1. Introduction
Ocean water masses of different origins have distinct contents of salt, heat, nutrients, and chemicals. Currents transport them, and energetic mesoscale features are responsible for most of their mixing with surrounding waters.
Vortices are the most well studied of the ocean structures. Frequently, they are long lived, and the water trapped inside could maintain its biogeochemical properties for a long time, being transported with the vortex. In steady horizontal velocity fields, the presence of closed streamlines around them is the mathematical reason for the isolation of the vortex core from the exterior fluid. Jets and strong currents are also important ocean features. They can be rather persistent and, because it is difficult for particles to cross them, water at both sides can keep different physical properties.
When the velocity field changes in time, closed streamlines are replaced by more complex structures, some of which can be related in idealized cases to the Kolmogorov–Arnold–Moser tori of dynamical systems theory (Jones and Winkler 2002; Wiggins 2005; Samelson and Wiggins 2006; Mancho et al. 2006b). For slowly varying velocity fields, vortex cores remain coherent during some time, but there is vigorous stirring of the surrounding fluid that finally leads to water mixing. To understand this mixing process, one should focus on features of the velocity field different from the elliptic points characterizing the vortex centers or from the smooth curves giving the average direction of jets or currents.
Since some time ago (Ju et al. 2003; Ide et al. 2002; Mancho et al. 2004; Wiggins 2005; Samelson and Wiggins 2006; Mancho et al. 2006b), hyperbolic trajectories (with saddlelike stability properties that are solutions to a dynamical system) have been recognized as the structures responsible for most of the stretching and generation of intertwined small scales that finally lead to mixing. In particular, there are distinguished hyperbolic trajectories (DHTs), characterized by their special persistence as compared with the other hyperbolic structures, which act as the organizing centers of the fluid- stirring processes. Despite the great amount of attention devoted to the identification and characterization of vortices, jets, and their dynamics in oceanographic contexts (Olson 1991; Puillat et al. 2002; Ruiz et al. 2002; Isern-Fontanet et al. 2006; Lozier et al. 1997), there are few studies focused on hyperbolic objects, and most of them in idealized settings. In Miller et al. (2002), fluxes in and out of a vortex are described in terms of hyperbolic objects in a model for an island recirculation. The work by Miller et al. (1997) quantifies transport associated with jets obtained from a numerical computation of a model. In the work by Mancho et al. (2006b), fluxes in and out of a vortex obtained from a computational model are computed from the lobes associated with the stable and unstable manifolds of DHTs. Fluxes through a turbulent jet are also calculated. Lobe dynamics supplies detailed information by locating volumes of water that evolve in time escaping from the interior of a vortex or crossing a jet, or by doing both things sequentially in time. Because simplified ocean models are used in the mentioned works, no link can be established with realistic transport properties of scalar quantities such as heat or salt.
In this paper we identify relevant hyperbolic trajectories present at the surface velocity field of the western Mediterranean Sea, obtained from a three-dimensional model simulation under climatological atmospheric forcing. Our aim is, first, to find the location and nature of the Lagrangian structures associated with the so-called Balearic Current that serve to maintain the North Balearic Front. And, second, to show that transport mechanisms, in particular the so-called turnstile mechanism previously identified in abstract dynamical systems (Channon and Lebowitz 1980; Bartlett 1982; MacKay et al. 1984; Wiggins 1992) and discussed in the context of rather simple model flows (Rom-Kedar et al. 1990; Beigie et al. 1994; Samelson 1992; Duan and Wiggins 1996), are also at work in this complex and rather realistic ocean flow. A close link between abstract concepts such as material lobes and transported scalar quantities such as temperature or salt is found. More broadly, nonlinear dynamics techniques are shown to be powerful enough to identify the key geometric structures in this part of the Mediterranean.
Western Mediterranean surface layers (up to a depth of about 150 m) contain modified Atlantic waters of different characteristics (Millot 1999). Fresher waters (salinity about 36.5 psu), having recently entered from the Atlantic, occupy the Algerian Basin at the south, and older and saltier waters (salinity above 38 psu) occupy the northern part of the area. The dynamics of the contact zone between these two surface water masses, and eventually their mixing, is important for understanding the physical and biogeochemical properties of the western Mediterranean. We focus on one of the oceanographic structures known to be of importance in these processes, the so-called North Balearic Front (López-García et al. 1994; Pinot et al. 1995; Millot 1999). It extends roughly along the southwest–northeast direction at the north of the Balearic Islands (see Fig. 1), with significant displacements and deformations. It is characterized by a strong salinity jump of about 0.6 psu (37.4 psu to the south and 38.0 to the north) down to a 150-m depth. This identifies the front as the main transition zone between the two water masses. In winter, a weak but detectable temperature gradient of 0.5–1 K (5 km)−1 can be observed in satellite images.
