## 1. Introduction

This study is continuation of the work by Wang and Yuan (2012), which studied the nonlinear dynamics of two equal-transport western boundary currents (WBCs) colliding at a gap. That work is extended in this paper by considering two unequal-transport WBCs colliding at a gap.

Existing studies of the colliding nonlinear WBCs have focused mostly on the dynamics of the collision at a nonpermeable wall (Jiang et al. 1995; Cessi and Ierley 1995; Dijkstra 2005). Nof (1996) considered the problem of two boundary currents colliding at a gap of the western boundary and found an analytic solution for the generation of two outgoing jets forced by the collision on an *f* plane. The *β* effects and the nonlinear bifurcations of the system were not considered. Wang and Yuan (2012) investigated the bifurcation and hysteresis of two equal-transport WBCs colliding at a gap on a *β* plane using a 1.5-layer, quasigeostrophic (QG) ocean circulation model. They adopted the approaches of Sheremet (2001) and Yuan and Wang (2011) and investigated the dependence of the solutions of the nonlinear system on the nondimensional parameters. In this paper, we study the case of two WBCs of unequal transports colliding at a gap.

A typical example of the problem in the real ocean is the meeting of the Mindanao Current (MC) and the South Equatorial Current (SEC) at the Pacific entrance of the Indonesian Throughflow (ITF). ITF is one of the most important ocean currents to global ocean circulation and climate variations (e.g., Godfrey 1996; Wajsowicz and Schneider 2001). It is a pathway for warm, fresh waters from the Pacific warm pool to the Indian Ocean and is the only low-latitude ocean current connecting two major ocean basins. Recent studies suggest that ITF could be an important ocean channel for Indian Ocean dipole to force on the Pacific ENSO variability (Yuan et al. 2011, 2013; Xu et al. 2013). Thus, the study of the origin of this current is very important to modern oceanography and climate studies. Observations suggest that the stronger MC has a retroflection that penetrates more deeply into the Sulawesi Sea than the SEC does (Gordon et al. 2012; Fig. 1a). The trajectories of satellite-tracked surface drifters of Lukas et al. (1991) have shown that the MC and SEC sometimes meet outside the entrance of the ITF as if there was an invisible wall connecting the Mindanao and the New Guinea islands (Fig. 1b). So far, there have been no dynamical explanations of these flow patterns at the entrance of ITF. In addition, recent observations of the circulation at the entrance of ITF show a significant cyclonic eddy in the Sulawesi Sea likely shed from the MC retroflection. The dynamics of this feature are yet to be explained.

The model configuration is described in section 2. Section 3 compares linear and nonlinear solutions of the colliding WBCs with unequal transport, the dynamics of which are discussed using controlled numerical experiments. The theory is compared with the latest drifter observations in the Sulawesi Sea in section 4. Section 5 contains the conclusions of this study.

## 2. Model description

*β*-plane and quasigeostrophic approximations are adopted. The governing equation is

*L*

_{R}is the Rossby deformation radius,

*ψ*is the streamfunction of a depth-averaged flow,

*A*

_{H}is the horizontal viscosity coefficient, and

*β*is the gradient of the Coriolis parameter with respect to the northward displacement. The

The computational domain of the model is a rectangular basin of 3300 km × 1800 km, separated by a thin (one grid wide) meridional wall into a western and an eastern basin of 2000 km × 1800 km and 1300 km × 1800 km, respectively. A 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* is northward). 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 northern, southern, and eastern 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*_{S} from −900 km to *−L*_{M} and −*Q*_{N} from *L*_{M} to 900 km. Both *Q*_{S} and *Q*_{N} are positive values. Between *−L*_{M} and *L*_{M}, the streamfunction decreases sinusoidally in a monotonic manner from *Q*_{S} to −*Q*_{N}. For the equal-transport case in Wang and Yuan (2012), *Q*_{N} equals *Q*_{S}; for the unequal-transport case, *Q*_{N} = *λQ*_{S}, where *λ* is the ratio between the northern and southern WBC transports. The width of the northern outflow at the eastern boundary is changed to *λL*_{M} correspondingly (because *Q*_{N} = *λQ*_{S}). These boundary conditions force a northward flow through the southern boundary and southward flow through the northern boundary, the transports of which exit the model domain through the middle of the eastern 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 standard finite difference method, which includes the Arakawa Jacobian operator for

## 3. Collisions of two WBCs with unequal transport

To generalize the problem, we used the nondimensional parameter *λ*, which is the ratio between the northern and southern WBC transports, to describe the asymmetry of the collision. Because the governing Eq. (1) is symmetric under the transformation *y* → −*y* and *ψ* → −*ψ* the solution is antisymmetric with respect to the *x* axis. Without losing generality, we will only consider the cases of *λ* < 1. As examples, we take *λ* = ¼, ½, and ¾, respectively, in the experiments to be discussed below.

