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Statistics and Physical Oceanography
techniques. Examples of weak turbulence include two-dimensional and geostrophic turbulence and surface gravity waves. Weak turbulence theory in its present form (Zakharov et al., 1992) permits derivation of kinetic equations describing energy exchanges (and exchanges of other quantities) among Fourier components, as well as derivation of higher-order spectra (bispectra, etc.) representing Fourier transforms of various statistical moments. Initially, this theory was developed for surface gravity and capillary waves (Hasselmann, 1962; Zakharov, 1984). However, statistical phenomena in waves (e.g., the existence of Kolmogorov-type spectra, the intermittency of breaking waves, and so on) have analogies in other oceanographic fields. The elegant Hamiltonian formulation of nonlinear wave dynamics (Zakharov, 1984; Zakharov et al., 1992) is a powerful tool for studies of fundamental statistical properties of turbulent fields.
To better characterize the scope of statistical issues that the weak turbulence theory or alternative statistical approaches could address, a brief review of some issues related to wind-generated surface gravity waves is in order. Until recently, statistical studies of field geometry were dominated by the work on Gaussian fields. Longuet-Higgins (1957, 1962, 1984) studied a large variety of geometrical properties of such fields with application to sea surface waves. Among other problems, he considered statistics of specular points (the points at which the gradient of the field is either zero or is specified depending on a viewing angle) and of the wave envelope, which play an important role in wave dynamics and analysis of sun glitter and radar backscatter from a wind-disturbed sea surface. A rigorous mathematical analysis of envelope statistics, high-level excursions, field maxima, and other geometrical properties of random two- and multi-dimensional Gaussian fields is presented by Adler (1981). Some of these results have been successfully employed in sea wave studies. Specifically, the theory of level crossings by two- and three-dimensional Gaussian- and Rayleigh-distributed fields was employed to estimate statistics of whitecaps (breaking waves) and of wave trains (Glazman, 1986; Glazman and Weichman, 1989; Glazman, 1991). Observations indicate that whitecaps occur in clusters. Hence, the use of a simple Poisson distribution (Glazman, 1991) for whitecap occurrence, which is known from the theory of high-level excursions by the (Gaussian) wave slope field, may be insufficient. The statistical theory of cluster point processes may be of great help here.
Linear methods are intrinsic for Gaussian stationary processes, and Fourier analysis is a natural tool to use in the resolution of stationary random fields. These yield a global resolution. However, in many situations, a resolution that is better adapted to local behavior would be more appropriate and interesting. This could be local behavior in time or local spatial behavior. One attempt in this direction makes use of wavelet transforms, which are in effect local filters of the field (Farge, 1992). Such a method amounts to a linear analysis of the field, although it could presumably be adapted to types of nonlinearity.
In the last few years, significant research effort in probability and statistics has been directed toward the development of models of non-Gaussian and time-varying random fields. Examples include stable fields; functionals of Gaussian, stable, and other fields represented via multiple integrals; density processes and measure-valued diffusions; and fields described by nonlinear stochastic differential equations. Applications of this research to oceanographic phenomena would be of interest to oceanographers since the fields they study are frequently non-Gaussian and time-varying random fields.