V. RESULTS DURING ON-AXIS NBI
Plasmas with on-axis NBI provide an opportunity to probe conditions (e.g., ) in which energetic ion diffusion due to microturbulence is theoretically expected. In addition, radial beam ion diffusion occurs always in the outward direction for these centrally peaked energetic ion profiles, which may increase any profile flattening. The magnetic equilibrium for the on-axis NBI shots is shown in Fig. 4. It is not possible to obtain FIDA, BES, ion temperature and plasma rotation profile data simultaneously given the constraint on the total beam power (MW), therefore, the results presented here involve different measurements during repeat shots. These are L-mode plasmas (T and MA) in which the ratio is lowered by holding beam energy fixed (keV) and increasing roughly a factor of two using electron cyclotron heating (ECH) deposited at . This results in one set of measurements for the condition of and another set at . The time evolution of these paired and repeated shots is shown in Fig. 11, indicating the well-matched behavior. The higher value of is achieved in shots 142358 & 142380, while the lower ratio is achieved in 142371 & 142381. Radial profiles for these shots, during the MHD-quiescent period of analysis, are shown in Fig. 12. Applied ECH in shot 142371 produces significantly higher electron temperatures [Fig. 12(a)], while the ion temperature and electron density varies little [Figs. 12(b,c)]. The toroidal rotation profiles are shown in Fig. 12(d), where the dashed lines represent the statistical uncertainty range (in the other plots this range is very small, so those dashed lines are removed for clarity).
The turbulence is well characterized in these plasmas. Figure 13 shows the measurable fluctuation amplitude profiles for density [Fig. 13(a)] and temperature [Fig. 13(b)]. While the density fluctuation amplitude is the same in both cases and peaking near , the electron temperature fluctuations increase by nearly a factor of two in the lower case. The temperature fluctuation increase is consistent with a transition from ion temperature gradient (ITG)-dominated turbulence (in the higher case without ECH) to TEM-dominated turbulence (in the lower case employing ECH to raise the value of ).  The error bars in Fig. 13(b) represent statistical uncertainty, while the sensitivity limit is equivalent to the reported uncertainty of the temperature fluctuations noted in Fig. 7. Further evidence for the difference in microturbulence character between these cases is given by the cross-phase angle between the density and temperature fluctuations, , shown in Fig. 14. The measured cross-phase angle is indicated by the darker lines, while the lighter portions of the trace indicate frequencies for which the density/temperature fluctuation coherency is too small to accurately resolve . In the higher case of Fig. 14(a), , while in the lower case of Fig. 14(b) the value is . These values are accurately reproduced by GYRO simulations that are plotted as dash-dot lines, which further indicates that the turbulent electron heat transport is approximately five times larger in the lower case. Detailed analysis and simulation of the microturbulence in these paired shots is given in Ref. 112. TGLF is used to calculate the real frequency of the most unstable mode, which is shown in Fig. 15, indicating that the lower has a larger region dominated by electron mode turbulence, while the higher case has a larger region dominated by ion mode turbulence. The radial correlation length of the turbulence is determined using BES measurements of density fluctuations and found to be cm in both cases. Across the pitch angle and energy ranges of energetic ions in these T plasmas, this radial length can be comparable to the energetic ion gyroradius.
Energetic ion profiles for the higher case are shown in Fig. 16. These FIDA density profiles are calculated over the energy range keV. Expected profiles based on classical transport, and calculated with the FIDASIM synthetic diagnostic, are indicated by the dashed lines and ensuing uncertainty ribbons. In Fig. 16(a), the profiles are taken from the plasma current ramp during which Alfvén eigenmodes were observed. Scaling the measured profile by 1.8 produces the minimum . This value is times as large as those values determined for all of the profile comparisons made during MHD-quiescent time periods when microturbulence is expected to be the dominant transport mechanism. In contrast to the Alfvénic time period, the classical and experimental profiles of the MHD-quiescent period in Fig. 16(b) agree very well. Here, a scaling factor of 1.15 produces .
