## 1. Introduction

Because of its potential impacts on local climate, the South China Sea (SCS) circulation has been a focus of research in recent years (e.g., Fang et al. 2005; Cai et al. 2005; Qu et al. 2005; Yu et al. 2007, 2008). An important aspect of that circulation is the existence of a mean current through the SCS, the SCS throughflow (SCSTF), consisting of inflow from the Kuroshio through Luzon Strait and outflow primarily through the Mindoro, Karimata, and Taiwan Straits (Fig. 1) (see, for example, Qu et al. 2000; Lebedev and Yaremchuk 2000; Sen and Chao 2003). The outflow from the Karimata and Mindoro Straits enters the Java and Sulu Seas, respectively, and has been hypothesized to be important for determining the thermohaline structure of the Indonesian Throughflow (Gordon 2005; Qu et al. 2006a). Thus, in addition to its local effects, the SCSTF remotely impacts large-scale circulations in both the Pacific and Indian Oceans.

The strength of the SCSTF has been difficult to determine observationally, owing to the scarcity of direct observations and vigorous eddy activity. It has also been difficult to determine with numerical models because of the need for very high resolution in order to represent adequately the topography of the narrow and shallow outlets and the mixing processes there. Currently, most SCS models are limited to a resolution of 15–20 km, resulting in a large scatter of SCSTF estimates. For example, using variable-grid, global, ocean general circulation models (OGCMs) with 0.16° resolution in the Indonesian Seas, Lebedev and Yaremchuk (2000) determined the annual-mean SCSTF transport to be 5.4 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}) and Fang et al. (2005) obtained 3.9 Sv. Xue et al. (2004) diagnosed a value of 2 Sv, using a regional, (Princeton Ocean Model) POM-type, SCS model with a grid resolution of 0.11°–0.33°. In earlier, coarser-resolution models (e.g., Miyama et al. 1995; Metzger and Hurlburt 1996, 2001) SCSTF estimates varied from 2 Sv to 5 Sv with a mean of about 3 ± 1.5 Sv. Most recently, the SCSTF magnitude was found to be 3.3 Sv in the climatological run of the global OGCM for the Earth Simulator (OfES; Masumoto et al. 2004) with a resolution of 0.1° [see Fang et al. (2005) for additional estimates].

The SCSTF is driven remotely by the large-scale winds over the Pacific. As noted by Qu et al. (2005), their impact can be understood using Godfrey’s (1989) “Island Rule,” which predicts cyclonic flow around the Philippine and Kalimantan islands (i.e., a southward SCSTF). (The island rule was originally used to predict successfully the transport of the Indonesian Throughflow, providing a value of the order of 15 Sv.) On the other hand, the SCSTF transport predicted by the *inviscid* island rule is an order of magnitude too large, pointing to the first-order effects of mixing within the shallow or narrow outflow ports (Wajsowicz 1996). Indeed, in an OGCM solution with an unrealistically broad and deep Mindoro Strait, almost all of the water for the Indonesian Throughflow entered the Indonesian Seas through the Mindoro Strait, rather than from near the equator (R. Furue 2006, personal communication).

Based on climatological data from the *World Ocean Atlas 2001* (*WOA01*) (Conkright et al. 2002), a prominent feature of the SCS is a subsurface salinity maximum at a depth of 150–200 m (Fig. 2; thin dashed curve), which results from the presence of North Pacific tropical water (NPTW) within the SCS (Qu et al. 2000). Its vertical structure is also not easy to reproduce in OGCMs. For example, Fig. 2 also plots salinity from the OfES solution (thick dashed curve). Although the solution qualitatively captures the depth structure of subsurface salinity, the core of the salinity minimum is located near 250 m, considerably deeper than the observed one; moreover, biases are large from 70 to 700 m, where they can be two–three standard deviations (shading) for both temperature and salinity.

A number of errors could account for the model–data discrepancies in Fig. 2. One possibility is that vertical mixing is too strong in OfES, resulting in its surface mixed layer being too thick. Another is that the surface fluxes forcing the model are inaccurate. Yu et al. (2008) used a regional model to explore the sensitivity of the SCS subsurface salinity maximum to a variety of processes, including the magnitude of the SCSTF transport, model parameters, and forcing fields. In one solution with closed outflow straits, so that there was no SCSTF, the subsurface waters became much too fresh and the subsurface salinity maximum disappeared, indicating the essential role of the SCSTF in its generation. Other solutions demonstrated the sensitivity of the salinity maximum to mixing of salty NPTW with the overlying fresher SCS waters, which are generated by the large precipitation and river runoff in the SCS. The authors demonstrated that it is possible to remove salinity and temperature biases, like those in Fig. 2, by tuning the model’s across-boundary transports, surface fluxes, and internal parameters. They were unable, however, to obtain a quantitative estimate of the SCSTF transport because solutions were so sensitive to both forcings and parameterizations.