### a. Linear cases

The steady linear solutions of the problem are characterized by the balance between the *β* term and the lateral friction term in Eq. (1). The patterns of the two WBCs depend only on a single nondimensional parameter *γ*, and the solution is calculated by setting the nonlinear advection terms to zero and transport small. For example, for *γ* = 8.11, the streamfunction of steady flows of the standard case (STD) with *λ* = 1, and of *λ* = 0.75, 0.5, and 0.25, is shown in Fig. 2. The wall is marked by a thick solid line along the *y* axis in Fig. 2. The horizontal coordinates are scaled by *L*_{M}.

Streamfunction of steady linear solutions for equal-transport WBCs [(a) STD] and unequal-transport WBCs [(b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25 for *γ* = 8.11] colliding at a wide gap. For the equal-transport case, *X*_{P} is the westernmost extent of the streamline *ψ* = *Q*/2. For the unequal-transport cases, *ψ* = *Q*_{S}/2 (*ψ* = *Q*_{N}/2), respectively. The *Q*_{S} (*Q*_{N}) is the transport of southern (northern) WBC, with *Q*_{N} = *λQ*_{S}. The contour interval is 0.1*Q*. The horizontal coordinates *x* and *y* are scaled by *L*_{M}.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Streamfunction of steady linear solutions for equal-transport WBCs [(a) STD] and unequal-transport WBCs [(b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25 for *γ* = 8.11] colliding at a wide gap. For the equal-transport case, *X*_{P} is the westernmost extent of the streamline *ψ* = *Q*/2. For the unequal-transport cases, *ψ* = *Q*_{S}/2 (*ψ* = *Q*_{N}/2), respectively. The *Q*_{S} (*Q*_{N}) is the transport of southern (northern) WBC, with *Q*_{N} = *λQ*_{S}. The contour interval is 0.1*Q*. The horizontal coordinates *x* and *y* are scaled by *L*_{M}.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Streamfunction of steady linear solutions for equal-transport WBCs [(a) STD] and unequal-transport WBCs [(b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25 for *γ* = 8.11] colliding at a wide gap. For the equal-transport case, *X*_{P} is the westernmost extent of the streamline *ψ* = *Q*/2. For the unequal-transport cases, *ψ* = *Q*_{S}/2 (*ψ* = *Q*_{N}/2), respectively. The *Q*_{S} (*Q*_{N}) is the transport of southern (northern) WBC, with *Q*_{N} = *λQ*_{S}. The contour interval is 0.1*Q*. The horizontal coordinates *x* and *y* are scaled by *L*_{M}.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

We use *X*_{P}, which is the westernmost extent of the streamline *ψ* = *Q*/2, as a measure to quantify the penetration of the WBCs. Computations have shown that the linear steady solutions of the system are always symmetric about the midlatitude (*y* = 0) of the computational domain when the transports of the two WBCs are equal (Fig. 2a; STD); thus, the *X*_{P}/*L*_{M} of the southern and northern WBCs are both at −20.7. The streamlines *ψ* = 0 separating from the tips of the wall extend westward to infinity, which are called “the *β* plume” by Stommel (1982). For the WBCs with unequal transport, the *X*_{P}) and *X*_{P}) are not equal. Furthermore, the zero-value streamline is no longer coincident with the *x* axis. For example, for *λ* = 0.75, the symmetry of the two WBCs about *y* = 0 is broken, with the 0 streamline lying north of *y* = 0 (Fig. 2b). The variable *λ* decreases to *λ* = 0.5 and 0.25,

To quantitatively describe the penetration of the currents into the western basin with different nondimensional gap width *γ*, the penetration measures *X*_{P} *γ* in Fig. 3. All quantities are scaled by *L*_{M}. For small *γ*, viscous force prevents the WBCs from penetrating into the gap. We call this the choking state. As *γ* increases, the WBCs turn from the choking state into the penetrating state and the stronger WBC always penetrates further into the western basin than the weaker one. For example, for *λ* = 0.25, the southern WBC turns from choke into penetration at *γ* = 5.7. By the time the northern WBC penetrates into the western basin at *γ* = 8.4, the southern WBC has already been in the penetrating regime. Here, the transition from choke to penetration is marked by the *X*_{P} *L*_{M}) and the gap half-width (i.e., *a*). Thus, the stronger WBC always penetrates further into the western basin than the weaker one. Figure 3 also indicates that the more unequal of the two WBCs is, the deeper the penetration of the stronger WBC is.

