## 1. Introduction

Since Veronis (1963) found multiple equilibrium solutions existing in the western boundary current (WBC) of the ocean subtropical gyre forced by strong winds, the study of nonlinear features of ocean circulation has been the subject of modern geophysical fluid dynamics study (Holland and Haidvogel 1981; Chao 1984; Cox 1987; Moro 1988, 1990). Ierley and Sheremet (1995) investigated the bifurcation map of the barotropic quasigeostrophic (QG) vorticity equation for a single subtropical gyre. Jiang et al. (1995) studied multiple equilibria of the double-gyre wind-driven circulation, showing that a perturbed pitchfork bifurcation and Hopf bifurcations exist so that the separation of the WBC from a solid wall and the formation of the eastward jet could occur either north or south of the maximum wind stress line. Speich et al. (1995) followed up the above work to determine the Hopf bifurcations more precisely, while Cessi and Ierley (1995) determined the attractors and basins of attraction of the double-gyre system. A more complete bifurcation diagram of the double-gyre system was given by Dijkstra and Katsman (1997) at a later time. Detailed history and evolution of the multiple steady states and bifurcation diagrams of the WBCs can be found in Dijkstra (2005).

The above studies have focused on the nonlinear bifurcations of the WBCs near a nonpermeable wall. However, the Pacific western boundary is porous and gappy. In particular, the collision of the South Equatorial Current and Mindanao Current forms the origin of the North Equatorial Counter Current and the Indonesian Throughflow. The validity of the above bifurcation theory to the Pacific gyre circulation has not been investigated. This article represents the first attempt to address this problem from a nonlinear dynamical system point of view.

Nof (1996) studied the problem of two WBCs colliding at a gap of the western boundary and found an analytic solution for the generation of two outgoing jets forced by the collision. Nof’s solution is on an *f* plane with the Coriolis parameter independent of the latitude. The nonlinear bifurcations of the system are not considered. Sheremet (2001) studied the problem of a unidirectional WBC flowing by a gap in a midlatitude *β* plane and found hysteresis in the evolution of the WBC path. Yuan and Wang (2011) studied the dependence of the hysteresis on the baroclinic deformation radius and have shown that the dynamics of the hysteresis result from multiple equilibrium balances among the inertia advection of the relative vorticity, the horizontal friction, and the *β* effect, which are subject to the perturbations of mesoscale eddies to shift regimes. This study extends the Yuan and Wang (2011) study to the case of two WBCs meeting at a gap of the western boundary. The hysteresis of the meeting WBCs has not been studied before.

In this study, we use a reduced-gravity, quasigeostrophic ocean model to study the nonlinear collision of two WBCs at a gap of the western boundary. Following the configuration of Nof (1996) and Sheremet (2001) studies, we adopt the midlatitude *β* plane and quasigeostrophic approximation. The two WBCs are assumed to be of equal strength for simplicity. The model configuration is described in section 2. The simple linear solution of the problem is first shown in section 3. Section 4 deals with the nonlinear solutions of the system, showing the existence multiple equilibrium states and hysteresis of the WBC paths. Section 5 contains discussions of the vorticity balance and the regime shift under the perturbations of mesoscale eddies. Section 6 contains conclusions of this study.

## 2. Model description

*L*is the Rossby deformation radius,

_{R}*ψ*the streamfunction of a depth-averaged flow,

*A*the horizontal viscosity coefficient, and

_{H}*β*the gradient of the Coriolis parameter with respect to the northward displacement. Here,

*J*() stands for the Jacobian operator.

The computational domain of the model is a rectangular basin of 3300 km × 1800 km, separated by a thin meridional wall with a gap in the middle into a western and eastern basins of 2000 km × 1800 km and 1300 km × 1800 km, respectively. The gap is located in the middle of the barrier wall with a width of 2*a*. The location of the wall is defined as *x* = 0 and the center of the gap is defined as *y* = 0 (*x* positive is eastward and *y* northward).

