Modeling Temperature Profiles in the Pedestal of NSTX with Reduced Models

Abstract: This paper describes new modeling capabilities for predicting H-mode pedestal profiles in spherical tokamaks. Temperature profiles for NSTX discharges 132543 and 132588 are modeled by coupling the \textsc{astra} transport solver with neoclassical transport and gyrokinetic-based reduced models for electron temperature gradient (ETG) and kinetic ballooning mode (KBM) instabilities. A quasi-linear surrogate model for ion-scale transport is developed using linear \textsc{gene} simulations, requiring only a single free parameter calibrated to one discharge. Time-evolving the temperatures with fixed density yields good agreement with experiments for both discharges. Systematic analysis of the transport mechanisms reveals that neoclassical transport is huge across the entire pedestal region for the ion channel. ETG turbulence is large in the plasma edge and low density gradient region, contributing substantially to the electron channel. However, KBM/MHD-like modes also drive significant transport in both the ion and electron thermal channels, making them essential for accurate pedestal modeling. Further refinements, including explicit $E \times B$ shear suppression and scaled ETG transport, produce quantitative but not qualitative improvements. This work lays the foundation for predictive modeling of future devices.

• The paper develops a time-dependent ASTRA model coupling ETG turbulence, neoclassical transport, and a gyrokinetic KBM/MHD-like surrogate to predict NSTX electron and ion pedestal temperatures.
• The combined model reproduces profiles in wide, ELM-free and narrow, ELMy discharges using one calibrated parameter, while sensitivity tests show stiff ion-scale transport constrains the pedestal near marginal stability.
• The results show that electron-ion energy exchange buffers profile changes, density shape strongly controls transport partitioning, and coupled two-species modeling is essential for predictive pedestal simulations.

Overview and modeling strategy

This paper develops a predictive modeling capability for H-mode pedestal temperature profiles in the National Spherical Torus Experiment (NSTX), coupling reduced transport models into the time-dependent astra 1.5D transport solver. The framework combines three transport channels: an algebraic reduced model for electron temperature gradient (ETG) turbulence, neoclassical transport computed with NCLASS, and a quasi-linear surrogate for kinetic ballooning mode (KBM) and MHD-like ion-scale instabilities built from a database of linear gene gyrokinetic simulations. Density profiles are held fixed at experimental values; only $T_e$ and $T_i$ are evolved. This restriction is deliberate—particle source uncertainties make density prediction difficult—and it is an explicit limitation of the present capability.

Two contrasting discharges anchor the validation: shot 132543, an ELM-free, lithium-conditioned plasma with a wide pedestal (analyzed at 650 ms in steady state), and shot 132588, a non-lithiated ELMy discharge with a narrow pedestal (analyzed at 700 ms during post-ELM recovery). A key methodological point is that the framework contains only one free parameter—the KBM surrogate prefactor $c_0 = 0.0008$, calibrated to one discharge—while all other coefficients derive from first-principles gyrokinetic calculations.

ETG-only predictions and the role of electron-ion exchange

The baseline ETG model follows the algebraic form of Hatch et al., scaling as $\omega_{Te}^2(\eta_e - 1)\eta_e^{b_0}\tau^{c_0}$ with coefficients fixed from conventional-aspect-ratio nonlinear databases. Comparison against nonlinear gene and cgyro simulations for these NSTX conditions shows the formula systematically underpredicts heat flux by approximately a factor of two; however, because the model is stiff in the temperature gradient (scaling roughly as $\omega_{Te}^{4.5}$), this magnitude error produces only small profile differences.

In single-species evolution with fixed $T_i$ and $n_e$, the unscaled ETG model alone reproduces the qualitative shape of the experimental $T_e$ pedestal, with diffusivity sharply localized where $\eta_e \gg 1$. A notable finding is the strong buffering provided by collisional electron-ion power exchange $P_{ei}$: scans varying density gradients by ±20–30% at two pivot radii leave the predicted profiles nearly unchanged, because changes in the turbulent drive through $T_i$0 cancel against density-dependent changes in the collisional energy transfer. Doubling the ETG transport likewise barely alters profile magnitude—a direct consequence of this cancellation. The authors draw a general lesson here: fixed-profile transport analysis is substantially limited because inter-species thermal exchange can be large, so coupled two-channel modeling is required.

