the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Connection between encounter volume and diffusivity in geophysical flows

### Stefan G. Llewellyn Smith

### Larry J. Pratt

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
*potential*, if not necessarily the act, of transmission of germs,
toxins, heat, salinity, etc.

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,
*V*, the volume of fluid that comes in contact with the reference fluid
parcel over a finite time interval. The increase in *V* over time is one
measure of the mixing potential of the element, “mixing” being the
irreversible exchange of properties between different water parcels. Thus,
fluid parcels that have large encounter volumes as they move through the flow
field have large mixing potential, i.e., an opportunity to exchange
properties with other fluid parcels, and vice versa.

In order to formally define the encounter volume *V*, Rypina and Pratt (2017)
subdivide the entire fluid into infinitesimal fluid elements with volumes
d*V*_{i}, and define the encounter volume for each fluid element to be the
total volume of fluid that passes within a radius *R* of it over a finite time
interval ${t}_{\mathrm{0}}<t<\phantom{\rule{0.125em}{0ex}}{t}_{\mathrm{0}}+T$, i.e.,

In practice, for dense uniform grids of trajectories, ${\mathit{x}}_{\mathit{k}}\left({\mathit{x}}_{\mathrm{0}k};{t}_{\mathrm{0}},T\right),k=\mathrm{1},\mathrm{\dots},K$, where *t*_{0} is
the starting time, *T* is the trajectory integration time, and
*x*_{0k} is the trajectory initial position satisfying $\mathit{x}\left({\mathit{x}}_{\mathrm{0}};{t}_{\mathrm{0}},T=\mathrm{0}\right)={\mathit{x}}_{\mathrm{0}}$, both the limit and the subscript
in the above definition Eq. (1) can be dropped. In this case, the encounter
volume can be approximated by

where the *encounter number*,

is the number of trajectories that come within a radius *R* of the reference
trajectory, ${\mathit{x}}_{\mathrm{ref}}\left({\mathit{x}}_{\mathrm{0}\mathrm{ref}};{t}_{\mathrm{0}},T\right)$, over a time ${t}_{\mathrm{0}}<t<{t}_{\mathrm{0}}+T$. Here the indicator function *I*
is 1 if true and 0 if false, and *K* is the total number of particles.
As in Rypina and Pratt (2017), we define encounter volume based on the number
of encounters with different trajectories, not the total number of encounter
events (see the schematic diagram of trajectory encounters in Fig. 1). Rypina
and Pratt (2017) discuss how the encounter volume can be used to identify
Lagrangian coherent structures (LCS) such as stable and unstable manifolds of
hyperbolic trajectories and regions foliated by the KAM-like tori surrounding
elliptic trajectories in realistic geophysical flows. A detailed comparison
between the encounter volume method and some other Lagrangian methods of LCS
identification, as well as the dependences on parameters, ${t}_{\mathrm{0}},T,R$, and
on grid spacing (or on the number of trajectories, *K*), and the relative
advantages of different techniques, was given in Rypina and Pratt (2017). The
interested reader is referred to that earlier paper for details. The current
paper is concerned only with the question of finding the connection between
the encounter volume and diffusivity, rather than identifying LCS.

Given the seemingly fundamental importance of close encounters, it is of
interest to relate metrics such as *V* to other bulk measures of
interactions within the system. For example, in some cases it may be more
feasible to count encounters rather than to measure interactions or property
exchanges directly, whereas in other cases the number of encounters might be
most pertinent to the process in question but difficult to measure directly.
In many applications, including ocean turbulence, the most commonly used
metric of mixing is the eddy diffusivity, *κ*, a quantity that relates
transport of fluid elements by turbulent eddies to diffusion (LaCasce, 2008;
Vallis, 2006; Rypina et al., 2012; Kamenkovich et al., 2015). The underlying
assumption is that the eddy field drives downgradient tracer transfer,
similar to molecular diffusion but with a different (larger) diffusion
coefficient. This diffusive parameterization of eddies has been implemented
in many non-eddy-resolving oceanic numerical models. The diffusivity can be
measured by a variety of means, including dye release (Ledwell et al., 2000;
Sundermeyer and Ledwell, 2001; Rypina et al., 2016), surface drifter
dispersion (Okubo, 1971; Davis, 1991; LaCasce, 2008; La Casce et al., 2014;
Rypina et al., 2012, 2016), and property budgets (Munk, 1966). In numerical
models *κ* is often assumed constant in both time and space, or
related in some simplified manner to the large-scale flow properties
(Visbeck et al., 1997).

