High-resolution heat flux width measurements at reactor-level magnetic fields and observation of a unified width scaling across confinement regimes in the Alcator C-Mod tokamak

Power exhaust remains one of the greatest unsolved challenges towards the safe operation of reactor-class tokamaks (e.g. ITER, ARIES, Demo, ARC, SPARC). The peak parallel heat flux entering the divertor chamber is set by the power entering the scrape-off layer (P_SOL) and the heat flux width of the scrape-off layer (SOL): ${{q}_{\parallel }}={{P}_{{\rm SOL}}}B/\left(4\pi R{{\lambda }_{\rm q}}{{B}_{{\rm p}}} \right)$. Progress is being made towards a fundamental understanding of what sets the heat flux width [1–3], however it remains largely an empirical science with projections informed by empirical scaling laws [4].

A large improvement in empirical projections was achieved in assembling a 6-machine international database of heat flux widths in H-mode [4]. It was found that the heat flux width scaled approximately with the inverse of the poloidal magnetic field (B_p) and was remarkably independent of machine size: both Alcator C-Mod (major radius R  =  0.67 m) and JET (R  =  2.96 m) had similar heat flux widths at the same poloidal magnetic field. One of the major limitations of the multi-machine database was that the largest poloidal magnetic field was ~0.8 T. This required a 1.5×  extrapolation to the 15 MA Q  =  10 scenario in ITER (B_p  =  1.2 T) [5] and a nearly 2×  extrapolation to ARC (B_p  =  1.5 T) [6]. Such extrapolations project to unmitigated heat flux densities parallel to the magnetic field in reactor-class tokamaks on the order of 10 GW m⁻² and greater [7].

Alcator C-Mod has been the only divertor tokamak that has operated at reactor-level poloidal magnetic fields, up to 1.3 T. JET is the next closest operating tokamak, able to go up to B_p ~ 0.9 T at its maximum plasma current I_p  =  4.5 MA. High poloidal magnetic fields have allowed Alcator C-Mod to approach reactor-class parallel heat flux densities, up to 1–2 GW m⁻². Given these unique capabilities, it was a major focus of the final campaign of Alcator C-Mod in 2016 to measure divertor heat flux profiles and other plasma conditions at reactor-level magnetic fields and near reactor-level parallel heat flux densities.

This work presents a first look into a database of ~300 plasma shots on Alcator C-Mod spanning nearly the entire operational range of engineering parameters: toroidal magnetic field 2.7–8.0 T, poloidal magnetic field 0.43–1.3 T, line-averaged core density 0.44–5.2  ×  10²⁰ m⁻³, input power 0.52–5.5 MW. However, due to the need to keep the strike point on the lower outer divertor sensors, it was over a somewhat narrow range in geometry: elongation 1.5–1.8 and triangularity 0.48–0.61. The database spans the three main steady-state confinement regimes on C-Mod: (1) L-mode, which lacks both an edge particle and heat transport barrier [8]; (2) EDA H-mode (enhanced D-alpha) a steady ELM-free regieme, which has both an edge particle and heat transport barrier [8, 9]; and (3) I-mode, which has an edge heat transport barrier, but lacks an edge particle transport barrier [10, 11]. All H-modes were with the $B\times \nabla B$-drift direction towards from the x-point ('favorable'), all I-modes were with the $B\times \nabla B$-drift direction away from the x-point ('unfavorable'), and approximately half of the L-modes were in the favorable and the other half in the unfavorable direction. Although all C-Mod H-modes here are free of ELMs, EDA H-modes still follow the inter-ELM heat flux width scaling of ELMing H-modes from other experiments [4]. All plasmas lacked internal transport barriers [12]. All plasmas had deuterium as the main fuel, minimal impurity seeding and low levels of divertor radiation (low to moderate recycling regime) such that the heat flux footprint on the divertor target is representative of the scrape-off layer power exhaust.

