Hamiltonian estimation of island width threshold for stochasticity onset on edge pedestal top in presence of a resonant magnetic perturbation

TL;DR

The paper tackles predicting the onset of large-scale magnetic stochasticity in tokamak edge regions under resonant magnetic perturbations (RMPs). It develops a Hamiltonian description of toroidal magnetic field lines, reduces the problem to a two-wave paradigm, and directly ties island widths to the chaos threshold through an $(M,P)$ parameterization. Three independent chaos diagnostics—the renormalization method, Lyapunov exponents, and weighted Birkhoff averages—show consistent thresholds, placing the large-scale stochasticity onset near $\epsilon_c \approx 0.014$ and linking it to the destruction of the last KAM torus. Applying the framework to a DIII-D-like q-profile reveals that RMP-induced islands generally require larger widths than edge MHD islands for stochasticity, while edge-edge island couplings can trigger stochastic layers independently, providing a quantitative basis for optimizing RMP-based ELM control.

Abstract

This study applies the Hamiltonian method to analyze the nonlinear magnetic topology induced by Resonant Magnetic Perturbations (RMPs) in tokamaks. We investigate the system's chaotic behavior by comparing three methods: the renormalization method, Lyapunov exponents (LE), and weighted Birkhoff average (WBA). A strong consistency is found among these methods in predicting the large scale stochasticity threshold. The magnetic island width threshold provides a quantitative criterion for optimizing RMP-based ELM control, bridging a critical gap in plasma control strategies.

Figures (14)

• Figure 1: Radial profiles of Hamiltonian perturbations $H_{mn}$ as a function of normalized toroidal flux $\psi$, calculated using the simulation parameters described in the text. Top: (2,1) perturbation, including RMP and MHD components; Bottom: (3,2) MHD perturbation.
• Figure 2: q-profile as function of the normalized toroidal flux $\psi$.
• Figure 3: SVM and theoretical Boundary in $(2\sqrt{M}, 2\sqrt{P})$ space, separating the stable region (yellow) and unstable region (blue) with sample points for $\epsilon = 0.010, 0.014, 0.020$. Large scale stochasticity is predicted to occur as $\epsilon$ crosses the boundary.
• Figure 4: Comparison of Poincaré maps for different perturbation magnitudes $\epsilon$. (a) $\epsilon=0.010$, (b) $\epsilon=0.014$, (c) $\epsilon=0.020$.
• Figure 5: Evolution of Lyapunov exponents for different perturbation magnitudes $\epsilon$. (a) $\epsilon=0.010$, (b) $\epsilon=0.014$, (c) $\epsilon=0.020$. The gray curve clusters illustrate the evolution process of the LEs $\lambda(N)$ for each magnetic field line trajectory between $\psi = 0$ and $\psi = 1$ surface as a function of the number of iterations $N$. The maximum LE, $\lambda_{\max}(N)$, is indicated by a red dashed line, while the system-averaged LE, $\langle\lambda(N)\rangle$, is represented by a blue solid line.