Introduction

As worldwide consumption of energy and resources steadily increases, it is becoming more and more probable that traditional, non-renewable energy resources such as coal, oil, and natural gas will be depleted, and that the resulting carbonaceous products could have a significant impact on the global warming and environment. Fusion is a possible pathway to provide carbon-free, clean energy sources1. International Thermonuclear Experimental Reactor (ITER), an international collaboration project, is under construction for the purpose of validating the scientific and technological feasibility of fusion. One of the great challenges for ITER and future fusion reactors is handling the divertor target heat load while maintaining high core confinement. The divertor is a vessel component that extracts heat and ash produced by the fusion reaction. The design limit for the maximum ITER divertor heat load is 10 MW/m2[ 2, which requires a sizeable heat load width to handle the fusion power flux on the divertor plasma-facing materials. Multi-machine divertor heat flux width scalings3 imply a very narrow heat flux width \({\lambda }_{q}\) ( < 1 mm) for the ITER Q = 10 baseline scenario. To explain the present-day narrow heat flux widths in high-confinement mode (H-mode) plasmas, the heuristic drift (HD) model4 was developed. The HD model ignores the turbulence in scrape-off-layer (SOL), as this is predicted to be quenched by the \(E\times B\) shear. The HD model estimate combines cross-field magnetic drift velocity \({v}_{d}\) with the parallel streaming dwell time \({\tau }_{\parallel } \sim 2{qR}/{C}_{s}\) to obtain \({\lambda }_{q} \sim {v}_{d}{\tau }_{\parallel } \sim 2\epsilon {\rho }_{\theta i}\). The result implies a narrow heat flux width ( ~ 2 mm) for ITER, where \(q\), \(R\), \({C}_{s}\), \(\epsilon\) and \({\rho }_{\theta i}\) are the safety factor, major radius, ion sound speed, aspect ratio and poloidal ion gyro radius. Such a narrow \({\lambda }_{q}\) could cause severe damage to the plasma-facing materials and threaten the safety of the ITER operation. Thus, there is considerable interest in broadening \({\lambda }_{q}\) for ITER and future fusion devices.

Wider divertor heat flux widths for ITER are predicted by state-of-art numerical modeling using the XGC15 and BOUT + + 6,7 simulation codes, due to edge turbulence which was ignored in the HD model. These studies claim that the edge turbulence could contribute to a stronger cross-field transport, and lead to a similar heat flux width near the marginally stable pedestal state, \({\lambda }_{q}\) = 5.9 mm for XGC1 and \({\lambda }_{q}\) ~ 5 mm for BOUT + +. However, the underlying physical mechanisms for SOL turbulence production and its implications for SOL broadening remained unexplored, making it essential to address these questions for ITER. In this paper, we identify turbulence spreading as the key player and illuminate the physics of edge-to-SOL spreading. Theoretical models have studied the effect of turbulence spreading on the divertor heat flux width8. These predicted that both drift wave and ballooning mode turbulence in the pedestal could lead to an enhanced turbulence intensity flux across the separatrix, thus broadening the SOL and heat flux widths. Experimental evidence relating turbulence spreading and the divertor heat flux width has also been observed in the large helical device (LHD)9. These works provide further insights into the physics that determines the divertor heat flux width. We also note that the transient pedestal events, such as edge localized modes (ELMs), could be triggered by the pedestal peeling-ballooning (PB) mode, and deposit large heat and particle flux to the divertor target, and so damage the plasma-facing components. Thus, developing a scenario with good confinement without ELMs, while maintaining a sizeable divertor heat flux width is a critical goal for the safe operation of future fusion reactors, like ITER and beyond.

