Comparison of MHD and gyrokinetic simulations of linear instabilities at the q = 1 surface

TL;DR

This work assesses linear instabilities at the tokamak $q=1$ surface by comparing gyrokinetic (ORB5/LIGKA) and MHD (JOREK/CASTOR3D) models. It shows that gyrokinetic equations without collisions reproduce the ideal MHD limit, and that instability outcomes depend strongly on simulation setup, including parallel magnetic-field fluctuations and diamagnetic effects. The collisionless tearing mode emerges in GK at low $ eta$ and larger electron mass, while the ideal internal kink mode can be stabilized by diamagnetic rotation, with MHD and GK results converging when comparable physics are included. The findings demonstrate that GK and MHD descriptions can be reconciled under careful model matching, but some instabilities like collisionless tearing require extended or kinetic treatments beyond pure MHD. The study outlines pathways for incorporating energetic particles and nonlinear physics (e.g., fishbone) in future work, enhancing relevance to burning plasmas.

Abstract

Accurate modeling of core instabilities in tokamak plasmas is essential to understand the underlying physical mechanisms and their impact on plasma confinement. The ideal stability of the internal kink mode and the m = 1 collisionless tearing mode are analyzed numerically both with gyrokinetic and MHD codes. We compare the different models implemented in the codes and show that the gyrokinetic equations without collisions inherently contain the ideal MHD limit. The simulation results show that the stability of the internal kink mode strongly depends on the choice of several setup parameters like the inclusion of parallel magnetic field fluctuations, the tokamak aspect ratio, the drift- or gyrokinetic treatment of the ions and the electron mass. Furthermore, we demonstrate the stabilization of the instabilities by diamagnetic effects. Our results indicate that gyrokinetic and MHD models can be reconciled in the description of the internal kink mode by careful consideration of the simulation setup and model assumptions, but instabilities like the collisionless tearing mode require a more advanced treatment beyond MHD.

Figures (18)

• Figure 1: Safety factor and pressure profile for the collisionless tearing mode case. Note that the pressure profile is flat. The radial coordinate used is $s = \sqrt{\psi_\mathrm{N}}$.
• Figure 2: (a) Poloidal and (b) radial mode structure in the electrostatic potential $\Phi$ (absolute value) and real part of the magnetic vector potential component $A_\parallel$ (c) obtained with ORB5 for the case with flat pressure profile and $m_\mathrm{e}/m_\mathrm{i} = 0.001$.
• Figure 3: (a) Time trace of the $n=1$ component of the electrostatic potential $\Phi$ for the ORB5 simulation with $\beta_\mathrm{ORB5} = 0.001385$ and $m_\mathrm{e}/m_\mathrm{i} = 0.005$. The maximum of the absolute value of $\Phi$ (in real space) is taken over the whole computational domain. $\Phi$ is normalized to $T_\mathrm{e} / q_\mathrm{i}$. (b) The helical flux $\Psi_\mathrm{he}$ at $t=148000 \,\omega_\mathrm{ci}^{-1}$ . This time point marks the end of the linear phase when the $n=1$ mode starts to saturate and is represented by the red dashed line in (a). A magnetic island is visible.
• Figure 4: The growth rate $\gamma$ of the unstable $m/n=1/1$ mode as a function of the electron-to-ion mass ratio $m_\mathrm{e}/m_\mathrm{i}$ obtained with ORB5 for three different values of $\beta_\mathrm{ORB5}$. The dashed lines represent the value $m_\mathrm{e}/m_\mathrm{i} = \beta_\mathrm{ORB5}$. The green and orange solid lines represent the theoretical scalings $\gamma \propto m_\mathrm{e}^{1/2}$ and $\gamma \propto m_\mathrm{e}^{1/6}$ which have been fitted to the data points. The blue solid line is a fit $\propto m_\mathrm{e}^{\alpha}$, where $\alpha \approx 0.311$ was found as optimal fit parameter.
• Figure 5: (a) Safety factor and pressure profile for the stable kink mode case. (b) The electron skin depth $\delta_\mathrm{e}$ and the ion Larmor radius $\varrho_\mathrm{i}$ as a function of the normalized poloidal flux $\psi_\mathrm{N}$ for a scenario where both, the collisionless tearing mode and the internal kink mode are stable. Over the whole domain $\varrho_\mathrm{i} > \delta_\mathrm{e}$.