H-mode grade confinement in L-mode edge plasmas at negative triangularity on DIII-D

An economically viable fusion reactor will need to sustain high pressure and energy confinement while maintaining the plasma relatively free of impurities to maximize the Lawson triple product. In the case of tokamaks, it is also desirable to generate a large fraction of the plasma current from the pressure driven mechanism, i.e., bootstrap, at a high pressure normalized to that of the poloidal magnetic field, β_p. The MHD theory defines limits to the pressure and current levels a given plasma can sustain, beyond which instabilities cripple or even zero performance, respectively, in the case of minor or major events. Restricting our attention to the research line of tokamaks, in the early 1980s much research effort was devoted by the community to understanding the maximum pressure that could be achieved; Troyon derived a limit in a simple form that closely matched that measured in existing machines¹ and generated real concern in the community because, if true, its low value would have hardly allowed operation in a reactor. It was later realized that plasma shaping, which when described by the formula commonly used for Dee-shaped plasmas,² is characterized by mainly two parameters, elongation κ and triangularity δ, can strongly affect the Troyon limit. In particular, it was found that the stability limit in elongated plasmas exceeds that predicted by the Troyon limit thanks to the increased maximum current, at a fixed edge safety factor, which in turn increases the maximum achievable pressure normalized to that of the confining magnetic field.^3,4 Subsequent studies tackled the task of optimizing plasma triangularity; in particular, it was found that large positive values allow one to maximize the pedestal height of H-mode regimes,^5–10 and thus increase confinement.

In parallel to the stability analysis, the impact of shaping on turbulence was, at first, investigated mainly through linear Gyrofluid (GF) and Gyrokinetic (GK) simulations, which focused on the effect of elongation on the Ion Temperature Gradient (ITG) driven^11,12 and Trapped Electron (TEM) modes.¹³ Later studies confirmed the stabilizing effect of elongation on ITG and mixed ITG/TEM modes also in the nonlinear regime both with local, or flux tube, simulations^14,15 as well as with global modeling.¹⁶ The impact of triangularity on turbulence in ITG dominated plasmas was found by GK modeling to be negligible, at least compared to the effect of elongation. It must be pointed out, however, that most of those studies were restricted to positive triangularity values, most likely due to the fact that ballooning modes are predicted to be more virulent at negative triangularity. Nevertheless, it was predicted in the early eighties that negative triangularity has a stabilizing effect on TEM turbulence,¹³ although the effect was predicted to be smaller than that exerted by elongation. In the mid 2000s, experiments from the TCV tokamak showed that in collision-less inner-wall limited plasmas dominated by TEM turbulence, the confinement time could by doubled simply by reversing triangularity with the other parameters held fixed.¹⁷ Those results, although experimentally demonstrating the stabilizing effect of negative triangularity on TEM, were obtained in a regime of low pressure and pure electron heating for which the electron temperature was much higher than that of ions. This motivated the experiments on the DIII-D tokamak presented in this work, which extends the TCV experiments to a more reactor relevant regime at higher β and where T_e ≃ T_i. A high level overview of this experiment is described in Ref. 18, while this paper discusses procedures and results in more detail.

The paper is organized as follows: Section II explains how the experiments were performed; Sec. III reports on low heating (P_aux < 6 MW) phases, during which matched discharges at negative and positive triangularity in the L-mode phase are compared in terms of the transport properties, intensity of fluctuations, and gyrokinetic modeling; Sec. IV describes the high heating phase, during which L-mode edge negative triangularity plasmas reach H-mode grade confinement; conclusions and perspectives are offered in Sec. V.

Experiments were carried out on the DIII-D tokamak¹⁹ to validate and extend earlier negative triangularity results obtained on the TCV tokamak.²⁰ To better connect with the TCV plasmas analyzed in Ref. 17 the plasma shape was chosen to be up-down symmetric. Since the DIII-D tokamak was not originally designed to operate plasmas at negative triangularity, some of the shaping parameters and the plasma positioning in the vacuum vessel could not be chosen arbitrarily; viz., because of the location of the poloidal field coils F7A-F7B elongation was kept fixed at κ = 1.33 and the Last Closed Flux Surface (LCFS) on the outboard midplane had to be positioned at R = 2.18 m, which is about 10 cm radially inward from the location typical of DIII-D discharges. The unusual position of the separatrix made plasmas unreachable by the reciprocating Langmuir probe and measurements of the heat flux width in the near Scrape-Off Layer (SOL) were not possible. Plasmas were limited on the central column of the vacuum vessel because the DIII-D wall is not designed to withstand flux resulting from strike-points located on the low field side; a control system was implemented to prevent the plasmas from inadvertently diverting or, failing that, to rapidly quench the discharge in case X-points had appeared in the vacuum vessel.