The northern, southern boundaries of the eastern basin are open boundaries. The WBCs in the model are driven by the streamfunction along the open boundaries as specified in the following: along the northern and southern boundaries in the eastern basin, *Q* at (−∞, −*a*) and −*Q* at (*a*, ∞), and decreases linearly from *Q* to −*Q* between −*a* and *a*. These boundary conditions force a northward flow through the southern boundary and southward flow through the northern boundary; along the barrier wall, the no-slip boundary condition is applied so that both the streamfunction and the velocity are zero. Along the northern, southern, and western boundaries of the western basin, the streamfunction and the relative vorticity are set to zero for simplicity. The partial differential equation of the quasigeostrophic vorticity is solved using the finite difference method, with a grid size of 10 km in both the zonal and meridional directions.

## 3. Linear solution

The linear solution of the system is always symmetric about the middle latitude (*y* = 0) of the computational domain (Fig. 1) because the boundary conditions and the configuration of the model are strictly symmetric about that latitude. Since the Munk thickness of the WBC is *L _{M}* = (

*A*/

_{H}*β*)

^{1/3}, the nondimensional width of the gap, is defined as

*γ*

*= a/L*.

_{M}The streamfunction of steady flows for *γ *= 6.5 and *γ* = 9.3 are shown in Fig. 1b and Fig. 1c, respectively. The wall is marked by a thick solid line along the *y* axis. Only a portion of the computational domain is shown. The horizontal coordinates are scaled by *L _{M}*. Computations have shown that, for

*γ*= 6.5, viscous force restricts the WBCs from penetrating into the western basin. Most of the streamlines of the two WBCs make only a small wiggle near the gap and turn eastward. We call this kind of state the choking state.

The streamline *ψ* = 0 extends westward to infinity and bounds the so-called “*β*-plume” region (Stommel 1982). In comparison, the steady *β*-plume solution for a larger gap, *γ *= 9.3, is shown in Fig. 1c to intrude into the western basin significantly. In this case, all the streamlines originating in the Munk boundary layer in the eastern basin have entered into the western basin. To describe the penetration quantitatively, we follow Sheremet (2001) and adopt the definition of *X _{P}*, which is the westernmost extent of the streamline Ψ =

*Q*/2, as a measure of penetration into the western basin. Figure 1a shows the penetration extent of

*X*plotted against the nondimensional gap width

_{P}*γ*. For

*γ*≤ 7.3, the two WBCs meet at the gap and then flow eastward, making nearly no penetration into the western basin; for

*γ*> 7.3, the

*X*has been into the western basin by a distance equal or larger than

_{P}*L*. We define the WBCs with

_{M}*X*less than −

_{P}*L*as penetrating into the western basin.

_{M}## 4. Nonlinear solution

### a. Bifurcation of the system

In the nonlinear regime, the solutions are more complicated and are dependent on some of the parameters of the system. To illustrate this, the following experiments are carried out using the QG model. For a fixed value of *γ*, the evolution of the *X _{P}* of the two WBCs is computed in the following way for different forcing of the open boundary conditions in the eastern basin. Starting from a small Reynolds number, defined as Re

*= Q/A*, the QG model is integrated into a steady state so that a value of

_{H}*X*is determined, which corresponds to the point in the bottom-left corner of Fig. 2. Because of the symmetry about the central latitude, the westward extents of the southern and northern WBCs are nearly the same. For simplicity, the southern

_{P}*X*is used to represent the WBC penetration into the western basin. After the

_{P}*X*is calculated, the open boundary condition is changed so that Re increases, and a new integration is implemented until a new steady state or periodic steady state is reached so that a new

_{P}*X*is calculated. For periodic eddy-shedding states, the

_{P}*X*is time dependent. In the following text, only the farthest and closest positions of the first

_{P}*X*from the gap during one period of the eddy formation cycle are plotted.

_{P}The westward extent *X _{P}* of the southern WBC streamline Ψ =

*Q*/2 as a function of Re for several values of

*γ*. The double curves for the time-dependent solution indicate the furtherest and nearest positions of the first

*X*from to the gap.

_{P}Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The westward extent *X _{P}* of the southern WBC streamline Ψ =

*Q*/2 as a function of Re for several values of

*γ*. The double curves for the time-dependent solution indicate the furtherest and nearest positions of the first

*X*from to the gap.

_{P}Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The westward extent *X _{P}* of the southern WBC streamline Ψ =

*Q*/2 as a function of Re for several values of

*γ*. The double curves for the time-dependent solution indicate the furtherest and nearest positions of the first

*X*from to the gap.