In the present study, we use an inverse modeling approach to estimate the SCSTF structure and transport quantitatively. Specifically, we extend the regional, 4½-layer model of Yu et al. (2007, 2008) to include four-dimensional variational data assimilation (4dVar), and obtain an optimized solution that “best fits” *WOA01* temperature and salinity fields within the SCS. The low vertical resolution of the model has the advantage that the number of internal model parameters to be adjusted during the data assimilation is relatively small; at the same time, the model is still able to capture the major stratification features in Fig. 2. During the optimization, external forcing, inflow and outflow boundary conditions, and model parameters are all adjusted. Particularly important for our purposes, the approach treats the transports through the inflow and outflow ports as unknowns to be reconstructed by the data assimilation, so in the optimized solution they are internally determined rather than externally prescribed. Thus, our indirectly forced (inverse) model does not require knowledge of any mixing processes over shallow shelves and within straits in order to determine strait transports, an advantage over solutions to directly forced OGCMs (see section 2a below).

The paper is organized as follows. In section 2, we describe the ocean model and the data-assimilation technique. In section 3, we report results, comparing our optimized solution against independent observations and the OfES solution, and discussing possible reasons for the differences among them. In section 4, we summarize our main results.

## 2. Methodology

In this section, we describe the ocean model (section 2a) and the procedures used to obtain the first-guess (section 2b) and optimized (section 2c) solutions. Section 2a also presents the initial (unadjusted) model parameters, surface fluxes, and across-boundary transports. The equilibrium solution in response to these parameters and forcings is heavily biased with respect to *WOA01* data; these biases are removed by adjusting a limited number of the key model parameters to produce the first-guess solution (section 2b) which is then optimized with respect to both surface and lateral boundary forcing fields (section 2c).

For convenience, we label variables *q* differently according to their type. Model variables and observational data that have been vertically averaged to correspond to layer *i* from the model, are designated *q _{i}* and

*q**

_{i}, respectively. First-guess and optimized variables are labeled

*q̂*and

*q̃*, respectively. Monthly averages of variables are designated

*q*(

^{m}*m*= 1, …, 12). All variables without a superscript are “instantaneous,” that is, defined at each time step. Variables defined at the five boundary ports (Fig. 1) include an additional subscript

*q*(

_{n}*n*= 1, …, 5). Annual-mean and basin-averaged variables are labeled

*.*q

### a. Numerical model

The ocean model is a reduced-gravity 4½-layer system in which temperature and salinity are allowed to vary within each layer. It is nearly identical to the one described by Yu and Potemra (2006) and used by Yu et al. (2007, 2008) for the SCS, differing only in that it has lower horizontal resolution and different specifications for river runoff and surface forcing. Details of a similar version of the model can be found in Han et al. (1999) and Han and McCreary (2001). The model consists of four layers with thicknesses *h _{i}*(

*x*,

*y*), velocities

**v**

*(*

_{i}*x*,

*y*), salinities

*S*(

_{i}*x*,

*y*), and temperatures

*T*(

_{i}*x*,

*y*) (

*i*= 1–4), overlying the deep ocean where pressure gradients vanish. Each of the layers represents water of a specific origin: Layer 1 is the surface mixed layer, determined by Kraus and Turner (1967) physics; layer 2 is the seasonal (upper) thermocline; and layers 3 and 4 represent lower thermocline and upper-intermediate waters, respectively.

Basin boundaries are defined by the locations of the 200-m isobath (Fig. 1), and the horizontal resolution of the grid is 0.5°. With this definition, the model domain extends over the deep part of the SCS, but neglects shelf regions along the western boundary and southern portion of the basin. The southern shelf separates the model Karimata Strait (port 3 in Fig. 1) from the actual one by a considerable distance. Since our interest is only the transport through the Karimata Strait (not details of the shelf and strait circulations themselves), the lack of the southern shelf is not a problem: Mass conservation ensures that in a model with the southern shelf, the transport off the shelf to the south must be essentially the same as the flow onto it from the north.

How might the lack of shelves and straits in our model impact the SCSTF? In a directly forced model, one expects that the SCSTF transport will be strongly impacted by both shelf and strait processes, as they provide the “drag” that limits the overall throughflow driven by the Pacific winds. In our inverse model, however, the strait transports are determined by the *T* and *S* properties within the SCS basin and do not require any knowledge at all about either mixing or circulations in the surrounding regions. In addition, our model does include several key coastal processes: It allows for coastal (shelfbreak) upwelling from both layers 3 and 4 into the upper layers and includes the spreading of the river outflow into the basin. Furthermore, shelf-exchange processes primarily affect the thermohaline structure of surface layers, which have a smaller impact on the strait transports in the inverse solution than the deeper layers do (see section 3c). Finally, it is possible that there are shelf-confined circulations that connect the SCS straits (e.g., the Karimata and Taiwan Straits), a property that could weaken several of our conclusions; however, such flows are not likely because the Vietnamese Shelf is very narrow from 11°–15°N, and currents there are characterized by vigorous eddy activity (Hwang and Sung-An 2000), which tends to mix shelf waters with those in the deep basin. For all of these reasons, we believe that our model’s lack of the shallow regions surrounding the SCS is not a severe limitation.