Making measurements in the intense boundary plasma of Alcator C-Mod is very challenging. To ensure confidence in the measurements, the Alcator C-Mod team has installed and validated an extensive set of diagnostics to measure the time-evolution of surface heat flux as well as the shot-integrated energy flux across the divertor target. These include: (1) tile-embedded thermocouples to measure the shot-integrated energy flux [13–15]; (2) thermally-isolated calorimeters to measure the shot-integrated energy flux at a finer spatial resolution [13–15]; (3) refractory-metal surface thermocouples to measure the time-evolution of the surface temperature and calculate the surface and parallel heat fluxes [14–17]; (4) Langmuir probes with two different geometries to measure the time-evolution of plasma temperature and density by which the surface heat flux can be calculated through sheath theory [14–16, 18, 19]; (5) an IR thermographic camera to measure time- and spatially-resolved images of surface IR emission that are used to calculate the surface heat flux [13, 20] (the IR camera was viewing the inner divertor in 2016, thus its data is not used here).

Figure 1 shows an example of the excellent agreement in shot-integrated energy flux from these sensors over the whole divertor target. A shot-integrated energy flux plot was done for every shot in the database to identify and remove sensors that significantly deviated from other nearby sensors. This is needed because sometimes a surface thermocouple open circuits and reports an erroneous temperature measurement or a Langmuir probe melts and the density it measures is systematically high due to a distorted collection area. The good agreement across five different measurements of shot-integrated energy flux gives strong confidence in the time-resolved heat flux measurements. A final check is performed by overlaying the heat flux measurements of the sensors to check for outliers before fitting an analytic profile, figure 2. The agreement between thermal and probe based measurements across the divertor profile indicates that the surface heat flux to the divertor in the regimes examined is dominated by the plasma, while the surface heat flux from neutrals and photons is relatively insignificant.

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The predominant technique used in recent years to extract λ_q from measurements has been to fit an analytic equation to the data that is based on simple physics ansatzes [4, 21]: Upstream of the magnetic x-point there is a competition between heat transport along and across the magnetic field. This results in an exponential heat flux profile across the field with the characteristic width λ_q, which is the main quantity of interest. Downstream of the x-point there is further dissipation across the magnetic field, due in part to plasma transport and dissipation with neutrals. This cross-field transport is assumed to be the same both radially inward into the private flux zone as well as radially outward into the common flux zone and described by a convolution of the upstream exponential profile (truncated at $\rho =0$, where $\rho$ is the poloidal magnetic flux coordinate mapped to the outer midplane) and a Gaussian profile with characteristic scale length S (the 'spreading' parameter). Taking this Gaussian spreading into account was thought to improve fits of the exponential part of the profile. An additional uniform 'background' heat flux (q_BG) was also included in the profile:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{q}_{\parallel }}\left(\rho \right)=\frac{{{q}_{0}}}{2}{\rm exp}\left({{\left(\frac{S}{2{{\lambda }_{\rm q}}} \right)}^{2}}-\frac{\rho }{{{\lambda }_{\rm q}}} \right){\rm erfc}\left(\frac{S}{2{{\lambda }_{\rm q}}}-\frac{\rho }{S} \right)+{{q}_{{\rm BG}}}.\nonumber \end{align} \tag{ 1 }$$

This is commonly called the 'Eich' fit or equation. But for simplicity in this work it is called the 'single-λ fit' (the reason for this will be clear shortly). The single-λ fit did a satisfactory job of reproducing the divertor heat flux profiles measured with IR thermography for the multi-machine database across a wide variety of plasma conditions [4].

However, with the aid of the increased spatial resolution (~0.1 mm, mapped to the outer midplane) and heat flux dynamic range (~4 orders of magnitude) of the probe-based sensors, the single-λ fit no longer accurately describes the experimental measurements, figure 2. On a log-linear plot, the analytic heat flux profile (equation 1) in the private flux region for the single-λ fit is shaped like an inverted parabola. Whereas the measurements are clearly closer to linear, like the common flux profile, rather than parabolic. Over the limited heat flux dynamic range available from IR thermography, ~1–2 orders of magnitude, this differentiation likely was not possible. But the nearly 4 orders of magnitude dynamic range with the probe-based sensors shows it to clearly be the case. Additionally, we find no evidence for a uniform background heat flux, even on the purely thermal sensors (also evident in the energy flux plot, figure 1), which is likely due to the minimal divertor radiation in these shots and the very narrow field-of-view from the vertical target plate to radiation from the core plasma. The second exponential region in the far common flux region could easily have been interpreted as a background heat flux for a system with limited dynamic range. There is also sometimes evidence for a second exponential region in the far private flux region.

