# Lagrangian transport in a microtidal coastal area: the Bay of Palma, island
of Mallorca, Spain

Ismael Hernández-Carrasco  Cristóbal López  Emilio Hernández-García  IFISC,
Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), 07122
Palma de Mallorca, Spain  Alejandro Orfila  IMEDEA, Instituto Mediterráneo de
Estudios Avanzados (CSIC-UIB), 07190 Esporles, Spain

###### Abstract

Coastal transport in the Bay of Palma, a small region in the island of
Mallorca, Spain, is characterized in terms of Lagrangian descriptors. The data
sets used for this study are the output for two months (one in autumn and one
in summer) of a high resolution numerical model, ROMS, forced atmospherically
and with a spatial resolution of 300 m. The two months were selected because
its different wind regime, which is the main driver of the sea dynamics in
this area. Finite-size Lyapunov Exponents (FSLEs) were used to locate semi-
persistent Lagrangian coherent structures (LCS) and to understand the
different flow regimes in the Bay. The different wind directions and
regularity in the two months have a clear impact on the surface Bay dynamics,
whereas only topographic features appear clearly in the bottom structures. The
fluid interchange between the Bay and the open ocean was studied by computing
particle trajectories and Residence Times (RT) maps. The escape rate of
particles out of the Bay is qualitatively different, with a 32$\%$ more of
escape rate of particles to the ocean in October than in July, owing to the
different geometric characteristics of the flow. We show that LCSs separate
regions with different transport properties by displaying spatial
distributions of residence times on synoptic Lagrangian maps together with the
location of the LCSs. Correlations between the time-dependent behavior of FSLE
and RT are also investigated, showing a negative dependence when the stirring
characterized by FSLE values moves particles in the direction of escape.

##  I Introduction

The study of transport and mixing in coastal flows is of major interest
because of their economic and ecological importance. Due to the
particularities that they present, like influence of complex topography,
coastline shape and the direct driving at the surface by highly variable wind
forcing, coastal flow dynamics remains still poorly understood.

Recently, coastal observations and modeling efforts in different regions have
been addressed from the Lagrangian point of view: Lekien et al. (2005) showed
that Lagrangian Coherent Structures (LCSs) computed from velocity fields
obtained from HF Radar measurements can be used to predict pollutant
dispersion in the coast of Florida; Gildor et al. (2009) and Shadden et al.
(2009) detected LCSs with HF Radar data in the Gulf of Eliat, Israel, and in
Monterey Bay, respectively. Haza et al. (2010) studied small-scale properties
of dispersion measurements obtained from HF Radar data in the Gulf of La
Spezia, Italy. Also Nencioli et al. (2011) have detected LSCs in a coastal
region with a Lyapunov method based on in situ observations. Besides Radar
measurements, LCSs obtained from velocity data of high resolution numerical
models have been used to analyze the effect of the waves on LCS in the Bay of
Palma, Spain (Galan et al., 2012), to study the transport in the tidal flow of
Ria de Vigo, Spain (Huhn et al., 2012), or to study the water quality of a
very small coastal region, the Hobie Beach, USA (Fiorentino et al., 2012)
Also, data from drifters released in the Santa Barbara Channel were used by
Ohlmann et al. (2012) to characterize relative dispersion, very useful to
improve Lagrangian stochastic models. The application of Lagrangian techniques
to study the dynamics in a shallow lake (small closed basin) has been
performed in Pattantyús-Ábrahám et al. (2008).

Palma is the largest city in the Balearic Islands. Human activities, in
particular recreational ones, give to water quality in the Bay of Palma a
large economic value. A proper analysis of transport can be useful to
understand the fluid dynamics in the Bay and therefore help protect the
coastal water. Previous studies performed in the Bay of Palma used Eulerian
techniques to understand the coastal dynamics (Jordi et al., 2009, 2011). In
this work we study some transport properties in the Bay of Palma using
Lagrangian techniques developed from dynamical systems theory. Computing both
LCSs and residence times the Bay of Palma can be sorted in regions of
different properties, for example having more or less connectivity with the
open ocean. This kind of studies have demonstrated to be useful to identify
pollution pathways or conditions for red tides (Lekien et al., 2005;
Fiorentino et al., 2012). The Bay is a semi-enclosed basin located in the
southwest of the island of Mallorca (western Mediterranean sea), whose coastal
flow is mainly induced by the wind (Jordi et al., 2009, 2011). Forcing by
tides is almost negligible with a tidal amplitude of less than 0.25$m$. This
makes the dynamics here different form other locations (e.g. Shadden et al.
(2009); Huhn et al. (2012)) where tides are dominant, and then provides the
opportunity to test the performance of dynamical systems tools in this
situation in which forcing only acts directly on the sea surface, and in which
there are rather different forcing regimes depending on the season. The
Lagrangian diagnosis will be obtained from velocity data of a realistic
numerical model at high resolution, which resolves spatial scales of a few
hundred of meters. We investigate the surface horizontal transport during two
months corresponding to different seasons (autumn and summer), and therefore
to different wind conditions, in order to highlight the effect of the wind on
transport. In the case of July we also study the deepest bottom layer. We
compute the barriers and avenues to transport (LCS) from lines of high values
of Finite-Size Lyapunov Exponents (FSLE). We also present calculations of
residence times and show synoptic Lagrangian maps (SLM) of these times
(Lipphardt et al., 2006), which will allow us a detailed visualization of the
interchange of fluid particles between the Bay and the open sea. The
relationship between LCSs and areas of different residence times will be
analyzed.