_{P}Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The above procedure is followed for each fixed value of *γ* while Re of the two WBCs increases from 2 to 46 and then decreases back to 2. Figure 2 shows how *X _{P}* varies as a function of Re for different

*γ*. The results have shown that, for narrow gaps of

*γ*≤ 7.3, no penetration of the WBCs into the western basin is possible, as suggested by the linear solution in Fig. 1. For 7.3 <

*γ*< 9.0 (

*γ*= 7.5 shown as an example), the

*X*is a single-valued monotonic function of Re, showing penetration of the WBCs at low Re and choke at higher Re. For

_{P}*γ*≥ 9.0 (

*γ*= 9.3 shown as an example); however, the

*X*becomes multivalued for some range of Re, meaning that at least two equilibria exist for the same Re of the open boundary forcing. These different equilibria can be searched by integrating the QG model from different initial states of the WBCs (penetration with small values of Re or periodic with large values of Re).

_{P}A sequence of steady-state or periodic steady streamfunction patterns for *γ *= 9.3 forced by different WBC transports at the open boundaries in the eastern basin are shown in Fig. 3. For a fixed value of *A _{H}*, the change of the WBC transport is equivalent to the change of Re. At small Re, for example at Re = 12, both WBCs penetrate into the western basin through the gap and then retroflect eastward. The two flows are nearly symmetric to the central latitude (

*y*= 0), with

*X*/

_{P}*L*= −22.9, where

_{M}*X*is the westernmost coordinate of the

_{P}*ψ*=

*Q*/2 streamline. As the transport through the open boundaries increases, the steady state

*X*increases from negative values to near zero (Fig. 2), so that the main streams of the WBCs retreat gradually from penetrating into the western basin. At Re = 30 (Fig. 3b), the steady state

_{P}*X*/

_{P}*L*is −1.0, indicating choking state for both WBCs at the gap. As Re increases further, at Re = 40, alternating periodic penetration and eddy shedding of the WBCs take place. Figures 3c and 3d show the cases of northern and southern WBCs shedding eddies at different times at Re = 40. This alternating eddy-shedding states of the WBCs persist for even larger Re until the model becomes numerically unstable. The numerical solutions thus indicate a Hopf bifurcation at Re = 37 of the equal-strength WBCs colliding at the gap of

_{M}*γ*= 9.3.

(a)–(d) Streamfunction patterns illustrating the transition from the penetrating regime to the leaping regime, then to the periodic eddy shedding regime, at *γ* = 9.3, associated with the increase of the Reynolds number. (a) Re = 16, steady penetration; (b) Re = 30, steady-leaping state; (c),(d) Re = 40, periodic solutions at different time; (e) Re = 30, periodic eddy shedding; and (f) Re = 12, steady-penetrating state. The contour interval is 0.1*Q*. Dashed contours indicate negative streamfunction values.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

(a)–(d) Streamfunction patterns illustrating the transition from the penetrating regime to the leaping regime, then to the periodic eddy shedding regime, at *γ* = 9.3, associated with the increase of the Reynolds number. (a) Re = 16, steady penetration; (b) Re = 30, steady-leaping state; (c),(d) Re = 40, periodic solutions at different time; (e) Re = 30, periodic eddy shedding; and (f) Re = 12, steady-penetrating state. The contour interval is 0.1*Q*. Dashed contours indicate negative streamfunction values.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

(a)–(d) Streamfunction patterns illustrating the transition from the penetrating regime to the leaping regime, then to the periodic eddy shedding regime, at *γ* = 9.3, associated with the increase of the Reynolds number. (a) Re = 16, steady penetration; (b) Re = 30, steady-leaping state; (c),(d) Re = 40, periodic solutions at different time; (e) Re = 30, periodic eddy shedding; and (f) Re = 12, steady-penetrating state. The contour interval is 0.1*Q*. Dashed contours indicate negative streamfunction values.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

In the decreasing leg of the loop, for example, as Re decreases from Re = 40 to Re = 30, the WBCs keep their eddy-shedding state (Fig. 2e), which are in contrast to the states in Fig. 2b in the increasing leg of the loop, when both WBCs are choked at the gap at Re = 30. The only difference between the experiments in Figs. 2b and 2e is in the initial conditions of the model integration. The multiple equilibria of the nonlinear system indicate a reverse Hopf bifurcation at Re = 14 (for *γ *= 9.3). As Re decreases further, at Re = 12, the flows return to their steady-penetrating state as in Fig. 2a (Fig. 2f), suggesting the disappearance of the multiple equilibrium states for the WBCs.

