## 1. Introduction

### a. General background

Recent investigations of the role of the Agulhas system in the variability of the Atlantic meridional overturning circulation (AMOC) and the global climate show that the Agulhas leakage variability can impact the strength of the AMOC on several time scales (Weijer et al. 2002; Knorr and Lohmann 2003; Biastoch et al. 2008a; Beal et al. 2011). In turn, the Agulhas leakage itself strongly depends on the position of retroflection (van Sebille et al. 2009).

According to Doglioli et al. (2006) and van Sebille et al. (2010), 35%–45% of the Agulhas leakage is carried within rings (the remainder is direct leakage and leakage carried by filaments and cyclonic eddies). Observations indicate that the leakage flux into the South Atlantic is about 10–15 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}) (see Table 1.1 from van Sebille 2009; see also Gordon 1986; Gordon et al. 1987, 1992; Ganachaud and Wunsch 2000; Garzoli and Goni 2000; Boebel et al. 2003; Richardson 2007). Therefore, the outflow via anticyclonic eddies is about 4–6 Sv.

Typically, Agulhas rings are shed at a frequency of 5–6 yr^{−1}. As a rule, the retroflection protrudes westward before shedding a recurrent eddy and abruptly shifts eastward after shedding (Lutjeharms and van Ballegooyen 1988a; Dencausse et al. 2010a). According to Dencausse et al. (2010a), the position of the Agulhas retroflection typically moves from 15° to 17°E during the eddy shedding to 20° to 22°E just after the shedding.

There were also periods of almost 6 months when no shedding event was observed (e.g., Gordon et al. 1987; Byrne et al. 1995; Schonten et al. 2000; Lutjeharms 2006; van Aken et al. 2003; Dencausse et al. 2010a,b). This increased length of the shedding period may be associated with a retroflection farther to the east (de Ruijter et al. 2004). Lutjeharms and van Ballegooyen (1988b) and Lutjeharms (2006) showed anomalous and more occasional eastward shifts of the Agulhas retroflection occurring two to three times per year with durations of 3–6 weeks. Also, very irregular, so-called early (upstream) retroflection events were observed in 1986 (Shannon et al. 1990) and 2000/01 (Quartly and Srokosz 2002; de Ruijter et al. 2004), when the Agulhas Current retroflected east of the Agulhas Plateau. However, these events are uncommon.

Often, the Agulhas Current protrudes westward from the Cape of Good Hope. Such a necklike protrusion is connected with the formation of a local “zonalization” of the Agulhas influx after it passes the Agulhas Bank between 17.5° and 20°E, as shown in Fig. 1 [adapted from Beal et al. (2011, their Fig. 1) as a fragment], where the zonalization and retroflection areas are indicated by dashed ellipses [see also a similar configuration in Lutjeharms (2006, their Fig. 1.2)]. This configuration is the most favorable for Agulhas ring shedding because the necklike protrusion is almost free of topographic effects, and the eddies are shed directly into the South Atlantic. Therefore, it is important to know under what conditions this protrusion is stable, and when the Agulhas Current is more likely to retroflect directly after passing the slanted eastern slope of the Agulhas Bank.

### b. Numerical background

The Agulhas retroflection dynamics have been studied in realistic numerical models. According to Biastoch et al. (2009), the retroflection occurs at 17°E during shedding and 23°E (i.e., east from the Agulhas Bank) afterward. Similar scenarios can be seen in the Backeberg et al. (2009) and Tsugawa and Hasumi (2010) simulations.

In the

We note, however, that the fine westward protrusion configuration was simulated recently by Loveday et al. (2014, see their Fig. 3).

### c. Theoretical background

Widely accepted theoretical models indicate that Agulhas rings are shed primarily because of inertial and momentum imbalances. Nevertheless, the exact mechanism of shedding is still under discussion. For example, according to Ou and de Ruijter (1986), the Agulhas Current front, moving slowly southwestward, forms a loop soon after separating from the coast, due to the coastline curvature. The loop occludes and forms a ring, whose shedding is accompanied by the instantaneous eastward retreat of the front. This mechanism was further discussed by Lutjeharms and van Ballegooyen (1988a) and Feron et al. (1992).

