy in the 2D (kr,kθ) spectrum of high-k turbulence, the tilt of turbulent eddies in the perpendicular direction to B (and the associated breaking of symmetry47,48) and, maybe most importantly, the first experimental demonstration of the presence of radially elongated and poloidally thin streamer fluctuations, which to the best of our knowledge, remain elusive to present. Inspired by previous works,52–55 we have identified the peak of the density fluctuation wavenumber spectrum in kθ and kr and the spectral indices in kθ and kr as critical validation metrics on the turbulence that could be used to validate future electron-scale gyrokinetic simulations. Turbulence characteristics, as well as turbulence fluxes, should be matched in order to fully validate our current turbulence and transport models. This in turn will guide the development of synthetic diagnostics, which yield a profound understanding toward the interpretation of fluctuation measurements. High-k measurements in NSTX-U will be possible from the core r/a0.1 to the pedestal at r/a0.99, and in higher performance plasmas, which will reveal intrinsic characteristics of the underlying turbulence in yet unexplored parameter regimes to date.

Care should be taken to extrapolate the results shown here to higher performance ST scenarios in the upcoming NSTX-U and MAST-U as well as for future reactors based on the ST concept, both from the machine specifications and the expected nature of the turbulence. Machine specifications such as doubling of the magnetic field in NSTX-U could decrease the size of the turbulent structures since their characteristic length scales with ρs (ρsTe1/2/B). Although higher temperatures will likely be reached in full-field NSTX-U experiments, Te will need to increase fourfold in order to maintain the same ρs as in NSTX. Assuming the turbulence characteristics unchanged, full-field NSTX-U experiments are likely to exhibit smaller ρs values, translating to smaller ion and electron-scale turbulent structures, i.e., characterized by higher k (m−1). Another factor impacting the measured values of kr and kθ is the flux surface geometry. Equation (1) shows how the flux-surface geometry can impact the mapped kr+ and kθ+ for a given k+. For clarity, Eq. (1) is simplified at the outboard midplane to kr+kR+/|r| and kθ+κq/(α/θ)kz+, showing how |r| [related to Shafanov shift |r|1/(1+Δ)] and κ (elongation) can strongly impact the measurement at the outboard midplane, in particular, for strongly shaped plasmas such as STs (and will also vary radially). However, the high-k scattering capabilities for NSTX-U should comfortably allow the measurement of even smaller streamer structures than those presented here, due to the large spectral range in k accessible [k>40 cm−1 (Ref. 27)]. Sampling the kr spectrum might be most limiting factor in future experiments (recall the spectral range for the present projections is |krρs|6 while kθρs27,  Appendix B), but the spectral peak at krρs1 and kθρs56 should comfortably be accessible in NSTX-U.

Care should also be taken to extrapolate the present high-k projections to higher performance plasmas accessing different turbulence regimes. NSTX H-mode 141767 is a relatively low β, modest collisionality spherical tokamak plasma. Ion-scale turbulence was stabilized by strong E × B shear and ETG was the dominant electron heat transport mechanism. However, as mentioned in Sec. I and in Fig. 1, the kinetic ballooning mode and micro-tearing modes are predicted to be the dominant instability in higher β regimes. High performance ST plasmas will also have lower collisionality, which following Fig. 1 would point to the kinetic ballooning mode as the dominant instability predicted in high performance ST scenarios. The presence of electron-scale turbulence in a different turbulent regime might peak at different krρs and kθρs values, which would have implications in the measurement (e.g., more weakly driven ETG turbulence would likely exhibit smaller eddies, i.e., higher k). The presence of electromagnetic turbulence also motivates further work to detect electromagnetic modes in NSTX-U and MAST-U devices, using diagnostics such as polarimetry, cross-polarization scattering, or additional techniques.

Further down the road, the goal is to predict equilibrium plasma profiles of future high performance ST reactors. This work serves as a stepping stone to understanding electron thermal transport, but more emphasis should be placed on developing reliable predictive transport simulations coupled with reduced transport models such as TGLF,56 which is still at its early stage for the spherical tokamak. Developing reduced transport models accurate to the high beta, high flow, and low-aspect ratio ST regime should be a main research priority of future confinement studies in NSTX-U and MAST-U. These efforts would drive the prediction and optimization of the performance of future fusion ST reactors.

