## 1. Introduction

### a. The problem

Though the separation of western boundary currents in the ocean is a well-known feature, it is not yet totally understood. In the cases of the North Brazil (Fig. 1) and Agulhas Currents (Fig. 2), the separation takes place abruptly and leads to a curve of the coastal current of more than 90° over a few hundred kilometers. This extreme change in the direction of the flow is accompanied by the shedding of eddies, which can travel thousands of kilometers and thus play a key role in the interbasin transfers of water masses (Gordon et al. 1992). Although the cases of the Agulhas and North Brazil Currents retroflections remain the most striking and energetic examples, they are not isolated: in the Sea of Japan, the East Korean Warm Current, which flows northward along the coast of Korea, suddenly veers eastward and leads to the generation of the Ulleung warm eddy (Katoh 1994; Preller and Hogan 1998). The Great Whirl, a large anticyclonic eddy, which results from the abrupt veering of the Somali Current in the Arabic Sea, constitutes another interesting example.

In this study, the mechanisms that lead to the formation and shedding of eddies during the separation of a western boundary current were investigated. The separation was taken for granted and only the parameters liable to influence the generation and shedding of eddies were examined within the framework of a reduced-gravity model. None of the mechanisms responsible for the separation of the western boundary current was considered. Moreover, despite their known influence upon the dynamical structure of the current, neither external forcing nor irregular topography was taken into account (Veronis 1973; Marshall and Tansley 2001; Boudra and Chassignet 1988; de Ruijter et al. 1999).

### b. Discussion of previous theoretical studies

Several studies have dealt with the separation of western boundary currents within the framework of a reduced-gravity model (Moore and Niiler 1974; Ou and de Ruijter 1986; Nof and Pichevin 1996; Arruda et al. 2004). To us, it sounded worth highlighting their similarities and differences through a presentation in a common framework, namely, by using an integrated momentum budget generalized from Nof and Pichevin (1996).

*u*,

*υ*, and

*h*, respectively;

*g*′ =

*g*Δ

*ρ*/

*ρ*is the reduced gravity; and

*f*is the Coriolis frequency. Let

*ϕ*so that

*ϕ*=

*ABCD*, where

*A*is the point where the flow detaches from the coast;

*B*and

*C*are taken far enough offshore so that the momentum of the flow at this longitude is purely eastward; and

*D*is located at the coast, at a latitude where there is no flow. On condition that a steady state is reached, the reduced-gravity steady

*x*-momentum and continuity equations can be written as

*h*× (1) +

*u*× (2) in space over domain

*ψ*so that ∂

*= −*

_{y}ψ*hu*and ∂

*=*

_{x}ψ*hυ*, and application of the Stokes theorem lead toThese calculations were detailed in Nof and Pichevin (1996). Because either

*u*,

*υ*, or

*h*is null along some parts of contour

*ϕ*, Eq. (3) can be simplified toThe use of the

*y*-momentum equation allows one to express the second term on the left-hand side of Eq. (4):By performing

*h*× (5) and then using −

*hu*= ∂

*, defining*

_{y}ψ*ψ*= 0 where

*h*= 0, and integrating twice along

*BC*, one getsIntroduction of (6) into (4) leads to

One should note that, in (7), terms I and II are both positive. Term IV can take several types of values: (i) it becomes negligible either for a meridional coast (*u* = 0) or when *u* is very small (e.g., Moore and Niiler 1974); (ii) it is negative when the coast is tilted toward the northwest (e.g., Ou and de Ruijter 1986); and (iii) it is null once again when there is no meridional velocity along *AA*′ because the coast is zonal and yields a 180° retroflection (e.g., Nof and Pichevin 1996). In these cases, a negative (or null) value for term IV yields another positive (or null) contribution to Eq. (7). The balance of that equation then relies on the analysis of term III, which is not alike in the cited studies.

From the observation in the nature of the closeness to geostrophy of the eastward-flowing current, Nof and Pichevin (1996) suggested to neglect the nonlinear term III. As Eq. (7) cannot be satisfied because it becomes the sum of three positive terms, they considered that the steady assumption had to be relaxed and that the negative contribution required to balance the positive contribution by the zonal current could be provided by the westward shedding of eddies.