Dependence of the westernmost position *X*_{P} of the streamline *ψ* = *Q*/2 on the nondimensional parameter *γ* = *a*/*L*_{M}, where *a* is the half-gap width of the western boundary and *L*_{M} is the Munk thickness of the WBC, for several *λ* in the steady linear solutions. The thin solid line is *X*_{P}/*L*_{M} = −*γ*. The intersections of this line with the *X*_{P} lines give points of transition from the choking to the penetrating flow regimes. NB stands for northern WBC and SB stands for southern WBC. The variable *λ* is the ratio between the northern and southern WBC transports. The *λ*25, *λ*50, and *λ*75 stand for *λ* = 0.25, 0.5, and 0.75.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Dependence of the westernmost position *X*_{P} of the streamline *ψ* = *Q*/2 on the nondimensional parameter *γ* = *a*/*L*_{M}, where *a* is the half-gap width of the western boundary and *L*_{M} is the Munk thickness of the WBC, for several *λ* in the steady linear solutions. The thin solid line is *X*_{P}/*L*_{M} = −*γ*. The intersections of this line with the *X*_{P} lines give points of transition from the choking to the penetrating flow regimes. NB stands for northern WBC and SB stands for southern WBC. The variable *λ* is the ratio between the northern and southern WBC transports. The *λ*25, *λ*50, and *λ*75 stand for *λ* = 0.25, 0.5, and 0.75.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Dependence of the westernmost position *X*_{P} of the streamline *ψ* = *Q*/2 on the nondimensional parameter *γ* = *a*/*L*_{M}, where *a* is the half-gap width of the western boundary and *L*_{M} is the Munk thickness of the WBC, for several *λ* in the steady linear solutions. The thin solid line is *X*_{P}/*L*_{M} = −*γ*. The intersections of this line with the *X*_{P} lines give points of transition from the choking to the penetrating flow regimes. NB stands for northern WBC and SB stands for southern WBC. The variable *λ* is the ratio between the northern and southern WBC transports. The *λ*25, *λ*50, and *λ*75 stand for *λ* = 0.25, 0.5, and 0.75.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

### b. Nonlinear cases

Wang and Yuan (2012) found multiple equilibria of two equal-transport WBCs colliding at a gap. Their numerical solutions indicate that, when the gap (of width 2*a*) is narrow, *a* ≤ 7.3*L*_{M}, and neither of the WBCs can penetrate into the western basin. When 7.3*L*_{M} < *a* < 9.0*L*_{M}, both WBCs penetrate into the western basin for small transport and choke for large transport. When 9.0*L*_{M} ≤ *a* ≤ 9.6*L*_{M}, the two WBCs penetrate for small transport, choke for intermediate transport, and shed eddies for large transport. When *a* > 9.6*L*_{M}, the WBCs have two equilibrium states: the penetrating and the eddy-shedding states. No steady choking state is found for *a* > 9.6*L*_{M}.

For the unequal-transport case, the two opposing WBCs present quite different flow patterns compared to those of the equal-transport case. To illustrate this, a sequence of steady (including periodic steady) streamfunction patterns for *γ* = 9.3 for the equal- (STD; left) and unequal-WBC (*λ* = 0.5; right) transports are shown in Fig. 4. For *γ* = 9.3, the two equal-transport WBCs have three different steady states (penetration, choke, and eddy shedding), depending on the transport of the WBCs.

Sequences of steady or periodic steady streamfunction *ψ* (*x*, *y*) for *γ* = 9.3 with (left) *λ* = 1 (STD) and (right) *λ* = 0.5 case (*λ*05) as the Re increases and decreases, illustrating the difference between the equal- and unequal-transport WBCs. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Sequences of steady or periodic steady streamfunction *ψ* (*x*, *y*) for *γ* = 9.3 with (left) *λ* = 1 (STD) and (right) *λ* = 0.5 case (*λ*05) as the Re increases and decreases, illustrating the difference between the equal- and unequal-transport WBCs. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Sequences of steady or periodic steady streamfunction *ψ* (*x*, *y*) for *γ* = 9.3 with (left) *λ* = 1 (STD) and (right) *λ* = 0.5 case (*λ*05) as the Re increases and decreases, illustrating the difference between the equal- and unequal-transport WBCs. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The evolution of the *X*_{P} of the two WBCs is computed in the following way. For a set of the open boundary conditions in the eastern basin, starting from a small Reynolds number, defined as Re *= Q*_{S}/*A*_{H}, the QG model is integrated into a steady state so that an *X*_{P} is determined. The steady state is determined if the change of the kinetic energy in the western basin due to the linear trend in the last 1000 outputs, each output representing 10 steps of model integration, is less than 0.01% of the total kinetic energy. Then the open boundary conditions are changed so that Re increases, and a new integration is conducted until a new steady or periodic steady state is reached so that a new *X*_{P} is calculated. For periodic eddy-shedding states, the *X*_{P} is time dependent. In the following text, only the farthest and closest positions of the first *X*_{P} from the gap during one period of the eddy-shedding cycle are plotted. The Reynolds number is gradually increased from a small value to a large enough value and then decreased back to a small value. We call the first procedure the increasing Re course (indicated by “_I” for simplicity) and the second procedure the decreasing Re course (indicated by “_D” for simplicity).