After selecting an interval of time in our simulation during which the front is well formed, we explore the transport properties of the surface velocity field in the region and find the relevant Lagrangian structures and the hyperbolic points and their manifolds responsible for the geometric characterization of the front in a manner that allows both qualitative and quantitative analyses of transport properties. The location of the front is identified as a “Lagrangian barrier,” across which transport is small (as quantified with the tools of lobe dynamics) and which occurs via filaments that entrain water along transport routes that we identify. The presence of eddies strongly affects the Lagrangian structures in a process that can be interpreted as the breakdown of the front.
The paper is organized as follows: in section 2, we introduce some of the dynamical systems concepts that will be used in the following. Section 3 describes our numerical ocean model, and section 4 addresses the adaptation of the standard algorithms to the kind of oceanographic data provided by the model. Section 5 contains our main results, and the conclusions are summarized in section 6.
2. Distinguished hyperbolic trajectories and their manifolds
In recent years, there have been many applications of the dynamical systems approach to transport in oceanographic flows (see the reviews by Jones and Winkler 2002; Wiggins 2005; Samelson and Wiggins 2006; Mancho et al. 2006b). In this section, we describe the basic ideas that we will use in our analysis. Although the same concepts are of use in three-dimensional flows, we describe here just the two-dimensional situation, because, as discussed below, the structures we identify can be considered two-dimensional to a good approximation during the time scales relevant here.
Stagnation points are well-known features of steady flows that generally play an important role in “organizing” the qualitatively distinct streamlines in the flow; for example, saddle-type stagnation points can occur on boundaries where streams of flow tangential to the boundary coming from opposite directions meet and then separate from that boundary. Saddle-type stagnation points can occur in the interior of a flow at a point where fluid seems to both converge to the point along two opposite directions and diverge from it along two different directions. In steady flows, the stagnation point is a trivial example of a fluid particle trajectory (i.e., it is a “solution,” in this case a fixed point solution, of the equations for fluid particle motions defined by the velocity field) and the saddle point nature is manifested by the fact that there are directions for which nearby trajectories approach the stagnation point at an exponential rate and move away from the stagnation point at an exponential rate. These directions are sometimes referred to as “stagnation streamlines.” They define material curves that “cross” at the stagnation point and typically form the boundaries between qualitatively distinct regions of flow.
A related time-dependent picture as that described above exists for unsteady flows, with similar as well as much more complex implications for transport. In unsteady flows a stagnation point at a given time—or rather, an instantaneous stagnation point (ISP)—is a location at which velocity vanishes at that time. The sequence of locations is generally not a fluid particle trajectory (Ide et al. 2002; Mancho et al. 2006b). The true analog of the saddle-type stagnation point of steady flows is a DHT (Ide et al. 2002). “Hyperbolic” is the dynamical systems terminology for the saddle type in incompressible flows. These are fluid particle trajectories that have (time dependent) directions for which nearby trajectories approach and move away from the DHT at exponential rates. “Distinguished” is a notion that is discussed in detail in Ide et al. (2002) and Mancho et al. (2006b), but the idea is that these are the key, isolated hyperbolic trajectories that serve to organize the transport behavior in a flow because they remain substantially more localized [in a well-defined sense, see Ide et al. (2002); Mancho et al. (2006b)] than neighboring hyperbolic trajectories. Ide et al. (2002), Ju et al. (2003), and Mancho et al. (2004) develop the algorithms that allow us to compute DHTs in a given flow. They are iterative methods that start with a first guess for the DHT positions in an interval of time (i.e., an initial curve in space and time) and then refine the space–time curve by imposing the criteria of hyperbolicity and localization. In the flows considered here, a good first guess is the location of ISPs, because it seems that a DHT is often found in the neighborhood of an Eulerian ISP (Ide et al. 2002; Ju et al. 2003; Mancho et al. 2004, 2006b).
Just as in the steady case, there are analogs to the stagnation streamlines: in the dynamical systems terminology, these are referred to as the stable and unstable manifolds of the DHT, and they are time-dependent material curves. In dynamical systems terminology, the fact that they are material curves means that they are invariant; that is, a fluid particle trajectory starting on one of these curves must remain on that material curve during the course of its time evolution. “Stable manifold” means that trajectories starting on this material curve approach the DHT at an exponential rate as time goes toward infinity, and “unstable manifold” means that trajectories starting on this material curve approach the DHT at an exponential rate as time goes toward minus infinity. Mancho et al. (2003, 2004) develop the algorithms that enable us to compute the stable and unstable manifolds of hyperbolic trajectories.
