1. Introduction
In a marginal sea adjacent to open ocean, there are many interesting physical issues associated with the circulation, jet, eddy, and interactions between them. Besides the local dynamical and thermal processes, the external forcing due to the intrusion of the circulation system from the adjacent open ocean also has a remarkable impact on the circulation of the marginal sea. One of the most notable examples is the interaction between the South China Sea (SCS) and the Kuroshio, which is the western boundary current (WBC) of the subtropical gyre in the northwestern Pacific. As the largest, semienclosed, marginal sea in the western tropical Pacific Ocean, the SCS has been studied widely for several decades (Wyrtki 1961; Shaw 1989, 1991; Farris and Wimbush 1996; Shaw et al. 1999; Qu 2000; Qu et al. 2000; Metzger and Hurlburt 2001a,b; Wang et al. 2003; Caruso et al. 2006; Yuan and Li 2008; Yuan et al. 2007; Nan et al. 2011a,b,c). The SCS, with a maximum depth reaching 5000 m, has several passages to other waters (Fig. 1). Among the passages, the Luzon Strait, a meridional gap with a depth exceeding 2000 m and a width exceeding 300 km, is the deepest and widest one. To the east of the Luzon Strait, the energetic WBC of the Pacific, the Kuroshio, usually flows northward via the strait with sometimes part or all of it entering the SCS through the strait. Hydrographic evidence has shown that the Kuroshio water mainly intrudes into the northern part of SCS from the Luzon Strait (Shaw 1989, 1991). The intrusion is not only confined to the upper layer that is directly driven by surface wind (Wang and Chern 1987; Shaw 1991; Farris and Wimbush 1996) but also at different depths of the Luzon Strait (Qu et al. 2000; Zhang et al. 2015). Zhang et al. (2015) found strong baroclinic structure of zonal transport in the Luzon Strait from full-depth velocity measurements, which exhibit the so-called sandwich structure of the zonal flow in the Luzon Strait, that is, westward transport in upper (<500 m) and deep (>2000 m) layers but eastward transport in the intermediate layer (500–2000 m).
Maps of northern Pacific and the SCS with bathymetry (m).
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
Multiple states of the WBC intrusion have been concluded according to the shape of the Kuroshio pathway at the Luzon Strait based on observations and numerical simulations (Farris and Wimbush 1996; Caruso et al. 2006; Nan et al. 2011b). The types of the Kuroshio pathways include the leaping/normal path, in which the Kuroshio leaps over the Luzon Strait with only a slight westward bending or eastward displacement; the penetration/bifurcation/leaking path, in which part or the whole body of the Kuroshio flows westward into the northern SCS, mainly along the shelf break of the Southern China; and the anticyclonic Loop Current path, in which the Kuroshio flows westward into the SCS through the southern edge of the Luzon Strait and turns back to the Pacific via the northern edge (Farris and Wimbush 1996; Caruso et al. 2006; Nan et al. 2011b). Besides these states, the eddy-shedding process associated with anticyclonic and cyclonic eddies can sometimes be found in the transition between different paths (Wang and Chern 1987; Sheremet 2001; Caruso et al. 2006). Additionally, the Kuroshio more frequently intrudes into the SCS in winter, when the northeast monsoon dominates the SCS and the adjacent western Pacific (Shaw 1991; Qu et al. 2000).
A variety of mechanisms have been proposed to explain the Kuroshio intrusion and its seasonal variation, which include Ekman transport due to the seasonally reversing wind stress (Wang and Chern 1987; Shaw 1991; Farris and Wimbush 1996; Qu et al. 2000), the Luzon Strait transport driven by pressure gradient due to the effect of wind stress pileup (Shaw 1991; Metzger and Hurlburt 1996; Metzger 2003), the balance between inertia and the β effect (Sheremet 2001; Sheremet and Kuehl 2007); the representation of coastlines and islands in numerical models (Metzger and Hurlburt 1996; Metzger and Hurlburt 2001a), the tropical gyre of the northern Pacific (Metzger and Hurlburt 1996), and the mesoscale flow instabilities (Metzger and Hurlburt 2001b).
Just like the Kuroshio in the Pacific, the Gulf Stream in the Atlantic also intrudes into the marginal sea (the Gulf of Mexico) in the form of the Loop Current. Seasonal and annual variations of the Loop Current of the Gulf of Mexico have been well documented by many studies (Behringer et al. 1977; Maul 1977; Hurlburt and Thompson 1980; Vukovich 1988; Vukovich 1995; Sheinbaum et al. 2002). By employing the dynamical systems theory, Lugo-Fernández (2007) suggests that the Loop Current behaves as a nonlinear-driven and dampened oscillator. Observations exhibit that the Loop Current in the Gulf of Mexico sheds eddies in a period ranging from 0.5 to 18.5 months (Vukovich 1995, 2007; Sturges and Leben 2000). For the mechanism of eddy shedding, large discrepancies still exist. The numerical simulations by Hurlburt and Thompson (1980) show the time-independent inflow can generate annual eddy shedding. Their numerical results support that the Yucatan Current inflow is the key factor. On the contrary, observations (Sturges et al. 2010) show the pulses of increased transport in the Florida Straits trigger the eddy shedding, although some studies (Candela et al. 2002; Oey et al. 2003; Lugo-Fernández and Badan 2007) have shown that the associated vorticity and transport fluctuations affect eddy shedding. The interaction between the Loop Current and deep eddies induced by the baroclinic instability is also thought as the eddy shedding mechanism (Oey 2007; Donohue et al. 2016). It seems that the Loop Current variation due to eddy shedding is also a multifactor-determining problem, which involves uncertainty related to the eddy and instability.
From the numerical experiments by Metzger and Hurlburt (2001b), the Kuroshio intrusion does not appear to be a wind-forced, deterministic event, at least, on the yearly mean time scale. This nondeterministic nature also leads to a weak relationship between the Kuroshio penetration and the yearly mean transports at the Luzon Strait and the Kuroshio (Metzger and Hurlburt 2001b). High variability of sea surface height (SSH) exists west of the Luzon Strait (Metzger and Hurlburt 2001b), which suggests that the strong nondeterminism is mainly due to the strong eddy activity therein (Wang et al. 2000; Wu and Chiang 2007; Nan et al. 2011c; Yang et al. 2013). There are many evidences to document the role of mesoscale eddies in the northern SCS (Wang and Chern 1987; Wang 1987; Li et al. 1998; Wang et al. 2003; Jia and Liu 2004; Xiu et al. 2010; Nan et al. 2011a,c; Zu et al. 2013).