It is worth mentioning that, for *γ *≥ 9.0, no steady-choking state was detected during the integration of the reduced Re experiments. Numerical experiments have shown that, for *γ *≥ 9.0, integration of the QG model in the reduced Re experiments from periodic initial states will result in the flow to transit from the eddy-shedding regime directly to the steady-penetrating regime, without going through the choking regime. In addition, this transition takes place at a value of Re much lower than that of the transition from penetration to periodic eddy shedding (Fig. 3), hence the multiple equilibria of the nonlinear system between the two Res. The numerical results suggest that the equilibrium solutions of the QG model in this parameter domain are dependent on the history of the WBC paths, hence the hysteresis of the path evolution of the WBCs at the wide gap.

For example, for *X _{P}* is shown in Fig. 2. When the transport is small, both of the two WBCs penetrate into the western basin. The penetration extent decreases as the transport increases, until Re > 30, when both currents choke at the gap and retroflect eastward. This kind of choking state is kept as the open boundary transport is increased further, until at Re ≥ 38, when both boundary currents become unstable near the gap and start to shed eddies periodically to the western basin (Fig. 3). From there, if the transport of the WBCs is reduced, the equilibrium solutions of the QG model stay in the periodic eddy shedding regime (Re = 30 shown as an example in Fig. 3) until Re < 14, when both WBCs will penetrate into the western basin steadily.

### b. Hysteresis at large gaps

As the gap width is increased further, periodic eddy shedding is more frequently reached in the search of the equilibrium solutions. Figure 4 summarizes the onset of transitions among penetrating, choking, and eddy-shedding regimes for various nondimensional gap widths of *γ*. For narrow gaps, *γ* < 9.0, flows penetrate for small Re and choke for large Re, and the transition is single valued. The curve *R*_{PC} (*R*_{CP}) is defined as the critical Re for transitions from the steady penetrating (choking) to the choking (steady penetrating) states. Similarly, curve *R*_{PE} (*R*_{EP}) is defined as the critical Re for transitions from the penetrating (eddy shedding) to the eddy-shedding (penetrating) states. For *γ *< 9.0, the curves of *R*_{CP} and *R*_{PC} are nearly identical, suggesting essentially no hysteresis of the WBC path evolution. For larger *γ*, 9.0 ≤* γ *≤ 9.6, the distinction between *R*_{CP} and *R*_{PC} starts to be obvious, which suggests multiple equilibria and hysteresis of the WBC paths. Because of the eddy shedding at large Re, the continuation of *R*_{PC} and *R*_{CP} is terminated at about *γ *= 9.6. Evidently, there is no steady-choking state of the WBCs for larger *γ*. Instead, only the steady penetration and periodic eddy shedding of the WBCs may exist. The curves *R*_{PE} and *R*_{EP} mark the transitional Re from the penetrating to the eddy shedding regimes and vice versa, respectively. Evidently, multiple equilibrium states and hysteresis exist between these two curves. Notice that for 9.0 < *γ* < 9.6, if the initial state is an eddy shedding state, the eddy shedding persists with reduced Re until *R*_{EP}, whereas the choke persists with reduced Re only down to *R*_{CP} if the initial state is a choking state.

The curves showing Re at which the choking, eddy-shedding, and penetrating regimes terminate as a function of *γ*. The penetrating branch persists up to Re < *R*_{PC} (*γ **<* 9.6) and Re < *R*_{PE} (*γ **≥ *9.6). The choking branch persists down to Re > *R*_{CP} (*γ *< 9.6) if the initial state is choking state, while the eddy shedding state persists down to Re > *R*_{EP} (*γ **≥ *9.6).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The curves showing Re at which the choking, eddy-shedding, and penetrating regimes terminate as a function of *γ*. The penetrating branch persists up to Re < *R*_{PC} (*γ **<* 9.6) and Re < *R*_{PE} (*γ **≥ *9.6). The choking branch persists down to Re > *R*_{CP} (*γ *< 9.6) if the initial state is choking state, while the eddy shedding state persists down to Re > *R*_{EP} (*γ **≥ *9.6).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The curves showing Re at which the choking, eddy-shedding, and penetrating regimes terminate as a function of *γ*. The penetrating branch persists up to Re < *R*_{PC} (*γ **<* 9.6) and Re < *R*_{PE} (*γ **≥ *9.6). The choking branch persists down to Re > *R*_{CP} (*γ *< 9.6) if the initial state is choking state, while the eddy shedding state persists down to Re > *R*_{EP} (*γ **≥ *9.6).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

