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

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.

💡 Research Summary

This paper presents a comprehensive analysis of thermalisation dynamics in the three-dimensional, incompressible Euler equation subjected to a Galerkin truncation. The core idea is that projecting the original inviscid partial differential equation (PDE) onto a finite-dimensional subspace of Fourier modes (with wavenumbers |k| ≤ K_G) transforms it into a finite-dimensional system of ordinary differential equations (ODEs). This truncated system conserves both kinetic energy and phase-space volume exactly, forming a bona fide Hamiltonian system. Consequently, its long-time behavior evolves towards a thermalised, chaotic state known as an absolute equilibrium, which can be analyzed using tools from statistical mechanics.

The study employs high-resolution Direct Numerical Simulations (DNS) using a fully-dealiased pseudo-spectral method on a triply-periodic domain, exploring a range of truncation wavenumbers K_G from 21 to 341. Starting from initial conditions where energy is concentrated at large scales, the evolution of the energy spectrum E(k) reveals a complex multi-stage process. At early times, a turbulent cascade develops, exhibiting a Kolmogorov-like k^{-5/3} scaling at low wavenumbers. As the cascade reaches the truncation limit K_G, a distinct transitional “pre-thermalised” phase emerges. During this phase, an intermediate range of wavenumbers (K_I ≤ k ≤ K_th) displays a steeper k^{-3} scaling, termed the “pseudo-dissipation range,” even though no true dissipation exists in the system. This range eventually gives way, at later times, to a thermalised k^2 spectrum at high wavenumbers (k ≳ K_th), characteristic of energy equipartition among Fourier modes. The transition wavenumber K_th between the turbulent and thermalised regions decreases over time, signifying the progressive thermalisation of smaller scales.

The paper investigates the physical origin of the robust k^{-3} scaling observed in the pre-thermalised phase. It offers a heuristic explanation based on an eddy-viscosity concept, where the thermalising small scales act as an effective viscous bath for the larger turbulent scales. Assuming a constant energy flux Π through this range leads to the relation E(k) ~ Π ν_eff^{-1} k^{-3}. However, the authors caution that DNS data shows a convincingly constant flux Π(k) only in the fully thermalised state, not consistently during the pre-thermalised phase, indicating the need for a more complete theoretical understanding.

A significant portion of the analysis is dedicated to identifying and characterizing the key timescales governing the thermalisation pathway. Three distinct timescales are identified: (1) the cascade completion time (t_c), when nonlinear interactions have sufficiently transferred energy to excite modes near the maximum wavenumber K_G; (2) the birth time (t_b), when the first tangible signatures of truncation—manifested as localized damped oscillations in physical space and an upturned spectrum tail in Fourier space—appear; and (3) the thermalisation time (t_th), when a clear k^2 equipartition spectrum is established at high wavenumbers. The paper draws parallels with the one-dimensional Burgers equation, where similar timescales have been analytically derived to scale as powers of K_G (e.g., t_b ~ K_G^{-2/3}). For the 3D Euler case, however, the precise scaling of t_c, t_b, and t_th with K_G remains an open question due to the system’s greater complexity, highlighting a direction for future research.

In summary, this work positions the Galerkin-truncated Euler equation as a fruitful minimal model for studying the statistical mechanics of turbulence. It provides detailed numerical evidence of the spectral signatures of thermalisation, explores the genesis of an intermediate k^{-3} range, and frames the dynamical process in terms of characteristic timescales, thereby connecting fundamental aspects of Hamiltonian dynamics, fluid turbulence, and non-equilibrium statistical physics.