To ensure numerical stability, horizontal smoothing is included at minimal strength. In the temperature, salinity, and momentum equations, smoothing is by biharmonic mixing with a coefficient of 10^{21} cm^{4} s^{−1}. In the layer thickness equations, it is by Laplacian mixing with a coefficient of 10^{7} cm^{2} s^{−1}, a process analogous to the Gent and McWilliams (1990) mixing parameterization used in ocean GCMs.

*w*between layer

_{αi}*i*and the underlying layer

*i*+ 1. There are three primary types of transfer. The first type,

*w*

_{k1}, is based on the Kraus and Turner (1967) surface mixed layer model, which relates entrainment into and detrainment from layer 1 to forcing by wind stirring and convective overturning. The strengths of these processes are proportional to parameters commonly labeled

*m*and

*n*, respectively, and their initial values are

*m*= 2 and

*n*= 0.2. The second type,

*w*, specifies entrainment into layer

_{ri}*i*whenever

*h*becomes thinner than either prescribed minima

_{i}*h*(

_{ci}*h*

_{c1}=

*h*

_{c2}= 10 m and

*h*

_{c3}= 50 m) or

*g*= 9.81 cm s

^{−2},

*ρ*is the density of layer

_{i}*i*,

*ρ*

_{0}= 1 g cm

^{−3}is a background density, and

*r*= 0.75 is the Richardson number;

*h*represents the thickness below which the flow becomes supercritical in the sense of the Richardson number criterion. Velocities

_{r}*w*are then given by

_{ri}*h*′

_{i}= max (

*h*,

_{ci}*h*),

_{r}*t*= 0.05 days is a relaxation time scale, and

_{r}*θ*is a step function [

*θ*(

*ξ*) = 1 for

*ξ*> 0, zero otherwise]. The third type,

*w*

_{d2}, is detrainment from layer 2 to layer 3,

*H*= 80 m is a thickness parameter and

_{d}*t*= 180 days is the corresponding relaxation time scale. It simulates a gradual erosion of the seasonal thermocline after the mixed layer retreats during the spring.

_{d}At its surface, the model is forced by climatological, monthly-mean, wind stress ** τ*** = (

*τ**

*,*

^{x}*τ**

*), precipitation*

^{y}*P**, river runoff

*R**, evaporation

*E*, and heat flux

*Q*=

*Q**

_{sw}+

*Q**

_{lw}+

*Q*

_{lh}+

*Q*

_{sh}fields, the four heat flux components being shortwave and longwave radiation, latent heat flux, and sensible heat flux; in addition, wind stirring in the Kraus and Turner (1967) mixed layer model requires an estimate of the friction velocity

*u**. The

*Q*

_{lh},

*Q*

_{sh}, and

*E*fields are determined from air temperature

*T**

_{a}, specific humidity

*q**

_{a}, and wind speed

*V** fields by bulk formulae using model SST,

*T*

_{1}(McCreary and Kundu 1989; McCreary et al. 1993). The

*P**,

*Q**

_{sw},

*Q**

_{lw}

*T**

_{a}and

*q**

_{a}fields are taken from the Comprehensive Ocean–Atmosphere Dataset (COADS) (da Silva et al. 1994), and

***,**

*τ**u**, and

*V** are obtained from the operational surface winds of the European Centre for Medium-Range Weather Forecasts (ECMWF). River runoff

*R** is derived from the reanalysis of Dai and Trenberth (2002) combined with the Global Runoff Data Center database. River transports are specified as point freshwater sources distributed along the rigid boundary. In the first-guess solution the monthly-mean transports of the Mekong, Zhujiang (Pearl), and Hong (Red) Rivers are included. To obtain forcing fields at every model time step, their monthly-mean values are linearly interpolated in time.

The model is also forced by transports through the inflow and outflow ports (Fig. 1). Let the transport through boundary port *n* in layer *i* be *M _{ni}*, where index

*n*= 1, …, 5 indicates the Kuroshio inflow port at 18°N, the Mindoro, Karimata, and Taiwan Straits, and the Kuroshio outflow port at 23°N, respectively. For convenience, we also assume that

*M*is positive (negative) for flow into (out of) the basin.

