## 1. Introduction

Vortices are an essential part of the oceanic circulation: they are long-lived, they can migrate over long distances compared with their size, and they trap water in their cores. Because of these characteristics, vortices are important in the transport and dispersion of physical and biological properties in the ocean. Two examples are the vortices advected by the Caribbean Current toward the Yucatan Channel and the vortices that after being ejected by the North Brazil Current (NBC) drift toward the Lesser Antilles.

In the Caribbean Sea, vortices with radii between 50 and 130 km and swirl speeds between 30 and 70 cm s^{−1} are advected westward by the Caribbean Current at speeds ranging from 2 cm s^{−1} in the west to 15 cm s^{−1} in the east (Richardson 2005). Some of these vortices reach the Yucatan Channel, where their further evolution is still unclear. The Canek observing program has exposed the passage of eddies with sizes comparable to the channel width. This is indicated both by the empirical-mode structure of the along-channel flow (Abascal et al. 2003) and by the looping streamlines computed using the full measured velocities **u**(*x*, *t*), where *x* is the distance across the channel and the time *t* is treated as equivalent to a spatial coordinate (Badan et al. 2005). Numerical simulations to study the connection between the variability in the Atlantic Ocean, the Caribbean Sea, and the Gulf of Mexico, also showed the passage of eddies through the Yucatan Channel (Murphy and Hulburt 1999).

The NBC flows to the northwest along the northern Brazilian coast and between 6° and 8°N separates sharply and curves back on itself. From this retroflection, about eight rings are shed yearly (Johns et al. 2003). With a diameter of about 400 km, these vortices are among the largest ones in the ocean. By traveling toward the Lesser Antilles they contribute to the transport of South Atlantic upper waters into the Northern Hemisphere. Altimetry data suggest that some of them enter the Caribbean Sea through the passages between the Lesser Antilles (Goni and Johns 2003). In situ observations, however, show only the passage of filaments of the original vortices (Fratantoni and Richardson 2006). Numerical simulations reproduce well the vortex shedding and give a detailed picture of the vortex evolution once they reach the Antilles (Garraffo et al. 2003). Thus, according to these simulations, some rings enter the Caribbean nearly intact, some rings move northward on the east side of the Antilles, and some rings split—a fraction of which enters the Caribbean while the rest remains east of the Antilles.

The evolution of vortices in the ocean is affected by the currents they are immersed in, which vary both in space and time, as well as by the irregular coastline and bottom topography. Nevertheless, the use of idealized models (either analytical or experimental) is a first step toward a better understanding of real vortices.

Simmons and Nof (2002) studied the squeezing of vortices through gaps both analytically, using integral constraints, and numerically, using a reduced-gravity primitive equations model. Their vortices move toward the gap either because they are advected by a current or because of the drift induced by the *β* effect. In the former case, the behavior of the vortex depends on the Rossby number (Ro) and the ratio of the speed of the advecting flow through the center of the gap to the swirl speed of the lens: *μ* = *U*/( *f*Ro*R _{i}*) (where

*R*is the Rossby deformation radius,

_{i}*f*the Coriolis parameter, and

*U*the flow speed at the center of the gap). Weak vortices compared to the system’s rotation (Ro = 1/4) and with strong advection (

*μ*> 0.3) were squeezed through the gap, whereas intense vortices (Ro = 1) were partially drained by symmetric wall jets until they shrank enough to pass. This last process is similar to the one undergone by vortices advected by the

*β*effect.

Johnson and McDonald (2004) studied the motion of two-dimensional vortices near a gap in an infinitely long straight barrier. They first used an analytical model to calculate the trajectories of point vortices embedded in simple ambient flows, such as a uniform flow parallel to the barrier and a flow converging toward the gap. It was found that the vortex may or may not pass through the gap depending on its initial position and the ambient flow. Then they used a numerical model (contour dynamics with conformal mapping to handle the nontrivial flow domain) to compute the evolution of circular patches of uniform vorticity (Rankine vortices). They found that the trajectories of the vorticity centroid were in good agreement with those of the point vortex, except when a flow through the gap forced the vortex toward the tip of the barrier and made it split.