The organization of this paper is as follows. The data set used in the
computations and the area of study is described in Section II. Section III
presents a brief overview of the Lagrangian tools that are used. Before
presenting the Lagrangian results, we show in Section IV a short summary of
Eulerian results by studying the velocties in the Bay. We present in Section V
a characterization of stirring in the Bay of Palma in terms of FSLE and
residence times. Using the definition of LCS given in Section III, Lagrangian
barriers are identified in the domain of interest. We compute escape rates and
residence times of fluid particles to describe the transport relation between
the Bay and the open ocean. We provide possible mechanisms to explain
differences in the residences times and FSLE between different seasonal
months. Finally we summarize the main results in Section VI.

Figure 1: Bathymetry contours (in meters) of the model domain. The black box
indicates the Palma Bay and the inset graphics give the geographical location
of Mallorca Island in the western Mediterranean Sea.

##  II Data and characteristics of the study region

###  II.1 Area of study

The island of Mallorca (Fig. 1) is part of the Balearic Islands Archipelago
and is located in the center of the western Mediterranean (between 39∘ and
40∘N and 2.50∘ and 3.50∘ E). The Bay of Palma is a nearly semi-circular and
semi-enclosed basin located in the southwest coast of Mallorca and it can
reach depths of more than 60 $m$. The Bay of Palma is defined as the water
mass inside the square in Fig.1, consisting of a northern limit at 39∘34′N, a
southern limit at 39∘24′N, and 2∘30′E and 2∘45′E as the western and eastern
limits, respectively. The open boundary to the sea is in the southern part and
it is 20 km wide.

The size of the Bay is smaller than the Rossby radius of deformation at these
latitudes, and the main circulation is determined by the bathymetry at the
bottom layer and by local and remote winds at the surface layer. In particular
the studies by Jordi et al. (2009, 2011) have shown that the major forcing
mechanisms come from wind-induced island trapped waves (ITW) propagating at an
island scale and by locally wind-induced mass balance. The intense ITW can
produce new instabilities which can generate coastal gyres at submesoscale
(see Jordi et al. (2011)). During summer there are persistent sea breeze
conditions. In July and August, the weather is often almost identical from one
day to the next. In the vicinity of the Bay and along the southern coast of
Mallorca the breeze blows from the south-west. Several studies (Ramis and
Alonso, 1988; Ramis and Romero, 1995), have pointed out that the
meteorological conditions of Mallorca (intense solar radiation, clear skies,
soil water deficit, dryness, weak surface pressure gradients, etc.) favors the
development of sea breeze, often from April to October, and almost every day
during July and August. Winds in autumn, and particularly in late September
and October are more irregular, with episodes of strong storm activity (Tudurí
and Ramis, 1997).

###  II.2 Data

The velocity data sets were obtained from the numerical model ROMS (Regional
Ocean Model System). ROMS is a free surface, hydrostatic, primitive equation
ocean model. The model uses a stretched, generalized nonlinear coordinate
system to follow bottom topography in the vertical, and orthogonal curvilinear
coordinates in the horizontal (Song and Haidvogel, 1994; Haidvogel et al.,
2000). At each grid point, horizontal resolution $\Delta_{0}$ is the same in
both the longitudinal, $\phi$, and latitudinal, $\theta$, directions.

We run the simulation with a resolution of $\Delta_{0}=0.0027^{\circ}$
($\sim$300$m$, ROMS300), which is itself nested into a larger and coarser grid
with $\Delta_{0}$ =1/74∘ ($\sim$1500$m$). Boundary conditions for the coarser
domain were taken from daily outputs of the Mediterranean Forecasting System
(Dobricic et al., 2007; Oddo et al., 2009). The ROMS300 domain covers
$39^{\circ}$12′N - $39^{\circ}$36′N (latitude), and $2^{\circ}$24′E -
$3^{\circ}$6′E (longitude). The total number of grid nodes is 260 $\times$
148\. Vertical resolution is variable with $10$ layers in total. All domains
were forced using realistic winds provided by the PSU/NCAR mesoscale model
MM5. The initial vertical structure of temperature and salinity was obtained
from the Levitus database (Locarnini et al., 2006; Antonov et al., 2006).