In unsteady flows, stable and unstable manifolds of DHTs can intersect in isolated points different from the DHTs. This is a fundamental difference with respect to steady flows and gives rise to moving regions of fluid bounded by pieces of stable and unstable manifolds, the so-called lobes. Because the manifolds are material lines, fluid cannot cross them by purely advective processes, and thus they are perfect Lagrangian barriers (diffusion or motion along the third dimension can, however, induce cross-manifold transport). Motion of the lobes is thus the mechanism responsible for mediating Lagrangian transport between different regions. References describing lobe dynamics in general include Rom-Kedar and Wiggins (1990), Beigie et al. (1994), Wiggins (1992), Malhotra and Wiggins (1998), Samelson and Wiggins (2006), and Mancho et al. (2006b). Examples of applications of lobe dynamics to oceanographic flows are Ngan and Shepherd (1997), Rogerson et al. (1999), Yuan et al. (2001, 2004), Miller et al. (2002), and Deese et al. (2002). We will describe these ideas more fully in the context of transport associated with the Balearic Front in the Mediterranean.
3. The ocean circulation model
In this work we analyze velocity fields obtained from an ocean model, DieCAST (Dietrich 1997), adapted to the Mediterranean Sea (Dietrich et al. 2004; Fernández et al. 2005). The 3D primitive equations are discretized with a fourth-order collocated control volume method. In zones adjacent to boundaries, a conventional second-order method is used. A fundamental feature of control volume–based models is that the predicted quantities are control volume averages, while face-averaged quantities are used to evaluate fluxes across control volume faces (Sanderson and Brassington 1998). These quantities are computed using fourth-order approximations, and numerical dispersion errors are further reduced in the modified incompressibility algorithm by Dietrich (1997).
Horizontal resolution is the same in both the longitudinal (ϕ) and latitudinal (λ) directions, with Δϕ = (⅛°) and Δλ = cosλΔϕ, thus making square horizontal control volume boundaries. Vertical resolution is variable, with 30 control volume layers. The thickness of control volumes in the top layer is 10.3 m, and they are smoothly increased up to the deepest bottom control volume face at 2750 m. Thus, 5-minute gridded elevations/bathymetry for the world (ETOPO5) is truncated at a 2750-m depth and is not filtered or smoothed.
Horizontal viscosity and diffusivity values are constant and equal to 10 m2 s−1. For the vertical viscosity and diffusivity, a formulation based on the Richardson number that was developed in Pacanowski and Philander (1981) is used, with background values set at near-molecular values (10−6 and 2 × 10−7 m2 s−1, respectively). We use monthly mean wind stress reanalyzed from 10-m wind output from the European Centre for Medium-Range Weather Forecasts (ECMWF), as chosen for the Mediterranean Sea Models Evaluation Experiment (Beckers et al. 2002). The heat and the freshwater fluxes used to force the model are model determined from monthly climatological SST and sea surface salinity (SSS), as described in Dietrich et al. (2004). The only open boundary is the Strait of Gibraltar, where inflow conditions are set similar to observations and outflow is model determined by upwind (see Fernández et al. 2005). Everywhere else, free-slip lateral boundary conditions are used. All bottom dissipation is represented by a conventional nonlinear bottom drag with a coefficient of 0.002. Lateral and bottom boundaries are thermally insulating. The model is initialized at a state of rest with the annual mean temperature and salinity fields taken from the climatological data. The spinup phase of integration is carried out for 16 yr. Each year is considered to be composed of 12 months, 30 days each (i.e., 360 days). The climatological forcings we use are adequate to identify the mechanisms and processes occurring under typical or average circumstances. With this approach, high-frequency motions are weak in our model and a daily sampling is adequate. The impact on transport of disturbances containing high frequencies, such as storms or wind bursts, is not the focus of the present paper and would need specific modeling beyond climatological forcing.
We focus on velocity fields obtained at the second layer, which has its center at a depth of 15.93 m. This is representative of the surface circulation and is not as directly driven by wind as the top layer. We have recorded velocities, temperatures, and salinities in this model layer for 5 yr. Dynamical systems approaches have already been applied to this dataset: in particular, Lyapunov techniques to quantify mixing strength (d’Ovidio et al. 2004) and the “leaking” approach (Schneider et al. 2005) to quantify escape and residence times in several areas. Here, we concentrate on the northwestern region and apply the methods of lobe dynamics to characterize transport processes in the North Balearic Front area.
Figure 1 shows an example of the output of the model for the velocity field in the selected layer of the western Mediterranean Sea at day 649 (the 19th day of the 10th month—October—of the second year). Two well-known currents, the Northern Current flowing southward close to the Spanish coast, and the Balearic Current, associated with the North Balearic Front and flowing northeastward north of the Balearic Islands, are observed, although significantly deformed by the presence of eddies. Figure 1 also shows the ISPs of the velocity field: circles for elliptic and crosses for the saddle type.