Two up-down symmetric equilibria at negative and positive triangularity (δ_LCFS = ±0.4) were developed to compare the transport coefficients and fluctuation characteristics between the two values of triangularity. It was decided that the two plasma shapes would have the same form of the LCFS, when mirrored around their vertical central line, with matched values for major and minor radii, elongation, and absolute value of triangularity; this resulted in differences in volume and toroidal flux within 8%. The impact of triangularity on confinement and fluctuations was evaluated by comparing discharges with matched shape, plasma current, toroidal field, and auxiliary power. This choice was dictated by the need to minimize the number of trial discharges needed to complete the experiment, and is indeed a much less discharge-demanding procedure than tweaking the actuators to try matching the volume, safety factor, and pressure profiles in an effort to eliminate their impact on the transport coefficients; additionally, it is a legitimate choice when comparing different scenarios from the perspective of operating a reactor at a given cost, i.e., at a fixed auxiliary power, plasma current, toroidal field, aspect ratio, and major radius. The edge safety factor in plasmas at positive triangularity was 15% higher, on average, than their negative triangularity counterparts; no attempt was made to evaluate the impact of shape on confinement at a fixed q-profile. The safety factor on-axis was allowed to decrease below unity, providing the opportunity to evaluate the impact of triangularity on saw-teeth: the results of these measurements will be presented in a separate publication. Efficient pumping is obtained only when field lines in the SOL reach pumping ducts; DIII-D is designed for plasmas at positive triangularity, for which such regions are on the high field side of the vessel, thereby causing inefficient pumping in plasmas at negative triangularity. As a result, when comparison discharges at positive triangularity were executed after the negative triangularity counterparts, the gas rate had to be usually increased by 50%–100%, for the same value of preplasma fill, to match the line averaged density. Discharge parameters are summarized in Table I for the two shapes.

Two heating phases were considered: Pure electron and mixed electron-ion heating, obtained with Electron Cyclotron (EC) and Neutral Beam Injection (NBI) combined with EC, respectively. The former, which included sporadic balanced torque beam blips to allow the measurement of ion pressure and velocity profiles, was used to study the electron dominated regimes to connect with the TCV results; the latter featured EC throughout to explore relevant reactor regimes where the ion temperature is close to that of electrons. Ion heating dominated scenarios, although interesting from a model validation standpoint, were not prioritized in this study and are deferred to future work. The EC power deposition, calculated with the linear ray-tracing code TORAY-GA,²¹ was strongly radially localized slightly outside the q = 1 surface to minimize sawtooth activity. For the explored line averaged densities, ranging from 3–5 × 10¹⁹ m⁻³, the EC power absorbed in the first pass through the electron cyclotron resonance is typically larger than 98%, with the current drive negligibly within 2 kA. The poloidal injection angle had to be adjusted for the two values of triangularity to guarantee the deposition of EC power at the same location in plasma coordinates. Real-time controlled gas puffing was generally used to control the line averaged density. The electron temperature was measured by Thomson Scattering (TS)²² as well as by Electron Cyclotron Emission (ECE)²³ diagnostics; the density was gauged by TS, CO₂ interferometry,²⁴ and microwave reflectometry;²⁵ the ion temperature along with toroidal and poloidal velocities were obtained by the Charge Exchange Recombination diagnostic (CER);²⁶ fluctuations were monitored by the Phase Contrast Imaging (PCI) diagnostic,²⁷ Beam Emission Spectroscopy (BES)²⁸ along with Correlation Electron Cyclotron Emission (CECE).²⁹ The normalized radial coordinate used throughout this paper is defined as the square root of the normalized toroidal flux. Throughout this paper, figures illustrating direct comparisons of either measurements or modeling results for the two shapes use blue and red colors for negative and positive triangularity, respectively.