The steady (or quasi steady) solutions for both equal- and unequal-transport WBCs at different Re are shown in Fig. 4. Figure 4 (left) is the equal-transport case (STD) and Fig. 4 (right) is the unequal-transport case of *λ* = 0.5. At Re = 10 in STD_I, nearly all of the streamlines from the Munk boundary layers penetrate into the western basin. The westernmost extent of the *ψ* = *Q*/2 streamline for the northern and southern WBCs *λ*05_I, the streamlines presents asymmetrical *β* plumes, with

At Re = 20, symmetry is preserved for the equal-transport case (STD_I Re = 20), with both WBCs choked at the gap, with *X*_{P}/*L*_{M} = −8.9. The main streams of the WBCs only make small wiggles at the gap and retroflect to flow eastward. The penetrations of the unequal-transport WBCs at Re = 20 in *λ*05_I also decrease at the gap, with

At Re = 30 in STD_I, the two equal-transport WBCs are further choked at the gap, with *X*_{P}/*L*_{M} = −0.98, whereas the two unequal-transport WBCs for *λ*05_I have transited into a periodic eddy-shedding state, with

At Re = 40 and larger, the WBCs of both equal- (STD) and unequal-transport (*λ* = 0.5) experiments are in eddy-shedding states near the gap.

We then use these eddy-shedding states as the initial condition and conduct the decreasing Re experiments, with Re decreasing from 40 gradually to 1. It is noticed that, at Re = 20, the WBCs of both the STD_D and the *λ*05_D experiments are in eddy-shedding states, which are different from the results in the increasing Re experiments.

At Re = 16 in *λ*05_D, the two WBCs have transited back to the same steady state as in the increasing Re experiment, while the WBCs in STD_D at Re = 16 are still shedding eddies periodically to the western basin. The WBCs do not stop shedding eddies until Re decreases to below 14 in the equal-transport case. Thus, both in the STD and *λ* = 0.5 cases, multiple steady states and hysteresis for some ranges of Re are identified.

It is noticed that, for the STD case, the WBCs transit from steady penetration first to the choking state and then to the periodic eddy-shedding state as Re increases. For comparison, in the *λ* = 0.5 experiments, the WBCs transit from steady penetration to periodic eddy-shedding states directly as Re increases, without going through the choking state.

Thus, when the transports of the two WBCs are unequal, the range of multiple equilibria and hysteresis loop could be quite different. To quantitatively describe this phenomenon, the hysteresis loops of the southern WBC *X*_{P} for several values of *λ* as a function of Re are shown in Fig. 5. We focus on the flow patterns of the stronger (southern) one as the Reynolds number varies. For equal-transport WBCs (Fig. 5a; STD), *λ* = 1, and both of the two WBCs penetrate into the western basin when the transport is small; the penetration decreases as the transport increases, until Re > 20, when both WBCs are choked at the gap and retroflect eastward. This kind of choking state is maintained as the open boundary transport is increased further, until at Re ≥ 38, when both WBCs began to shed eddies periodically to the western basin. Then, if the transport of the WBCs is reduced, the equilibrium solutions of the QG model stay in the periodic eddy-shedding regime (STD_D Re = 20 shown as an example in Fig. 4a) until Re < 14, where both WBCs penetrate into the western basin steadily. Notice that *X*_{P} becomes multivalued for 14 < Re < 38, which means that there exists two steady states for the same Re, which can be achieved by integrating from different initial conditions.

The westward extent *X*_{P} of the streamline *ψ* = *Q*/2 of the southern WBC as a function of Re for different values of *λ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. Solid lines are experiments with increasing Re, and dashed lines are experiments with decreasing Re. Shadings indicate the parameter domains of multiple equilibria.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The westward extent *X*_{P} of the streamline *ψ* = *Q*/2 of the southern WBC as a function of Re for different values of *λ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. Solid lines are experiments with increasing Re, and dashed lines are experiments with decreasing Re. Shadings indicate the parameter domains of multiple equilibria.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The westward extent *X*_{P} of the streamline *ψ* = *Q*/2 of the southern WBC as a function of Re for different values of *λ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. Solid lines are experiments with increasing Re, and dashed lines are experiments with decreasing Re. Shadings indicate the parameter domains of multiple equilibria.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

For *λ* = 0.75, 0.5, and 0.25 (Figs. 5b–d), the two WBCs also penetrate steadily into the western basin when the transport is small. As Re increases, they become eddy shedding at much smaller Re than in the equal-transport case. During the Re decreasing course, the two WBCs transit from eddy shedding back to the steady penetrating state at around Re = 16, which is nearly the same with the equal-transport case (*λ* = 1).

