Transport and stability in sustained high qmin , high βN discharges on DIII-D

To address the needs for a fusion pilot plan design, DIII-D/EAST joint experiments on DIII-D have demonstrated high normalized beta β N ∼ 4.2, toroidal beta β T ∼ 3.3% with q min > 2, q 95 ⩽ 8 sustained for more than six energy confinement times in high poloidal beta regime. The excellent energy confinement quality (H 98y2 ∼ 1.8) is achieved with an internal transport barrier at high line-averaged Greenwald density fraction f Gr > 0.9. The trapped gyro-Landau fluid (TGLF) modeling of the transport characteristics shows that the beam-driven rotation does not play an important role in the high confinement quality. The modeling also captures very well several transport features, giving us confidence in using integrated modeling to project these experimental results to future machines. The high-performance phase is terminated by fast-growing modes triggered near the n = 1 ideal-wall kink stability limit. New radio frequency (RF) capabilities for off-axis current drive could remove the residual ohmic current to achieve a fully non-inductive state, and improve the mode–wall coupling to increase the ideal-wall β N limit, enabling sustainment of the fully non-inductive high performance plasma in stationary conditions.


Introduction
Steady-state tokamak research aims at efficient low-cost fusion reactors.It requires high fusion gain Q together with * Author to whom any correspondence should be addressed.
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the bootstrap current fraction f bs to be maximized for high fusion power and continuous operation [1].The fusion gain is defined as the ratio of the fusion power produced P fus to the auxiliary heating power P aux as Q ≡ P fus /P aux .Here, P fus is roughly proportional to ⟨p⟩ 2 in the temperature regime of interest to burning plasma experiments, and for the stationary conditions, P aux is given by the plasma power losses P loss (∝ ⟨p⟩/τ E ) offset by the self-heating P α , then Q Q+5 = P fus P loss ∝ ⟨p⟩τ E ∝ β t τ E B 2 0 [1][2][3].Therefore, improving the fusion plasma performance requires maximizing simultaneously the thermal energy confinement time τ E , and the toroidal beta β t defined as 2µ 0 ⟨p⟩ /B 2 0 , at high toroidal field.β p .For steady-state operation, the high β p approach (first proposed by Kikuchi [4]) has three key attractive features: (1) since f bs ∝ ε 0.5 β p ∝ ε −0.5 qβ N [3], where β p is the poloidal beta defined as 2µ 0 ⟨p⟩/B 2 p , β N is the normalized toroidal beta defined as β t (I P /aB 0 ) −1 , q is the safety factor, the high β p scenario can maximize f bs and minimize the need for the expensive external auxiliary current drive; (2) since β p ∝ ⟨P⟩/I 2 p , higher β p with lower I P has a lower risk of damage from disruptions [5] and can reduce the edge localized modes (ELMs) challenge [6,7]; (3) since the Shafranov shift ∆ ∝ β p ∝ α MHD ∝ dβ p /dr, high β p plasma enables strong α MHD -stabilization of turbulence, and therefore improves the energy confinement quality [8,9].
To fulfill both the requirements of bootstrap current dominated steady-state plasma and high fusion gain, high β N and optimized q are needed and bounded by the magnetohydrodynamic (MHD) β N stability limits [10].In particular, the β N limit depends on profiles shaping and wall stabilization [11].Taking advantage of wall stabilization, the maximum β N achieved on JT-60U is up to ∼2.7, close to the ideal wall limit [12].On DIII-D, based on previous high β p experiments [13], with the optimization of the plasma-wall separation and with broader pressure profile the experimental maximum β N was extended to ∼4, approaching the ideal wall limit at high q 95 ∼ 12 [14,15].There are two paths to ITER Q = 5 steadystate performance [16] in the high β P regime on DIII-D.One is the low collisionality path to fully-noninductive plasma at higher plasma current and lower q 95 (I P ∼ 0.9-1 MA, q 95 ∼ 6-7) [17], requiring strong external off-axis current drive.The other is the high β N path toward comparable normalized fusion performance to ITER Q = 5 but at higher q 95 .Following this latter path, the performance of the high β p regime on DIII-D has been extended to β N ∼ 4.2 with q min > 2 at moderate q 95 ∼ 8.
The rest of the paper is organized as follows: section 2 presents the details of the high β N path achievement of high β N ∼ 4 at high q min > 2. Transport analysis is discussed in section 3, which shows consistency between turbulence measurements and gyrokinetic simulations, shedding light on the key mechanisms to sustain high energy confinement.Section 4 investigates the limiting instability preventing the stationary sustainment of high performance.Section 5 summarizes the results and future plan to extend to fully non-inductive plasma sustained in stationary conditions.