In this section, we evaluate the impact of triangularity on confinement and fluctuations by comparing two discharges at opposite values of edge triangularity (δ_LCFS = −0.4 and δ_LCFS = +0.4) that have matched actuators throughout the shot length. Two heating phases are evaluated: pure electron heating achieved with EC, and mixed ion-electron heating through combined EC-NBI. Plasmas were operated at values of plasma current, line averaged density, and confining magnetic field listed in Table I. The averaged density, which at low power is typically in the low end of the reported range, is maintained the same between the two shapes in the EC-only heating phase, while it tends to differ when the NBI system is used. Gas puff rates were higher at positive triangularity by 50%–100%. Both discharges operate with an L-mode edge and are thus free of Edge Localized Modes (ELMs).

It is worth mentioning that, since about 35% of the NBI power couples to electrons, plasmas are in an electron-flux dominated regime even during the combined EC-NBI heating phase. In the discharges analyzed in this section, MHD activity is either limited or absent. Kinetic equilibria were reconstructed using the ONETWO³⁰ transport code and the EFIT³¹ equilibrium solver constrained by measurements from the Motional Stark Effect (MSE)³² diagnostic. MSE data from the core region were more strongly weighted than those from the edge to ensure accuracy in the position of the magnetic axis, as was verified by comparing electron temperature profiles from the ECE high-field-side and low-field-side systems. In order to estimate confidence intervals on the resulting heat diffusivities, a series of one hundred simulations per time slice were carried out with the ONETWO code, with random variations of kinetic profiles, radiated power, and wall recycling within their experimental uncertainties. The volume integrated radiated power is less than 25% of the total heating power and is typically equal for the two shapes within experimental uncertainties. In Fig. 1(a), where we display kinetic profiles during the EC-only phase, it can be seen that both discharges feature closely matched electron density, ion temperature, and toroidal velocity profiles, while the electron temperature is higher for the discharge at δ_LCFS = −0.4 by roughly 10% in the core and 35% at the edge. A reconstruction of the kinetic equilibrium shows that, by merely reversing edge triangularity, the plasma stored energy content increases by 24 ± 7%. The resulting electron heat diffusivities decrease by 20%–80% in the region outside the EC deposition location, with the difference increasing monotonically with the minor radius; in contrast, the ion heat diffusivity is much less sensitive to shape and, although its expected value is lower at negative triangularity across most of the minor radius, the amount of reduction is well within one population uncertainty (Fig. 2). In the combined EC-NBI heating phase shown in Fig. 1(b), as compared to the discharge at δ_LCFS = +0.4, the plasma profiles for the negative triangularity case feature a 20% increase for the on-axis electron density with about the same edge value, while T_e, T_i and Ω_ϕ are higher in the radial region 0.4 < ρ < 0.9, with cross-overs on the axis and at the edge. A kinetic analysis shows that, even in this regime where T_e ≃ T_i, the effect of reversing the triangularity causes the stored energy to increase by 21 ± 8%, while the shape dependence of the electron and ion heat diffusivities are similar to those computed in the EC-only phase, namely large and small reductions in the electron and ion channels, respectively, at negative triangularity (Fig. 2). It is important to mention that, for both heating schemes, the midradius inverse effective collisionality 1/ν_eff, defined as the inverse of the electron collision frequency normalized to the bounce time, was typically less than two. This might partly explain why the confinement improvement at negative triangularity is observed to be significantly lower than that measured on TCV, where 1/ν_eff was lower by a factor of two to three as compared to the DIII-D experiments reported in this work. Technical issues probably related to error field correction did not allow lower densities to be explored, which will be a topic for future experiments.

The intensity of electron density fluctuations was monitored by the Phase Contrast Imaging (PCI) diagnostic that images line-integrated density fluctuations onto a linear array of 32 detector elements. The vertical line of sight, shown in Fig. 3, has its center point located at R = 1.98 m, which corresponds to the laser beam line being tangent to the flux surfaces roughly around midradius. The gaussian full-width at 1/e points in the electric field is 7 cm, which, together with the minimum aperture stop in the optical system, sets the detectable wave-number range to 1.5–35 cm⁻¹. The linear array is such that the diagnostic can reconstruct wavelets in the range k_R < 25 cm⁻¹ and 10 kHz < f < 2 MHz, with the lower frequency cut-off set by a high-pass filter aimed at reducing the impact of mechanical vibrations on the measurements.²⁷ Since the system performs a line integral of density fluctuations along a vertical direction in the laboratory frame, it is sensitive to scattering wavelets propagating in the horizontal plane; this translates into measured wave-vectors having varying poloidal and radial components in the plasma coordinates along the line of sight. In both heating phases, the intensity of density fluctuations is seen to decrease at all f > 20–30 kHz at negative triangularity, as shown in Fig. 4(a) for combined EC-NBI heating. A quantitative comparison of the rms values shows that the total intensity of fluctuations, normalized to the line-averaged density along the PCI line of sight, decreases for negative triangularity by 40% and 30%, respectively, in the EC-only and EC-NBI heating phases.