A purely inertial shedding mechanism was proposed by Nof and Pichevin (1996) and discussed by Pichevin et al. (1999). According to this mechanism, ring shedding from the retroflection area is necessary to circumvent the so-called retroflection paradox. Specifically, the “rocket force” caused by the westward-propagating eddies balances the nonzero, combined zonal momentum flux (or flow force) of incoming and outgoing currents. Nof and Pichevin (1996) did not address the formation of the ring in the retroflection area and focused instead on the detaching phenomenon.

An alternative idea is that the momentum fluxes of currents are compensated by the Coriolis force, which is caused by the “ballooning” of the basic eddy (a generic term to designate the retroflection area), a mechanism suggested by Nof and Pichevin (2001) for the outflows and elaborated further by Nof (2005) for the case of a

However, Zharkov and Nof (2008a) pointed out a “vorticity paradox”; in the case of retroflecting currents, both the momentum and mass conservation equations can only be satisfied for small Rossby numbers. To circumvent this paradox, the authors considered the incoming current retroflecting from a coastline with a slant greater than a threshold value of ~15°. Zharkov and Nof (2008b) and Zharkov et al. (2010) elaborated on the effect of coastal geometry on ring shedding for 1.5-layer models with slanted and “kinked” coastlines and showed that, in the case of a rectilinear coast, there is a critical slant above which there is almost no shedding. These results are in agreement with the numerical runs in Pichevin et al. (2009).

Nevertheless, recent theoretical models still have shortcomings. The basic equations are (i) the mass conservation, (ii) the momentum balance, and (iii) the Bernoulli’s principle. It is assumed that these three equations can be satisfied together when (i) the basic eddy radius is much larger than the widths of upstream/downstream flows and significantly grows, and (ii) the potential vorticity (PV) of the basic eddy is constant. In fact, when the near-linear dynamics of approximately geostrophic incoming/outgoing currents are transformed into the dynamics of a radially symmetric basic eddy of constant PV, the Bernoulli integral cannot be conserved because an additional portion of energy is required to raise the basic eddy’s nonlinear advection [this forcing term is as strong as the Coriolis force in the basic eddy momentum equation; see Eq. (17) below]. As a consequence, there are a number of contradictions in the previous analytical models. Two of these contradictions are reported in appendix A.

### d. Present approach

To address these contradictions, we develop a new model of current retroflection and eddy shedding. Contrary to previous models of retroflection, in which the vorticity of the basic eddy, as well as the vorticity of the incoming and outgoing currents, were treated as constants, we allow the vorticity to vary. (Actually, in the text, we consider the Rossby numbers characterizing the relative vorticities, although, in principle, it does not matter which type of vorticity we mean because the Coriolis force is assumed almost constant at the scale of retroflection area.) By doing so, we aim to overcome the aforementioned paradoxes.

The main goal of this study is to use this model to understand why the Agulhas Current retroflection protrudes westward from the Agulhas Bank. We also want to clarify the relation of this westward protrusion to the ring-shedding event and establish a simple criterion for the stability of each configuration of the Agulhas Current.

The paper is organized as follows: In section 2, we introduce the governing equations for the basic eddy development and derive an analytical model of a zonal retroflecting current. We also discuss the reasonable intervals for the initial value of the current’s Rossby number. In section 3, we compare this theoretical model with a numerical model for several values of the Rossby number. On the basis of this investigation, we propose a criterion for the stability of the westward-protruding configuration of the Agulhas Current being dependent on its relative vorticity. In section 4, we confirm this hypothesis using data from eddy-resolving numerical models. Last, we summarize and discuss our results in section 5. For convenience, we define all the variables both in the text and in appendix E.

## 2. Theoretical model

In this section, we derive an analytical model of Agulhas Current retroflection. We first describe the incoming and outgoing currents, then the basic eddy. All these elements are then connected using three equations expressing Bernoulli’s principle, the momentum balance, and the conservation of volume. We allow the main variables of the model to vary in time and attempt to describe their temporal evolution. Finally, we consider the stability of this system and establish a criterion for eddy shedding.

