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

TL;DR

This work examines how the inviscid, Galerkin-truncated 3D Euler equation in a triply periodic domain thermalises, revealing a coexisting turbulent large-scale regime and a rapidly equilibrating small-scale bath. Through direct numerical simulations across multiple Galerkin cutoffs $K_G$, the authors identify a robust $E(k) ~ k^{-3}$ pseudo-dissipation range, a persistent large-scale $E(k) ~ k^{-5/3}$ range, and a late-time $E(k) ~ k^2$ equipartition tail, with three characteristic timescales: $t_c$ (cascade completion), $t_b$ (birth of pre-thermalised structures), and $t_{th}$ (thermalisation). They find $t_c$ and $t_b$ scale roughly as $\log N$, while $t_{th}$ is largely independent of $K_G$, and observe a near-plateau in the energy flux $\Pi(k)$ at late times, consistent with energy transfer to a thermalised bath. The results shed light on the statistical mechanics of truncated Euler flows and have implications for turbulence modeling and the fundamental understanding of dissipation-like behavior in Hamiltonian systems.

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.

Figures (3)

• Figure 1: (a) Representative plot of the energy spectra, for different values of $K_{\rm G}$ (see legend) at $t = 3$. The (left) vertical dashed line corresponds to the wavenumber $K_{\rm I}$ marking the transition from a Kolmogorov-like $k^{-5/3}$ scaling (indicated by the thick line as a guide to the eye) to one which follows a $k^{-3}$, indicated by a thick and a dashed line as guides to the eye, respectively. The $k^{-3}$ scale terminates and a pre-thermalised spectrum, with a positive slope, starts to develop. In the inset of panel (a), we confirm the pseudo-dissipation range $k^{-3}$ scaling through a compensated spectrum. At later times, such as $t = 6$ as shown in panel (b), the right tail follows a $k^2$ scaling as indicated by the thick line.
• Figure 2: Representative plots of the energy flux, for different values of $K_{\rm G}$ (see legend), at (a) $t = 3$ and (b) $t = 6$. The solid black curve, which overlay the data points in panel (b), are obtained from a theoretical estimate of $\Pi$ (see text) and provide a reasonable approximate of the flux for large $k$ and $K_{\rm G}$.
• Figure 3: (a) A plot of the enstrophy $\Omega$ versus time for different Galerkin-truncation wavenumbers $K_{\rm G}$ showing the growth and $K_{\rm G}$ dependent saturation at $\Omega_\infty$, indicated by the horizontal dashed lines (see text). At short times the curves for different $K_{\rm G}$ collapse, and as smaller scales get excited, they separate. (b) The timescales for cascade completion $t_c$ (triangles), and the birth$t_b$ (stars) of the localised oscillatory structures which eventually trigger thermalisation plotted against the logarithm of $N$. The linear curves suggest that both these timescales depend logarithmically on $N$. The inset shows the energy spectrum, for different $N$, or equivalently for different $K_{\rm G}$, at $t_b$ showing the departure at the ultraviolet end from the $k^{-3}$ scaling before the onset of an equipartition $k^2$ spectrum. (c) The variation of the spectral slope for wavenumbers in the boundary layer $K_{\rm BL} = 10\%$. The horizontal dashed line indicates a slope of 2 corresponding to the equipartition $E(k) \sim k^2$ spectrum. We note a $K_{\rm G}$-independent convergence to the thermalised regime suggesting that the thermalisation time $t_{\rm th}$ is, unlike the 1D Burgers problem as well as $t_c$ and $t_b$, independent of $K_{\rm G}$. In the inset we show the analogous result for $K_{\rm BL} = 15\%$ leading to the same conclusion.