For example, at *γ *= 9.3, numerical experiments have shown that, starting from eddy-shedding states, the WBCs keep their periodic penetration regime as Re is reduced to *R*_{EP }= 14, where both WBCs shift into the penetration regime. In contrast, if starting from a choking state, the WBCs will only be able to maintain its choking regime until *R*_{CP} = 28.

For large *γ*, numerical experiments have shown that the hysteresis of the WBC paths in the vicinity of the gap is different from that at the intermediate *γ* (Fig. 2). For example, at *γ *= 11.6, as Re increases from 10 to 46, the WBCs have experienced penetration into the western basin and retreat of the penetration until a Hopf bifurcation leading to periodic eddy shedding at Re = 23. As the Re decreases from 46 to 10, the WBCs keep their eddy-shedding states past the Hopf bifurcation point at Re = 23, until at Re = 14, when both WBCs penetrate into the western basin in a steady symmetric pattern. The numerical solutions thus indicate clearly the existence of hysteresis of the WBC paths between Re = 14 and Re = 23, whereby the search of the respective equilibrium solutions (the steady penetration or the periodic solutions in the present case) are dependent on the history of the WBC evolution. The search of the respective regime solutions is also independent of the rate of the change of the parameter (the Re in this case), suggesting the hysteresis nature of the nonlinear system.

### c. The effect of a gap

Jiang et al. (1995) used a confluence point (C point) to study the multiple equilibria and low-frequency variations of the WBC near a solid western boundary. The C point is defined as the merging point of the northward and southward moving currents near the western boundary. Following their work, two kinds of numerical experiments (with and without a gap) were carried out to examine how the evolution of the C point is changed by the presence of a gap. The C point is plotted against the Re for *γ *= 9.3 in Fig. 5. If the western boundary has no gap, the C point lies on the *y* axis for as long as Re < 54 (dashed line). Here, Re = 54 is a Hopf bifurcation, beyond which the C point loses its stability to a periodic solution. Numerical experiments indicate that the Hopf bifurcation at Re = 54 is supercritical and the amplitude of C-point oscillation grows as Re increases. No pitchfork bifurcation is identified in our calculation, in comparison with the results of Jiang et al. (1995) study, because of the southern and northern open boundaries and the different driving force. To compare with the case of a nonpermeable western boundary as in Jiang et al. (1995), a C′ point is defined in this study as the intersection of the meridian of the gap with the merging latitude of the two flows separating from the western boundary. The solid line in Fig. 5 shows that the C′ point loses its stability at a much lower Re, with a Hopf bifurcation occurring at Re = 31 (solid line). The amplitude of the periodic solutions also increases with Re up to Re = 40.

Bifurcation diagram for the positions of the C point without a gap (dashed line) and of the C′ point with a gap (solid line) as a function of Re.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Bifurcation diagram for the positions of the C point without a gap (dashed line) and of the C′ point with a gap (solid line) as a function of Re.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Bifurcation diagram for the positions of the C point without a gap (dashed line) and of the C′ point with a gap (solid line) as a function of Re.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Although the C′ point of the two flows become unstable at Re = 31 near the western boundary, this instability is so trivial that the two flows are still choked, and the *X _{P}* is nearly unchanged (figures omitted). It is not until Re = 38 that the two WBCs in the vicinity of the gap become prominently periodic eddy shedding. At the equilibrium state of the model, both the

*X*and C′ points at Re = 38 exhibit clearly a purely periodic oscillation with a period of about 280 days. This oscillating behavior of the two WBCs can be confirmed by monitoring the kinetic energy time series (not shown). Figure 6 shows five snapshots of the total (left column) and anomalies of streamfunction (right column) for a period of the eddy-shedding cycle at Re = 38 and

_{P}*γ*= 9.3.