In the SCS, the eddy formed near the Luzon Strait generally has a diameter of approximately 150–250 km (Wang and Chern 1987; Wang 1987) and a vertical depth reaching 1000 m with part of the waters originating from the Kuroshio (Li et al. 1998). Mesoscale eddy shedding caused by frontal instability during the Kuroshio intrusion is considered to be a crucial mechanism for the eddy formation to the west of the Luzon Strait (Metzger and Hurlburt 2001b; Jia and Liu 2004; Yuan et al. 2006; Wu and Chiang 2007). In addition, some eddies could also be generated by strong Ekman pumping due to enhanced local wind stress curl in the Luzon Strait (Chi et al. 1998; Yuan et al. 2007) or instability in the baroclinic Rossby wave from the western Pacific (Chelton and Schlax 1996; LaCasce and Pedlosky 2004; Sheu et al. 2010). Meanwhile, through eddy–flow interaction, the mesoscale eddy in the vicinity of the Luzon Strait can induce the variation of the Kuroshio path as well as the intrusion state of it (Yuan and Li 2008; Sheu et al. 2010; Yuan and Wang 2011). On the other hand, eddy energy analyses (Zhuang et al. 2010; Yang et al. 2013) indicate that the strong eddy energy is fluxed from the Pacific Ocean to the SCS through the Luzon Strait. Intraseasonal variability in the vicinity of the Luzon Strait is thought to be induced by the fluctuation of mesoscale eddy energy released by the barotropical and baroclinic processes induced by the Kuroshio intrusion (Zhuang et al. 2010; Yang et al. 2013; Zhang et al. 2015). Therefore, to a great extent, the issue of the Kuroshio intrusion into the SCS may be reduced to the problems related to jet instability and jet–eddy interaction.
Theoretical and numerical studies have extensively explored the interaction mechanisms between stable/unstable jets and eddies in quasigeostrophic (QG) or shallow-water layered models (Stern and Flierl 1987; Bell 1990; Bell and Pratt 1992; Vandermeirsch 2003; Vandermeirsch et al. 2003). In terms of potential vorticity (PV), the relative strength between jet and eddy is considered to be a key factor in determining whether the eddy crosses the jet or not (Vandermeirsch 2003; Vandermeirsch et al. 2003). Sheu et al. (2010) further concluded that a weak/strong PV gradient across the Kuroshio tends to form a looping/leaping state of intrusion and westward passage/blocking of eddies in the Luzon Strait.
As mentioned above, the mesoscale eddy as well as the interaction between eddy and large-scale flow strongly regulates the variability of the Kuroshio pathway near the Luzon Strait. However, because of the lack of regular observation, the information obtained from the limited observations cannot provide the whole dynamical picture of the intrusion behavior of the Kuroshio, much less show the detailed mechanism due to the mesoscale eddy. The numerical simulation including full thermal and dynamical processes with realistic coastline, bathymetry, and atmospheric forcing is a good alternative to study the Kuroshio intrusion. But the uncertainties resulted from multiple mechanisms still exist, such as those caused by atmospheric forcing and the spatial resolution of the model (Metzger and Hurlburt 2001a,b). Although the nondeterministic nature of the Kuroshio intrusion is attributed to the mesoscale flow instability (Metzger and Hurlburt 2001a), the question of how the mesoscale eddy influences the WBC intrusion through the intrinsic ocean dynamics has not been well answered because much of the external variability is simultaneously included in the realistic model. From a more fundamental perspective, some mechanism research based on the ideal models that only include time-independent jets and eddies provides many meaningful and heuristic results for the understanding of the Kuroshio intrusion into the SCS (Sheremet 2001; Vandermeirsch et al. 2003; Vandermeirsch 2003). However, as is well known, even in the regime of the wind-driven circulation, strong variability associated with the WBC can be induced by the nonlinear processes purely through the intrinsic ocean dynamics (Berloff and McWilliams 1999b; Chang et al. 2001; Simonnet and Dijkstra 2002; Simonnet 2005; Berloff et al. 2007). Therefore, understanding the dynamical role of the mesoscale eddy in the time variation process, even in a relatively simple scenario only including intrinsic ocean variability, can greatly improve our understanding of the mechanism of Kuroshio intrusion into the SCS. So, to do this, the present study explores the influence of the eddy on a time-varying WBC near an ideal strait from the points of view of PV forcing and transport. To reduce the problem, a QG ocean driven by steady wind forcing is used. To generate a dynamical environment for the Kuroshio intrusion, the problem is studied in ideal basins with almost the simplest coastline. Our aim is to derive the relationship between eddy forcing and the WBC intrusion in the case that only includes the internal dynamical processes of the ocean.
This paper is organized as follows: The model and the eddy–flow decomposition methods used in the study are described in detail in section 2. In section 3, the role of eddy forcing is preliminarily evaluated from the ideal WBC intrusion process through coupling the eddy-resolving (ER) solution and noneddy-resolving (non-ER) solutions. The parametric sensitivity of intrusion behavior is also discussed in this section. In section 4, the detailed characteristics of eddy PV forcing and the PV transport due to the grid-scale and subgrid-scale eddies are analyzed during typical WBC intrusion. Further comparisons between different components of eddy PV forcing and flux derived by the eddy–flow decomposition are presented. Finally, a summary of the key findings and associated discussions are provided in section 5.
2. Model and methods
a. Model equations
b. Flow decomposition methods
The eddy–flow decomposition is critical to the discussion of the role of the eddy. In this study, the eddy effects at both grid scale and subgrid scale are included in the discussion. To obtain the grid-scale eddy, the traditional decomposition is used: the mean flow and eddy are represented by the time mean and temporal variation, that is, 
But for the subgrid-scale eddy, a dynamically consistent decomposition presented by Berloff (2005) is used. Other than the statistical approach, the dynamical method utilizes a preexistent high-resolution ER solution to provide detailed information of small-scale eddies. According to the work of Berloff (2005), the dynamical decomposition extracts the eddy effect by integrating a non-ER model forced by the PV forcing of eddy, which is interactively extracted from preexistent ER solution. Based on this approach, the non-ER solution is forced by unresolved eddy forcing (UEF) to accurately approximate the ER solution (Berloff 2005). At the same time, the dynamically consistent decomposition provides the whole history of the unresolved eddy with respect to the time-varying, large-scale, non-ER flow. This is completely different from the time-mean decomposition method. Thus, we can easily diagnose the subgrid-scale eddy in a time-varying process.
c. Experimental configuration
The main goal of this study is to address the role of the eddy in the interaction between the marginal sea and open ocean. To reduce the problem, the variability from external forcing should be excluded as much as possible. Two connected, ideal, flat-bottom basins are used to remove the impacts from the complex coastal boundary and bottom topography. The modeling domain is shown in Fig. 2b, which consists of two flat-bottom rectangle basins connected by a narrow gap. The scales of the small and large basins are assumed to be close to those of the SCS and the northwestern Pacific. The narrow gap, having a meridional width of 300 km and zonal width of 50 km, is similar to the scale of the Luzon Strait. To eliminate the variability due to the variation of surface wind, only the time-independent wind stress curl is considered. To further simplify the problem, the local, wind-driven circulation in the small-basin sea is excluded through setting the wind stress curl therein to be zero, although the variability of the SCS circulation is significantly modulated by the summer and winter monsoon (Shaw 1991; Qu et al. 2000). As a result of it, the circulation in the small basin is purely driven by the intruding flow from the large basin. The small basin only serves as a PV and energy sink for the large basin. This does not mean the circulation in the SCS is only driven by the intrusion of the Kuroshio. This extremely simple dynamical scenario gives us a new insight into the internal oceanic dynamics in the WBC intrusion process. Under this ideal experimental setting, the circulation of the small-basin sea is isolated from interaction with the intruding large-basin WBC so as to focus on the impact of eddy on the WBC intrusion.