_{ni}*M**

_{n}^{m}for the Kuroshio inflow port and the SCS outflow straits are specified by averaging several, independent estimates (Fig. 3), OfES output (Fang et al. 2005; Lebedev and Yaremchuk 2000; Yaremchuk and Qu 2004; and others). Initial values of

*M*for the Karimata and Taiwan Straits are then given by

_{ni}^{m}*M*for ports 1–4. Their values are linearly interpolated to provide transports at each time step.

_{ni}^{m}*M*

_{5i}for each layer.

The boundary transports are assumed to be spread uniformly across each port, thereby specifying the normal velocity field at every grid point within the port. No-slip and slip (zero normal derivative) conditions are imposed on the tangential velocity components across the inflow (port 1) and outflow ports (ports 2–5), respectively.

Temperatures *T*_{1i} and salinities *S*_{1i} at the inflow port are prescribed by *T*_{1i} = *T**_{1i}, *S*_{1i} = *S**_{1i}, where *T**_{1i} and *S**_{1i} are obtained from the *WOA01* climatology by averaging over the depth ranges of each model layer. Zero normal derivatives are imposed on *h _{ni}* at all the ports and on

*T*and

_{ni}*S*at the outflow ports (

_{ni}*n*= 2–5).

### b. First-guess solution

A first step in optimizing the SCS seasonal cycle is obtaining a “good” first-guess solution for the main data-assimilation run (section 2c)—one with an annual-mean stratification that is not too far from the data. Because the equilibrium response depends nonlinearly on model parameters, external forcing, and port transports, this task is challenging. (As illustrated in Fig. 2, even a state-of-the-art model may easily differ significantly from climatology owing to errors in parameterization and surface forcing.) Solutions are particularly sensitive to the initial layer thicknesses *H _{i}*, through their impact on the vertical distribution of thermohaline fluxes at the inflow port.

*D*between a model solution and the annual-mean

*WOA01*climatology is

*WOA01*dataset, determined using layer depths from the initial model solution and further refined after preliminary assimilation experiments; final results are not sensitive to their values. Spatial and temporal averaging is obtained during the last year of the integration. The model − data misfits (

*q*−

_{i}*q**

_{i}) in

*D*are nonlinear functions of

*h*(

_{i}*x*,

*y*,

*t*) because the vertical averaging of the

*WOA01*data

*q** to obtain

*q**

_{i}is performed over the depth range of model layer

*i*at each space–time location. Finally, only the layer thickness misfit for layer 1 appears in (10) because the mixed layer thickness is the only layer thickness that can be objectively estimated from an arbitrary T–S profile. We use the methodology of Kara et al. (2003) for estimating

*h**

_{1}.

*D*, the key forcing and model parameters that determine the equilibrium state are adjusted. Specifically, the model parameters

*P*= {

_{j}*m*,

*n*,

*t*,

_{d}*H*,

_{d}*r*,

*t*,

_{r}*H*} and the annual-mean amplitudes of the surface forcing fields

_{i}*S*= {

_{k}^{m}*τ**

_{x}^{m},

*τ**

_{y}^{m},

*Q**

_{sw}^{m},

*Q**

_{lw}^{m},

*P**

^{m},

*T**

_{a}^{m},

*q**

^{m},

*V**

^{m},

*u**

^{m}}

*k*= 1, …, 9 are varied, the latter by forcing the model with scaled fields

*α*. Constraints (4)–(9) reduce the number of independent port transports from 20 (5 ports × 4 layers) to 9. Let the set of independent transports from the initial solution be

_{k}S_{k}^{m}*β*. Finally, the inflow temperatures and salinities

_{l}B_{l}^{m}*T**

_{1i}and

*S**

_{1i}are varied by replacing them with

*T**

_{1i}+

*δ*(

*T*

_{1i}) and

*S**

_{1i}+

*δ*(

*S*

_{1i}); both

*δ*(

*T*

_{1i}) and

*δ*(

*S*

_{1i}) are adjusted.

Altogether, there are 36 adjustable parameters: 18 external forcing coefficients *α _{k}* and

*β*, 10 internal parameters

_{l}*P*, and 8 corrections to the inflow temperatures and salinities

_{j}*δ*(

*T*

_{1i}) and

*δ*(

*S*

_{1i}). To minimize

*D*, parameters are adjusted in the standard way by obtaining a sequence of pairs of integrations: a model solution run forward in time from a state of rest to equilibrium (15 years), followed by a solution to the adjoint model run backward in time. The integration pair allows the gradient of

*D*with respect to each of the parameters to be evaluated, and the gradient is then used to determine updated parameters for the next step in the sequence via the limited-memory, quasi-Newtonian algorithm of Byrd et al. (1995). This algorithm also allows limits of variation for the adjusted variables to be preset. Values of

*α*are bounded by the inequality 0.9 ≤

_{k}*α*≤ 1.1. Values of the other parameters are less well known and, hence, are allowed to vary over wider ranges. Values for

_{k}*β*satisfy the inequality 0 ≤

_{l}*β*≤ 2, and minima and maxima for the model parameters and for the inflow temperatures and salinities are listed in Table 1. These range constraints can be interpreted as additional “data” (with non-Gaussian error statistics) that regularize the nonlinear optimization problem.