Failure of ETG plus neoclassical transport

When both species evolve with only ETG and neoclassical transport, both temperature profiles are systematically overpredicted. Neoclassical transport provides a large baseline across the entire pedestal for the ion channel but cannot hold the ion temperature pedestal for either discharge; the deficit is most severe for 132588, where modeled $T_i$1 vastly exceeds measurement. The paper states plainly that another anomalous ion-scale channel is required, and that no amount of improved electron-channel accuracy can substitute for it.

Quasi-linear surrogate for KBM/MHD-like transport

To supply the missing channel, the authors map KBM stability space via linear gene simulations over equilibria reconstructed self-consistently with SPIDER and CHEASE, scanning temperature profiles within experimental error bars and global toroidal mode numbers $T_i$2. The resulting $T_i$3-$T_i$4 diagrams show the experimental pedestals operating directly at or near the KBM/MHD-like stability boundaries, motivating marginal-stability-constrained transport.

The surrogate adopts the Giacomin-style quasi-linear form $T_i$5, with a field-weighted perpendicular wavenumber including Bessel-function FLR weighting. An automated Python routine classifies the dominant instability and interpolates transport via radial basis functions over the simulation database, exporting JSON tables read by astra at each timestep. With the single prefactor $T_i$6 calibrated to 132588, the combined model (unscaled ETG + neoclassical + surrogate) achieves good agreement for 132588 and near-agreement for 132543, where residual overprediction is attributed to the non-steady post-ELM recovery state rather than model deficiency. Sensitivity scans varying $T_i$7 by ±10% produce little variance, concentrated at the pedestal top—evidence that ion-scale transport is stiff and the pedestal is constrained by marginal stability.

Refinements yield quantitative but not qualitative improvements. An explicit $T_i$8 shear suppression factor (typically 0.8–0.9 in the pedestal) slightly raises predicted temperatures rather than lowering them, since stiff transport compensates a 10–20% diffusivity reduction through imperceptible gradient steepening. Similarly, applying the $T_i$9 ETG scaling in the full coupled model adjusts profile shape without changing magnitudes. The final configuration—scaled ETG plus neoclassical plus the KBM surrogate with fixed $c_0 = 0.0008$0—matches experiment well across both thermal channels for both discharges.

Physical interpretation

The transport balance decomposes cleanly by mechanism and location. Neoclassical transport is large across the whole pedestal for ions. ETG turbulence dominates the edge and low-density-gradient regions where $c_0 = 0.0008$1, contributing substantially to the electron channel. KBM/MHD-like modes, driven by total pressure gradient $c_0 = 0.0008$2, provide anomalous transport in steep-density-gradient regions and couple to both species. Because each mechanism occupies a distinct parametric regime, the pedestal structure is fundamentally determined by the density profile shape—an important implication given that density is prescribed here rather than predicted.

Limitations and open questions

Several limitations are acknowledged explicitly. Density prediction is deferred pending better particle-source estimates. The KBM surrogate relies on a heuristic prefactor and a database restricted to equilibria near the two experimental discharges; replacing it with physics-based quasilinear closures remains open. Microtearing modes are not included, though the authors argue they are secondary given that ETG and KBM capture most transport. Validation beyond NSTX—including MAST and standard-aspect-ratio devices—is incomplete, and the steady-state solver treatment of transient post-ELM phases introduces known bias. Whether the single-parameter calibration generalizes across wider parameter space is untested.

Conclusion

The paper demonstrates that a tightly constrained reduced-model hierarchy—one free parameter, otherwise determined by gyrokinetics and neoclassical theory—can reproduce pedestal temperature profiles from pedestal top to separatrix in two very different NSTX discharges. The central physical result is the complementary partitioning of transport among neoclassical, ETG, and KBM/MHD-like channels, and the demonstration that coupled two-species evolution is essential because $c_0 = 0.0008$3 exchange buffers profile responses. The framework is positioned as a foundation for predictive studies of NSTX-U, STEP, and low-recycling scenarios linking separatrix conditions to pedestal confinement.