Geodesic extended modes in low magnetic shear tokamaks and stellarators

Abstract

Theories of ion-scale microinstabilities in tokamaks and stellarators typically assume that the passing electrons respond adiabatically due to their fast propagation speed. However, when the magnetic shear becomes sufficiently small, ion-scale modes can extend far along the magnetic field and the non-adiabatic response of passing electrons becomes important. We derive a theory of extended modes at low magnetic shear through a multiscale expansion of the gyrokinetic equation. The theory elucidates the physics of the geodesic extended mode, a new type of microinstability. The new mode couples the non-adiabatic physics of both electrons and ions, unlike extended modes at magnetic shear of order unity. The theory is validated against gyrokinetic simulations and the parameter dependences of the new mode are studied.

Figures (12)

• Figure 1: Complex frequency (a) and eigenmode structure for $k_y \rho_i = 0.1$ (b) at small and large magnetic shear, $\hat{s} =0.01$ and $\hat{s} = 0.8$. The data is obtained from gyrokinetic simulations using the stella code barnes_stella_2019 for a tokamak with circular cross-section. The safety factor is $q=2.38$, the electron-to-ion mass ratio is $m_e/m_i = 1/1836$, and a pure ion temperature gradient drive is considered, $a/L_{Ti}=4$ and ${a/L_{Te}=a/L_n = 0}$. All other parameters are identical to those presented in Section \ref{['sec:longtail_sims_numerical']}.
• Figure 2: Meijer G-functions for the evaluation of the electron propagator \ref{['eq:Pe_MeijerG']}. Only Re$(z_c)>0$ values are considered, corresponding to growing modes (Im $\omega=\omega_i>0$). One-dimensional cuts are projected on the sides as contours.
• Figure 3: Extended tail eigenmode obtained from stellabarnes_stella_2019 gyrokinetic simulations (black curve) and from the dispersion relation \ref{['eq:tail_modes_disp_integral_eq']}, for the general electron propagator \ref{['eq:Pe_MeijerG']} (red curves) and for the slow electron propagation limit (blue curves). The thick transparent curves are the averages $\langle \delta\hat{\varphi}\rangle_{\theta_f}$ of the eigenmode in the fast coordinate, found as solutions to the dispersion relation \ref{['eq:tail_modes_disp_integral_eq']}. The variation of $B$ along the magnetic field is plotted for reference (grey curve) and the mode frequencies are provided in the legend. We only show the eigenmode for positive values of $\theta$ as it is symmetric in $\theta$. We model a Deuterium plasma ($\sqrt{m_i/m_e} = 60.6, Z_i = 1$) on the flux surface $r/a=0.3$ of a tokamak with circular cross-section, aspect ratio $R/a = 6$, safety factor $q=5$, and magnetic shear $\hat{s} = 0.1$. We consider a binormal wavenumber $k_y \rho_i = 0.1$ and a pure ion-temperature-gradient drive $a/L_{Ti}=4$ and $a/L_n = a/L_{Te}=0$. The vertical black dashed lines indicate the distance along the field line where the radial and binormal wavenumbers are of similar size ($\theta = \hat{s}^{-1} \Leftrightarrow k_x \approx k_y$) and where the ion FLR effects become significant ($\theta = (k_y \rho_i \hat{s})^{-1} \Leftrightarrow k_\perp \rho_i \approx 1$). The horizontal left-right arrow gives the distance of parallel propagation of a thermal electron within the characteristic timescale of the mode ($\Delta\theta = \omega_{{ \|} e} 2\pi/|\omega|$).
• Figure 4: Complex frequency for varying magnetic shear -- other parameters are identical to the reference case of Figure \ref{['fig:longtail_phi_theta_ref_case']}. The solid and dashed curves correspond to the real and imaginary frequencies, respectively, obtained from solving the dispersion relations \ref{['eq:tail_modes_disp_integral_eq']} (red), \ref{['eq:longtail_disp_rel_weak_Pe']} (blue), and \ref{['eq:longtail_disp_rel_weak_Pe_NR_ions_approximate']} (green). The circles and markers correspond to the real and imaginary frequencies, respectively, obtained from gyrokinetic simulations.
• Figure 5: Complex frequency for varying electron mass (a) and safety factor (b) -- other parameters are identical to the reference case of Figure \ref{['fig:longtail_phi_theta_ref_case']}. The solid and dashed curves correspond to the real and imaginary frequencies, respectively, obtained from solving the dispersion relations \ref{['eq:tail_modes_disp_integral_eq']} (red), \ref{['eq:longtail_disp_rel_weak_Pe']} (blue), and \ref{['eq:longtail_disp_rel_weak_Pe_NR_ions_approximate']} (green). The circles and markers correspond to the real and imaginary frequencies, respectively, obtained from gyrokinetic simulations.