This paper presents one possible realization of this goal in the Quiescent H-mode(QH mode). QH mode10,11 and wide-pedestal QH mode12,13 show the compatibility of high core confinement14 and a quiescent edge, without ELMs. In recent QH mode experiments on the DIII-D tokamak, divertor heat flux width is observed to broaden following the increase of the pedestal broadband turbulence without driving edge harmonic oscillations (EHO)15. Separate experiments presented in this paper study the roles of the pedestal modes and the physics of turbulence spreading in setting the divertor heat flux width of DIII-D QH mode plasmas. The pedestal turbulence consists of a low-frequency ion diamagnetic direction (IDD) mode, which is radially extended from the pedestal top to the SOL; and a high-frequency electron diamagnetic direction (EDD) mode, which is radially peaked at the upper pedestal and localized in the pedestal location. In order to connect the upstream pedestal turbulence and downstream heat load width, this paper studies the roles of the pedestal modes and the physics of turbulence spreading in setting the divertor heat flux width of DIII-D QH mode plasmas. Here, we demonstrate how pedestal turbulence can expand into, or entrain, the stable scrape-off layer and so broaden the heat flux width beyond predicted neoclassical values. We show that the turbulence spreading is the mechanism, indeed the ‘missing link’, which connects pedestal turbulence to the heat load distribution on the downstream divertor.

Results

Experimental observations

The essential ingredients in the core-edge-boundary plasmas consist of the core region, SOL region, and divertor region. The separatrix sets the boundary between core and SOL plasmas. A schematic illustration of a QH mode cross-section on the DIII-D tokamak is shown in Fig. 1a. Broadband turbulence is widely observed in the pedestal region of wide-pedestal QH mode16,17 and broadband QH mode18 on DIII-D, which could have an impact on the heat flux width broadening as suggested by previous work15. Figure 1b, c shows the upstream midplane pedestal turbulence in a typical QH-mode and downstream divertor heat load pattern. Their relationship will be investigated in the following paper.

Fig. 1: Schematic of DIII-D turbulent quiescent-H(QH) mode.
figure 1

a Illustration of DIII-D plasma cross-section; b Frequency spectrum of density turbulence perturbation as a function of time measured by Beam Emission Spectroscopy (BES) on the outboard-mid-plane, blue shaded b region in a. The color code gives the intensity of the turbulence; c Downstream divertor heat load pattern measured by Divertor Langmuir Probes, which shows the parallel heat load as a function of the distance to the separatrix mapped back to the outboard-mid-plane, green shaded c region in a.

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The broadband turbulences in the edge of QH mode have complex characteristics16,17,18,19 as illustrated in Fig. 2, which contains the density fluctuation measured by the Beam Emission Spectroscopy (BES)20 in one of the QH mode discharge #184902. In the upper pedestal region, i.e. \({\psi }_{N}\) = 0.90 (blue) and 0.93 (green), broadband turbulence has dual bands (Fig. 2a, b). One band is a low-frequency (10–60 kHz) mode rotating in the ion diamagnetic direction (IDD) and the other band is in the high-frequency (60–200 kHz) range rotating in the electron diamagnetic direction (EDD), in the lab frame. At the pedestal peak gradient \({\psi }_{N}\) = 0.95 (red), it is a combination of IDD and EDD mode in the mid-frequency mode (60–80 kHz) range, whose \({k}_{\theta }\) is kept almost constant regardless of the frequency. In the lower part of the pedestal \({\psi }_{N}\) = 0.98 (purple) and SOL, the turbulence is dominated by the IDD mode for the whole frequency range (10–200 kHz). The radial mode structures resulting from the integration of different frequency ranges of IDD and EDD modes are depicted in Fig. 2c. The IDD mode, characterized by density perturbation \(\delta n/n\), exhibits a larger mode amplitude, with its peak situated at the separatrix (\({\psi }_{N}\) = 1.0), and a broader structure that extends farther into the SOL. Conversely, the EDD mode peaks at the upper pedestal, \({\psi }_{N}\) = 0.93, and its influence barely reaches the SOL. The fundamental mechanism linking the pedestal turbulence and divertor heat flux width will be further analyzed numerically and theoretically in the latter part of this paper.