(a) Wind stress curl at x = 3000 km and (b) time-mean, upper-layer stream field driven by it. The contour interval (CI) of Fig. 1b is 4000 m2 s−1 with dashed lines for negative values. The modeling domain consists of a large basin (3800 km × 3800 km in an x × y plane and the same convention below) and a small basin (900 km × 1800 km). The two basins are connected by a narrow gap with the scale 50 km × 300 km. The zonally uniform wind stress curl [Eq. (7)] is only imposed on the large basin.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
Model parameters of reference experiment.
d. Numerical implementation
In this work, Eq. (9), as well as its decoupled counterpart Eq. (11), is discretized on a uniform mesh of grid points and solved by a classic, second-order, finite-difference scheme. The Jacobian operator is approximated by the Arakawa scheme (Arakawa 1966), which conserves kinetic energy and enstrophy. For the time discretization, the second-order leapfrog technique is used. Every 50 time steps, a forward Euler step is adopted to eliminate the time-splitting problem due to the leapfrog scheme. To avoid linear instability, the dissipative term is placed at the time level that is lagging other terms by one step (Holland 1978).
After obtaining the solution of modal Eq. (11), the streamfunction at each layer can be easily recovered by the relation VΦ = Ψ, where V is the eigenvector of the coupled matrix 
An ER solution containing abundant eddy activities, as well as eddy–flow interaction, is used as the reference solution in the following discussion. To allow appropriate amounts of mesoscale motions, the ER solution uses a low eddy viscosity coefficient AH of 80 m2 s−1 that is similar to the magnitude frequently used in the realistic regional ocean model. For the two-grid (13 km) wave, the damping time of this magnitude of AH is 0.29 days (Holland 1978). This is enough for the model to inhibit the noise growth. In fact, lower and higher values of AH (40 and 120 m2 s−1), corresponding to the damping time ranging from 0.19 to 0.57 days, are also tested; the results are qualitatively similar. If we use AH lower than 30 m2 s−1, the damping time will rapidly increase due to the hyperbolic relationship between the damping time and AH (Holland 1978). That will result in too much smaller-scale motions and instability problems. The sensitivity to AH can be found in section 3c. So, without loss of generality, the eddy viscosity of the reference solution is set to be 80 m2 s−1. Under the density stratification used by the ER solution shown in Table 1, the thickness of the Munk layer, δM = (AH/β)1/3, is about 16 km. And the first and second baroclinic deformation radii are about 34 and 19 km, respectively. For the ER experiment, the fine grid interval of 6.25 km can marginally resolve those three characteristic scales by at least two grid points. Under this set of parameters, the modeled ER circulation reaches a quasi-stable state at about 3000 model days. After that time, the total kinetic energy of the flow fluctuates around a mean value. In the parametric space discussed in this work, this spinup behavior is common for different sensitivity experiments and only differs in the length of the spinup time and amplitude of kinetic energy fluctuation. The main numerical results shown in subsequent parts are derived from the simulations after the total kinetic energy reaches a statistically steady state.
3. Eddy-induced WBC intrusion
a. Intrusion behaviors of WBC in ER and non-ER solutions
Besides the reference experiment, another two experiments are carried out under the same wind forcing but with a quadrupled grid interval (25 km) and much higher viscosity 
Snapshots of the upper-layer stream field ψ1 of the ER solution from model day 17 112 to 17 210 with dashed lines for negative streamfunction (m2 s−1).
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
As in Fig. 3, but for the non-ER solution.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
As in Fig. 3, but for the non-ER solution with eddy forcing.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
From the ER solution shown by Fig. 3, an energetic, narrow, and highly variable WBC of subtropical gyre prominently features the instantaneous stream fields. Near the western boundary, as well as the WBC extension between the two gyres, the curving pathway of the WBC with eddies around it suggests strong eddy–flow interaction therein. Another impressive feature is the variation of the flow at the ideal “strait.” The WBC dramatically distorts its pathway west of the gap and forms a Loop Current. The intruding WBC has strong temporal variability that is reflected by the variable shape and meridional displacement of the Loop Current. The intrusion process presented here may be divided into three stages according to the status of the Loop Current, that is, the growing Loop Current (stage I, days 17 119–17 132), the quasi-stable Loop Current (stage II, days 17 132–17 188), and the WBC retreat (stage III, day 17 188–17 210). In stage I, the state of the WBC rapidly changes from the leaping state to Loop Current in about two weeks. In the stage II, the WBC intrusion persists as a relatively stable Loop Current until eddy detachment. The second stage has a longer duration spanning about 50 days. Between stages II and III, the eddy-shedding process cuts off the intruding Loop Current (at about day 17 188). After the instability energy releases through the anticyclonic eddy detachment, the WBC rapidly retreats to the east of the gap in about another two weeks in stage III. In total, the WBC intrusion in the way of the unstable Loop Current has a time period of about 30–90 days and can be easily observed throughout the whole 1200 model months. The time period of the intrusion process has a similar scale to the intraseasonal variability in the Luzon Strait (Zhuang et al. 2010; Zhang et al. 2015), although the variability of the wind stress is not included here. Just as that reported by Mu et al. (2011), low-frequency modes can be carried into the small-basin sea through the above intrusion behavior.
An eddy-shedding process is also found for the non-ER solution (Fig. 4) even under the quadrupled spatial interval (25 km) and higher viscosity (600 m2 s−1). But, apparently, the lower grid-scale Reynolds number of the non-ER solution has much more straight WBCs both in the WBC extension and the gap due to the coarse grid capturing much less small-scale energy from the eddy feedback. The reduction of eddy activities weakens the eastward jet in the WBC extension (Berloff et al. 2007) as well as the variation of the intruding WBC in the small-basin sea.
The difference between non-ER and ER solutions is greatly reduced by the UEF, that is, fe in Eq. (2). Figure 5 presents the instantaneous, non-ER solutions with the correction of UEF. Although the circulation in Fig. 5 shows somewhat wider WBC in comparison to the ER solution (Fig. 3), it almost reproduces the ER dynamical behaviors in the western boundary and WBC extension. Driven by the time-varying UEF, part of the WBC intrudes into the small-basin sea as a north–south swinging Loop Current until it detaches an anticyclonic eddy. The whole intrusion process is dynamically consistent to that shown in ER solution.
