Fast-ion transport in $qmin>2$⁠, high-$β$ steady-state scenarios on DIII-D^a)

Understanding superthermal fast-ion transport caused by interaction with Alfvén eigenmodes (AEs) is an important aspect of steady-state scenario development in existing tokamaks heated by neutral beam injection (NBI). The DIII-D program is testing a range of potential high-β, fully noninductive operating scenarios for next-step long pulse tokamaks leading up to the development of a fusion power plant. Primarily, the shape of the current density profile distinguishes the scenarios from each other, and this ranges from strongly peaked to very hollow. Access to this range of profiles is provided by adjustable NBI and electron cyclotron heating (ECH) and current drive (ECCD).¹ 

Presently and for the near future, the available NBI power on DIII-D is at least 3 times greater than the available ECH power, so most high-$β$ scenario exploration is dominated by NBI power. In such cases, the classically predicted $βfast$ can approach ∼40% of $βtotal$⁠. Low-density operation is often chosen, because this is expected to maximize the non-inductive current drive.² There are plans to increase both the NBI and ECH power, including more off-axis NBI. The classical fast-ion slowing down time on thermal electrons is proportional to $Te3/2/ne$⁠, where $Te$ is the electron temperature and $ne$ is the density.³ If significant increases in NBI and ECH power were applied to future plasmas then the classical slowing-down time would increase, especially at low density, and the classically predicted fast-ion fraction $βfast/βtotal$ could increase. Such changes would increase the possibility for interaction between fast-ions and AEs. This may cause enhanced (i.e., higher than classical prediction) fast-ion transport that may take the form of redistribution to higher radius or loss to the wall. Enhanced fast-ion transport can reduce the power that would otherwise be absorbed by the thermal plasma, and it can reduce or redistribute the NBI current drive.⁴ Therefore, understanding how candidate steady-state operating scenarios can affect the drive and damping of fast-ion modes is essential.

The prototypical high-$β$ steady-state scenario envisioned for many future devices uses an elevated $q$-profile. Various ITER steady state scenario models have $qmin>2$ with $βN$ near 2.5–3.^5,6 A Fusion Nuclear Science Facility design (FNSF-AT) uses $qmin>1.4$ with strong negative central shear at $βN=3.7$⁠.⁷ The ARIES-AT design uses $qmin=2.4$ and $βN=5.4$⁠.⁸ All of these designs are based on several predicted advantages for steady-state operation at high $qmin$⁠. These include (i) a high ideal-wall kink mode $β$ limit when the current density is sufficiently broad or hollow to produce strong coupling to a nearby conducting wall,⁹ (ii) a high bootstrap current fraction, which scales as $β$-poloidal (⁠$βP$⁠) or $qβN$⁠,¹⁰ (iii), avoidance of low-order tearing modes by exclusion of rational surfaces from the plasma, e.g., the $m/n=2/1$ tearing mode when $qmin>2$⁠,¹¹ and (iv) thermal confinement exceeding typical H-mode levels enabled by weak or negative magnetic shear.¹² 

Present experiments can test physics that goes into predictive models for next-step steady-state devices. For example, one can test if a proposed current profile is capable of sustaining the target value of $βN$ without encountering resistive tearing modes. Also of interest is whether or not the current profile is consistent with the expected transport and energy confinement.