Because the purpose of the diffusivity coefficient *κ* is to quantify
the intensity of the eddy-induced tracer transfer, i.e., the intensity of
mixing, it is tempting to relate it to the encounter volume, *V*, which
quantifies the mixing potential of a flow and thus is closely related to
tracer mixing. Such an analytical connection between the encounter volume
and diffusivity could potentially also be useful for the parameterizations
of eddy effects in numerical models. The main goal of this paper is to
develop a relationship between *V* and *κ* in one and two dimensions.
Specifically, we seek an analytical expression for the encounter volume,
*V*, i.e., the volume of fluid that passed within radius *R* from a
reference particle over time, as a function of *κ*. The relationship
is not as straightforward as one might first imagine, but can nevertheless
be written down straightforwardly in the long-time limit. This is opportune,
since the concept of eddy diffusivity is most relevant in the long-time
limit.

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.

## 2.1 Main idea for 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 Δ*x* in
random directions along the *x*-axis in 1-D or along both the *x*- and
*y*-axes in 2-D, respectively, at fixed time intervals Δ*t*.

The single particle dispersion, i.e., the ensemble-averaged square
displacement from the particle's initial position, is ${D}_{\text{1-D}}=\langle {\left(x-{x}_{\mathrm{0}}\right)}^{\mathrm{2}}\rangle $ and ${D}_{\text{2-D}}=\langle {\left(x-{x}_{\mathrm{0}}\right)}^{\mathrm{2}}+{\left(y-{y}_{\mathrm{0}}\right)}^{\mathrm{2}}\rangle $ in 1-D or 2-D, respectively. For a diffusive process, the
dispersion grows linearly with time, and the constant proportionality
coefficient is related to diffusivity. Specifically, *D*_{1-D}=2*κ*_{1-D}*t*
with ${\mathit{\kappa}}_{\text{1-D}}=\mathrm{\Delta}{x}^{\mathrm{2}}/\left(\mathrm{2}\mathrm{\Delta}t\right)$, and
*D*_{2-D}=4*K*_{2-D}*t* with ${\mathit{\kappa}}_{\text{2-D}}=\mathrm{\Delta}{x}^{\mathrm{2}}/\phantom{\rule{0.125em}{0ex}}\left(\mathrm{4}\mathrm{\Delta}t\right)$.

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, *κ*^{moving}=2*κ*^{stationary} (Rypina and Pratt,
2017).

The problem of finding the encounter number is then reduced to counting the
number of randomly walking particles (with diffusivity
*κ*^{moving}) that come within radius *R* of the origin in the
moving frame. This is related to a classic problem in statistics – the
problem of a random walker reaching an absorbing boundary, usually referred
to as “a cliff” (because once a walker reaches the absorbing boundary, it
falls off the cliff), over a time interval from 0 to *t*.

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, *p*(** x**,

*t*), of random walkers in 1-D or 2-D:

We place a cliff, *x*_{c}, at the perimeter of the encounter
sphere, i.e., at a distance *R* from the origin, and impose an absorbing
boundary condition at a cliff,

which removes (or “absorbs”) particles that have reached the cliff (see
Fig. 2 for a schematic diagram). We then consider a random walker that is
initially located at a point *x*_{0} outside the cliff at *t*=0,
i.e.,

and we write an analytical solution for the probability density function satisfying Eqs. (4)–(5),

that quantifies the probability of finding a random walker initially located
at *x*_{0} at any location ** x** outside of the cliff at a later
time

*t*>0. In mathematical terms,

*G*is Green's function of the diffusion equation.