The velocity field has small vertical components, so that this is not strictly a two-dimensional flow. With vertical velocities on the order of 10−5 m s−1, particles in the second model layer require about 13 days to traverse the layer. But during that time, this vertical velocity is not constant. Averaging over the relevant time scales, we find effective velocities of 0.1–0.7 m day−1, depending on location and season (d’Ovidio et al. 2004; Schneider et al. 2005), and thus residence times in the second layer are between two weeks and several months. As a rule of thumb, we can consider that trajectories preserve two-dimensionality during time intervals of about 20 days. Because most of our trajectory integrations will be restricted to time intervals below that duration, they can be considered two-dimensional to a good approximation.
By inspection of the temperature and salinity model outputs, we identify several time intervals during which gradients are strong and thus the North Balearic Front is well defined. An example of this situation (see Fig. 2) is a 4-month interval starting in October of the second simulation year (i.e., autumn and early winter). We select this interval of time as our study case for which dynamical systems structures will be calculated. Our choice reveals that as expected for aperiodic flows, the mechanisms are not acting indefinitely for all times. They are persistent enough, however, to allow detailed description and to strongly influence the dynamics in the area.
4. Computation of trajectories and manifolds in Mediterranean datasets
Once trajectories are integrated from these equations, for presentation purposes one can convert μ values back to latitudes λ just by using (6).
Distinguished hyperbolic trajectories and their unstable and stable manifolds are the main dynamical systems objects that we use to describe and quantify transport. Our velocity dataset lasts only for a finite time, is highly aperiodic in time, and is turbulent. Algorithms to compute hyperbolic trajectories for this kind of flow are discussed in Ide et al. (2002) and Ju et al. (2003). A novel technique to compute stable and unstable manifolds of hyperbolic trajectories in aperiodic flows is developed in Mancho et al. (2003). In Mancho et al. (2004), these hyperbolic trajectory and manifold computation algorithms are combined into a unified algorithm and successfully applied to a turbulent, wind-driven, quasigeostrophic double gyre. We will now apply these same algorithms to compute DHTs and their stable and unstable manifolds, which will be used to describe and quantify transport associated with the North Balearic Front. One important aspect is to determine whether the Lagrangian structures found from the model velocity data are sensitive to resolution, interpolation, or measurement imprecisions. This issue has been addressed by some authors. Haller (2002) concludes that even large velocity errors in the velocity field lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. The works by Lehahn et al. (2007) and d’Ovidio et al. (2008) characterize finite-size Lyapunov exponents in the North Atlantic and in the Algerian Basin, respectively, from altimetry data, which has a resolution poorer than the simulated dataset. The structures found, however, are very similar to chlorophyll and temperature features observed from satellite at the same time, but independently, of the altimetric data. These results support the assertion that Lyapunov exponents, lobes, manifolds, and chaotic advection structures in general are determined mainly by the large-scale advection field, which is appropriately captured by altimetry or by our simulated velocity field.
5. Lagrangian structures and transport in the Balearic Sea
We focus on the region north of the Balearic Islands: the Balearic Sea. The main oceanic structures known to be present there are the Balearic Current and the associated North Balearic Front (López-García et al. 1994; Millot 1999). This last feature is known to represent a transition zone between saltier and fresher waters in the western Mediterranean. The salinity fields obtained from our numerical simulation display significant gradients (there are also temperature gradients in winter—during summer the surface layer is heated in a rather homogeneous way) in the area (see Fig. 2). Our aim here is to interpret the presence of the gradients and the front in terms of a semipermanent Lagrangian barrier, across which little transport occurs. This construction would also reveal the routes along which this transport happens. Topological changes in that picture, which are associated with the crossing of the strong current by eddies and which may be interpreted as the breakdown of the front, are also observed during the simulation.
An example of situations at which gradients are rather well defined is during autumn of the second simulation year and early winter of the third simulation year. During this period we find a long interval (from day 649 to 731) in which a Lagrangian structure constructed using stable and unstable manifolds of DHTs remains persistent and acts as a partial barrier to transport. Its location is well correlated with the salinity gradients so that it can be interpreted as a Lagrangian identification of the North Balearic Front. The weak transport across the structure can be described and quantified in terms of lobe dynamics. The situation resembles that in Coulliette and Wiggins (2001) in which quasigeostrophic dynamics has been used to model a double-gyre situation and the central jet between the two gyres in that problem plays a similar role as that of the Balearic Current in our problem. Lobe dynamics was successfully applied there to quantify transport across the jet, occurring by the so-called turnstile mechanism. However, in the more realistic dataset analyzed here, we need to generalize some ideas used there.