To display the intrusion behavior more clearly, an intrusion index Ia is adopted to represent the strength of the WBC intrusion. The index is defined as 
Figure 6 presents the comparison of the daily indices of Ia between ER and non-ER solutions. As shown by Fig. 6a, the intrusion index of the UEF-forced, non-ER solution almost coincides with that of ER solution except for relatively smaller fluctuation amplitude throughout the whole 100 modeling years. The extremely high correlation coefficient of about 0.8 above the 99% confidence level is found between the two 100-yr time series. But if the UEF is excluded from the non-ER experiment (Fig. 6b), the intrusion index only shows fluctuation in large time scale, which is completely out of phase with the ER series. It is easily conjectured that the small-scale fluctuation of the WBC intrusion is caused by UEF.
The comparisons of intrusion area index. The intrusion index is defined as 
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
Additionally, the turbulence oscillator can induce the low-frequency variability of the large-scale circulation (Berloff et al. 2007). Through spectral analyses (not shown) for small-basin flows, the shape of the power spectrum of the UEF-forced, non-ER solution well approximates that of the ER solution. Furthermore, both of the high-pass and low-pass filtered Ia of the eddy-forced, non-ER solution highly correlate to those of the ER solution at seasonal, interannual, and interdecadal time scales (not shown), which suggests that the eddy–flow interaction is extremely important for the intrusion behavior of the WBC.
As is known, mesoscale eddies in the northern Pacific mainly concentrate in two bands: one in the Kuroshio Extension and the other in the area of the subtropical countercurrent (STCC). To display the remote and local impacts of the eddies on the flow at the gap, two additional sensitivity experiments are carried out by, respectively, including the UEF in boxes S1 (the WBC extension) and S2 (gap region), as shown by the first subgraph of Fig. 5. The WBC intrusion indices Ia of these two experiments are presented in Figs. 6c and 6d. Although the most active eddy–flow interaction exists in the WBC extension (region S1), there is only weak correlation (r = 0.024) between the indices of ER and non-ER solutions. That suggests the UEF in the WBC extension only impacts the flow locally, such as maintaining the eastward jets (Berloff 2005). But by only including the UEF in region S2, the non-ER series will highly correlate (r = 0.769) to the ER one, which is almost the same as the full UEF-forced experiment in Fig. 6a. That is, the local eddy forcing around the strait (region S2) is key for the model to produce the “correct” intrusion behavior of WBC. At the realistic ocean circumstance, the region of STCC locates just east of the Luzon Strait. That is one of the most eddy-concentrating regions in the global oceans (Qiu 1999). The westward-propagating eddies from STCC may move close to the Kuroshio and interact with it near the Luzon Strait. Although the model used here lacks the mechanisms of STCC and eddies therein, the results of the sensitivity experiments in Figs. 6c and 6d still suggest that the eddy to the east of the Luzon Strait is very important for the Kuroshio intrusion into the SCS. So, the model that has spatial resolution high enough to resolve the mesoscale eddy to the east of the Luzon Strait has more chances to correctly simulate the dynamical behavior of the Kuroshio at the strait.
b. Multiple intrusion states induced by eddy forcing
As is mentioned in section 1, multiple states of the intrusion have been determined based on observation and numerical simulation (Caruso et al. 2006; Farris and Wimbush 1996; Nan et al. 2011b). The different intrusion states and the transitions between them can also be found in the instantaneous fields shown in Figs. 3–5. Several typical states are concluded in Fig. 7, together demonstrating the upstream WBC transport and the stream at the entrance of the small basin. As is shown in Figs. 7c and 7d, in total, eight states can be found from left to right as a leaping state, southern Loop Current, northern eddy shedding, deep penetration, strong eddy shedding, southern eddy shedding, eastward displacement, and northern Loop Current. For the states having a stronger bending pathway, small-scale eddies are easily found around the jet. This is more clearly shown by the PV field (Fig. 7d). And the evolution of positive (cyclonic) PV anomalies to the west of the gap well matches the meridional displacement of the Loop Current and eddy shedding as well as the eastward retreat of the intruding WBC. The origin of this positive PV will be discussed in detail in section 4.
Multiple states of the WBC intrusion of ER solution: (a) the instantaneous (curved line) and time-mean (horizontal line) upper-layer transports at y = 1200 km and 950 ≤ x ≤ 1200 km; (b) the meridional time section of upper-layer streamfunction ψ1 to the west of the gap (x = 850 km and 1200 ≤ y ≤ 1500 km); (c) the snapshots of the upper-layer stream fields at the times marked by dotted lines in (a) and (b); and (d) the snapshots of upper-layer potential vorticity q1 in the rectangle region defined in the first snapshot of plot (c).
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
Figure 7b shows a cross section of the stream field on the west side of the gap, where the positive stream represents the WBC’s intrusion into the small-basin sea, and the transition from positive to negative generally means the eddy detachment or eastward retreat of WBC. For the ER reference solution, the WBC shows aperiodic intrusion behavior that is displayed by positive-valued blocks at different time scales (Fig. 7b). The eddy-shedding process shown in Fig. 3 just corresponds to the positive-valued block spanning from days 17 100 to 17 200. Besides this relatively long-period (about 90 day) intrusion process, some intrusion processes have much shorter periods, such as those during days 17 210–17 250 and days 17 370–17 400. In general, the period of the intrusion process is about 30–90 days, approximately an intraseasonal time scale (Zhuang et al. 2010; Zhang et al. 2015).
The upstream WBC transport is also shown in Fig. 7 because it has been considered as a key dynamical parameter in determining the intrusion state (Sheremet 2001). During each intrusion process, an obvious consistence between the variations of the WBC transport and intrusion strength is observed. High, instantaneous WBC transport approximately corresponds to the strong intrusion strength. But nondeterministic nature still exists in the relationship between the WBC transport and its intrusion behavior, particularly shown by the fact that sometimes there is no WBC intrusion taking place, but the WBC transport magnitude remains comparable to that during intrusion process. In other word, there seems not to be a simple linear relationship between upstream WBC transport and the WBC intrusion in the highly nonlinear regime discussed here.
As shown by Figs. 8b–d, the non-ER experiment with UEF also simulates the intrusion behavior as highly in concert with the ER one, including the multiple intrusion states and transitions between them. Because of the higher viscosity (600 m2s−1) and lower spatial resolution (25 km) used in the non-ER solution, the mean transport of WBC, as well as the intrusion strength, is relatively weaker than that of ER. As seen in Fig. 8a, although the time-mean WBC transport is about two-thirds (10 Sv; 1 Sv ≡ 106 m3 s−1) that of the ER solution (14 Sv) (Fig. 7a), the non-ER WBC fluctuates consistently in similar amplitude as the ER solution. So, the UEF seems to dominate the fluctuation of WBC transport that further regulates the time period of the WBC intrusion. This is confirmed by the non-ER solution without UEF shown in Fig. 9. It should be noted that the scale of the time axis of Fig. 9 is much longer than that of Fig. 7. Under the same wind forcing, viscosity, and spatial resolution, but excluding UEF term, the non-ER model simulates a greater strength of intrusion at a much longer time period (Fig. 9b), which approximately reaches 400–500 days. This kind of WBC intrusion corresponds to the small amplitude but long-period fluctuation of the WBC transport (Fig. 9a), although the time-mean transport is almost the same as the ER one (Fig. 7a). Similar features are also shown by some extra experiments on parameter sensitivity in section 3c. As the quasi-stable Loop Current shows in Fig. 9d, some numerical simulations, by using more realistic numerical model, have also modeled the stable bend of Kuroshio to the west of the Luzon Strait (Metzger and Hurlburt 1996, 2001b). Without the forcing from UEF, fewer eddy activities are modeled by the non-ER experiment (Fig. 9). This produces a less-variable WBC, fewer intrusion states (Figs. 9c,d), and quasi-stable intrusion at annual or even longer time scales.