(left) Streamfunction and (right) anomalies evolution at Re = 38 and *γ *= 9.3. The contour interval is *Q*/4 for the streamfunction fields and *Q*/12 for the anomalies. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

(left) Streamfunction and (right) anomalies evolution at Re = 38 and *γ *= 9.3. The contour interval is *Q*/4 for the streamfunction fields and *Q*/12 for the anomalies. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

(left) Streamfunction and (right) anomalies evolution at Re = 38 and *γ *= 9.3. The contour interval is *Q*/4 for the streamfunction fields and *Q*/12 for the anomalies. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

At day 0, the northern branch begins to intrude into the western basin (Fig. 6; day 0). At this stage, the intrusion of the northern branch is stronger than that of the southern one, with its intruding tongue extending in the southwestward direction. The southern branch retroflects eastward because of the blocking of the northern tongue and experiences no intrusion into the western basin. As time goes by, on day 80, the northern branch sheds an eddy to the western basin and retreat from intruding into the western basin through the gap. The intrusion of the southern branch then dominates the gap circulation at day 120, forming a tongue in the northwestward direction. The intrusion of the northern WBC into the gap is blocked at this time. The southern tongue starts to grow, which then repeats the growth and demise of the northern tongue. At day 280, a very similar pattern of the streamfunction to day 0 reappears.

Shown in the right column of Fig. 6 are the instantaneous anomalies of the streamfunction, corresponding to the difference between the snapshot streamfunction and the average of the streamfunction over a cycle. The anomalies disclose a wave pattern, with cyclonic and anticyclonic eddies propagating westward alternately through the gap. When the northern branch begins to intrude into the western basin, the anomaly wave is a cyclonic eddy propagating through the gap. Similarly, the propagation of the anticyclonic eddy corresponds to the intrusion of the southern WBC into the western basin.

## 5. Discussion

### a. Vorticity balance of the hysteresis

For steady states (penetrating and choking states) at *γ *= 9.3 and Re ≤ 37, the dynamics of the two WBCs path are determined by the vorticity balance among the inertial term, the beta term and horizontal viscosity term (Fig. 7). All terms of the QG equation are moved to the left side of the equation and are extracted at a grid point nearest to the point *X _{P}*. Because of the definition of

*X*, the zonal advection of the vorticity is very small. The inertia term is dominated by the meridional advection. For small Re ≤ 21, the beta term dominates and is balanced by the inertial advection and horizontal viscosity. For Re > 21, the primary balance of vorticity is between the beta term and the inertial term (Fig. 7). As suggested by the scale analysis of transition between steady states (Sheremet 2001), the branch

_{P}*R*

_{PC}(corresponding to his curve Re

_{P}) satisfies the relationship

*R*

_{PC}~

*γ*

^{3}, while the branch

*R*

_{CP}(corresponding to his curve Re

_{L}) follows the relationship of

*R*

_{CP}~

*γ*. Indeed, the

*R*

_{CP}curve in Fig. 4 is almost straight, whereas the

*R*

_{PC}curve bends upward for large

*γ*. If not for the curve

*R*

_{PE}, we can imagine that the difference between these two curves will grow larger as

*γ*increases. At the curve of

*R*

_{PE}, the vorticity balance of the steady states is broken by instability so that the two WBCs shift to the eddy shedding regime (see later in the discussions of Figs. 8 and 10).

Vorticity balance at *X _{P}* during the interactions of the two WBCs at

*γ*= 9.3 as the Re increases from 3 to 37, before periodic solutions should occur. Vis, Bet, and Jac represent horizontal viscosity term, beta term, and inertial term in the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Vorticity balance at *X _{P}* during the interactions of the two WBCs at

*γ*= 9.3 as the Re increases from 3 to 37, before periodic solutions should occur. Vis, Bet, and Jac represent horizontal viscosity term, beta term, and inertial term in the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Vorticity balance at *X _{P}* during the interactions of the two WBCs at

*γ*= 9.3 as the Re increases from 3 to 37, before periodic solutions should occur. Vis, Bet, and Jac represent horizontal viscosity term, beta term, and inertial term in the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Percentage contributions to the total vorticity balance averaged in the D region during the interactions of the two colliding WBCs at Re = 38 and *γ *= 9.3. Vis, Bet, Jac, and Tde represent horizontal viscosity term, beta, term inertial term, and the time dependent term of the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Percentage contributions to the total vorticity balance averaged in the D region during the interactions of the two colliding WBCs at Re = 38 and *γ *= 9.3. Vis, Bet, Jac, and Tde represent horizontal viscosity term, beta, term inertial term, and the time dependent term of the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Percentage contributions to the total vorticity balance averaged in the D region during the interactions of the two colliding WBCs at Re = 38 and *γ *= 9.3. Vis, Bet, Jac, and Tde represent horizontal viscosity term, beta, term inertial term, and the time dependent term of the QG equation, respectively.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