Moore and Niiler (1974) and Ou and de Ruijter (1986) chose other scales for the separated current; indeed, they used *R _{d}* for the crosscurrent scale and (a large)

*fR*/

_{d}*β*

^{1}Nevertheless, the important point here is that the balance of Eq. (7) becomes possible a priori with such a scaling.

Finally, the separation occurs very differently in Arruda et al. (2004; see Fig. 4), where a huge eddy is trapped between the approaching and the separated currents. The pressure force exerted by the coast on that eddy provides an additional term (not shown) in Eq. (7) that can balance the budget.

In conclusion, according to these studies, in a reduced-gravity configuration, the zonal momentum balance of a western boundary current that veers eastward toward the ocean could be balanced by (i) the generation of isolated coherent vortices propagating westward, (ii) holding strong nonlinearities (e.g., large and intense meanders), or (iii) trapping an eddy that relies on the coast.

These considerations drove us to perform a new set of numerical experiments in order to survey the parameter space and gain more insight into what happens. Section 2 presents briefly the numerical model and describes the configuration in use in our experiments. The general behavior of the flow is described in section 3, where it is shown that eddies are formed in the separation region. Section 4 examines their possible shedding. Finally, section 5 summarizes the results of our investigations and discusses some possible applications with their limitations. Note that we chose the Northern Hemisphere as the area for the simulations.

## 2. Model description

### a. Numerical model

*f*is calculated by using the

*β*-plane approximation so that

*f*=

*f*

_{0}+

*β*(

*y*−

*y*

_{0}), where

*f*

_{0}is the Coriolis frequency at the southern boundary,

*y*is the meridional position, and

*y*

_{0}is the latitude of the domain southern boundary. Finally,

**F**(

*F*,

_{x}*F*) represents the diffusion of momentum along isopycnals and is modeled by a Laplacian operator.

_{y}### b. Configuration

Figure 5 provides a schematic view of the configuration under study. A structure termed “continent” was designed in a rectangular basin so that the angle between its eastern coast and the east–west direction is *γ*. The initial condition consists of a northward coastal western boundary current and an eastward zonal current connected by a somewhat artificial piece of circularly turning current. For both currents, the potential vorticity *Q* is uniform. The coastal current starts veering eastward at the latitude *Y _{S}*, set at the distance

*d*from the northern cape. The separated current is at the distance

*L*from the extremity of the coast. In the simulations,

*d*was always set smaller than

*L*; this means that the separation always occurs southward of the latitude of the eastward flow. The calculations of the initial field and the inflow are detailed in the appendix. The intensity and velocity profile of the coastal current depend on its potential vorticity

*Q*and on its latitude of separation, which is associated to the volume transport of the current.

At every time step of the numerical integration, the western boundary current is forced by specifying a constant-in-time inflow at the basin southern boundary. This inflow has the same constant potential vorticity *Q* as the initial field and is in geostrophic balance. Application of Orlansky-type boundary conditions at the eastern and western sides of the domain allows the propagation, in a wavelike form and out of the domain, of any disturbance of the interior flow. To enhance the performance of the radiation conditions, the viscosity coefficient is increased by up to 10 times its interior value, and the meridional velocity is nudged toward 0 in a band of 10 grid points along the open boundaries. The size of the domain is adjusted to avoid side-boundary effects. In practice, none of our numerical simulations showed accumulation of energy or blocking of meanders and eddies at the borders. Besides, few simulations performed with a larger domain showed only minor differences of behavior in the interior of the domain. These findings make us think that the interior solution evolves freely and depends only on the imposed inflow at the southern boundary. Finally, the basin is closed at the northern and southern borders (with an exception at the location of the inflow), and a free-slip boundary condition is used along them and along the continent coastline.

The growth of barotropic instability is inhibited by the lack of gradient of potential vorticity in the current (Charney and Stern 1962). The current profile stability on a purely zonal current was checked prior to the running of numerical simulations and showed neither eddy generation nor meandering. This confirmed us in our feeling that, in our simulations, any forthcoming eddy generation should be related to the separation process.