The survival probability, which quantifies the probability that a random
walker initially located at *x*_{0} at *t*=0 has “survived” over
time *t* without falling off the cliff, is

where the integral is taken over all locations outside of the cliff. The encounter, or “non-survival”, probability can then be written as the conjugate quantity,

which quantifies the probability that a random walker initially located at
*x*_{0} at *t*=0 has reached, or fallen off, the cliff over time
*t*. This allows one to write the encounter volume, i.e., the volume
occupied by particles that were initially located outside of the cliff and
that have reached the cliff by time *t*, as

where the integral is taken over all initial positions outside of the cliff.

## 2.2 1-D case

Consider a random walker initially located at the origin, who takes, with a
probability of 1∕2, a fixed step Δ*x* to the right or to the left
along the *x*-axis after each time interval Δ*t*. Then the
probability of finding a walker at a location *x*=*n*Δ*x* after (*m*+1)
steps is

Using a Taylor series expansion in Δ*x* and Δ*t*, we can write
down the finite-difference approximation to the above expression as

yielding a diffusion equation

with diffusivity coefficient $\mathit{\kappa}=\frac{\mathrm{\Delta}{x}^{\mathrm{2}}}{\mathrm{2}\mathrm{\Delta}t}$.

Green's function for the 1-D diffusion equation without a cliff is a solution with initial condition $p(x,t=\mathrm{0};{x}_{\mathrm{0}})=\mathit{\delta}(x-{x}_{\mathrm{0}})$ in an unbounded domain. It takes the form

Green's function with the cliff (see Fig. 2 for a schematic diagram), i.e., a solution to the initial-value problem with $p(x,t=\mathrm{0};{x}_{\mathrm{0}})=\mathit{\delta}(x-{x}_{\mathrm{0}})$ in a semi-infinite domain, $x\in \left[-\mathrm{\infty};{x}_{\mathrm{c}}\right]$, with an absorbing boundary condition at a cliff, $p\left(x={x}_{\mathrm{c}},t;{x}_{\mathrm{0}}\right)=\mathrm{0}$, can be constructed by the method of images from two unbounded Green's functions as

It follows from Eqs. (7) to (9) that the survival or non-encounter probability is

the encounter probability is

and the encounter volume is

The above formula accounts for the randomly walking particles that have
reached the cliff from the left over time *t*. By symmetry, if the cliff was
located to the right of the origin, the same number of particles would be
reaching the cliff from the right, so the total encounter volume is

Note that formula (18) gives the encounter volume, i.e., the volume of fluid
coming within radius *R* from the origin, in a reference frame moving with
the reference particle, so the corresponding diffusivity on the right-hand
side of Eq. (18) is *κ*^{moving}=2*κ*^{stationary}.

## 2.3 2-D case

Consider a random walker in 2-D, who is initially located at the origin and
who takes, with a probability of 1∕4, a fixed step of length Δ*x* to
the right, left, up or down after each time interval Δ*t*. Then the
probability of finding a walker at a location $x=n\mathrm{\Delta}x,y=m\mathrm{\Delta}y$ at
time $t=(l+\mathrm{1})\mathrm{\Delta}t$ is

Using a Taylor series expansion in Δ*x*, Δ*y* and Δ*t*,
the finite-difference approximation leads to a diffusion equation

with diffusivity coefficient $\mathit{\kappa}=\frac{\mathrm{\Delta}{x}^{\mathrm{2}}}{\mathrm{4}\mathrm{\Delta}t}$.

To proceed, we need an analytical expression for Green's function of Eq. (20)
with a cliff at a distance *R* from the origin, i.e., a solution to the
initial-value problem with $p(\mathit{x},t=\mathrm{0};{\mathit{x}}_{\mathrm{0}})=\mathit{\delta}(\mathit{x}-{\mathit{x}}_{\mathrm{0}})$ for the above 2-D diffusion equation on a
semi-infinite plane ($r\ge R,\phantom{\rule{0.125em}{0ex}}\mathrm{0}<\mathit{\theta}\le \mathrm{2}\mathit{\pi}),$ bounded internally by
an absorbing boundary (a cliff) located at *r*=*R*, so that $p\left(r=R,\mathit{\theta},t;{\mathit{x}}_{\mathrm{0}}\right)=\mathrm{0}$ (see Fig. 2 right for a schematic
diagram). Here (*r*,*θ*) are polar coordinates.