As in Fig. 7, but for the non-ER solution with subgrid-scale eddy forcing.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
As in Fig. 7, but for the non-ER solution without subgrid-scale eddy forcing during model days 13 000–15 000.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
From the above analysis, the intrusion of the WBC into the adjacent sea may be concluded as the eddy-shedding process associated with transitions between the multiple states of the WBC pathway, which closely relates to the variation of the WBC transport due to eddy activity. Although the eddy-shedding processes exist in the above ER and non-ER solutions, additional sensitivity experiments are still needed to verify it in parametric space.
c. Parametric sensitivity
1) Strength of wind stress curl
Many previous studies have shown that the wind forcing, especially the seasonally reversed monsoon (Wang and Chern 1987; Shaw 1991; Farris and Wimbush 1996; Qu et al. 2000), is a dominant factor impacting the Kuroshio intrusion. Here, different strengths of wind stress curl are used to investigate its impact on the WBC intrusion. Figure 10a shows four ER solutions driven by different strengths of wind stress curl ranging from 0.5 to 1.5 times that used in the reference solution in Fig. 7. In all four ER experiments, the WBC intrudes into the small basin in the way of the Loop Current with eddy detachments, just as the reference solution doing. It is also obvious that the strength of the wind stress has great influences on the intrusion period. More short-period and strong intrusion events can be found for stronger wind stress curl imposed upon the large-basin ocean. Under the parameters used here, it seems that stronger wind forcing results in higher-frequency variability of the WBC and then the increase of the intrusion intensity and the decrease of intrusion period.
The sensitivities of the 1000-day series of the upstream WBC transport (blue line) (defined in Fig. 7) and the upper-layer streamfunction (ψ1) (red line) averaging over the gap to (a) the strength of wind stress curl τ0, (b) eddy viscosity AH, (c) non-ER eddy viscosity 
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
2) Eddy viscosity
To explore the impact of the eddy viscosity, several experiments are carried out by varying AH from 40 to 120 m2 s−1 (Fig. 10b). Loop Current intrusion and the eddy-shedding process are also general phenomena near the gap. To some extent, lowering eddy viscosity is equivalent to enhancing the wind forcing; that is, lower AH resulting in a shorter intrusion period. Under the same wind forcing, but only changing eddy viscosity, the total energy input for the ocean remains unchanged. Because of the energy conservation constraint, lower eddy viscosity results in higher eddy energy but a lower large-scale current energy. Therefore, higher-frequency variation of WBC transport due to eddies results in the WBC intruding in the small sea basin at shorter time scales with even the time-mean WBC transport remaining at a relatively lower level.
3) Strength of unresolved eddy forcing
The dynamical flow eddy decomposition method used here provides us a chance to evaluate the sensitivity to the UEF strength by simply changing the eddy viscosity 
4) Bottom friction
Although bottom friction is only imposed on the lowest layer, it is the primary mechanism for mesoscale energy dissipation (Holland 1978). Therefore, if bottom friction is present, the modeled circulation will be rather insensitive to the lateral friction (Holland 1978), which is the parametric eddy effect imposed on the whole body of water in this work. If the model has already included lateral friction, further presenting bottom friction may cast a cover over the truth of the eddy effect except when the very weak bottom friction coefficient is used. This is why the reference experiments in this work excludes the bottom friction, just as what Berloff (2005) has done. In spite of this, there is no harm in discussing the sensitivity of the WBC intrusion to the bottom friction. Figure 10d shows the results under four bottom friction coefficients ϵ increasing at a tenfold rate. An obvious smoothing effect can be found in the four experiments. Although small-scale fluctuation is still found in the first two lower ϵ solutions, the eddy activities are significantly inhibited compared to other solutions mentioned before. Because of the lack of eddy activity, the WBC only has large and long-period fluctuations, which accordingly result in low-frequency WBC intrusion. The intrusion WBC still detaches eddies in the small basin (figures omitted) but in a much longer period compared to the circulation solely dissipated by lateral friction.
5) Boundary condition
The boundary condition is critical for the PV conservation of global circulation (Pedlosky 1996). To evaluate the impact of the boundary condition, five ER experiments are carried out under α = 0 (free-slip boundary), α = 1.0 × 10−4, 1.5 × 10−4, 2.0 × 10−4 (reference experiment), and α → ∞ (no-slip boundary). As the lowermost limit of the parameter α, the free-slip boundary produces a quasi-stable intrusion state where the whole WBC flows into the small basin along the boundary (figures omitted). Under this situation, the small-basin circulation has become as a part of subtropical gyre in the large basin. At the other end of the parametric axis, the no-slip boundary condition almost completely prevents the WBC from intruding into the small-basin sea because of the extremely high viscous stress of the boundary (figures omitted). These two extreme solutions have almost no Loop Current intrusion or eddy shedding. Between these two limits, the rest of the three solutions show similar WBC intrusion behaviors including eddy-shedding phenomenon (Fig. 10e). Generally, the WBC intrusion becomes weaker but long period with the enhancing viscosity stress of the boundary, which is consistent with the sensitivity experiments presented by Mu et al. (2011).
6) Gap width
Previous research (Sheremet 2001; Mu et al. (2011)) has proven that gap width has a strong impact on the WBC intrusion. Generally, the narrow gap tends to prevent the WBC from bending into the small basin. So, the solution with the narrow gap width (D = 200 km) (Fig. 10f) displays the relatively longer intrusion period and consequently longer eddy-shedding process when compared to the reference solution (D = 300 km). In fact, an even narrower gap may finally prevent WBC from intruding into the small basin, just as in the experiments presented by Mu et al. (2011). On the contrary, when enlarging the gap width to 400 km (Fig. 10f), the problem changes completely. The WBC as a whole crosses the gap and forms a basin-scale circulation (figure omitted). Under this gap width, the small-basin circulation is only an eastward extended subtropical gyre from a large basin. Without the dynamical blocking effect of WBC at the gap, mesoscale eddies can directly pass through the gap, which also results in large-amplitude and short-period fluctuations in the stream field (Fig. 10f).
4. Role of eddy forcing in the process of the WBC intrusion
a. Time-mean eddy forcing at subgrid scale
According to Eq. (4), the UEF serves as a PV source on the right-hand side of Eq. (2). Thus, the positive/negative UEF produces positive/negative PV and drives cyclonic/anticyclonic flow. As mentioned at the end of section 2a, the UEF can be further decomposed into the temporal mean and fluctuation component. At first, Fig. 11 presents the temporal means of the 100-yr eddy PV fluxes and their divergences. On average, the spatial distribution of total UEF 
Time-mean PV fluxes (streamlines) and PV flux divergences (shaded contours) due to subgrid-scale eddy. The PV flux divergence has been normalized to have a value ranging from −1 to 1. The eddy flux and its divergence have been decomposed into the relative vorticity component R and isopycnal thickness component (B) [Eq. (6)]. All the fluxes are further decomposed into scale interaction components according to Eq. (4).
