A continuous model for turbulent energy cascade

TL;DR

This paper introduces a new PDE model in frequency space for turbulent energy cascade that captures classical turbulence scaling laws and the dissipation anomaly, offering a continuous scale perspective.

Contribution

It presents a novel PDE model based on continuous scales, differing from traditional discrete models, and demonstrates existence of unique stationary solutions.

Findings

Model reproduces Kolmogorov's scaling laws

Shows existence of unique stationary solutions

Replicates classical dissipation anomaly

Abstract

In this paper we introduce a new PDE model in frequency space for the inertial energy cascade that reproduces the classical scaling laws of Kolmogorov's theory of turbulence. Our point of view is based upon studying the energy flux through a continuous range of scales rather than the discrete set of dyadic scales. The resulting model is a variant of Burgers equation on the half line with a boundary condition which represents a constant energy input at integral scales. The viscous dissipation is modeled via a damping term. We show existence of a unique stationary solution, both in the viscous and inviscid cases, which replicates the classical dissipation anomaly in the limit of vanishing viscosity. A survey of recent developments in the deterministic approach to the laws of turbulence, and in particular, to Onsager's conjecture is given.