At Re = 38, the two WBCs near the gap start to oscillate periodically. This oscillation induces the northern and the southern WBCs to shed eddies alternatively to the western basin through the gap, the dynamics of which are analyzed in Fig. 8. For periodic eddy shedding states, the positions of *X _{P}* are sometimes difficult to locate because the flow may shrink back to the east of the gap. For this reason, we selected a rectangular region diagonalled by the points (−

*a*, −

*a*) and (

*a*,

*a*), which we call the

*D*region in the following text, near the gap and calculate the absolute value of each term in the vorticity equation averaged in the

*D*region. The ratio of each term contributing to the sum of the absolute values of the vorticity equation terms is then obtained and plotted in Fig. 8. Between Day 0 and Day 40, the northern WBC moves southward on the western side of the southern WBC in the gap and passes the central latitude, forming a loop of the northern WBC. The ensuing evolution of this loop (increase of tendency term) is evidently forced by the

*β*effect and the frictional force (decrease of the two terms), which are balanced to a large extent by the inertial term, until an eddy is shed from the loop at around Day 80. The eddy takes away a part of the momentum of the northern WBC and reduces the intrusion of the northern WBC into the western basin, which results in the southern WBC to intrude into the gap on the western side of the northern WBC in the gap, forming a southern loop. The southern loop then grows under the forcing of the beta effects and friction balanced by the inertial advection until another eddy is shed from the southern WBC at around Day 240. The above procedure repeats, resulting in the northern and southern WBCs to shed eddies alternatively to the western basin through the gap.

From the vorticity balance analysis, it is clear that, as the transport of the flows, also known as Re, exceeds a certain critical value, the steady state vorticity balance cannot hold between the beta, the inertial, and the viscosity terms. From time to time, the time-dependent relative vorticity becomes important because of the instability of the current system, which then breaks the equilibrium to result in periodically shed eddies. The emergence of this oscillation, that is, the occurrence of the Hopf bifurcation with respect to the increasing Re, is influenced by the magnitude of the baroclinic deformation radius, according to numerical experiments. This can be understood from Eq. (1), in which the relative importance of *γ* = 9.3. The results of the numerical experiments indicate that for *L _{R}* = 50, 40, and 30 km, the Re of transitions from steady penetrating to periodic eddy shedding occur at

*R*

_{PE}= 37, 41, and 46, respectively, suggesting that the Hopf bifurcation occurs at larger Re for smaller deformation radii. In contrast, the reverse Hopf bifurcations of the three experiments all occur at about Re = 14, suggesting its independence to the magnitude of the baroclinic deformation radius.

Hysteresis map of two WBCs colliding at a gap of *γ *= 9.3 in the western boundary with different baroclinic deformation radii, *L _{R}*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Hysteresis map of two WBCs colliding at a gap of *γ *= 9.3 in the western boundary with different baroclinic deformation radii, *L _{R}*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

Hysteresis map of two WBCs colliding at a gap of *γ *= 9.3 in the western boundary with different baroclinic deformation radii, *L _{R}*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

### b. Perturbation by mesoscale eddies

The important role of the relative vorticity in disrupting the steady vorticity balance between the beta, the inertia, and the viscosity terms can be further demonstrated by an eddy perturbation experiment. It is known that, for a large *γ* and intermediate Re, the choking state of the two WBCs can be obtained if the driving force and the initial condition are perfectly symmetric. Owing to the existence of the multiple equilibrium states, however, these symmetric steady states can be easily perturbed to transit into a periodic eddy shedding state by a small yet finite perturbation such as that from a mesoscale eddy approaching the gap from the east. In other words, the steady states of the two WBCs are prone to shift into periodically eddy shedding states. Figure 10 shows how this could happen. At *γ *= 9.3 and Re = 31, two WBCs have reached to the steady choking state initially. A cyclonic eddy is introduced into the model initial condition by superposing streamfunction of the Gaussian distribution at a distance of 2.5*a* east of the barrier wall in the eastern basin, which then migrates westward on a *β* plane afterward. The maximum streamfunction of the eddy is *ψ*_{0} = 3.0*Q* at the center, with the maximum azimuthal velocity at about 0.4*a* from the center. The perturbation of the eddy has induced a transition of the two WBCs into periodic eddy-shedding state. The numerical experiment thus demonstrates the importance of the relative vorticity in perturbing the nonlinear current system into periodic states. At high Re, the Hopf bifurcation occurs because the threshold to such perturbations reduces to infinitesimal.

