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 wavenumbers 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). Tygers appear when complex-space singularities come within one Galerkin wavelength $\lambdag = 2\pi/K_G$ from the real domain and 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 in 1952. The sudden dissipative anomaly - the presence of a finite dissipation in the limit of vanishing viscosity after a finite time $t_\star$ -, which is well known for the Burgers equation and sometimes conjectured for the 3D Euler equation, has as counterpart, in the truncated case, the ability of tygers to store a finite amount of energy in the limit $K_G\to\infty$. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and 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.