### a. Description of the currents

Consider the situation depicted in Fig. 2; a boundary current flows along a zonal coast (in the Southern Hemisphere) and retroflects at some point. To describe this system, we consider the Cartesian coordinate system

^{−1}and

^{−1}s

^{−1}, we have

^{−1}and

^{−2}), we obtain

### b. Description of the basic eddy

### c. System of equations

At this stage, there are three unknowns that may vary in time:

#### 1) Bernoulli’s principle

#### 2) Momentum balance equation

#### 3) Conservation of volume

### d. Stability of the system

We just derived a system of equations describing the evolution of a retroflecting current and the basic eddy. Let us recall the hypothesis made to obtain the final set of equations.

The transport is accurate if the value of

### e. Eddy-shedding criteria

We finish our theoretical analysis with some considerations on the conditions of the basic eddy detachment from the retroflecting current.

#### 1) First criterion

#### 2) Second criterion

We end the development of the theoretical model here. In summary, the time evolution of the system of incoming and outgoing currents and the basic eddy in the retroflection area are characterized by Eq. (8) for

## 3. Numerical simulations

In this section, we compare the theoretical model developed in the previous section with several numerical simulations. We use a modified version of the Bleck and Boudra (1986) reduced-gravity isopycnal model with a passive lower layer and the Orlanski (1976) second-order radiation conditions for the open boundary.

### a. Model setup

The parameters are ^{−2}, ^{−1}, and *regular* value for ^{−1} s^{−1}) or a *magnified* value for ^{−1} s^{−1}). The initial condition

Parameters used in the model.

The northeastern section of the domain is filled by land, and the position of the corner of the coastline is given in Table 1. We use the free-slip boundary condition at the “coastal” walls.

The spatial resolution is ^{2} s^{−1} (the minimal value for the stability of long time simulations). Such a choice of parameters is adequate because the diffusion speed (^{−1}, which is small compared to the rings’ orbital speed (of the order 1 m s^{−1}). Our simulations are significantly nonlinear since, according to Dijkstra and de Ruijter (2001a,b), viscous effects become dominant when ^{2} s^{−1} (for their forced model of Agulhas retroflection). This conclusion is also in agreement with Boudra and Chassignet (1988).

All the simulations are run for more than 700 days and initialized with a fully retroflecting current at

### b. Snapshots of the numerical simulations

In Figs. 4–7, we plot the snapshots of the four experiments listed in Table 1. The time intervals between these snapshots are 150, 50, 50, and 10 days, respectively. We adjust these intervals according to the eddy-shedding periods in each experiment (several hundred days for

The figures show the upper-layer contours through 100-m increments. Some contours are marked in meters. The scales on the coordinate axes are in kilometers.

In all these experiments, at least one eddy forms and detaches from the current. We note that the position and the propagation speed of detached eddies do not change significantly as we move the meridional wall relative to the retroflection, even in the case when the retroflection is strongly protruding into the “open ocean.”

### c. Comparison with the analytical model

Figures 8–11 show the comparison of the results of our theoretical model with the outputs of the corresponding numerical model for the four experiments (see Table 1).

The curves of

Similarly, we plot with solid lines (theoretical model) and dashed lines (numerical model) for

In Fig. 8, we compare the two models for the configuration of expt 1 (

In Fig. 9, we compare the results of expt 2

The results of expt 4

### d. Physical interpretation

On the basis of our comparisons, we suggest that, without loss of generality, the agreement of our theoretical model with numerics can be characterized by the values of *I*, *II*, and *III* denote reasonable, transient, and unreasonable intervals, respectively. Based on Fig. 12, we conclude that

We suggest that, when

Our main hypothesis is that the unsteadiness of the zonal current retroflection with insufficiently small relative vorticity could cause the Agulhas retroflection to shift eastward from the open South Atlantic basin to some location east of the Agulhas Bank, where the incoming current flows along the slanted continental shelf. This hypothesis is based on the fact that, in our numerical simulations, the retroflection propagates upstream when the Rossby number falls into the unstable interval.

## 4. Comparison with more realistic numerical models

In this section, we apply the results obtained in the previous sections to the interpretation of the realistic numerical model simulations.

We first consider the study by van Sebille et al. (2009). The authors investigated the relation between the Agulhas retroflection position and the magnitude of the Agulhas leakage to the South Atlantic. The

Assuming that approximately 40% of Agulhas leakage is carried by rings (Doglioli et al. 2006; van Sebille et al. 2010), we multiply the values of *I*, *II*, and *III* denote stable, transient, and unstable intervals, respectively, for the regular value of *I*); the regression line shows that characteristic values of the Rossby number are in the stable interval when the location is west of 16°E and in the unstable interval when the location is east of 19°E, which is close to the Agulhas Bank. This confirms our hypothesis; the retroflection is likely to shift toward a slanted coast when the Rossby number gets into the unstable interval.