It is interesting to note that the on-axis NBI cases produce a spatial profile of energetic ions and microturbulence similar to that in ITER. As shown in Fig. 14 of Ref. 63, the -particle profile in ITER is expected to be core-localized. Microturbulence, driven by pressure gradients, begins to reach appreciable fluctuation amplitude near mid-radius, and then peaks further out towards the edge. The result is that only for the region is there both a sizable -particle density and large amplitude turbulent fluctuations. Figure 17 reproduces this situation using measurements and calculations from the higher case. On-axis NBI is core-localized as indicated by the FIDA density profile [taken from Fig. 16(b)]. The growth rate of the most unstable turbulent mode is calculated by TGLF and plotted as the dashed line. This scenario should be favorable to measuring a turbulent transport effect because any enhanced diffusion should serve to move energetic ions out of the plasma center. Our results indicate that turbulent diffusion is below an observable limit in this scenario.
The lower case results in comparatively large values of , as shown in Fig. 18. Figure 18(a) shows that m$^2$/s is achieved up through keV. The radial distribution peaks nearer to the plasma center compared to the off-axis NBI case discussed in Sec. IV. The trapped ion exhibits a similar radial profile, but is nearly a factor of two smaller than the passing component for all energies. These levels of diffusion lead to a large modeled redistribution of energetic ions. Figure 19(a) plots the NUBEAM-modeled energetic ion density for this case for both the classical and Pueschel treatments. The large values of for in the Pueschel formulation result in a significant depletion of energetic ions in the plasma center. This transport is large enough to affect the beam current drive profile as shown in Fig. 19(b). Integrating over this profile indicates that the total beam-driven current of the Pueschel case (kA) is smaller than the classical expectation (kA). This is a unique result, as on-axis beam injection modeling  in ASDEX Upgrade produced no change in beam-driven current profiles when setting m/s.
FIDA measurements from this case are shown in Fig. 20. The range for FIDA analysis is determined by review of the individual spectra that are plotted in Fig. 20(a). Lower Doppler shifted energies are resolved in this shot, and the vertical dashed lines mark the values of and keV. This wide range of analysis should provide a better opportunity to identify transport due to microturbulence since that effect increases as the energetic ion energy decreases. The resulting FIDA density profiles are shown in Fig. 20(b). The -symbols mark the measured values, which are plotted alongside the expectations based on the classical and Pueschel models. Absolute comparisons between the modeled results are valid, and this shows that the energetic ion profile modification is as expected: enhancing diffusion at mid-radius leads to a reduction of the core energetic ion density. The measured FIDA density fits equally well to either modeled result. For the classical comparison, at the best-fit data scaling factor of 1.33, and for the Pueschel comparison, at a scale of 1.13. Motivated by the strong energy dependence in for this case, we performed a separate set of fitting analyses based on the FIDA spectra. Since FIDA density is integrated spectra, it may be smoothing out significant differences between the measured and modeled spectra as a function of energy. The quality of the fit is calculated based on radiance only over the FIDA density integration range. These results are shown in Fig. 21. Figure 21(a) is a plot of the for each radial chord. The measured spectra are a decent fit to each model, producing . The classical spectra are systematically better fits than the Pueschel spectra, though this is a small improvement. In Fig. 21(b) a radial profile of the scaling factor corresponding to the best-fit spectrum is shown. There is little variation in the scaling factor across the radius. Two example spectra are shown in Fig. 21(c). Here, the modeled results are divided by the best-fit scaling factor in order to show the level of agreement between these spectra. Representative error bars on the measured spectra indicate the statistical uncertainty (the systematic uncertainty is addressed by the process of fitting). At , the best-fit classical profile is within the statistical uncertainty across the spectrum, while the Pueschel model produces too low a radiance at the lower energy limit. This suggests that the modeled turbulent transport of lower energy ions is greater than experimentally observed. For both models produce excellent agreement with the shape of the measured FIDA spectrum.