We will manage velocity data from the surface layer and the bottom layer for
the grid of $\Delta_{0}$ $\approx$ 300$m$. This domain allows us to analyze
the fluid interchange between the Bay and the open ocean, using a high
resolution velocity field. Only horizontal velocities are considered, so that
vertical displacements are neglected in the surface layer, and particles in
the bottom remain in the bottom layer. This is justified by the small
integration times we will use. Nevertheless, close to the coast they can have
an impact that will be the subject of future work. The output of the model was
compared with data from drifters (see Galan et al. (2012)) and a reasonable
agreement was found, although it improved when adding the influence of wave
intensity. Thus the present study should be considered as a simplified
baseline case against which to compare the future consideration of the full 3d
dynamics, or the influence of small scale process such as waves (Galan et al.,
2012). We will study two different intervals of time corresponding to two
different wind regimes: one starting on October $5$th, $2008$ and finishing on
October $29$th, $2008$; and the other extending from July $1$st, $2009$ until
July $26$th, $2009$. Temporal resolution is 15 minutes and 10 minutes for
October and July, respectively, resulting in a total of $2375$ snapshots of
the velocity field for October, and $3744$ for July.

##  III Methodology

###  III.1 LCSs and particle dispersion from FSLE

Our methodology is based on the Lagrangian analysis of marine flows. In the
Lagrangian view, particles are advected by the flow and their horizontal
motion (neglecting motions between model layers) is governed by the
differential equations

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle{v_{x}(x,y,t)},$
(1) $\displaystyle\frac{dy}{dt}$ $\displaystyle=$
$\displaystyle{v_{y}(x,y,t)},$ (2)

where ($x(t),y(t)$) are the west-east and the south-north coordinates of the
trajectories and ($v_{x},v_{y}$) are the eastwards and northwards components
of the velocity. Because of the small sizes involved, we will use a Cartesian
coordinate system.

LCSs (Haller and Yuan, 2000; d’Ovidio et al., 2004; Shadden et al., 2005), are
roughly defined as the material lines organizing the transport in the flow.
They are the analogs, for time-dependent flows, of the unstable and stable
manifolds of hyperbolic fixed points. Among other approaches (Mancho et al.,
2006; Mendoza and Mancho, 2010; Mezić et al., 2010; Rypina et al., 2011;
Haller and Beron-Vera, 2012), ridges of the local Lyapunov Exponents provide a
convenient tool to locate them. In our case, we use the so-called Finite-Size
Lyapunov Exponents (FSLEs) which are the adaptation of the asymptotic
classical Lyapunov Exponent to finite spatial scales (Aurell et al., 1997;
Boffetta et al., 2001). FSLEs are a local measure of particle dispersion and
thus of stirring and mixing, as a function of the spatial resolution, serving
to isolate the different regimes corresponding to different length scales of
the oceanic flows, very useful in coastal systems (Cencini et al., 2010). In
fact the first applications of the FSLE technique in oceanography were for
closed or semi-closed basins (Buffoni et al., 1996, 1997).

For two particles of fluid, one of them located at x, the FSLE at time $t_{0}$
and at the spatial point x is given by the formula:

$\lambda(\textbf{x},t_{0},\delta_{0},\delta_{f})=\frac{1}{|\tau|}\ln{\frac{\delta_{f}}{\delta_{0}}},$
(3)

where $\delta_{0}$ is the initial distance of the two given particles, and
$\delta_{f}$ is their final distance. Thus, to compute the FSLEs we need to
calculate the minimal time, $\tau$, needed for the two particles initially
separated $\delta_{0}$, to get a final distance $\delta_{f}$ (in this way the
FSLE represents the inverse time scale for mixing up fluid parcels between
length scales $\delta_{0}$ and $\delta_{f}$). To obtain this time we need to
know the trajectories of the particles (from Eqs. (1) and (2)) which gives the
Lagrangian character to this quantity. The FSLEs are computed for the points x
of a square lattice with lattice spacing coincident with the initial
separation of fluid particles $\delta_{0}$. We can obtain a good estimation of
the minimal $\tau$ at each site by selecting the trajectory which diverges the
first among the four trajectories starting at the neighbors of the given site
in the grid of initial conditions. Numerically we integrate the equations of
motion using a standard, fourth-order Runge-Kutta scheme, with an integration
time step corresponding to the time resolution of the velocity data: $dt=15$
minutes in October and $dt=10$ minutes in July. We have checked in selected
trajectories that using in July the same time step $dt=15$ as in October does
not alter the trajectories. Since velocity information is provided just in a
discrete space-time grid, spatiotemporal interpolation of the velocity data is
achieved by bilinear interpolation. For the spatial scales that define FSLEs,
we take $\delta_{f}=0.1^{\circ}$, i.e., final separations of about $10\ km$,
because of the size of the Bay. On the other side, we take $\delta_{0}$ equal
to $75\ m$, four times smaller than the resolution of the velocity field,
$\Delta_{0}=300\ m$. Since we are interested only in fast time scales, our
integrations are restricted to 5 days. Locations for which the final
separation at the end of this period has not reached the prescribed
$\delta_{f}=10\ km$ (or for which particles have been trapped by land) are
assigned a value $\lambda=0$.