Here is the summary of the results of all three unequal-transport cases (*λ* = 0.75, 0.5, and 0.25) for *γ* = 9.3 in Fig. 5. First, for small Re, the penetration of the stronger WBC is a single-value function of Re. The smaller the *λ*, the smaller the value of *X*_{P}, which means deeper penetration in the more unequal-transport case. Second, the unequal WBCs transit from the steady penetrating state directly to the periodic eddy-shedding state without going through the choking state, which is different from the equal-transport case. Third, eddy-shedding (the Hopf bifurcation) in the increasing Re experiments occurs at much smaller Re than in the equal-transport case. Meanwhile, the critical Re for the transition from periodic eddy-shedding state to the steady penetrating state (the reverse Hopf bifurcation) is nearly unchanged. Thus, the range of multiple equilibria and hysteresis becomes smaller for the unequal-transport WBCs than for the equal-transport WBCs. For instance, for equal-transport WBCs *λ* = 1, the hysteresis range is 24 (from Re = 14 to Re = 38), whereas the hysteresis range of *λ* = 0.75 is only 6 (between Re = 16 and Re = 22), which is dramatically shorter than in the equal-transport case.

### c. Bifurcation diagrams of two unequal-transport WBCs

The above illustration is made for a nondimensional gap width of *γ* = 9.3. Without losing generality, a more comprehensive bifurcation map of the unequal-transport collision problem is given in Fig. 6. The curves of Fig. 6 are obtained in the following way. For each fixed *λ* (*λ* = 1, 0.75, 0.5, and 0.25), we run both the Re increasing and decreasing experiments for different width (i.e., *γ*) of the gap. Three kinds of curves are obtained. Curve *R*_{PE} represents the transition from penetration to eddy shedding, curve *R*_{EP} represents the transition from eddy shedding to penetration, and curve *R*_{PC} represents the transition from penetration to choke.

Bifurcation curves showing critical Re*,* at which the penetrating (i.e., *R*_{PC}), choking (i.e., *R*_{PE}), or eddy-shedding (i.e., *R*_{EP}) branches terminate as a function of different *γ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. The penetrating branches persist up to Re < *R*_{PC} or Re < *R*_{PE} (if there is no *R*_{PC}), and the choking branches persist up to Re < *R*_{PE}, while the eddy-shedding branches persist down to Re > *R*_{EP}. Domain E represents the eddy-shedding states, domain P represents the penetrating states, and domain C represents the choking states.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Bifurcation curves showing critical Re*,* at which the penetrating (i.e., *R*_{PC}), choking (i.e., *R*_{PE}), or eddy-shedding (i.e., *R*_{EP}) branches terminate as a function of different *γ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. The penetrating branches persist up to Re < *R*_{PC} or Re < *R*_{PE} (if there is no *R*_{PC}), and the choking branches persist up to Re < *R*_{PE}, while the eddy-shedding branches persist down to Re > *R*_{EP}. Domain E represents the eddy-shedding states, domain P represents the penetrating states, and domain C represents the choking states.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Bifurcation curves showing critical Re*,* at which the penetrating (i.e., *R*_{PC}), choking (i.e., *R*_{PE}), or eddy-shedding (i.e., *R*_{EP}) branches terminate as a function of different *γ* [(a) *λ* = 1, (b) *λ* = 0.75, (c) *λ* = 0.5, and (d) *λ* = 0.25]. The penetrating branches persist up to Re < *R*_{PC} or Re < *R*_{PE} (if there is no *R*_{PC}), and the choking branches persist up to Re < *R*_{PE}, while the eddy-shedding branches persist down to Re > *R*_{EP}. Domain E represents the eddy-shedding states, domain P represents the penetrating states, and domain C represents the choking states.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

There are several domains divided by the three curves. The area above the curve *R*_{PE} is the eddy-shedding domain, the area under the curve *R*_{EP} is the penetration domain, and the area between the two curves *R*_{PE} and *R*_{PC} is the choke domain. The area between the two curves *R*_{PE} and *R*_{EP} is the multiple equilibrium domain that depends on the initial conditions.

