# Lagrangian flow networks for passive dispersal:
tracers versus finite size particles

Deoclécio Valente  Ksenia Guseva  Ulrike Feudel  Carl-von-Ossietzky
Universität Oldenburg D-26111 Oldenburg,
Germany
Centre for Microbiology and Ecosystem Science, University of Vienna, A-1090,
Vienna, Austria.  Carl-von-Ossietzky Universität Oldenburg D-26111 Oldenburg,
Germany.

###### Abstract

The transport and distribution of organisms like larvae, seeds or litter in
the ocean as well as particles in industrial flows is often approximated by a
transport of tracer particles. We present a theoretical investigation to check
the accuracy of this approximation by studying the transport of inertial
particles between different islands embedded in an open hydrodynamic flow
aiming at the construction of a Lagrangian flow network reflecting the
connectivity between the islands. To this end we formulate a two-dimensional
kinematic flow field which allows the placement of an arbitrary number of
islands at arbitrary locations in a flow of prescribed direction. To account
for the mixing in the flow we include a von Kármán vortex street in the wake
of each island. We demonstrate that the transport probabilities of inertial
particles making up the links of the Lagrangian flow network depend
essentially on the properties of the particles, i.e. their Stokes number, the
properties of the flow and the geometry of the setup of the islands. We find a
strong segregation between aerosols and bubbles. Upon comparing the mobility
of inertial particles to that of tracers or neutrally buoyant particles, it
becomes apparent that the tracer approximation may not always accurately
predict the probability of movement. This can lead to inconsistent forecasts
regarding the fate of marine organisms, seeds, litter or particles in
industrial flows.

##  I INTRODUCTION

The dynamics of passive advection of small finite-size particles are
ubiquitous to many processes within marine ecology and industrial systems.
Precise predictions about the trajectories and final fate of these particles,
representing organisms, seeds, waste or plastic particles are crucial for
understanding central phenomena in both settings. In marine systems, passive
transport by ocean currents significantly impacts the dispersion of nutrients,
pollutants, seeds, marine organisms and marine aggregates. Models of this
transport contribute to our understanding of ecological dynamics such as e.g.
the transport of harmful algal blooms [1], the connectivity of habitats [2,
3], biogeochemical cycles [4, 5], and the spread of contaminants in the marine
environment [6, 7]. Similarly, in industrial settings, passive transport is a
key factor in the movement of particles within engineered systems, where the
objective is to manipulate trajectories, for example, to achieve size-based
segregation [8, 9], to tune the outcome of mixing or aggregation-fragmentation
processes [10, 11] or even to guide targeted transport [12, 13]. Moreover, for
industrial applications it is important to study the dependence of the
rheological properties of the flow on the volume fraction of the particles and
their density [14, 15, 16]

Of particular interest is the long-distance passive transport in marine
environments. It ranges from the distribution of seeds of plants among islands
and coastal areas [17, 18, 19, 20] via the transport of larvae of different
organisms between habitats [21, 22, 23] up to the fate of litter which is
transported through the sea [24] or into the ocean from coastal areas [25].
This transport is determined by ocean flows possessing preferred directions
which are influenced by prevailing wind directions and mesoscale hydrodynamic
structures like vortices, jets and fronts [26, 27, 28]. One of the challenges
is the identification of the major pathways of particles such as the pathways
of micro-and macro-plastics [29] and the locations where this litter
concentrates in the ocean [30, 31], the pathways of seeds to understand the
distribution of plant species on different archipelagos in the world’s ocean
[32] or the pathways of larvae to get from spawning areas back to their reefs
[33].

The simplest way to study this transport is to consider the particles as point
like tracers neglecting their size and densities and use a Lagrangian approach
for the transport of these particles [34, 35, 36]. In this case the particle
trajectories follow exactly the trajectories of the fluid parcels. To quantify
the transport between different locations it is sufficient to follow a large
number of tracers from a release location to a target location and count the
number of particles which reach the latter. This approach leads to Lagrangian
flow networks which have been constructed for several regions in the world
like the Caribbean Islands [37] or connections between hydrodynamic provinces
in the Mediterranean [2]. These studies are mostly based on ocean flows, while
more theoretical approaches are based on kinematic flows [38, 39, 40, 41]. We
note that the use of the tracer approach in general offers very good
predictions to the long-distance dispersal dynamics of particles in marine
systems [42].

A more sophisticated approach is to approximate those particles as spheres
with a certain size and density and treat them as inertial particles
experiencing various forces in the flow. The trajectories of inertial
particles in a given flow field can then be computed by solving the Maxey-
Riley-Gatinol equation [43, 44, 45]. Those trajectories deviate notably from
the ones of tracers leading to preferential concentrations, deviations in
lateral transport [46] as well as [47, 48, 49, 50, 51] sedimentation [52, 53,
54]. The flow fields employed in works using this description range from
simple two-dimensional kinematic flows mimicking e.g. the flow in the wake of
an island [55, 27] to random and turbulent flows [56, 48, 57].

Motivated by the many aforementioned possible applications, we provide here a
theoretical study in which we combine the idea of constructing Lagrangian flow
networks, which has been so far restricted to tracer transport between release
and target regions with the transport of inertial particles. Therefore we use
the Maxey-Riley equations to build up such Lagrangian flow networks for
inertial particles and compare them with the ones obtained for tracers. We
show that depending on the size and density of particles those networks
deviate from the ones realized by tracers. The most striking observation is
that the tracer transport overestimates the probabilities for certain pathways
leading to flow network with many links, while the network obtained for
particles with large Stokes numbers are much sparser. It has been
demonstrated, that particles having a density larger than the fluid (aerosols)
are expelled from the vortices while particles with a density lighter than the
fluid (bubbles) are gathered within vortices [58, 56]. These properties of
inertial particles influence their paths and, hence, the development of
Lagrangian flow networks. To study systematically the transport probabilities
between different locations in the flow under the influence of mesoscale
hydrodynamic vortices, we construct analytically a two-dimensional kinematic
velocity field mimicking major properties of an ocean flow like a preferred
flow direction and the emergence of a von Kármán vortex street in the wake of
islands. Our flow field allows for an arbitrary number of islands embedded in
the flow in an arbitrary geometry. Those islands are considered as the release
and target regions for the inertial particles. We quantify the differences in
the connectivity, i.e. the probability of transport from one island to another
depending on the properties of the particles like size and density and the
properties of the flow like vortex strength and background velocity.

