Statistical mechanics of the 3D axi-symmetric Euler equations in a Taylor-Couette geometry

TL;DR

This paper develops a statistical mechanics framework for 3D axially symmetric Euler flows in Taylor-Couette geometry, revealing two equilibrium regimes and complex fluctuation behaviors influenced by energy levels.

Contribution

It introduces microcanonical measures for 3D axially symmetric flows and uncovers a novel phase diagram with distinct equilibrium states based on energy.

Findings

Existence of two equilibrium regimes depending on energy

High-energy states exhibit large-scale dipole and jet structures

Poloidal fluctuations are unbounded despite conservation laws

Abstract

In the present paper, microcanonical measures for the dynamics of three dimensional (3D) axially symmetric turbulent flows with swirl in a Taylor-Couette geometry are defined, using an analogy with a long-range lattice model. We compute the relevant physical quantities and argue that two kinds of equilibrium regimes exist, depending on the value of the total kinetic energy. For low energies, the equilibrium flow consists of a purely swirling flow whose toroidal profile depends on the radial coordinate only. For high energies, the typical toroidal field is uniform, while the typical poloidal field is organized into either a single vertical jet or a large scale dipole, and exhibits infinite fluctuations. This unusual phase diagram comes from the poloidal fluctuations not being bounded for the axi-symmetric Euler dynamics, even though the latter conserve infinitely many ''Casimir invariants''. This shows that 3D axially symmetric flows can be considered as intermediate between 2D and 3D flows.