Trajectory encounter volume – the volume of fluid that passes close to a reference fluid parcel over some time interval – has been recently introduced as a measure of mixing potential of a flow. Diffusivity is the most commonly used characteristic of turbulent diffusion. We derive the analytical relationship between the encounter volume and diffusivity under the assumption of an isotropic random walk, i.e., diffusive motion, in one and two dimensions. We apply the derived formulas to produce maps of encounter volume and the corresponding diffusivity in the Gulf Stream region of the North Atlantic based on satellite altimetry, and discuss the mixing properties of Gulf Stream rings. Advantages offered by the derived formula for estimating diffusivity from oceanographic data are discussed, as well as applications to other disciplines.

The frequency of close encounters between different objects or
organisms can be a fundamental metric in social and mechanical systems. The
chances that a person will meet a new friend or contract a new disease during
the course of a day is influenced by the number of distinct individuals that
he or she comes into close contact with. The chances that a predator will
ingest a poisonous prey, or that a mushroom hunter will mistakenly pick up a
poisonous variety, is influenced by the number of distinct species or variety
of prey or mushrooms that are encountered. In fluid systems, the exchange of
properties such as temperature, salinity or humidity between a given fluid
element and its surroundings is influenced by the number of other distinct
fluid elements that pass close by over a given time period. In all these
cases it is best to think of close encounters as providing the

In cases of property exchange within continuous media such as air or water,
it may be most meaningful to talk about a mass or volume passing within some
radius of a reference fluid element as this element moves along its
trajectory. Rypina and Pratt (2017) introduce a trajectory encounter volume,

In order to formally define the encounter volume

Schematic diagram of trajectory encounters, showing trajectories of
nine particles, with dots indicating positions of particles at three time
instances, at the release time,

Given the seemingly fundamental importance of close encounters, it is of
interest to relate metrics such as

Because the purpose of the diffusivity coefficient

This problem was framed in mathematical terms in Rypina and Pratt (2017), who outlined some initial steps towards deriving the analytical connection between encounter volume and diffusivity but did not finish the derivation. In this section, we complete the derivation.

Let us start by considering the simplest diffusive random walk process in one
or two dimensions, where particles take steps of fixed length

The single particle dispersion, i.e., the ensemble-averaged square
displacement from the particle's initial position, is

It is convenient to consider the motion in a reference frame that is moving
with the reference particle. In that reference frame, the reference particle
will always stay at the origin, while other particles will still be involved
in a random walk motion, but with a diffusivity twice that in the stationary
frame,

The problem of finding the encounter number is then reduced to counting the
number of randomly walking particles (with diffusivity

In the next section we will provide formal solutions; here we simply outline
the steps to streamline the derivation. We start by deriving the appropriate
diffusion equation for the probability density function,

Schematic diagram in 1-D

The survival probability, which quantifies the probability that a random
walker initially located at

Consider a random walker initially located at the origin, who takes, with a
probability of

Green's function for the 1-D diffusion equation without a cliff is a solution
with initial condition

Consider a random walker in 2-D, who is initially located at the origin and
who takes, with a probability of

To proceed, we need an analytical expression for Green's function of Eq. (20)
with a cliff at a distance

Carlslaw and Joeger (1939) give the answer as

The survival probability (from Eq. 7) is

The explicit connection between the encounter volume and diffusivity is thus
given by the inverse Laplace transform of the above expression (28),

small-

In the small-

large-

In the large-

For practical applications, it is sufficient to only keep the leading-order
term of the expansion, yielding a simpler connection between encounter volume
and diffusivity,

Before applying our results to the realistic oceanic flow, we numerically tested the accuracy of the derived formulas in idealized settings by numerically simulating a random walk motion in 1-D and 2-D, as described in the beginning of Sects. 2.1 and 2.2, respectively. We then computed the encounter number and encounter volume using definitions (2)–(3), and compared the result with the derived exact formulas (18) and (28)–(29) and with the asymptotic formulas (31) and (38). Note that although formulas (28)–(29) are exact, the inverse Laplace transform still needs to be evaluated numerically and thus is subject to numerical accuracy, round-off errors, etc.; these numerical errors are, however, small, and we will refer to numerical solutions of (28)–(29) as “exact,” as opposed to the asymptotic solutions (31) and (38).