Such questions motivated several experiments on DIII-D to identify β_N limits in steady-state scenario plasmas with $qmin$ sustained above 2. The experiments identified unexpected 20%–30% differences in the normalized global energy confinement between two different regimes. Both regimes were tested with toroidal field 1.75–2.1 T, and in double-null shapes. The “standard high-$qmin$” regime has $qmin=$ 2–2.5, $q95=$ 5–7, line-density = $3.5–4.5×1019 m−3$⁠, $IP=$ 0.8–1 MA, and Greenwald fraction¹³ near 0.5. These conditions were chosen to target fully noninductive operation with a significant fraction of externally driven current, and projected fusion gain sufficient for next-step tokamaks like FNSF. In these plasmas, $βN$ is usually limited to about 3 by the available power, and the normalized global energy confinement $H89P$¹⁴ is typically 1.6–1.8.¹⁵ A “high-$βP$” regime developed for testing long-pulse operation on the EAST tokamak¹⁶ has $qmin=$ 3–5, $q95=$ 11–12, line-averaged density = $5–6.5×1019 m−3$⁠, $IP=$ 0.6–0.65 MA, and Greenwald fraction near 1. $βN$ is limited to about 4 by instability rather than by confinement, and $H89P=$ 2–3.

This paper discusses measurements and modeling that confirm that differences in AE stability and enhanced fast-ion transport account for the observed large variation in normalized confinement. To summarize, in standard high-$qmin$ plasmas many AEs are unstable across a majority of the plasma radius, and these lead to fast-ion transport greater than that predicted by classical theory. This limits the absorbed heating power and limits the achievable $βN$⁠. Similar plasmas having $qmin$ just above 1 have less AE activity and approximately classical fast-ion transport. A detailed paper on fast-ion measurements in these discharges was recently published.²⁸ High-$βP$ plasmas have better than expected total energy confinement because of the presence of either a high-radius transport barrier or a high pedestal pressure. In these cases, the fast ion transport approaches the classical level. This is due to lower drive for AEs near the axis caused by a lower gradient in the fast ion pressure. These results suggest that increasing global confinement and $βN$ in high-$qmin$ plasmas at lower $q95$ might be achieved by broadening the NBI and ECH heating profiles to reduce core AE activity, and further optimization of thermal transport.

The current density or $q$-profile is a key attribute of the equilibrium that may be adjusted to optimize steady-state operation and projected fusion power gain. Most potential steady-state scenarios tested on DIII-D for use in a next-step reactor have $q95$ in the range of 5–7 during the high-$βN$ phase of the discharge when the inductive current fraction is minimized. By adjusting the L-to-H-mode transition time and the starting time of high-power NBI and ECH, $qmin$ during the high-$βN$ phase can be set to a value between ∼1 and ∼2.5. At the low end of this range are scenarios like the “high-$li$”¹⁷ and “steady-state hybrid”¹⁸ that have potential applicability in next-step reactors. These tend to have good total energy confinement, i.e., $H89P≥2$⁠, and approximately classical fast-ion confinement. This paper focuses on fast-ion transport in high-$qmin$ plasmas, but comparison of similar high- and low-$qmin$ plasmas serves to elucidate many of the effects.

Figure 1(a) compares the $q$-profiles obtained during the high-$βN$ phases of a pair of consecutive discharges. During the time range t = 3.2–3.7 s the case with $qmin≈1$ obtained $βN=2.9$ using 8 MW of NBI and 1.3 MW of ECH at $H89P=2.2$⁠. During the same time period, the case with $qmin≈2$ obtained $βN=2.6$ using 9.4 MW of NBI and 1.3 MW of ECH at $H89P=1.8$⁠. During the analysis time period, both discharges use the same boundary shape and have density profiles with the same shape, but the q_min ≈ 2 case is about 10% lower density at all radii. The ion and electron temperature profiles outside ρ = 0.5 are the same within 10%, but inside ρ = 0.5 the q_min≈2 case has significantly lower temperature, i.e., 20% lower Te(0) and 25% lower Ti(0). Due to these differences, the q_min≈2 case has about a 20% shorter classical fast-ion slowing-down time at the magnetic axis than the q_min≈1 case. While a shorter on-axis slowing-down time might be expected to result in lower Alfvénic activity, the observations show otherwise. The cross-power density-fluctuation spectra between the two most central interferometer chords shows significantly more toroidal Alfvén eigenmode (TAE)²⁰ activity in the plasma with $qmin≈2$ than the plasma with $qmin≈1$ [Figs. 1(b) and 1(c)]. To quantify this, the average amplitude of coherent modes with frequency in the TAE frequency band²⁸ is about four times higher in the high-$qmin$ plasma. A plastic scintillator neutron counter measured a ∼43% lower signal in the high-$qmin$ case compared to the low $qmin$-case.

