Multi-Machine Scaling Laws for Fuel and Impurity Puffing Rates Sufficient for Detachment Access: a Systematic Review of Magnetic Confinement Fusion Devices

TL;DR

The paper develops cross-device, 0D scaling laws that relate edge detachment access to controllable puffing actuators across tokamaks, stellarators, and linear devices, using an open 457-entry database. Fuel puffing is found to be strongly tied to divertor geometry, especially divertor volume, with opaqueness providing a secondary, physics-informed refinement; impurity puffing requires a nonlinear composite parameter $\gamma_{\text{DZ}}$ to capture device- and size-dependent trends. The results yield simplified bounds for tokamaks (HDL/GES) and stellarators (Sudo limit) and demonstrate validation against unseen $\text{DOD}=1$ data within factors of about 2–3 on average, offering practical, macroscopic design tools for reactor fuel-cycle and edge-plasma modelling. Overall, the work demonstrates that physics-based 0D laws can reliably connect detachment access to engineering actuators across machines, with implications for reactor design and future machine-specific extensions.

Abstract

An open-source database of 457 experimental and numerical entries representing 32 machines$-$including tokamaks, stellarators, and linear plasma devices$-$is assembled. From this dataset, we derive multi-machine scaling laws that predict the fuel and impurity puffing rates sufficient to edge plasma detachment$-$the leading reactor-relevant solution to the challenge of plasma-wall interaction. Validation against up to 40 L- and H-mode plasmas shows agreement within a factor of 1.5 in about 50\% of cases, and within a factor of 2 on average. Divertor volume alone is found to strongly correlate with the fuelling rate. Inclusion of plasma opaqueness leads to $Γ_{\text{D}} \propto [n_{\text{sep}}\, a\, (S_{\text{div}}/V_{\text{div}})^{-1.5}]^{1.05}$, valid across all toroidal devices. Its H-mode simplification, $Γ_{\text{D}}^{\text{HDL}} \propto 0.43\, a^{1.58}\, λ_q^{-0.89}$, avoids explicit dependence on $n_{\text{sep}}$ and carries intrinsic physical meaning through the H/L density limit and the power fall-off length. The impurity seeding rate is captured by a general non-linear law, from which the Greenwald-Eich-Scarabosio simplification, $Γ_{\text{Z}}^{\text{GES}} \propto a^{1.51}\, λ_q^{-0.27}$, is obtained. Similar relationships are defined for stellarators, consistent with tokamak trends but still awaiting validation$-$an opportunity for further study. These results have immediate relevance for reactor fuel-cycle design and edge plasma modelling. More broadly, they demonstrate that physics-based 0D laws can reliably link detachment access to engineering actuators, offering practical tools for reactor design. Our laws represent macroscopic patterns across machines rather than microscopic variations within an individual device$-$providing the basis for our forthcoming studies aimed at extending this framework to machine-specific behaviour.

Figures (8)

• Figure 1: View in the poloidal plane of the DTT's diverted single null plasma of Moscheni et al.Moscheni_2022Moscheni_2025: wall in black; interior of the chamber in light grey; separatrix in dark grey; simplistic rectangular representation of the divertor in yellow; segment connecting inner, left, and outer, right, strike points in dash-dotted red (of length $L_{\text{div}}$, with the radial location of its mid-point being $R_{\text{div}}$); $H_{\text{div}}$ is the distance of the X-point from it (orange).
• Figure 2: Illustrative instances with dummy data where the strategy devised in section \ref{['sec:methods_statistics_validation']} does (green tick) and does not (red crosses) work. Top left: the number of DOD=1 points (black dots) is too small for validation. Top right: DOD=1 points are too clustered within the DOD$>$1 points in white. Bottom left: DOD=1 points are systematically below the rest. Bottom right: actual case, where DOD=1 points are well distributed (see text for details).
• Figure 3: Results for the fuel puffing rate $\Gamma_{\text{D}}$ from: (a) the geometry-based equation (\ref{['eq:Vofl_vs_Qpuff']}) (VOL), exclusively depending on the divertor volume $V_{\text{div}}$; (b) equation (\ref{['eq:V_OFL_S_OFL_1p5_times_opacity_vs_Qpuff']}) (OPQ), where the opaqueness at the outer mid-plane separatrix $n_{\text{sep}} \times a$ appears alongside empirical calibration factors (exponents). Insets in the bottom right corners highlight points at DOD=1 (black) and DOD$>$1 (white) within the main plot. Scaling laws are derived from DOD$>$1 points only (training set).
• Figure 4: (a) Lack of correlation of the impurity puffing rate $\Gamma_{\text{Z}}$ with the divertor volume $V_{\text{div}}$. (b) Result of equation (\ref{['eq:gammaDZ_aux']}) for the non-linear function $\gamma_{\text{DZ}} = (\Gamma_{\text{D}} + \Gamma_{\text{Z}}) \times (\Gamma_{\text{D}} / \Gamma_{\text{Z}})$.
• Figure 5: Pictorial interpretation with dummy data of a prototypical scaling law---that is a macroscopic model of the trends across devices (a "barycentric average", dashed black line), but without necessarily being representative of each individual machine (coloured segments).