Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations

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.