For example, in Coulliette and Wiggins (2001), the DHTs relevant to intergyre transport remained on one-dimensional boundaries for all times (as boundaries are invariant). This meant that the search for the relevant DHTs was reduced to a one-dimensional problem, which is relatively straightforward (Ide et al. 2002). However, in our scenario, the relevant current neither starts nor ends on any obvious one-dimensional boundary. Rather, the most appropriate DHTs for defining the front are in the interior of the flow. This introduces some new computational difficulties, and for some times there may even be some ambiguity in the identification of the relevant DHTs [and of the saddle-type ISPs used as starting positions in the iterative algorithm of Mancho et al. (2004) that we use to determine the DHTs] from which to compute the manifolds that will define the Lagrangian barrier. Most of the time, however, pieces of manifolds computed from DHTs obtained from different ISPs in the same area rapidly converge toward each other, thus indicating that the location of the dominant hyperbolic curve is a robust property of the flow. These difficulties and ambiguities, and how to deal with them computationally, are discussed in detail in Mancho et al. (2006b). Changes in the topology of the flow cause the convergence property to be lost. This happens at the end of the time interval chosen in our study case and will be commented on below.
a. Cross-frontal transport: The turnstile mechanism
The geometric objects used to characterize transport by lobe dynamics methods are constructed by the rules discussed in Malhotra and Wiggins (1998), Coulliette and Wiggins (2001), and Samelson and Wiggins (2006). Here we describe in some detail these procedures in a way that is particular to our flow situation. Figures 2 –4 illustrate the construction at days 649 and 657. Figure 2 contains all the dynamical systems structures superimposed on a salinity field, whereas for clarity only the boundary and the lobes are displayed in Figs. 3 and 4, respectively.
1) Two DHTs should be identified, one in the western part of the front to be characterized and another in the eastern part, which persists close to their initial positions during the whole time interval of interest (denoted by [t0, tN]). Because our algorithm to locate DHTs uses saddle-type ISP positions as first guesses, figures displaying ISPs such as Fig. 1 are used to estimate these positions and the temporal persistence of the ISPs. The positions calculated for the selected DHTs are plotted in Figs. 2 –6 as black dots and labeled as HW (western) and HE (eastern).
2) As the mean current flows eastward, we proceed as in Coulliette and Wiggins (2001) and compute the unstable manifold of the western DHT and the stable manifold of the eastern DHT (denoted by the red and blue curves in Fig. 2, respectively). For clarity in the presentation, of the two branches of each manifold (one at each side of the DHT from which they emanate), we display in Fig. 2 only the one pointing in the direction of the other DHT. Along both pieces of manifolds, in the region between the two DHTs, the dominant direction of motion is from west to east. As is characteristic in unsteady flows, both manifolds intersect repeatedly. Some of the intersection points are marked with cyan dots. It is also typical that the unstable manifold (red) of a DHT (HW in this case) emerges relatively straight [which is a consequence of the fact that behavior of trajectories near a DHT is dominated by the linearization of the flow about the DHT; see Samelson and Wiggins (2006)], but it fluctuates wildly when approaching the vicinity of the opposite DHT. In the same way, the stable manifold (blue) of HE is relatively straight near HE, but it displays large oscillations when close to the western DHT. The stable and unstable manifolds displayed in Fig. 2a have been computed after backward- and forward-time integration of a small segment in the direction of the stable and unstable linear subspaces in the neighborhood of the DHT for time periods of 14 and 19 days, respectively. Integrations for longer time periods provide longer manifolds. However, because of the restrictions of the two-dimensional approximation, a time period beyond 20 days is not completely trustworthy. In practice, this means that the parts of the displayed manifolds that are far from the DHT may deviate from the true manifold because those parts are obtained with longer time integrations. For instance, the unstable manifold in Fig. 2b has been computed for an integration time of 27 days. This means that unstable structure in the far east is not completely reliable. However the piece of manifold governing the turnstile mechanism, which is closer to the DHT, is obtained with numerical integrations below the validity limit of 20 days, and predictions obtained from them are correctly given by the two-dimensional approximation.
3) We choose a sequence of times within the chosen time interval at which to analyze the manifold positions and compute objects relevant for Lagrangian transport. The sequence of observation times is denoted by t0 < t1 < t2 < . . . < tN−1 < tN. We note that the ti variables do not need to be equally spaced. To illustrate the construction of boundaries and the turnstile mechanism for crossing the boundaries, we need only two times; for this purpose we will choose to show days 649 and 657.
4) At each of the selected times ti, a boundary is constructed by choosing a finite piece of the unstable manifold of the western DHT, starting at HW and ending at bti, denoted by [HW, bti], and a finite piece of the stable manifold of the eastern DHT, starting at HE and ending at bti, denoted by [bti, HE] so that these two segments intersect at precisely one point, bti, which is called the boundary intersection point. The points {bti}, in addition to satisfying an ordering constraint specified below, should be selected in such a way that a relatively straight boundary (i.e., free of the violent oscillations displayed by each of the manifolds when approaching the opposite DHT) is obtained. Because the boundary is pinned at the points HW and HE, we obtain a sequence of boundaries that fluctuate in time but remain approximately in the same place. Figure 2 shows the selection of the boundary intersection points b649 and b657 at days 649 and 657, respectively. For clarity, Fig. 3 displays just the finite segments of the unstable manifold of HW and the stable manifold of HE that intersect at precisely one point and form the boundaries at these two times.