FSLEs can be computed from trajectory integration backwards and forward in
time. Their highest values as a function of the initial location, x, organize
in filamental structures approximating relevant manifolds: ridges in the
spatial distribution of backward (forward) FSLEs identify regions of locally
maximum compression (separation), approximating attracting (repelling)
material lines or unstable (stable) manifolds of hyperbolic trajectories,
which can be identified with the LCSs (Haller and Yuan, 2000; d’Ovidio et al.,
2004; Shadden et al., 2005; Tew Kai et al., 2009; Hernández-Carrasco et al.,
2011), and characterize the flow from the Lagrangian point of view (Joseph and
Legras, 2002; Koh and Legras, 2002). Attracting LCSs associated to backward
integration (the unstable manifolds) have a direct physical interpretation
(Joseph and Legras, 2002; d’Ovidio et al., 2004, 2009). Tracers (chlorophyll,
temperature, …) spread along these attracting LCSs, thus creating their
typical filamental structure (Tél and Gruiz, 2006; Lehan et al., 2007; Tew Kai
et al., 2009; Calil and Richards, 2010). When not stated explicitly, by FSLE
we will mean the backwards FSLE values. In addition to locate spatial
structures, time-averages of FSLE give an indication of the intensity of
stirring in given areas, which we analyze in Sect. V.1.

We close this section by noting that the relationship between LCSs and
Lyapunov exponents is based on heuristic arguments which may not be correct in
some cases (see for example Haller (2011)). We identify as possible LCSs only
the locations having the largest values of FSLE, which align in linear
structures. In this way we effectively select only the highest FSLE ridges
which are more likely to organize the flow. Even in this case, it is possible
that the FSLE technique identifies regions of high shear which are not
hyperbolic and then may lack some of the properties of bona fide LCSs. Thus,
direct inspection of particle trajectories and comparison with complementary
techniques would be needed to confirm the validity of the FSLE approach in
this situation. One of such complementary techniques is residence time maps
that we present in the following section.

###  III.2 Escape and residence times

Another characteristic time-scale for transport processes in open flows is the
so-called escape rate (Lai and Tel, 2011). This quantity measures how quickly
particles trajectories escape from a domain. If we initiate $N(0)$ particles
in a flow, we can measure how the trajectories escape the preselected region.
In the case in which the decay in the number of particles remaining in the
region up to time $t$, $N(t)$, decays exponentially with time, $N(t)/N(0)\sim
e^{-\kappa t}$, there is a well-defined escape time defined as the inverse of
the escape rate $\kappa$: $\tau_{e}=1/\kappa$. For the range of times explored
in our work, we will see that the particle escape is close to exponential and
then we can estimate the value of $\tau_{e}$.

$\tau_{e}$ is a global quantity associated to the whole basin. A more detailed
description of the transport processes can be obtained by other suitable
Lagrangian quantities such as residence times (Buffoni et al., 1996, 1997;
Falco et al., 2000; Orfila et al., 2005). The particle residence time (RT) is
defined as the interval of time that a fluid particle remains in a region
before crossing a particular boundary. For each fluid particle inside the Bay
at an initial time, we need to compute two times: the forward exit time,
$t_{f}$, computed as the time needed for a particle to cross the line
delimiting the Bay, taking the forward-in-time dynamics; and the backward exit
time, $t_{b}$, the same but in the backward-in-time dynamics. The residence
time is defined as $RT=t_{f}+t_{b}$. RTs can be displayed in plots named
Lagrangian Synoptic Maps (Lipphardt et al., 2006), in which the residence time
of each fluid particle is referenced to its initial position on the grid.

##  IV Preliminary Eulerian description

A first approach to the transport process in the Bay can be a description from
the Eulerian point of view, by studying averages of the velocity field. To do
this we consider separately the meridional $v_{y}$ and zonal $v_{x}$
components of the surface flow, and we analyze the time evolution of their
spatial averages.

Figure 2: a) Complete time series throughout October of the zonal (top panels)
and meridional (bottom panels) of the spatial average of the surface velocity
field (black line). The red line is a running daily average. b) the same as a)
but for July.  Figure 3: Spectra for the zonal (left panels) and meridional
component (right panels) of the surface velocity field ($m^{2}/s$)in October
(top) and July (bottom)

Figures 2 a) and b) show the time series of data taken every 15 min in October
and 10 min in July (black lines), and daily average time series (red lines) of
$v_{x}$ and $v_{y}$ for October and July, respectively. The impact of the more
variable and stormy weather in October is clear in the high frequency
variability of the time-series. During the two months both components of the
flow present daily variability related to the presence of land and sea
breezes. In July the zonal fluctuations are much more noticeable and regular
than the meridional ones, being $\langle v_{y}\rangle$ very small. We have
computed the power spectra for both months (see Fig. 3). In October, in
addition to higher power at high frequencies, there are also stronger low-
frequency fluctuations. From such features in their spectra of ADCP-derived
velocities Jordi et al. (2011) identified wind-induced island trapped waves as
the main source of variability in the Bay dynamics, in addition to the local
wind (essentially sea breeze). In contrast, the dominant role of sea breeze in
July is seen as the very strong dominance of the daily frequency peak at the
July zonal spectrum.