The whole study was carried out on considering a background stratification with a layer thickness *H* = 700 m at rest and a reduced-gravity *g*′ = 1.5 × 10^{−2} m s^{−2}. These values for *g*′ and *H* give a separation latitude close to 50°N and an internal deformation radius *R _{d}* =

*g′H*

*f*(

*Y*) that is roughly equal to 35 km. In most of the numerical simulations, the western boundary current was characterized by

_{s}*Q*= 0.8 ×

*f*(

*Y*)/

_{s}*H*and the transport was roughly equal to 35 Sv (1 Sv ≡ 10

^{6}m

^{3}s

^{−1}).

The numerical model was integrated with a time step of 800 s and a grid space of 15 km, which, for numerical stability reasons, requires a Laplacian viscosity coefficient such as *ν* = 5 × 10^{2} m^{2} s^{−1}.

### c. Parameters

To minimize the number of parameters in the analysis, the variables under study were nondimensionalized, as done in Nof and Pichevin (1996). The seven remaining nondimensional parameters, which totally define the flow, are as follows: the potential vorticity of the current *P** = *QH*/*f* (*Y _{S}*); the volume transport of the current

*M** =

*Mf*(

*Y*)/(

_{S}*g*′

*H*

^{2}); the tilt of the coast

*γ*; the distance between the tip of the coast and the separation point

*d** =

*d*/

*R*; the distance between the cape and the eastward zonal separated current

_{d}*L** =

*L*/

*R*;

_{d}*β** =

*βR*/

_{d}*f*(

*Y*); and the Reynolds number Re =

_{S}*R*

^{2}

*(*

_{d}f*Y*)/

_{S}*ν*, which governs the viscosity. This last parameter was not modified in these simulations because the viscosity was set to be as small as possible to ensure numerical stability.

## 3. General behavior of the flow

Numerous numerical simulations were run to gain more insight into the possible behaviors of the flow and its dynamics according to the nondimensional parameters under study. As shown here through a reference simulation, these experiments highlighted the keeping of several aspects in the behavior of the current.

### a. Description of the reference simulation

The configuration of the reference simulation is close to the one used by Moore and Niiler (1974) and Arruda et al. (2004): the angle of the coast is *γ* = 90° and the coast extends northward through the whole domain. At midlatitude, the current separates from the coast and veers eastward. The potential vorticity of the flow is uniform so that *Q* = 0.8 *f* (*Y _{S}*)/

*H*and the volume transport is 35 Sv. The parameter

*β*is set equal to 6 × 10

^{−11}m

^{−1}s

^{−1}to accelerate the experiments; as shown later, it has no impact upon the general behavior of the flow. The time length chosen for the running of the model is 4000 days to allow the development of an overall behavior.

Figure 6 illustrates the evolution of the layer thickness. It shows that, over the first days of the experiment, the separation point is moving northward along the coast. This motion goes along with the development of meanders within the eastward flow. Their propagation is at first eastward. Then, while they are growing, they change direction and travel westward. The first meander eventually “hits” the western boundary current and forms an eddy.

As discussed later, the fate of this eddy proved to greatly differ between the experiments. In the present simulation, the eddy stays at the location where it was generated. After the formation of the eddy, the separation point of the western boundary current is rejected farther south and then starts moving northward again until the eventual recapture and absorption of the eddy by the coastal current. In the meantime, new meanders develop within the eastward flow and the whole scenario continues to reproduce indefinitely. In this simulation, the generation of eddies is approximately periodic and the period is *T* = 400 days. A steady state is never reached.

### b. Analysis

It is now worth considering the processes at play in this reference simulation. In our opinion, the flow is mainly driven by the growth of meanders in the eastward current. The behavior of meanders that grow out of eastward flows was described by Cushman-Roisin et al. (1993). When meanders are small, their travel is eastward, but while they are growing they start to be more affected by the planetary vorticity gradient rather than by the inertial effect and they eventually travel westward. Cushman-Roisin et al. (1993) showed that the critical amplitude *A*_{0}, which separates the two regimes, is about *A*_{0} ∼ *f* (*Y _{S}*)

*R*/

_{d}*β*

*A*

_{0}, its migration becomes westward. This observation is in reasonably good agreement with the analysis by Cushman-Roisin et al. (1993). When the meander reaches the coast, it closes upon the approaching current and forms a cell point in the flow; as shown in Capet and Carton (2004), this is a prerequisite to the formation of eddy.