Carlslaw and Joeger (1939) give the answer as

where ${r}_{\mathrm{0}}(\ge R),{\mathit{\theta}}_{\mathrm{0}}$ denote the source location, and

are the inverse Laplace transforms of

with $q=\sqrt{\frac{s}{\mathit{\kappa}}}$.

The survival probability (from Eq. 7) is

Next, we take the Laplace transform of the survival probability and write it
in terms of a Laplace variable *s* as

Using $\int r{I}_{\mathrm{0}}\left(r\right)\mathrm{d}r=r{I}_{\mathrm{1}}\left(r\right)$ and $\int r{K}_{\mathrm{0}}\left(r\right)\mathrm{d}r=-r{K}_{\mathrm{1}}\left(r\right)$, and ${lim}_{x\to \mathrm{\infty}}x{K}_{\mathrm{1}}\left(x\right)=\mathrm{0}$ we find

But ${I}_{\mathrm{1}}\left(x\right){K}_{\mathrm{0}}\left(x\right)+{I}_{\mathrm{0}}\left(x\right){K}_{\mathrm{1}}\left(x\right)=\frac{\mathrm{1}}{x}$, so

From Eq. (8), the encounter probability ${P}_{\mathrm{en}}\left(t;{\mathit{x}}_{\mathrm{0}},R\right)=\mathrm{1}-S\left(t;{\mathit{x}}_{\mathrm{0}},R\right)$, and from Eq. (9) the encounter volume is

We now take the Laplace transform of the encounter number to get

where we used ${\int}_{\mathrm{0}}^{\mathrm{\infty}}{e}^{-\mathrm{st}}\mathrm{d}t=\frac{\mathrm{1}}{s}$, $\int {K}_{\mathrm{0}}\left(z\right)z\phantom{\rule{0.125em}{0ex}}\mathrm{d}z=-z{K}_{\mathrm{1}}\left(z\right)$, and ${lim}_{z\to \mathrm{\infty}}{K}_{\mathrm{1}}\left(z\right)=$0.

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

Although numerically straightforward to evaluate, a non-integral analytic
form does not exist for this inverse Laplace transform. To better understand
the connection between *V* and *κ* and the growth of *V* with time, we
next look at the asymptotic limits of small and large time. The small-*t*
limit is transparent, while the long-*t* limit is more involved.

- a.
small-

*t*asymptoticsIn the small-

*t*limit, the corresponding Laplace coordinate*s*is large, giving$$\begin{array}{}\text{(30)}& \stackrel{\mathrm{\u203e}}{V}(s;R)\phantom{\rule{0.125em}{0ex}}\sim \mathrm{2}\mathit{\pi}R{\mathit{\kappa}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{{s}^{\mathrm{3}/\mathrm{2}}}}\end{array}$$because ${lim}_{z\to \mathrm{\infty}}\frac{{K}_{\mathrm{1}}\left(z\right)}{{K}_{\mathrm{0}}\left(z\right)}=\mathrm{1}$. Noting that ${L}^{-\mathrm{1}}\left\{{s}^{-\frac{\mathrm{3}}{\mathrm{2}}}\right\}=\frac{\mathrm{2}\sqrt{t}}{\sqrt{\mathit{\pi}}}$, the inverse Laplace transform of the above gives the following simple expression connecting the encounter volume and diffusivity at short times:

$$\begin{array}{}\text{(31)}& V(t;R)\stackrel{t\to \mathrm{0}}{\mathit{\u27f6}}\mathrm{4}R\sqrt{\mathit{\pi}}\phantom{\rule{0.125em}{0ex}}\sqrt{\mathit{\kappa}t}.\end{array}$$ - b.
large-