_{l}### c. Optimization of the seasonal cycle

*Q*=

^{m}*Q̂*+

^{m}*δQ*,

^{m}

*τ*^{m}=

*τ̂*^{m}+

*δ*

*τ*^{m},

*P*=

^{m}*P̂*+

^{m}*δP*, and

^{m}*R*=

^{m}*R̂*+

^{m}*δR*, and port transports

^{m}*M*=

_{ni}^{m}*M̂*+

_{ni}^{m}*δM*, where the

_{ni}^{m}*q̂*fields are taken from the first-guess solution and the

^{m}*δq*are adjustable error fields. The mixing parameters in Table 1 (top block) are kept fixed to their first-guess values:

^{m}*J*that penalizes the magnitudes of the forcing errors, and the

*W*coefficients in (12) are inverse variances of these errors estimated from the corresponding data.

_{αi}Note that *J _{r}* does not contain a term involving

*δM*, so the port transport corrections are not restricted. In this case, a measure of the distance of the solution from the

_{ni}^{m}*T*–

*S*data is

*D*′ =

*J*−

*J*, a version of

_{r}*D*using instantaneous variables.

Cost function *J* is minimized using essentially the same procedure as for *D*, except in this case the total number of control variables is 28 890 with approximately 80 000 data points. During each step of the iteration, the model is integrated forward for one year, with the first-guess solution as its initial state. The short (1 yr) integration was found to be adequate because the model did not deviate too far from a seasonally cyclical state during the optimization: The relative difference between model states at the beginning and the end of 1-yr integrations never exceeded 6%. Minimization is again performed using the Byrd et al. (1995) algorithm, with no limits on variations except for the river transport and rainfall errors *δR ^{m}* and

*δP*, which are limited from below by −

^{m}*R̂*and −

^{m}*P̂*, respectively, to ensure that the river runoff and rainfall

^{m}*R*and

^{m}*P*remain nonnegative.

^{m}### d. Error estimation

Formal error bars can be obtained for control variables by interpreting *J* to be the argument of a Gaussian probability distribution function in the vicinity of the optimal state (e.g., Thacker 1989): In theory, a rigorous error estimate requires multiplication of the Hessian inverse by the operator, projecting the control vector on the quantity of interest (e.g., annual-mean total transport through a strait). Because of the large dimension of the Hessian matrix, we took a simplified approach, estimating the second derivatives of *J* with respect to total transports via finite differentiation: The optimal transports were perturbed by adding a steady uniform flow of 0.01 Sv in the cross-sectional areas of the straits, and the finite-difference second derivatives of *J* were computed. Uncertainties in transports were then estimated as the reciprocals of these second derivatives. Such treatment, although approximate, gives a rough idea of the accuracy of our reconstruction. In general, if *J* is sensitive to a particular control variable, the resulting error bars tend to be smaller and vice versa. The error bars quoted below for the outflow transports were determined in this simplified way.

## 3. Results

In this section, we first discuss the convergence of the first-guess and optimized solutions to the *WOA01* data and the resulting adjustments to parameters and forcings (section 3a). Then, we describe the open port transports from the optimized solution in detail, comparing them to observations and OfES model output (section 3b). Finally, we note the robustness of the optimized solution, discussing the sensitivity of its thermohaline structure to forcings and parameters (section 3c).

### a. Convergence to the data

#### 1) First-guess solution

There are nine independent terms in *D*, so an acceptable value is *D* < 9, that is, a value on the order of the inherent variability in the data itself or less. As expected, the distance of the initial solution from the data is unacceptably large (*D*_{in} = 56.8), whereas for the first-guess solution it is much reduced to an acceptable level (*D̂* = 1.13). Figure 4 illustrates the bias reduction in the top three layers, plotting basin-averaged salinity differences of the initial (solid curves) and first-guess (dashed curves) solutions from the *WOA01* observations, both normalized by *σ _{Si}*. Biases of the first-guess solution are reduced to acceptable levels in all three layers. The bias in layer 3 salinity is particularly reduced, accounting for much of the overall reduction of

*D*; this sensitivity results from the normalization

*σ*

_{S}_{3}being relatively small (0.044 psu), so biases of

*S*

_{3}in the initial solution often exceed 3

*σ*

_{S}_{3}(Figs. 2 and 4). The layer 4 bias (not shown) is reduced from an annual-mean value of −1.92 to −0.43.