The author would like to thank the whole NSTX team for providing the profile data used for the analysis of the plasma discharge presented here. This work was supported by U.S. D.O.E. Contract No. DE-AC02-09CH11466 and in part by the Engineering and Physical Sciences Research Council (EPSRC) [EP/R034737/1]. Computer simulations were carried out at the National Energy Research Scientific Computing Center, supported by the Office of Science of the U.S. D.O.E. under Contract No. DE-AC02-05CH11231, and at the MIT-PSFC partition of the Engaging cluster at the MGHPCC facility (www.mghpcc.org), which was funded by D.O.E. Grant No. DE-FG02-91-ER54109. N.F.L. was partially funded by U.S. D.O.E. Grant No. DE-FG02-91ER54109. The digital data for this paper can be found in http://arks.princeton.edu/ark:/88435/dsp011v53k0334.

In this appendix, we clarify the difference in coordinates and notation employed for the synthetic diagnostic throughout this manuscript with respect to a previous publication.19 

In Ref. 19, the synthetic diagnostic was described in terms of a mapping between wavenumber components written in a Cartesian coordinate system (kx,ky,kz)+ [noted (kx,ky,kz)+ (Ref. 19) in what follows] mapped to toroidal and radial mode numbers (n,p)+ in GYRO [Eq. (8) in Ref. 19]. We emphasize that the Cartesian coordinates from Eq. (8) in Ref. 19 do not correspond to the Cartesian coordinates defined here. In Ref. 19, the Cartesian components employed a Cartesian system centered at the scattering location, where kx (Ref. 19) was along the major radial direction, ky (Ref. 19) was along the toroidal direction, and kz (Ref. 19) along the vertical direction. In the case of the 2D description of the synthetic diagnostic [in the (R, Z) plane], these are, respectively, equivalent to the kR, kφ, and kZ components defined here. However, special care must be taken when implementing the 3D version of the synthetic diagnostic including toroidal variation. The cylindrical coordinate system varies with toroidal angle, which introduces variations in the (kR,kφ,kZ) components for a fixed k+. Using fixed kR+,kφ+, and kZ+ components in the synthetic diagnostic would result in selecting a different k+ in different toroidal locations, which would be inconsistent. Using the Cartesian coordinate system from Ref. 19 alleviates this issue. However, in the limit of the 2D synthetic diagnostic description, both coordinate systems are strictly the same.

Throughout this manuscript, the Cartesian components (kx,ky,kz) correspond to a Cartesian system centered at the center of the tokamak domain, where kx, ky, and kz are the k-components along the X, Y, and Z-directions, respectively (Fig. 9), and should not be confused with the Cartesian coordinate system in Ref. 19. Figure D2(b) in Ref. 19 shows the Cartesian component definitions (kx,ky,kz).19 

With respect to the (n, p) mode numbers used in the synthetic diagnostic description in Ref. 19, these have a direct relationship to the (kθ,kr) defined in this manuscript, namely, (kθ,kr)=(nqr,2πpLr) where Lr is the radial box dimension of the gyrokinetic simulation.

In this appendix, we provide additional details concerning geometry definitions and specific parameter values needed in the wavenumber mapping [Eq. (1)]. The values of k, the cylindrical components (kR,kZ,kφ) and field-aligned (kr,kθ) components used in the projection scans for high-k scattering in NSTX-U are also included. These are related via Eq. (1), repeated here for reference

krrqαrkθ=RrkR+ZrkZ1qαθkθ=1rRθkR+1rZθkZ.
(A2)

In Eq. (A2), R=R(r,θ) and Z=Z(r,θ) are the flux functions describing the shape of the flux surface. Here we employed the Miller flux surface parametrization.26 α=α(r,θ) is the field-line label, related to the equilibrium magnetic field via B=α×ψ, where ψ is poloidal flux divided by 2π. The functions R, Z and α can be computed numerically from knowledge of the magnetic equilibrium reconstruction such as provided by EFIT.

The first row of Table I shows the Miller geometry parameters κ,sκ,δ,sδ,R0/a,Δ,q,ŝ,β*26 necessary to reconstruct the flux surface functions R(r,θ),Z(r,θ) (here we have chosen to provide β*=8π/Bunit2dpdr instead of αMHD). The second row of Table I shows the radial and θ derivatives of R, Z and α necessary for the wavenumber mapping, along with the normalized minor radius r/a, the θ location of scattering θs and |r| the gradient of r. The cylindrical coordinates of the scattering location are Rs=1.3547 m, Zs=0.05771 m, and φs=115.71°. Normalization parameters are Bunit=1.4461 T, ρs=0.1975 cm (using Bunit), ρsexp0.7 cm, and a =0.601 2 m minor radius at the last closed flux surface.