The measured heat flux profile appears to be best described by 4 exponential profiles, one each for the near/far common/private flux regions (λ_q,cn, λ_q,cf, λ_q,pn, λ_q,pf) and 3 heat flux magnitudes (the peak q₀, the contribution from the far-common profile q_cf, and the contribution from the far-private profile: q_pf):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{q}_{\parallel }}\left(\rho \right)=\left\{\begin{array}{@{}cccccccccccccccccccc@{}}\left({{q}_{0}}-{{q}_{{\rm pf}}} \right){{{\rm e}}^{\rho /{{\lambda }_{\rm q,{\rm pn}}}}}+{{q}_{{\rm pf}}}{{{\rm e}}^{\rho /{{\lambda }_{\rm q,{\rm pf}}}}}, & \rho <0 \nonumber \\ \left({{q}_{0}}-{{q}_{{\rm cf}}} \right){{{\rm e}}^{-\rho /{{\lambda }_{\rm q,{\rm cn}}}}}+{{q}_{{\rm cf}}}{{{\rm e}}^{-\rho /{{\lambda }_{\rm q,{\rm cf}}}}}, & \rho \geqslant 0 \nonumber \\ \end{array} \right..\nonumber \end{align} \tag{ 2 }$$

For simplicity, we call this the 'multi-λ fit'. Fitting the multi-λ fit to the profile in figure 2, we find excellent agreement across the entire profile. Like the single-λ fit, the multi-λ fit can be convolved with a Gaussian to account for any additional spreading or dissipation at the heat flux peak. The far private flux e-folding (λ_q,pf) is not always resolved. So, a triple-λ fit was used for all profiles in this database, neglecting the fourth e-folding in the far private flux region. This reduced the number of free parameters by 2 (λ_q,pf and q_pf) and increased the reliability of fits of the other parameters (λ_q,cn, λ_q,cf, λ_q,pn, q₀, and q_cf).

A similar profile was pointed out in a recent examination of divertor power sharing around double-null in C-Mod [22]. Also, this is not the first time the heat flux profile in the private flux region has been characterized with an exponential profile; other cases include the heat flux [23] as well as electron temperature and density [24] divertor profiles in DIII-D.

The observation that the heat flux profile in the private flux region is better represented by an exponential decay indicates that the ansatz in the single-λ fit of cross-field heat transport in the divertor leg being uniform in both directions is likely wrong. In these low dissipation plasmas, the cross-field heat transport is asymmetric. Such a situation is similar to upstream measurements of transport out of the core and into the boundary, which indicates a very high degree of poloidal asymmetry: nearly all of the fluctuation-induced fluxes occur on the low-field side [25]. This is indicative of ballooning-like transport where the pressure gradient is stabilized by magnetic curvature on the high field side and destabilized on the low field side. Similar curvature-driven cross-field heat transport may be present along the divertor legs. A more detailed study of the scaling of the exponential decay length in the private flux region from this database is forthcoming.

The Gaussian spreading has been used to account for rounding of the heat flux peak in addition to transport into the private flux region. In nearly all of the cases examined for this database, it was found that the Gaussian spreading parameter was not needed (i.e. it is much less than either λ_q,cn or λ_q,pn). The exponential profile (λ_q,pn) takes care of the private flux region and, unlike the IR-based measurements, relatively little rounding of the peak heat flux is seen on the probe-based measurements. This suggests that the apparent appropriateness of the Gaussian spreading in fitting the analyzed IR data, at least on C-Mod, may be due in part to instrumental effects (i.e. insufficient dynamic range and/or spatial resolution—possibly due to blurring from periscope shaking or a coated first mirror) rather than divertor transport physics.

