
Marie FARGE, Eric GOIRAND, Yves MEYER, Frédéric
PASCAL and Mladen Victor WICKERHAUSER
M.F.: Laboratoire de Météorologie Dynamique, École
Normale Supérieure, 24, rue Lhomond, 75231 Paris cedex 05, France
E.G.: Laboratoire de Météorologie Dynamique, École
Normale Supérieure, 24, rue Lhomond, 75231 Paris cedex 05, France
Y.M.: CEREMADE, Université ParisDauphine, Paris, France
F.P.: Laboratoire d'Analyse Numérique, Université Paris
XI, Orsay, France
M.V.W.: Department of Mathematics, Washington University at St. Louis,
MO 63130, USA

Abstract 
We propose to use new orthonormal wavelet packet bases, more
efficient the the Fourier basis, to compress twodimensional turbulent flows.
We define the "best basis" of wavelet packets as the one which,
for a given enstrophy density, condenses the L^{2 }norm
into a minimum number of nonnegligible wavelet packet coefficients. Coefficients
below a threshold are discarded, reducing the number of degrees of freedom.
We then compare the predictability of the original flow evolution with several
such reductions, varying the number of retained coefficients, either from
a Fourier basis, or from the bestbasis of wavelets packets. We show that
for a compression ratio of 1/2, we still have a deterministic predictability
using the wavelet packet bestbasis, while it is lost when using the Fourier
basis. Likewise, for compression ratios of 1/20 and 1/200 we still have
statistical predictability using the wavelet packet bestbasis, while it
is lost when using the Fourier basis. In fact, the significant wavelet packet
coefficients in the bestbasis appear to correspond to coherent structures.The
weak coefficients correspond to vorticity filaments, which are only passively
advected by the coherent structures. In conclusion, the wavelet packet bestbasis
seems to distinguish the lowdimensional dynamically active part of the
flow from the highdimensional passive components. It gives us some hope
of drastically reducing the number of degrees of freedom necessary to the
computation of twodimensional turbulent flows. 