The tiger phenomenon for the Galerkin-truncated Burgers and Euler equations

Urial Frisch, Lab. Cassiopee, Observatoire de la Cote d'Azur

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 KG exhibit unexpected features. The study is carried out

for both the one-dimensional Burgers equation and the 2D incompressible Euler equation. At large KG, for

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 complexspace

singularities come within one Galerkin wavelength λG = 2π/KG from the real domain and typically arise

far away from preexisting small-scale structures at locations whose velocities matches 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 amplitude and width are proportional to KG−2/3 and KG−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_, which is well known for the Burgers equation and sometimes conjectured for the 3D Euler

equations, has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in

the limit KG → ∞. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and

thus prevents the flow from converging to the zero-viscosity limit solution. There are indications that it may be

possible to purge the tygers and thereby to recover the correct inviscid-limit behaviour.