We next consider the origin of the generation of such meanders. As previously mentioned, numerical experiments initialized with a strictly zonal current characterized by a constant potential vorticity value showed no signs of instability. Therefore, the cause for the growth of meanders needs to be searched for in the dynamics of the separation process itself. The initial state in use in our reference simulation is not balanced, mainly because of the artificial connection between the western alongshore boundary and eastward detached currents. However, the finding of a periodic global behavior for the whole flow suggests that the importance of the initial state is limited. In fact, our experimental results are in favor of the initiation of meander formation by the periodic migration of the separation point along the coast. Such a migration can be explained by a meridional momentum budget integrated over the

In the analysis by Arruda et al. (2004), the balance of the northward current momentum can also be achieved steadily when the separation point is north enough and when a stationary “intrusion” anticyclonic eddy is formed at the coast, but none of our simulations reached such a stationary state. In their simulations, Arruda et al. (2004, see their Fig. 7) indeed reached a steady solution, but we suspect this is because of the unrealistically large value (1500 m^{2} s^{−1}) of their diffusion coefficient. Moreover, within our surveyed parameter regimes, we never reached any of the steady solutions of Moore and Niiler (1974), Ou and de Ruijter (1986), or Arruda et al. (2004). Figure 6 shows that, at the time at which the eastern meander switches from an eastward migration to a westward one, the current goes through a state that looks like a steady meandering solution but without getting close enough to this particular state to hold it. Figure 8 highlights that, even in a numerical experiment starting from the steady analytical solution proposed by Moore and Niiler (1974), the meanders keep growing until they eventually drift westward to hit the coast and form eddies. This finding is in agreement with the results of the study by van Leeuwen and de Ruijter (2009), which shows that the solution proposed by Moore and Niiler (1974) is not balanced. Nevertheless, as the focus of the present study is the behavior of the generated eddies, our search for steady solutions was stopped at this stage.

The periodic movement of the separation point suggests that *d**, the initial distance between the separation point and the tip of the coast, is not a useful parameter. Indeed, according to several experiments (not shown) carried out on varying *d** while keeping the other parameters constant, *d** proved to have some impact only over the first steps of the experiments. The next section of this study will therefore focus on the roles played by the five remaining independent parameters (*L**, *P**, *M**, *β**, and *γ*) and their impacts on the fate of eddies.

## 4. The fate of eddies

Because our numerical experiments had evidenced the generation of eddies in the separation area and great differences in their behaviors between the experiments, in this section it is worth investigating the parameter regime that allows the eddies to exit the separation region. What can make them drift away from the generation area? In reduced-gravity models, Northern Hemisphere anticyclones tend to follow a westward route because of the planetary vorticity gradient (Nof 1983; Killworth 1983; Sutyrin and Morel 1997). Thus, anticyclones generated well beyond the tip of the coast may escape from their region of formation and migrate westward without bumping into the coast. When Northern Hemisphere anticyclones interact with a western boundary wall, their drift results from the alongshore component of the *β*-induced velocity and the image effect (Shi and Nof 1994; Sansón et al. 1998). In our reference simulation, carried out in the case of a meridional coast and a zero *β*-induced component, the eddies show no drift away from their generation area, which suggests a small image effect. It is likely that tilting the wall toward the west through a decrease of *γ* will have a very limited impact upon the image effect, which is known to depend only on the type of boundary condition and on eddy characteristics. On the other hand, it will obviously give rise to an alongshore component of the *β* drift and thus allow northwestward migration of the eddy along the coast.