*t*asymptotics

In the large-*t* limit, the Laplace coordinate *s* is small and the
asymptotic expansions for *K*_{0} and *K*_{1} take the form

where *γ* is the Euler–Mascheroni constant, giving

where

We now need to take an inverse Laplace transform of $\stackrel{\mathrm{\u203e}}{V}$. The
second term on the right-hand side gives ${L}^{-\mathrm{1}}\left\{\frac{\mathit{\pi}{R}^{\mathrm{2}}}{s}\right\}=\mathit{\pi}{R}^{\mathrm{2}}$. Llewelyn Smith (2000) discusses the literature for
inverse Laplace transforms of the form (*s*^{α}ln *s*)^{−1} for
small *s*. For our problem, the discussion in Olver (1974, Chap. 8,
Sect. 11.4) is the most helpful approach. His result (11.13), discarding the
exponential term which is not needed here, shows that the inverse Laplace
transform of ${\left({s}^{\mathrm{2}}\mathrm{ln}s\right)}^{-\mathrm{1}}$ has the asymptotic expansion

Using ${L}^{-\mathrm{1}}\left\{F\left(\mathit{\tau}s\right)\right\}=\frac{\mathrm{1}}{\mathit{\tau}}f(t/\mathit{\tau})$, we thus obtain the desired connection between the encounter volume and diffusivity at long times:

Physically, the timescale *τ* (Eq. 35) defines the time at which the
dispersion of random particles, *D*=4*κ**τ*, is comparable to the
volume of the encounter sphere, i.e., *R*^{2}*e*^{2γ}≅*π**R*^{2} in
2-D. Thus for *t*≫*τ*, particles are coming to the encounter sphere
“from far away.”

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,

Note again that the diffusivity on the right-hand side of Eqs. (28)–(29),
(31) and (38) is *κ*^{moving}, which is equal to 2*κ*^{stationary}.

## 2.4 Numerical tests of the derived formulas in 1-D and 2-D

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).

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(100Δ*t*)) timescales, the
agreement between numerical simulation and theory is poor at those times in
both 1-D and 2-D. Once the random walkers have executed > 100
time steps, however, the dispersion reaches the diffusive regime, and the
agreement between the theory (red) and numerical simulation (black) rapidly
improves for both the 1-D and 2-D cases, with the two curves approaching each
other at long times. In 2-D, the long-time asymptotic formula (38) works well
at long times, *t*≫*τ*, as expected. The 2-D short-time asymptotic
formula (green) agrees well with the exact formula (red) at short times but
not with the numerical simulations (black) for the same reason as discussed
above, i.e., because the numerically simulated random walk has not yet
reached the diffusive regime at those times.

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 (http://marine.copernicus.eu/), both along satellite tracks and as a gridded mapped product in both near-real and delayed time. Here we use the delayed-time gridded maps of absolute geostrophic velocities with 1∕4 deg spatial resolution and a temporal step of 1 day, and focus our attention on the Gulf Stream extension region of the North Atlantic Ocean. There, the Gulf Stream separates from the coast and starts to meander, shedding cold- and warm-core Gulf Stream rings from its southern and northern flanks. These rings are among the strongest mesoscale eddies in the ocean. However, their coherence, interaction with each other and with other flow features, and their contribution to transport, stirring and mixing are still not completely understood (Bower et al., 1985; Cherian and Brink, 2016).

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 $\mathrm{d}x=\mathrm{d}y\cong \phantom{\rule{0.125em}{0ex}}\mathrm{10}$ km on 11 January 2015 and
were integrated forward in time for 90 days using a fifth-order variable-step
Runge–Kutta integration scheme with bi-linear interpolation between grid
points in space and time. The encounter radius was chosen to be *R*=30 km in
both the zonal and meridional directions, i.e., about a third of a radius of
a typical Gulf Stream ring. Similar parameter values were used in Rypina and
Pratt (2017), although our new simulation was carried out using more recent
2015 velocities instead of 1997 as in that paper.

