Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations

Abstract

It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold K(G) exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. For large K(G) and smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a "tyger," is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc.). These tygers appear when complex-space singularities come within one Galerkin wavelength λ(G)=2π/K(G) from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first-in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to K(G)(-2/3) and K(G)(-1/3), respectively-but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee [Q. J. Appl. Math. 10, 69 (1952)]. The sudden dissipative anomaly-the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t(⋆)-which is well known for the Burgers equation and sometimes conjectured for the three-dimensional Euler equation, has as counterpart, in the truncated case, the ability of tygers to store a finite amount of energy in the limit K(G)→∞. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid-limit behavior.

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

“…However, as shown in Ref. [17], at t = t , at points which have the same velocity as the shock(s), Ψ = 0 and a symmetric, localised, monochromatic bulge, called tyger by Ray, et al [17], forms as shown in Fig. 2 (inset).…”

“…We begin by performing direct numerical simulations of equations (1) and (4), without any loss of generality [17,18], with initial conditions u 0 (x) = v 0 (x) = sin(x + φ), where φ = 0.7 is a phase which shifts the location of the cubic singularity away from x = π. For such an initial condition, it is easy to show that t = 1.0 [28,29].…”

“…As a result of these very recent developments [5,6,8,9,[14][15][16][17] (see also Ref. [18] for a recent review), the last few years have seen a furthering in our understanding of the possible existence of equilibrium solutions in inviscid equations of hydrodynamics.…”

“…[18] for a recent review), the last few years have seen a furthering in our understanding of the possible existence of equilibrium solutions in inviscid equations of hydrodynamics. Specifically, equipartition solutions to the Galerkin-truncated Gross-Pitaevskii [19], magnetohydrodynamic [20], Burgers [5,17], and Euler equations [6,21] have been studied extensively in recent years. Alongside the very important theoretical under-pinnings of such studies, thermalised or partially thermalised states have been shown [22][23][24] to be a possible explanation of the ubiquitous bottleneck [25] in the energy spectrum of turbulent flows.…”

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

“…As soon as we introduce the fractal decimation projector, several sharp oscillatory structures appear in the solution for v(x, t), even for mild decimation (D 1), as seen in Fig 1(b). Such structures, although reminiscent of features (tygers) of the Galerkin-truncated Burgers equation [22], are crucially different because they are spatially much more delocalized.…”