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
To further investigate the flow driven by the time-mean UEF, we remove the wind forcing upon the large-basin ocean and let the time-mean UEF solely drive the circulations. As shown by Fig. 12, the 
The time-mean flows purely driven by the time-mean divergences of total eddy PV flux F′, relative vorticity-type eddy flux R′, and isopycnal thickness-type eddy flux B′.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
As is shown in Fig. 11, features of 
Unlike the distribution of R′ divergence, the time-mean B′ divergence has a south–north structure in the gap with linkages farther away from the interior of circulations. From Fig. 11, a strong positive 
It can also be found from Fig. 11 that the time-mean eddy PV fluxes show obvious differences between relative vorticity-type component R and isopycnal thickness-type component B. The R flux is characterized by small-scale PV transport. On the contrary, the B flux transports PV at a much larger scale. This suggests that the B flux related to the baroclinic process plays a more important role in connecting the boundary and ocean interior (Berloff 2005). As is expected, the time-mean thickness flux 
Another interesting problem is the role of the scale interaction component of the eddy flux. For the eddy–flow interaction components 
b. Composite analysis of eddy-shedding process
What are the roles of the eddy PV forcing at different stages of WBC intrusion? Analysis based on individual event seems not convinced due to the high nondeterminism of the eddy. To answer this question, analyzing the intrusion events based on a large sample is necessary. From the instantaneous analysis presented in section 3, a typical WBC intrusion process can be generally divided into three stages: preshedding, eddy-shedding, and postshedding stages. This characteristic provides us with a chance to study the common features of the intrusion events by compositing the intrusion events with respect to shedding eddies. To derive an individual intrusion event from the 1200-month modeling fields, a simple algorithm is developed to identify eddy-shedding events. Details of the technique of eddy detection and eddy-shedding identification are present in the appendix.
1) Basic statistics
Through the use of the eddy-shedding identification method, 527 and 585 eddy-shedding events were found in total for ER and UEF-forced, non-ER solutions, respectively. The basic statistics of the eddy-shedding process are illustrated in Fig. 13. From Figs. 13a and 13e, the typical eddy-shedding event generally has a period ranging from 30–90 days with a mean period of about 60–70 days. Although a small difference exists between ER and non-ER solutions, the shapes of the period distributions coincide well. This time scale of the intrusion period is consistent with the conjecture drawn from the instantaneous fields in section 3. From Figs. 13b and 13f, the non-ER solution has slightly larger shedding eddies. This feature can also be found in the instantaneous fields in Figs. 3 and 5, which show the eddy in the ER solution has a tighter structure but with greater strength. Both the ER and non-ER solutions shed eddies at about 200 km west of the gap and slightly south of the gap center. Those statistics about the eddy-shedding process further confirm the dynamical consistence between the ER and the UEF-forced, non-ER solutions.
The statistics of eddy-shedding events: (a),(e) period of intrusion event; (b),(f) radius of the shedding eddy; (c),(g) the zonal position of the eddy detachment; and (d),(h) the meridional position of the eddy detachment. In each plot, the quantities Q1, Q2, and Q3 are the first, second, and third quantiles of the statistics, respectively. The plots in the left column are derived from ER solutions, and those in the right column are from UEF-forced non-ER solutions. All the statistics are based on 1200-month simulations.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
2) Eddy PV forcing in mean intrusion cycle
Having the eddy-shedding sample described above, the composite analysis is necessary to obtain the common features of the eddy-shedding process. Because of the nondeterministic feature of the eddy motion, the composite method that adopts the simple time average in a fixed spatial region will smooth out the “footprint” of the eddy. Thus, the fixed region composite can hardly reflect the life cycle of the eddy-shedding process, particularly for a large sample with high variation like this study. In consideration of this point, a rectangle box that moves with the eddy is adopted to obtain the information of physical fields around the eddy in a fixed, relative, spatial region. Considering the mean distance of the eddy to the gap (Figs. 13g,h), we choose a box with its four edges 275, 375, 325, and 325 km from the eddy center on the west, east, south, and north, respectively. For the UEF-forced, non-ER solution, the overall 585 eddy-shedding events are projected to a mean cycle possessing 63 phase positions, which approximately correspond to the mean period of shedding process (Fig. 13d). Because shedding events have different lengths of periods, the projection procedure adds an intrusion snapshot on the mean cycle according to its phase position in its own event. For example, if we want to project a 21-day intrusion event to the 63-phase mean cycle, the snapshot at the seventh day will be added to the twenty-first phase (T/3) of the mean cycle because the seventh day is just at the one-third of the 21-day period.
Figure 14 demonstrates the composites of eddy PV flux and its divergence for both of the subgrid-scale and grid-scale eddies. In this and the following figures, the eddy detachment takes place at lag-0 phase. In Fig. 14, the composite PV (thick black contour) well reflects a typical cycle of the WBC intrusion, in which the WBC changes from a leaping state to Loop Current and then retreats eastward by the detaching eddy from it. For the subgrid-scale eddy (Fig. 14a), at the early stage of the intrusion (lag −23 to lag −6), the negative PV is transported from the subtropical gyre in the large basin to the gap area by the PV flux with the structure of the counterrotating vortex pair. This kind of structure can also be found in the time-mean eddy PV flux shown in Fig. 11. The cross-WBC PV transport induces the jet to bend its pathway westward until it forms a strong Loop Current at lag −6. At this phase, the intruding negative PV is enclosed entirely by the Loop Current, with strong positive PV locating north of it. Then the positive PV gradually moves southeastward and cuts off the negative PV transport from the east at lag 0, when the eddy detaches and the WBC retreats eastward. After that, PV enclosed in the eddy gradually changes from negative to positive. Accordingly, the anticyclonic eddy starts diminishing and eventually disappears. In a word, the PV forcing and PV transport induced by the subgrid-scale eddy correspond well to the WBC intrusion evolution. As mentioned above, the positive eddy PV forcing east of the eddy greatly regulates the Loop Current displacement and eddy detachment. And, for the grid-scale eddy, a similar relationship between eddy forcing and intrusion is also found (Fig. 14b), except that the REF does not evolve in concert with the intrusion as well as the UEF.
