Overview
This study examines the emergence of cross-scale fluxes associated with energy and enstrophy within a stochastic version of Zeitlin's $SU(N)$ approximation of 2-D Euler dynamics. The research interprets the nonlinear transport as motion along coadjoint orbits, linking it to an underlying symmetry that preserves Casimir invariants. A fast-slow stochastic framework is employed to model rapid mixing via structured fast stochastic forcing in selected directions. The study identifies conditions under which nontrivial fluxes arise.
Research Context
The Euler-Arnold formulation provides a framework for understanding the system's dynamics, particularly the nonlinear transport, as a movement along coadjoint orbits. This formulation highlights an inherent symmetry that ensures the conservation of all Casimir invariants. The investigation specifically focuses on a stochastic variant of Zeitlin's $SU(N)$ approximation, which is relevant to 2-D Euler dynamics. The concept of cross-scale fluxes refers to the transfer mechanisms of quantities like energy and enstrophy across different scales within the system. The research delves into how these fluxes are influenced by stochastic forcing and underlying symmetries.
Approach
The researchers developed a fast-slow stochastic framework. This framework incorporates rapid mixing through a structured fast stochastic forcing. This forcing is applied along specific directions within the system. The fundamental hypothesis tested is how the preservation or breaking of coadjoint orbit symmetry, afforded by the natural symplectic structure, impacts the presence and characteristics of averaged nonlinear flux. The approach involves analyzing the system under two primary scenarios:
Scenario 1: Coadjoint Symmetry Preservation: The fast dynamics maintains the full coadjoint orbit symmetry, which is defined by the natural symplectic structure.
Scenario 2: Coadjoint Symmetry Breaking: This symmetry is disrupted by the introduction of tangential non-Hamiltonian vector fields acting on the coadjoint orbit.
The study sought to identify distinct outcomes regarding flux generation and direction in these contrasting scenarios.
Findings
The investigation yielded two primary findings concerning the behavior of fluxes in the stochastic Euler-Arnold system:
When the fast dynamics preserves the full coadjoint orbit symmetry, which is conferred by the natural symplectic structure, the averaged system exhibits no nontrivial flux.
Conversely, when this coadjoint symmetry is broken by tangential non-Hamiltonian vector fields acting on the coadjoint orbit, specific conditions were identified that lead to a nonzero flux. In this case, the Euler-Arnold nonlinearity carries these fluxes.
These findings suggest that the presence of coadjoint symmetry acts to suppress averaged nonlinear flux. In contrast, the breaking of this symmetry can select a preferred direction for cross-scale transfer.