Extension to higher normalized performance with high β N > 4
The high β N path is explored to extend the normalized performance of the high β P scenario on DIII-D toward reactorrelevant steady-state operation targets.Figures 1(a)-(d) shows the time history of discharge 185959.During the discharge, the plasma current I P has a second, slower ramp starting at 2 s from 730 kA to 800 kA, and a temporary (∼1 s long) B T rampdown starting at 1.3 s from 2.1 T to 1.7 T. The β N target is set to 4 at 2.76 s by the feedback control using neutral beam power.
The plasma cross section is an upper-biased double-null divertor shape.This plasma achieves high normalized performance with β N ∼ 4.2, β t ∼ 3.3% at q min > 2 sustained for 0.7 times current relaxation time (more than six energy confinement times).The excellent energy confinement quality (H 98y2 ∼ 1.8) is achieved at high (reactor level) density (n e ) >7 × 10 19 m −3 , with the line averaged Greenwald fraction f Gr up to ∼1, and with bootstrap current fraction f BS ∼ 80%.No core impurity accumulation is observed, and the radiated power fraction and line-averaged Z eff are nearly constant.Despite the formation of an internal transport barrier (ITB) in all channels.Figures 1(e)-(f ) shows the ITB is formed at a large minor radius (ρ ∼ 0.7) for n e , T e , T i and toroidal rotation profiles.The safety factor profile is nearly flat in the core, with q min sustained above 2. Here, sustainment of the ITB at a large radius is not tied to the use of off-axis external current drive.Figure 1(g) shows the current density profile from the equilibrium fitting code (EFIT) reconstruction, including kinetic and motional Stark effect (MSE) measurements.The non-inductive fraction f NI is ∼90%.The bootstrap current fraction f bs up to ∼80% is predicted by the Sauter model [18].

Transport features during high-energy confinement
In this section, we will map the overall turbulence picture in regulating the kinetic profiles in the high energy confinement phase.In particular, the consistency between gyrokinetic simulations, fluctuation measurements, and transport fingerprints [19] is verified, which helps consolidate our transport physics understanding.Notably, it will be shown that a kinetic ballooning mode (KBM) plays a critical role in saturating the ITB.

Transport fingerprint analysis
Transport fingerprints of this discharge across the whole core region are shown in figure 2. The turbulent flux (red in figure 2) is obtained by subtracting the neoclassical flux (blue in figure 2), calculated by the first-principle drift-kinetic code NEO [20], from the experimental fluxes estimated via ONETWO code [21].As can be seen, the ion energy flux (Q i ) is dominated by the neoclassical transport inside ρ < 0.4, the role of turbulent transport undergoes a fast transition region between ρ = [0.4,0.65], during which region turbulent Q i changes from subdominant to dominant, and then turbulent Q i fully dictates for ρ > 0.65.A similar trend is observed for the electron particle channel (Γ e ), which shows the divergence between turbulent and neoclassical transport is gradually enhanced toward the outer core region.Based on the fingerprints analyzed above, it is inferred that the longwavelength turbulence, which drives both turbulent Q i and Γ e , is likely to be weak inside ρ < 0.4.Its significance/intensity will undergo a transition region between ρ = [0.4,0.65], e.g. from weak to moderate, after which (ρ > 0.65) the turbulence will become robustly unstable.Such an inference will be verified against linear gyrokinetic simulations, as discussed below.

