Improved predictability of two-dimensional turbulent flows using wavelet packet compression
	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 Paris-Dauphine, 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 two-dimensional turbulent flows. We define the "best basis" of wavelet packets as the one which, for a given enstrophy density, condenses the L2 norm into a minimum number of non-negligible 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 best-basis of wavelets packets. We show that for a compression ratio of 1/2, we still have a deterministic predictability using the wavelet packet best-basis, 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 best-basis, while it is lost when using the Fourier basis. In fact, the significant wavelet packet coefficients in the best-basis 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 best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components. It gives us some hope of drastically reducing the number of degrees of freedom necessary to the computation of two-dimensional turbulent flows.