The apparent correlation between higher AE activity, lower neutron rate, and lower normalized global energy confinement suggests that enhanced fast-ion transport could be a significant energy loss channel. All of the direct fast-ion loss detectors on DIII-D only work when $IP>0$ and $BT<0$ because this is the configuration used by most experiments. All experiments described in this paper use $IP>0$ and $BT>0$ to maximize the off-axis NBI current drive that helps maintain elevated $qmin$⁠, and therefore no direct measurements of fast-ion loss exist. Instead, enhanced fast-ion transport above the classically predicted level is inferred by comparing two ways to compute the plasma total stored energy. First, the EFIT code²¹ is used for equilibrium reconstructions by fitting to magnetic measurements outside the plasma. This method accurately determines the total stored energy $WEFIT$ to better than 3% over the range of $βP$ obtained in this study. This is based on a Monte-Carlo statistical analysis of magnetic reconstructions with a 3% uncertainty in external magnetic measurements as described in Sec. 3.2 of Ref. 21. The second method is to use the ONETWO transport code²² to compute the total stored energy $WC$⁠, using the magnetic geometry, measured density and temperature profiles, and NBI power and injection angles as inputs. The NUBEAM Monte Carlo code²³ is used to compute the fast-ion distribution function assuming that the fast-ion transport is completely classical. NUBEAM does not account for possible neutral beam ionization in the scrape-off layer. This is estimated to be up to 3%–4% of the beam power depending on the scrape-off layer density. Figure 2 shows the values of $WEFIT$ and $WC$ determined from both approaches for the two discharges in Fig. 1. Figure 2(b) shows the percentage that $WC$ overestimates $WEFIT$—a value that will be referred to as the “stored energy overestimate” throughout this paper. The plasma with $qmin≈1$ has good agreement between the measured and computed total stored energies—the average stored energy overestimate during the high $βN$ phase shown is 3%. Therefore, within the uncertainties, classical fast-ion transport is an accurate description for this plasma. In contrast the high-$qmin$ case $WC$ overestimates $WEFIT$ by an average of 17%. This overestimate is interpreted as an indication of fast ion redistribution and/or loss.

Fast-ion $Dα$ (FIDA) diagnostic measurements²⁴ confirm that high-$qmin$ plasmas can have a deficit of fast ions compared to the level expected classically. Several FIDA views are available on DIII-D, but in steady-state scenario plasmas these do not routinely make useful measurements without careful design of the various NBI source waveforms. This must be done to provide background subtraction and temporally isolate “signal” beams the diagnostic is tuned to from “polluting” beams that it is not. This was done for the pair of discharges in Figs. 1 and 2. Figure 3 compares measurements from a vertical-viewing FIDA array with synthetic diagnostic predictions using the FIDASIM code.²⁵ The solid lines are the FIDASIM predictions for the high- and low-$qmin$ cases using a classical fast-ion transport assumption. The classical prediction for the high-$qmin$ case exceeds that of the low-$qmin$ case. However, the FIDA measurements are at approximately the same level for both cases, so that the classical prediction roughly matches most measurements for $qmin≈1$⁠, but the classical prediction is higher than the measurements for $qmin≈2$⁠. The dashed line is the FIDASIM prediction for the $qmin≈2$ case after applying an ad-hoc uniform fast-ion diffusion profile. A value of 1.3 $m2/s$ was chosen because this makes the computed stored energy and neutron rate approximately match the experimental values. This value improves the match between prediction and experiment, although no attempt was made to find a fast-ion diffusion profile that might make a best fit to the FIDA data.