In a laboratory Cenedese et al. (2005) studied the evolution of vortices impinging on a pair of circular islands. In these experiments the vortices drifted because of a topographic *β* effect. They found that only small vortices pass wholly through the gap between the islands (*D/R* > 3.6, where *D* is the gap span and *R* is the radius of maximum azimuthal velocity); vortices of intermediate sizes pass only partially (2.3 < *D/R* < 3.6); and large vortices do not pass through the gap (*D/R* < 2.3).

Here, we study the two-dimensional flow of a vortex embedded in a channel flow obstructed by a partial barrier. Our objectives are to characterize the behavior of the vortex as a function of the parameters of the problem and to describe stirring and transport in the flow. We use a vortex-in-cell model to solve numerically the two-dimensional Euler equations and perform a straightforward vorticity census in the up- and downstream sides of the barrier to characterize the behavior. We analyze transport and stirring in the flow with a dynamical systems approach: we determine the Lagrangian flow geometry and compute residence times of fluid particles and finite-size Lyapunov exponents (FSLEs). We performed laboratory experiments in a homogeneous, rotating fluid to compare with the numerical results.

The rest of the paper is organized as follows: after defining the problem in section 2, we find the paths of point vortices in section 3. In section 4 we present the evolution of finite-area vortices as a function of the problem parameters. In section 5 we use a dynamical systems approach to study the transport and stirring properties of the flow. In section 6 we present the experimental results. Finally, we summarize and discuss our results in section 7.

## 2. Definition of the problem

*R*), circulation (Γ), and initial position (

*x*

_{0},

*y*

_{0}) of the vortex; the speed of the current (

*U*); the gap span (

*D*); the channel width (

*b*); and the barrier thickness (

*d*). To simplify the problem, we keep fixed

*x*

_{0},

*b*, and

*d*by making the following assumptions (a posteriori justifications of these are given in sections 3 and 4c): (i) the vortex initial location is far from the barrier, (ii) the channel width is much larger than the vortex radius, and (iii) the barrier is thin. Therefore, we use

*x*

_{0}= −15

*R*(the origin of the coordinate axes is at the center of the gap),

*b*= 20

*R*, and

*d*=

*R*/2. Under these assumptions, the following dimensionless parameters determine the flow evolution:

*D/Dt*= ∂/∂

*t*+

*u*∂/∂

*x*+

*υ*∂/∂

*y*is the material derivative, and

*u*is the velocity in the

*x*direction (along the channel) and

*υ*in the

*y*direction (across the channel). Note that in this model a negative vortex evolves as the mirror image, with respect to the channel axis, of a positive vortex. Therefore, in the rest of the paper we will discuss only positive vortices.

## 3. Motion of point vortices

The evolution of finite-area vortices will be better understood if we discuss first the simpler case of a point vortex. The motion of a point vortex is the result of the advection by the steady current and the velocity induced by the presence of the boundaries. The latter is easily explained by invoking the “method of images” in which the impenetrability condition is enforced by placing an “image” vortex of opposite circulation at the mirror point. Thus, a vortex that is close to a wall moves parallel to it and in the same direction as the fluid located between the vortex and the wall (Helmholtz 1858).

*z*=

*x*+

*iy*be the position in an infinite channel with walls given by

*y*= 0 and

*y*=

*b*, which at

*x*= 0 has a perpendicular, infinitely thin barrier with a gap of length

*D*in the middle; and let

*ζ*=

*ξ*+

*iη*be the position in a similar channel but without the barrier. Then the following transformations map one domain into the other:

*a*= (

*b*−

*D*)/2.

*ζ*channel, the complex potential of the flow due to a uniform stream and a point vortex located at

*ζ*

_{0}=

*ξ*

_{0}+

*iη*

_{0}is given by

*χ*in this domain is found by solving ∂

*χ*/∂

*η*

_{0}=

*u*,

*ζ*

_{0}denotes that the result is to be evaluated at the point

*ζ*

_{0}.

The trajectories in the original *z* channel are given by *χ*′ = constant after substituting for the value of *ζ* given by Eq. (2). Figure 2 shows several examples for different values of the nondimensional parameter Γ/*U b* (note that we use *b* as length scale here because a point vortex has zero radius). Figure 2a corresponds to Γ/*U b* = 0; it therefore represents streamlines of a flow that is uniform far away from the barrier; Fig. 2d corresponds to Γ/*U b* = ∞; it thus represents the trajectories of the point vortex when there is no stream. Note the fixed point of saddle type in the middle of the gap. The trajectory that asymptotically moves toward this point divides the flow domain in qualitatively different regions: trajectories on the right-hand side (looking downstream) pass through the gap, whereas trajectories on the left-hand side end up moving upstream. Figures 2b,c show two cases where Γ/*U b* has intermediate values (i.e., both the stream speed and the vortex circulation are different from zero).

