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

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 tudied 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 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 [29], 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. 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 [22,23] 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. [23]). 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 [39,40], 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 [47].

B. 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 [15,44]. At each grid point, horizontal resolution ∆ 0 is the same in both the longitudinal, φ, and latitudinal, θ, directions.
We run the simulation with a resolution of ∆ 0 = 0.0027 • (∼300m, ROMS300), which is itself nested into a larger and coarser grid with ∆ 0 =1/74 • (∼1500m). Boundary conditions for the coarser domain were taken from daily outputs of the Mediterranean Forecasting System [8,35]. The ROMS300 domain covers 39 • 12 N -39 • 36 N (latitude), and 2 • 24 E -3 • 6 E (longitude). The total number of grid nodes is 260 × 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 [1,30].
We will manage velocity data from the surface layer and the bottom layer for the grid of ∆ 0 ≈ 300m. 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 [13]) 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 [13]. We will study two different intervals 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 [9,18,42], 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 [17,[31][32][33]41], 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 [2,3]. 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 [7]. In fact the first applications of the FSLE technique in oceanography were for closed or semi-closed basins [4,5].
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: where δ 0 is the initial distance of the two given particles, and δ f is their final distance.
Thus, to compute the FSLEs we need to calculate the minimal time, τ , needed for the two particles initially separated δ 0 , to get a final distance δ f (in this way the FSLE represents 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 [9,18,20,42,46], and characterize the flow from the Lagrangian point of view [24,25]. Attracting LCSs associated to backward integration (the unstable manifolds) have a direct physical interpretation [9,10,24]. Tracers (chlorophyll, temperature, ...) spread along these attracting LCSs, thus creating their typical filamental structure [6,27,45,46]. 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 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 [16]  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 τ e . τ 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 [4,5,11,37]. 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 RTs can be displayed in plots named Lagrangian Synoptic Maps [29], 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.

B. Coastal LCSs
The temporal averages computed in Sec. V A 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. October is well inside the Bay, aligned with the Lagrangian structure identified from the FSLE analysis, as will be discussed in the next section.

C. Transport between the Bay of Palma and the open sea
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.

D. Relation between LCSs and RTs
We now examine the connection between regions of different residence times with LCSs. To compare RT and FSLE we have superimposed in Fig. 7  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.