Gyrokinetic simulation and validation against fluctuation diagnostics
To understand the transport physics across the entire core region, linear gyrokinetic simulations are performed with the CGYRO code [22], which is a flux-tube model optimized for electromagnetic calculations and multiscale physics in the presence of collisions.With high density, the fast-ion population and pressure is low as n f ne = 0.017 and p f ptot = 0.10.Thus, only the dilution effect of fast ions is considered in the simulation, and the goal is to focus on understanding the background micro-turbulence.Linear simulations are done with E × B shearing off.The results are presented in figure 3, with the γ/k y ρ s × sign (ω) of the dominant instabilities in both the poloidal wave number (k y ρ s ) and radial space (ρ).Here, γ is the growth rate normalized by c s /a (c s is the ion sound speed), positive and negative sign denoted modes drifting in ion and electron direction, respectively, while the amplitude is represented by color.k y = nq/r is the poloidal wave number with n the toroidal mode number, q local safety factor and r the minor radius, and ρ s is the ion gyro-radius with the sound speed.
The poloidal wavenumber, k y ρ s , is scanned from 0.1 to 100, which covers the spatial range of major micro-instabilities that can lead to transport events.The radial range covers from ρ = 0.2-0.8,almost representing the whole core region, and can be divided into three regions for the convenience of interpretation in the following context: the inner core region at 0.2 ⩽ ρ < 0.4, the ITB region at 0.4 ⩽ ρ < 0.65, and the outer core region at 0.65 ⩽ ρ < 0.8, which is consistent with fingerprint analysis in section 3.1.In the inner core region, the low-k turbulence (k y ρ s < 1) is marginally unstable, while the high-k modes (k y ρ s > 1) is linearly stabilized.In the ITB region, low-k turbulence starts to emerge, and KBM is predicted to be unstable.In the outer core, both low-k (ion temperature gradient (ITG) driven mode) and high-k (trapped electron mode (TEM)/electron temperature gradient (ETG) driven mode) turbulence becomes robustly unstable.Here, trapped gyro-Landau fluid (TGLF) code can also reproduce the drift wave instabilities (e.g.ITG, TEM and ETG) well.However, KBM in the ITB region is not predicted to be unstable by TGLF.The reason is that from CGYRO simulation, the KBM is unstable only when the compression magnetic component (δB || ) is included, while TGLF cannot treat δB || pretty well in the current version.

Experimental validation of KBM.
While CGYRO predicts the KBM to be unstable around ρ = [0.45,0.60], this section will show that experimentally there exists an electromagnetic fluctuation whose characteristics are consistent with the theoretical expectation of KBM.Specifically, (1) its A high frequency (130-220 kHz) coherent mode is observed from the cross power between the line-integral measurement of density and magnetic fluctuations of the radial interferometer polarimeter (RIP) in the midplane chord shown in figure 4(a), suggesting that this mode has electromagnetic nature.Figure 4(b) shows this fluctuation is associated with the β p saturation, and the performance saturation is induced by the halt of ITB strength (P ITB ) growth, rather than the pedestal height (P ped ), indicating that the fluctuation excitation is related to core dynamics [23].
Those fluctuations are also observed in the local beam emission spectroscopy (BES) array, from which the eigenfunction (more specifically, the radial perturbation displacement ξ r ) of such a mode can be obtained (figure 5).Here, |ξ r |=| δne ne | ne −∇ne , with the normalized density fluctuation δne ne coming from BES and the density profile n e measured via Thomson scattering.As can be seen, such a mode has its eigenfunction peaked at ρ ∼ 0.55. Figure 5 also shows that flux-tube CGYRO predicts the KBM to be robustly unstable across the ITB region and has its growth rate peaks at ρ ∼ 0.55, consistent with the experimentally measured eigenfunction.

