Simulations of internal kink modes and sawtooth crashes for SPARC baseline-like scenarios using the M3D-C1 code

Abstract

A relaxed baseline case, based on the SPARC Primary Reference Discharge (PRD) design point, is used to conduct a thorough investigation for the most unstable low-$n$ MHD instabilities for the first time. The simulations use the high-fidelity 3D extended-MHD code M3D-C1. The linear simulation, by scanning over the resistivity, identifies a dominant internal kink mode at the $q=1$ surface with a toroidal mode number $n=1$. Both the current and the pressure profiles are strongly affecting the kink instability in the baseline case. The linear growth rate is sensitive to the keV-level temperature profile and the on-axis $q_0$ around unity. A simplified 1D eigenvalue solver shows a good qualitative agreement for the observed pressure effects. In 3D nonlinear simulations, the marginally unstable case gives a moderate sawtooth crash soon after $q_0$ drops below unity, likely because of the lack of stabilizing effects in our simulations, such as heating and energetic particles. When both the current and the pressure drives exist (the baseline case), a strong sawtooth is observed, which features a magnetic reconnection event and a hollowed pressure profile. This can be explained by mixing both the Kadomtsev and Wesson models. The actual sawtooth crash may occur in SPARC before $q_0$ drops far below unity due to the sensitive changes of the instability around $q_0\sim 1$. The sawtooth-like oscillations shown in low-$β$ simulations also provides an opportunity to investigate periodic sawtoothing timescales in SPARC. This work forms a basis for understanding particle and heat transport under the influence of MHD instabilities, which can be essential for properly assessing the performance of the SPARC tokamak and future fusion pilot plants.

Figures (13)

• Figure 1: Relaxed equilibrium (labeled "Relaxed") profiles based on SPARC PRD design point: Panel (a) is the safety factor $q$, where the $\rho_1=0.53$, $\rho_2=0.91$ and $\rho_3=0.97$ are $q=1,2,3$ locations respectively (labeled with vertical dashed lines). Panel (b) is the density profile $n_i=n_e$. Panel (c) is the temperature profile, for SPARC baseline case in this paper $T_i=T_e$. The PRD H-mode profiles (dashed lines labeled "H-mode") are also provided for reference. All profiles are functions of the radial coordinate, $\rho=\sqrt{(\psi-\psi_0)/\psi_{\mathrm{max}}}$, where the poloidal flux $\psi$ is normalized by its maximum magnitude $\psi_{\mathrm{max}}=\psi_w-\psi_0$ between the LCFS and the magnetic axis.
• Figure 2: Mesh geometries used in M3D-C1 simulations: (a). high resolution mesh region including the first wall and the conducting vessel walls (solid lines). The coil locations are marked in red rectangulars and the poloidal flux geometry is shown by the dashed blue contours. The LCFS is shown by the black dashed line. (b). simplified mesh region with plasma region only. The actual simulation boundary (black solid line) is different from the first wall (grey solid line) but contains a comparable plasma volume. The blue contours are the poloidal flux surfaces and the LCFS is the black dashed line.
• Figure 3: (a) Linear growth rate $\gamma$ for a single toroidal mode $n$ in the linear simulation. The dashed line is a fitting line to show the decreasing trend of $\gamma$. The normalization is $\tau_A = 2.10\times 10^{-7}$ s. (b) Poloidal mode structure of the perturbed toroidal current $\delta j_\Phi$ for $n=1$ with the SPARC baseline equilibrium. The dashed lines indicates the flux surfaces at $q=1,2,3$. The linear mode is normalized to its maximum amplitude. The solid lines are schematic contours of the first wall and the vacuum vessel in the whole-device M3D-C1 simulations.
• Figure 4: Linear growth rate $\gamma$ with respect to the Lunquist number $S$, by scanning over a constant resistivity $\eta$. The time unit is $\tau_A = 2.10\times 10^{-7}$ s. The relaxed baseline (PRD) case is labeled far to the right of the figure (ideal MHD limit) considering its typical resistivity level in the core region. The vertical lines mark the region where two scaling trend can be compared to the kink mode theory with plasma $\beta_0=0$. The grey data points are beyond the linear fit for scaling, which approach the extremely resistive case (to left) and the ideal MHD case (to right). The dashed fitting lines show different scaling trends.
• Figure 5: Linear growth rate $\gamma$ with respect to the on-axis safety factor $q_0$, with fixed plasma $\beta_0=2.41$%. The time unit is $\tau_A = 2.10\times 10^{-7}$ s. The baseline case is labeled as "PRD". The vertical dashed line marks the $q_0=1$.