Topological Arrest of Ballooning Modes in Non-Axisymmetric Plasmas

TL;DR

The paper addresses why non-axisymmetric plasmas often avoid global ballooning crashes despite linear instability. It maps the flux-surface problem to a network of Anderson-localized ballooning modes and derives a weakly nonlinear Ginzburg-Landau equation on that network, with stability governed by connectivity. A key result is the percolation threshold eta_c = 1.128, which separates benign, localized scintillations from global, rapid growth; subcritical regimes correspond to fizz-like saturation, while supercritical regimes enable spanning clusters and disruptive dynamics. The work provides a topological framework for confinement in 3D devices and suggests design strategies that preserve aperiodicity to maintain nonlinear stability against ballooning crashes in future reactors.

Abstract

Nonlinear ballooning instabilities in non-axisymmetric equilibria exhibit spatial localization along field lines that mirrors Anderson localization in disordered lattices. We demonstrate that this localization fundamentally alters nonlinear stability, transforming a global ballooning crash into a connectivity phase transition on a Ginzburg-Landau network. We identify a dimensionless number, denoted by ηc, as a topological threshold derived from continuum percolation theory. Below this threshold, global instability is topologically arrested as isolated scintillations, providing a rigorous explanation for the robust, benign saturation observed in experiments. Above this threshold, a spanning cluster path forms, unlocking rapid and potentially disruptive nonlinear growth.

Figures (2)

• Figure 1: Topological connectivity of ballooning modes on a toroidal flux surface. The spatial distribution and interaction of localized ballooning mode structures are shown for three characteristic magnetic geometries. Each circle represents the interaction radius of a mode. (a) Subcritical regime (W7-X): High magnetic aperiodicity (large Lyapunov exponent) leads to short localization lengths. The resulting connectivity parameter $\eta$ is well below $\eta_c$, preventing global synchronization and resulting in benign topological arrest. (b) Critical Regime (LHD): Reduced geometric disorder increases the interaction radius, bringing the system to the marginal percolation threshold ($\eta_c=1.128$). In this regime, a global-spanning path (orange line) intermittently forms, thereby facilitating the bursty MHD activity observed experimentally. (c) Supercritical Regime (DIII-D): Near-perfect axisymmetry leads to effectively infinite localization lengths. The system is in a highly connected state, where global phase-locking of modes enables the explosive, singular growth characteristic of major profile crashes.
• Figure 2: Schematic estimates of the connectivity parameter $\eta=\rho\,\pi R_*^2$ for representative operating regimes. The dashed line indicates the continuum percolation threshold $\eta_c \simeq 1.13$. The tokamak point is based on measured inter-ELM radial correlation lengths of edge density fluctuations obtained from reflectometry and BES diagnostics Yan2011Rhodes1992Barada2021. The stellarator points reflect shorter spatial scales associated with filamentary structures observed in W7-X and LHD Zoletnik2020Tanaka2012. These values are not direct extractions of $\rho$ and $R_*$ from experimental event statistics but proxy estimates based on published spatial scales, intended to illustrate the separation of regimes. A program for quantitative determination of $\rho$ and $R_*$ from spatiotemporal diagnostics is described in the text.