Optimal thermalization in a shell model of homogeneous turbulence

TL;DR

This paper develops an optimal statistical closure for turbulence that accurately predicts energy relaxation in a shell model, validated against DNS and compared with EDQNM closure.

Contribution

It introduces an exact optimal closure approach for turbulence, capturing damping effects through a Gaussian model and distinguishing ensemble averages from proxies.

Findings

Optimal closure accurately predicts energy relaxation.

Closure captures intrinsic damping effects.

Validated against direct numerical simulations.

Abstract

We investigate the turbulence-induced dissipation of the large scales in a statistically homogeneous flow using an "optimal closure," which one of us (BT) has recently exposed in the context of Hamiltonian dynamics. This statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. We focus on a simple problem for which the optimal closure can be fully and exactly worked out: the relaxation arbitrarily far-from-equilibrium of a single energy shell towards Gibbs equilibrium in an inviscid shell model of 3D turbulence. The predictions of the optimal closure are validated against DNS and contrasted with those derived from EDQNM closure.