In all cases the trajectories are almost parallel to the channel walls, except in the neighborhood of the barrier. That is to say, the initial position of the vortex along the channel is unimportant if it is sufficiently far from the barrier (at distances larger than one channel width). The initial position across the channel, on the other hand, critically determines the vortex evolution.

## 4. Motion of finite-area vortices

### a. Vortex-in-cell model

We solve Eq. (1) with a vortex-in-cell model written in MATLAB. The outline of the method is the following: the initial vorticity field *ω*(*x*, *y*) is approximated by a set of point vortices distributed on the numerical domain. The area *s* represented by each point vortex is assumed equal [the circulations are thus *γ _{k}* =

*sω*(

*x*,

_{k}*y*), where (

_{k}*x*,

_{k}*y*) is the point-vortex position]. The flow domain (Fig. 1) is covered by a Cartesian grid, and the vorticity on grid points is calculated by adding the contributions of all the point vortices within neighboring cells with a bilinear scheme. The streamfunction is obtained by inversion of the Poisson equation, ∇

_{k}^{2}

*ψ*= −

*ω*, using a five-point finite-difference Laplacian and MATLAB standard functions to solve the set of algebraic equations. The velocity field is evaluated from the streamfunction using second-order centered differences. Then, the velocity of each point vortex is determined using a bilinear interpolation, and the positions of the point vortices are advanced in time with a second-order Runge–Kutta scheme. This process is repeated every time step.

We use free-slip conditions on the solid boundaries (i.e., *ψ* = *A* along the left-hand- side boundary—looking downstream—and *ψ* = *B* along the right-hand-side boundary, where *A* and *B* are constants) and uniform flow on the entrance and exit of the channel (i.e., *ψ* varies linearly from *A* to *B* on these boundaries). The vortex is always far from the entrance and exit to minimize the effect of the open boundaries.

### b. Numerical experiments

Two different resolutions are used in the numerical experiments. Low-resolution simulations (525 × 150 grid points, with 7 grid points per vortex radius) were used to characterize the vortex behavior in the parameter space (Γ′, *D*′, *y*′_{0}). High-resolution simulations (875 × 250 grid points, with 12 grid points per vortex radius) were used to analyze stirring and transport in a few points of the parameter space. Although details differ between low- and high-resolution simulations, large-scale features and integrated quantities are roughly the same; for instance, the vorticity passage across the gap differs by only 1%–2% if computed with a high- or low-resolution run.

*ω*

_{0}is the peak vorticity,

*r*

_{0}is the radius where the vorticity becomes zero,

*n*is an integer number, and

*r*is the radial coordinate. This kind of distribution has been used in previous works (e.g., Velasco Fuentes 2001). We consider two different profiles: (i) steplike,

*n*→ ∞, and (ii) smooth,

*n*= 2. The radius of the vortex

*R*is taken as the radius where the maximum velocity is reached. Therefore,

### c. Numerical results

Because the only vorticity present in the flow is that of the vortex, a vorticity census on the upstream and downstream sides of the barrier suffices to identify different types of behavior. We identified three:

(i) Total passage, when at the end of the simulation all the vorticity is found downstream of the barrier. The left column of Fig. 3 shows an example: the vortex (black blob) is advected toward the barrier; it touches one tip of the barrier as it reaches the gap and filamentation occurs, but finally the whole vortex passes.

(ii) Partial passage, when at the end of the simulation some vorticity is found downstream of the gap and some is found upstream. The middle column of Fig. 3 shows an example: the vortex splits into two parts when it touches the tip of the barrier, one part is advected by the ambient flow and the other moves along the barrier against the flow.

(iii) Total blockage, when at the end of the simulation no vorticity is found downstream of the gap. The right column of Fig. 3 shows that this occurs when the drift induced by the barrier is so large that the vortex misses the gap, hits the left side of the barrier, and moves along this against the ambient flow.