Wavenumber spectra displayed in Fig. 4(b) indicate that the frequency integrated intensity of fluctuations at positive and at negative wave-vectors are different in the two shapes. More specifically, while at positive triangularity, the wave-number spectra are roughly symmetric, at negative triangularity the total power at positive wave-vectors is lower than that in the negative part of the spectrum. Since both configurations are up-down symmetric and this imbalance is observed even in the EC-only phases during which flow shear is much lower, this feature suggests that turbulent eddies might be oriented in different ways in the two shapes; this would be consistent with exploratory linear simulations with the LORB5 code,¹⁷ which showed that profile shearing effects are more prominent at negative triangularity and induce a poloidal tilt to turbulent eddies, resulting in larger k_⊥ and reduced mixing-length diffusivities. The asymmetry in the PCI response is due to the line integral, which strongly weights turbulent eddies aligned orthogonally to the laser beam direction of propagation, while it averages out eddies aligned along it. This mechanism is equivalent to asymmetries detected in flow-sheared plasmas: low-shear regimes, such as the L-mode, typically create spectra symmetric in k_R, while the highly-sheared H-mode edge typically create asymmetric wave-number spectra.³³ 

Ion density fluctuations were monitored by the Beam Emission Spectroscopy (BES) diagnostic, which is sensitive to wave-vectors in the range k_⊥ < 2 cm⁻¹ at frequencies less than 1 MHz.²⁸ The diagnostic measures density of fluctuations on a radially movable array of 64 elements, which for the present experiments were disposed in a poloidal-radial configuration of an 8x8 matrix. As it is shown in Fig. 5(a), where fluctuation data from two repeat discharges at positive triangularity offer a wide coverage in a minor radius, the intensity of fluctuations in the combined NBI-EC heating phase is seen to decrease at negative triangularity by an amount typically larger than error bars. It is interesting to report that the absolute reduction of the relative intensity of fluctuations is an increasing function of the minor radius, which approximately follows the radial penetration length of triangularity.

Electron temperature fluctuations were monitored with the Correlation Electron Cyclotron Emission system (CECE), which simultaneously samples eight locations along a radial line of sight whose center is located 7.6 cm above the plasma midplane.²⁹ The system samples fluctuations in the frequency range f < 1 MHz, is sensitive to wave-vectors k_⊥ < 2 cm⁻¹ and has a floor noise of approximately 0.3% when sampling data for 1 s duration. In both heating regimes, the relative intensity of fluctuations is seen to decrease at negative triangularity by approximately 10%–15%. It is important to underscore that, as opposed to the BES diagnostic, the observed reduction does not seem to depend on the radial location, at least in relative terms, as displayed in Fig. 5(b). It is an open question whether such behavior, recently observed also on TCV with a CECE and a PCI system^34,35 is due to global effects.^36,37 Detailed nonlinear gyrokinetic simulations are deemed necessary to interpret the above observations through synthetic diagnostics. Preliminary nonlinear simulations did not obtain a satisfactory agreement between experimental and simulated power fluxes when modeling ion and electron scale dynamics separately, thus suggesting the need for multiscale modeling. Such simulations, however, require resources beyond those available to this work and are deferred to a future publication. Nevertheless, basic physics mechanisms at play in these discharges can be grasped with linear modeling alone, which is therefore the subject of Sec. III C.