We have also found that fitting the multi-λ fit to the data reduces systematic biases induced by the single-λ fit. The Gaussian spreading parameter (S) fit is strongly influenced by the decay in the private flux region for the single-λ fit: When S is comparable to or larger than λ_q, it starts to compete with λ_q in the common flux region, reducing the value of λ_q by up to a factor of 2 relative to the multi-λ fit in the cases examined. On the other hand, when S is much smaller than λ_q, it plays a minimal roll in affecting the fitted λ_q. But in this regime the fitted λ_q for the single-λ fit can be up to a factor 2 over λ_q,cn from the multi-λ fit. This is presumably because the single-λ fit can be skewed by the 2nd e-folding in the far common flux region, which can be poorly accounted for by the background heat flux term in the single-λ fit. This 2nd e-folding in the far common flux region is well known from upstream measurements [26, 27] but less frequently considered on divertor heat flux measurements.

The figure from the multi-machine heat flux width database [4] showing the scaling of the common flux near scrape-off layer heat flux width with poloidal magnetic field is reproduced here in figure 3, along with the measurements in H-mode from the new high-field C-Mod database. Extrapolation of the multi-machine trend line:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda }_{\rm q}}\ \left({\rm mm} \right)=\left(0.63\pm 0.08 \right)\times {{\left({{B}_{{\rm p}}}\ \left({\rm T} \right) \right)}^{-1.19\pm 0.08}},\nonumber \end{align} \tag{ 3 }$$

fits the new C-Mod data well. There is a slight systematic under prediction of the high field points, likely due to the stronger than inverse scaling from the spherical tokamaks, which dominate the low-field measurements [4].

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Figure 4 shows the scaling of the heat flux width against poloidal magnetic field in all confinements regimes (L-, I-, and H-mode) along with power law fits to each regime. In isolation from the multi-machine database, the C-Mod-only H-mode data scales almost perfectly inversely with the poloidal magnetic field and has a ~20% higher amplitude factor than the purely poloidal field scaling from the multi-machine database. Both of these differences are presumably due to not including the effects of the low-field, low-aspect ratio machines, MAST and NSTX. However, it is in-line when including aspect ratio in the scaling (regression #15 in [4]): ${{\lambda }_{\rm q}}\ \left({\rm mm} \right)\approx 0.79\times {{\left({{B}_{{\rm p}}}\ \left({\rm T} \right) \right)}^{-0.92}}$ for C-Mod geometry.

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L-mode heat flux widths are ~2 times as wide as H-mode and have a slightly shallower scaling with poloidal magnetic field. This is consistent with previous scalings on JET [28] and ASDEX-U [29] that showed L-mode heat flux widths to be ~2–3 times wider than H-mode. I-modes have an approximately inverse square root dependence on poloidal magnetic field but with very large scatter. The I-modes, consistent with their name of 'improved L-mode', span the entire space between L- and H-mode heat flux widths for a given poloidal magnetic field. Past measurements with IR thermography on C-Mod are consistent with this, indicating that I-modes tended to have somewhat wider heat flux widths than H-modes [30].

Given the large scatter of heat flux widths against poloidal magnetic field across confinement regimes, the C-Mod database was searched for other global parameters which could better-unify the data. Figure 5 shows the tightest single-parameter scaling across confinement regimes. Remarkably, the heat flux width organizes well across all confinement regimes (L-, I-, and H-modes), scaling with the inverse square root scaling of volume-averaged core plasma pressure $\bar{p}$ (or equivalently, at constant geometry parameters, total plasma stored energy):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda }_{{\rm q,cn}}}\ \left({\rm mm} \right)=\left(0.91\pm 0.01 \right)\times {{\left(\bar{p}\ \left({\rm atm} \right) \right)}^{-0.48\pm 0.01}}.\nonumber \end{align} \tag{ 4 }$$

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Simplifying the coefficients results in the rule-of-thumb that the divertor heat flux width in millimeters is the inverse square root of the volume-averaged core plasma pressure in atmospheres across confinement regimes. In SI-units equation (4) (${{\lambda }_{{\rm q,cn}}}\ \left({\rm m} \right)\approx {{\left({{C}_{{\rm f}}}/~\bar{p}\ \left({\rm Pa} \right) \right)}^{0.5}}$) the fitted constant has the value of ${{C}_{{\rm f}}}\approx 0.08\ \left({\rm N} \right)$. The significance of the magnitude and dimension of this constant are not clear at this time.

