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

2025/12/11 by Rajarshi, P. Anand, Khan, Mohammad Saif +3
Mathematics · Engineering · Earth and Planetary Sciences · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2512.10788

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} ≤ \bf k ≤ \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.

Citations

• Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations
• Onsager's "Ideal Turbulence" Theory
• Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics
• Turbulent flows are not uniformly multifractal
• The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalised to Kardar-Parisi-Zhang scaling
• Numerical evidence of anomalous energy dissipation in incompressible Euler flows: Towards grid-converged results for the inviscid Taylor-Green problem
• Inertial Ranges in Two-Dimensional Turbulence

Related

• Intermittency and Thermalization in Turbulence
• Euler Turbulence and thermodynamic equilibrium
• Equilibrium and nonequilibrium properties of Euler turbulence
• Finite Time Blow-up of a 3D Model for Incompressible Euler Equations
• Numerical evidence of anomalous energy dissipation in incompressible Euler flows: Towards grid-converged results for the inviscid Taylor-Green problem