The dynamics of thermalisation in the Galerkin-truncated, three-dimensional Euler equation

TL;DR

This paper reviews how Galerkin-truncated 3D Euler equations thermalise, exploring the emergence of pseudo-dissipation and the associated time-scales, providing insights into turbulence and equilibrium states in finite-mode systems.

Contribution

It offers a comprehensive review of the thermalisation process in Galerkin-truncated Euler equations and its implications for turbulence and energy spectrum behavior.

Findings

Thermalisation leads to absolute equilibrium states in truncated Euler systems.

An emergent pseudo-dissipation range appears in the energy spectrum.

Time-scales of thermalisation are characterized and linked to turbulence phenomena.

Abstract

The inviscid, partial differential equations of hydrodynamics when projected via a Galerkin-truncation on a finite-dimensional subspace spanning wavenumbers $-{\bf K}_{\rm G} \le {\bf k} \le {\bf K}_{\rm G}$, and hence retaining a finite number of modes $N_{\rm G}$, lead to absolute equilibrium states. We review how the Galerkin-truncated, three-dimensional, incompressible Euler equation thermalises and its connection to questions in turbulence. We also discuss an emergent pseudo-dissipation range in the energy spectrum and the time-scales associated with thermalisation.