There were some indications of this from past data: Earlier C-Mod H-mode data showed an inverse relation between the boundary heat flux width and the core confinement [14]. But the trend is much clearer with the expanded dataset presented here. It is also similar to observations that the integral heat flux width decreased with increasing power onto the divertor target in L-mode in ASDEX-U [31] (H-mode results may have been affected by ELMs): For otherwise constant conditions, increasing input power tends to both increase plasma pressure and increase the power to the divertor target. Thus, a higher plasma pressure results in narrower heat flux widths. Also, using gas puffing to decrease the confinement quality in H-mode in ASDEX-U tended to show wider integral heat flux widths at lower confinement [31]. ASDEX-U also found an inverse dependence on the heat flux width with stored plasma energy in L-mode [29], although this was also normalized to the pedestal density to transform it into an approximate scaling with pedestal temperature. The addition of lithium to NSTX both reduced the divertor heat flux width [32] and increased the stored plasma energy [33].

To predict the heat flux width scaling in other machines requires identifying whether the volume-averaged core plasma pressure is the proper size normalization for stored plasma energy. As an initial exploration into the size scaling, figure 6, we unify data at common poloidal fields from the multi-machine heat flux width database (taken from figure 3 in [4]) and the ITPA H-mode confinement database [34] (DB4v5 obtained from [35]). Volume-averaged core plasma pressure is calculated using the plasma stored energy and plasma volume in the confinement database. Each ellipse (the apparent distortion is due to the log–log plot) represents one machine at a given poloidal magnetic field ($\pm 10\%$). The left and right extrema of each ellipse is taken as the mean volume-averaged core plasma pressure $\pm 2$ standard deviations for the subset of the data from that machine at that poloidal field. The top and bottom extrema of each ellipse is taken as the maximum and minimum heat flux width for that machine at that poloidal field. For such simple scaling relationship, there is remarkably close agreement between the combined datasets and the C-Mod data, especially at higher poloidal magnetic field.

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Included in in figure 6 are core pressure-based projections (equation (4)) for heat flux widths in ITER (0.52 mm) [36], ADX (0.49 mm) [7], ARC (0.39 mm) [6], and SPARC V0 (0.26 mm) (unpublished). These are very close to those projected by the poloidal field-based projections (equation (3)): ITER (0.51 mm), ADX (0.42 mm), ARC (0.39 mm), and SPARC V0 (0.28 mm). This result should perhaps not come at too much of a surprise. Multi-machine confinement scalings (e.g. ITER H98y2 [37]), much like the divertor heat flux width, are dominated by the plasma current (which is closely related to the poloidal magnetic field through geometry factors). What is remarkable is the relative unity of the scaling across different confinement regimes in C-Mod as well as across different machines. The consequences of the inverse square root scaling of the boundary heat flux width with the volume-averaged core plasma pressure across regimes has an important consequence for reactors: good plasma core confinement will only come at the expense of narrow boundary heat flux widths. Apparently, the transport barrier that defines the pedestal region—and thus confinement quality—extends at least partially into the scrape-off layer, an observation that has been apparent for some time now [38].

Future work with this database will include a more detailed examination of the scaling of the private flux region heat flux width to assess what physics is setting the asymmetric cross-field transport in the divertor leg. It will also examine detailed profiles of plasma temperature and density from the core, through the pedestal, and into the boundary to assess by what physics mechanisms the core confinement and boundary heat flux width are related. These will provide a critical benchmark with recent simulations and models of the boundary heat flux width [1–3, 39, 40] and improve projections to reactor-class machines. Additionally, these results motivate dedicated investigations of the heat flux widths in plasmas with different mixes of edge and internal transport barriers but otherwise similar average pressures. This could possibly indicate ways of optimizing the core confinement for fusion power performance and the boundary for power exhaust.

Thanks to the entire Alcator C-Mod team, it was only by their hard work and dedication to the pursuit of fusion energy that these measurements were possible. Thanks to M.L. Reinke for his comments and encouragement to carry out these experiments and build this database. Thanks to S.M. Wolfe for his expertise in running C-Mod and especially for setting up the high B_p shots. This work was supported by US DoE cooperative agreements DE-SC0014264 and DE-FC02-99ER54512 on Alcator C-Mod, a DoE Office of Science user facility.