Fig. 2: Broadband turbulence measured by Beam Emission Spectroscopy (BES) in the pedestal region of quiescent-H (QH) mode discharge #184902.
figure 2

a Cross power and b poloidal wave number \({k}_{\theta }\) as a function of frequency at different pedestal radial locations. The positive wave number shows the mode rotates in the ion diamagnetic direction (IDD) in the lab frame and the negative wave number shows the mode rotates in the electron diamagnetic direction (EDD) in the lab frame. c the radial envelope of density fluctuation (\(\delta n/n\)) associated with the IDD (red) and EDD (blue) mode. The error bar for the density fluctuation \(\delta n/n\) is derived from the reciprocal of the square root of the coherence in the related frequency range.

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The divertor heat flux width of broadband QH mode for different plasma poloidal magnetic field \({B}_{p}\) was measured in the recent QH mode experiments. The divertor strike point sweep technique was applied to obtain radially-resolved divertor heat load data from Divertor Langmuir Probes and Infrared Imaging Systems (IR camera). In Fig. 3a, heat flux widths are measured, and the comparison with the current multi-machine database is shown in Fig. 3a. The divertor heat flux width of broadband QH-mode also decreases with increasing poloidal magnetic field \({B}_{p}\) along the hyperbola. The trend is due to the lowered safety factor \(q\) and thus shortened parallel dwell time \({\tau }_{\parallel } \sim 2{qR}/{C}_{s}\). The ratio of measured heat flux width to Eich scaling \({\lambda }_{q}/{\lambda }_{q,{Eich}}\) increases with the edge turbulence amplitude, shown in Fig. 3b. The horizontal axis shows the low-frequency (5–18 kHz) IDD turbulence measured by BES at the separatrix. Compared with the higher current discharges, the divertor heat flux widths still follow the trend of hyperbolic decrease with increasing \({B}_{p}\), but are broadened by stronger edge turbulence for the low-\({B}_{p}\) discharges(yellow). Note that the magnitude of \({\lambda }_{q}\) was derived from the multi-scale scaling by increasing it by 40%

Fig. 3: Divertor heat flux width with different out-mid-plane poloidal magnetic field Bp and turbulence at the separatrix.
figure 3

a Comparison of measured divertor heat flux width with experimental multi-machine scaling regression #1452 (black); b Ratio of measured divertor heat flux and Eich scaling as a function of BES measured low-frequency IDD turbulence amplitude \(\delta n/n\) close to the separatrix. Note that both ion diamagnetic direction (IDD) and electron diamagnetic direction (EDD) modes coexist inside the separatrix, and only IDD mode exists in the open field line region. The solid bullets are the heat flux width measured from Infrared Imaging Systems (IR camera), and the empty bullets are measured from Divertor Langmuir Probes. The errorbars are estimated by performing bootstrapping simulations for the heat flux widths measured by Divertor Langmuir Probes.

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Numerical modeling

The role of broadband turbulence in broadening the divertor heat flux width is numerically investigated using the BOUT + + 21,22 six-field two-fluid model, which is widely utilized to study the edge physics and divertor heat flux width \({\lambda }_{q}\) for several different current and future machines, C-Mod23, DIII-D24, EAST25, ITER5,6 and CFETR26. BOUT + + modeling achieved a reasonable agreement with experimental measurements on current machines23,24,25, which normally follow Eich’s scaling law and Goldston’s heuristic drift model. For future reactors, like ITER and CFETR, the reduced radial drift velocity ( ~ \(1/R\)) could make the cross-field transport fall in a turbulence-dominant regime5,6, while the underlying mechanism of how the SOL turbulence is generated and impacts heat load width must be determined. The BOUT + + six-field model with carbon impurity is implemented for the recent DIII-D QH mode discharges, which intends to understand the long-intermediate wave length edge turbulence within the fluid framework. Similar to previous work19 the peeling-ballooning mode (PBM) and drift Alfvén wave (DAW) dominate for different toroidal mode numbers n, for the QH-mode discharge. The linear growth rate and linear mode structure are shown in Fig. 4. Linearly, PBM dominates at low to intermediate n, n = 5-30, and extends radially from the pedestal to the SOL; DAW turbulence dominates at higher n, n > \({n}_{{crit}}\) = 30 and is localized radially at the upper pedestal \({\psi }_{N}\) = 0.93–0.95. These results are consistent with the characteristics of the experimentally observed IDD and EDD mode shown in Fig. 2. A more detailed and quantitative comparison of linear and nonlinear simulation to the turbulence spectrum can be found in ref. 19

