Fluid parcels can exchange water properties when coming into contact with each
other, leading to mixing. The trajectory encounter mass and a related
simplified quantity, the encounter volume, are introduced as a measure of the
mixing potential of a flow. The encounter volume quantifies the volume of fluid
that passes close to a reference trajectory over a finite time interval.
Regions characterized by a low encounter volume, such as the cores of coherent
eddies, have a low mixing potential, whereas turbulent or chaotic regions
characterized by a large encounter volume have a high mixing potential. The
encounter volume diagnostic is used to characterize the mixing potential in three flows of increasing complexity: the Duffing oscillator, the Bickley jet and
the altimetry-based velocity in the Gulf Stream extension region. An
additional example is presented in which the encounter volume is combined
with the

Mixing is an irreversible exchange of properties between different water
masses. This process is important for maintaining the oceanic large-scale
stratification and general circulation, and it plays a key role in the
redistribution of biogeochemical tracers throughout the world oceans.
Mixing occurs between different water masses when they come into direct
contact with each other. Thus, the mixing potential of the flow, i.e., the
opportunity for mixing to occur, is generally enhanced in regions where
water parcels meet or encounter many other water parcels and are thus
exposed to a large amount of fluid passing by them as the flow evolves. This
would be the case, for example, for a parcel within a chaotic zone, which is a
region of the flow that is in a state of chaotic advection. There, the
separation between initially nearby water parcels grows exponentially in
time and, in the infinite time limit, each water parcel encounters all the
other water parcels within the same zone and comes into contact with the entire
volume of the chaotic zone. Similarly, high encounter volumes will exist in
turbulent regions. In contrast, the mixing potential and encounter volume is
expected to be smaller in regions where water parcels do not experience many
encounters with other water parcels and remain close to their initial
neighbors as the flow evolves. This would be the case, for example, for a
water parcel that is located inside a coherent eddy. If the eddy is in a
state of solid body rotation, the water parcel would forever stay close to
its initial neighbors and will not have any new encounters at all. If some
amount of azimuthal shear is present within the eddy, then for a water
parcel located at a radius

Of course, the presence of a mixing potential does not guarantee that the
mixing of a tracer will occur; it is also essential that the tracer
distribution is nonuniform so that irreversible property exchange can take
place between the different water parcels during their encounters. This exchange
happens through diffusion and therefore relies on a concentration difference
between the two parcels. Thus, the intensity of the mixing would depend on both the
tracer distribution and the flow, whereas the mixing potential is the property
of only the flow field alone. In this work, we introduce the concept of an
encounter mass,

For a given reference trajectory,

To this end, we subdivide the entire fluid at

We define

For a numerical implementation of the trajectory encounter volume-based
mixing characterization, one would need to start at a chosen time

In the infinite time limit,

Since the locations of the hyperbolic trajectories and manifolds generally evolve in
time,

The radius

Finally, the approximation

Once the timescale

We proceed to test the performance of the encounter volume technique in quantifying the mixing potential of several geophysically relevant sample flows of increasing complexity, starting from a simple analytically prescribed periodically perturbed double-gyre Duffing oscillator system, followed by a dynamically consistent solution of the PV conservation equation on a beta plane known as the Bickley jet and finishing with an observationally based geostrophic velocity field in the North Atlantic derived from the sea surface height altimetry.

The Duffing oscillator flow with its figure-eight geometry has become a
standard test case for emerging techniques related to the dynamical systems
theory. This flow consists of two gyres with the same sign of rotation
(clockwise in our case) with elliptic centers that oscillate in time around
their mean positions. A hyperbolic point is located at the origin between the
two gyres, and a pair of stable and unstable manifolds emanate from it,
forming a figure eight in the absence of the time-dependent perturbation or
forming a classic homoclinic tangle in the presence of the perturbation. The
velocity field is two-dimensional, incompressible and given by

The trajectory segments for the different integration times

The encounter volume for the Duffing oscillator for the various integration
times from

The encounter volume was computed for a range of trajectory integration
times from

The Poincaré section (the black dots; same as in Fig. 1b) superimposed onto the encounter volume (in color; same as the top and middle right panels in Fig. 2). Only select trajectories from the Poincaré section are shown.

In order to more clearly highlight the link between high values of

The encounter volume (in color; the same as the second row and the second column subplot of Fig. 2) and the stable (black) and unstable (white) manifolds for the Duffing oscillator flow computed using the direct method.