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-*V* (blue) cores and larger high-*V* (yellow) peripheries.
Although the azimuthal velocities and vorticity-to-strain ratio are large
within the rings, the coherent core regions with inhibited mixing potential
are small, suggesting that the *coherent* transport by these rings
might be smaller than anticipated from the Eulerian diagnostics such as the
Okubo–Weiss or closed-streamline criteria (Chelton et al., 2011; Abernathey
and Haller, 2018). On the other hand,
the rings' peripheries, where the mixing potential is elevated compared to
the surrounding fluid, cover a larger geographical area than the cores. Thus,
while rings inhibit mixing within their small cores, the enhanced mixing on
the periphery might be their dominant effect. This is consistent with the
results from Rypina and Pratt (2017), but a more thorough analysis is needed
to test this hypothesis. Notably, the encounter volume is also large along
the northern and southern flanks of the Gulf Stream jet, with two separate
yellow curves running parallel to each other and a valley in between (although
the curves could not be traced continuously throughout the entire region).
This enhanced mixing on both flanks of the Gulf Stream extension current is
reminiscent of chaotic advection driven by the tangled stable and unstable
manifolds at the sides of the jet (del-Castillo-Negrete and Morrison, 1993;
Rogerson et al., 1999; Rypina et al., 2007; Rypina and Pratt, 2017), and is
also consistent with the existence of critical layers (Kuo, 1949; Ngan and
Sheppard, 1997).

We now apply the asymptotic formula (38) to convert the encounter volume to
diffusivity. Because Eq. (38) is not invertible analytically, we converted
*V* to *κ* numerically using a look-up table approach. More
specifically, we used Eq. (38) to compute theoretically predicted *V* values
at time *T*= 90 days for a wide range of *κ*s spanning all possible
oceanographic values from 0 up to 10^{9} cm^{2} s^{−1}, and we used
the resulting look-up table to assign the corresponding *κ* values to
*V* values in the third row of Fig. 4. Note that, instead of the long-time
asymptotic formula (38) (as in the third row of Fig. 4), it is also possible
to use the exact formulas (28)–(29) to convert *V* to *κ* via a
look-up table approach. The resulting exact
diffusivities, shown in the second row of Fig. 4, are similar to the
long-time asymptotic values (third row). Because both exact and asymptotic
formulas were derived under the assumption of a diffusive random walk,
neither should work well in regions with a non-diffusive behavior. The
asymptotic formula has the advantage of being simpler and it also provides
for a numerical estimate of the “long-time-limit” timescale, *τ*, shown
in the bottom row of Fig. 4.

As expected, the diffusivity maps in the second and third rows of Fig. 4,
which resulted from converting *V* to *κ* using (28)–(29) or (38),
respectively, have the same spatial variability as the *V*-map, with large
*κ* at the peripheries of the Gulf Stream rings and at the flanks of
the Gulf Stream and small *κ* at the cores of the rings, near the Gulf
Stream centerline and far away from the Gulf Stream current, where the flow
is generally slower. The diffusivity values range from O(10^{5}) to O(10^{7}) cm^{2} s^{−1}. Using the 1971
Okubo's diffusivity diagram and scaling law,
*κ*_{Okubo}[cm^{2} s^{−1}]= 0.0103 L[cm]^{1.15}, our
diffusivity values correspond to spatial scales from 10 to 650 km, thus
spanning the entire mesoscale range. This is not surprising considering the
Lagrangian nature of our analysis, where trajectories inside the small
(<50 km) low-diffusion eddy cores stay within the cores for the entire
integration duration (90 days), whereas trajectories in the high-diffusivity
regions near the ring peripheries and at the flanks of the Gulf Stream jet
cover large distances, sometimes >650 km, over 90 days.