Figure 4 shows the amount of vorticity found downstream of the barrier on the parameter plane (*D*′, Γ′) for three values of *y*′_{0} = −5, 0, 5. Full passage is more common if *D*′ is large and Γ′ is small (note that we show a different range of Γ′ values with the different values of *y*′_{0}). The regions of parameter space where total passage or total blockage occur are usually contiguous: partial passage is restricted to a small area of the parameter space (it is found for small *D*′; and the smaller the *y*′_{0}, the smaller the range of *D*′ where it occurs).

Figure 5 shows the influence of the vorticity profile on the vortex evolution. The distribution of the three regimes in the parameter space is almost equal for vortices with steplike and smooth vorticity profiles. If a steplike vortex completely passes to the downstream side of the barrier, then a smooth vortex with the same initial conditions also passes completely or leaves a tiny fraction of its mass on the upstream side. If a steplike vortex stays on the upstream side, then a smooth vortex also stays or forms a tiny filament that passes to the other side. The largest differences occur when a steplike vortex only partially passes through the gap; in this case, a smooth vortex does the same, but the fraction that passes can be up to 20% smaller or 10% larger than for steplike vortices.

The effect of the barrier’s thickness on the flow evolution was assessed by computing the passage of a vortex through a barrier of thickness 15*R* (where *R* is the vortex radius). Note that this is 30 times the thickness used in the rest of the simulations discussed here. This geometry now resembles more a narrow channel than a gap in a wall, but it was so chosen to enhance the effect of the barrier’s thickness. We found that the flow in the upstream part was not significantly modified by the thickness of the barrier and that the difference in the vorticity passage was between 1% and 2%. Obviously, for total and partial passage, the flow in the downstream side had important differences because the vortex takes longer to pass through the narrow channel and it is deformed in the process.

## 5. Transport and stirring

The study of transport and stirring in fluid dynamics with a dynamical systems approach has become standard in the past 25 yr [see Aref (2002) for the history and Ottino (1989) for an introduction to this subject]. Initially, these tools were used to study industrial and idealized flows (e.g., Aref 1984), and more recently they have been applied to oceanic flows (for a review, see Wiggins 2005).

*ψ*is the streamfunction. This system of equations is known in dynamical system theory as a Hamiltonian system; the streamfunction

*ψ*represents the Hamiltonian.

The data produced by the numerical model have characteristics—such as finite time and aperiodic time dependency—that render it impossible to analyze transport with traditional techniques such as Poincaré maps, Melnikov function calculations, and so on. Therefore, the analysis was made using three different tools: the Lagrangian flow geometry, the residence time of fluid particles, and the finite-size Lyapunov exponents.

For this analysis, we used four high-resolution simulations (875 × 250 grid points) with a vortex initially centered with respect to the gap (*y*′_{0} = 0), and (Γ′, *D*′) = (38, 5), (40, 3), (46, 3), and (60, 3). The first two simulations correspond to total passage, the third to partial passage, and the fourth to total blockage. The difference between the two total passage simulations is that the vortex underwent filamentation in the second simulation only.

### a. Lagrangian flow geometry

The Lagrangian geometry of a two-dimensional, incompressible flow consists of three elements: (i) elliptic particles are material particles that while moving induce a swirling motion on a set of particles; (ii) hyperbolic particles are material particles that while moving attract a set of particles exponentially and repel another set of particles exponentially; and (iii) invariant manifolds are the material lines formed by the particles that are attracted (stable manifold) or repelled (unstable manifold) by the hyperbolic particles. Note that the terms *induce*, *attract*, and *repel* are used here only to describe the fluid motion in the vicinity of the elliptic and hyperbolic particles, and hence they do not imply a cause–effect relationship.

The stable and unstable manifolds constitute the geometrical template that governs transport between different flow regions (see, e.g., Wiggins 2005). The following property will serve to interpret the results of this section: two particles located on different sides of the stable manifold, however small the distance between them, evolve very differently in the future. Similarly, two particles located on different sides of the unstable manifold, however small the distance between them, evolved very differently in the past.