A comparison between the FTLEs

The encounter volume,

With a variety of dynamical systems techniques available, it is important to
understand the advantages and limitations of the different methods. We
compared the encounter volume to two well-established and commonly used
methods, the Poincaré section (Fig. 3) and the FTLEs (Fig. 5). Since the
Poincaré section requires stroboscopic sampling of the trajectories in time, it
can only be applied to time-periodic flows and requires that the trajectories
are computed over a long integration time, typically thousands of periods
of the perturbation. On the other hand, it generally requires only a few
parcels to be released at some key locations rather than releasing a dense
grid of initial positions to map out the entire phase space. The encounter
volume and FTLEs, on the other hand, are not limited to time-periodic flows
and also work with significantly shorter segments of trajectories (the longest
integration time in our simulations in Fig. 2 is only 50 periods of the
perturbation). They are also better suited for identifying the manifolds than
the Poincaré sectioning as they do not require any a priori knowledge about the
location of the hyperbolic trajectory. On the other hand, they require many
more parcels to be released in order to map out the phase space. When
applied to the same set of trajectories (the same initial positions and
integration times), the FTLEs and the encounter volume methods produced
similar results (Fig. 5), with

Related to the issue of computational cost is the question of a sufficient grid
size. We have carried out numerical simulations (Fig. 6) to investigate the
dependence of the encounter volume on the grid size and to come up with a
rule-of-thumb recommendation regarding the appropriate grid spacing. Our
simulations suggest that the encounter volume values (approximated by

The meandering Bickley jet flow is an idealized, but linearly dynamically
consistent, model for the eastward zonal jet in the Earth's stratosphere
(del-Castillo-Negrete and Morrison, 1993; Rypina et al., 2007, 2011). This flow consists of a steady eastward zonal jet on which two
eastward propagating Rossby-like waves are superimposed. All flow parameters
used here are identical to those used in our previous 2007 and 2011 papers.
In the reference frame moving at a speed of one of the waves, the flow
consists of a steady background velocity subject to a time-periodic
perturbation. The background looks like a meandering jet, with three
recirculation gyres to the north and south of the jet core. Between the
recirculation gyres, there are three hyperbolic points with the associated
stable and unstable manifolds. Under the influence of the time-periodic
perturbation imposed by the second wave, heteroclinic tangles are formed by
the manifolds emanating from different hyperbolic regions between the
recirculations, and a chaotic zone emerges on either side of the jet. The
manifolds, however, cannot penetrate through the jet core, which remains
regular and acts as a transport barrier separating the northern and southern
chaotic zones. All of these features are clearly visible in the Poincaré
section shown in Fig. 7 (top). The bottom subplot shows the

Past its separation point from the coast at Cape Hatteras, the strong and narrow Gulf Stream current turns offshore, where it loses its coherence, broadens and weakens and then starts to meander. Some of the meanders then grow and eventually detach from the current, forming strong mesoscale eddies known as the Gulf Stream rings. On 11 July 1997, a number of such Gulf Stream rings of various strengths and sizes at different stages of their lifetimes were clearly present both north and south of the Gulf Stream extension current (Fig. 8).

The flow in the Gulf Stream extension region, with a nonsteady meandering
jet, the Gulf Stream rings and the recirculations to the north and south of
the jet core, has a lot in common, at least qualitatively, with the Bickley
jet example. Unlike the idealized model, however, the real Gulf Stream rings
have finite lifetimes, and the jet is not periodic in the zonal direction.
Nevertheless, many of the qualitative features of the Bickley jet's

The Poincaré section

The velocity field that we used was downloaded from the AVISO website
(

The encounter volume was estimated for three different integration times,

The encounter volume for the AVISO velocities in the Gulf Stream
extension region for the trajectories released on 11 July 1997 and integrated over 30 days

The overall leakiness of the Gulf Stream rings and the small extent of their
coherent Lagrangian core regions suggests that the coherent transport by the
Gulf Stream rings (and maybe by mesoscale eddies in general) over time
intervals of a few months or longer may be significantly smaller than what
is generally anticipated from Eulerian diagnostics based on closed
streamlines or Okubo–Weiss criteria. Interestingly, the prominent red
rings (the large

To visualize the Lagrangian evolution of the core regions and to illustrate
the eddy leakiness, we extracted the trajectories from the core of the northern
eddy in Fig. 8a (i.e., the trajectories with

The jet region, although noisy, seems to suggest higher

The positions of the trajectories that were initially located within the eddy core on 11 July 1997 (blue patch) after 30 days (green), 60 days (red) and 90 days (yellow) of integration. The background shows the kinetic energy of the flow as a snapshot on 11 July 1997.

