Fast electrostatic microinstability evaluation in arbitrary toroidal magnetic geometry using a variational approach

TL;DR

The authors develop a fast, field-line global dispersion relation for electrostatic ITG and TEM in arbitrary toroidal geometry by applying a lowest-order gyrokinetic expansion and integrating the local response along magnetic field lines to construct $D_{\mathrm{glob}}(\omega)$. This variational formulation captures nonlocal geometry effects and resonances with magnetic drifts, while enabling analytical FLR approximations (notably Padé) to reduce computational cost. Validation against linear gyrokinetic simulations across DIII-D, HSX, and W7-X shows good quantitative agreement at transport-relevant long wavelengths, with some discrepancies in low-shear stellarators and in regimes requiring non-adiabatic passing-electron physics (UI, ETG). The paper also introduces reduced-fidelity models that preserve key drive/damping physics, enabling geometry-aware microstability screening for optimization, and highlights future work to achieve self-consistent eigenfunctions and extended kinetic-electron treatments. Overall, the work provides a practical, physics-informed proxy for fast microinstability evaluation in complex magnetic geometries with potential relevance for fusion-device optimization.

Abstract

Small-scale turbulence originating from microinstabilities limits the energy confinement time in magnetic confinement fusion. Here we develop a semi-analytical dispersion relation based on lowest-order solutions to the gyrokinetic equations in an asymptotic expansion in the ratio of transit (bounce) frequency to the mode frequency for ions (electrons), capable of describing two common instabilities: the ion temperature gradient (ITG) mode and trapped-electron mode (TEM), in the electrostatic limit. The dispersion relation, which is valid in arbitrary toroidal geometry, takes into account resonances with the magnetic ion and bounce-averaged electron drifts, incorporates non-local effects along the magnetic field line, is valid for arbitrary sign of the growth rate and magnetic curvature, and is shown to satisfy a variational property. Several common approximation models are introduced for both the magnetic drift and finite Larmor radius (FLR) damping, with the Padé approximation for FLR effect in particular resulting in remarkable agreement with the baseline dispersion relation model at significantly reduced costs. The baseline model is verified by comparing solutions of the dispersion relation model to high-fidelity linear gyrokinetic simulations, where the exact eigenfunction of the electrostatic potential from simulations is used as a trial function, showing good quantitative agreement for ITGs and TEMs in (shaped) tokamaks as well as low-magnetic-shear stellarators.

Figures (20)

• Figure 1: Variation along the magnetic field line of the normalised local eigenfrequency $\omega_{\mathrm{loc}}(l)$ solution to \ref{['eq:Dloc']} in the adiabatic-electron limit, for a toroidal ITG instability characterised by radial and bi-normal wavenumbers $k_x \rho_{\mathrm{ss}} = 0, k_y \rho_{\mathrm{ss}}=0.3$ driven by density and temperature gradients of $a/L_{ni} = 2$ and $a/L_{T\mathrm{i}}=4$, respectively, in the flux tube of the DIII-D tokamak considered in this work (see \ref{['sec:numerics']}). The plasma consists of only a single ion species with $Z_\mathrm{i} = 1$ and $T_\mathrm{i}=T_\mathrm{e}$. Contrasted against the eigenfrequency (solid lines; split into growth rate and propagation frequency) are the variation of the magnetic geometry (dashed lines) focussing on the magnetic field strength in blue, the magnitude of the (normalised) perpendicular wavevector in green and the bi-normal component of the $\grad{B}$ drift operator (see \ref{['sec:fluxtubes']}) in maroon. For reference, the corresponding eigenfrequency obtained by Gene is $\omega a/c_{s}=-0.048+0.1820i$, being within the extremes of the local solutions.
• Figure 2: Flux-tube geometries for (a) the HSX stellarator and (b) high-mirror configuration of the W7-X stellarator showing the variation of the magnetic field strength (blue), bi-normal component of the $\grad{B}$ drift (maroon) and magnitude of the perpendicular wavenumber (green) along the field line. In the latter we have taken $k_x \rho_{\mathrm{ss}}=0 , \ k_y \rho_{\mathrm{ss}}= 0.3$, such that a direct comparison to $\norm{\bm{k_\perp}} \rho_{s}$ in the DIII-D geometry from \ref{['fig:loc-freq-sol-visual']} is facilitated. Note that the W7-X flux-tube spans only a single poloidal turn, whereas the HSX flux-tube contains the geometric data of four poloidal turns.
• Figure 3: Eigenfrequency solutions of the global dispersion relation model (solid lines) contrasted with Gene simulations (symbols) for adiabatic-electron ITG in the presence of $\mathrm{C}^{6+}$ impurities, while varying the impurity concentration by modifying $Z_{\mathrm{eff}}$ (darker colours indicate larger impurity concentration). Shown are results for (a) the DIII-D tokamak, (b) the HSX stellarator and (c) the high-mirror configuration of the W7-X stellarator. In all cases, a density- and temperature gradient of $a/L_{n_s}=2$ and $\ a/L_{T_s}=4$, respectively, are considered for both deuterium ions and the carbon impurity.
• Figure 4: Eigenfrequency solutions of the global dispersion relation model (solid lines) contrasted with Gene simulations (symbols) including kinetic electrons while varying the electron temperature gradient $a/L_{T_e}$ (lighter colours indicate a stronger temperature gradient). Shown are results for (a) the DIII-D tokamak, (b) the HSX stellarator and (c) the high-mirror configuration of the W7-X stellarator. In all cases, the density gradient is fixed at $a/L_n=3$ whilst the ion temperature gradient is suppressed ($a/L_{T\mathrm{i}} = 0$).
• Figure 5: Eigenfrequency solutions of the global dispersion relation model (solid lines) contrasted with Gene simulations (symbols) including kinetic electrons while varying the ion temperature gradient $a/L_{T\mathrm{i}}$ (lighter colours indicate a stronger temperature gradient). Shown are results for (a) the DIII-D tokamak, (b) the HSX stellarator and (c) the high-mirror configuration of the W7-X stellarator. In all cases, the density gradient is fixed at $a/L_n=3$ whilst the electron temperature gradient is suppressed ($a/L_{T\mathrm{e}} = 0$).