An investigation of possible global effects due to turbulent transport is shown by Fig. 22. An autopower spectrum of density fluctuations measured by an interferometer is given in Fig. 22(a). The interferometer chord lies along the midplane and identifies apparent Alfvén cascades through ms. The MHD-quiescent period used for analysis of transport due to microturbulence, ms, is enclosed by the dashed rectangle. The total plasma stored energy is plotted in Fig. 22(b). The measured values for this pair of lower shots (142371 & 142381) are shown to be equivalent. The results from TRANSP analysis based on the classical and Pueschel energetic ion transport anomalous diffusivities are also plotted. In this case, the classically expected stored energy is slightly larger than measured during the time range for which MHD is observed. This is to be expected since the thermal plasma energy is essentially an input to TRANSP (by way of the plasma profiles and equilibrium), while the energetic ion contribution is determined based on NUBEAM calculations. After ms, as the MHD activity decays away, the modeled stored energy traces approach the measured value. Even if the Pueschel model is perfectly describing energetic ion transport due to microturbulence, that transport has a smaller effect on stored energy than the spectrum of Alfvén eigenmodes driven in this low beam heating scenario of MW.
A. Trends in the present results
A survey of all the shots studied is presented by the plot of in Fig. 23. This plot concerns the fitting results of FIDA density and, in the case of the main ion system, the FIDA brightness as a function of major radius. In the presence of weak Alfvén eigenmode activity (these AEs are driven by MW, while dedicated AE experiments  typically feature MW), the fit is poor and . The FIDA profile at , where we may theoretically expect to observe the most dramatic energetic ion transport by microturbulence, is fairly well described by a classical model using . Furthermore, a turbulence model including a treatment tends to produce a poorer fit. The on-axis beam injection case with is a suitable condition for testing models of turbulent energetic ion diffusion because the highest temperatures overlap with the location of the beam ions. Both the DEP and Pueschel models presented in this work determine based on the local temperature, though previous experimental results and Fig. 23 report using the peak . These are nearly the same in the on-axis case for which the lowest value of is achieved. The value of in all cases is taken as the highest injection energy from a neutral beam during the time of interest (meaning much lower values of are achieved during the slowing down process). The off-axis beam injection case at produces the worst fit to the classical model, though in this case the inclusion of turbulent transport does not improve the agreement with experiment.B. Consideration of previous results
Three experimental results are often cited as evidence for energetic ion transport by microturbulence. Given that the present work demonstrates that microturbulence-induced energetic ion transport is insignificant, it is useful to revisit the previous works. Results from AUG intended to show  that microturbulence enhances beam ion diffusion sufficiently well that off-axis NBCD is reduced. The plasma parameters are well characterized, and the experimental method successfully maintains comparable scenarios amongst changes in applied heating and plasma shape. Perhaps one challenge for this work is that there were no current drive or FIDA (or equivalent energetic ion profile diagnostic) measurements during the off-axis beam injection period. Current drive profiles were measured before and after the off-axis period, with TRANSP simulations used to infer the behavior during off-axis NBCD. Agreement between simulated and measured behavior in these before and after time periods was achieved by setting m/s constant in radius and energetic ion phase space. This value, especially when applied to ions of keV, is much larger than the modeling performed with the DEP and Pueschel methods in the plasmas shown here. Electron temperature data from the AUG cases implies keV and shows keV, which is similar to the off-axis NBI shot of Sec. IV. While FIDA was not available during those experiments, a system has since been commissioned  and observes classical energetic ion profiles during cases of MW in both on-axis and off-axis beam injection. 
Results from DIII-D claim to demonstrate a measurable energetic ion transport by microturbulence in both off-axis [44,45,48] and on-axis  scenarios. The crucial aspect of these experiments is that the plasmas are MHD-quiescent. The largest NBCD differences between classical expectation and measurement are identified in the highest shots. In Ref. 45 it is noted that the highest shots feature either intermittent tearing modes or weak Alfvénic activity. Since any beam ion transport resulting from these fluctuations is not quantified, the result cannot be considered a strong argument for microturbulence-induced transport. Specifically, tearing modes are well known  to reduce NBCD. Additional analysis of these high-power shots is presented in other publications, [44,48] but still without quantifying the effects of coherent fluctuation activity. Another challenge for these experiments is that the off-axis NBCD is achieved by vertically shifting the plasmas. Since plasma profile diagnostics are fixed in real space, profile data is available only for . This leads to a great variability in the profiles as used for current drive and power balance calculations. An accounting of the variability in current drive across the possible range of central profiles should be provided. An on-axis beam injection example is presented in Ref. , where decreasing beam energy is reported to enhance the energetic ion transport by microturbulence because the injected beam ions feature smaller gyroradii. In this example, additional characterization of the beam injection and deposition may be necessary. For example, changing the neutral beam voltage can affect the injected power by producing a change in the fractional values of the 1/2 and 1/3 energy components. The development of advanced systems allows for the measurement (or, technically, the inference) of neutral beam performance  and greater confidence in the realized .
