From the statistical standpoint, a random field is a stochastic process with multidimensional parameters (e.g., time and position) or a more complicated parameter such as a function. The fields of primary interest have four parameters: one dimension of time and three dimensions of space. Examples of such time-varying fields include fluid velocity, pressure, water density, temperature, and salinity. Fields with only two spatial dimensions include sea surface height (sea level), wind velocity and wind stress at the surface, sea surface temperature (SST), ocean color, and sea ice. Wavenumber spectra of these fields are usually very broad, covering several decades of wavenumbers (e.g., Fu, 1983; Freilich and Chelton, 1986), and the spectral density function can be approximated by a power law. Characteristic values of exponents in the power laws indicate a fractal regime in the geometry of the fields. For instance, the sea surface elevation field, for scales related to wind-generated surface gravity waves (from a decimeter to several hundred meters), is characterized by a two-dimensional wavenumber spectrum that falls off roughly as k-7/2. This corresponds to a cascade pattern in surface topography (a hierarchy of randomly superimposed waves with decreasing amplitude and wavelength). A characteristic property of this field is its statistical self-affinity (Glazman and Weichman, 1989). The corresponding Hausdorff dimension, for an assumed Gaussian distribution, is 2.25.
The fluid velocity field, whose kinetic energy spectrum is characterized by k-5/3, exhibits a Hausdorff dimension of 2.666. A typical geometrical feature of such fields is a hierarchy of eddies. Such cascade patterns in a field’s geometry are related to the cascade nature of the energy transfer along the spectrum through nonlinear interactions among different scales of fluid motion. Other physical quantities, e.g., momentum, enstrophy (i.e., half the square of vorticity), and wave action, may also be transferred either up or down the spectrum. The spectral cascades of these quantities are not necessarily conservative: interactions between different oceanographic fields (occurring within certain limited ranges of scales—the “generation and dissipation subranges”—and resulting in energy and momentum exchange) provide energy sources or sinks in various spectral bands. For instance, at meter scales wind provides the energy input into surface gravity waves that in turn exchange momentum and energy with larger-scale motions (e.g., mesoscale eddies, Langmuir circulations, internal waves). Mesoscale oceanic eddies are caused by the barotropic instability of basin-scale currents. Seasonal heating and cooling of the ocean surface causes convection and vertical mixing, while differential (across the oceanic basins) heating, evaporation, precipitation, and ice melting cause density-driven currents. Ocean circulation on basin scales is caused by large-scale curl of the wind stress. This multiplicity of the energy sources and sinks and the interactions between different scales and individual components of ocean dynamics are responsible for the extreme complexity of patterns of ocean circulation, sea surface temperature, sea level, and so on as observed both in satellite images and in highly complicated trajectories of free-drifting floats. Apparently, the interaction of motions with different scales implies statistical dependence between corresponding Fourier components or between corresponding eigenvectors in the empirical orthogonal functions (EOF) series (Karhunen-Loeve expansion; see, e.g., Lorenz, 1956; Davis, 1976; Preisendorfer, 1988). Identifying and accounting for such correlations in statistical models are important problems of oceanographic data analysis.