Orthonormal Wavelet Expansion and Its Application to Turbulence

Abstract

Orthonormal wavelet expansion is applied to experimental data of turbulence. A direct relation is found between the wavelet spectrum and the Fourier spectrum. The orthonormal wavelet analysis with conditional sampling is applied to data of wind turbulence, yielding Kolmogorov's spectrum and the dissipation correlation with the intermittency exponent ,11'0.2. Fourier transform method is a fundamental and indispensable tool in data analysis since it enables us to decompose data into components with different scales. Many fundamental properties of physical systems have been described in terms of Fourier spectrum, that is, the amplitude of Fourier coefficients. However, since Fourier spectrum totally ignores the phase of each Fourier coefficient, it lacks infor­ mation about positions of local events which underlie the characteristics of the spectrum. The Fourier spectrum analysis therefore encounters difficulty in ana)yzing data in temporal or spatial intervals which include different kinds of local events. The method of scale analysis applicable even to such complicated situations should enable us to identify the origin of characteristics of the spectrum with local events occurring in physical space. This requirement would be satisfied at least partially by an expansion in terms of basis functions which are local both in physical and Fourier space, although the locality is limited in its extent by the uncertainty principle. In this paper, we adopt an expansion method' in terms of orthonormal wavelets (discrete wavelet analysis) as one of such types of expansion method. The orthonormal wavelet expansion is a discrete version of continuous wavelet analysis.!) The latter is an integral transform method with kernel functions obtained by translating and dilating a l~calized function (analyzing wavelet). The continuous wavelet transform of a square integrable function is an isometric transform between a Hilbert space (V space on Rn) and V space on a locally compact topological group (a group of translation and dilation) with its Haar measure: 2H ) The continuous wavelet is a useful tool especially for studying a singularity or a fractal structure of a given function. 5H ) In particular, the energy cascade process in fully-developed turbulence has been captured remarkably in such an analysis. 7 ) However, it is not very advantageous if one is interested in the energetic aspect because the kernel functions are not mutually orthogonal and no physically immediate meaning can be associated with the expansion coefficients.