When the flow is stationary, hyperbolic particles coincide with saddle-type stagnation points and manifolds coincide with separatrix streamlines. It is then to be expected that in a slightly or even moderately time-dependent flow, the hyperbolic particles are close to the instantaneous saddle-type stagnation points (see, e.g., Velasco Fuentes 2001, and references therein). Therefore, we construct the Lagrangian geometry using the Eulerian, instantaneous geometry as a starting point. We first take the streamfunction *ψ* computed by the numerical model and correct it for the vortex translation; we thus obtain a modified streamfunction Ψ with a reduced temporal dependency. Then we find the fixed points of Ψ at different times with the method presented in Velasco Fuentes and Marinone (1999). With these elements, the Lagrangian geometry is computed as follows: the unstable manifolds at time *t _{m}* are found by taking a short segment that, at the start of the simulation (

*t*= 0), crosses the saddle-type stagnation point along the unstable direction and computing its evolution under

**u**(

*x*,

*y*,

*t*) from time

*t*= 0 to time

*t*=

*t*. The stable manifold is computed in the same way, but now taking a short segment that at the end of the simulation (

_{m}*t*=

*t*), crosses the saddle-type stagnation point along the stable direction and computing its evolution under

_{f}**u**(

*x*,

*y*,

*t*) from time

*t*=

*t*to time

_{f}*t*=

*t*. The evolution of these passively advected segments is obtained by computing the evolution of a set of particles (nodes) that lies along the line. As the flow evolves, the nodes move apart from one another due to stretching of fluid elements; new nodes must therefore be added between the old ones to guarantee an accurate description of the contour.

_{m}We calculated the Lagrangian geometry at every time step of the four high-resolution simulations mentioned above. In the initial condition one hyperbolic particle, located somewhere between the vortex and the left side of the barrier, is observed in all cases. In the final condition, the number of hyperbolic particles observed depends on the regime of evolution. In the total passage regime, only one hyperbolic particle is observed and this is located somewhere between the vortex and the left side of the barrier. In the total blockage regime, two hyperbolic particles are observed, one on the front and one on the rear side of the advancing vortex. In the partial passage regime, three hyperbolic particles are observed, one on the downstream and two on the upstream side of the barrier.

Figure 6 shows the stable manifold at time *t* = 0 for the four simulations. This manifold has a spiral shape that goes from the barrier toward the vortex; at some point the manifold folds back, surrounds a parcel of fluid, and then moves back toward the barrier almost along the same path. The turning point of the manifold lies outside the vortex in the total passage and total blockage regimes, whereas it lies inside the vortex in the cases of partial passage and in total passage with filamentation (cf. Velasco Fuentes 2005). The area of the vortex surrounded by the manifold grows from zero in the total passage regime to a maximum equal to half the vortex area in the partial passage regime, then it decreases to zero again in the total blockage regime.

### b. Residence time

We define the residence time as the interval during which a particle remains on the upstream side of the channel. We computed the evolution under **u**(*x*, *y*, *t*) of a regular mesh of particles located upstream of the barrier and measured the time that each particle takes to go from its initial position to the downstream side of the channel. We plotted these results to obtain residence time maps (Fig. 7). These maps are characterized by a strong gradient with a spiral shape, which approximately matches the stable manifold at time *t* = 0. The curve of strong gradient divides the flow domain in regions of short, long, and infinite residence times of fluid particles. This is a manifestation of the property of the stable manifold discussed above; namely, that nearby particles located on different sides of the stable manifold will have a very different future: one passes to the downstream side in a short time while the other remains in the upstream side for a long time, even indefinitely. Figure 7 shows that the region of short residence time is found to the right and in front of the vortex (when looking downstream), the region of long residence time is found to the left of the vortex, and the region of infinite residence time corresponds to the part of the vortex that does not pass through the gap. This region may extend to the ambient flow if the vortex is strong enough.

### c. Finite-size Lyapunov exponents (FSLEs)

*τ*is the time it takes for two particles, initially separated a distance

*δ*

_{0}, to get separated by a distance

*δ*(Aurell et al. 1997).

_{f}The method used to calculate the FSLE is based on that of Shadden et al. (2005). A Cartesian grid of particles is placed over the part of the domain that is going to be studied. Each of these particles has four satellites: two in the *x* direction and another two in the *y* direction. The distance between each pair of satellites is *δ*_{0}. Then the particles are advected during one time step, and the distance between the satellites is measured. The interval *τ* is the time that any pair of satellites takes to reach a separation *δ _{f}.* The value of

*λ*is calculated using Eq. (12) with

*δ*

_{0}= 2 and

*δ*= 4. The values for the FSLE at the end of the simulation are an average of the

_{f}*λ*computed for each particle. (Please note: below we use a dimensionless form of the FSLE

*λ*′ =

*λ*

*R*/

*U*.)