Early nonlinear local gyrokinetic simulation of TCV discharges found that the increased confinement at negative triangularity was due to a weakening of Trapped Electron Modes, which were the dominant instability in those cases, caused by a modification of the toroidal precession drift exerted by negative triangularity.³⁶ The CGYRO code³⁸ was used to evaluate the linear modes at play in these discharges. Simulations were executed by obtaining kinetic equilibria reconstructed with the EFIT equilibrium code and subsequently read directly by the CGYRO input parser, which expands flux surfaces in a truncated 16-element Fourier series to compute particle drifts and geometric coefficients. Eigenvalues were modified by less than 10% when using a Miller³⁹ parametrization of the flux surface shape, rather than the numerical fit using the truncated series, which suggests that the numerical equilibrium is correctly evaluated by CGYRO even in this unusual shape. Simulations evolve electron, deuterium, and carbon as kinetic species and perturbations to the electrostatic and the parallel vector potential. Collisions were modeled using the Sugama operator. The effective charge radial profile was set to a flat value that matched the edge loop voltage; including radial variations as measured by the CER system, or a Lorentz collision operator, did not result in any remarkable difference in the overall trends. In both heating regimes and across the minor radius, TEMs are found to be the dominant instability in the wave-number region k_θρ_s < 1, consistent with the discharges being in an electron-flux dominated regime. Sensitivity studies show that TEMs in these plasmas respond almost equally to the same relative perturbation in the electron temperature and in the electron density scale lengths. By artificially reducing collisionality, TEMs are predicted to become more virulent, as expected, while the difference in growth rates between the two shapes slightly increases (which would be consistent with the collisionality dependence of electron heat diffusivities reported in Ref. 17). At electron scales, Electron Temperature Gradient (ETG) driven modes are predicted to be unstable at all radii and heating phases. Depending on the radius, heating phase and profiles uncertainties, ITG modes, which are usually subdominant to TEMs, are seen at times become the dominant instability only in the low-end of the wavenumber spectrum, i.e., in the k_θρ_s < 0.3 region. Linear simulations carried out with the same plasma profiles on the two equilibria at opposite values of triangularity indicate that, as expected from previous work described in Sec. III C, growth rates are reduced at negative triangularity in the TEM dominated region while ETGs are almost insensitive to the shape change. A survey of growth rates was done on actual profiles along with ten random variations thereof within experimental uncertainties; although the stabilization at ion scales exerted by negative triangularity is robust across the radius and heating phases, the reduction in growth rates is an involved function of radius, wavenumber, and heating phase and reflects the complex radial structure of the measured inverse scale lengths and ion to electron temperature ratios. In particular, as can be seen in Fig. 6, even though the reduction in the linear growth rates is usually larger than the population error bar associated with profile uncertainties, it is not rare to find subregions in the radius and wavenumber space in which differences in growth rates are within uncertainties. Main parameters used in the CGYRO input files for the simulations shown in Fig. 6 are summarized in Table II. The trapped particle fraction increases at negative triangularity, therefore it cannot explain the stabilizing effect on TEM and the confinement increase. During both heating phases, growth rates at electron scales are modeled to decrease at all radii in the negative triangularity equilibrium due to a reduced inverse electron temperature scale length. Indeed, consistent with observations reported in Ref. 40, in the plasma region ρ > 0.9, the electron temperature radial profile is seen to increase more rapidly at negative triangularity, while similar gradients are measured in the region 0.5 < ρ < 0.9; as a result, in this radial region the inverse electron scale length is lower at negative triangularity. The difference in the edge gradient between the two shapes is postulated in Ref. 40 to be due to a larger critical gradient for the onset of electron-scale fluctuations at the very edge of negative triangularity plasmas, which would be consistent with nonlinear flux-tube gyrokinetic simulations reported in Ref. 37. The predicted presence of strong ETG modes also in the core of the plasmas described in this report constitutes a major difference with respect to the TCV experiments, in which such modes were stable everywhere inside ρ ≃ 0.9 partly due to the much higher T_e/T_i ratio.

The high heating phase of these experiments was obtained mainly through ion heating, up to an administrative limit of 12 MW, including 2–3 MW of EC heating deposited slightly inside midradius. Plasmas were operated at values of plasma current, line averaged density, and confining magnetic field listed in Table I, with the averaged density typically being in the high end of the reported range, i.e., about 5 × 10¹⁹ m⁻³. Gas valves were not operated in either shape.