Comparing the velocity components of both months we observe quantitative
differences. The values of $v_{y}$ in the case of October range from -1.0 to
1.5 $m/s$ (bottom panel of Fig. 2 a), while in the case of July, $v_{y}$ is
two orders of magnitude smaller, ranging from -0.1 to 0.02 $m/s$ (bottom panel
of Fig. 2 b). On other hand, $v_{x}$ are similar during October and July. In
October, $v_{x}$ ranges from -1.5 to 0.5 $m/s$ (top panel of Fig. 2 a), the
same order of magnitude than the meridional velocity, resulting in circular
motions (clockwise along the Bay). In July the situation is significantly
different. The zonal velocity ranges from -1.5 to 0.7 $m/s$ (top panel of Fig.
2 b), much larger than the meridional velocity, resulting in a flow consisting
on oscillations along the zonal direction. In October the mean values (and the
standard deviations given in parenthesis) of the time series are
$<v_{x}>=-0.0704~{}(0.1897)m/s$ and $<v_{y}>=0.0440~{}(0.1982)m/s$. In July we
have $<v_{x}>=0.0013~{}(0.3052)m/s$ and $<v_{y}>=-0.0140~{}(0.0134)m/s$. The
large standard deviation in the zonal velocity in July is an indicator of the
large (breeze induced) daily fluctuations in this month, but restricted to a
single direction of motion.

##  V Lagrangian Results

###  V.1 Average characterization of stirring

We now describe our Lagrangian results. First we compute the temporal average
(over the months of October and July) of the FSLEs for the surface layer, and
for July in the bottom layer. This calculation helps us to unveil areas of
different stirring and the differences between layers and months.

Figure 4: Spatial distribution of the time average of 6-hourly FSLEs maps over
different months and at different layers: a) October at surface layer, b) July
at surface layer, c) July at bottom layer.

The surface computations for the different seasonal months, October and July
(Fig. 4 a, b) show different values and spatial distributions of stirring. We
use the same colorbar to compare the stirring in both months. The Bay of Palma
appears to be an area with important activity. Average FSLE field looks more
homogeneous in July than in October. During October filamental structures of
high values of FSLE are accumulated over the northeast side of the Bay,
forming a linear structure running from north to south-east which comes from
similar structures in the instantaneous (non-averaged) fields that can act as
barriers, therefore dividing the Bay in two flow regions of qualitatively
different dynamics. The difference in wind regularity and intensity between
these months, and the fact that local and remote winds are the main drivers of
the bay dynamics, explains the difference in mean stirring distribution among
the two months. The importance of wind will be replaced by bottom topography
when going to the deep layers. The effect of the terrain topography on
stirring is clear in Fig. 4 c), where FSLEs are computed at the deepest layer
for July. The high values of time-averaged FSLEs are located close to a region
of high bathymetry gradient, which seems to act as a barrier along which the
flow is stretched.

Figure 5: Evolution of the locations of two sets of particles in the Palma Bay
during a night of October, superimposed on the spatial distributions of high
values of backward FSLEs. The colorbars specify FSLE values in days-1. Zero
FSLE values, displayed as white, are assigned to locations for which the
particles do not attain the prescribed $\delta_{f}=10km$ separation after
5-days integration. Note the highest values of FSLEs (green lines) act as a
barrier practically dividing the Bay in two parts. The two sets of particles
are deployed from both sides of the barrier. (a) Initial conditions of the
particles on October 8 at 20:00 GMT, 2008.; (b) October 9 at 00:00 GMT, 2008 ;
(c) October 9 at 04:00 GMT, 2008 ; (d) October 9 at 08:00 GMT, 2008. Particles
marked by black dots were released in the right side (northeast) of the
barrier while the particles marked with red were released on the left side of
the barrier.

###  V.2 Coastal LCSs

The temporal averages computed in Sec. V.1 give us a rough idea of stirring in
the Bay. More detailed information is obtained by looking at non-averaged
quantities, that may reveal the existence of barriers to transport. Figure 5
shows the location of the high backward FSLE values (LCSs), appearing as a
network of lines, computed at successive instants of time in October. These
temporary structures can remain for one or more days, as happens in October,
or they can appear in the same location periodically (not shown). We stress
here the appearance of a clear barrier, from north to south-east, that divides
the Bay in two areas that correlates with the temporal average in Fig. 4 a).
This barrier appears in almost the same position in different days, remaining
without displacing too much. To effectively see that it acts as a barrier we
have considered the evolution of virtual particles released at both sides of
the barrier. Red and black particles do not mix and they tend to spread along
the barrier (confirming that, as expected, it is an attracting line).