The third experiment concerns off-axis NBCD measurements  from JT-60U. Here, however, the anomalous behavior actually suggests a lack of microturbulence-induced transport. The measured beam driven current profiles are overly peaked compared to broader profiles produced in classical transport calculations. Furthermore, the measured current profiles peak for , and the shots featured MW. Taken together, these previous experimental results represent excellent work from three different facilities exercising wide arrays of diagnostic and modeling coverage. The ability of energetic ions to excite difficult-to-observe modes (e.g., multiple small amplitude Alfvén eigenmodes are known to enhance energetic ion transport,  along with the inability to measure energetic ion diffusion profiles directly, suggests that a particularly discerning review of such experimental results is warranted.
A final note in review of previous experiments concerns improvements to neutral beam current drive modeling. The choice of beam current shielding model [117,118] in NUBEAM is known  to vary the calculated beam driven current by up to 20%-30% for the plasma parameters of existing tokamaks. For consistency across shots, all of the NUBEAM calculations presented here are performed with the Honda shielding model.  The atomic physics options of NUBEAM have also been updated in recent years, and the present work uses the ADAS310 option.  Future review of the DIII-D cases mentioned above will be conducted with these same options.
C. Consideration of “energetic” ions
While the shots presented here cover a wide range of parameter space, including that which should produce a measurable effect, it remains possible that other discharges will provide evidence for a turbulent transport effect. Theoretical scalings with suggest that, eventually, an energetic ion slows down to a low enough energy that it experiences diffusion due to microturbulence. Indeed, it is known that in L-modes the turbulent transport of the thermal plasma is orders of magnitude larger than neoclassical expectations, so we should expect that energetic ions experience turbulent diffusion before thermalizing. We therefore consider whether our diagnostic suite is capable of measuring such effects throughout the slowing down evolution of a beam ion population. Figure 24 is a contour of the difference between the energetic ion distributions calculated in the Pueschel and classical cases. Negative values indicate a reduced number of beam ions within that phase space compared to the classical calculation. The solid outline labeled “FIDA Signal Region” is the contour level of the FIDA phase space weighting from Fig. 3. The largest change in the distribution (recalling this is the increased model) occurs below keV, where most of the change is found outside of the FIDA sensitivity range. It is reasonable to conclude that perhaps a transport enhancement due to microturbulence occurs, but that it manifests at energies only slightly above the thermal plasma and therefore produces no relevant or observable effects on the high-energy ion distribution. If so, then we should expect that measurements of this effect require the application of full spectrum fitting, such as is possible with the main-ion charge exchange diagnostic  that acquired the result shown in Fig. 10.
The overwhelming indication from experiments, including those discussed in Sec. IIA and the present work, is that microturbulence is an insignificant transport mechanism for energetic ions in tokamaks (limitations are detailed in the next paragraph). In much of the presently achievable tokamak parameter regime, in addition to the scenarios from DT-operation at TFTR and JET, energetic ion transport due to coherent modes is always dominant. For example, trapped energetic ions are theoretically expected to suffer less diffusion from microturbulence, yet they are easily transported out of confinement by ripple effects or neoclassical tearing modes.  The experimental results presented in this work demonstrate that energetic ion diffusion due to microturbulence is a small effect in tokamak plasmas. Energetic ion profiles, as measured by FIDA systems, remain classical in truly MHD-quiescent plasmas across a wide range of plasma parameters (e.g., ) and the character of microturbulence (e.g., ITG versus TEM). During off-axis neutral beam injection, the driven beam current is accurately modeled by NUBEAM for both strong turbulence in L-mode as presented here, and for high performance shots as shown in Ref. 7. These results are consistent with updated modeling that predicts  no significant energetic ion transport by microturbulence in ITER, while also identifying  plasmas for further study in DEMO and TCV.