Experimental validation of drift wave turbulence.
A Doppler backscattering system (DBS) [24] has been employed to detect the density turbulence in this study.In this discharge, the targeted poloidal wavenumber is about 3-5cm -1 , and its normalized value, k θ ρ s , ranges from 0.8 to 1.5.Such a wavenumber range falls in the so-called intermediate-k scale.The measured density turbulence propagates in the electron diamagnetic drift (EDD) direction at the ITB foot but moves in the ion diamagnetic drift direction at the steep gradient region  of the ITB (contour plot in figure 6(a)).Here, the turbulence power is plotted in terms of its radial location and poloidal phase velocity.That is, darker spots stand for more turbulence power measured by DBS systems at specific radii and phase velocities.The power-weighted poloidal phase velocity can be calculated based on the Doppler shifts of the measured density turbulence (cyan circles in figure 6(a)).By comparing against the background E × B mean flow, measured by the charge exchange recombination spectroscopy (CER) diagnostic, we can determine that turbulence at the ITB foot also propagates in the EDD direction in the plasma rest frame, indicating its electron-mode nature.
As mentioned in the previous section, the TEM/ETG modes are predicted by the CGYRO simulation to be linearly unstable.A more detailed scan has been performed and validated with the experimental measurements of turbulence at the ITB foot (ρ ∼ 0.7).As shown in figure 6(b), given the experimental value of the local gradients, for a single mode of k θ ρ s = 1.2, the linear simulation suggests that the dominant mode at this scale has a negligible poloidal phase velocity in the plasma rest frame.A nonlinear simulation is undergoing to shed light on the underlying mechanism that drives the instabilities.

Key mechanisms in the sustainment of high energy confinement quality
The analysis presented in section 3.2 suggests that the core transport is dominated mainly by drift-wave turbulence except in the ITB region, where KBM plays an important role.According to the analysis in [23], the KBM is mostly efficient in particle transport, while its effect on the thermal transport is limited, justifying TGLF analysis for the transport in the temperature channel, which is a theory-based reduced model specified in drift-wave turbulence simulations.In particular, the goal is to understand the key mechanism in sustaining the strong ITBs with excellent confinement (H 98y2 ∼ 1.8).TGLF modeling suggests that the E × B shear plays a critical role in the high confinement, which can also be achieved with only the self-generated E × B mean flow shear through the diamagnetic contribution, suggesting the potential extrapolability of such a discharge toward fusion reactors, which are expected to have low toroidal rotation.
To dedicatedly study the role of E × B shear, our approach is to implement different components of E × B shear in the TGYRO code [25], which calls TGLF [26] and NEO for the calculation of turbulent and neoclassical transport flux, respectively, and compare the predicted kinetic profiles with experimental ones.From the ion force balance equation, the E r can be expressed in the heuristic form as shown below: For convenience, the subscript is omitted.Here Z and P are the ion charge number and pressure respectively.v ϕ (v θ ) and B ϕ (B θ ) are the toroidal (poloidal) ion velocity and toroidal (poloidal) magnetic field, respectively.The three components on the right-hand side are termed diamagnetic, toroidal, and poloidal term hereafter.The diamagnetic term is proportional to the kinetic gradient and is self-generated, and the toroidal term is related to the toroidal rotation, which depends strongly on external torque.From CER measurements [27], different components are obtained in the form of rotation frequency ω 0 = Er RBp with R the major radius in figure 7(a), and the shearing rate γ E = r q * dω0 dr in figure 7(b).Predicted profiles considering different components of E r profiles are summarized in figures 7(c) and (d).When the nominal (full) Er shear profile is retained, TGYRO predicted kinetic profiles (red line in figures 7(c) and (d)) match well with experiments (black line), further justifying the application of the TGYRO code here.Here, the spectrum shift model is employed when accounting the E × B shearing effect on the turbulence.When the whole E × B shear is removed, the predicted kinetic profiles drop drastically (purple), suggesting the important role of E r shear in the ITB sustainment, even though the E r shearing rate is close to zero in the region ρ = [0.5, 0.7].However, the strong ITB can be restored (or even a bit improved) with the self-generated diamagnetic component only (blue).It should be noted that under similar q 95 discharges, previous literature [17] argues the ITB sustainment is mostly enabled by the so-called alpha stabilization of drift wave turbulence, which is a spontaneous geometrical stabilization effect that can be strongly leveraged by the negative or flat magnetic shear.Considering the temporal dynamics of this discharge shown in figure 4(b), the strong ITB is already formed before 2850 ms.Here, as discussed in [23], together with the negative magnetic shear, the alpha stabilization is enough to sustain the ITB without E × B shear.At 3350 ms, the ITB region is aligned with positive shear (e.g.ρ ∼ 0.55 in figure 2), so relying on the alpha stabilization alone is unlikely to reproduce the experimental profile.The Grad-P E r has an important role here while the toroidal rotation shear has a weak or no effect on sustaining ITB, suggesting the excellent confinement in this discharge might be self-consistently extrapolated to future machines.