Last, we consider the

## 5. Conclusions and discussion

### a. Summary

In this paper, we have presented a new model of a zonal current retroflection and eddy shedding. In contrast to the preceding models of retroflecting currents, this one satisfies, (i) the mass conservation equation, (ii) the momentum balance equation (i.e., to resolve the retroflection paradox), and (iii) the time-dependent, altered Bernoulli’s principle (i.e., the continuity of velocity along the free streamline bounding the currents and the retroflection area). Our main assumption is that the net mass flux going into the eddies is only due to the

As shown, the system of aforementioned equations yields reasonable solutions for small values of the Rossby number (defined in the text as the stable interval). In this case, we obtained good agreement between our theoretical solutions and numerical simulations (Figs. 8, 10). Otherwise, the Rossby number is in the unstable interval. In the theoretical model, this unstable interval is characterized by an unphysical increase of the width of the incoming and outgoing currents (remember that, in our problem statement, we assumed the incoming mass flux and the current velocity to be constant). In this case, the agreement between our model and numerics significantly deteriorates (Figs. 9, 11). In numerics, we observe a strong eastward shift of the retroflection and the formation of a chain of eddies in its path (Figs. 5, 7). Based on these findings, we hypothesize that the retroflection of a zonal current becomes unstable for larger Rossby numbers and therefore the retroflection point is expected to shift toward the slanted coast (in the case of the Agulhas Current and the eastern slope of the Agulhas Bank).

This hypothesis is in agreement with the data of van Sebille et al. (2009). Using their data and Eq. (15), we show (Fig. 13) that the Rossby number most likely gets to the stable interval (defined in our theoretical model) when the Agulhas retroflection is located west of the Cape of Good Hope and to the unstable interval when it is located near and east of the Agulhas Bank. It is possible that some eddy-resolving models do not simulate the local “zonalization” of the Agulhas incoming flux after it passes the Agulhas Bank (which, in turn, can lead to underestimation of the leakage into the Atlantic) because the Rossby number is overestimated numerically, possibly because of an underestimation of mesoscale processes or a low viscosity value. The viscous friction is probably the main cause of the vorticity variation—we leave this mechanism as a subject for future investigations.

### b. Another physical interpretation

^{−1}and

^{−1}s

^{−1}is shown in Fig. 14. The gray area corresponds to the unstable domain. We hypothesize that for these values of

^{−1}for the Agulhas Current parameters used in our model. Overall, this value is reasonable. We do not consider the entire current as a part of Indian Ocean Gyre, but focus only on the area not far upstream from the retroflection. According to Duncan and Schladow (1981), the current velocities not far from the Agulhas retroflection are in excess of 1.5 m s

^{−1}. Gordon et al. (1987) gave an estimate of about 1.1 m s

^{−1}. Therefore, the velocities are usually only a bit above the upper boundary of the stable interval, implying that the westward protrusion of the retroflection area is not as common as the retroflection east of Agulhas Bank but nevertheless is not rare.

Second, the viscosity in numerical models can affect the Agulhas leakage in different ways. Although, as we mentioned before, the viscosity increases the widths of the currents in our simulations, the main effect we consider is protrusion of the retroflection toward the South Atlantic, possibly resulting in the increase of Agulhas leakage, which is carried by the rings. We do not talk here about any other parts of leakage. On the other hand, Weijer et al. (2012) also show that, in their CCSM4 numerical model, the high viscosity simulation widens the current and indeed overestimates the Agulhas leakage. However, the reason is that a simulation of the Agulhas Current that is too viscous lacks the inertia necessary for retroflection and ring shedding. Instead, outflow to the South Atlantic is mainly due to direct leakage through a viscous sublayer. Such a scenario has no relation to the subject of our paper.

Overall, the current widening is due to an internal process that regulates the vorticity, which is not necessarily the effect of viscosity.

