(3.3)

where is the probability density of a random vector ξ_{1},...,ξ_{p}, 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

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.