The paper is organized as follows. In Sec. II we develop the velocity field
which mimicks the transport between $n$ islands embedded in the flow and
recall briefly the Maxey-Riley equations which are used to compute the
transport of spherical inertial particles in this flow. Then, in Sec. III we
start with the characterisation of the main differences in the advection
dynamics between tracers and inertial particles. Next, we analyse the spatial
distribution of the particles depending on the flow and particle properties
and demonstrate the large heterogeneities in the locations to which particles
are transported. Furthermore, we measure the connectivity between the
different islands by computing the probabilities of transport between the
islands and work out the substantial differences depending on the size and
density in comparison to the usual tracer approach. Finally, we quantify the
deviations in the transport probabilities when modelling inertial particles as
point like tracers. We discuss the results in Sec. IV.

##  II Methods and theoretical background

###  II.1 The hydrodynamic flow

To study the connectivity among different islands embedded in a hydrodynamic
flow in detail, it is useful to have an analytical 2D flow field which allows
for an arbitrary choice of the number of islands and their location in the
flow. This way one can systematically change the geometry of the setup and the
properties of the flow field. To achieve this goal we extend a well-known
kinematic flow to an arbitrary number of islands. Starting point is the flow
field developed by Jung et al. [55] which considers one island embedded in the
flow with a given flow direction in which the island acts as an obstacle
giving rise to a von Kármán vortex street in its wake. As an extension of this
framework we consider a two-dimensional kinematic flow with $n$ embedded
islands of circular shape with radius $r_{i}$ ($i=1,\cdots,n$) and of the same
size $r_{i}=r\;\;\forall i$. The position of the islands is fixed according to
an arbitrary but prescribed geometry. We consider a two-dimensional
observation area in which a main flow is assumed to flow from left to right
along the ${x}$-direction with different islands along is path. Far away from
the position of the islands in the ${y}$-direction the flow is expected to be
uniform. Behind each island, a pair of counter-rotating vortices is created,
at locations slightly below and above of the center of the corresponding
island, with a time difference of half a period $T_{c}/2$. They move a
distance L along the ${x}$-direction during one period $T_{c}$, until they
disappear due to dissipation. This process mimics the emergence of a von
Kármán vortex street in the wake of each island. Since the flow under
consideration is incompressible, a stream-function
${\varphi}={\varphi}(x,y,t)$ can be defined such that the two velocity
components can be computed as derivatives of this stream function

${u}_{x}=\frac{\partial{{\varphi}}}{\partial y},\,\,\,\,\,\,\,\
{u}_{y}=-\frac{\partial{{\varphi}}}{\partial x}.$ (1)

The stream function for a velocity field with an arbitrary number of islands,
say $n$ islands, at arbitrary positions ${[(h_{i},k_{i}),\,i=1,\cdots,n]}$ is
given by

${\varphi}=\sum_{i=1}^{n}(f_{i}(x,y)g_{i}(x,y,t))-u_{0}(n-1)y.$ (2)

The terms from left to right are explained as follows: the first factor
$f_{i}(x,y)$ ensures the presence of a boundary layer around each island such
that the velocity field goes to zero at the boundary of each island,

$\begin{split}f_{i}(x,y)=1-e^{-a_{i}(\sqrt{(x-h_{i})^{2}+(y-k_{i})^{2}}-r_{i})^{2}},\\\
i=1,\cdots,n,\end{split}$ (3)

where $(h_{i},k_{i})$ are the spatial coordinates of the centres of each
island $i$. The coefficient $1/\sqrt{a_{i}}$ mimics the width of the boundary
layer. This assures that at the boundary of each island the tangential
velocity tends linearly to zero, while the radial velocity component decreases
quadratically with the distance from the boundary.

The second factor $g_{i}(x,y,t)$ models the von Kármán vortex street and the
main background flow $u_{0}$ reads:

$\begin{split}g_{i}(x,y,t)=\omega_{1i}H_{1i}g_{1i}+\omega_{2i}H_{2i}g_{2i}+u_{0}ys_{i}(x,y),\\\
i=1,\cdots,n.\end{split}$ (4)

The first two terms in Eq. (4) describe the presence of the counter-rotating
vortices in the wake of each island. These vortices are of equal strength, but
opposite in sign $\omega_{1i}=-\omega_{2i}$. They are of Gaussian shape,
formulated by the following function

$\begin{split}g_{pi}(x,y,t)=e^{-R_{0i}[-(x-x_{pi}(t))^{2}+\alpha_{i}^{2}(y-y_{pi}(t))^{2}]},\\\
p=1,2\;\text{and}\;i=1,\cdots,n\end{split}$ (5)

with $1/\sqrt{R_{0i}}$ and $\alpha_{i}$ being the radius and the
characteristic ratio between the elongation of the vortices in $x$\- and
$y$-direction, respectively. The strength of the vortices
$\omega_{pi}\;(p=1,2;\;i=1,\cdots,n)$ is modulated by

$\displaystyle H_{1i}(t)$ $\displaystyle=$ $\displaystyle\sin^{2}(t\pi),$ (6a)
$\displaystyle H_{2i}(t)$ $\displaystyle=$ $\displaystyle\cos^{2}(t\pi).$ (6b)

The centers of the vortices move, along the horizontal; direction $x$
according to

$\displaystyle x_{1i}(t)$ $\displaystyle=$ $\displaystyle
h_{i}+r_{i}+\text{L}_{i}\;\text{mod}\;(t,T_{c}),$ (7a) $\displaystyle
y_{1i}(t)$ $\displaystyle=$ $\displaystyle k_{i}+y_{0i},$ (7b) $\displaystyle
x_{2i}(t)$ $\displaystyle=$ $\displaystyle x_{1i}(t-T_{c}/2),$ (7c)
$\displaystyle y_{2i}(t)$ $\displaystyle=$ $\displaystyle
k_{i}-y_{0i},\;i=1,\cdots,n.$ (7d)