An intriguing observation of this experiment is that, even for discharges at matched density and actuators, the L → H power threshold appears to be significantly higher at negative triangularity. Indeed, while plasmas at positive triangularity usually exhibit a regular transition to an inner-wall limited H-mode phase when the auxiliary power exceeds 6–7 MW, no L → H transition in plasmas at negative triangularity was observed within the administrative power limit; consequently, all plasmas at negative triangularity described hereafter maintain edge pressure profiles typical of an L-mode phase. The higher L → H power threshold does not appear to be due to a difference in the power flux passing through the separatrix, nor to an imbalance between ion and electron energy fluxes between the two shapes. Although the two equilibria are no longer comparable in terms of diffusivities and fluctuations because they are in two distinct confinement regimes, especially near the edge where plasmas at positive triangularity feature a regular H-mode pedestal, it is still legitimate to compare their global, or volume integrated, properties. In Fig. 7, we display a comparison of two pairs of shots at matched actuators, for which the discharges at positive triangularity transition to the H-mode. It is remarkable to note that discharges at negative triangularity, despite maintaining edge pressure profiles typical of an L-mode phase that do not trigger any ELM, display global confinement values quantitatively comparable to those featured by positive triangularity discharges sustaining pressure and confinement values typically observed in H-mode plasmas. These discharges reach β_N > 2.5, confinement factor H_98y,2 > 1 and, at least on the DIII-D tokamak, set a record for L-mode performance at full field. The H-mode grade confinement level obtained in L-mode edge plasmas at negative triangularity is particularly apparent when visualizing the functional dependence of stored energy on the coupled power computed by the ONETWO code, which uses TORAY-GA and NUBEAM⁴¹ for EC and NBI, respectively. As shown in Fig. 8, where such dependence is compared to that from the ITER-89P⁴² power scaling law computed at the experimental value of line averaged density, plasma current, and toroidal field, the stored energy in negative triangularity plasmas progressively exceeds that of the scaling law, reaching, at the highest power, values corresponding to a rough definition of the H-mode in terms of pure global confinement. Within the power window available for this experiment, the computed values of stored energy correspond to confinement factors in the range H_89P = 1.05–1.6 and H_98,y2 = 0.75–1.2. The global confinement measured in these negative triangularity discharges is superior to that in other experiments in which plasmas are maintained in L-mode at high heating power, usually using a limiter, which typically feature confinement levels equal or below that predicted by the ITER-89P scaling law.^43–45 It is worth mentioning the existence of other scenarios that, despite not developing a typical H-mode phase, feature improved confinement with respect to the ITER-89P scaling law while displaying an approximately linear dependence between the stored energy and absorbed power: The I-mode⁴⁶ and the hot ion mode regimes.⁴⁷ However, such cases are in a different regime than that considered in this paper because the I-mode features a temperature pedestal, while the hot ion mode is characterized a strong transport barrier that typically yields T_i/T_e in the range 2–3. Studies at low or zero auxiliary torque were not possible in this experiment due to time constraints and are deferred to future work. As a side note, the increased confinement displayed by negative triangularity plasmas is consistent with a numerical exercise recently published in Ref. 48, in which the elongation and the triangularity of the last closed flux surface are left free to evolve while the algorithm maximizes the expected volumetric fusion power with radial profiles predicted, at each step, by a nonlinear mixed gyrofluid/gyrokinetic model.

The H-mode level pressures sustained by these negative triangularity discharges are always reached in the high power phase, regardless of the heating trajectory, as can be understood by looking at Fig. 7, and were not limited by stability. Indeed, as it can be seen in Fig. 9, the high pressure phase can be sustained for several energy confinement times without any ELM event. In particular, the normalized pressure rises in response to the power staircase increase without any sign of saturation or collapse, which indicates that the achieved values for normalized pressure are not due to plasma stability, rather to the administrative power limit imposed on these experiments. The ideal β limit for this up-down symmetric shape at negative triangularity was estimated to be approximately three by using a procedure developed in Ref. 49. A number of kinetic equilibria corresponding to various time slices of one of the negative triangularity discharges were first reconstructed with the equilibrium code EFIT. Since sawtooth oscillations are generally active in these plasmas, the β limit for the onset of an internal pressure driven kink mode is evaluated by estimating, as a function of pressure, a threshold in β_N above which growth rates and edge displacement increase sharply. Such threshold is compared to the onset of an n = 1 mode in a modified equilibrium in which the safety factor is artificially raised above unity to avoid the sawtooth oscillation; ideally, the two β_N limits should approximately coincide. The equilibrium from EFIT was read by the equilibrium code CORSICA⁵⁰ that was used to compute an inverse equilibrium after a slight modification of the q-profile to make it larger than unity at every radius. Subsequently, various equilibria corresponding to a pressure scan were generated by CORSICA and evaluated by the stability codes GATO⁵¹ and DCON.⁵² In Fig. 10(a), we display an example of the β_N dependence of growth rates for an n = 1 mode predicted by GATO for the original, or sawtoothing, equilibrium, and a modified equilibrium. It is interesting to note that, as shown in Fig. 10(b), the beta dependence of the growth rates features a break in slope across the predicted β limit, which for the case shown is equal to β_N = 3. This indicates that the pressure dependence of the sawtooth is rather weak, while that of the internal pressure driven kink is substantially higher. Sensitivity studies on the effect of the modified q-profile on the estimated β limit were carried out by varying the minimum of the modified q-profile, its radial location, and the flux surface at which the modified safety factor merges with the original profile smoothly up to the second radial derivative, all of which are arbitrary parameters. Pressure scans were carried out on forty different time instants and twenty modified q-profiles per time slice with the DCON code, resulting in a 7% relative change in the estimated β limit; the reason for such low sensitivity is likely due to the relative flatness of the pressure radial profiles in the regions where the safety factor was modified.