Because the boundary is made of material curves, no fluid can cross it by horizontal advection processes. However, at the observation times t1, t2, . . . , tN, the boundary is redefined. Therefore, fluid may appear at the opposite side of the boundary at ti+1 because the boundary is changed at each observation time. Moreover, the only way fluid at one side of the boundary at one observation time can be transferred to the other side at the next observation time is by the turnstile mechanism described and quantified below. If this transport amount is small (as will be shown to be the case), the boundary can be characterized as a barrier, and gradients will be maintained across it. As seen in Fig. 2, the position of the boundary is well correlated with the position of the salinity front, thus confirming that the dynamical systems techniques developed here are useful to identify the North Balearic Front in terms of a Lagrangian object.
5) Construct turnstiles at ti. At time ti we consider the point, denoted by b−ti+1 that will evolve into the boundary intersection point bti+1 (clearly, we cannot do this at tN). Because the stable and unstable manifolds are invariant, this point is also on both the stable and unstable manifolds. In the same way, bti−1+ is the location at time ti of the boundary intersection point that was located at bti−1 at the previous time ti−1. The additional constraint that needs to be imposed when choosing the sequence {bti} is that b−ti+1 results “upstream” (i.e., closer to HW along its unstable manifold, or farther from HE along its stable manifold) from btt. This introduces a restriction on the choice of bti+1 once bti is chosen. Because ordering of points along manifolds is preserved by time evolution, it turns out that bti−1+ would be “downstream” from bti+1. Figures 2 –4 display some of these intersection points, showing that our selection satisfies the ordering constraints. The segments of stable and unstable manifolds between bti and bti+1− (and between bti and bti−1+) trap regions of fluid, and these regions of fluid defined in this way are referred to as lobes. Some of them can be seen in Fig. 2, and more clearly in Fig. 4.
6) Consider the evolution of the turnstile lobes from ti to ti+1. In this case b−ti+1 evolves to the boundary intersection point, bti+1, and the boundary intersection point at ti evolves to a point b+ ti, which is also on both the stable and unstable manifolds. Then the lobes between bti+1 and b+ ti represent the time evolution of the turnstile lobes from ti to ti+1. Note that, because of the way in which the boundary is redefined at each observation time, and because of the different shape of the manifolds when close or far from the DHT from which they emanate, each lobe is on opposite sides of the boundary at the two considered times. These turnstile lobes contain all the fluid that has crossed the boundary between ti and ti+1. This transport process is illustrated more clearly in Fig. 4, where the lobes experiencing the turnstile mechanism are plotted before and after crossing the boundary.
The geometric construction performed at every observation time ti, as explained above, allows us to calculate the amount of transport occurring across the boundary during each time interval.
For day 649, as shown in Fig. 4, the two turnstile lobes have areas of 493.2 km2 (the lobe below the boundary, to the east) and 716.9 km2 (the lobe above the boundary, to the west). At day 657, the eastern lobe is above the new boundary and the western lobe is below it. Assuming that the divergence of the surface flow can be neglected so that the areas are unchanged—in numerical experiments we have never observed more than a 3% change—we can calculate the flux across the barrier to be (716.9 − 493.2) km2 = 223.7 km2, in the southerly direction.
Modified Atlantic Waters occupy the surface layers of the area for an average depth of about 150 m. If we take for the mean horizontal speed of this water mass the values at the second model layer considered here, multiplication of the 150-m depth by the area of the lobes gives 33.56 × 109 m3 in 8 days, or an average flux over this interval of 0.049 Sv (1 Sv = 106 m3 s−1). The average flux obtained from area calculations of further turnstile lobes until mid-November remains below that value (the average is 0.025 Sv, always southward). This should be compared to the 0.75–0.5 Sv that are transported by the Balearic Current. In terms of salt transport, multiplying the area of the lobes by a 150-m depth and by their salt content (about 38 kg m−3 for the lobe coming from the north, and 37.4 kg m−3 for the one coming from the south), we obtain a transport of 2.24 × 1010 kg in the whole 8-day duration of the event. The vertical velocity profile is, however, not constant, because the average speeds are smaller at deeper layers. Examination of the simulation data at the bottom of the Modified Atlantic Water mass gives average velocities about one-half of the values in the second layer we have analyzed in detail, and average velocities across the whole 150-m-wide water mass are about 70% of the ones in our layer. Thus, our above estimations for transport across the front, already small, are in fact upper bounds for the actual transport. We see, then, that cross-boundary transport is small during this time interval, and thus the Lagrangian boundary acts as a barrier to transport that permits only small amounts of mixing between northern and southern waters. It will maintain a salinity (and thus density) front that we identify with the observed North Balearic Front. There is some indeterminacy in the definition of the boundary that we identify as the front, arising from some freedom in the selection of the intersections {bti}. But other choices can only displace the boundary by a distance on the order of the size of the lobes, which we see is small when not too close to the DHTs, and in fact also on the order of the width of the transition region in salinity distributions such as the one in Fig. 2 (i.e., the width of the front).
