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

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.