Global linear drift-wave eigenmode structures on flux surfaces in stellarators: ion temperature gradient mode

TL;DR

This work investigates linear electrostatic ITG eigenmodes in nonaxisymmetric stellarators and demonstrates that their structures are nonuniform along flux surfaces, localizing downstream of the ion diamagnetic drift. Using global gyrokinetic simulations with GTC, the authors quantify the localization and growth, and explain it with a simple model where a nonzero imaginary binormal wavenumber $k_y$ induces a phase-driven localization that scales with $ ho_*^{-1}$. They further show that a surface-global mode can be constructed from local gyrokinetic results (Stella and GX) by solving the local dispersion with real wavenumbers and analytically continuing to complex wavenumbers, underscoring the necessity of complex-$k$ spectra to capture global drift-wave structure. This framework advances understanding of linear drift-wave modes in stellarators and provides a path toward interpreting nonlinear ITG turbulence in these devices.

Abstract

Turbulent transport greatly impacts the performance of stellarator magnetic confinement devices. While significant progress has been made on the numerical front, theoretical understanding of turbulence in stellarators is still lacking. In particular, due to nonaxisymmetry, different field lines couple within flux surfaces, the effects from which have yet to be adequately studied. In this work, we numerically simulate the linear electrostatic ion-temperature-gradient modes in stellarators using the global gyrokinetic particle-in-cell code GTC. We find that the linear eigenmode structures are nonuniform across field lines on flux surfaces and are localized at the downstream of the ion diamagnetic drift. Based on a simple model from Zocco et al. [Phys. Plasmas 23, 082516 (2016); 27, 022507 (2020)], we show that the localization can be explained from the nonzero imaginary part of the binormal wavenumber. We further demonstrate that a localized surface-global eigenmode can be constructed from local gyrokinetic codes stella and GX, only if we first solve the local dispersion relation with real wavenumbers on each field line, and then do an analytic continuation to the complex-wavenumber plane. These results suggest that the complex-wavenumber spectra from surface-global effects are required to understand the linear drift-wave eigenmode structures in stellarators.

Figures (10)

• Figure 1: The linear global ITG eigenmode structures ${\rm Re}\,\delta\Phi_{\omega_{\rm r}}$ calculated from eq:GTC_delta_Phi at $\zeta=0$. Lengths $(R,z)$ are normalized by $R_0$ of each configuration.
• Figure 2: (a-d) The linear ITG eigenmode structures ${\rm Re}\,\delta\Phi_{\omega_{\rm r}}$ calculated from eq:GTC_delta_Phi in field-line following coordinates $(\alpha,\theta)$ at $r/a=0.5$, with $\alpha_{\max}=\iota\pi/N_{\rm fp}$. (e-g) The normalized magnetic-field strength $B/B_a$ versus $(\alpha,\theta)$ at $r/a=0.5$. Note that the range in the colorbar is different from (a-d).
• Figure 3: Comparison of numerical and analytical solutions of eq:theory_global. Here, $\Phi(\alpha)$ is the numerical solution, $S(\alpha)$ and $\hat{S}(\alpha)$ are analytical results eq:theory_S and eq:theory_Shat, and $\hat{\Phi}(\alpha)=\Phi\rme^{-\rmi S/\rho_*}$.
• Figure 4: (a) and (b): the real and imaginary part of the global wavenumber versus $y$ at $\omega=0.66+0.89\rmi$. (c) and (d): the local frequency and growth rate versus real $k_y$ at $y=0$ and $y=\pi$. The global eigenvalue $\omega$ is indicated by the yellow circle.
• Figure 5: The local linear eigenmode real frequencies (a,b) and growth rates (c,d) versus $k_y$ at $k_x=0$ at two different field lines: $\alpha=0$ and $\alpha=\iota\pi/N_{\rm fp}$. Shown are results from stella using the zero-incoming boundary condition, and from both stella and GX using the periodic boundary condition.