MHD stability limits
Figures 8(a)-(c) show time traces of shot 185959 during the high-performance period until the discharge experiences a fast stored energy collapse around 3507 ms.Even though the β N target is set as 4.0 in the NBI feedback control system, the actual β N has transient excursions above the feedback target, excursions that worsen as confinement slightly improves over time.
Using space-domain analysis for δB p , figures 8(d) and (e) shows the process of the rapid precursor as a limiting instability prevents stationary sustainment of higher performance: (i) before disruption, MHD bursts that are a mixture of n = 1, 2, 3, with the n = 2 being the strongest; (ii) between ELMs, n = 1 amplitude becomes the largest indicating a growing non-rotating n = 1 precursor before the last ELM; (iii) after the last ELM, a rotating n = 1 mode with 2-3 toroidal rotations begins to grow exponentially with the observed growth time of τ g ∼ 550 µs; (iv) before thermal quench, the growth of the rotating n = 1 mode accelerates with the growth time of τ g ∼ 100 µs and δB p increases up to ∼25 G.This rapid growth is not typical for tearing modes, but consistent with previous observations of n = 1 kink mode near the idealwall stability limit on DIII-D [14,28].From a model [29] of an ideal mode slowly driven through the stability boundary, the growth time can be expressed as 2 h , combining the expected growth time τ MHD of the ideal MHD mode with the characteristic time τ h of the instability drive.In this case, the observed growth time is τ g ∼ 100 µs, and the instability drive is τ h ∼ 0.5 s, evaluated from calculations of the n = 1 ideal MHD kink growth rate from the stability code GATO [30].This combination yields τ MHD ∼ 1.3 µs, which is a growth rate typical of ideal-MHD instabilities.Therefore, this 'hybrid' growth time τ g ∼ 100 µs is consistent with an ideal kink mode that is slowly driven through the stability limit.
Figure 8(f ) shows more details of the n = 1 kink mode stability calculated by GATO.The plot shows the variation of the instability growth rate (normalized to the Alfven frequency) as a perfectly conducting wall is moved from the position of the DIII-D wall (dashed vertical line) to infinitely far away.The growth rate behavior is shown for equilibrium reconstructions calculated at three different times before the beta collapse: 3200 ms, 3300 ms and 3480 ms.GATO predicts that all three equilibria are stable with a perfectly conducting wall at the position of the DIII-D graphite tiles.However, comparing the different time slices shows that the no-wall growth rate is gradually increasing with time, and the plasma is gradually approaching the ideal-wall n = 1 kink limit before the disruption.The no-wall growth rates are also shown by the solid circles in figure 8(b).Thus, the high-performance phase is terminated by a fast-growing mode triggered near the n = 1 idealwall kink stability limit, likely destabilized by the β N transient overshoot of the feedback target.In the future study, we will aim at fully non-inductive high β P plasma at β N > 4 with q min > 2 sustained in stationary conditions on DIII-D.Thus, we need to improve stability around the ideal-wall limit by good mode-wall coupling.New radio frequency (RF) capabilities with more off-axis current drive (110 GHz ECCD launched nearly vertically [31], 476 MHz helicon waves launched from above the midplane on the low field side [32], 4.6 GHz lower hybrid wave launched from the high field side launch [33]) are shown in figure 9(a).Based on the plasma for shot 185959, for new RF current driven capabilities, preliminary GENRAY predictions in figure 9(b) show that: (1) the helicon wave drives current ∼43.9 kA MW −1 at ρ ∼ 0.4; (2) EC wave drives current ∼13.2 kA MW −1 at ρ ∼ 0.6; (3) low hybrid wave drives current ∼45.8 kA MW −1 at ρ ∼ 0.75.These could remove the residual ohmic current to realize fully non-inductive conditions and also improve the mode-wall coupling to increase the ideal-wall β N limit.Together with improved β N feedback control to prevent β N overshoot, this technique allows exploring the sustainment of the fully non-inductive high-performance plasma in a stationary regime.
New RF capabilities with more off-axis current drive could remove the residual ohmic current to realize fully non-inductive conditions and broaden the current profile to increase the mode-wall coupling and thus the ideal-wall β N limit.