The third term in Eq. (4) provides the contribution of the main background
flow $u_{0}$. The factor

$\begin{split}s_{i}(x,y)=1-e^{[-(x-h_{i}-r_{i})^{2}/\alpha_{i}^{2}-(y-k_{i})^{2}]},\\\
i=1,\cdots,n,\end{split}$ (8)

is the shielding factor associated with each island. This factor suppresses
the impact of the background flow in the wake of the corresponding island. We
display the non-dimensional values of the parameters used in the model in
Table 1,

Table 1:  The representative parameters used in the model unless specified and their respective non-dimensional values. Parameter | Symbol | Dimensionless
---|---|---
Radius of the island | $r_{i}$ | 1
Period of the flow | $T_{c}$ | 1
Main background flow | $u_{0}$ | 12
Strength of the vortex | $|\omega_{pi}|$ | $40\;\text{or}\;60$
Characteristic ratio | $\alpha_{i}$ | 1
y-coordinate of the vortex | $y_{0i}$ | 0.5
Travel distance of the vortices | $\text{L}_{i}$ | 4
Width of the boundary layer | $a_{i}$ | 1
Linear size of the vortices | $R_{0i}$ | 1

a possible parametrization can be found in [40]. Note that Eq. (2) is an
extension of the velocity field developed in [55] to an arbitrary number of
islands at arbitrary positions. The presence of the last term in Eq. (2) is a
direct consequence of the linear superposition principle assumed for the
construction of the flow field. This term is necessary to assure that the main
background flow $u_{0}$ is not counted as often as the number of islands.
Furthermore, it is important to note that the linear superposition principle
is only valid provided that the shielding areas, $s_{i},s_{j}$, associated
with two neighbouring islands, respectively, do not overlap. This condition
poses a restriction on the distance between neighbouring islands, such that
they can not be placed too close to each other.

Nonetheless one of the main advantages of our extended velocity field is the
generality which allows for the study of an almost arbitrary location of the
islands. This enables us to vary systematically the flow properties and the
geometry of the arrangement of the islands to find out the relationship
between properties of the particles and the flow determines the connectivity
between the islands. We nondimensionalize the stream-function Eq. (1) by
measuring the length in terms of units of the radius of the island $r$. As a
unit of time, we take the period of the flow $T_{c}$.

###  II.2 Advection of particles

The simplest approach to describe the motion of particles advected by fluid
flows is to assume that these particles are just passive point-like tracers
that take on the same velocity as the surrounding fluid parcels. The velocity
V of these point-like tracer particles, or shortly _tracers_ , is therefore
the same as the velocity of the flow field u at the particle’s position X:

${\bf\dot{X}}={\bf V}({\bf X},t)={\bf u}({\bf X},t).$ (9)

However, there are many cases where this assumptions above does not hold, and
one has to consider that particles have a considerable size and density [59].
Such, finite-size particles have a delayed reaction response to the flow
dynamics. Their trajectories deviate substantially from the fluid parcels due
to the action of several forces and for this reason they are also known as
_inertial_ particles in the literature. For spherical particles these forces
are described using the Maxey-Riley-Gatinol equations [43, 44] with inclusion
of a correction made by Auton [45]. In this work we use an simplified version
of these equations, neglecting the Faxén corrections [44] and the history
force [27, 11]. This approximation in its dimensionless form is given by

$\displaystyle{\bf\dot{X}}$ $\displaystyle=$ $\displaystyle{\bf V}({\bf
X},t),$ (10a) $\displaystyle{\bf\dot{V}}$ $\displaystyle=$
$\displaystyle{\beta}D_{t}{\bf u}({\bf X},t)-\frac{1}{\text{St}}[{\bf V}({\bf
X},t)-{\bf u}({\bf X},t)],$ (10b)

where $D_{t}=\partial_{t}+\bf{u}\cdot\nabla$ represents the derivative along
the flow trajectory. The first term in this equation includes the added mass
effect accounting for the necessary displacement of fluid by a moving particle
and the acceleration of the fluid. The second term is the Stokes drag
proportional to the difference between the velocities of the particle and the
fluid. The two dimensionless parameters are:
$\beta=3\rho_{f}/(\rho_{f}+2\rho_{p})$, the ratio between fluid $\rho_{f}$ and
particle $\rho_{p}$ densities, and the Stokes number given by
$\text{St}=\frac{\alpha_{p}^{2}}{3\beta\nu\tau_{f}}$ where $\alpha_{p}$ is the
size of the particle, $\nu$ is the kinematic viscosity and $\tau_{f}$ is the
characteristic flow time. If the particles are denser than the fluid, i.e.
$\beta<1$, they are known as aerosols, if they are lighter than the fluid,
i.e. $\beta>1$, we call them bubbles, if $\beta\equiv 1$ they are known as
neutrally buoyant particles. The density of the fluid ($\rho_{f}$) equals the
density of water, approximated by the value $\rho_{f}=1000kg/m^{3}$. Note that
Eq. (9) is valid as long as the particle Reynolds number is small, $Re_{p}\ll
1$.

For our simulations, we choose two different Stokes numbers which are
$\text{St}=10^{-3}$ and $\text{St}=0.03$, and held the following parameters
fixed $\beta=0.495$ for aerosols, $\beta=1.875$ for bubbles, and $\beta=1$ for
neutrally buoyant particles Fig. 2 illustrates that the chosen $\beta$ values
lie in a realistic range. However those values as well as the selected Stokes
numbers are quite far away from Stokes numbers of real seeds or organism in an
ocean flow. According to our aim to provide a theoretical study, we made a
choice which better illustrates the impact of inertia.

###  II.3 The geometric setup for particle transport

To study the transport of particles in a general framework of an arbitrary
number of islands embedded in a flow we employ two different geometrical
setups composed of six islands each. One of them is a Chain of Islands (COI),
where all islands are arranged in a chain similar to the setup in [60, 56].
The other one is a Random Geometry (RG) with randomly chosen positions of the
islands. While in the COI, the islands are equally spaced, in the RG, they are
not (see Fig. 1). The main flow in both setups is directed along the $x$-axis
from left to right.

