Modification of ion-temperature-gradient turbulence by impurities in stellarator plasmas

TL;DR

This work analyzes how impurities modify ion-temperature-gradient (ITG) turbulence in stellarator plasmas. By solving the toroidal ITG dispersion relation in selective limits (notably $k_{||}=0$, large $\eta_i$, highly charged impurities, and small impurity concentration), the authors derive a compact analytical expression for the impurity-induced change in ITG growth rate, decomposed into three physically distinct contributions: impurity density gradient, impurity temperature gradient, and dilution. Linear gyrokinetic simulations validate the predicted scalings and signs, and a remarkable correlation is found between the analytical growth-rate modification and the nonlinear ion heat flux change, linking linear theory to nonlinear transport. The results offer physical insight and practical guidance for interpreting impurity effects in experiments and for designing reactor-operation scenarios with reduced turbulent transport.

Abstract

Recent nonlinear gyrokinetic simulations have shown that impurities can strongly modify the turbulent heat flux in stellarator plasmas. Here, the ion-temperature-gradient (ITG) dispersion relation in a plasma containing impurities is analytically solved in certain limits and an expression for the modification of the ITG growth rate by impurities is derived. The analytical expression is the sum of three terms corresponding to three different physical causes (impurity density gradient, impurity temperature gradient and dilution) of the change in the growth rate. The scalings predicted analytically for the modification of the growth rate are shown to be reproduced by linear gyrokinetic simulations. The conditions for reduction or increase of the ITG growth by impurities are also correctly predicted by the analytical solution to the dispersion relation. Finally, a remarkable correlation is found between the analytical expression for the modification of the growth rate and the modification of the turbulent heat flux obtained from nonlinear gyrokinetic simulations.

Figures (15)

• Figure 1: Flux surfaces $r/a = 0.7$ of the standard configuration of W7-X (left) and an inward-shifted configuration of LHD (right). The color represents the value of $B$, with red corresponding to the largest values and blue to the smallest values.
• Figure 2: Representation of flux tubes like the ones employed in the gyrokinetic simulations of this paper in the standard configuration of W7-X (left) and an inward-shifted configuration of LHD (right).
• Figure 3: Magnetic field strength and bad curvature regions along three poloidal turns of the flux tubes represented in figure \ref{['fig:flux_tubes_W7-X_LHD']}. The gray background corresponds to one poloidal turn. Bad curvature regions for $k_r = 0$ are those for which $\mathcal{K}_\alpha > 0$, with $\mathcal{K}_\alpha = (a^2 B_r / B^3) (\mathbf{B} \times \boldsymbol{\nabla}\space B) \cdot \boldsymbol{\nabla}\space \alpha$. Here, $B_r = 2\Psi_t(a)/a^2$ is a reference value for the magnetic field strength.
• Figure 4: $\overline{\Delta\gamma}$ versus $a/L_{n_z}$ for different impurities obtained from complete linear gyrokinetic simulations (black), the exact solution of the toroidal ITG linear gyrokinetic equation (red) and the analytical approximation to the solution of the toroidal ITG dispersion relation (blue). Here, $a/L_{T_z}=0$ and $n_z/n_i$ is chosen so that $\varepsilon = Z_z^2n_z / (Z_i^2 n_i) = 0.4$.
• Figure 5: $\overline{\Delta\gamma}$ versus $a/L_{T_z}$ for different impurities obtained from complete linear gyrokinetic simulations (black), the exact solution of the toroidal ITG linear gyrokinetic equation (red) and the analytical approximation to the solution of the toroidal ITG dispersion relation (blue). Here, $a/L_{n_z}=0$ and $n_z/n_i$ is chosen so that $\varepsilon = Z_z^2n_z / (Z_i^2 n_i) = 0.4$.