Optimized values *q̂* for the model parameters determined by obtaining the first-guess solution are listed in Table 1. Adjustments to the inflow temperatures and salinities *δ*(*T*_{1i}) and *δ*(*S*_{1i}) lie within the limits prescribed by the corresponding variabilities east of Luzon, with a tendency for cooling the inflow in layers 2 and 3 and freshening it in layers 1–3. There is not much change to *H _{i}*, so the initial guesses were good.

The Kraus and Turner (1967) mixing parameters, *m̂* and *n̂* essentially double their initial values, thereby acting to thicken the mixed layer. Indeed, in a test run using the initial values of *m* and *n* with other parameters and forcings as their first-guess values, *t̂ _{d}* reduced by half, strengthening detrainment from layer 2 to layer 3 (the erosion of the seasonal thermocline). These changes to upper-ocean mixing act to pump more freshwater downward from layer 1 into layer 3. We confirmed this property in a test run using the initial values of

*m*,

*n*, and

*t*with other parameters and forcings as their first-guess values: The value of

_{d}*σ*

_{S}_{3}= 0.044 psu, so the model − data distance increased from

*D̂*= 1.13 to

*D*= 2.15 (see the discussion of these parameters in section 3c).

Optimized values for the surface forcing amplitudes are *α̂ _{k}* = {1.08, 1.07, 0.95, 1.02, 1.09, 0.97, 0.98, 1.07, 0.99}: none of them reached their limiting 10% deviations from the initial guess. The largest excursions occurred for wind stress (

*α̂*

_{1}and

*α̂*

_{2}) and rainfall (

*α̂*

_{5}), indicating that their climatological values are possibly underestimated by 8% and 9.5%, respectively. Most importantly, the total annual-mean surface freshwater flux (

*E*−

*P*) reversed sign from 0.16 m yr

^{−1}in the initial solution to −0.19 m yr

^{−1}in the first-guess solution; the optimized value corresponds well with the mean estimate derived from the average of the ECMWF and the National Centers for Environmental Prediction (NCEP) reanalyses and COADS climatology (−0.23 ± 0.08 m yr

^{−1}).

Values for the port transport amplitudes determined by obtaining the first-guess solution *β̂ _{l}* = {1.44, 1.52, 0.07, 0.48, 1.84, 0.35, 0.94, 1.46, 1.58} exhibit several distinctive features. The coefficient for the layer 4 Kuroshio inflow (

*β̂*

_{3}) is reduced significantly from 1 to 0.07, drastically weakening the flow in the depth range from 400 to 700 m, and there is a compensating increase in the transports in layers 1–3 so that the total inflow transport remains almost unchanged (i.e.,

*β̂*

_{1}+

*β̂*

_{2}+

*β̂*

_{3}≈ 3). The Kuroshio outflow is redistributed similarly, with a decrease in the layer 4 transport balanced by an increase in the shallower layers (

*β̂*

_{8}and

*β̂*

_{9}). There is also an extreme vertical redistribution in the Mindoro Strait outflow (

*β̂*

_{4}and

*β̂*

_{5}), so approximately 80% [

*β̂*

_{5}/(

*β̂*

_{4}+

*β̂*

_{5})] of the first-guess Mindoro Strait transport occurs in layer 3. Finally, the first-guess transport through Karimata Strait (

*β̂*

_{6}) is only 35% of that in the initial solution. These prominent features are retained in the optimized solution and discussed further below.

#### 2) Optimized solution

As for *D*, distance *D*′ also shows a remarkable decrease among the solutions with *D*′_{in} = 65.6, *D̂*′ = 8.6, and *D̃*′ = 3.7. The final value, *D̃*′, is considerably less than 9, indicating that a larger part of the spatiotemporal variability of the monitored fields (temperature, salinity, and mixed layer thickness) can be captured by the model. It is not as small as *D̂* because the errors in *D̃*′ are calculated at each time step.

Figure 5 shows basin-averaged temperature and salinity differences of the initial, first-guess, and optimized solutions from the corresponding *WOA01* observations. The improvement of both assimilation runs is significant, especially for salinity for which differences decrease from values of about 3 in the initial solution to about 1 and 0.5 in the first-guess and optimized solutions, respectively. As noted above, the improvement is largely controlled by *S*_{3} because *σ*_{S3} is small and initial *S*_{3} biases are large. The initial temperature errors are somewhat smaller, primarily because values of *σ _{Ti}* are larger throughout the water column in comparison with the initial temperature biases.