We fix four different locations as sources of particles $S_{i}$ with
$i=0,1,2,3$, at which we release particles. One of these locations is upstream
at the line $(x=-5,y\in[-3,3])$ and $(x=-5,y\in[-5,5])$ for COI and RG,
respectively, from where particles advected by the main background flow
$u_{0}$ enter the observation area at $x=-5$. The other three locations are in
a ring of width $0.4$ around each of the first three islands, respectively,
representing other locations from where particles start (see Fig. 1). In
addition, we fix seven target locations $O_{j}$ with $j=1,...,7$, which we
check for particle visits. Six of these locations are in a ring of width $0.4$
around each one of the islands. The other location is downstream the
observation area at $(x=55,y\in[-3,3])$ and $(x=45,y\in[-5,5])$ for COI and
RG, respectively.

Figure 1: Two geometrical setups are considered. (a): Chain of Islands (COI).
(b): Random Geometry (RG). For both geometries we schematically represent the
release area of particles at the start of the observation area $S_{0}$ (shown
as a strip dark strip (red/gray) at x=-5), and three additional sources around
each one of the first islands: $S_{1}$ (dark blue/dark gray ring around O1),
$S_{2}$ (cyan/very light gray ring around O2) and $S_{3}$ (green/light gray
ring around O3). We also show the target region at the end on the observation
area at x = 45 marked as EOA in yellow/very light gray. In addition (not
shown), all of the islands have a target region around them, they are
equivalent to the sources regions described above for the first three islands.
(Color online)

For our particle advection experiments we start $150$ particles in each
$S_{i}$, $i=0,...,3$ whose initial positions ${(x,y)}$ are randomly chosen
within the specified areas.The particles enter the system with the same
velocity as the fluid particles at their release position. During each flow
period, particles are released at a randomly selected time, denoted as $\tau$
which is uniformly distributed. On average, approximately $10000$ particles
are released per period. After being released, they are advected by the flow
until they reach either one of the other target regions $O_{j}$ with
$j=1,2,3,4,5,6$ or they escape at $O_{7}$ which is the ”End of Observation
Area” (EOA) located at the right boundary at $x=55$ (COI) and $x=45$ (RG),
respectively. Importantly, we only consider that the particle has reached the
given target when this particle has crossed the boundary of this region (the
ring of width 0.4 around each island). Once inside the ring, few particles can
get very close to the island and would stay there for an extremely long time,
since the flow velocity goes to zero at the boundary of the island. We check
for those particles with another ring of width $0.014$ around each island. All
particles which enter this very small ring are removed. Although their number
is small we still substract it from the total number of particles released and
we discard them from further statistics. In the $y$-direction the boundaries
of the observation area are open. With this setup we are able to count how
many particles are advected from each of the source regions $S_{i}$
($i=0,...,3$) to each of the target regions $O_{j}$ ($j=1,...,7$). Measuring
the time interval from release to reaching one of the targets allows us to
determine the time needed to be advected from one location to another. Since
many particles visit different islands on their way from the source to the end
of observation area we can follow their whole path in the flow.

Figure 2: Comparative Analysis of finite-size particle properties based on
density and relative size. This graph plots various particles, including air,
bubbles in our simulations, wood, water, sand, aerosols in our simulations,
and iron, showcasing the diversity in density ($\rho_{p}$) and dimensionless
parameter $\beta$. The density of the fluid ($\rho_{f}$) which enters the
parameter $\beta$ corresponds to the density of water, approximated by the
value $\rho_{f}=1000$ kg/m3.

##  III Results

###  III.1 Comparing trajectories of inertial particles with tracers

####  III.1.1 General aspects of advection dynamics

We start our analysis by characterizing the main differences in the advection
dynamics of our four types of particles (tracers, neutrally buoyant particles,
bubbles and aerosols). As described in the methods, we have a continuous
influx of particles from four source areas and the escape of these particles
at the right boundary of the observation area (see Fig. 1). After a short
transient, the picture becomes periodic due to the periodic flow. From this
time on, the ensemble of the injected particles spreads through the whole
observation area forming dynamic fractal patterns, which strongly vary for the
four particle types. We illustrate these patterns by taking a snapshot of the
spatial distribution of particles at $t=23.48\text{T}_{\text{c}}$, see Fig. 3,
and we compare the different particle types. We can see that while the
differences to tracers are more subtle for $\text{St}=10^{-3}$, they are very
pronounced for large $\text{St}=0.03$. As expected, the strongest similarity
can be observed in the spatial distribution of tracers and neutrally buoyant
particles. We also observe the well known trapping of bubbles in the vortices
and the ejection of aerosols from the vortices in the flow field [56]. These
effects are already present at $\text{St}=10^{-3}$ and especially visible for
$\text{St}=0.03$ (see Fig. 3). These differences in the advection of tracers
and inertial particles should clearly translate into contrasting transport
properties for different particle types.

Before we quantify the transport between the different locations,

Figure 3: Spatial distribution of tracers and inertial particles in the wake
of six islands. From the top to the bottom, (a): ideal tracers. (b) and (c):
neutrally buoyant particles for $\text{St}=10^{-3}$ and $\text{St}=0.03$,
respectively. (d) and (e): aerosols for $\text{St}=10^{-3}$ and
$\text{St}=0.03$, respectively. (f) and (g): bubbles for $\text{St}=10^{-3}$
and $\text{St}=0.03$, respectively. Those snapshots were taken at a time
$23.48\text{T}_{\text{c}}$ for vortex strength $\omega=60$. Colors correspond
to the colors in Fig. 1 (Color online).

we study the distribution of aerosols and bubbles along the observation area
depending on the strength of the vortices, the geometrical setup with respect
to the position of the islands and the place where the particles are released.

