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 *K*G exhibit
unexpected features. The study is carried out

for both
the one-dimensional Burgers equation and the 2D incompressible Euler equation.
At large *K*G, 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 *λ*G = 2*π/K*G 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 *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 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 *K*G *→ ∞*. 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.