Some interesting features are found in Fig. 6. First, the position of curve *R*_{PE} moves to the lower left as *γ* decreases, so the eddy-shedding domain is increasing as *γ* decreases. This also means that the two WBCs are prone to eddy shedding for large Re when the asymmetry increases. Second, the position of curve *R*_{PC} also moves to the left as *γ* decreases (asymmetry increase). In other words, the area between curve *R*_{PE} and *R*_{PC} becomes smaller as the asymmetry increases, which means that the two WBCs are difficult to choke at intermediate Re when the asymmetry is strong. Meanwhile, the shift to the left of curve *R*_{PC} also means that the two WBCs are easier to penetrate into the western basin for small Re when the asymmetry is strong. Third, the position of curve *R*_{EP} is nearly unchanged, which means that the transitions from eddy shedding back to steady penetration during the decreasing course occur at nearly the same Re for different asymmetry. Fourth, the domain between the curve *R*_{PE} and *R*_{EP} becomes smaller as *γ* decreases, suggesting that the range of multiple equilibria and hysteresis decreases as the asymmetry increases.

Figure 6 can be comprehended in another way. For narrow gaps (*γ* < 5.7), the two WBCs could not penetrate into the western basin. The two flows choke at the gap even in the linear case Re = 0. For wide enough gaps (*γ* > 9.6), only two kinds of equilibrium states exist: the two WBCs penetrate into the western basin for small transport and become eddy shedding for large transport. No choking states are found for *γ* > 9.6, hence no *R*_{PC} curve. The penetrating branches persist up to Re < *R*_{PE}, while the eddy-shedding branches persist down to Re > *R*_{EP}. Within the domain between the two curves of *R*_{PE} and *R*_{EP}, there exist two equilibrium flow regimes (penetration and eddy shedding). Because the curve *R*_{PE} is larger than that of *R*_{EP}, hysteresis exists. For intermediate gaps (5.7 ≤ *γ* ≤ 9.6), there exist three steady states: penetration, choke, and eddy shedding. The penetration persists up to Re < *R*_{PC} or Re < *R*_{PE} (if no choking branch exists), while the choke persists up to Re *< R*_{PE} (if choking branch exists) and the eddy shedding maintains down to Re > *R*_{EP}.

### d. Discussion

To understand the dynamics of the asymmetric collision, let us first consider the problem of two WBCs meeting along a solid western boundary. The problem is different from the classical double-gyre solutions with bifurcations due to the specification of the boundary conditions. The comparison of the symmetric collision with the double-gyre solutions has been discussed by Wang and Yuan (2012). The asymmetric double-gyre solutions have not been studied before.

We used the northernmost position of the zero streamline near the western boundary *Y*_{P} as a measurement of the asymmetric collision. A series of numerical experiments were carried out, under exactly the same model configuration as in section 3b, but with the gap closed, to examine how *Y*_{P} changes with *λ* and Re (Fig. 7). First, we fixed Re and changed *λ*. For small Re, *Y*_{P} is at the central latitude (*y* = 0) if the two WBCs have equal transport (see Wang and Yuan 2012). For two unequal WBCs of small transports (Re = 2 for the southern branch), the position of *Y*_{P} moves to the north as *λ* decreases (Figs. 7a,b). The movement of *Y*_{P} is determined by a complicated nonlinear momentum balance outside the Munk frictional layer, where the analytic solution is difficult to find. In general, the more asymmetric the WBC transports are, the more deviated from the central latitude the zero streamline is.

Positions of *Y*_{P} for (a) *λ* = 0.25 and Re = 2; (b) Re = 2 and *λ* = 0.25, 0.5, and 0.75; (c) *λ* = 0.25 and Re = 15; and (d) *λ* = 0.25 and Re = 30. Contours indicate the streamfunction, with the contour interval of 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Positions of *Y*_{P} for (a) *λ* = 0.25 and Re = 2; (b) Re = 2 and *λ* = 0.25, 0.5, and 0.75; (c) *λ* = 0.25 and Re = 15; and (d) *λ* = 0.25 and Re = 30. Contours indicate the streamfunction, with the contour interval of 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Positions of *Y*_{P} for (a) *λ* = 0.25 and Re = 2; (b) Re = 2 and *λ* = 0.25, 0.5, and 0.75; (c) *λ* = 0.25 and Re = 15; and (d) *λ* = 0.25 and Re = 30. Contours indicate the streamfunction, with the contour interval of 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Second, we fixed *λ* and changed Re. For a fixed asymmetry (*λ* = 0.25) of the two WBCs, numerical experiments suggest that the *Y*_{P} tends to shift northward as Re increases (Figs. 7c,d). Figure 8 summarizes the changes of *Y*_{P} with respect to *λ* and Re. Evidently, *Y*_{P} shifts more northward with decreasing *λ* and increasing Re. In terms of the magnitude, *Y*_{P} seems more sensitive to the changes of *λ* than to the changes of Re.