In July the situation is rather different. Lines of high Lyapunov exponents
(forward and backwards) are mainly oriented zonally in the bay (except close
to the opening to the sea), which is also the dominant direction of motion.
Thus, it does not seem that they represent hyperbolic LCS, but rather lines of
intense shear between zonally moving strips.

Figure 6: Average of 15 subsequent (started at $t_{0}$ values separated 18
hours) estimations of $N(t)$, the number of particles remaining in Palma Bay
at least for a lapse of time $t$ after release at $t_{0}$. Black and red lines
are for surface layer in October and July respectively. Dashed lines are the
measured averages, and the solid lines are the indicated exponential fits.

###  V.3 Transport between the Bay of Palma and the open sea

In this section we study the surface transport of particles in and out of the
Bay. To have an idea of the time scales involved in this interchange we
proceed by computing the number of particles remaining in the Bay, $N(t)$,
averaged over different starting times (separated by $18$hours in order to
collect the information of diurnal and nocturnal signal; this gives us 15
different simulations to be averaged for each month) as a function of the
integration time $t$. A particle is considered to leave the Bay when crossing
the red open-sea boundary in Fig. 1, so that particles landing on the coast
are considered not escaped. Fig. 6 shows the different average decays for
October and July. In both cases $N(t)$ is reasonably fitted by an exponential
in the considered time-range, thus identifying the escape rates $\kappa$ =
0.62 and 0.47 $days^{-1}$, respectively. The corresponding escape times, given
by the inverse of the escape rate, are, respectively, $\tau_{e}$ = 1.61 and
2.12 $days$. The relative difference of the escape rates of July with respect
to October, $(\kappa_{October}-\kappa_{July})/\kappa_{July}$, is 0.32. Thus
the exchange of fluid particles between the Bay and the open ocean is a 32$\%$
more active in autumn than in summer.

Next we compute synoptic maps of the residence times. As was indicated in Sec.
III.2 the residence time of the particles throughout the study area is
considered as the sum of the entry time ($t_{b}$) and the escape time
($t_{f}$). To compute $t_{f}$ and $t_{b}$ particles are initialized every 6
hours in a regular grid of 75$m$ spacing and they are integrated forward and
backward in time during 5 days. We consider that 5 days is a proper
integration time according with the time scales associated with the coastal
processes of this small Bay, and also owing to the short period of the
available data. In these computations we assign the maximum possible value of
$t_{f}$ and $t_{b}$ (5 days) to the fluid particles that remain in the pre-
selected area after the 5 days of integration.

Figure 7: Lines are the locations of top values of FSLE (greater than
$0.5days^{-1}$ in October and greater than $1.5days^{-1}$ in July). Backward
FSLE lines are colored in black and forward FSLEs in white. They are
superimposed on spatial distributions of residence times in Palma Bay for
different dates. The colorbars give the residence times in days. a) and b)
correspond to two different days in October, and c) and d) in July.

In Fig. 7 we color the initial positions of particles in the Bay attending to
the time they transit through the Bay ($RT=t_{f}+t_{b}$) for different days.
Initial positions of particles with short residence times are indicated in
blue in Fig. 7. Regions from where particles have longer residence time (i.e.
take more time between entry and escape) are marked in red/brown.

These maps show that the spatial distribution of residence times of particles
can be complex and time depending, presenting different patterns at different
times. A number of small structures can be observed, including thin filaments
or small lobes. Comparing both months, one can see differences in the RT
distributions. The most noticeable is the approximate east-west alignment of
the zones of similar RT in July, which is not seen in October. Also, in
October the values of residence times of the particles are smaller, in
agreement with the global rate estimations showed before. A common feature is
the southwest region with low values of residence times, because in this
region there are not coastal boundaries and it is totally open to the ocean.

Figure 8: Spatial distributions of time averages of 60 snapshots of 6-hourly
RT values collected over 15 days in Palma Bay for a) October and b) July. The
colorbar units are $days$.

In order to reveal regions with different persistent transport properties we
compute time averages of the spatial distributions of residence times. We
average 6-hourly snapshots of RT during 15 days (i.e. 60 snapshots) for each
month. The results, plotted in Fig. 8 a) and b), show the common features
that, in general, the low values of RT are for particles initiated close to
the open ocean, specially in the southwest part, and high values are for
particles started near the coast, as expected. However, on average, the
residence time is larger in July than in October (3.25 days in July and 1.51
days in October) consistent with the behavior of the corresponding values of
$\tau_{e}$. Also, in July there is a clear boundary between the interior of
the Bay to the north, with large average residence times and the open sea to
the south, whereas the boundary between high and low residence times in
October is well inside the Bay, aligned with the Lagrangian structure
identified from the FSLE analysis, as will be discussed in the next section.

