Title: The tyger phenomenon for the Galerkin-truncated Burgers and Euler equations
Speaker: Prof. Uriel Frisch Observatoire de la Cote d'Azur Nice, France
Date: 24 December 2010
Time: lh 1, department of mathematics indian institute of science, bangalore
Venue: 4:00

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 two-dimensional 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 complex-space singularities come within one Galerkin wavelength $\lambdag = 2\pi/\kg$ 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 $\kg ^{-2/3}$ 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 $\ts$-, 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 $\kg\to\infty$. 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 be possible to purge the tygers and thereby to recover the correct inviscid-limit behaviour.

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Last updated: 06 Mar 2020