Figure 8 shows the FSLE map when the vortex passes completely through the gap without undergoing any filamentation; the parameter values are (Γ′, *D*′, *y*′_{0}) = (38, 5, 0). The largest values of the FSLE are found around the vortex, and local maxima are arranged along a spiral curve that approximately coincides with the stable manifold at *t* = 0.

Figure 9 shows the FSLE average values inside the vortex (*r* < *R*) and around it (*R* < *r* < 1.5*R*). These quantities were calculated for the four simulations used to obtain the Lagrangian geometry as well as for another four high-resolution simulations. In all simulations, the average FSLE inside the vortex is close to zero up until the vortex collision with the wall. After this event, the value increases: the increment is larger in the cases of partial passage and total passage with filamentation than in the cases of total passage without filamentation and total blockage. The more vorticity is shed, the bigger the FSLE becomes at the end. Also, in all simulations, the highest FSLE values are found in a ring surrounding the vortex. Before filamentation starts, this values tend to be constant and to depend linearly on Γ′ and to be practically independent of the value of *y*′_{0} or *D*′.

## 6. Laboratory experiments

The experiments were performed in a rectangular tank with a 100 cm × 60 cm base and a depth of 30 cm mounted on a rotating table. We simulate the current by moving the barrier toward a vortex in an otherwise quiescent fluid. A moving bridge across the tank sustains two plates that span the whole water depth (Fig. 10). The thickness of the plates is 1 cm, and their width is varied depending on the desired gap span.

We filled the tank up to a depth *H* = 20 cm and set the rotation period of the table to 7.5 s (the Coriolis parameter is therefore *f* = 1.68 s^{−1}). Once the water inside reached a state of solid body rotation, we generated the vortex by withdrawing 2 L of water during 15 s, that is, two rotation periods of the table. This was done using a thin tube (1.0-cm diameter) placed at the center of the tank and immersed 4 cm. When the vortex was formed, the plates were moved at a constant speed along the tank using a pulley system and a dc motor fed with a power supply mounted on the table.

We measured, using particle image velocimetry (PIV), the velocity field of the vortices generated with the forcing described above and with the plates being at rest. A theoretical velocity profile, obtained from Eq. (7) with *n* = 2, was fitted to the measured profile using a least squares method. The values obtained for the radius and the circulation of the vortex were *R* = 2.6 ± 0.2 cm and Γ = 111 ± 6 cm^{2} s^{−1}. Because the vortices are much smaller than the Rossby deformation radius [(*gH*)^{1/2}/*f* = 83 cm] and the Coriolis parameter is constant, the dynamics observed in this experimental setting are equivalent to those represented by Eq. (1).

The values of the parameters that were kept constant in the numerical simulations are similar in the laboratory experiments: the thickness of the barrier (*d* ≈ *R*/2.6 in the laboratory versus *d* = *R*/2 in the numerical simulations), the channel width (*b* ≈ 23*R* versus *b* = 20*R*), and the initial along channel position of the vortex (*x*_{0} ≈ −10*R* versus *x*_{0} = −10*R*).

Several experiments were performed varying the gap span (i.e., *D*′) and the wall speed (i.e., Γ′), always leaving the vortex approximately centered with respect to the gap (*y*′_{0} ≈ 0). Figures 11 –13 show the flow evolution in three experiments whose parameter values correspond, respectively, to the three different regimes found in the numerical simulations. The vortex is marked with dye. The lines drawn on the bottom are 20 cm apart, so it can be seen that the dye blob (about 15 cm in diameter) is much bigger than the vortex, which has a radius of approximately 2.6 cm. For comparison, Fig. 14 shows the paths of the vortex centroids in these experiments as well as the trajectories of point vortices with matching initial conditions.

