Speaker
Prof.
Uriel Frisch
Description
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. (Based on a paper by
Samriddhi Sankar Ray, Uriel Frisch, Sergei Nazarenko and
Takeshi Matsumoto, available at http://arxiv.org/abs/1011.1826)