The performances of the exact and asymptotic diffusive formulas vary greatly
throughout the domain, with better/poorer performances in high-/low-*V*
areas. This is because in the low-*V* areas, the behavior of fluid parcels is
non-diffusive, so the diffusive theoretical formulas work poorly. The
breakdown of the long-time asymptotic formula is evident in the fourth row of
Fig. 4, which shows the corresponding long-time scales, *τ* (from
Eq. 35), throughout the domain. As suggested by our 2-D random walk
simulations, the long-time asymptotic diffusive formula only works well when
*t*≫*τ*, but in reality *τ* values are < 9 days (1∕10 of
our integration time) only in the highest-*V* regions, and are much larger
everywhere else, reaching values of ≅90 days within the cores of the
Gulf Stream rings. More detailed comparison between theory, both exact and
asymptotic, and numerical *V*(*t*) is shown in Fig. 5 for three reference
trajectories that are initially located inside the core, on the periphery,
and outside of a Gulf Stream ring (black, red, and blue, respectively)
centered at approximately 36.8^{∘} N and 60^{∘} W. Clearly, the diffusive theory works poorly for the trajectory
inside the eddy core (black curve). The agreement is better for the blue
curves and even better for the red curves, corresponding to trajectories
outside and on the periphery of the eddy, although deviations between the
theory and numerics are still visible, raising questions about the general
validity of the diffusive approximation in ocean flows on timescales of a few
months.

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.,
*D*∝*T*^{2} and *V*∝*T* (Rypina and Pratt, 2017). At longer times,
when velocity shear can no longer be assumed constant in space and time, the
regime may transition to a local Richardson regime (i.e., *D*∝*t*^{3}),
where separation at a given scale is governed by the local features of a
comparable scale (Richardson, 1926; Bennett, 1984; Beron-Vera and LaCasce,
2016), or to a non-local chaotic-advection spreading regime (i.e., *D*∝exp(*λ**t*)), where separation is governed by the large-scale
flow features (Bennett, 1984; Rypina et al., 2010; Beron-Vera and LaCasce,
2016). The kinetic energy spectrum of a flow indicates whether a local or
non-local regime will be relevant. The chaotic transport regime is generally
expected to occur in mesoscale-dominated eddying flows, such as, for example,
AVISO velocity fields, over timescales of a few eddy winding times. At times
long enough for particles to sample many different flow features, such as
Gulf Stream meanders or mesoscale eddies in the AVISO fields, the velocities
of the neighboring particles become completely uncorrelated, and transport
finally approaches the diffusive regime. With the mesoscale eddy turnover
time being on the order of several weeks, it often takes longer than 90 days
to reach the diffusive regime.

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
2×10^{4} m^{2} s^{−1} in the most energetic regions in the
immediate vicinity of the Gulf Stream and its extension, to
10^{3} m^{2} s^{−1} in less energetic areas, to
200 m^{2} s^{−1} in the coastal areas of the Slope Sea. Diffusivity
estimates vary significantly depending on the initial tracer distribution
used (Abernathey and Marshall, 2013) and depend on whether the suppression by
the mean current has been taken into account (Klocker and Abernathey, 2014).
The diffusivity tensor has also been shown to be anisotropic, with a large
anisotropy ratio near the Gulf Stream (Rypina et al., 2012). Data resolution
and coverage, as well as the choice of timescales and lengthscales also play
a role in defining *κ* value (Cole at al., 2015). All of these issues
complicate the reconciliation of different diffusivity estimates.
Nevertheless, ignoring these complications for a moment, and avoiding the
smallest diffusivities in those geographical areas of Fig. 4 where the
diffusive approximation is invalid, our O(10^{3} m^{2} s^{−1})
encounter-volume-based diffusivity estimates tend to be in the middle of the
range of available estimates for the western North Atlantic. Although not
inconsistent with other estimates, the encounter volume method did not
predict diffusivities to reach values of 10^{4} m^{2} s^{−1}
anywhere within the considered geographical domain.