TABLE I.

First row provides the Miller geometry parameters26 necessary to reconstruct R(r,θ),Z(r,θ): κ elongation, sκ=rdln(κ)/dr, δ triangularity, sδ=rdδ/dr,R0/a aspect ratio, Δ=dR0/dr Shafranov shift, q safety factor, ŝ=(r/q)dq/dr magnetic shear, β*=8π/Bunit2dpdr. Second row provides additional geometry parameters necessary in the wavenumber mapping [Eq. (A2)]: r/a normalized minor radius, θs poloidal angle of scattering location, along with r and θ derivatives of R, Z and α. |r| is the gradient of r.

κsκδsδR0/aΔqŝβ*
2.11270.15400.24830.32401.5227−0.30413.41032.16560.02167
r/a θs(deg) Rr Rθ Zr Zθ αr αθ |r| 
0.7084 −3.5552 0.6913 0.0427 −0.1512 0.9289 −0.2031 1.1551 1.4337 
κsκδsδR0/aΔqŝβ*
2.11270.15400.24830.32401.5227−0.30413.41032.16560.02167
r/a θs(deg) Rr Rθ Zr Zθ αr αθ |r| 
0.7084 −3.5552 0.6913 0.0427 −0.1512 0.9289 −0.2031 1.1551 1.4337 

Next we provide the set of values of k, the cylindrical wave-vector components (kR,kZ,kφ), and the mapped values to the field-aligned (kr,kθ) used for scanning the kθ (Table II), kr < 0 (Table III), and kr>0 dependence (Table IV) of the high-k wavenumber spectrum in Sec. IV of the main text. Since k is intrinsically a local quantity defined by the measurement, in Tables II–IVk is normalized to the local, experimental ρsexp (using local BT), which can prove useful for the experimental diagnostician. On the other hand, kr and kθ are normalized to ρs using Bunit as internally used in GYRO, consistent with making “apples-to-apples” comparisons with simulated spectra. This will prove useful to the gyrokinetic simulation expert and turbulence theoretician.

TABLE II.

The set of wavenumber components used for scanning the kθ dependence of the spectrum in Sec. IV. All k+ have a constant value of ky=4 cm−1.

Wavenumber components in kθ scan
k (cm−15.2 8.5 11 14 18 25 
kρsexp 3.6 4.2 4.9 5.9 7.7 9.8 12.6 17.5 
kR (cm−1−4.83 −5.09 −5.38 −5.76 −6.35 −7.02 −7.90 −9.40 
kZ (cm−11.76 2.87 4.04 5.62 8.07 10.86 14.50 20.75 
kφ (cm−1−0.80 −1.36 −1.94 −2.74 −3.96 −5.36 −7.18 −10.30 
krρs −0.67 −0.70 −0.74 −0.79 −0.87 −0.96 −1.07 −1.26 
kθρs 2.00 3.42 4.93 6.96 10.09 13.69 18.35 26.38 
Wavenumber components in kθ scan
k (cm−15.2 8.5 11 14 18 25 
kρsexp 3.6 4.2 4.9 5.9 7.7 9.8 12.6 17.5 
kR (cm−1−4.83 −5.09 −5.38 −5.76 −6.35 −7.02 −7.90 −9.40 
kZ (cm−11.76 2.87 4.04 5.62 8.07 10.86 14.50 20.75 
kφ (cm−1−0.80 −1.36 −1.94 −2.74 −3.96 −5.36 −7.18 −10.30 
krρs −0.67 −0.70 −0.74 −0.79 −0.87 −0.96 −1.07 −1.26 
kθρs 2.00 3.42 4.93 6.96 10.09 13.69 18.35 26.38 
TABLE III.

The set of wavenumber components used for scanning the kr<0 dependence of the spectrum in Sec. IV. All k+ have a constant value of kz = 4 cm−1.

Wavenumber components in kr<0 scan
k (cm−14.5 6.5 10 15 20 28 36 45 
kρsexp 3.1 4.6 10.5 14 19.6 25.2 31.5 
kR (cm−1−0.46 −4.74 −8.97 −14.35 −19.52 −27.67 −35.75 −44.80 
kZ (cm−1
kφ (cm−1−2.01 −1.94 −1.86 −1.77 −1.68 −1.53 −1.39 −1.23 
krρs −0.06 −0.65 −1.24 −2.00 −2.72 −3.86 −4.99 −6.25 
kθρs 5.17 4.91 4.66 4.34 4.03 3.54 3.06 2.52 
Wavenumber components in kr<0 scan
k (cm−14.5 6.5 10 15 20 28 36 45 
kρsexp 3.1 4.6 10.5 14 19.6 25.2 31.5 
kR (cm−1−0.46 −4.74 −8.97 −14.35 −19.52 −27.67 −35.75 −44.80 
kZ (cm−1
kφ (cm−1−2.01 −1.94 −1.86 −1.77 −1.68 −1.53 −1.39 −1.23 
krρs −0.06 −0.65 −1.24 −2.00 −2.72 −3.86 −4.99 −6.25 
kθρs 5.17 4.91 4.66 4.34 4.03 3.54 3.06 2.52 
TABLE IV.