Fig. 4: Linear modeling results for the DIII-D quiescent-H (QH) mode discharge #184902.
figure 4

a Linear growth rate \(\gamma /{\omega }_{A}\) as a function of toroidal mode number n, where \({\omega }_{A}\) is the Alfvén frequency. The red-shaded region is the peeling-ballooning mode (PBM) dominant, and the blue-shaded region is drift Alfvén wave (DAW) dominant. The critical \({n}_{{crit}}\) = 30 differentiates the PBM and DAW; b Radial mode structure of PBM and DAW.

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Simulations suggest that the spatially extended PBM turbulence plays a key role in broadening the divertor heat flux width \({\lambda }_{q}\) beyond the Eich scaling prediction. The role of PBM and DAW in setting the divertor heat flux width in QH plasmas are investigated in numerical experiments where the pedestal ion temperature \({T}_{i}\) and electron temperature \({T}_{e}\) profiles are scaled up and down to change the relative amplitude of PBM and DAW. As illustrated in Fig. 5a, c, the pedestal profile inside the separatrix is scaled by the factor \({F}_{i,e}\), which ranges from 0.6 to 1.8. The SOL profiles are kept fixed to exclude the SOL profile effects on divertor heat load and its width. So the SOL remains linearly stable for all these cases. The linear growth rate of PBM increases with the \({T}_{i}\) scale factor \({F}_{i}\), as shown in Fig. 5b. In contrast, DAW, destabilized by parallel passing electrons, is insensitive to the pedestal \({T}_{i}\) profile. The critical transition toroidal mode number \({n}_{{crit}}\) between PBM and DAW increases with the pedestal scale factor \({F}_{i}\) increases. PBM linear growth rate dominates over DAW in the whole n = 5-60 range for \({F}_{i}\ge\)1.2, and DAW dominates the whole n = 5–60 range for \({F}_{i}\le\)0.6 in Fig. 5b. In Fig. 5d, we found that DAW becomes unstable with a steepened pedestal \({T}_{e}\) profile, and PBM linear growth also increases slightly due to the change of ballooning drive with total pressure. In this sense, the magnitude of PBM tracks \({F}_{i}\), while DAW increases with \({F}_{e}\). The divertor heat flux width \({\lambda }_{q}\) is broadened remarkably with an increase of \({F}_{i}\) (Fig. 5e) while shows much less sensitivity to the electron temperature and the upper pedestal localized DAW suggesting that DAW does not spread to the SOL, and so does not broaden \({\lambda }_{q}\) directly. The results of this numerical “experiment” are summarized in Table 1.

Fig. 5: The changes in divertor heat flux widths correspond to the variations in pedestal conditions.
figure 5

Variation of pedestal ion temperature \({T}_{i}\) a and electron temperature \({T}_{e}\) b to change the linear growth rate of most unstable mode c, d. The critical transition \({n}_{{crit}}\) increases with \({F}_{i}\) while remains as, \({n}_{{crit}}\) = 30, with \({F}_{e}\) increases; e Divertor heat flux width \({\lambda }_{q}\) as a function of pedestal scale factor \({F}_{i}\) and \({F}_{e}\).

