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- W1551101025 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 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 $lambdag = 2pi/kg$ 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 $kgtoinfty$. 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." @default.
- W1551101025 created "2016-06-24" @default.
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- W1551101025 date "2010-11-08" @default.
- W1551101025 modified "2023-09-27" @default.
- W1551101025 title "The tyger phenomenon for the Galerkin-truncated Burgers and Euler equations" @default.
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