A composite cycle of WBC intrusion event: (a),(b) for the total eddy PV fluxes (
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
Comparing the R′ (Fig. 14c) and B′ (Fig. 14e) components of the UEF, it is found that negative 
Additionally, the above relationships between eddy forcing and WBC intrusion are true as well for other intrusion subsets derived according to different statistics, such as the eddy-shedding location, shedding eddy radius, eddy-shedding period, and so on. Figure 15 presents the north intrusion (Figs. 15a,b) and south intrusion (Figs. 15c,d) subsets of the intrusion events. Here, we define the events with eddy shedding taking place farther north than y = 1400 km, which is the third quantile (as shown in Fig. 13h) of the y locations for all 585 events. Similarly, the south intrusion subset is obtained by deriving the events having eddy shedding locating south of the first quantile (y = 1175 km). It is found that the history of the eddy PV forcing as well as its relationship with the eddy-shedding process is qualitatively similar to that of the full-sample composite in Fig. 14. From the two composites shown in Fig. 15, it can also be concluded that the relative locations of the negative and positive eddy PV forcing, especially that produced by the UEF, determine the intrusion position of the WBC. In the gap, when the positive eddy PV forcing locates south of the negative one, the WBC will bend and detach the eddy near the northern part of the gap, and vice versa. Other composites based on different statistical categories of intrusion events also confirm the extreme importance of eddy PV forcing for the WBC intrusion behavior (figures omitted).
As in Fig. 14, but for the composites of (a),(b) north eddy-shedding subset and (c),(d) south eddy-shedding subset of intrusion events. The two subsets are obtained by extracting the events with the meridional eddy-shedding position north of the Q3 = 1400 km and south of Q1 = 1175 km (as is shown in Fig. 13h), respectively. The two subsets have 173 (for north-shedding subset) and 164 (for south-shedding subset) members, respectively.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
3) Relative importance of eddy forcing components
From the above composite analysis, it is found that the eddy PV forcing enclosed in the Loop Current or eddy determines its strength. And the negative-to-positive variations of the eddy PV forcing to the east of the eddy can tell us when and where the eddy detaches from the Loop Current. As a result of it, these two key regions provide us the convenience to diagnose the relative importance of different eddy forcing components during the typical cycle of WBC intrusion.
Figure 16 demonstrates the variations of the terms in the QG equation and eddy PV forcing components during a typical cycle shown in Fig. 14. The series in this figure are obtained by spatially averaging the quantities in the two regions: one approximately in the Loop Current/eddy and the other to the east of it, as the blue and red boxes show in the lag-0 subgraph of Fig. 14a. Figures 16a and 16b present the variations of the main balance terms of the QG equation [Eq. (2)] in two boxes. Because there is no wind forcing upon the small-basin sea, the term of wind stress curl is excluded. Obviously, the UEF dominates the dynamics in the eddy/Loop Current (Fig. 16a), particularly during the period near the eddy shedding (lag 0). After the eddy detaches from the Loop Current (lag 0 to lag 18), the balance between lateral friction and UEF determines the eddy evolution, which results in the gradually weakening of the eddy. In the region to the east of eddy (Fig. 16b), the UEF is generally still the maximal term, which produces anticyclonic flows at the preshedding (lag −23 to lag −2) and postshedding (lag 18 to lag 38) stages but cyclonic flows at the eddy-shedding stage (lag −1 to lag 17). The upstream WBC transport is also present to find the possible connection to the evolution of the Loop Current. As is shown in the instantaneous field in Fig. 8, eddy shedding generally corresponds to a minimum of low WBC transport. But the variation of UEF enclosed in the eddy as well as the intensity of the shed eddy/Loop Current seem to have no direct correlation to the upstream WBC transport. However, the UEF variation to the east of eddy well corresponds to the fluctuation of the upstream WBC. Because the eddy forcing in this region determines the intrusion position and eddy detachment, the upstream WBC transport seems to have the effect of regulating the eddy shedding.
The spatial means of QG terms and components of eddy PV forcing during the composite WBC intrusion cycle shown in Fig. 14. The plots in the left column are the means in the Loop Current or eddy, that is, the green box shown in the lag-0 subgraph of Fig. 14a. The right-column series are obtained from averaging the quantities over the region east to the shedding eddy (red box in lag-0 subgraph of Fig. 14a). (a),(b) The major terms in the Eq. (2) together with the upstream WBC transport, which has the same definition as that in Fig. 7; (c),(d) relative vorticity and isopycnal thickness components of UEF and REF; and (e),(f) the scale interaction components of UEF and REF.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
For different components of eddy PV forcing in the eddy/Loop Current (Figs. 16c,d), the most prominent feature is the isopycnal thickness components (
The spatial means of eddy PV flux components during the composite WBC intrusion cycle shown in Fig. 14: (a) for relative vorticity and isopycnal thickness components and (b) for the scale interaction components. The lines in each plot are obtained by using the averages over the northern half part of a rectangle (700 ≤ x ≤ 900 km and 1200 ≤ y ≤ 1500 km) to minus the averages over the southern part of the same rectangle. The rectangle locates approximately between shedding eddy and the gap.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
The PV transport due to the eddy flux between the shedding eddy and the gap is also projected on a typical intrusion cycle. As shown by Fig. 17a, the isopycnal thickness fluxes play leading roles in transporting and mixing the PVs from the two basins. For the scale interaction components, the two grid-scale fluxes, FLE + FEL and FEE, counteract each other throughout the cycle. But for the PV flux due to the subgrid-scale eddy, both the eddy–eddy and eddy–flow fluxes flow in the same direction during the whole intrusion cycle. All three scale interaction components are important for the total PV transport, particularly at the eddy-shedding stage (Fig. 17b).
5. Discussion and conclusions
The western boundary current (WBC) intrusion problem in the interaction between the northwestern Pacific and the South China Sea (SCS) is still in dispute, particularly in the aspects of the intrusion way, intrusion mechanism, seasonal variation, and so on. This article is intended for the role of the eddy–flow interaction in the process of the WBC intruding into the adjacent marginal sea. To do this, the problem is reduced to the interaction between a time-dependent WBC driven by an asymmetric, time-independent wind stress and a resting marginal sea in an ideal dynamical scenario. Because the work is targeted to discuss the intrinsic ocean dynamics, the influences due to coastal geometry, bottom topography, wind forcing variability, and the local circulation in the marginal sea are excluded to simplify the problem as much as possible. For this case, a three-layer quasigeostrophic (QG) model is used and solved numerically in a domain with two, rectangular, flat-bottom basins. The two basins have similar scales as the SCS and northwestern Pacific and are connected through a narrow gap that has a similar meridional scale as the Luzon Strait.
High spatial resolution of 6.25 km and low eddy viscosity of 80 m2 s−1 are used for the model to resolve the mesoscale eddies as well as their interplay with the mean flow. The eddy-resolving (ER) experiment simulates a classic, asymmetric, double-gyre circulation with a strong variable WBC to the east of the ideal strait. In the ER solution, the WBC intrudes into the adjacent small-basin (ideal marginal) sea in the manner of the Loop Current with multiple states in a period of 30–90 days. Generally, the Loop Current intrusion terminates by detaching the eddy from it. And this kind of WBC intrusion is ubiquitous in the whole 100-yr modeling time of the ER solution. The Loop Current and eddy-shedding process are also produced by many other simulations with different strengths of friction, wind forcing, and so on. Through changing the nonlinearity of the flows, the time period of the WBC intrusion event may vary from a monthly scale to interannual scale.