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Table 1 Characteristics of different modes of DIII-D QH mode in BOUT + + modeling
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On how the pedestal modes link to the divertor heat flux, we argue it is through the turbulence spreading. The SOL of these plasmas is linearly stable, so turbulence there is necessarily dominated by turbulence spreading from the pedestal, across the separatrix. Over the past few decades, significant progress has been made in understanding turbulence spreading27 by theory, simulations28,29,30,31 and experiments32,33,34,35,36. The flux of turbulence intensity is calculated to quantify the spreading into the SOL in the turbulent QH plasma studies here. Figure 6a, b shows turbulence spreading of normalized pressure perturbation and normalized turbulence intensity flux. Turbulence is generated inside of the pedestal, at \({\psi }_{N}\) = 0.96 (as shown in Fig. 6a). The negative and positive turbulence intensity fluxes in Fig. 6b indicate the inward and outward spreading of turbulence intensity. The turbulence spreads from the pedestal peak gradient to the separatrix and enters the SOL, from the linearly most unstable zone to the linearly stable zone. The time-averaged pressure perturbation and turbulence intensity flux are shown in Fig. 6c, d. The skewness of pressure perturbation is also plotted in Fig. 6d. The skewness of pressure perturbation is studied to clarify the physics of turbulence spreading, which is associated with the birth and propagation of blob (filament) and void structures37,38,39,40,41. Positive skewness indicates the higher-than-average pressure ‘blobs’ and negative skewness indicates ‘holes’. One sees that the zero-crossing point of skewness and turbulence intensity flux are well-correlated, and situated at the location of maximal turbulence generation, \({\psi }_{N}\) = 0.96–0.97. Both negative signs of skewness and turbulence intensity inside of \({\psi }_{N}\) = 0.96 indicate inward turbulence spreading, and vice versa. The large positive skewness (\({{{{{\mathscr{S}}}}}}\) > 2) in the SOL shows a strong, positively skewed pressure perturbation and indicates the outward spreading feature of the turbulence. Further study of the impact of turbulence spreading on divertor heat flux width will be discussed in the following sections.

Fig. 6: Nonlinear simulation of the DIII-D turbulent quiescent-H (QH) mode.
figure 6

a normalized pressure perturbation; b normalized turbulence intensity flux. Here \(\delta p,{p}_{{ped}}\) and \({\delta v}_{r}\) are the perturbed pressure, equilibrium pedestal top pressure, and perturbed radial velocity in BOUT + + modeling. The horizontal axis is the time in Alfvén time, and the vertical axis is the normalized poloidal magnetic flux. The horizontal dashed line indicates the location of the separatrix. c The time-averaged pressure perturbation (blue) in the saturation phase. d The time-averaged turbulence intensity flux (blue) in the saturation phase, and normalized by its radial maximum absolute value; radial skewness of the pressure perturbation (red).

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The effect of turbulence spreading on the divertor heat flux width is also studied numerically with SOL \({E}_{r}\) scans where a stronger \(E\times B\) shearing rate at the separatrix is found to obstruct pedestal turbulence spreading40 and thus narrow the divertor heat flux width. In Fig. 7a, the SOL \({E}_{r}\) profiles are modified accordingly, so as to change the \(E\times B\) shearing rate at the separatrix, thus simulating situations with different sheath potentials. To maintain the pedestal turbulence unchanged, \({E}_{r}\) profiles inside the separatrix are kept fixed. Separatrix \(E\times B\) shearing rate is defined as the form of Hahm–Burrell \(E\times B\) shearing rate42, \({\omega }_{E}=-\frac{{\left(R{B}_{p}\right)}^{2}}{B}\frac{\partial }{\partial \psi }\left(\frac{{E}_{r}}{R{B}_{p}}\right)\), where \(R,{B}_{p},B,\psi\) and \({E}_{r}\) are major radius, poloidal magnetic field, total magnetic field, poloidal magnetic flux and radial electric field, respectively. The separatrix \(E\times B\) shearing rate ranges over ~3–7 MRad/s for these scans. As the separatrix \(E\times B\) shearing rate increases, the pressure perturbation skewness \({{{{{\mathscr{S}}}}}}\) decreases from ~0.7 to ~0.3, which is consistent with the expectation that less turbulence spreads into the SOL. The downstream divertor heat flux width \({\lambda }_{q}\) also narrows accordingly, as shown in Fig. 7b. The turbulence intensity flux is also calculated for different separatrix \(E\times B\) shearing rates in Fig. 7b. The turbulence intensity flux decreases with the increase of the separatrix \(E\times B\) shearing rate, and it also follows the trend, as the red circles shown in Fig. 7b. All these findings indicate less turbulence spreading into the SOL when the separatrix \(E\times B\) shearing rate increases, thus resulting in a narrower heat flux width.