The set of wavenumber components used for scanning the kr>0 dependence of the spectrum in Sec. IV. All k+ have a constant value of kz = 4 cm−1.

Wavenumber components in kr>0 scan
k (cm−14.5 6.5 10 15 20 28 36 45 
kρsexp 3.1 4.6 10.5 14 19.6 25.2 31.5 
kR (cm−10.39 4.67 8.90 14.28 19.45 27.60 35.68 44.73 
kZ (cm−1
kφ (cm−1−2.03 −2.10 −2.17 −2.27 −2.36 −2.50 −2.64 −2.80 
krρs 0.06 0.66 1.25 2.00 2.73 3.87 5.00 6.26 
kθρs 5.22 5.47 5.73 6.05 6.36 6.84 7.32 7.87 
Wavenumber components in kr>0 scan
k (cm−14.5 6.5 10 15 20 28 36 45 
kρsexp 3.1 4.6 10.5 14 19.6 25.2 31.5 
kR (cm−10.39 4.67 8.90 14.28 19.45 27.60 35.68 44.73 
kZ (cm−1
kφ (cm−1−2.03 −2.10 −2.17 −2.27 −2.36 −2.50 −2.64 −2.80 
krρs 0.06 0.66 1.25 2.00 2.73 3.87 5.00 6.26 
kθρs 5.22 5.47 5.73 6.05 6.36 6.84 7.32 7.87 

To end, in Fig. 14 are given the values of the collection point (X, Z) coordinates of the central scattered rays for each of the scans presented in Sec. IV. These could prove useful to understand the practical feasibility of the measurements as well as guide future experiments.

FIG. 14.

Collection point coordinates (X, Z) taken to be at Y =1.8 m for the different schemes developed in Sec. IV to scan the kθ and kr dependence of the high-k density spectrum. The Cartesian (X, Y, Z) coordinates are the same as in Fig. 9.

FIG. 14.

Collection point coordinates (X, Z) taken to be at Y =1.8 m for the different schemes developed in Sec. IV to scan the kθ and kr dependence of the high-k density spectrum. The Cartesian (X, Y, Z) coordinates are the same as in Fig. 9.

Close modal

Table V summarizes the parametric scans performed for the five simulations corresponding to the strong ETG condition that were employed in the validation work of Sec. II C. The uncertainties in a/LTe,a/Lne were computed from uncertainties in the background electron density and temperature profiles followed by a Monte Carlo analysis approach, and the values of 10%q and +20%ŝ were taken as reasonable estimates of the equilibrium reconstruction uncertainty.

TABLE V.

The sensitivity scans performed using electron-scale simulation for the strong ETG case are summarizing, including the value of the drive terms scanned (a/LTe,a/Lne,q,ŝ) used in the input, as well as the predicted value of the electron thermal power with respect to the experimental level Pesim/Peexp. The scaled parameter values with respect to the nominal, base case are shown in bold gray.

Baseσ(T,n)σ(T)q,ŝσ(n),q,ŝσ(T,n),q,ŝ
a/LTe 3.3626 4.2032 4.2032 3.3626 4.2032 
a/Lne 1.0048 0.5024 1.0048 0.5024 0.5024 
q 3.7892 3.7892 3.4103 3.4103 3.4103 
ŝ 1.8047 1.8047 2.1656 2.1656 2.1656 
Pesim/Peexp 29% 70% 95% 91% 170% 
Baseσ(T,n)σ(T)q,ŝσ(n),q,ŝσ(T,n),q,ŝ
a/LTe 3.3626 4.2032 4.2032 3.3626 4.2032 
a/Lne 1.0048 0.5024 1.0048 0.5024 0.5024 
q 3.7892 3.7892 3.4103 3.4103 3.4103 
ŝ 1.8047 1.8047 2.1656 2.1656 2.1656 
Pesim/Peexp 29% 70% 95% 91% 170% 
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