Abstract

Two fundamental properties of turbulence are intermittency and non-linearity. They imply that the standard Fourier spectral techniques are inadequate for its analysis. Spectral analysis based on wavelets provides a means to handle intermittency. New tools are required to handle non-linearity.

In this chapter, we redesign spectral analysis in terms of wavelet methods, paying particular attention to statistical stability, error estimates and nonlinearity. The application to both computer simulations and measurements carried out in fusion plasmas provide some illustrative examples.

Introduction

Although the phenomenon of turbulence is only partially understood, there seems to be consensus on several aspects. First, that intermittency is a basic property of turbulence. This means that the characteristics of the turbulence (spectral distribution, amplitude etc.) vary on a short time scale. Analysis techniques that rely on the accumulation of data over time scales larger than this characteristic time scale will then average out much of the dynamics and obliterate relevant information (as may occur with Fourier analyses). Wavelet analysis provides an interesting starting point for redesigning the standard analysis techniques in order to tackle this problem. In this chapter we shall redefine some basic Fourier analysis techniques in terms of wavelets, such as cross coherence. We shall emphasize the need for statistical stability and provide noise level estimates. Finally, we provide some examples of these techniques.

Second, it is generally accepted that turbulence only arises in non-linear systems. Therefore, to understand the nature of turbulence, it is essential to employ analysis tools that are capable of handling this non-linearity.