Another feature observed in movies of particle trajectories (not shown) is
that in October fluid particles tend to circulate mostly clockwise, while in
July they are oscillating along the zonal direction (see Section IV). This
difference, arising as discussed in Sect. IV from the different regimes of
wind forcing, is likely to be responsible for most of the different behavior
between both months.

###  V.4 Relation between LCSs and RTs

Figure 9: Snapshots of top values of FSLE (greater than $0.5days^{-1}$)
plotted over residence times. a) is for backward integration in time, and b)
is for forward integration. Both plots correspond to October 19th at 00:00h
(GMT) The colorbar units are $days$.

We now examine the connection between regions of different residence times
with LCSs. To compare RT and FSLE we have superimposed in Fig. 7 the filaments
of high values of forward (white) and backward FSLE (black) values on the
spatial distribution of residence times. The Figure shows a good
correspondence between many structures of RT and FSLE. Note that RT is the sum
of a forward and a backward exit time, so that some of the strong gradients of
RT will correspond to forward and some others to backwards LCSs. Fig 9 shows
clearer the correspondence between FSLE backwards in time with entry times,
and FSLE forward with escape times. At least in October, the LCSs given by
high values of FSLE clearly separate regions with different values of
residence times, confirming the value of the FSLE technique to identify
boundaries between different flow regions and barriers to transport. There are
however some lines of FSLE that are not associated to gradients of RT, and
viceversa. For the first this happens mainly because we only integrate 5 days
backward and 5 days forward in time, and the time assigned to the particles
that not cross the open-ocean boundary in the 5 days of integration
corresponds to the maximum time integration (5 days). This makes the spatial
distribution of the residence time more homogeneous. We need to integrate the
trajectories longer to unveil more areas with different RT. In the same way
not all abrupt changes in RT are captured by FSLE lines since we only plot the
highest ridges. This illustrates that both techniques have limitations and
that the complementary use of both could give a rather complete overview of
the geometric structure of flow in marine areas. Pattantyús-Ábrahám et al.
(2008) studied the relation between residence time and FSLE for a wind-forced
hydrodynamical model of a shallow lake. They found that areas with long
residence time visualize the stable manifolds of the so-called chaotic saddle,
a structure controlling the escape properties at long times. In our case, our
integrations are restricted to times too short to characterize long-time
chaotic behavior, but still there is a good correspondence between the FSLE
structures characterizing attracting or repelling trajectories, and escape or
residence times.

Fig. 8 a) shows a time average over October of the spatial distribution of RT,
to be compared with the corresponding average figure (Fig. 4 a) for FSLE. It
is evident that the region in the north and east side of the Bay with high
values of RT is separated from the rest by a region of high values of FSLE.
This can be explained by the presence of persistent barriers which do not
allow particles to escape from the northeast side of the Bay, and thus
separating the Bay in regions with different residence times. In July the
situation is different, because the spatial distribution of FSLE (Fig.4 b) and
RT (Fig. 8 b) is almost homogeneous, with higher values over the whole area of
the Bay, and lower values in the small region bordering the open ocean. This
indicates that the instantaneous configurations of high FSLE lines (Fig. 7)
are not persistent, and that only there is a persistent large difference
between the interior of the Bay and its opening to the ocean. The
predominantly zonal direction of particle motion in the Bay is consistent with
the orientation of the boundaries between areas of different RTs in July.

###  V.5 Variability of RT and FSLE

The differences of residence times and FSLEs in the two considered months
indicate that the dynamics of the flow is qualitatively different, as
anticipated by the different wind regimes that are the main drivers of the
Bay. Now we analyze the time evolution of their spatial averages.

Figure 10: a) Time series throughout October of the spatial average of
residence times (top) and spatial average of FSLE (bottom) for the surface
layer of Palma Bay. b) the same that a) but for July.

Figures 10 a) and b) show the time series of the spatial mean of residence
times (top panel) and backward FSLE (bottom panel) for October and July,
respectively.

Comparison between time evolution of spatial average of the RT for the
different months confirms, again, that particles tend to stay longer times in
the Bay during July than in October. The values of RT vary approximately from
0.25 days to 3 days in October (Fig. 10 a, top), and from 2 days to 6 days in
July (Fig. 10 b, top). The same happens with FSLE, higher values correspond to
July and lower ones to October. Diurnal fluctuations, likely related to the
effect of the sea breeze, are evident in RT and FSLE for both months. In
October there are some large fluctuations of low frequency in RT, probably
induced by the variability of remote forcing winds. On the other hand, during
October, minima of RT correspond to maxima of FSLE, and maxima of RT
correspond to minima of FSLE. In July, the relationship between FSLE and RT is
looser and only observed in the high-frequency fluctuations. To be more
quantitative, Pearson correlation coefficient between RT and FSLE time series
is -0.526 in October, but just -0.223 in July. This difference in behavior
probably arises from the fact that high values of FSLE determine clear and
well-defined barriers to transport only in the case of October (see Sect. V.3
and V.4). In July lines of high FSLE remain nearly zonal and parallel to
dominant particle direction of motion. Thus, as commented in Sect. V.2, they
probably represent regions of high zonal shear everywhere in the Bay. Their
high or low values indicate large or small differences in East-West velocities
but by themselves they do not imply stronger or weaker escape towards the
South.