Fig. 7: The impact of separatrix E×B shearing rate on the divertor heat flux width.
figure 7

a Scan of scrape-off-layer SOL \({E}_{r}\) profile; b Divertor heat flux width \({\lambda }_{q}\) (green), turbulence intensity flux \({\varGamma }_{{turb}}\)(red) and skewness \({{{{{\mathscr{S}}}}}}\) of pressure perturbation (blue) at the outboard-mid-plane separatrix as a function of separatrix \(E\times B\) shearing rate.

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Theoretical model

Different sets of scans discussed above of changing pedestal PBM/DAW turbulence are summarized in Fig. 8. The horizontal axis in Fig. 8 is the turbulence intensity flux across the separatrix8,43,

$${\varGamma }_{{turb}}={c}_{s}^{2}{\left\langle {\left(\frac{\delta P}{{P}_{{sep}}}\right)}^{2}{\delta v}_{r}\right\rangle }_{\theta ,\zeta ,t}$$
Fig. 8: The divertor heat flux width broadens with the turbulence intensity increases.
figure 8

Normalized divertor heat flux width as a function of flux surface and time-averaged turbulence intensity flux \({\varGamma }_{{turb}}\) across the separatrix. Different symbols demonstrate the different scans on the profiles.

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\({c}_{s}\) is the ion sound speed, \(\delta P\), \({P}_{{sep}}\) and \({\delta v}_{r}\) are the perturbed pressure, equilibrium separatrix pressure and perturbed radial velocity form BOUT + + nonlinear simulation, bracket represents the flux-surface and time average in the nonlinear saturation phase, respectively. The vertical axis is the ratio of BOUT + + simulated heat flux width to the Eich scaling value. We find that the divertor heat flux width is close to the Eich scaling when the turbulence intensity flux is weak. \({\lambda }_{q}\) exceeds the Eich scaling prediction when the turbulence intensity flux exceeds a threshold, \({\varGamma }_{\varepsilon ,{crit}}\) ~ 1.0e11m3/s3. A similar trend of \({\lambda }_{q}\) is observed for different sets of PBM/DAW mode shearing rate scans. The threshold behavior is similar to Fig. 6 in ref. 8 and Fig. 2 in ref. 6 which show that the magnetic drift dominates the cross-field SOL heat transport when turbulence intensity is sufficiently weak. When the turbulence intensity flux is sufficiently large, turbulence will overtake the neoclassical cross-field transport and the divertor heat flux width increases monotonically with increasing turbulence intensity flux \({\varGamma }_{{turb}}\). Figure 8 demonstrates the connection between the pedestal turbulence characteristics and the downstream heat flux width.

The proportionality between the heat flux width and SOL mode scale length is demonstrated in Fig. 9. Normalized pressure perturbations with different \({T}_{i}\) pedestals are shown in Fig. 9a. The normalized pressure perturbation increases in the midplane pedestal and SOL region, and turbulence spreads deeper into the far SOL region with a steepened \({T}_{i}\) pedestal. A more extended ‘tail’ of the turbulence intensity profile in the SOL is observed when stronger PBM turbulence is excited in the pedestal, as shown in Fig. 9a. In Fig. 9b), the radial scale lengths of pressure perturbations at the outboard-mid-plane are quantified by e-folding estimation. The turbulence intensity scale length sets the effective mixing length in the SOL. A linear relation between the upstream turbulence intensity scale length \(\delta l\) for the SOL and downstream divertor heat flux width \({\lambda }_{q}\) is demonstrated for different sets of scans, including pedestal \({T}_{i}\), \({T}_{e}\) and SOL \({E}_{r}\). These findings reinforce the importance of the turbulence-broadening effect on the divertor heat flux width.