The positions of *Y*_{P} for three different *λ* (*λ* = 0.25, 0.5, and 0.75) as Re increases.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The positions of *Y*_{P} for three different *λ* (*λ* = 0.25, 0.5, and 0.75) as Re increases.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The positions of *Y*_{P} for three different *λ* (*λ* = 0.25, 0.5, and 0.75) as Re increases.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

We then used above solutions as initial conditions to search for the steady-state solutions of the two WBCs colliding at a gap. Without the support of the wall, the two WBCs enter the gap and penetrate into the western basin due to the beta effect. The continuous evolution of the streamfunction for *γ* = 9.3 and Re = 2 is shown in Fig. 9. After a long enough integration to reach the steady state, the numerical experiment shows that both the northern and southern WBCs penetrate into the western basin, with the extent of the southern (stronger) penetration much deeper than the northern (weaker) one. The latitude of the zero streamline *Y*_{P} is shown to change in a similar manner with *λ* and Re as in the solid wall experiments (Fig. 10).

Continuous evolution of the two colliding WBCs after a gap (of width *γ* = 9.3) is opened in the western boundary. The transport of the southern WBC corresponds to Re = 2. The ratio of the northern and southern WBC transport is *λ* = 0.25. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Continuous evolution of the two colliding WBCs after a gap (of width *γ* = 9.3) is opened in the western boundary. The transport of the southern WBC corresponds to Re = 2. The ratio of the northern and southern WBC transport is *λ* = 0.25. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Continuous evolution of the two colliding WBCs after a gap (of width *γ* = 9.3) is opened in the western boundary. The transport of the southern WBC corresponds to Re = 2. The ratio of the northern and southern WBC transport is *λ* = 0.25. The contour interval is 0.1*Q*. Negative streamfunction is plotted in dashed curves.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The change of the northernmost position *Y*_{P} of streamline *ψ* = 0 near the gap with respect to Re and *λ* for *γ* = 9.3.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The change of the northernmost position *Y*_{P} of streamline *ψ* = 0 near the gap with respect to Re and *λ* for *γ* = 9.3.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The change of the northernmost position *Y*_{P} of streamline *ψ* = 0 near the gap with respect to Re and *λ* for *γ* = 9.3.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The *Y*_{P} variations in Figs. 8 and 10 are similar, suggesting that the dynamics are nearly the same. During the evolution of the penetration in Fig. 9, the *Y*_{P} in the far field in the eastern basin is nearly unchanged, which determines the partition of the penetration in the vicinity of the gap. The relatively stronger (southern) WBC pushes the weaker (northern) WBC northward, hence the northward shift of the zero streamline. As the southern WBC occupies a wider latitudinal range of penetration near the gap, its penetration is deeper than that of the northern WBC (Wang and Yuan 2012). The more asymmetric the two WBCs are, the more northward the *Y*_{P} shifts and the wider part of the gap the southern (stronger) WBC occupies. Therefore, the larger transport it needs to get choked. This is why the curves *R*_{PC} move to the left as *λ* decreases in Fig. 6.

Another phenomenon of the asymmetric collision is that periodic eddy shedding occurs at much smaller Re than in the symmetric collision (Fig. 6). This is because the stronger WBC occupies a wider range of the gap and is more likely to shed eddies than the weaker WBC is. The eddy shedding of the stronger WBC gives up the gap to the weaker WBC, until another round of the full penetration of the stronger WBC is developed, resulting in the latter to shed eddies as well, hence the periodic solutions at smaller Re. The smaller the *λ* is (more asymmetric), the smaller the Re is needed for the eddy shedding to occur. This is why the curve *R*_{PE} moves to the lower left with smaller *λ* in Fig. 6.

## 4. Application to ocean circulation study in the western Pacific

An application of this theory to the ocean circulation study is the nonlinear collision of MC with SEC at the Pacific entrance of the Indonesian Throughflow, with a branch of MC penetrating more deeply into the Sulawesi Sea than SEC (Fig. 1a). This can be explained by the fact that the transport of MC is much larger than that of SEC. The situation is in analogy with the case of unequal-transport WBCs colliding at a gap, in which the stronger northern WBC makes a deeper penetration into the western basin than the weaker southern one does. Sometimes, both the MC and the SEC retroflect outside the entrance of the Indonesian Seas (Fig. 1b), which can be explained by the existence of the choking state of the colliding WBCs.

In the winter of 2012, a research cruise was conducted by the National Natural Science Foundation of China to study the ocean circulation in the western Pacific Ocean. During the cruise, 18 satellite-tracked surface drifters were deployed, 11 of which were released along 8°N section, in the Sulawesi Sea, in the Maluku Strait, and along the 130°E section (Fig. 11). The surface drifters manufactured by the Chinese Xiaolong Instruments Co., Ltd., use the global positioning system to fix locations and the Globalstar commercial satellites to return the geolocations back to the ground base. The drifter consists of a floating ball and a tube-shaped drogue deployed at 15 m below the sea surface. The configuration of the drifters follows the World Ocean Circulation Experiment (WOCE) standards, with the sizes in proportion to the diameter square of the WOCE drifters. The ratio between the surface area of the drogue and of the semisphere of the surface floating ball exposed in the air is 82. The design guarantees that the drifter follows the ocean surface currents instead of the winds.