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,

with a constant strain coefficient *α*. The particle trajectories are
given by $x={x}_{\mathrm{0}}{e}^{\mathit{\alpha}\mathrm{t}},\phantom{\rule{0.125em}{0ex}}y={y}_{\mathrm{0}}{e}^{-\mathit{\alpha}\mathrm{t}}$
where *x*_{0}, *y*_{0} are particles' initial positions. The dispersion of a
small cluster of particles that are initially uniformly distributed within a
small square of side length 2d*x* is

where $X=x-{x}_{\mathrm{0}}$ and $Y=y-{y}_{\mathrm{0}}$ are displacements of particles from their initial positions and the overbar denotes the ensemble mean. Since the linear strain velocity remains unchanged in a reference frame moving with a particle, without loss of generality we can restrict our attention to a cluster that is initially centered at the origin, so $\stackrel{\mathrm{\u203e}}{X}=\stackrel{\mathrm{\u203e}}{Y}=\mathrm{0}$. In the long-time limit, when ${e}^{\mathit{\alpha}\mathrm{t}}\gg \mathrm{1}\gg {e}^{-\mathit{\alpha}\mathrm{t}}$, the dispersion becomes

If one is using a diffusive fit,

to approximate diffusivity, then the resulting diffusivity is

On the other hand, the encounter volume for the linear strain flow is

whereas the long-time diffusive fit is

yielding

where the function ProductLog(*z*), also known as the Lambert function, is a solution to *z*=*w**e*^{w}. Because
*κ*_{D} is exponential in time, while *κ*_{V} is not,
*κ*_{D} always becomes larger than *κ*_{V} at large
*t*.

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 *R*= 30 km as a cluster radius for consistency) to
estimate their dispersion and infer diffusivity from its slope. Consistent
with our arguments above, the resulting dispersion-based diffusivities in
Fig. 6 are larger than the encounter-volume-based diffusivities in Fig. 4 and
reach values of O(10^{4} m^{2} s^{−1}) in the energetic regions of
the Gulf Stream and its extension, in agreement with the previous diffusivity
estimates from the literature. In applications where the number of encounters
is a more important quantity than the spread of particles, the
encounter-volume-based diffusivity might be a more appropriate estimate to
use.

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,
*V*_{mean}, from the full encounter volume, *V*. The result
characterizes the contribution of eddies, although strictly speaking
${V}_{\mathrm{eddy}}\ne V-{V}_{\mathrm{mean}}$ because of nonlinearity. Note also
that because we are interested in the Lagrangian-averaged effects of eddies
following fluid parcels, *V*_{eddy} cannot be estimated by simply
advecting particles by the local eddy field alone (see an extended discussion
of this effect in Rypina et al., 2012). Not surprisingly, the eddy-induced
encounter volumes (upper right panel of Fig. 4) are smaller than the full
encounter numbers, with the largest decrease near the Gulf Stream current,
where both the mean velocity and the mean shear are large. In other
geographical areas, specifically at the peripheries of the Gulf Stream rings,
the decrease in *V* is less significant, so the resulting map retains its
overall qualitative spatial structure. The same is true for the diffusivities
in the second and third rows of Fig. 4. The overall spatial structure of the
eddy diffusivity is preserved and matches that in the left panels, but the
values decrease, with the largest differences near the Gulf Stream, where
some diffusivity values are now O(10^{6}) instead of
O(10^{7}) cm^{2} s^{−1}. In contrast, *κ* only
decreases, on average, by a factor of 2 (instead of an order of magnitude)
near the peripheries of the Gulf Stream rings. The long-time diffusive
timescale *τ* generally increases, and the ratio *t*∕*τ* generally
decreases throughout the domain, but the long-time asymptotic formula (38)
still works well in high-*V* regions, specifically on the peripheries of the
Gulf Stream rings where *τ* is still significantly less than *t*.

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 *t* – then the resulting encounter volume could be converted to
diffusivity using the derived connection. This would allow estimation of
diffusivity using a single instrument.

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: http://marine.copernicus.eu/services-portfolio/access-to-products/?option=com_csw&view=details&product_id=SEALEVEL_GLO_PHY_L4_REP_OBSERVATIONS_008_047 (CMEMS, 2018).

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

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