The perturbation of two choke WBCs (left) by a cyclonic eddy from the east to transit to periodic eddy-shedding states (right) at Re = 31 and *γ *= 9.3. The contour interval is 0.1*Q*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The perturbation of two choke WBCs (left) by a cyclonic eddy from the east to transit to periodic eddy-shedding states (right) at Re = 31 and *γ *= 9.3. The contour interval is 0.1*Q*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

The perturbation of two choke WBCs (left) by a cyclonic eddy from the east to transit to periodic eddy-shedding states (right) at Re = 31 and *γ *= 9.3. The contour interval is 0.1*Q*.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-12-05.1

## 6. Conclusions

Numerical experiments are conducted using a 1.5-layer, quasigeostrophic model to study the bifurcation and hysteresis of two WBCs of equal transport colliding at a gap of the western boundary. In the linear case, two steady states are identified, which are the penetrating state of the WBCs for large *γ* and choking state for small *γ*. In the nonlinear case, the steady paths of the two WBCs in the vicinity of the gap depend on the nondimensional width of the gap *γ* and on the Reynolds number (Re) of the current system and are subject to nonlinear bifurcation and hysteresis in certain ranges of the nondimensional parameters. When the gap is narrow, *γ *≤ 7.3, neither of the WBCs can penetrate into the western basin because of the restriction of the viscous force. When 7.3 < *γ *< 9.0, both WBCs penetrate into the western basin for small transport and choke for large transport. When 9.0 ≤ *γ *≤ 9.6, the two WBCs penetrate for small transport, choke for intermediate transport, and shed eddy periodically for larger transport. When *γ *> 9.6, no steady choking state is found. Instead, the WBCs can either be the penetrating or the periodic eddy shedding states in equilibrium, depending on the open boundary transport and the initial conditions. Analyses have shown that, if the beta effect dominates the *X _{P}* position and is balanced by the other terms, the two WBCs take the penetrating path; if the meridional inertial advection dominates and is balanced by the other terms, the two WBCs take the choking path. When the transport of the two WBCs is large, shear instability destabilizes the steady choking state and forces the current system to oscillate and shed eddies periodically.

In the increased Re experiments, a Hopf bifurcation is identified for the current system with *γ *> 9.0, which is sensitive to the magnitude of *γ* (*a/L _{M}*) and the baroclinic deformation radius. The Re of the bifurcation decreases for larger gaps and deformation radii. In addition, reverse Hopf bifurcations are identified in the decreased Re experiments, which occur at a smaller Re than that of the Hopf bifurcation. The Re of the reverse Hopf bifurcation is found not sensitive to the magnitude of the baroclinic deformation radius.

Hysteresis behaviors of the WBCs are found for *γ *> 9.0 because of the existence of the multiple equilibrium states. Between *R*_{EP} and *R*_{PE}, steady penetrating or choking states coexist with eddy-shedding states. The equilibrium solutions of the current system depend on the history of the evolution in the vicinity of the gap, hence the hysteresis. The steady states of the nonlinear current system are found sensitive to perturbations of relative vorticity. Impingement of mesoscale eddies can induce the transition of the current system from the steady states to the periodic eddy-shedding state.

## Acknowledgments

This work is supported by the 973 project (2012CB956000) of China, by NSFC Grants 40888001 and 41176019, by CAS Grant KZCX2-YW-JS204, and by Qingdao Municipal Grant 10-3-3-38jh. Z. Wang is supported by NSFC Grant 41206019 and by an open grant of the State Key Laboratory of Tropical Oceanography of SCSIO/CAS, and open grant of LASG of IAP/CAS. Discussions with Henk A. Dijkstra and with William K. Dewar are helpful for the improvement of the manuscripts.

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