A universal extension of helicity to topological flows

TL;DR

This paper extends the concept of helicity, a key invariant in fluid and plasma flows, to topological flows without rest points using $C^0$ Hamiltonian structures, connecting fluid dynamics with recent symplectic geometry advances.

Contribution

It provides an affirmative answer to Arnold's question by extending helicity invariance to volume-preserving homeomorphisms for flows without rest points, using $C^0$ symplectic geometry.

Findings

Helicity invariance extends to topological flows without rest points.

Utilizes $C^0$ Hamiltonian structures to analyze volume-preserving homeomorphisms.

Connects fluid dynamics invariants with recent developments in $C^0$ symplectic geometry.

Abstract

Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows, modeled by the three-dimensional Euler equations, and conducting plasmas, described by the magnetohydrodynamics (MHD) equations. A key property of helicity is its invariance under volume-preserving diffeomorphisms. In an influential article from 1973, Arnold, having provided an ergodic interpretation of helicity as the "asymptotic Hopf invariant", posed the question of whether this invariance persists under volume-preserving homeomorphisms. More generally, he asked whether helicity can be extended to topological volume-preserving flows. We answer both questions affirmatively for flows without rest points. Our approach reformulates Arnold's question in the framework of what we call $C^0$ Hamiltonian structures. This perspective enables us to leverage recent developments in $C^0$ symplectic geometry, particularly results concerning the algebraic structure of the group of area-preserving homeomorphisms.