To this end, we have fixed two lines along the y axis, one in the middle of
the observation area ($x=25$) and the other at the end ($x=55$). We were
interested in registering the y coordinates of all particles to create spatial
distributions when particle cross the middle and the end of observation area.
We show the results for COI geometry, which is easier to interpret due to its
symmetry, for two vortex strength ($\omega=40$ and $\omega=60$) and different
release sources (i.e., main flow and first island). To make the distributions
along those lines comparable, we normalize each of them by the total number of
particles crossing the corresponding line. In Fig. 4, we represent the
normalized distributions $\delta_{p}$, which show the proportion of particles
crossing the lines $x=25$ and $x=55$ during a simulation time. In what
follows, we will only show and discuss the distribution for the most extreme
case of aerosols and bubbles. Since inertial effects are larger for large
Stokes numbers, we consider only $\text{St}=0.03$, for $\text{St}=10^{-3}$ the
influence of inertia is less pronounced.

The results quantify the observations already presented in Fig. 3 (d and e).
Due to the centrifugal forces which aerosols experience in the vortices they
move further away from the symmetry axis at $y=0$. This behaviour is observed
for particles released with the main flow (dashed line in Fig. 4 a and b) as
well as the ones released at the first island (dashed line in Fig. 4 c and d).
With increasing distance from the release location the maximum concentration
of particles moves away from the symmetry axis as expected (compare the black
($x=25$) and magenta/gray ($x=55$) curves). Increasing the vortex strength
from $\omega=40$ (Fig. 4 left panels) to $\omega=60$ (Fig. 4 right panels)
intensifies the centrifugal forces and leads again to a further movement away
from the symmetry axis.

By contrast, bubbles are trapped in the vortices as already illustrated in
Fig. 3 (f, g). Therefore, the maximum of their concentration (solid line Fig.
4) is around the symmetry axis because the location of the two vortex centers
are at $y=-0.5$ and $y=0.5$. Next to the maximum, we find a region of almost
zero particles since those released with the main flow are also trapped. Due
to the trapping phenomena, the maximum from $x=25$ to $x=55$ does not change.
In addition, we find a non-zero distribution function $\delta_{p}$ in regions
far away from the symmetry axis (Fig. 4 a and b) as it corresponds to the
particles transported with the main flow unaffected by the vortices. These
parts of $\delta_{p}$ are of course absent for particles released at the first
island.

Overall, we observe a clear segregation of aerosols and bubbles. While
aerosols move away from the symmetry axis due to centrifugal forces, bubbles
approach it. The larger the vortex strength, the stronger is the segregation.
Other geometries break this symmetry and lead to a more mixed distribution of
aerosols and bubbles.

Figure 4: The distribution of aerosols and bubbles along the $y$ axis after
crossing half of the observation area at $x=25$ (black lines) and the whole
observation area at $x=55$ (magenta/gray lines). Particles where released
within the set up with a chain of islands (COI) in the observation area with
dimensions $[-5,55]{\times}[-3,3]$. The line styles represent the particles:
aerosols (dashed lines), bubbles (solid lines). The different rows identify
the source locations: (a), (b) (main flow, $S_{0}$) and (c),(d) (first island,
$S_{1}$). We only show the results for the large Stokes number
$\text{St}=0.03$. (Color online)

###  III.2 Measuring connectivity between islands

In this section, we study the connectivity between islands by measuring the
probability of transport of particles between source and target regions. To
this end we use the setups described in Subsec. II.3 and introduce a
probability measure to evaluate the transport between a source $S_{j}$ and a
target $O_{i}$ by computing the probability $P_{ij}$ as the ratio of the total
number of particles transported from $S_{j}$ to $O_{i}$ and the total number
of particles $N_{j}$ released at $S_{j}$:

$P_{ij}\approx\frac{\text{Prob}\\{S_{j}\,\to\,O_{i}\\}}{N_{j}}.$ (11)

This probability measure depends crucially on the physical properties of the
particles like their Stokes number as well as on the properties of the
hydrodynamic flow like the background velocity $u_{0}$ and the strength of the
vortices. Note that, the probability to reach a location depends on the
geometry of the setup, i.e. the positions of the islands in the flow. Although
the choice of the target ring area influences the quantitative aspects, it
does not affect the qualitative differences between the particle types. Once
the transport between $S_{j}$ and $O_{i}$ has been quantified for all $j$ and
$i$, we can represent the connectivity as a weighted directed graph in which
the nodes are given by the islands as source and target regions. We add the
main flow (left boundary of the observation in Fig. 1) as an additional source
and the end of the observation area (right boundary of the observation area)
as an additional target region. The edges of the graph are given by the
transport probabilities, thereby we only establish a link, if the transport
probability is larger than $0.01$. This procedure constructs a Lagrangian flow
network serving as a representation of the connectivity between different
locations in a hydrodynamic flow. The matrix of probabilities $P_{ij}$
contains the strength of the connections as weights that are multiplied with
the adjacency matrix in graph theory which usually contains only ones and
zeros depending on whether a connection exists or not.

We now investigate how the strength of the links depends on the properties of
the particles and the geometry. The differences in the spatial particle
distribution in the flow depending on the Stokes number revealed already an
important impact of the vortices (see Fig. 3 in Subsec. III.1), since we
observed, as expected, that aerosols are expelled from vortices while bubbles
are attracted by them.

Let us first discuss the probability of the transport for the $4$ particle
types. Though we expect the largest impact for larger Stokes numbers we choose
to discuss first the smaller Stokes number $\text{St}=10^{-3}$, since we can
show that even for small Stokes numbers where inertial effects are small,
there is a considerable impact. Fig. 5 shows the entries in the probability
matrix color coded by their strength. As expected, there are large
similarities for tracers and neutrally buoyant particles, which are also
reflected in the transport probabilities. Their values are close but not equal
as shown in Fig. 5, where we present the matrix of transport probabilities in
a logarithmic scale to emphasize smaller differences. More striking are the
differences for aerosols and bubbles, as inertial effects play a bigger role.
However, it is rather surprising that even for the case of such small Stokes
numbers, where the impact of inertia is still quite small, the transport is
different.

