Nonlinear anisotropic equilibrium reconstruction in axisymmetric magnetic mirrors

TL;DR

This paper addresses nonlinear equilibrium reconstruction for axisymmetric plasmas with anisotropic pressure at high $\beta$, extending beyond conventional isotropic, low-$\beta$ methods. It couples the Pleiades free-boundary Grad-Shafranov solver with a semi-analytic kinetic basis for anisotropic mirror pressure and a scalable constrained Bayesian optimization (SCBO) framework to fit to limited diagnostics while providing uncertainty quantification, via the objective $\chi^2=\sum_i (M_i-\mathcal{M}_i)^2/\sigma_{M_i}^2$. Verification with Maxwellian, hybrid, and kinetic sloshing-ion synthetic data shows the method can recover $\langle\beta_0\rangle$, $\langle E_i\rangle$, and $W_{tot}$ to within roughly 20%, and application to WHAM experimental data identifies sloshing-ion pressure peaks in moderate-density shots, distinguishing them from fast-electron effects. The work demonstrates robust, non-Maxwellian, uncertainty-aware equilibrium reconstructions in axisymmetric magnetic mirrors and suggests broad applicability to high-$\beta$ devices with limited diagnostics, including potential extension to multiobjective optimization and 3D/MHD regimes.

Abstract

Magnetic equilibrium reconstruction is a crucial simulation capability for interpreting diagnostic measurements of experimental plasmas. Equilibrium reconstruction has mostly been applied to systems with isotropic pressure and relatively low plasma $β= 2μ_0p/B^2$. This work extends nonlinear equilibrium reconstruction to high-$β$ plasmas with anisotropic pressure and applies it to the Wisconsin High Temperature Superconducting Axisymmetric Magnetic Mirror experiments to infer the presence of sloshing ions. A novel basis set for the plasma profiles and machine learning algorithm using scalable constrained Bayesian optimization allow accurate nonlinear reconstructions with uncertainty quantification to be made more quickly with fewer experimental diagnostics and improves the robustness of reconstructions at high $β$. In addition to WHAM and other mirrors, such reconstruction techniques are potentially attractive in high-performance devices with constrained diagnostic capabilities such as fusion power plants.

Figures (8)

• Figure 1: Normalized parallel pressure $p_\parallel/p_{\parallel 0}$, perpendicular pressure $p_\perp/p_{\parallel 0}$, and density $n/n_0$ for $m_i=2~\mathrm{u}$, $R_m = 20$, $T_e=1~\mathrm{keV}$, $Z_\mathrm{eff} = 1.0$, $E_\mathrm{NBI} = 25~\mathrm{keV}$, and $\theta_\mathrm{NBI} = 45^\circ$.
• Figure 2: A comparison between CQL3D-m/Pleiades simulation results (blue crosses) and the physics parameters reconstructed from the simulations using synthetic measurements (orange dots) for the "hybrid" case described in Section \ref{['sec:hyb']}.
• Figure 3: Measurements from the "high-density" 250305121--43 (dark) and "moderate-density" 250306045--64 (light) WHAM experiments. On the left are the Thomson scattering measurements of $n_e$ in (a) and $T_e$ in (b) as well dashed lines showing their Gaussian fits. On the right are the time traces from high-density shot 250305132 (dark) and moderate-density shot 250306062 (light) showing the interferometer density measurement in (c), the flux loop measurements for flux loops 1 (blue), 2 (orange), and 3 (green) in (d), and forward ECH power (blue) and NBI power (orange) in (e). The black dashed vertical line on the right hand side plots denotes 10.3 ms when the Thomson scattering data was collected.
• Figure 4: Plots of the surrogate model for $\chi^2$ versus the normalization on the fast ion ion distribution $\mathcal{A_0}$ and the Maxwellian ion pressure $p_{\mathrm{GD},0}$ for experiments 250305121--43 (a) and 250306045--64 (b). The joint $\sigma_1$ confidence region is shown by the red contour, samples used to construct the model are shown by white dots and the best fit point is shown with a red star. Sloshing ions can be inferred with much greater confidence in the 250306045--64 experiments.
• Figure 5: A plot of the reconstructed moderate-density 250306045--64 WHAM equilibrium. Contours of constant $\psi$ are shown by the dashed white lines, the pressure profiles with sloshing ion peaks at $Z\approx\pm0.4~\mathrm{m}$ are shown by the colored contours, the flux loop locations are shown by the blue circles, the Thomson measurement locations are shown by the orange crosses, the interferometer sight-line is shown by the dotted grey line, and the HTS magnets are shown in black at $Z\approx\pm1~\mathrm{m}$.