Sawtooth crash in tokamak as a sequence of Multi-region Relaxed MHD equilibria

TL;DR

The paper treats sawtooth crashes in tokamaks as a sequence of Multi-region Relaxed MHD (MRxMHD) equilibria constructed with the Stepped-Pressure Equilibrium Code (SPEC). It demonstrates that non-axisymmetric equilibria containing magnetic islands have slightly lower energy than axisymmetric states and that islands grow as reconnection proceeds, with a final axisymmetric after-crash state. The MRxMHD sequences are validated against nonlinear MHD simulations from M3D-C1, showing strong agreement in field-line topology, the $q$-profile, and current distribution, while recognizing that SPEC cannot resolve the detailed current-sheet structure or full time evolution. Overall, MRxMHD offers a computationally efficient, topology-aware framework to capture the reconnection path during sawtooth crashes and complements time-dependent MHD approaches, especially at low $\beta$.

Abstract

This study examines the sawtooth crash phenomenon in tokamak plasmas by modelling it as a sequence of Multi-region Relaxed Magnetohydrodynamic (MRxMHD) equilibria. Using the Stepped-Pressure Equilibrium Code (SPEC), we constructed a series of equilibria representing intermediate states during the sawtooth crash, with progressively increasing reconnection regions. Numerical results demonstrated that the system prefers the lower energy non-axisymmetric equilibria with islands and is eventually back to an axisymmetric state, capturing key features of the reconnection process. Comparisons with the nonlinear MHD code M3D-C1 showed remarkable agreement on the field-line topology, the safety factor, and the current profile. However, the simplified MRxMHD model does not resolve the detailed structure of the current sheet. Despite this limitation, MRxMHD offers an insightful approach and a complementary perspective to initial-value MHD simulations.

Figures (12)

• Figure 1: A schematic view of the plasma regions $\mathcal{R}_i$ and ideal interfaces $\mathcal{I}_i$. Reproduced from [Dennis et al. Physics of Plasmas 20(3), 032509 (2013)]dennis_infinite_2013, with the permission of AIP Publishing.
• Figure 2: The $q$ profile of the initial equilibrium.
• Figure 3: Resistive instability growth rate calculated by PHOENIX, showing a transition between a tearing mode ($\gamma \sim \eta^{3/5}$) to a resistive-kink ($\gamma \sim \eta^{1/3})$ as the plasma resistivity increases.
• Figure 4: The helical flux used to subdivide the plasma volume and the location of the interfaces (indicated by the blue diamonds. The vertical dashed line stands for the $q=1$ surface and the horizontal dashed line stands for $\chi=0$.
• Figure 5: The Poincaré map for the initial SPEC equilibrium with 15 subregions. The interfaces are represented by the red solid lines.