For the larger Stokes number those differences in transport probabilities are
much more pronounced and depend strongly on the geometry of the islands’
positions. For illustration, we plot the graphs of connectivity for the two
geometries (see Fig. 6). To visualize the Lagrangian flow network we use the
following convention for the strength of the connections: If the transport
probability $P_{ij}<0.01$ we do not draw any link, for $0.01\leq P_{ij}<0.1$
the connectivity is weak and drawn by a thin line, for $0.1\leq P_{ij}<0.5$
the connectivity is moderate reflected by a medium size line, and finally, for
$0.5\leq P_{ij}\leq 1$ the strong connectivity is visualized by a thick line.
Using this classification the results for vortex strength of $\omega=40$ are
shown in Fig. 6 for aerosols (a,c) and bubbles (b,d), respectively. We find
that the impact of the geometry is quite different for aerosols and for
bubbles. For aerosols (Fig. 6 a, b), we observe relatively sparse connectivity
compared to RG in the case of COI. This is due to the fact that aerosols are
expelled from the vortices and this happens mostly already in the first von
Kármán vortex street which the particles enter after their release. As a
consequence, they are expelled from the whole region where the islands
together with the vortex streets are located along the $x$-axis and move
further outside this region without hitting any other island’s influence until
they reach the end of the observation area. This effect is even more
pronounced for the stronger vortex strength (not shown). By contrast, in the
RG geometry, there is no confined area in which the von Kármán vortex street
is located and a particle can experience different islands and the vortices in
their wake on their way through the observation area. Therefore the
connectivity is much larger with different probabilities connecting the
different islands.

For bubbles (Fig. 6 b, d) the situation is exactly opposite. Here a large
connectivity is observed for the COI but only a sparse one for RG. Since
bubbles are entrained and transported by the vortices until the vortex
disappears they are moving inside the vortices from island to island in the
chain leading to a large connectivity. By contrast, in RG they will also be
entrained by a vortex, but when the vortex disappears due to dissipation, then
they will be transported mainly by the background flow to the end of the
observation area and only be influenced by other vortices if the geometry of
islands permits the impact of another vortex. Since the impact of vortices
decays exponentially with distance, such an influence is rather unlikely, if
the distance between islands is large enough.

Those results underline the paramount importance of the geometry of the
islands’ locations when inertial particles are considered and compared to
tracers. While tracers and neutrally buoyant particles exhibit a large
connectivity for all considered vortex strength and geometry, this is not the
case for inertial particles with larger Stokes numbers. In certain geometries
some locations can be reached only by one type of inertial particles (e.g.
aerosols) but not by the other (e.g. bubbles). This fact leads to a separation
of different types of particles in terms of transport. High connectivity
between islands we observed only for the chain geometry COI for bubbles and
for the random geometry RG for aerosols, while the connections for bubbles in
RG and aerosols in COI are rather sparse. It is important to emphasize that in
case of low connectivity we observe a number of islands which are not
reachable at all. This is fundamentally different from the tracer and
neutrally buoyant particle approach where all islands can be reached.

Figure 5: The transport probabilities for tracers and inertial particles.
Geometry, COI. Observation area $[-5,55]{\times}[-3,3]$. (a): tracers. (b):
the neutrally buoyant particles. (c): aerosols. (d): bubbles. Vortex strength
$\omega=40$. Figure 6: Connectivity for different geometrical setups for the
position of the islands. (a) and (b), COI and observation area
$[-5,55]{\times}[-3,3]$. (c) and (d), RG and observation area
$[-5,45]{\times}[-5,5]$. In (a) and (c), we display the connectivity for
aerosols. In (b) and (d), we display the connectivity for bubbles. The vortex
strengths and the Stokes number are $\omega=40$ and $\text{St}=0.03$,
respectively.

###  III.3 Estimation of the error in forecasting particle transport

In the previous section we have shown that certain types of particles
(aerosols and bubbles) cannot reach some islands in the flow, while tracers
would do. However, many applications studying the spatial distributions of
larvae, seeds and litter like plastic particles in the ocean use tracers
models. The same applies to particle laden flows in industrial settings, where
the Stokes numbers could be larger. Usually the argument is that the Stokes
numbers of these objects are small to justify this approach. But there is no
study to what extent this argument is really valid. In this section, we
quantify the deviations caused by approximating inertial particles by tracers.
These deviations will be given by the factors $\lambda_{ij}$ computed from the
ratio of transport probabilities $P_{ij}$ (see Subsec. III.2) for inertial
particles and tracers:
$\lambda_{ij}=\frac{P^{\text{inertial}}_{ij}}{P^{\text{tracer}}_{ij}}$. In
such a case, $\lambda_{ij}$ will describe how much the probability of
transport of particles between source and target regions of inertial particles
deviate from those computed for tracers. Since $P^{\text{inertial}}_{ij}$ (see
Subsec. III.2) is sensitive to the strength of the vortices (i.e., $\omega$),
the geometry (i.e., COI and RG) and the source of the particles (i.e.,
$S_{j}$), all these factors are reflected in $\lambda_{ij}$. Once
$\lambda_{ij}$ has been computed, we say that $P^{\text{inertial}}_{ij}$ will
be overestimated by the tracer approach (i.e., $P^{\text{tracer}}_{ij}$) if
$\lambda_{ij}<1$ and underestimated if $\lambda_{ij}>1$. In the following, we
compute the ratio of $\lambda_{ij}$ for aerosols
$\lambda^{\text{a}}_{ij}=\frac{P^{\text{a}}_{ij}}{P^{\text{t}}_{ij}}$, bubbles
$\lambda^{\text{b}}_{ij}=\frac{P^{\text{b}}_{ij}}{P^{\text{t}}_{ij}}$ with
respect to tracers for both $\text{St}=10^{-3}$ and $\text{St}=0.03$ applying
the vortex strength $\omega=60$.

