In the last two decades the use of Lagrangian (i.e., current-following) devices has become very popular in oceanography (for a review, see Davis, 1991a). Drifting buoys have been developed that can follow the ocean currents with good accuracy, moving either at the surface of the ocean or in the interior on surfaces of equal pressure or density. These drifting buoys are tracked acoustically or via satellite for extensive time after deployment (up to a year or more). They report their position at discrete times, with an interval that can vary from hours to days depending on the specific purpose of the measurements made. From these positions, an estimate of the horizontal velocity along the buoy trajectory can be made. In addition to their position, drifting buoys are often equipped to measure other physical quantities, such as temperature or pressure.

Data from drifting buoys are used both for understanding the dynamics of ocean circulation (e.g., Price and Rossby, 1982; Bower and Rossby, 1989) and for describing its statistical properties (e.g., Kraus and Boning, 1987; Figueroa and Olson, 1989). This chapter focuses on this second aspect. An appropriate statistical description of ocean circulation includes two main parts. One is the statistics of the velocity field, and the other is the statistical description of the transport mechanisms. The ocean plays a fundamental role in the transport of such quantities as heat, salinity, or chemical substances (both natural and anthropogenic) that are fundamental for environmental and climatic studies. Before going into the details of how the Lagrangian data are actually utilized to obtain the statistical information, it is useful to point out that there is a direct connection between Lagrangian trajectories and transport properties in a flow (e.g., Davis, 1983). This can be seen by considering the equation for the evolution of the concentration of a substance released and transported in an incompressible fluid: (∇, * u*)=0 (see, e.g., Pedlosky 1987). Assuming that the substance concentration is a scalar function

*∂*_{t}*c+*(*u,*∇)*c*=0, *c*(0,*x*)=*c*_{0}. (3.1)

Note that equation (3.1) is the same as equation (1.4) of Chapter 1, except that the molecular diffusivity is neglected because here the concern is large-scale flows, and for simplicity no sources or sinks are considered. The solution of equation (3.1) by the method of characteristics takes the form

*c(t,r)*=*c*_{0}(*X*^{−1}(*t,r*)), (3.2)

where *X*^{−1} is the inverse of the function *r*→*X(t, r)* that represents the position reached at time *t* by a particle that was at *r* at *t*=0.

From (3.2) one can calculate statistical moments of the concentration *c(t,r)* by the formula

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Statistics and Physical Oceanography
3
LAGRANGIAN AND EULERIAN DATA AND MODELS
In the last two decades the use of Lagrangian (i.e., current-following) devices has become very popular in oceanography (for a review, see Davis, 1991a). Drifting buoys have been developed that can follow the ocean currents with good accuracy, moving either at the surface of the ocean or in the interior on surfaces of equal pressure or density. These drifting buoys are tracked acoustically or via satellite for extensive time after deployment (up to a year or more). They report their position at discrete times, with an interval that can vary from hours to days depending on the specific purpose of the measurements made. From these positions, an estimate of the horizontal velocity along the buoy trajectory can be made. In addition to their position, drifting buoys are often equipped to measure other physical quantities, such as temperature or pressure.
Data from drifting buoys are used both for understanding the dynamics of ocean circulation (e.g., Price and Rossby, 1982; Bower and Rossby, 1989) and for describing its statistical properties (e.g., Kraus and Boning, 1987; Figueroa and Olson, 1989). This chapter focuses on this second aspect. An appropriate statistical description of ocean circulation includes two main parts. One is the statistics of the velocity field, and the other is the statistical description of the transport mechanisms. The ocean plays a fundamental role in the transport of such quantities as heat, salinity, or chemical substances (both natural and anthropogenic) that are fundamental for environmental and climatic studies. Before going into the details of how the Lagrangian data are actually utilized to obtain the statistical information, it is useful to point out that there is a direct connection between Lagrangian trajectories and transport properties in a flow (e.g., Davis, 1983). This can be seen by considering the equation for the evolution of the concentration of a substance released and transported in an incompressible fluid: (∇, u)=0 (see, e.g., Pedlosky 1987). Assuming that the substance concentration is a scalar function c(t, r), and that the substance does not interact with the flow while it is advected (i.e., it is a passive scalar, or “tracer”), the equation is
∂tc+(u,∇)c=0, c(0,x)=c0. (3.1)
Note that equation (3.1) is the same as equation (1.4) of Chapter 1, except that the molecular diffusivity is neglected because here the concern is large-scale flows, and for simplicity no sources or sinks are considered. The solution of equation (3.1) by the method of characteristics takes the form
c(t,r)=c0(X−1(t,r)), (3.2)
where X−1 is the inverse of the function r→X(t, r) that represents the position reached at time t by a particle that was at r at t=0.
From (3.2) one can calculate statistical moments of the concentration c(t,r) by the formula

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Statistics and Physical Oceanography
(3.3)
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.