Comparison between theoretical expression (red, green, blue) and
numerical estimates (black) of the encounter volume for a random walk in 1-D

The comparison between numerical simulations and theory is shown in Fig. 3.
Because the numerically simulated random walk deviates significantly from the
diffusive regime over short (< O(100

Sea surface height measurements made from altimetric satellites provide
nearly global estimates of geostrophic currents throughout the World Oceans.
These velocity fields, previously distributed by AVISO, are now available
from the Copernicus Marine and Environment Monitoring Service (CMEMS) website
(

Encounter volume

Maps showing the encounter volume for fluid parcel trajectories in the
region, and the corresponding diffusivity estimates (Fig. 4), could be useful
both for understanding and interpreting the transport properties of the flow,
as well as for benchmarking and parameterization of eddy effects in numerical
models. In our numerical simulations, trajectories were released on a regular
grid with

The encounter volume field, shown in the top left panel of Fig. 4, highlights
the overall complexity of the flow and identifies a variety of features with
different mixing potential, most notably several Gulf Stream rings with
spatially small low-

We now apply the asymptotic formula (38) to convert the encounter volume to
diffusivity. Because Eq. (38) is not invertible analytically, we converted

As expected, the diffusivity maps in the second and third rows of Fig. 4,
which resulted from converting

The performances of the exact and asymptotic diffusive formulas vary greatly
throughout the domain, with better/poorer performances in high-/low-

Comparison between numerically computed

The non-diffusive nature of the parcel motion over 90 days is because ocean
eddies have finite lengthscales and timescales, so a variety of different
transport regimes generally occur before separating parcels become
uncorrelated and transport becomes diffusive, as in a random walk. At very
short times the motion of fluid parcels is largely governed by the local
velocity shear, so the resulting transport regime is ballistic, i.e.,

A number of diffusivity estimates other than Okubo's have been made for the
Gulf Stream extension region (e.g., Zhurbas and Oh, 2004; LaCasce, 2008;
Rypina et al., 2012; Abernathey and Marshall, 2013; Klocker and Abernathey,
2014; or Cole et al., 2015). These estimates are based on surface drifters
(Zhurbas and Oh, 2004; LaCasce, 2008; Rypina et al., 2012),
satellite-observed velocity fields (Abernathey and Marshall, 2013; Klocker
and Abernathey, 2014; Rypina et al., 2012), and Argo float observations (Cole
et al., 2015), and they use either the spread of drifters or the evolution of
simulated or observed tracer fields to deduce diffusivity. The resulting
diffusivities are spatially varying and span 2 orders of magnitude, from

Because the action of the real ocean velocity field on drifters or tracers is generally not exactly diffusive, all methods simply fit the diffusive approximation to the corresponding variable of interest, such as particle dispersion, tracer variance, or, in our case, encounter volume. The analytic form of the diffusive approximation is, however, different for different variables and different flow regimes. For example, for a diffusive random walk regime, dispersion grows linearly with time, whereas the growth of the encounter volume is nonlinear, as defined by Eq. (38). This generally leads to different diffusivity estimates resulting from different methods. In other words, the diffusivity value that fits best to the observed particle dispersion at 90 days does not necessarily provide the best fit to the observed encounter volume at 90 days, and vice versa.

To illustrate this more rigorously, we consider a linear strain flow,

Of course, real oceanic flows are more complex than the simple linear strain
example. However, for flows that are in a state of chaotic advection,
exponential separation between neighboring particles will occur and the
dispersion will grow exponentially in time, as in the linear strain example.
Although we do not have a formula for the encounter volume for a chaotic
advection regime, the linear strain example suggests that the encounter
volume growth will likely be slower than exponential. Thus, for a chaotic
advection regime, the dispersion-based diffusivity could be expected to be
larger than the encounter-volume-based diffusivity. This can potentially
explain the smaller encounter-volume-based diffusivity values in Fig. 4
compared to other available estimates from the literature. Numerical
simulations (not shown) using an analytic Duffing oscillator flow, which
features chaotic advection, indeed produced smaller encounter-volume-based
diffusivity than dispersion-based diffusivity, in agreement with our
arguments above. The AVISO velocities are dominated by the mesoscales rather than
submesoscales, and the 90-day time interval is about a few mesoscale eddy
winding times; thus, this flow satisfies all the pre-requisites for the
chaotic advection to occur. Finally, the particle trajectories that we used
to produce Fig. 4 can be grouped into small clusters (we are using the
encounter radius