b. Spatiotemporal structure of cross-frontal transport
Because the only way in which our constructed Lagrangian boundaries can be crossed is via the turnstile mechanism, the earlier and later locations of the turnstile lobes reveal the dominant routes along which the weak cross-front transport occurs. Figure 5 shows the time evolution of a pair of turnstile lobes, one initially above and the other initially below the boundary, as they evolve in time. The crossing of the boundary by the turnstile mechanism occurs between days 676 and 681, and the remaining figure shows the position of these lobes at some earlier and later times. The sequence illustrates that lobes move essentially along the boundary, except when close to HE, where they are ejected as filaments transverse to the front (the one initially in the south ejected toward the north, and reciprocally for the one started in the north) and when close to HW, where they also have the shape of transverse filaments and become entrained into the boundary region. Figure 5 also illustrates how lobes transport water of different salinity (coded in colors) and how the above routes for lobe motion and shape correlate with the salinity distribution in the area. Note that the length of the manifolds and the whole process depicted in Fig. 5 lasts only 21 days, so that the plotted manifolds remain approximately horizontal, with only small corrections from the vertical flow.
c. An eddy-front interaction: Disruption of the Lagrangian boundary
Not all flow configurations allow a geometric construction that identifies a Lagrangian boundary associated to two DHT and their turnstile lobes. It may happen that no pair of DHTs persists in a given area long enough to support the mechanism or that their manifolds fail to intersect. It may also happen that manifolds started at relatively close DHTs do not converge to the same curve but remain significantly distinct, so that a unique, well-defined boundary cannot be properly identified. In such cases, the turnstile transport mechanism loses relevance with respect to other phenomena present in the flow. At the end of the simulation interval analyzed here, we observe a change in the topology of the flow structure that signals the end of the predominance of the turnstile mechanism (described above) in a process that can be interpreted as the breakdown of the front by the interaction with an eddy. As a first symptom, calculations of turnstile lobe areas reveal an increase in cross-front transport starting at day 674 (mid-November). The average transport between days 674 and 700 is 0.303 Sv (southward), still smaller than the Balearic Current transport but significantly larger than the average cross-frontal transport during the previous month (0.025 Sv, also southward). At day 711, stable manifolds that emerge from HE and from another rather close DHT cease to converge into each other, signaling the end of a situation with an essentially unique, well-defined boundary. Later, at day 731, our algorithm is unable to find the location of HE starting from ISPs in the area. This probably means that HE has moved away from the area under study. The black dot in Fig. 6 is another DHT found in the region. But its stable manifold (i.e., the finite length of manifold that we are able to compute) does not intersect the unstable manifold from HW, thus revealing that it is in fact a DHT different from HE and that it cannot support the turnstile mechanism. The time evolution of the unstable manifold from HW suggests that the reason for the change in behavior is the breakdown of the current by the crossing of an eddy, identified by the rolling up of the manifold around an elliptic ISP (Fig. 6). Note that even in this situation, the manifold position is well correlated with the salinity distribution, thus indicating that Lagrangian structures are still relevant. But the transport mechanism is clearly different from the turnstile described above, being more appropriately described as water transport inside an eddy.
6. Conclusions
In this work we have applied in a systematic way tools developed in the context of dynamical systems theory and known generically as lobe dynamics. The velocity field obtained from the numerical simulation studied here contains a strong current more irregular than other velocity fields previously considered in this context, but we have found that one of the main mechanisms of transport by lobe motion—the turnstile—is still at work. The methodology includes the identification of a Lagrangian barrier and the subsequent computation of transport across it. We stress that our methodology simultaneously provides the location of the barrier and the transport across it. Standard Eulerian calculations of transport require the position of the front to be given a priori by a fixed boundary, and these calculations do not distinguish the transport across the Lagrangian front from the fluctuations of this object with respect to the fixed Eulerian boundary.
In our application to the transport in the northwestern Mediterranean region, the Lagrangian barrier has been identified with one of the main oceanographic structures present here, the North Balearic Front. Transport across it proceeds in the form of filaments that are entrained into the front close to a DHT upstream, and released also in the form of filaments close to another DHT located downstream. These dynamical structures drive the transport of quantities such as temperature or salt. For this reason, the ejection of these filaments at that location can explain recent observations of waters saltier than expected just east of the island of Menorca (M. Emelianov 2006, personal communication). The identification of the DHTs and the calculation of their locations is by itself an important subject because, along with their associated stable and unstable manifolds, they serve to organize the flow in the area. Moreover, this fact, coupled with the exponential divergence of trajectories near DHTs, implies that DHTs are candidates for launch locations for efficiently designed drifter release experiments (Poje et al. 2002; Molcard et al. 2006).