For the $qmin≈2$ case, if only the inferred absorbed NBI power (i.e., what is left after applying an ad-hoc fast-ion diffusion) is included in the calculation, then $H89P$ rises from 1.8 to 2.2. This again implies that enhanced fast ion transport is the primary reason for lower normalized confinement at $qmin>2$⁠. Power balance analysis with the inferred absorbed power finds that the thermal diffusivities do not differ significantly between the high- and low-$qmin$ cases. Thus, under the presumption that NBI power is redistributed by Alfvénic modes this particular high-$qmin$ plasma has thermal transport similar to the low-$qmin$ case. It is possible that any expected increase in thermal transport with $q$ is being offset by a decrease due to lower shear.²⁶ But without significantly improved thermal transport the $qmin≈2$ case has $H89P<2$ because the fast-ions are not classically confined.

Degraded fast-ion confinement in the presence of enhanced AE activity is reproducibly observed by all fast-ion diagnostics that are sensitive to the co-passing portion of velocity space. Figure 4 compares classical and measured signals for the stored energy, volume-averaged neutron rate, and vertically viewing FIDA diagnostic in several quasi-stationary discharges with $q95=$ 5–12. The classically predicted neutron signals are derived from a zero-dimensional prediction of the beam-plasma collision rate,²⁷ and the FIDA data and the fast-ion stored energy points are from the carefully analyzed discharges of Ref. 28. When there are many coherent AEs, the classical prediction overestimates the measured signals. The four discharges with a neutron overestimate greater than 100% are all at the lowest current (0.5–0.65 MA) in the group and they have quite strong AEs at ρ = 0.6–0.7.

Local fluctuation diagnostics show that the AE activity occurs throughout much of the plasma volume. Figure 5 shows electron cyclotron emission (ECE)³¹ measurements of the AE profile for three coherent modes in a quasi-stationary discharge with $βN=2.8$⁠, $qmin=2.5$⁠, $q95=6.9$⁠, and $H89P=1.4$⁠. The mode at 113 kHz is located near the $qmin$ radius, which is at a normalized minor radius of $ρ=0.37$ in this discharge; this is probably a reverse-shear AE.³² The mode at 137 kHz is localized near the top of the H-mode pedestal. The mode at 153 kHz is a global mode, probably a TAE.

Both the MHD code NOVA-K^37,38 and the gyrokinetic code GYRO³³ predict AE instability for high-$qmin$ plasmas. For example, GYRO was used to predict the eigenfunctions of AEs in the pair of discharges shown in Fig. 1. The calculations were performed in the range $ρ=$ 0.2–0.8. The fast-ion pressure profiles supplied as inputs are those that result after any fast-ion diffusion needed to match the measured total stored energy is applied, i.e., for the high-$qmin$ case the fast-ion pressure is lower than the classical prediction. Figure 6 compares the electrostatic potential of the most unstable eigenfunction for an $n=4$ mode. The case with $qmin=2$ has the potential for AEs extending out to higher radius. Figure 6(b) compares the growth rates for a range of $n$⁠. The high-$qmin$ case has larger growth rate than the low-$qmin$ case for $n=$ 2–13. If the classically predicted fast ion pressure is used for the high-$qmin$ case, then the predicted growth rates are significantly higher. (The low-$qmin$ case fast ion pressure is already approximately classical).

Theoretically, the many modes cause fast-ion transport through resonance overlap.^29,30 Figure 7 illustrates this last point by comparing phase space diagrams for a pair of steady-state scenario plasmas at high- and low-$qmin$⁠. In both, the possible fast-ion orbits are gray symbols. Black symbols show where the NBI sources used in the discharge actually populate phase space with fast ions. Red symbols denote possible fast ion orbits that can resonate with the various AEs that were observed in the discharge. Where red and black symbols overlap—i.e., where actual fast-ion orbits are capable of resonating with observed modes—there is the possibility for fast ion redistribution to other parts of phase space, including to loss orbits. While the case with $qmin=1$ has little overlap, the case with $qmin=2$ has substantial overlap.