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Statistics and Physical Oceanography
The theoretical problem of determining Eulerian statistics from Lagrangian statistics is quite difficult, and it is still open (e.g., Li and Meroney, 1985; Babiano et al., 1985). Oceanographers take the simplest possible approach. They consider a set of measurements taken in a certain geographical region and assume that the region can be divided into smaller subregions (boxes) characterized by a space scale L, where the statistics are approximately homogeneous and stationary. All the data present in each box at all times can then be considered as representative of the same spatial point, and can be used to compute averages of the quantities of interest. In this way, the Eulerian statistics are computed from a combination of space and time averaging. The important question is, What happens when the hypotheses of homogeneity and stationarity inside the boxes are relaxed, as is expected to occur in a realistic situation? An extensive analysis regarding this problem has recently been done by Davis (1991b) in the context of the elaborated advection and diffusion equation. The following paragraph briefly summarizes some important points.
Stationarity can be relaxed fairly realistically provided the ocean is characterized by slowly varying fluctuations so that time averages, even though not constant, are representative of the particular ocean climate present during the measurements. Inhomogeneity could in principle be reduced inside each box by increasing the resolution, i.e., by decreasing L, the scale of the boxes. In practice, though, the uncertainty in the estimate of the statistical quantities also depends on L, so that a trade-off must be found between resolution and accuracy. The scale L must be large enough to give a reasonable uncertainty and small enough so that the statistical quantities computed in the box are meaningful.
It is important to note that biases can occur in estimating the statistical quantities as a consequence of both inhomogeneity in the sampling (array bias) and in the turbulent velocity (diffusion bias). This last type of bias reflects the observed tendency of drifting buoys deployed at a point to migrate toward regions of high turbulent energy. As shown by Davis (1991b), the size of these biases can be identified for mean velocity, but it appears to be much harder to identify for diffusivity. The use of other model equations for transport (or equivalently for particle motion) may help in identifying this bias or possibly suggest better estimators for the quantities of interest.
Finally, in some special cases the inhomogeneity of the statistical quantities can likely be solved explicitly. This can happen when general information is available on the spatial structure of the quantities, so that they can be approximated by space-functions dependent on a discrete number of parameters. An approach of this type has thus far only been applied to simple linear flows (e.g., Davis, 1985), but it is likely to also be useful for more complex flows, such as strong vortices or meandering currents, which play an important role in oceanography. The technique consists of estimating the parameters by using the data in conjunction with a model equation, such as some form of the advection and diffusion equation or a stochastic model for particle motion. The use of a stochastic model also provides a natural and straightforward way to filter the data.

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Statistics and Physical Oceanography
PROSPECTIVE DIRECTIONS FOR RESEARCH
As is apparent from the preceding discussion, a number of key problems (e.g., the “closure” problem, determining Eulerian statistics from Lagrangian statistics, dealing with array bias and diffusion bias) are still open that relate to the use of Lagrangian data in the description of the ocean circulation. They suggest a variety of directions for statistical research, ranging from statistical analysis for oceanographic data to probabilistic modeling for processes in the ocean. Some specific considerations are the following:
Statistical methods for irregular and sparse observations, with emphasis on estimation of spectral and correlation characteristics (see Chapters 6 and 8);
Filtering and parameter estimation for random fields governed by randomly perturbed ordinary and partial differential equations, with emphasis on numerical methods for nonlinear filtering, spectral methods, and others;
The study of single-particle statistics in inhomogeneous and nonstationary turbulent flows;
The study of multiparticle statistics;
The Lagrangian approach to turbulence;
The derivation of closed-form equations for moments of passive scalars; and
The exploration of the time evolution of distributions of passive scalars, with emphasis on intermittence (“patchiness”).