The optimized layer thicknesses *WOA01* data, and the vertical distribution of the model layers corresponds well with their intended water masses. For example, the annual-mean mixed layer thickness determined from the *WOA01* data is

Figure 6 shows seasonal variations of the port transports in the optimized solution (top panel) and the resulting changes in the transports from their initial values (bottom panel). The transport curves (top panel) are similar to their counterparts from the initial solution in Fig. 3, but with several notable differences. The most pronounced changes (bottom panel) are the weakening of the annual-mean Karimata Strait transport from 1.4 Sv in the initial solution to 0.3 Sv in the optimized one and the strengthening of the annual-mean Mindoro Strait transport from 1.2 to 1.5 Sv. As a result, the Mindoro Strait outflow dominates the optimized solution, contributing more than 60% to the total annual-mean SCSTF transport of 2.4 Sv through Luzon Strait. On the seasonal scale, the major quantitative change is a noticeable reduction of Karimata outflow in winter, which causes (Fig. 6, bottom panel) a significant decrease in the Luzon Strait transport in December–March (from 6 to 4 Sv).

### b. Transports

The horizontal structure of the SCS circulation in all of our solutions is similar to those from observations and in other modeling studies (see Yu et al. 2007, 2008 for a detailed description). Here, then, we focus on the port transports that are determined by the data assimilation. We note, however, that the Kuroshio loops extensively into the northern SCS in our solutions, forming a prominent anticyclonic gyre there in August–December when the Kuroshio transport falls below 25 Sv. This feature happens because the model Kuroshio is too broad and, hence, too slow (Yu et al. 2007), a consequence of the model’s coarse resolution. On the other hand, it does not significantly impact the port transports determined by the data assimilation, which are primarily constrained by the thermohaline structure throughout the rest of the basin.

#### 1) Kuroshio inflow

It is noteworthy that the Kuroshio inflow transport is modified by the data assimilation at all, given that the adjustment is determined entirely by data within the SCS. To explore the sensitivity of our solutions to the Kuroshio transport, we obtained two additional first-guess solutions in which the initial Kuroshio transport curve (Fig. 3) was uniformly increased and decreased by 7 Sv (about 25% of its annual-mean value). In both tests *D̂*,*α̂ _{k}*,

*β̂*, and most of the parameters listed in Table 1 hardly changed, the exceptions being for some of inflow temperature and salinity adjustments

_{l}*δ*(

*T*

_{1i}) and

*δ*(

*S*

_{1i}) and a 15% (3 Sv) increase of the total Kuroshio transport in the experiment with the reduced initial transport. Since the

*β̂*amplitudes were essentially the same, the adjusted port transports were distributed much as they were in the main run; in particular, the Kuroshio inflow was almost eliminated in layer 4, it was strengthened in layers 1–3, and its overall transport was not much changed.

_{l}The likely reason for a strong constraint imposed by the SCS *T*/*S* data on the Kuroshio structure is that the model layer thicknesses at the inflow port entirely determine the advective SCSTF heat and salt flux (since temperature and salinity at the inflow port are obtained by averaging of the *WOA01* data over the model layers). In support of this idea, in a test first-guess solution with the layer 4 inflow fixed to its initial state (i.e., *M*_{14} = *M**_{14}), layer 4 at the inflow port thickened (380 versus 270 m), layer 3 thinned (260 versus 300 m) and extended over a shallower depth range (120–380 m versus 130–430 m), allowing warmer and saltier layer 3 water to advect into the SCS. Similarly, the transport increase in layers 1–3 (with the layer 4 transport being small) deepens the third layer at the inflow port and hence cools *T**_{3} and freshens *S**_{3}, making the layer 3 properties of the first-guess solution more consistent with the *WOA01* data in the SCS basin.

Finally, we note that the weak layer 4 transport in the first-guess and optimized solutions is consistent with the conclusions of Qu and Lukas (2003) and Yaremchuk and Qu (2004), who diagnosed weak (1–2 cm s^{−1}) currents in this depth range east of Luzon. In addition, hydrographic and current meter data east and south of Taiwan also show that more than 90% of the Kuroshio transport is concentrated in the upper 350 m of the water column (Lee et al. 2001; Gilson and Roemmich 2002).

#### 2) Luzon Strait

The seasonal cycle and vertical distribution of the Luzon Strait transport (LST), computed along 121°E and equal to the SCSTF transport in the optimized solution, are consistent with observational and modeling results with a maximum in winter and a minimum in summer (e.g., Qu 2000; Chu and Li 2000; Yaremchuk and Qu 2004; Fang et al. 2005) and with most inflow occurring in the upper 350 m (e.g., Qu et al. 2000; Tian et al. 2006). Its annual-mean value is 2.4 ± 0.6 Sv. This value is smaller than that in other numerical solutions [an average of 3.5 ± 2.0 Sv reported by Fang et al. (2005)], primarily because of the marked reduction in the Karimata Strait transport after the assimilation. Another possible reason for our smaller LST is that our model does not allow for Luzon Strait flow deeper than 750 m (Qu et al. 2006b), but it is difficult to understand how such a deep inflow can upwell enough to participate significantly to the transport through the shallow SCS outflow straits. Unfortunately, direct observations are not accurate enough to provide a reliable estimate of the annual-mean LST, a consequence of vigorous eddy activity and interannual variability across the Luzon Strait.