In summary, the standard high-$qmin$ steady-state scenario is observed to have lower than expected global energy confinement, substantial AE activity, and a deficit of confined fast ions compared to classical predictions. The negations of these statements are true for similar low-$qmin$ plasmas. AE stability calculations confirm that the high-$qmin$ plasmas have greater potential for instability and fast-ion transport than the low-$qmin$ plasmas. This implies an important question: is high-$qmin$ by itself necessarily bad for fast-ion transport and global energy confinement? Section III will show that the answer to this question is “no”.

Very high $qmin$ in the range of 3–5 has been sustained with $H89P≥2$ in the DIII-D “High-$βP$” regime^34–36 that is used to prepare for long-pulse operation on superconducting tokamaks like EAST. Fully noninductive plasmas sustained by 70%–80% bootstrap current fraction and the 20%–30% NBI + ECCD have been produced in the following parameter ranges: $βN=$ 3–4, $βP=$ 3–4, $βT=$ 1.5%–2.5%, $q95=$ 11–12.5, $IP=$ 600–700 kA, and $BT=2$ T. The line–density through the magnetic axis is typically in the range $5.5–6.5×1019 m−3$⁠. Successful shots with both thermal and total-energy confinement at or above levels expected for H-mode¹⁹ (i.e., $H98P≥1$ and $H89P≥2$⁠) have either a high radius (⁠$ρ≈0.7$⁠) transport barrier or infrequent edge localized modes (ELMs) and a high average pedestal pressure. Figure 8 shows time traces and profiles for an example High-$βP$ discharge.

High-$βP$ plasmas with $H89P≥2$ have inferred fast-ion transport near the classical level. Figure 9 compares a standard high-$qmin$ case and a high-$βP$ case. Whereas the standard case has $qmin≈2.4$⁠, $H89P≈1.8$ H, and a ∼15% stored energy overestimate, the high-$βP$ case has $H89P≈2.1$⁠, and a stored energy overestimate of ∼3% with almost double the value of $qmin$⁠. Figure 10 shows the FIDA profile acquired between 3900–6000 ms during ELM-free phases in the same high-$βP$ discharge. (The absence of ELMs facilitates accurate FIDA measurements despite the relatively high density of this discharge.) Within experimental uncertainty, the FIDA, neutron, and stored energy measurements are all consistent with classical theory. The high-$βP$ discharges demonstrate that very high values of $qmin$ in the range of 3–5 do not invariably result in enhanced fast-ion transport or lower than expected global energy confinement. In both high-$qmin$ regimes the strongest AEs observed are usually TAEs, and NOVA code^37,38 analysis (Fig. 11) of a pair of discharges finds that the core TAE gap structure inside of $ρ≈0.5$ is much wider for the standard high-$qmin$ case than the high-$βP$ case. In the former case, NOVA predicts a plethora of radially extended core TAEs that are minimally damped, while in the latter case the TAE gap only widens a similar amount outside of $ρ≈0.5$⁠. Measurements of density and temperature fluctuations associated with TAEs are in qualitative agreement with these predictions. Figure 12 compares profiles of the electron temperature fluctuation frequency-spectrum measured by the electron cyclotron emission diagnostic. The standard high-$qmin$ plasma has many more coherent modes inside of $ρ≈0.5$ than the high-$βP$ case, which has strong modes around 60 and 70 kHz only outside of $ρ≈0.6$⁠. Since in both cases the fast-ion profile is peaked on-axis, AEs near the magnetic axis are more likely to cause enhanced fast-ion transport than AEs at large radius.