Figure 7: The deviation of the transport probabilities of source and target
regions for aerosols and bubbles in relation to tracers. Geometry: COI.
Observation area: $[-5,55]{\times}[-3,3]$. The dashed and solid black lines
represent: ${\lambda^{\text{t}}_{ij}}=1$ and ${\lambda^{\text{t}}_{ij}}=0$,
respectively. The different colors/shades help to identify the source
location: $S_{0}$ (main flow) in red/black, $S_{1}$ (island 1) in blue/dark
gray, $S_{2}$ (island 2) in cyan/gray and $S_{3}$ (island 3) in green/light
gray. We represent aerosols in panels (a, b), while panels (c, d) show results
for bubbles. (Color online)

Let us first discuss the situation for the chain geometry COI (Fig. 7) since
there we have a systematic deviation due to symmetry of the set up. For
aerosols (Fig. 7 a,b) we find an overestimation of the transport modelled by
tracers, while transport is systematically underestimated for bubbles (Fig. 7
c,d). Even if the Stokes number is small like $\text{St}=10^{-3}$, the
deviation is considerable, namely the transport of aerosols is only half of
that modelled by tracers and for bubbles it is twice as large as the tracer
equivalent. For the large Stokes number $\text{St}=0.03$ the error is, as
expected, much larger: aerosols in many cases do not even reach the targets,
their transport becomes zero, while for bubbles the transport is up to $15$
times larger than estimated by tracers. For the random geometry RG with its
random island disposition, we do not find such large systematic deviations.
Over- and underestimation of transport in this geometry depends strongly on
the considered source and target locations. Particularly, for bubbles we
observe that the transport probabilities approach zero even for the small
Stokes number. For certain target locations the transport probabilities would
be larger by factors $2$ for aerosols or $5$ bubbles even for the small Stokes
number $\text{St}=10^{-3}$. For the large Stokes number $\text{St}=0.03$ the
effects are much more pronounced leading to factors of $5$ for aerosols and
$>12$ for bubbles.

From those findings we can conclude that one has to be very careful with
predictions of studies of transport of particles based on tracers, the real
objects should better be treated as inertial particles to obtain reasonable
estimates of particle transport.

##  IV Conclusions

We have employed the Maxey-Riley-Gatinol equations to describe the motion of
spherical inertial particles in a flow to establish the links in a Lagrangian
flow network connecting a certain number of islands. In order to study such
networks depending on the properties of the flow and the particles we have
introduced a kinematic two-dimensional flow field which allows for placing an
arbitrary number of islands at prescribed positions in an open background flow
with a given direction. Using this approach we have demonstrated that the
Lagrangian flow networks for inertial particles may quite deviate
substantially from the ones of tracer particles. As it is well-known that
inertial particles do not follow the flow exactly but have their own paths
depending on their size and density, deviation from Lagrangian networks
obtained with tracers had to be expected. Mesoscale hydrodynamic vortices in
the flow cause segregation of aerosols and bubbles. Bubbles are entrained and
transported within the vortex and released from it when the vortex disappear
due to dissipation. By contrast, aerosols are expelled from vortices. As a
consequence, some target regions can only be reached by bubbles and not by
aerosols and vice-versa depending on the geometry of the setup. Tracers and
neutrally buoyant particles spread trough the flow field and reach all parts
of the observation area in slightly higher or slightly lower concentrations.
Inertial particles, on the other hand, experience a much stronger effect of
the vortices and concentrate only in certain regions of the flow field. The
consequence is that they, in general, have a smaller probability to reach a
target region, however if they reach it they would do it in much higher
numbers. This is the reason why in a random geometry the connectivity is
larger for aerosols and sparse for bubbles because the latter get trapped in
the vortices. The opposite is true for the chain of island where we find
higher connectivity for bubbles. These distinct connectivity pattern cannot be
observed for tracers and neutrally buoyant particles, which in our setups
always realised all possible connections. We can conclude that the computation
of transport paths of inertial particles employing an approximation by tracers
can lead to considerable over- or underestimation of transport probabilities.

According to our aim to provide a theoretical study of the impact of inertia
on Lagrangian flow network, we have chosen Stokes numbers St and density
ratios $\beta$ which are rather different from seeds, larvae or plastic
particles in order to make the difference obvious. However, we could show that
even for small Stokes numbers, the deviations are not negligible. Therefore,
one should carefully check if whether a tracer approximation is suitable if
ones wants to predict transport probabilities. In our study we have restricted
ourselves to two dimensional flows which neglects the possible sinking of
particles having densities larger than the fluid. For such aerosols the
sinking process should also be taken into account. This applies also for the
density ratio $\beta$ which we used for our study. But our focus was also on
the impact of the geometry of the setup and this could only be achieved by
using 2D flow which is governed by a stream function. Therefore, we could not
take sinking velocities into account.

Finally we would like to address another simplification we made: The
generation of the vortices was uniform across all islands such that they were
synchronised in their behaviour. In an additional study (see the appendix) we
have taken randomised the times at which vortices detach from each island
which makes the setup more realistic. We find that, as we would expect, more
mixing is introduced in the flow filed and particle distributions for small
Stokes numbers look more similar to the ones of tracers and neutrally buoyant
particles (Appendix Fig. 8). However, qualitatively we obtain the same results
but with increasing transition probabilities and small and large numbers (Fig.
9) due to the increasing mixing and slightly different connectivity maps (Fig.
10).

##  V Acknowledgments

We want to thank Tamás Tél for the hospitality at Eötvös Loránd University
Budapest, Hungary, which led to valuable discussions as well as suggestions.
Furthermore, we would like to acknowledge the contributions of Bernd Blasius,
Dirk C. Albach, Matthias Schroeder, Michael Kleyer, Rahel Vortmeyer-Kley, and
above all, the DFG research unit DynaCom. Finally, we acknowledge the HPC
Cluster CARL with which the simulations were performed. It is located at the
University of Oldenburg (Germany) and funded by the DFG through its Major
Research Instrumentation Program (INST 184/157-1 FUGG) and the Ministry of
Science and Culture (MWK) of the Lower Saxony State, Germany. The authors
acknowledge financial support from the German Science Foundation (DFG) for the
project No.379417748, as subproject $6$, ’modelling hydrodynamics and passive
dispersal’, in the research group DynaCom (Spatial community ecology in highly
dynamic landscapes: from island biogeography to metaecosystems).

