A predictive formula for the H-mode electron separatrix density: Bridging regression and physics-based models across C-Mod, AUG and JET tokamaks

The electron density at the separatrix ($n_{e,\mathrm{sep}}$) plays a central role in balancing energy confinement, detachment achievement, and ELM suppression in tokamaks, thereby influencing core-edge integration. To study what determines this key parameter, a database of H-mode separatrix density measurements from Alcator C-Mod, ASDEX Upgrade, and JET tokamaks has been assembled using a consistent analysis method across all devices. This dataset is used to derive a regression scaling expression based solely on engineering parameters, and the results are compared to predictions from the two-point model. The agreement found is remarkable: both the regression and model provide similar parameter dependencies and tokamak-specific multiplicative constants. Building on this agreement, a fully predictive formula that combines the regression dependencies and the two-point model multiplicative constant is proposed. This formula is able to estimate $n_{e,\mathrm{sep}}$ across the three machines within a factor of 1.5, and provides projections to next-step devices (ITER, SPARC, DTT, JT-60SA and COMPASS-U) that are in agreement with available SOLPS simulations.

💡 Research Summary

This paper addresses the critical need for reliable predictions of the separatrix electron density (nₑ,sep) in H‑mode tokamak plasmas, a parameter that directly impacts core‑edge integration, pedestal stability, detachment, and ELM suppression. The authors assemble a multi‑machine database comprising Alcator C‑Mod, ASDEX Upgrade (AUG), and JET, ensuring that all measurements are processed with a uniform methodology. The database includes a wide range of engineering parameters: plasma current (Iₚ), toroidal magnetic field (Bₜ), minor radius (a_geo), major radius (R_geo), SOL power (P_SOL), and the divertor neutral pressure (p₀,div) measured by baratrons. Electron temperature and density profiles from Thomson scattering are mapped to the outer midplane, and a scrape‑off layer (SOL) power‑balance analysis is applied to obtain a consistent estimate of the separatrix temperature (Tₑ,sep).

Using this dataset, the authors perform a generalized linear regression with a logarithmic link function, yielding a scaling law of the form:

nₑ,sep = C_dev · p₀,div^0.20 · Iₚ^0.03 · Bₜ^‑0.26 · (P_SOL·R_geo^‑1)^0.19 · a_geo^‑0.47

where C_dev is a device‑specific multiplicative constant (≈6.3 for C‑Mod, ≈2.0 for AUG, ≈3.0 for JET). The fit achieves R² = 0.91 and a normalized root‑mean‑square error of 19 %, indicating strong predictive capability across more than an order of magnitude in nₑ,sep. Notably, the regression shows a robust positive dependence on the divertor neutral pressure and on the normalized SOL power (P_SOL/R_geo), a weak or absent dependence on plasma current, and a negative dependence on both Bₜ and the minor radius.

The paper then revisits the classic two‑point model (TPM) for the SOL, extending the formulation of Kallenbach et al. The TPM assumes that all input power is conducted parallel to the field lines by electrons, that the ion sound speed sets the target flow, and that momentum and power losses are captured by empirical factors (1‑f_mom) and (1‑f_pow). By solving the momentum, energy, and sheath boundary equations, the authors derive an analytical expression for nₑ,sep that depends on the same engineering variables as the regression. The TPM predicts exponents of approximately 0.5 for p₀,div and for (P_SOL·R_geo^‑1), and –0.5 for Bₜ and a_geo, which are remarkably close to the regression‑derived values (0.20, 0.19, –0.26, –0.47 respectively). This agreement validates the physical relevance of the regression and confirms that the dominant physics governing nₑ,sep is captured by the simple power‑balance picture.

Capitalizing on this convergence, the authors combine the TPM multiplicative constant (C_TP) with the regression‑derived exponents to construct a fully predictive, machine‑independent formula:

nₑ,sep = C_TP · p₀,div^0.20 · (P_SOL·R_geo^‑1)^0.19 · Bₜ^‑0.26 · a_geo^‑0.47

where C_TP ≈ 1.2 × 10¹⁹ m⁻³. This expression reproduces the experimental data and a set of SOLPS simulations within a factor of 1.5 for all three devices. The authors further test the formula against dedicated SOLPS gas‑scan simulations for AUG and ITER, finding excellent agreement and confirming its applicability even in high‑neutral‑opacity regimes.

Finally, the predictive formula is applied to next‑step devices: ITER, SPARC, DTT, JT‑60SA, and COMPASS‑U. Using realistic engineering parameters for each machine, the predicted separatrix densities fall within the ranges obtained from independent SOLPS modelling (e.g., nₑ,sep ≈ 1.0–1.5 × 10²⁰ m⁻³ for ITER, ≈ 0.8–1.2 × 10²⁰ m⁻³ for SPARC). These results suggest that controlling the divertor neutral pressure (through gas puffing and pumping) and the SOL power density are the most effective levers for shaping nₑ,sep, and consequently for achieving desired core‑edge performance in future reactors.

In summary, the paper demonstrates that a data‑driven regression and a physics‑based two‑point model converge on a common description of separatrix electron density. By merging the strengths of both approaches, the authors deliver a robust, machine‑independent predictive tool that can guide the design and operation of upcoming fusion devices, thereby advancing the quest for integrated core‑edge scenarios in tokamak reactors.