The drive for core AEs is lower in high-$βP$ plasmas than in standard high-$qmin$ plasmas because the former have a shorter slowing-down time and therefore a lower classically predicted $∇βfast$ [Figs. 13(a) and 13(b)]. This is consistent with recent work³⁹ that suggests that the strength of AE activity in L-mode, high-q_min plasmas has a critical-$∇βfast$ dependence whereby the modes and fast-ion transport increase quickly above a certain value of $∇βfast$⁠. This produces stiff fast-ion transport and limits the actual fast-ion profile peaking (i.e., actual profiles show less peaking than classically predicted profiles). In the high-$βP$ cases, the shorter slowing-down times across the whole profile result from higher density and lower electron temperature. Profiles of these are usually “step-like,” i.e., relatively flat inside a transport barrier, rapidly changing at the barrier, and then flat again out to the H-mode pedestal barrier. Between the transport barriers, the very short slowing-down time should be more effective at thermalizing fast ions before they can be expelled from the plasma relative to plasmas without this feature.

Many observations made in high-$βP$ plasmas lead to the conclusion that these operate very close to a critical fast-ion gradient for enhanced transport. Reducing the density in these plasmas results in an increase in AE activity, lower confinement time and lower achievable $βN$⁠. In the example shown in Fig. 14, only the density is changed between two consecutive discharges, from a maximum of about $6.5×1019 m−3$ to about $5×1019 m−3$⁠. Lowering the density increased the slowing-down time, raised the maximum classically predicted fast-ion gradient by 50%–60%, increased the number and amplitude of AEs, and reduced $H89P$ to less than 2. Figure 15 shows another discharge in which the line-density was again about $5×1019 m−3$⁠. As before, strong AE activity was observed. In the middle of the high-$βN$ phase, the total NBI power was stepped down by about 32%. After this change, the classically predicted maximum in $∇βfast$ is reduced, the interferometer shows a reduction in AE amplitude, the stored energy overestimate falls from 5%–10% to close to 0 (i.e., the classical level), and remarkably, $βN$ remains approximately constant at 3. These observations are consistent with the NBI power reduction causing a relaxation of $∇βfast$ to below a critical value and subsequently a reduction in fast-ion transport. Considering this case backwards in time (i.e., as if it were an NBI power step-up) shows that the effectiveness of NBI power to increase $βN$ falls rapidly once the critical gradient has been reached. Through variation of density and NBI power the classically predicted $∇βfast$ can be varied. Figure 16 shows a compilation of data from different times in three high-$βP$ discharges. The AE amplitude and stored energy overestimate increase with increasing $∇βfast$ while $H89P$ decreases. These observations are consistent with the picture of fast-ion transport increasing with $∇βfast$⁠.

The interplay between fast-ion transport and thermal transport is noteworthy for its complexity in high-$βP$ discharges. A correlation is observed between classical or nearly classical fast-ion confinement and the existence of either a transport barrier near $ρ=0.7$ or a transport barrier near $ρ=0.5$ with a higher pedestal pressure. Conversely, lower density discharges with enhanced fast-ion transport and lower $H89P$ do not have internal transport barriers or higher pedestals. Figures 17 and 18 compare pairs of discharges to illustrate these points. In Fig. 17, both discharges had the same density feedback target waveform, but in one case more NBI power was used at early times resulting in an earlier L-to-H-mode transition time and higher $qmin$⁠. Once in H-mode, the density of both discharges followed the same trajectory until about t = 3 s when the higher-$qmin$ case developed a transport barrier in all channels at $ρ=0.7$⁠. After this time, the density and temperature inside the barrier rose to higher values than what was observed in the discharge without a barrier. But, the slowing-down-time profiles and classically predicted fast-ion pressure profiles did not change significantly. Thus the stored energy overestimate remained near 10% until after the NBI power was reduced at about t = 4.3 s. In this case, the transport barrier apparently formed while the fast-ion transport was still higher than classical. The increase in thermal stored energy brought about by the barrier compensated for the enhanced fast-ion transport and achieved higher confinement and $βN$ than the case without a barrier.