To conclude, the eddies may escape from their region of formation on the condition that either (i) the separated eastward flow is located far enough beyond the northernmost point of the coast, which allows the eddies to escape westward without bumping into the coast, or (ii) the coast is sufficiently tilted toward the west (*γ* < 90°) and thus the *β*-induced westward drift speed has a nonzero northwestward alongshore component. Examining these two situations provides a rational way to survey the parameter space and minimizes the risk of missing an important flow regime.

### a. The role of the latitude of retroflection

Figure 9 presents the time evolution of layer thickness when the eastward flow of the separated current is located far beyond the northern cape of the coast. As previously noted, meanders grow within the eastward flow, eventually drift westward until they reach the coast, and then close upon themselves to form an eddy. The newly formed eddy can now escape westward thanks to *β*.

Figure 10 illustrates the results of the *β**–*L** parameter space survey when *Q* = 0.8 *f* (*Y _{S}*)/

*H*and

*M*= 35 Sv. When the values of

*L** are large enough, the generated eddies leave their region of formation and migrate westward without interacting with the coast. On the other hand, when

*L** values are small, the eddies are blocked. But, what is the critical value

*L**

*that separates the two regimes? Because eddies are formed when a westward-propagating meander finally hits the coast, their diameter should be roughly equal to the meander amplitude; a good order of magnitude for this amplitude, and thus for eddy diameter, was given in Cushman-Roisin et al. (1993), in which the critical size is*

_{c}*f*(

*Y*)

_{S}*R*/

_{d}*β*

*β**

*β**

*L**

_{c}=

*α*/

*β**

*α*= 0.4, for which the agreement is the best.

### b. The role of the angle made by the coast

When the tilt of the coast *γ* is less than 90°, the whole behavior of the flow becomes quite intricate: (i) the *β*-induced westward eddy drift speed now gives birth to a northward alongshore component, which may help the eddies drift away from the generation area; (ii) as the jet path separates from the coast, it may still move north and south in order to adjust the alongshore momentum budget (Arruda et al. 2004) and *β* may also bend it westward, because of its influence on the not-yet-separated eddy; (iii) the drift of eddies depends on their own size, which is unknown; (iv) the along-wall momentum budget is neither zonal nor meridional, and it becomes more complex; and (v) numerical viscous effects may be increased at the coast because of the staircase-like discretization of the tilted coast, so that the shed vortices may be quickly eroded, even within our higher-resolution simulations. For all of these reasons, going into the theoretical analysis of such flows would lead us far beyond the scope of this paper.

These considerations lead us to propose the following reasoning. Our numerical simulations showed that, in the case of a meridional coastline (*γ* = 90°), eddies cannot propagate westward; thus, they are always recaptured by the alongshore current. On the other hand, with a zonal coastline (*γ* = 0°), eddies always escape westward (Pichevin et al. 1999). The findings about these two extreme regimes are in favor of the existence of an in-between critical angle value at which the flow switches from a non-eddy-escaping behavior to an eddy-escaping one. To check this hypothesis, numerical experiments were carried out under the same conditions as before [*Q* = 0.8*f* (*Y _{S}*)/

*H*and

*M*= 35 Sv]. Figures 11 and 12 illustrate the evolution of the layer thickness at different

*γ*values (80° and 30°, respectively). When the tilt is large, Fig. 11 shows that the eddies are recaptured by the western boundary current before escaping from the generation area. On the other hand, when the tilt is small, Fig. 12 shows that they are released into the open ocean.

Figure 13 displays the results of the *γ*–*L** survey of the parameter space in order to determine whether or not eddies reach the open ocean. In the case of very negative *L** values (a very long coast), the critical angle is about 60°; when the angle *γ* is less than 60°, the behavior of the flow is the same as in the *γ* = 0° regime (eddies escape from their formation region). On the other hand, at higher *γ* values, the flow behaves as in the *γ* = 90° case (eddies are recaptured by the western boundary current). Whatever the value of *L**, eddies are released into the open ocean at low *γ* values and recaptured at large *γ* values. Higher *L** values only increase the critical *γ* value required for switching to an eddy-shedding regime; the proximity of the tip of the coast compensates for the more slowly induced drift speed. Finally, when *L** exceeds a critical value (between 0 and 3; i.e., when the current retroflects far enough beyond the tip of the coast), the flow always generates eddies.