Figure 11: a) Snapshot of spatial distributions of residence times at the
bottom layer in Palma Bay corresponding to July 17, 2009 at 18:00h (GMT).
Lines are the locations of top values of FSLE (greater than $0.3days^{-1}$).
Backward FSLE lines are colored in black and forward FSLEs in white. b)
spatial distribution of time average of 60 snapshots of 6-hourly maps of RT
collected over 15 days in July at the bottom layer.

###  V.6 Transport at the bottom layer

In this subsection we compare the main Lagrangian characteristics at the
bottom layer, not driven directly by wind, with those at the surface. In Fig.
11 a) we show an instantaneous map of the residence times in the bottom layer
one day of July, overlayed with lines of high FSLE values. Again, the spatial
distribution is inhomogeneous, and we find high values of RT over all the Bay
except very close to the ocean. The correlation of RT values with FSLE lines
is weaker than in the upper layer, but still we see that the relatively lower
values of RT in the western part of the Bay at that particular day appear
bounded by backwards FSLE lines, indicating a temporal escape route of
particles in that region towards the southwest. The spatial distribution of
the time average of RT plotted in Fig. 11 b) shows that highest average values
of RT are concentrated in the northwestern region of the Bay. Fig. 4 c)
displays high values of FSLE located precisely in the same region where the RT
qualitatively change to high values. This suggests the presence of persistent
barriers that separate this southeastern region from the rest in this bottom
layer. The formation of these persistent LCSs is associated to the gradient of
the bathymetry (see Fig. 1).

The time evolution of the spatial average of RT and FSLE are plotted in top
and bottom panel of Fig. 12 respectively. Contrarily to the surface, in the
bottom layer the diurnal fluctuations in the time series of RT disappear,
showing that the flow at this depth is not directly influenced by breeze. The
RT values are larger than in the surface, and therefore the interchange
between the ocean and the Bay is less intense at bottom layers. This is a
consequence of the slowness of the flow produced by the absence of direct wind
forcing at deepest levels. There is a strong negative correlation ($r=-0.803$)
between stirring and residence times: when the flow is more dispersive the
particles transit during less time over the Bay, so that maxima of RT
correspond to minima of FSLE and vice-versa.

Figure 12: Time series throughout July of (top) spatial average of residence
times and (bottom) spatial average of FSLE computed for the bottom layer.

##  VI Conclusions

Properties of coastal transport in the Bay of Palma, which is a small semi-
enclosed region of the Island of Mallorca, were studied in a Lagrangian
framework, by using model velocity data at high resolution. We have applied
two complementary Lagrangian methods (FSLEs and RT) to analyze the small
scales of these coastal currents. LCSs have been detected as high ridges of
FSLE, and virtual experiments with particle trajectories have shown that these
structures really act as barriers in most cases, organizing the coastal flow.
Global and average aspects of the transport in different seasonal months show
that, in the period studied, in autumn there is more exchange between the Bay
and the open ocean than in summer. This arises from the different wind regimes
in both months, that during July induce a flow that restricts motion of the
coastal marine surface to the zonal direction, preventing the flow to enter or
escape toward the open ocean. The transport of particles at the deepest layer
is less active than at the surface and not directly driven by wind, but
influenced by the bottom topography. Regions with different values of RT are
generally separated by ridges of FSLE, proving the fact that FSLE separate
regions of qualitatively different dynamics also in small coastal regions.
Thus, we think that these Lagrangian quantities can be used as key variables
able to determine the dynamics and health of other bays or estuaries,
particularly in relation with human activities. Future improvements include
the adaptation of these methods to three-dimensional spaces and capture three-
dimensional effects, such as upwelling and downwelling in coastal areas, and
analyzing longer periods of time.

## Acknowledgments

I.H-C, C.L and E.H-G acknowledge support from MICINN and FEDER through
projects FISICOS (FIS2007-60327), INTENSE@COSYP (FIS2012-30634), and ESCOLA
(CTM2012-39025-C02-01). AO acknowledges financial support from MICINN and EU-
Med Programme through projects BUS2 (CGL2011-22964) and TOSCA (2GMED-09-425).
We acknowledge the ICMAT Severo Ochoa project (SEV-2011-0087) for the funding
of the publication charges of this article.

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