Although the noneddy-resolving (non-ER) solution under much coarser spatial resolution and higher viscosity also show the Loop Current intrusion with eddy detachment, the non-ER intrusion process has a much longer duration time but fewer types of intrusion states compared to those of the ER solution. By using the dynamically consistent eddy–flow decomposition (Berloff 2005), the unresolved (subgrid scale) eddy forcing (UEF) is interactively extracted from the ER solution. By imposing UEF upon the non-ER model, the coarse grid solution reproduces the consistent intrusion behavior of the WBC as the ER solution, although the time-mean transport of upstream WBC decreases due to higher viscosity. So, the intrusion period is mainly regulated by the variation of WBC caused by the eddy. Further sensitivity experiments prove that the local effect of the UEF around the gap is extremely important for the non-ER model to simulate correct WBC intrusion behavior. This suggests that the mesoscale eddies produced east of the Luzon Strait, especially those from the subtropical countercurrent (STCC), may have a strong impact on the pathway of the Kuroshio near the Luzon Strait.
Further discussions are present to reveal the detailed mechanism of the impact of eddy–flow interaction on the WBC intrusion. At both the grid and subgrid scales, high-valued eddy forcing (eddy PV flux divergence) is found to be near or enclosed in the Loop Current. Although the marginal sea and open ocean are separated by the strong PV front due to the WBC, the PV exchange between them can be established by the PV transport due to mesoscale eddy. By decomposing the eddy flux into relative vorticity–type (R type) and isopycnal thickness–type (B type) components, the relative importance of the processes induced by shear instability and baroclinic instability in the WBC intrusion process can be further explored. Based on large-sample composite analysis, it is further confirmed that the eddy PV forcing, especially the UEF, regulates the WBC intrusion process. The strengthening/weakening of the Loop Current may be attributed to the negative/positive eddy PV forcing produced by small-scale baroclinic processes through eddy–flow and eddy–eddy interactions. Because the PV forcing enclosed in the Loop Current depends on large-scale PV transport, the R-type eddy forcing, which mainly concentrates in the boundary layer, contributes little to the intensity of the Loop Current.
It should be emphasized that the eddy PV forcing on the west side of the gap plays an important role in determining the intrusion position, Loop Current displacement, and eddy detachment. When the negative eddy PV forcing occurs there, the Loop Current will maintain or strengthen. But when a positive forcing occupies that area, the induced cyclonic flows will cut off the negative PV supply for the Loop Current, which will result in eddy detachment and eastward retreat of WBC. Therefore, the eddy PV forcing in this area greatly regulates the intrusion cycle. Because the gap region is the boundary of the marginal sea and open ocean, the strong shear flows make the R-type UEF have a magnitude comparable to the B-type UEF. Generally, the R-type UEF tends to produce negative PV to the west of the gap throughout the whole intrusion cycle. That is in favor of the westward bending of the WBC. But the B-type eddy forcing therein has a negative–positive–negative transition during the typical intrusion event, which corresponds to the intrusion cycle very well. That is to say, the B-type eddy forcing plays different roles at different stages of WBC intrusion; it favors cutting off the Loop Current at the eddy-shedding stage but inducing WBC westward bending at other stages. To the west of the gap, when the B-type UEF goes from negative to positive and dominates the total UEF by counteracting the negative R-type UEF, the Loop Current begins to detach the eddy. So, the processes related to baroclinic instability regulate the WBC intrusion period and Loop Current intensity more. And the mechanism of eddy shedding may be explained as the balance between barotropic (R-type UEF) and baroclinic (B-type UEF) processes.
To our knowledge, the detailed mechanisms of the eddy-driven intrusion for a time-dependent WBC into an adjacent marginal sea have not yet been reported. Therefore, the new findings from the above analysis can confirm that the eddy feedback upon the mean flow is one of the main factors to drive and regulate the WBC intrusion. The above results have an important implication for our understandings of the interaction between the Kuroshio and SCS, such as the mechanism of the intraseasonal variability of the Luzon Strait transport (Zhang et al. 2015), the transitions between different pathways of the Kuroshio near the Luzon Strait (Caruso et al. 2006; Nan et al. 2011b), and the origin of the nondeterministic nature of Kuroshio penetration and eddy shedding (Metzger and Hurlburt 2001b). Although all the analyses present in this work are based on the local PV analysis, the role of the small basin in the global PV budgets and its relationship with the WBC intrusion are worth exploring. Furthermore, it should be pointed out that the above results are obtained in an extremely ideal scenario, by using a simple model, and with almost the simplest vertical stratification. Recent ultra-high-resolution simulation (Shevchenko and Berloff 2015) shows that higher-order baroclinic modes become important. Even in the purely dynamical frame, many other factors, such as the seasonally reversing monsoon, local circulation of the SCS, and the variability of the North Equatorial Current, are not included in this model. So, to obtain more dynamical details of the mechanism for the Kuroshio intrusion into the SCS, a more realistic dynamical environment including multiple factors should be considered in further work.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China (Grant 41275064), the National Basic Research Program of China (Grants 2011CB403500 and 2013CB956203), and the Public Science and Technology Research Fund Projects of the Ocean (Grants 201005019).
APPENDIX
Eddy-Shedding Detection Methodology
For an instantaneous stream field, such as that shown in Fig. A1, a simple algorithm is designed to detect the eddy and identify the eddy-shedding event. The algorithm uses the information of instantaneous streamfunction ψ, relative vorticity ∇2ψ, and velocity V = (u, υ) to derive the detailed properties of eddies, such as the eddy location, radius, shape, shedding time, and so on. Taking the anticyclonic intrusion and eddy shedding as an example, the whole identification procedure can be summarized according to following steps:
Identify the maxima of ψ in a 100 km × 100 km moving box throughout the study domain. The maxima identified in this step are considered the candidates of the anticyclonic eddy center. We name this set of possible eddies set
Derive the closed eddy from the eddy candidate set
Identify the isolated eddy from the eddy set
Identify the eddy detaching from WBC. In the eddy set
set the distance from the first zero ψ location to eddy center as radius rψ;
set the distance from that first location, where Δψ reaches 0.7ψmax (ψmax is the streamfunction at eddy center), to the eddy center as rΔψ;
set the distance from the first location, where the relative vorticity changes from negative to positive (i.e., ∇2ψ = 0), to the eddy center as radius
project the velocity on the direction normal to the radius. Then identify the first location where the clockwise vector (red vector on the radii in Fig. A1) becomes the counterclockwise one (white vector on the radii in Fig. A1). Then the distance from that location to the eddy center is considered as the radius rV.
A snapshot of the eddy-shedding detection procedure. The shaded contour is relative vorticity. The black line is the zero streamfunction.
Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0220.1
In this work, the direction number n is set to be 64, that is, examining the eddy property every π/32 around the center. This parametric setting is fine enough to obtain the eddy information for this study. Additionally, the same algorithm is also suitable for the cyclonic eddy by simply reversing the criteria presented above.
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