A new formulation of the near-equilibrium theory of turbulence

TL;DR

The paper addresses formulating a near-equilibrium statistical theory of three-dimensional turbulence without restricting vorticity to discrete filaments. It develops a dense lattice vortex model and a continuum limit to interpret turbulence as a near-equilibrium phenomenon connected to critical points, encapsulated by a dimensionless parameter $\Lambda = L\sqrt{E}/\Gamma$. By fixing energy $E$ and a finite enstrophy-like bound $Z_2$ while tuning the mesh size $h$, the authors obtain intermittent, coherent-structure statistics and a Kolmogorov-like spectrum along the critical line, supported by qualitative numerical results on small lattices. The approach forges a link between turbulence and critical phenomena, offering a statistical-mechanical route to derive turbulence statistics, albeit with current computational limitations and qualitative outcomes. Overall, the work expands the near-equilibrium turbulence framework beyond discrete filament representations and highlights the potential of phase-transition concepts to capture intermittency and coherence observed in fully developed turbulence.

Abstract

We present a status report on a discrete approach to the the near-equilibrium statistical theory of three-dimensional turbulence, which generalizes earlier work by no longer requiring that the vorticity field be a union of discrete vortex filaments. The idea is to take a special limit of a dense lattice vortex system, in a way that brings out a connection between turbulence and critical phenomena. The approach produces statistics with basic features of turbulence, in particular intermittency and coherent structures. The numerical calculations have not yet been brought to convergence, and at present the results are only qualitative.