Bounce-Averaged Theory In Arbitrary Multi-Well Plasmas: Solution Domains and the Graph Structure of their Connections

TL;DR

This paper develops a general framework for bounce-averaged theories in complex electromagnetic fields, using graph structures to represent domain connections and enabling simulations in arbitrarily complicated plasma geometries.

Contribution

It introduces a systematic method to identify and connect multiple solution domains in bounce-averaged plasma models, extending applicability to complex and evolving field configurations.

Findings

Established conditions for domain identification in complex fields

Presented a graph-based representation of domain connections

Enabled bounce-averaged simulations in arbitrary plasma geometries

Abstract

Bounce-averaged theories provide a framework for simulating relatively slow processes, such as collisional transport and quasilinear diffusion, by averaging these processes over the fast periodic motions of a particle on a closed orbit. This procedure dramatically increases the characteristic timescale and reduces the dimensionality of the modeled system. The natural coordinates for such calculations are the constants of motion (COM) of the fast particle motion, which by definition do not change during an orbit. However, for sufficiently complicated fields -- particularly in the presence of local maxima of the electric potential and magnetic field -- the COM are not sufficient to specify the particle trajectory. In such cases, multiple domains in COM space must be used to solve the problem, with boundary conditions enforced between the domains to ensure continuity and particle conservation. Previously, these domains have been imposed by hand, or by recognizing local maxima in the fields, limiting the flexibility of bounce-averaged simulations. Here, we present a general set of conditions for identifying consistent domains and the boundary condition connections between the domains, allowing the application of bounce-averaged theories in arbitrarily complicated and dynamically-evolving electromagnetic field geometries. We also show how the connections between the domains can be represented by a directed graph, which can help to succinctly represent the trajectory bifurcation structure.