Fig. 9: The upstream turbulence scale length is proportional to the downstream heat flux width.
figure 9

a Normalized root-mean-square pressure perturbation as a function of distance from separatrix at outboard-mid-plane in the SOL; b Radial scale length of pressure perturbation \(\delta l\) by e-folding as a function of divertor heat flux width \({\lambda }_{q}\).

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Conclusions

In conclusion, this study investigates how the pedestal turbulence can broaden the divertor heat flux width. We focused on DIII-D QH mode plasmas in the absence of ELMs where heat flux widths that are broad relative to Eich scaling are experimentally observed. Heat flux widths in broadband QH-mode are measured to increase with stronger turbulence at the separatrix. The turbulence consists of an ion-directed peeling-ballooning mode extending radially to the SOL, and an electron-directed drift Alfvén wave localized at the upper pedestal. The pedestal ion/electron temperature profiles are varied to study the role of different scale modes in broadening the divertor heat flux width \({\lambda }_{q}\). We showed that the ion-directed peeling-ballooning mode, spreads turbulence intensity into the SOL and thus efficiently broadens the \({\lambda }_{q}\). The heat flux width \({\lambda }_{q}\) drops for stronger separatrix \(E\times B\) shearing rate, leading to less a positively skewed perturbation and reduced spreading into the open field line region.

All these results are consistent with the theory of turbulence spreading and the calculation of turbulence intensity flux \({\varGamma }_{{turb}}\) across the separatrix, which shows a broader heat flux width when the turbulence intensity flux exceeds a threshold. In particular, this crucial result establishes the connection between pedestal turbulence and downstream heat flux width. The divertor heat flux width is also found to show a positive correlation to the skewness of pressure perturbations and be proportional to the scale length of turbulence in the SOL at the outboard midplane. Note that, this paper focuses on the ion-directed peeling-ballooning turbulence and the electron-directed drift-Alfvénic turbulence with a reduced two-fluid model; further types of turbulences by including kinetic effects, i.e. trapped electron mode (TEM) turbulence5,15 and micro-tearing mode (MTM) turbulence could also broaden the heat flux width by the spreading and entrainment process in the SOL. More generally, this work, numerically-theoretically-experimentally, supports the merit of turbulent pedestals. These could have a broader heat flux width that exceeds that predicted by the Heuristic drift scaling law, while simultaneously maintaining good core confinement. However, it is worth noting that a downside of pedestal MHD/drift wave turbulence would lead to some degradation of pedestal performance. Note that current modeling doesn’t have a self-consistent flux originated from the core and the full physics of core-edge coupling. It is crucial, then to examine trade-offs, through dedicated experiments and flux-driven modeling, so as to find an optimum solution for future reactors, like ITER and beyond.

Methods

DIII-D tokamak

The DIII-D tokamak is the largest magnetic fusion experiment in the United States, supported by the U.S. Department of Energy Office of Science. The tokamak consists of a toroidal vacuum chamber surrounded by coils that produce the magnetic field to confine and shape the plasma. It has a major radius of 1.67 m and a minor radius of 0.67 m, with a toroidal magnetic field of up to 2.2 T. The plasma is created by applying a voltage to ionize a small amount of gas injected into the vacuum chamber and drive a large, toroidal electrical current. The plasma is then quickly heated to a high temperature by injection of high-power neutral beams ( < 16 MW), while additional gas fueling increases the density. More information can be found in ref. 44.

BES measurements for pedestal turbulence

The beam emission spectroscopy (BES)45 diagnostic is applied for the study of dynamics of long-wavelength (\({k}_{\perp } < 3{{cm}}^{-1}\)) density turbulence structures on DIII-D. A 64-channel BES system is configured with an 8\(\times\)8 grid of discrete channels that image an approximately 7\(\times \)9 cm region at the outboard midplane with the sampling frequency of 1 MHz, thus providing a modest spatial resolution, high throughput, high-time resolution turbulence imaging system.