Trajectories of satellite-tracked surface drifters released in the western Pacific Ocean during the winter of 2012. Diamonds and circles stand for the deployment and the final positions of each drifter, with the deployment and final observation dates marked. The trajectories are obtained at the 8-h intervals.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Trajectories of satellite-tracked surface drifters released in the western Pacific Ocean during the winter of 2012. Diamonds and circles stand for the deployment and the final positions of each drifter, with the deployment and final observation dates marked. The trajectories are obtained at the 8-h intervals.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

Trajectories of satellite-tracked surface drifters released in the western Pacific Ocean during the winter of 2012. Diamonds and circles stand for the deployment and the final positions of each drifter, with the deployment and final observation dates marked. The trajectories are obtained at the 8-h intervals.

Citation: Journal of Physical Oceanography 44, 7; 10.1175/JPO-D-13-0234.1

The trajectories of the satellite-tracked surface drifter during the winter of 2012 indicate a sizable eddy in the Sulawesi Sea likely shed from the intrusion of MC (Fig. 11). The elliptic eddy has its long axis of about 400 km oriented nearly in the zonal direction. The short axis in the meridional direction is about 300 km. The eddy appears to move to the west, where it interacts and merges with other smaller size eddies in the western Sulawesi Sea. The altimeter data, with an error bar of ~2–4 cm, cannot reveal these eddies because the Coriolis force is weak near the equator. However, the eddy is believed to be in geostrophic balance because the Rossby number of the eddy is *U*/(*fL*) = 0.1, assuming *U* = 0.5 m s^{−1}, *f* = 10^{−5} s^{−1}, and *L* = 400 km. This significant eddy in the Sulawesi Sea is observed for the first time in history. The dynamics of the eddy formation can be explained by the eddy-shedding state of the colliding WBCs.

It is worth mentioning that the QG model has limitations in the equatorial region. However, as the first approach and an extrapolation of the midlatitude solutions, the study is of some value, especially because the circulation we are studying is in geostrophic balance, even in the equatorial area. The QG model is essentially equivalent to the shallow-water equation model except that the fast gravity waves are filtered out in the former. For low-frequency, steady-state circulation, these fast processes are not important.

## 5. Conclusions

We have considered an idealized problem of two WBCs with unequal transport colliding at a gap of the western boundary. When the gap is narrow, *a* < 5.7*L*_{M}, the two WBCs cannot penetrate into the western basin. When the gap is wide, *a* > 5.7*L*_{M}, multiple states and hysteresis exist. The stronger WBC tends to penetrate more deeply into the western basin than the weaker one. For 5.7*L*_{M} ≤ *a* ≤ 9.6*L*_{M}, three kinds of equilibrium states are identified in the quasigeostrophic ocean circulation model: the penetrating state, the choking state, and the eddy-shedding state. The current system transits between these equilibrium states in hysteresis associated with the variations of the WBC transports. When the gap is very wide, *a* > 9.6*L*_{M}, the choking state disappears due to instability of the current system at the gap.

Based on numerical experiments, we found that the multiple equilibria and hysteresis are dependent on the asymmetry of the WBCs. When the asymmetry is strong, the two WBCs tend to penetrate into the western basin for small transport and are prone to eddy shedding for large transport. The range of Re for the unequal-transport WBCs to have hysteresis is found to be smaller than that of the equal-transport case.

The dynamics of the changes from the symmetric case are due to the deviation of the zero streamline from the central latitude, which results in wider latitudinal range for the stronger WBC to penetrate into the western basin. The wider latitudinal range for the stronger WBC also results in the instability of the penetration at smaller Re.

Our model is limited by the closed boundary condition in the western basin and by the quasigeostrophic assumption of the ocean circulation in the near equatorial oceans. A study of a similar problem using a shallow-water model of the ocean circulation is underway, the results of which will be reported in a separate paper in the near future.

## Acknowledgments

This work is supported by the NSFC Grant 41206019, National Basic Research Program of China (2012CB956000), and partially supported by LASG/IAP/CAS and LTO/SCSIO/CAS (Grant LT1102). D. Yuan is supported by the NSFC Project 41176019. Data and samples were collected onboard Research Vessel (R/V) *Kexue-1* during the NORC2012-09 cruise funded by the Shiptime Sharing Project of NSFC. We thank the captain and the crew for their hard work. Thanks also go to the High Performance Computing Center of IOCAS for the computing resources and technological support. The authors would like to thank two anonymous reviewers for their valuable comments.

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