By analogy with molecular diffusion, the eddy diffusivity,

Although we have not been able to find an analytical expression connecting

It is convenient to move to a reference frame that is tied to a reference
particle, which would then always stay at the origin, while the other particles
would be involved in a random walk motion. The problem of finding the
encounter number is then reduced to counting the number of particles that come
within the radius

This change in the step size between the stationary and moving frames leads
to a doubling of the diffusivity in the moving reference frame. To show this,
we write the dispersion in the moving frame as

We thus seek an expression for the number of particles involved in
a random walk process with a diffusivity of

The numerical Monte Carlo simulations of a random walk process suggest that the
dependence of the encounter number (and the encounter volume) on the integration
time

The ballistic spreading that is dominated by a local velocity shear is
another commonly encountered spreading regime. The separation between
particles grows in proportion to time. Ballistic spreading can often be
observed in nonsteady realistic oceanic flows at timescales that are much
shorter than the onset of diffusive spreading (which develops after a
trajectory samples multiple different eddies or other flow features). To
derive a connection between the encounter volume and the velocity shear, we consider a
trajectory that is advected by a flow field with a constant meridional
velocity shear,

A comparison between the numerically computed encounter volume (blue)
and the
analytical predictions (Eqs. 8 and 9; red) for the linear
strain

The steady linear saddle flow with a constant strain rate

The linear growth of the encounter volume with time in the linear shear and linear strain flows could be anticipated by noting that both flows are steady in a reference frame moving with a reference trajectory, and all particles only encounter the origin once and never come back. Thus, the flux through the encounter circle is constant in time, and the encounter volume, which is a time integral of flux, is proportional to time. The random walk flow seems to be different because the particles can encounter the reference trajectory more than once, leading to a nonsteady flux of first encounters and a nonlinear time dependence of the encounter volume.

The above examples are centered on the mixing potential of a flow field, but
there may be value in computing the encounter volume for swarms of
trajectories of biological organisms, drifting sensors and other
non-Lagrangian trajectories. For example, if one is interested in the actual
transport of scalar properties such as heat, salt or vorticity, then it may
be useful to calculate

A schematic diagram for estimating the encounter number for a linear saddle.

If indeed

Consider the Bickley jet flow with the same parameters as in Sect. 2.2 and
assume that one is interested in a tracer that, at initial time

In the spirit of Speetjens (2012), we regard

This behavior was further numerically validated in Fig. 12, where we first
numerically simulated the evolution of this tracer in the Bickley jet flow
and then
estimated

When water parcels come into direct contact with each other, they can exchange
water properties, leading to mixing. The trajectory encounter volume,

The

The encounter volume diagnostic was tested in three flows with increasing
complexity: the Duffing oscillator, the Bickley jet and the altimetry-based
velocity in the Gulf Stream extension region. In all cases,

Similar to finite-time Lyapunov exponents (FTLEs) that are commonly used to
delineate regions with qualitatively different motion (Haller, 2002; Shadden
et al., 2005; Lekien and Ross, 2010),

Finally, while

As with FTLEs, complexity measures (Rypina et al., 2011), Lagrangian
descriptors (Mendoza et al., 2014) and other techniques from the dynamical
systems theory (Beron-Vera et al., 2013; Budisic and Mezic, 2012; Froyland
et al., 2007; Haller et al., 2016),

For a ballistic spreading regime dominated by the velocity shear

An analytical connection between the encounter volume and a widely used
measure of mixing, the diffusivity

The mixing potential is the property of the flow field and characterizes the
intensity of stirring, whereas the actual tracer mixing depends on both the
flow and the tracer. For example, no tracer mixing will occur if the tracer
gradient is zero, even if the mixing potential of the flow is high. To
address this, we have proposed combining the encounter number diagnostic
with the

The encounter volume is a frame-independent quantity because it is based on the relative distances between water parcel trajectories rather than on the properties of isolated trajectories. The encounter volume values do not change under the orthogonal transformations of the coordinates, i.e., under the rotations and translations of a reference frame. This is a desirable property because the ability of a flow to mix tracers should not depend on the reference frame.

The encounter volume and, more generally, the encounter mass ideas presented in this paper are not restricted to two dimensions and can be used to quantify the mixing potential in three-dimensional flows. This framework also does not require incompressibility and can work with unstructured irregular grids. The investigation of the performance of the method in quantifying the mixing potential of a flow in more complicated cases is left for a future study.

The velocity field that we used in Sect. 2.3 is publicly
available from the AVISO website,

The authors declare that they have no conflict of interest.

This work was supported by NSF grants OCE-1154641, OCE-1558806 and EAR-1520825 as well as by ONR grant N00014-11-10087 and NASA grant NNX14AH29G. Publication of this article was supported by the Office of Naval Research, grant no. N00014-16-1-2492. Edited by: S. Wiggins Reviewed by: two anonymous referees