The Onset of Thermalisation in Finite-Dimensional Equations of Hydrodynamics: Insights from the Burgers Equation

TL;DR

This paper investigates how finite-dimensional approximations of hydrodynamic equations, specifically the Burgers equation, develop thermalisation over time, providing precise estimates for the onset time and implications for fluid dynamics research.

Contribution

It offers the first precise estimates of thermalisation onset time in Galerkin-truncated equations using the Burgers model, combining theoretical analysis and numerical simulations.

Findings

Thermalisation time scales as rac{1}{K_G^{4/9}}.

Sharp localized structures called tygers lead to thermalisation.

Results inform methods to detect blow-ups in Euler equations.

Abstract

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber $K_G$), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localised structures, called {\it tygers} [Ray, et al., Phys. Rev. E {\bf 84}, 016301 (2011)], which eventually lead to completely thermalised states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time $\tau_c$ at which thermalisation is triggered and show that $\tau_c \sim \kg^\xi$, with $\xi = -4/9$. Our results have several implications including for the analyticity strip method to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.