Langmuir probe measurements for divertor heat flux width

The probe46 tips are dome-type shape with 6 mm diameter, 1 mm height above tile surface, and ~1.5 cm separation at the outer divertor target plate. They are operated in a single-probe mode with a 1 kHz sweeping frequency. A multi-point median filter was applied to the data from each probe and the error bars were given by calculating the median absolute deviation.

IRTV infrared imaging measurements for divertor heat flux width

Heat flux measurements are performed via infrared thermography47. The IRTV diagnostic is used to measure the surface temperature of the lower divertor tiles in the proximity of the outer strike point. The THEODOR code is used to solve the heat conduction equation and derive the perpendicular heat flux. Parallel heat flux is derived from the perpendicular heat flux taking into account of the field line incident angle. IRTV measurements are acquired at 12 kHz with a spatial resolution of 2 mm per pixel. Note that the spreading factor sets an upper limit, S = 1.5 mm, to get reasonable fitting results.

Numerical modeling

The BOUT + + code19,21,24 is six-field module us used for numerical modeling, which is based on Braginskii equations and evolves the ion density, ion/electron temperature, ion parallel velocity, vorticity, and magnetic vector potential. The simulation domain covers the pedestal and SOL region in normalized poloidal flux \({\psi }_{N}\) = 0.85–1.10, with a grid resolution of radial \({n}_{\psi }=256\), field-aligned \({n}_{y}=64\) and binormal \({n}_{z}=64\) in the field-aligned coordinate on simulating a 1/5 of the torus. The simulation contains n = 5, 10, 15, …, 80 toroidal number modes. PVODE, an implicit time integration method, is used for time-stepping and the simulation continuous until the divertor heat flux is well saturated. Realistic Spitzer-Härm resistivity is used in the modeling. Sheath boundary conditions are applied on the divertor targets. In the radial direction, the main ion and carbon density and temperatures are are set to be Neumann, e.g. \(\partial n/\partial \psi =0\) to let them freely evolve. Neumann boundary conditions are also applied to the vorticity and parallel velocity and Zero-Laplacian boundary condition, \(\Delta {A}_{\parallel }\) = 0, is applied to the parallel vector potential. The pedestal plasma profiles are based on the well-converged reconstructed equilibrium using EFIT code48. The parallel thermal conductivity in the flux limited form is given by, \({\kappa }_{{eff},j}=\frac{{\kappa }_{\parallel ,j}^{{SH}}{\kappa }_{\parallel ,j}^{{FS}}}{{\kappa }_{\parallel ,j}^{{SH}}+{\kappa }_{\parallel ,j}^{{FS}}}\), where j signifies ion or electron, \({\kappa }_{\parallel ,i}^{{SH}}=3.9{n}_{i}{v}_{{th},i}^{2}/{\nu }_{i}\) and \({\kappa }_{\parallel ,e}^{{SH}}=3.2{n}_{e}{v}_{{th},e}^{2}/{\nu }_{e}\) are the Spitzer-Härm parallel thermal conductivities, \({v}_{{th},j}\) and \({\nu }_{j}\) are the thermal velocity and collision rate for the j species. \({\kappa }_{\parallel ,j}^{{FS}}={\alpha }_{j}{n}_{j}{v}_{{th},j}q{R}_{0}\) is the free streaming parallel thermal conductivity, where q is the safety factor and \({R}_{0}\) is the major radius. A realistic \({E}_{r}\) profile from the pedestal up to SOL calculated from the ion momentum balance equation based on Charge Exchange Recombination (CER) Spectroscopy System49,50 in DIII-D is used in the simulation. The diamagnetic \({E}_{r}\) arising from the zonal pressure component is retained, while the Reynolds Stress-induced zonal flow is omitted in the simulation, which is in line with the approach in ref. 51 and other publications. Carbon is the major impurity considered in the modeling: the equilibrium carbon density and temperature profiles are from the DIII-D CER system; carbon and main ion temperature are set to be the same. Divertor heat flux widths are measured in the fully saturated stage of the simulation. More details about the model can be found in Supplementary note 1 and on saturation in Supplementary note 2.