wavelet transforms and their applications to turbulence - Wavelets ...

Annual Reviewswww.annualreviews.org/aronlineWAVELET TRANSFORMS 397tions. Are there some elementary coherent structures? Do their mutualinteractions have a universal character? Is it possible to compute the flowevolution with a reduced number of degrees of freedom relative to the verylarge number of Fourier components otherwise necessary? This reductioncould correspond to a projection of Navier-Stokes equations on thosecoherent structures or on some related functional bases well localized inphysical space.Being very uncomfortable with these two separate descriptions of turbulence,I was immediately enthusiastic when, in 1984, A. Grossmann toldme about the wavelet transform theory he was developing from Morlet’soriginal ideas. His theory had the promise of a unified approach whichcould reconcile these two descriptions and allow us to analyze a turbulentflow in terms of both space and scale at once, up to the limits of theuncertainty principle. Another reason, rather naive, for the immediateappeal of wavelets was the fact that the Morlet wavelet evoked to me theshape of Tennekes and Lumley’s eddy (Tennekes & Lumley 1972) proposedto model turbulence, and of some coherent structures whose existencehad been conjectured by Ruelle (D. Ruelle, personal communication,1983) and then observed by Basdevant in his numerical simulations oftwo-dimensional flows (Basdevant & Couder 1986). Indeed, it seems muchbetter to decompose a turbulent field into such localized oscillations offinite energy as wavelets, rather than into space-filling trigonometric functionswhich do not belong to the L2(II n) functional space and therefore arenot of finite energy.In the context of turbulence, the wavelet transform may yield someelegant decompositions of turbulent flows (Section 5.1). The continuouswavelet transform offers a continuous and redundant unfolding in termsof both space and scale, which may enable us to track the dynamics ofcoherent structures and measure their contributions to the energy spectrum(Section 5.2). The discrete wavelet transform allows an orthonormal projectionon a minimal number of independent modes which might be usedto compute or model the turbulent flow dynamics in a better way thanwith Fourier modes (Section 5.3).2. WAVELET TRANSFORM PRINCIPLES2.1 HistoryThe wavelet transform originated in 1980 with Morlet, a French researchscientist working on seismic data analysis (Morlet 1981, 1983; Goupillaudet al 1984), who then collaborated with Grossmann, a theoretical physicistfrom the CNRS in Marseille-Luminy. They developed the geometricalformalism of the continuous wavelet transform (Grossmann et al 1985,