### c. Model applicability

This model can only be applied to the dynamics of Agulhas Current retroflection. Other retroflecting currents, such as the Brazil Current, North Brazil Current, and East Australian Current, have no zonalization because they retroflect directly from slanted coasts. Also, our present approach is not valid for meddies (Pichevin and Nof 1996) because, in that situation, the incoming and outgoing flows are separated by a wall, and the upstream retreat of the retroflection area is impossible.

The foremost shortcoming of the present model is that, for a reasonable Rossby number (getting into a stable interval), eddies are unrealistic (with radii about 300 km and more) and the period of shedding is extremely long (one or even several years). For a higher value of

As for now, it is not quite clear how to describe the direct leakage and filaments in the momentum equations, so, in our theory, we assume that the net mass flux

In our model, we focus only on the basic eddy detachment time scale and do not consider periodic solutions that would allow tracing the possible retreat at longer time scales. In the future, it would be interesting to develop the theoretical model aimed on the description of the upstream propagation of the retroflection.

Finally, our theory is developed only for lenses. Applying this theory to a finite-thickness upper layer is difficult in that, if the thickness is significant, increased eddy volume due to radial growth cannot be compensated for by the decrease in vorticity (and eddy thickness). This, in addition to the viscous effect, can cause the gradual collapse of the initialized 1.5-layer structure in numerical simulations.

## Acknowledgments

The study was supported by NASA Doctoral Fellowship Grant NNG05GP65H, LANL/IGPP Grant (1815), NSF (OCE-0752225, OCE-9911342, OCE-0545204, OCE-0241036, and OCE-1434780), BSF (2006296), and NASA (NNX07AL97G). Part of this work was done during Wilton Arruda’s visit to the Department of Earth, Ocean and Atmospheric Science at Florida State University supported by a post-doctoral fellowship from CNPq of the Ministry for Science and Technology of Brazil (Proc. 201627/2010-8). This research is also part of the activities of the Instituto Nacional de Ciência e Tecnologia do Mar Centro de Oceanografia Integrada e Usos Múltiplos da Plataforma Continental e Oceano Adjacente (INCT-Mar-COI, Proc. 565062/2010-7). We are grateful to Steve van Gorder for helping in the numerical simulations and to Erik van Sebille for presenting the data we used in Fig. 13. We also thank Donna Samaan for helping in preparation of the manuscript and Kathy Fearon and Meredith Field for assistance in improving the style.

## APPENDIX A

### Contradictions in Previous Models of Retroflecting Currents

In this appendix, we describe two paradoxes (or inconsistencies) that were encountered in previous models of retroflections.

#### a. Thickness paradox on an plane

#### b. Slowly varying problem statement

According to the mass conservation and momentum balance principles, the basic eddy develops by analogy with the ballooning outflow bulge. Its volume

## APPENDIX B

### Derivation of the Momentum Balance Equation

We use the slowly varying approximation, introducing the “fast” time scale *AB* (i.e., cross sections of both incoming and outgoing flows). Because both currents are approximately geostrophic (but not exactly geostrophic; see van Leeuwen and de Ruijter 2009; Nof et al. 2012),

With all the simplifications mentioned above, Eq. (B2) takes the form of Eq. (25).

## APPENDIX C

### Reasonable Interval Criterion Derivation

We want to assess the limit of validity of the theoretical model. This is done via the characterization of the initial evolution of the system. We compute the expansion rate of the current

## APPENDIX D

### Plotting of Numerical Data in Figs. 8–11

Here we clarify the calculation of numerical parameters for Figs. 8–11:

Values of

and are calculated from the average vorticity over the eddy and over the upstream currents far from the retroflection area, then fitted with polynomials; is estimated by averaging the distance from the center to four points at the eddy boundary (north, south, east, and west); is more complicated to directly measure from the model, so we calculate it from the maximum depth of the currents and use ; ; is calculated through , where is the *x*coordinate of the basic eddy center at time*t*;is the only parameter we differentiate from the polynomial for , since discrete differentiating of was noisy; is also directly calculated from the data, using the coordinate of the easternmost point with maximum depth; and is calculated according to Eq. (43) with using data for and .

## APPENDIX E

### List of Abbreviations and Symbols

AMOC Atlantic meridional overturning circulation

HYCOM Hybrid Coordinate Ocean Model

ODE Ordinary differential equation

PV Potential vorticity

Sv Sverdrup (^{3} s^{−1})

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