### c. The role of the remaining parameters

The above analysis emphasized the roles of *L** and *γ*. The impact of the last three parameters (*P**, *β**, and *M**) was investigated by numerical experiments (not shown) performed by varying *P** between 0.5 and 2.0, *β** between 0.001 and 0.04, and *M** between 1.7 and 3.4. They proved to exert only a quantitative effect by modifying, for example, the critical value at which the different regimes are separated.

## 5. Conclusions

The investigations reported in this study were aimed at gaining more insight into the separation of western boundary currents and the formation of eddies during the process, and at determining whether these eddies reach the open ocean or stay trapped in their formation area, where they are further reembedded into the boundary current that has given birth to them. They showed that, within the parameter space under consideration, the separation point of the western boundary current keeps migrating south and north along the coast, which leads to the formation of growing meanders within the eastward flow. When their size is large enough, they start migrating westward till they finally hit the coast, close upon themselves, and regularly form eddies.

The release of these eddies into the open ocean then strongly depends on the geometrical configuration. Indeed, when they are formed far enough from the northern tip of the coast, no obstacle blocks their westward drift, and thus they can escape westward whatever the set parameters. On the other hand, when they are formed at the south of the tip, their fate is governed by the tilt of the coast. Our experiments highlighted the existence of a critical angle of about 60°, below which the eddies are always shed into the open ocean. Above it, their release, or their trapping within their formation area, depends on both the angle of the coast and the position of the eastward flow relative to the northern cape.

As with any process-orientated study, these investigations have some limitations. By eroding the generated eddies, numerical viscous effects in turn affect their advection because of the *β* and image effects. In fact, scaling the eddy radius as 0.5 × *f* (*Y _{S}*)

*R*/

_{d}*β*

*f*(

*Y*) = 8 × 10

_{S}^{−5}s

^{−1},

*β*= 6 × 10

^{−11}m

^{−1}s

^{−1}and

*ν*= 500 m

^{2}s

^{−1}led to an eddy diffusive time scale of about 300 days, which is comparable to the periodicity of our flows. The paths of eddies were also affected by the lack of barotropic mode in our simulations through a decrease of the meridional component of their

*β*-induced velocity.

Nevertheless, the results of this study are in good agreement with available observations. Indeed, in the case of the North Brazil Current (Fig. 1), the coastline continues beyond the retroflected current; therefore, the tilt of the coast is a crucial parameter. Its value is far smaller than the critical value found here. Observations and this study both suggest that rings of the North Brazil Current drift fast enough to escape from their formation area. In the case of the Agulhas Current (Fig. 2), the retroflection occurs far beyond the tip of the coast, so that we are in the eddy-shedding parameter domain again.

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## APPENDIX

### Constant Potential Vorticity Coastal Jet

*υ*is the alongshore component of the velocity field,

*x*is the longitude of the coast,

_{c}*l*is the width of the current,

*H*is the layer thickness at rest,

*Y*is the separation latitude,

_{S}*y*

_{0}is the latitude of the domain southern boundary, and

*Q*is the current potential vorticity. Figure 5 shows the configuration. Analytical solutions are given below. They are the same as the ones derived by Ou and de Ruijter (1986) for a straight coastline:whereBecause the coastal jet is in geostrophic equilibrium and has a finite offshore extension

*l*, its volume transport can be easily calculated by following Ou and de Ruijter (1986):At the latitude of separation

*Y*, the layer thickness becomes null,

_{S}*h*(

*x*,

_{c}*Y*) = 0, and the jet volume transport becomesOne should note that, for a given stratification, the current velocity and the layer thickness profile are entirely defined by

_{S}*Q*and

*Y*.

_{S}The analytical solutions for the initial fields of the separated eastward current are calculated in the same way by matching the two regions with circular streamlines tangential to both the alongshore and eastward currents.

^{1}

During the final revision of our paper, we found a proof of this in an early online version of a paper by van Leeuwen and de Ruijter (2009). We were not aware of this paper while working on our manuscript.