The example in Fig. 18 compares the low-$H89P$ discharge from Fig. 17 to a case programmed to have higher density for the entire discharge. The higher density case is inferred to have nearly classical fast-ion transport for the entire high-$βN$ phase of the discharge. After about t = 2.3 s this discharge obtains a higher average pedestal pressure, $βN$⁠, and $H89P$⁠. The higher average pedestal pressure is a result of an unsteady, but unusually long (∼100–500 ms) ELM period. This coexists with a transport barrier near $ρ=0.5$ that is somewhat narrower and shorter than the barriers observed at $ρ=0.7$ in other discharges. In this case, it appears that the attainment of nearly classical fast-ion confinement precedes and possibly contributes to the formation of the high pedestal and internal transport barrier by more complete transfer of heat to the thermal plasma.

The comparisons in Figs. 17 and 18 show improved thermal confinement can either push the global energy confinement time back up to expected H-mode levels (i.e., $H89P≥2$⁠) despite enhanced fast-ion transport, or a mode of better thermal confinement (e.g., internal transport barrier or higher pedestal) can result from having classical fast-ion transport. In either case, better thermal confinement reduces the required NBI power to obtain a target $βN$⁠, which helps to reduce the drive for AE modes.

In summary, the high-$βP$ operating scenario shows that high-$qmin$ plasmas can have classical fast-ion confinement and good global energy confinement if $∇βfast$ is kept low in the core to avoid strong AE activity there. This is caused by a relatively low and flat slowing-down time profile inside $ρ∼0.5$⁠. There may be other ways to control the slowing-down time and fast-ion profiles to minimize AE activity and achieve good performance. These are discussed in Sec. IV.

On DIII-D, achieving higher $βN$ in high-$qmin$ plasmas with reactor-relevant $q95$ will require improved control of fast-ion modes. The L-mode studies in Ref. 39 showing that moving the same amount of NBI power from on- to off-axis reduces AE activity suggests that moving more NBI power from on- to off-axis may help in a high-$qmin$⁠, high-$βN$ scenario. Presently, ∼1/3 of the co-$IP$ NBI power is injected off-axis in these scenarios, but a planned upgrade will increase this to ∼2/3. This will enable further lowering of the fast-ion gradient near the axis, which will reduce the drive for AEs there. The high-$βP$ regime demonstrates cases where some mode of improved thermal confinement can either partially compensate for enhanced fast-ion transport, or permit classical fast-ion transport by lowering the NBI power requirement. In both cases, the normalized confinement is at or above the expected level for an H-mode plasma. Therefore, future experiments should seek to identify and optimize potentially useful regimes of improved thermal confinement. The most likely modes to be compatible with high-$qmin$ are a high-radius transport barrier or a “super-H-mode” pedestal.⁴⁰ 

A third lesson from the present experiments is that future experiments should seek to replace NBI heating with rf-electron heating. This will lower the drive for AEs and avoid the issue of reducing fast-ion confinement. This option is actually well aligned with future burning plasma reactors, which, unlike the experiments described here, will have chiefly electron heating, low input torque, and isotropic rather than anisotropic fast ions. To this end, the DIII-D program has plans for a significant increase in ECH power and will test a novel high-harmonic fast wave or “helicon” system for off-axis heating and current drive. Lastly, in DIII-D and elsewhere there have been observations of direct impacts on reverse-shear AEs and TAEs by the careful application of ECH. For example, modulation between ECH on- and off-axis has been shown to turn TAEs on and off at the same rate that is faster than a slowing down time.⁴¹ These results are not fully understood and from low-$βN$ plasmas, but their existence encourages continued exploration of novel AE-control methods.

