where is the probability density of a random vector ξ1,...,ξp, representing the probability distribution of Lagrangian trajectories in the fluid.
In oceanography, most of the work performed to date has focused on the first moment of c (i.e., on the mean concentration 〈c〉) and on the related probability density function for a single particle . A few studies have considered the statistics of particle pairs (e.g., Bennett, 1984; Davis, 1985). Even in the simplest case of a single particle, though, the data are not sufficient to compute , so that (3.3) cannot be used directly. Information on 〈c〉 can, in principle, be retrieved by combining the data with the equation for 〈c〉 obtained by averaging (3.1). The trouble with this approach is that the resulting equation for 〈c〉 involves terms such as 〈u ∇c〉; the equation for these terms in turn involves still higher order statistical terms, and so on in an unending hierarchy. This is the “closure” problem, one of the central problems in fluid dynamics. In practice, what is usually done is to “close” the equations for 〈c〉 at a chosen level using some kind of assumptions. The issue then becomes identifying the closed equations’ appropriate form for the specific context under examination (e.g., see Molchanov and Piterbarg, 1992). As discussed in Chapter 1, the simplest form of closure is given by the advection and diffusion equation (1.4) where molecular diffusivity is replaced with turbulent (“eddy”) diffusivity. An estimate of diffusivity can be obtained from the data, as a function of the velocity autocorrelation measured by buoys (e.g., Kraus and Boning, 1987). This form of closure is, strictly speaking, valid only if the flow is homogeneous in space and stationary in time, and if the time scales considered are longer than the time scales of the turbulence. Other more general and more widely valid equations have also been used in the literature. Examples are the elaborated form of the advection and diffusion equation proposed by Davis (1987) and stochastic models used to describe the motion of single particles (Thomson, 1986; Dutkiewicz et al., 1992).
One of the difficulties in using data from drifting buoys is that, whereas the data are inherently Lagrangian, the information oceanographers are interested in is often Eulerian (i.e., associated with a fixed point). Typically, oceanographers seek maps of simple statistics of the velocity, such as the mean flow and the variance, and of some turbulent transport quantities, such as the diffusivity. The knowledge of diffusivity as a function of space is of great importance for a number or reasons. First, it provides a direct picture of the nature of ocean turbulence, which is still not well understood (as discussed in Chapter 1). In particular, comparing diffusivity maps and maps of mean flow or velocity variance provides a way to test simple theories of turbulence, and eventually indicates how to improve them. Secondly, one must know diffusivity as a function of space, because it is an input of key importance for numerical models that simulate oceanic processes using equations (1.1)-(1.4) in Chapter 1.