Summary
On the path to addressing the needs for an economical fusion pilot plant, DIII-D/EAST joint experiments on DIII-D have demonstrated high normalized beta β N ∼ 4.2, toroidal beta β T ∼ 3.3% with q min > 2, f Gr ∼ 1.0 sustained for 0.7 times current relaxation time (more than six energy confinement times), expanding the high bootstrap current fraction regime (f BS ∼ 80%) with large radius ITB to higher normalized performance, relative to previous results.The excellent confinement quality in these plasmas (H 98y2 ∼ 1.8) is driven by the combination of alpha stabilization of turbulence and the Grad-P dominant E r , while does not require the NBI-driven rotation, as confirmed by transport modeling.In the ITB region, a high-frequency coherent mode (130-220 kHz) is observed by RIP during the high-performance period, with experimental features consistent with theoretical expectations for KBM at ρ ∼ 0.55.Outside the ITB foot region, TEM/ETG turbulence measured using the DBS is consistent with CGYRO calculations at ρ ∼ 0.7.A rapidly growing n = 1 mode appears as limiting instability, preventing stationary sustainment of the high-performance phase.GATO calculations indicate that the plasma is reaching the ideal-wall n = 1 kink limit right before the disruption.While the achieved bootstrap current fraction on DIII-D is high (up to 80%), it will be even higher in a reactor at the same β N , q 95 , and f Gr due to the lower collisionality from higher B T and I P .Future experiments will aim to make up the difference in bootstrap current and achieve fully non-inductive and stationary conditions by using new RF capabilities for the additional, efficient off-axis current drive at high density and beta.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Figure 2 .
Figure 2. Neoclassical and turbulence flux of (a) Q i and (b) Γe across the whole core region.

Figure 3 .
Figure 3. Linear dominant micro-instabilities in both the poloidal wave number space (x-axis) and radial space (y-axis).Positive (negative) value represents mode drifting in ion (electron) direction while the amplitude is denoted by color.

Figure 4 .
Figure 4. (a) The frequency spectrum of cross power between the line integral measurement of density and magnetic fluctuations of RIP; (b) the ITB strength (P ITB , black dotted), pedestal height (P ped , black dashed line) and βp (blue solid).

Figure 6 .
Figure 6.(a) Er × B velocity from CER measurements (V Er×B ), the power-weighted poloidal phase velocity inferred from DBS (V DBS ), positive (negative) sign denotes modes drifting in ion (electron) direction, turbulence power distribution as a function of ρ and poloidal phase velocity V θ with darker regions standing for more turbulence power in the phase space; (b) frequency of dominant instability (kyρs = 1.0) in the space of a/L Te and a/L Ti scan.

Figure 7 .
Figure 7. Profiles of different Er components (a) with each shearing rate (b): total, diamagnetic, toroidal and poloidal term denoted by red, blue, black and green, respectively; (c) electron and (d) ion temperature profiles from the experimental data fit (black) and the TGYRO predictions with full Er shear (red), with the self-generated diamagnetic component only (blue), without Er shear (purple).

Figure 8 .
Figure 8. Experimental time traces for shot 185959: (a) plasma current (Ip); (b) the β N target is set as 4.0 using NBI feedback control with NBI injected power P inj in (c), the calculated growth rates of the n = 1 kink mode normalized to the Alfven frequency for the ideal wall far away from the plasma; (d) time traces of n = 1, 2, and 3 amplitude of δBp at the outer midplane; (e) 2D contour of δBp versus time and toroidal angle with several dashed lines divided into different periods; (f ) the growth rate of the n = 1 kink mode normalized to the Alfven frequency against the ideal wall radius multiplier relative to the DIII-D vessel wall for 185959 at t = 3200 ms, 3300 ms, 3480 ms before the instability, calculated by GATO.

Figure 9 .
Figure 9. (a) New RF capabilities with more off-axis current drive with EC, Helicon and LH wave systems.(b) Based on the plasma for shot 185959, GENRAY predicts each new RF current driven capability.