Additional parameters of interest in assessing the effect of triangularity on the potential reactor performance are bootstrap current and neutron yield. While it is generally true that the pedestal region provides a large fraction of the total bootstrap current, due to the combined effect of large pressure gradients and the large area enclosed by flux surfaces near the LCFS, inner regions can significantly contribute to the total current, especially if large pressure gradients are sustained over a significant fraction of the minor radius. This is particularly apparent in Fig. 11, where we compare the radial profile of the bootstrap current density computed for the two discharges displayed in Fig. 7(a) at 2.2 s. While the positive triangularity plasma features an H-mode pedestal, as evidenced by the large peak in the current density near the LCFS, the plasma at negative triangularity maintains an L-mode edge and is therefore free of the edge current peak. However, due to the larger pressure gradients developed across the minor radius, which is allowed by lower anomalous transport at a fixed heating power, inner regions at negative triangularity contribute up to 50% more bootstrap current than at positive triangularity, resulting in the negative triangularity plasma generating 18% more bootstrap current than its positive triangularity counterpart. The pair of discharges from Fig. 7(b), for which the H-mode pedestal is steeper and offers 15% increase in β_N compared to its counterpart at negative triangularity, results in 23% more bootstrap current produced at positive triangularity as computed by the NEO⁵³ code. The difference in the bootstrap current at inner radii is not due to the trapped particle fraction because this quantity can be shown to be identical for the two shapes within 8% at the LCFS, where the largest difference takes place.

The neutron rate in high confinement L-mode edge plasmas at negative triangularity is measured to be 30%–60% higher than that in H-mode plasmas at positive triangularity at matched values of plasma current, toroidal field, line averaged density, and auxiliary power. Figure 12, where we display the time evolution of relevant quantities for the two shapes, suggests that the reason for the lower neutron rate at positive triangularity is main ion dilution caused by higher effective charge in the H-mode phase. This is confirmed by time dependent calculations of the expected neutron rate computed with the TRANSP⁵⁴ code; as it can be seen in Fig. 13, while the expected neutron yield based on experimental profiles is higher at negative triangularity, in agreement with the measurements, the two expected neutron rates almost reverse by merely swapping the carbon content between the two shapes with all other profiles held fixed. The much higher carbon content measured at positive triangularity is certainly partly due to the edge barrier that, due to low particle transport and inward convective velocity given by the large main ion density gradient, yields increased impurity retention. However, a larger carbon source from the wall cannot be excluded; indeed, as compared to negative triangularity plasmas that maintain an L-mode edge, the C^III line intensity from a DIII-D filterscope, that looks at a location close to the inner-wall region used as a limiter in this experiment, sharply decreases as plasmas enter the H-mode regime but then quickly saturates due to the first few ELMs. Data on the C^II line are not available. The Laser Blow-Off (LBO)⁵⁵ system, which was recently installed on DIII-D, was used to estimate the impurity confinement time in high β plasmas at negative triangularity. The injected impurity was aluminum because its charge exchange cross section data allow accurate analysis. The decay rate of the Al^XIII density was measured by the CER system to be 64 ± 3 ms (Fig. 14) and 47 ± 5 ms for, respectively, a CER chord close to ρ = 0.2 and the outermost chord with a visible signal at ρ = 0.7; such values are quantitatively similar to those extracted from previous measurements in regular L-mode plasmas. Although we were not able to employ the LBO system in matched discharges at positive triangularity, the impurity confinement time in H-mode plasmas on DIII-D has consistently been measured to be longer than 160 ms.⁵⁶ The ratio of particle to energy confinement times, τ_P/τ_E, is measured to be within 0.69–0.94, depending on the CER chord used to estimate τ_P, while it is usually in the range 2–3 in H-mode plasmas.⁵⁷ Initial analysis shows that the convective velocity of Al impurities is directed radially outward, which is a desirable property in future fusion reactors because it lowers impurity accumulation; however, as stated previously, we cannot compare the magnitude of the convective velocity to that in matched discharges at positive triangularity.