Despite the success of the approach described here, much work remains to be done in order to develop dynamical systems techniques into a collection of systematic tools for analyzing general oceanographic data. A classification and understanding of the different topological regimes leading to qualitatively different modes of transport and the transitions among them, similar to the existing ones for steady and time-periodic flows, would be desirable for the cases of turbulent aperiodic flows. A characteristic of the dynamical systems approach is that it provides an unusually highly detailed description of the spatiotemporal structure of Lagrangian transport. Therefore, it may well turn out to be the optimal tool for analyzing data from ocean models with much higher spatiotemporal resolution (e.g., higher-frequency atmospheric forcing and more resolved spatial scales), which captures more physical processes. This would enable us to better define, for example, the process of the destruction of a barrier. In addition, consideration of the impact of non-Lagrangian processes, including diffusion, vertical motion, and strong localized perturbations beyond climatological forcing such as storms, would be needed to gain a more complete picture of transport phenomena and mechanisms.
Acknowledgments
AMM acknowledges the MCyT (Spanish government) for a Ramón y Cajal Research Fellowship and financial support from CSIC Grant PI-200650I224, and the Royal Society–CSIC cooperation agreement reference B2003GB03. EH-G. acknowledges financial support from MEC and FEDER through Project CONOCE2 (FIS2004-00953). AMM and EH-G acknowledge support from CSIC Grant OCEANTECH (No. PIF06-059). DS and SW acknowledge financial support from ONR Grant No. N00014-01-1-0769. We also acknowledge M. Emelianov for communicating to us the results from a recent cruise before publication.
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Velocity field at a 15.93-m depth in the northwestern Mediterranean at simulation day 649 (October). The modulus of the velocity is indicated by the arrow length. The westernmost coast and islands are the Spanish coast and Balearic Islands, respectively. The easternmost islands are Corsica, France, and Sardinia, Italy. Circles indicate elliptic ISPs and crosses indicate saddle-type ISPs. The rectangle shows the area selected for study in this article. In this area, the maximum modulus of the velocity is ∼31 cm s−1 and the minimum is 0.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
Salinity front and manifolds at days 649 and 657. Salinity (psu) is color coded as indicated by the color bar. The black circles denote the DHT in the west (HW) and the DHT in the east (HE). These are the DHTs whose unstable (red) and stable (blue) manifolds, respectively, are used to construct the Lagrangian boundary of the front. The blue dots are boundary intersection points.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
Boundaries at days 649 and 657 constructed from a (finite length) segment of the unstable manifold of HW and a (finite length) segment of the stable manifold of HE. The boundary intersection points are denoted by b649 and b657.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
Turnstile lobes from day 649. There is precisely one intersection point between b649 and b−657, which implies that there are two lobes in the turnstile. The bottom figure shows the evolved position of these lobes at day 657. Comparing the two figures, one can see that the turnstile lobe to the north of the boundary at day 649 has moved to the south of the boundary at day 657, and the turnstile lobe to the south of the boundary at day 649 has moved to the north of the boundary at day 657 (the location of the boundary itself can be seen in Fig. 3).
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
The spatiotemporal structure of cross-frontal transport described by lobe dynamics. Salinity (psu) is color coded as indicated by the color bar. (a) Evolution of the turnstile lobes at day 674 backward in time to day 664, or, in other words, the location on day 664 of the lobes that are turnstile lobes at day 674. The notation b−−681 denotes the evolution of the boundary intersection point at day 681 backward two observation times (to day 674 and to the current time, day 664). (b) Turnstile lobes at day 674. (c) The turnstile lobes evolved to day 681. (d) The turnstile lobes evolved to day 688. The notation b++674 denotes the evolution of the boundary intersection point at day 674 forward two observation times (to day 681 and to the current time, day 688).
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
Disruption of the current by an eddy. Salinity (psu) is coded in colors as indicated by the color bar. Red dots are ISPs of elliptic character; green dots are saddle ISPs. Also shown as black dots are the DHTs HW and H′E. (a) Day 733. The unstable manifold of HW does not appear to intersect the stable manifold of H′E. The unstable manifold is strongly influenced by the circulation associated with the elliptic ISP immediately to the lower right of H′E. (b) Day 736. As the elliptic ISP moves farther away from H′E, its influence carries the unstable manifold with it. (c) Day 742. The earlier Lagrangian barrier now appears completely deformed as a result of the influence of the eddy.
Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3677.1