It was beyond the scope of the present paper to try to predict fast-ion confinement in next-generation steady state burning plasmas based on existing DIII-D results. But a few important dimensionless parameters can be compared to get some sense of how conditions will be different in ITER, for example. Taking parameters from a model $qmin>2$ steady-state scenario for ITER⁵ shows that it will have a mix of both more and less favorable conditions than DIII-D for fast-ion confinement. The analysis in this paper has focused mostly on the drive for AEs and resulting fast-ion transport. The quotient $βfast/βthermal$ serves as a proxy for the ratio of AE drive to damping. In the DIII-D high-$qmin$ plasmas with $q95=$ 5–7 all having enhanced fast-ion transport, $βfast/βthermal=0.2–0.4$⁠. The high-$βP$ plasmas having approximately classical fast-ion transport have $βfast/βthermal=0.15−0.35$⁠. The predicted ITER plasma from Ref. 5 would have $βfast/βthermal=0.15$⁠. This is at the low end of the DIII-D high-$βP$ range, and therefore, this is favorable for ITER. Next, the much larger size of ITER compared to DIII-D means that the ratio of the fast-ion orbit to machine size ρ_fast/a is much lower in ITER: 0.015 for beam ions and 0.02 for alphas compared to 0.03–0.04 in the DIII-D experiments. Therefore, the ability of AEs to move fast-ions all the way to the edge of the plasma is less in ITER. But ITER will have many more fast-ions that are capable of resonating with various AEs than DIII-D: the ratio of (on-axis) parallel fast-ion speed to Alfvén speed $vfast/vA$ is 0.3–0.45 in DIII-D but about 0.64 for beam ions in ITER and over 1 for alpha particles. Thus additional experiments and modeling are needed to improve confidence in predictions of fast-ion confinement in high-$qmin$ scenarios in ITER.

Recent DIII-D steady-state scenario experiments have shown the importance of optimizing the current density and fast-ion pressure profiles to minimize AE instability and fast-ion transport. High-$βN$ operation is dominated by co-I_P neutral beam injection heating, and therefore confined fast ions can make up a significant fraction of the total stored energy. Many experiments have had $q95$ in the range 5–7, because this corresponds to a value that would produce a reasonable level of fusion gain in a next-step reactor. Also, to maximize the externally driven noninductive current the line-density is often kept below ∼5 × 10¹⁹ m⁻³. With these conditions, an increase of fast-ion transport above classically predicted levels occurs when $qmin$ is raised from just above 1 to just above 2. This results in 20%–30% lower global energy confinement time at the higher value of $qmin$⁠. Taking the inferred lower absorbed NBI heating power into account, the thermal ion and electron diffusivities are found to be similar for both values of $qmin$⁠.

A high-$βP$⁠, high-$qmin$ scenario developed in a different regime is useful for clarifying the physics underlying enhanced fast-ion transport, and also for demonstrating paths to recovering good global energy confinement. When higher $q95$ (i.e., lower I_P) and line-density above $∼5×1019 m−3$ are used, this results in high-$βP$ plasmas (⁠$βP=$ 3–4) that have H_89P > 2 and classical fast-ion transport. Classical fast-ion transport is evident even with q_min ≈ 5 and ∼8–10 MW of NBI power because higher density and lower temperature lead to shorter slowing-down times and a smaller fast-ion pressure gradient. Compared to standard high-$qmin$ plasmas (⁠$q95=$ 5–7), the high-$βP$ plasmas have a narrower TAE gap structure near the magnetic axis and observed AEs are usually only outside $ρ=0.5$⁠. Good high-$βP$ plasmas operate just below but close to a critical fast-ion gradient for increased AE activity. The thermal plasma has either a high-radius transport barrier or a high average pedestal pressure, and these help to reduce the required NBI power and fast-ion gradient. These observations suggest that improved global energy confinement in standard high-$qmin$ plasmas might be achieved by using planned heating and current drive upgrades to broaden the NBI heating profile and further optimize thermal transport at high radius.

This material was based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award Nos. DE-FC02-04ER54698, DE-AC52-07NA27344, SC-G903402, DE-AC02-09CH11466, DE-AC05-00OR22725, and DE-FG02-04ER54761. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3D_DMP.