This study shows that tokamak plasmas with a negative triangularity shape, despite having been prematurely dismissed by the fusion community on poor MHD stability grounds, are expected to achieve volume averaged pressure levels that would be suitable in future reactors. Although future experiments with diverted negative triangularity shape need to be carried out in order to extrapolate results to larger machines, this experiment shows the potential of an L-mode edge at negative triangularity as the operational regime in future reactors. First, ELMs would not be excited thus removing the need for 100% reliable “active” ELM stabilization techniques. Second, the SOL heat flux width would likely be larger than that in comparable H-mode plasmas. Third, main ion dilution would be lower due to poor impurity confinement given by the absence of the edge transport barrier. Fourth, the bootstrap current would be quantitatively comparable to that generated in H-mode plasmas but, instead of being concentrated near the separatrix, would distributed more uniformly across the radius.

Finally, the authors stress that alternative configurations, such as those investigated in this work, offer significant opportunities for model validation in various fields, from MHD to GK theory.

Inner-wall limited discharges at negative triangularity have been created on the DIII-D tokamak and compared to discharges with the same shape of the LCFS but with positive triangularity at matched values of plasma current, confining magnetic field, and auxiliary power. In both pure electron heating (up to 3 MW of EC) and combined electron-ion heating (up to 6 MW evenly divided between EC and NBI), plasmas at negative triangularity feature 20%–30% increase in confinement and lower heat diffusivities, primarily on the electron channel, accompanied by decreased intensity of fluctuations in the electron density, ion density, and electron temperature as measured by Phase Contrast Imaging, Beam Emission Spectroscopy, and Correlation Electron Cyclotron Emission diagnostics, respectively. The overall reduction of the intensity of fluctuations is in the range 10%–30% depending on the diagnostic, on the heating scheme, and the coupled power. A linear gyrokinetic analysis shows that these plasmas, regardless of triangularity, are dominated at the ion-scale by Trapped Electron Modes (TEMs) in both heating schemes and across the minor radius, with the Electron Temperature Gradient (ETG) modes destabilized at electron scales. Depending on the actual coupled power as well as on the radial location being analyzed, Ion Temperature Gradient (ITG) modes, which are usually subdominant to TEMs, appear at times as the dominant instability in the low-end part of the ion-scale wave-vector spectrum. Negative triangularity is seen to reduce TEM growth rates at all radial locations examined, although the predicted stabilization is not uniform in the radius and wavenumber due to plasma profiles being different in the two shapes. While the amount of reduction usually exceeds the impact of experimental uncertainties on growth rates, it is predicted to be within error bars for a limited subset in the wavenumber and radius. In the high heating power phase, during which the NBI power was increased up to 9 MW while maintaining the ECH power constant, plasmas show near-zero power degradation and sustain, for several energy confinement times, volume averaged pressure levels typically observed in H-mode discharges. These plasmas reach β_N = 2.7 and H_98,y2 ≃ 1.2, despite maintaining edge pressure profiles typical of L-mode plasmas and therefore naturally free of Edge Localized Modes (ELMs). The ideal β limit for these discharges is predicted by the GATO code to take place at β_N ≃ 3; such a value would be suitable in a reactor provided that the L-mode edge can be maintained. High pressure plasmas in this shape offer low impurity retention and similar amounts of bootstrap current compared to regular H-mode plasmas for a similar energy confinement, which are desirable properties for a fusion reactor.

The authors wish to thank Dr. F. Turco for the valuable assistance with the CORSICA and DCON codes, as well as Dr. K. H. Burrell, Dr. T. C. Luce, Dr. T. W. Petrie, and Dr. T. L. Rhodes for the useful discussions. This material is based upon the work supported 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-FG02-94ER54235, DE-FC02-99ER54512, and DE-SC0016154. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3DDMP. TRANSP simulations were performed on the PPPL cluster using version 2018-1.0, DOI: 10.11578/dc.20180627.4. Part of the data analysis was performed using the OMFIT framework.

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