In experiment 1, the parameter values are Γ′ ≃ 34 and *D*′ ≃ 3; hence, the numerical model predicts that vortices with either steplike or smooth vorticity profiles make a full passage through the gap (see Fig. 5) and, indeed, Fig. 11 shows that the dye blob passes easily through the gap. The path of the vortex centroid (light gray line in Fig. 14) shows an initial drift to the left, which is reversed as the vortex approaches the gap. Experiment 3 has parameter values Γ′ ≃ 103 and *D*′ ≃ 3, and in this case the numerical model predicts that the vortex will end up moving upstream along the wall. Figure 13 shows that when the vortex reaches the barrier, it deflects to the left of the gap and impinges on one of the plates. Because water is a slightly viscous fluid, the flow must satisfy the no-slip condition at the solid boundaries; therefore, negative vorticity is generated on the plate (recall that the original vortex has positive vorticity) and, at *t* = 39 s, a dipole forms and it starts to move away from the plate (see, e.g., Doligalski et al. 1994). Because its positive half is stronger, the dipole moves along a curved path and, by time *t* = 51 s, it is headed directly toward the gap through which it finally passes. The whole process is clearly illustrated by the looping path of the vortex centroid (black line in Fig. 14). Finally, in experiment 2, the parameter values are Γ′ ≃ 50 and *D*′ ≃ 3; here, the numerical model predicts that the vortex will split in two when colliding with the tip of the barrier, then, one part will pass through the gap while the other moves upstream along the wall. Figure 12 shows that one fraction of the blob passes through the gap while the other fraction stays for a few seconds on the upstream side of the plate before also passing through the gap. The reason for this is the same as in experiment 3, except that in this case, the dipolar vortex is not well developed: the negative vorticity generated on the plate barely forms a bulge visible in the frames corresponding to *t* = 24 s and *t* = 25 s. The path of the vortex centroid (dark gray line in Fig. 14) shows a strong drift to the left, which is abruptly reversed before the vortex can advance along the barrier.

In all experiments, including the ones not shown here, the vorticity generated at the tips of the plates is continuously introduced in the downstream side. In most cases, this vorticity prevents the reorganization of the vortex after passing through the gap.

## 7. Discussion and conclusions

We analyzed the dynamics and transport properties of a vortex in a channel flow obstructed by a partial barrier. Three parameters are identified as determining the flow evolution: the relative circulation of the vortex (Γ′), the initial position of the vortex (*y*′_{0}), and the relative gap span (*D*′). The vortex trajectories were first analyzed with a point-vortex model; then the evolution of finite-area vortices was studied with a two-dimensional numerical code. The vortex behavior depends on a competition between the advecting stream and the effect of the boundaries. The former compels the vortex to pass through the gap. The latter compels the vortex to stay on the upstream side not only by the obvious mechanism of blocking the vortex motion but by inducing a drift (through the image vortex) that is ultimately in the upstream direction. We found three possible behaviors: total passage, partial passage, and total blockage. The boundaries between these regimes are irregular surfaces in the three-dimensional space (Γ′, *D*′, *y*′_{0}); hence, there are no simple “critical values” marking out the transitions between regimes. This notwithstanding, the following rules of thumb may be applied when a vortex is initially located on the axis of the channel (*y*′_{0} = 0): weak vortices (Γ′ < 45) always make a full passage, strong vortices (Γ′ < 55) make full passage only through wide gaps (*D*′ > 5), and vortices of intermediate strength (45 < Γ′ < 55) make a partial passage through narrow gaps (*D*′ > 5) and a full passage through wide gaps (*D*′ > 5).

A comparison with previous studies is not straightforward because of the differences in the geometry or in the models used. For instance, Simmons and Nof (2002) studied reduced-gravity lenslike vortices that only interact with the solid boundary through direct contact; that is, there is no image vortex, and thus the boundaries play no dynamic role until the vortex touches them. In spite of this major difference, they have a similar result when vortices are advected by a stream: weak vortices make a full passage, whereas strong vortices are destroyed and some of their mass remains on the upstream side. Johnson and McDonald (2004) studied, as we do, point and finite-area vortices in two dimensions, but the geometry of the boundaries and the ambient flow are different. Only in the neighborhood of the gap are the flow and the boundaries comparable, and here our results are consistent with theirs: the influence of the relative strength of the point vortex on the position of the fixed point of saddle type (see Fig. 2) and the splitting of the finite-area vortex as it collides with the tip of the barrier. The experimental study of Cenedese et al. (2005) differs from ours both in the geometry of the obstacles (circular cylinders instead of straight plates) and in the propulsion mechanism of the vortex (*β*-induced drift instead of an advecting current). This notwithstanding, let us assume that the vortex motion caused by the topographic *β* effect is completely analogous to the motion caused by the current. Then they have Γ′ ≈ 30, and thus all their vortices should make a full passage and not only those with *D*′ > 3.6. Any of the following factors may explain this contrasting result. Because Cenedese et al. (2005) generated the vortex with an ice cube placed on the free surface of the water, baroclinic effects may still be important when the vortex encounters the cylinders; this would lead to a rather complicated process whose outcome would be determined by the relative strengths of the bottom lens, the upper vortex, and the *β* effect (Shi and Nof 1994). On the other hand, if the vortex is already fully barotropic, then it should move to the northwest, heading toward the northern cylinder rather than toward the gap, thus producing the splitting of the vortex and only a partial passage through the gap.