While this paper demonstrates that energetic ion transport by microturbulence is a negligible transport channel in tokamaks, there are caveats to that conclusion. First, the full parameter space of tokamak operation is not included here. The summary of results in Fig. 23 does not include the operating point of at the location of off-axis beam injection. Such a scenario is not achieved in the other cited publications that have studied this problem, however, so this may not be an important issue. Second, it is possible that energetic ion profile diagnostics are not sensitive enough to accurately identify diffusion due to microturbulence. That could be true either because the diagnostics need to be improved, or because the transport effect itself is small. The plot of Fig. 24 shows that the largest turbulence-induced perturbation of the energetic ion distribution occurs at energies commonly below the FIDA resolvable limit. This is also demonstrated by the modeled FIDA density profile variation of Fig. 16(b), in which a very large (modeled) turbulent diffusion produces a reduction in the core () that is within the uncertainty of the synthetic diagnostic. From this, it is suggested that further development of synthetic FIDA diagnostics is absolutely necessary to gain confidence in any future identification of turbulent transport. The third caveat is that the principle of energetic ion interaction with small-scale turbulent fluctuations is well established, so the present lack of observable effect must be placed within that physical context. Basic plasma devices clearly demonstrate this transport effect, though since the fluctuation levels routinely reach , perhaps that is the most relevant difference compared to tokamaks where this level is closer to . Recent work on DIII-D finds that an energetic ion population (treated as a hot Maxwellian) must be included in gyrokinetic simulations of high performance QH-modes in order to more closely reproduce the experimentally measured energy fluxes.  It was noted that this inclusion is only necessary in those shots for which the energetic ion density reached large values of , however, and the interpretation is that this is a dilution effect (i.e., not indicative of an interaction between energetic ions and microturbulence).
The advances in physics understanding concerning the interaction between small-scale turbulence and energetic ions may find application in other areas. Energetic electrons have smaller gyroradii compared to energetic ions, and might therefore be expected to diffuse significantly in the presence of microturbulence. Electron cyclotron heating and current drive is capable of producing a narrower spatial distribution than a neutral beam, which might make it easier to measure broadening due to diffusion.  A recent study  examines this process for runaway electrons in the presence of magnetic fluctuations and identifies enhanced transport.
In summary, investigations of energetic ion transport by microturbulence have led to the development of models that predict the resulting diffusivity as a function of ion energy and pitch. This represents a greater modeling ability than is presently available for transport due to Alfvén eigenmodes or other MHD. Certainly, the possible identification of plasma modification due to turbulent energetic ion transport in other scenarios will require the application of these formulations. While the DEP and Pueschel models are readily accessible to experiments, it is also possible to conduct gyrokinetic simulations that simultaneously address the development of the turbulent field and the resulting energetic ion transport. These tools represent a major advance in the area of energetic particle transport in tokamaks. In terms of ITER, the results of this paper suggest that energetic ion concerns should give priority to MHD and other non-microturbulence effects.
This work was supported in part by the US Department of Energy under DE-FC02-04ER54698, DE-FC02-99ER54512, DE-FG03-97ER54415, DE-FG02-07ER54917, DE-AC02-09CH11466, SC-G903402, DE-FG02-08ER54984, DE-AC52-07NA27344, DE-FG02-89ER53296, DE-FG02-08ER54999, and DE-AC05-00OR22725. The authors are grateful to M. Albergante, A. Fasoli, S. Jardin, and the ITPA Energetic Particles Group for thoughtful discussions on this topic, to R.J. Groebner for assisting with the processing of CER data, and to B. Geiger and M. García-Muñoz for sharing their results from ASDEX Upgrade. The experimental work is made possible by the extraordinary personnel of the DIII-D National Fusion Facility.
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