Multiresolution methods, such as the wavelet decompositions, are increasingly used in physical applications where multiscale phenomena occur. We present in this paper two applications illustrating two different aspects of the wavelet theory.
In the first part of this paper, we recall the bases of the wavelets theory. We describe how to use the continuous wavelet decomposition for analyzing multifractal patterns. We also summarize some results about orthogonal wavelets and wavelet packets decompositions.
In the second part, we show that the wavelet packet filtering can be successfully used for analyzing two-dimensional turbulent flows. This technique allows the separation of two structures: the solid rotation part of the vortices and the remaining mainly composed of vorticity filaments. These two structures are multiscale and cannot be obtained through usual filtering methods like Fourier decompositions. The first structures are responsible for the inverse transfer of energy while the second ones are responsible for the forward transfer of enstrophy. This decomposition is performed on numerical simulations of a two dimensional channel in which an array of cylinders perturb the flow.
In the third part, we use a wavelet-based multifractal approach to describe qualitatively and quantitatively the complex temporal patterns of atmospheric data. Time series of geopotential height are used in this study. The results obtained for the stratosphere and the troposphere show that the series display two different multifractal behaviors. For large time scales (several years), the main Hölder exponent for the stratosphere and the troposphere data are negative indicating the absence of correlation. For short time scales (from few days to one year), the stratosphere series present some correlations with Hölder exponents larger than 0.5, whereas the troposphere data are much less correlated.