The presence of a partial barrier in the channel introduces a strong time dependency for both point and finite-area vortices. It thus enhances the stirring ability of the flow. Indeed, a point vortex embedded in a uniform channel flow is either stationary (when the drift induced by its image balances the advection produced by the flow) or it moves with constant speed. In any case, the flow observed in a system that moves with the point vortex is steady and no stirring occurs. If the point vortex is replaced by a circular vortex of small radius (compared to the channel width), the flow is still approximately steady and thus a poor stirrer.

The three methods employed to study transport and stirring give results that are both consistent and complementary. It was found that the geometry of the stable manifold at time *t* = 0 is associated with the regime of behavior. It also divides the flow field into areas of short, long, and infinite residence times. FSLE maps indicate where stirring is maximum: a comparison between the average FSLE values inside the vortex and the behavior of the vortex suggests that the larger the average, the more important the mass exchange between the vortex and the ambient flow. The average FSLE around the vortex, where an area of maximum stirring occurs, suggests that the vortex also serves as a stirrer for the ambient flow, and that this stirring initially depends only on the relative vortex circulation (Γ′).

Results of laboratory experiments in a homogeneous, rotating fluid are consistent with the analytical and numerical results only up to the moment when the vortex touches the barrier. Viscous effects then become dominant and the flow evolution differs from that computed with the inviscid models. In real oceanographic situations, the coastal sloping topography would probably have a greater influence than the no-slip condition. Because the final tendency of vortices moving over topography is to translate along isobaths (see, e.g., McDonald 1998), we conjecture that a coastal topography will inhibit the passage of vortices. Although this should be the subject of further investigations, in situ observations seem to support this hypothesis: NBC rings abruptly turn northward as they encounter the bottom topography between Tobago and Barbados (Fratantoni and Richardson 2006).

Even though we used highly idealized models (analytical, numerical, and experimental), our results have some implications to real oceanographic flows. The coastal geometry in the region of the Lesser Antilles is very different from the one used in the present study, yet the fact that the passages are much thinner than the average size of the NBC rings suggests that the Lesser Antilles constitute “an unsurmountable barrier to ring translation” (Fratantoni and Richardson 2006). The geometry of the western Caribbean Sea, on the other hand, is more akin to the one used here; therefore, conclusions derived from our results are more relevant for this region. The eddies of the Caribbean Sea, with radii between 50 and 130 km and swirl speeds between 30 and 70 cm s^{−1}, are immersed in a current that flows at speeds between 25 and 80 cm s^{−1} (Richardson 2005). The distance between the coasts of Cuba and Honduras (about 750 km) is taken as the channel width, and the distance between Cabo Catoche, Mexico, and Cabo San Antonio, Cuba (about 200 km), is taken as the gap span. Therefore, the dimensionless parameters are estimated to be 1.5 < *D*′ < 4, 2 < Γ′ < 16 and −4 < *y*′_{0} < 4. This region lies well below the transition between partial and full passage regimes shown in Figs. 4, 5; therefore, we conclude that Caribbean eddies should make a full passage through the Yucatan Channel. Numerical models of the circulation in the Caribbean Sea and the Gulf of Mexico support this result, whereas altimetric observations seem to show a less frequent passage of vortices (J. Sheinbaum 2008, personal communication). Further in situ observations should provide a definite answer.

## Acknowledgments

We are thankful to two anonymous reviewers for their suggestions and comments on an earlier version of this manuscript. This research was supported by CONACyT (México) through a postgraduate scholarship to MDM.

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