##  Appendix A Detailed analysis of randomized vortex release

To reproduce the results in appendix, we had to adjust the equations which
describes the presence of the counter-rotating vortices in the wake of each
island as follow

$\begin{split}g_{pi}(x,y,t)=e^{-R_{0i}[-(x-x_{pi}(t))^{2}+\alpha_{i}^{2}(y-y_{pi}(t))^{2}]},\\\
p=1,2\;\text{and}\;i=1,\cdots,n\end{split}$ (12)

with $1/\sqrt{R_{0i}}$ and $\alpha_{i}$ as the radius and the characteristic
ratio between the elongation of the vortices in $x$-and $y$-direction,
respectively. Here each pair of vortices moves a distance L during time
${T_{c}}$ and then disappears due to dissipation. But the centers of the
vortices are modulated by

$\displaystyle H_{1i}(t+\delta_{ti})$ $\displaystyle=$
$\displaystyle\sin^{2}((t+\delta_{ti})\pi),$ (13a) $\displaystyle
H_{2i}(t+\delta_{ti})$ $\displaystyle=$
$\displaystyle\cos^{2}((t+\delta_{ti})\pi),$ (13b)

move, along the horizontal; direction $x$ according to

$\displaystyle x_{1i}(t+\delta_{ti})$ $\displaystyle=$ $\displaystyle
h_{i}+r_{i}+\text{L}_{i}\;\text{mod}\;((t+\delta_{ti}),T_{c}),$ (14a)
$\displaystyle y_{1i}(t+\delta_{ti})$ $\displaystyle=$ $\displaystyle
k_{i}+y_{0i},$ (14b) $\displaystyle x_{2i}(t+\delta_{ti})$ $\displaystyle=$
$\displaystyle x_{1i}((t+\delta_{ti})-T_{c}/2),$ (14c) $\displaystyle
y_{2i}(t+\delta_{ti})$ $\displaystyle=$ $\displaystyle
k_{i}-y_{0i},\;i=1,\cdots,n.$ (14d)

Here the amplitude modulation and horizontal progression of vortices are
governed by harmonic functions, with a temporal adjustment factor
$\delta_{ti}$ reflecting the staggered release of vortices. The new component
$\delta_{ti}$ introduces variability in the vortices’ initiation. In case of
synchronization all $\delta_{ti}$ values are set to zero for all vortices,
which leads all of them to be released at the same time instant (the set up of
the simulation described in the main text).

###  A.1 Comparing trajectories of particles with inertia to tracers

Fig. 8 shows the distribution of particles in the flow with randomized
detachment of the vortices. We observe that the additional mixing introduced
by the randomization of the vortex detachment has an impact on the
distribution patterns for small Stokes numbers. They become more similar to
the ones of tracers and neutrally buoyant particles. For large Stokes numbers
this mixing effect is not small pronounced.

This additional mixing effects also the transition probabilities for particles
with small Stokes numbers. They become larger for all types of particles (Fig.
9). Finally we checked the resulting connectivity maps (Fig. 10). For aerosols
we observe a few more connections due to the increase in mixing indicating a
strong effect on expelling the particles. By contrast, the connectivity map
for bubbles with large Stokes number remain unchanged in the number of
connections only the transition probabilities are increased. This can again be
explained by the trapping of bubbles in the vortices.

Figure 8: Spatial distribution of tracers and inertial particles in the wake
of six islands, considering a randomized release of vortex. From the top to
the bottom, (a): ideal tracers. (b) and (c): neutrally buoyant particles for
$\text{St}=10^{-3}$ and $\text{St}=0.03$, respectively. (d) and (e): aerosols
for $\text{St}=10^{-3}$ and $\text{St}=0.03$, respectively. (f) and (g):
bubbles for $\text{St}=10^{-3}$ and $\text{St}=0.03$, respectively. Those
snapshots were taken at a time $23.48\text{T}_{\text{c}}$ for vortex strength
$\omega=60$ and $\delta_{1}=0$, $\delta_{2}=1.79393233$,
$\delta_{3}=1.35884153$, $\delta_{4}=1.40654021$, $\delta_{5}=0.11204869$ and
$\delta_{6}=1.90098945$.

###  A.2 Measuring connectivity between islands

The figure 9 presents transport probability matrix that indicates the
likelihood of particles transitioning from one island to another island. This
matrix design enables a comparative analysis of particle behaviour under
randomized vortex conditions, identifying differences in transport dynamics
due to particle properties such as inertia and buoyancy. The figure shows a
quantified visual representation of how the randomized release of vortices
affects the dispersal and eventual settling of various particle types,
offering insights critical for understanding particle transport phenomenon in
both natural fluid environments and industrial applications.

Figure 9: The transport probabilities for tracers and inertial particles,
considering a randomized release of vortex. Geometry, COI. Observation area
$[-5,55]{\times}[-3,3]$. (a): tracers. (b): the neutrally buoyant particles.
(c): aerosols. (d): bubbles. Vortex strength $\omega=40$ and $\delta_{1}=0$,
$\delta_{2}=1.79393233$, $\delta_{3}=1.35884153$, $\delta_{4}=1.40654021$,
$\delta_{5}=0.11204869$ and $\delta_{6}=1.90098945$.

In the Fig. 10, we illustrate the probability of particle transitions from one
island to another under a randomized vortex release timing, highlighting the
increased complexity and randomness introduced by the vortex release timing.

Figure 10: Connectivity for different geometrical setups for the position of
the islands, considering a randomized release of vortex. (a) and (b), COI and
observation area $[-5,55]{\times}[-3,3]$. (c) and (d), RG and observation area
$[-5,45]{\times}[-5,5]$. In (a) and (c), we display the connectivity for
aerosols. In (b) and (d), we display the connectivity for bubbles. The vortex
strengths and the Stokes number are $\omega=40$ and $\text{St}=0.03$,
respectively. With $\delta_{1}=0$, $\delta_{2}=1.79393233$,
$\delta_{3}=1.35884153$, $\delta_{4}=1.40654021$, $\delta_{5}=0.11204869$ and
$\delta_{6}=1.90098945$.

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