Dispersion-based diffusivity,

In the left panels of Fig. 4 we used the full velocity field to advect
trajectories, so both the mean and the eddies contributed to the resulting
encounter volumes and the corresponding diffusivities. But what is the
contribution of the eddy field alone to this process? To answer this
question, we have performed an additional simulation in the spirit of Rypina
et al. (2012), where we advected trajectories using the altimetric time-mean
velocity field, and then subtracted the resulting encounter volume,

With many new diagnostics being developed for characterizing mixing in fluid flows, it is important to connect them to the well-established conventional techniques. This paper is concerned with understanding the connection between the encounter volume, which quantifies the mixing potential of the flow, and diffusivity, which quantifies the intensity of the down-gradient transfer of properties. Intuitively, both quantities characterize mixing, and it is natural to expect a relationship between them, at least in some limiting sense. Here, we derived this anticipated connection for a diffusive process, and we showed how this connection can be used to produce maps of spatially varying diffusivity and to gain new insights into the mixing properties of eddies and the particle spreading regime in realistic oceanic flows.

When applied to the altimetry-based velocities in the Gulf Stream region, the encounter volume and diffusivity maps show a number of interesting physical phenomena related to transport and mixing. Of particular interest are the transport properties of the Gulf Stream rings. The materially coherent Lagrangian cores of these rings, characterized by very small diffusivity, are smaller than expected from earlier Eulerian diagnostics (Chelton et al., 2011). The periphery regions with enhanced diffusivity are, on the other hand, large, raising a question about whether the rings, on average, act to preserve coherent blobs of water properties or to speed up the mixing. The encounter volume, through the derived connection to diffusivity, might provide a way to address this question and to quantify the two effects, clarifying the role of eddies in transport and mixing.

Our encounter-volume-based diffusivity estimates are within the range of other available estimates from the literature, but are not among the highest. We provided an intuitive explanation for why the encounter-volume-based diffusivities might be smaller than the dispersion-based diffusivities, and we supported our explanation with theoretical developments based on a linear strain flow, and with numerical simulations. We note that in problems where the encounters between particles are of interest, rather than the particle spreading, the encounter-volume-based diffusivities would be more appropriate to use than the conventional dispersion-based estimates.

Reliable data-based estimates of eddy diffusivity are needed for
parameterizations in numerical models. The conventional estimation of
diffusivity from Lagrangian trajectories by calculating particle dispersion
requires large numbers of drifters or floats (LaCasce, 2008). It would be
useful to have a technique that would work with fewer instruments. The
derived connection between encounter volume and diffusivity might help in
achieving this goal. Specifically, one could imagine that if an individual
drifting buoy were equipped with an instrument that would measure its
encounter volume – the volume of fluid that came in contact with the buoy
over time

In the field of social encounters, it is becoming possible to construct large data sets by tracking cell phones, smart transit cards (Sun et al., 2013), and bank notes (Brockmann et al., 2006). As was the case for the Gulf Stream trajectories, some of the behavior appears to be diffusive and some not so. Where diffusive/random walk behavior is relevant, it may be easier to accumulate data on close encounters rather than on other metrics using, for example, autonomous vehicles and instruments that are able, through local detection capability, to count foreign objects that come within a certain range.

The velocity fields that we used in Sect. 3 are publicly available from the CMEMS website:

The authors declare that they have no conflict of interest.

This work was supported by NSF grants OCE-1558806 and EAR-1520825, and NASA grant NNX14AH29G. Edited by: Ana M. Mancho Reviewed by: two anonymous referees