The optimized LST has almost no layer 4 (350–700 m) inflow (0.1 Sv) with the Kuroshio inflow transport in layer 4 (0.6 Sv) being almost entirely balanced by outflow east of Taiwan (0.5 Sv). The primary reason for the weak layer 4 inflow is that there are no SCS outflow passages in layer 4; as a result, layer 4 water must first upwell into layer 3 to be able to exit the basin, but the deep mixing processes that could allow for such upwelling are weak in the model. The weak inflow agrees with our previous unassimilated solutions (Yu et al. 2007, 2008). It is also consistent with the observational studies, noted in the previous section, that show the weakness of the Kuroshio in the depth range of layer 4: If the layer 4 transports are weak east of Luzon and Taiwan, they must also be weak in Luzon Strait. There are also observational indications that the annual-mean LST may even change sign below 350 m (Qu et al. 2000; Qu and Lindstrom 2004).

It is noteworthy that an annual-mean LST develops at all in our regional model since in the real ocean its strength is determined remotely by a balance between forcing by Pacific winds and resistance in the outflow straits (section 1). In our model, the controlling influences of *T**_{i} and *S**_{i} within the SCS, particularly of *S**_{3}, are apparently strong enough to recover this remotely driven response.

#### 3) Mindoro Strait

The annual-mean transport through the Mindoro strait in the optimized solution is 1.5 ± 0.4 Sv. Since the Mindoro Strait sill is more than 400 m deep (Fig. 7), it is the only strait with outflow in three model layers. Interestingly, the major part of the annual-mean Mindoro outflow occurs in layer 3 (1.2 Sv), the remaining 0.3 Sv being almost evenly distributed between layers 1 and 2. Our sensitivity analysis (section 3c) indicates that this result is robust with the formal error associated with the layer 3 outflow being 0.3 Sv, a consequence of the constraint that *S̃*_{3} be close to *S**_{3}. Indeed, in an additional test experiment we obtained a first-guess solution keeping the initial Mindoro Strait outflow (*M*_{21}^{m}, *M*_{23}^{m}) intact (i.e., 1.2 Sv with even distribution in the vertical); as a result, the value of *D̂* increased from 1.13 to 3.5, mostly because of salinization of the model layers. It is interesting that similar bottom-intensified distributions for the Mindoro outflow occur in the OfES solution (Fig. 7) and a highly resolved Hybrid Coordinate Ocean Model (HYCOM) simulation (0.0833°).

The cause of the bottom trapping is that Rossby wave adjustments within the SCS require the SCSTF to take the westernmost possible pathway through the basin. Mindoro Strait provides the only pathway through the SCS at thermocline and subthermocline depths (layer 3), whereas most of the shallower waters (0.9 Sv) flow through the Karimata and Taiwan Straits.

#### 4) Karimata Strait

The optimized solution has an annual-mean outflow of 0.3 ± 0.5 Sv through Karimata Strait. As noted above, this annual-mean transport is low in comparison to other estimates (e.g., Fang et al. 2005), which typically vary between 1 and 2 Sv. In the OfES solution, for example, the Karimata transport has a similar temporal behavior, but a considerably larger southward annual-mean transport of 1.1 Sv. Our low annual-mean value is nevertheless a robust feature of the solution, being highly constrained primarily by the requirement that layers 1 and 2 are sufficiently fresh. To confirm our result, we obtained a first-guess solution with the Karimata Strait transports fixed to their initial monthly values *M*_{3}^{m}. The resulting salinization of the SCS was even larger than in the similar Mindoro Strait test experiment (*D̂* increased to 9.5). A stronger drain of surface freshwaters through Karimata Strait caused their faster replacement by saltier waters of Pacific origin; this salinization could not be driven to the observed salinities by adjustment of the freshwater fluxes within their error bars.

The flow through Karimata Strait and over the neighboring shallow shelves (Fig. 1) is determined by a balance between forcing (local wind stress, remotely generated pressure gradient) and dissipation (bottom form drag, horizontal, and vertical friction). As noted above, one *possible* reason for the wide range of values of Karimata transports in directly forced OGCMs is that they are determined to a large degree by their specification of dissipation and representation of the shallow bottom topography in the model (Fig. 8a). Therefore, it is reasonable to assume that the large scatter in Karimata transport estimates is caused by the high sensitivity of model solutions to subgrid parameterizations and bottom topography within the strait, which, unlike the forcing fields, vary considerably from model to model. As an example, *z*-coordinate OGCMs have a lower limit on the ocean depth set by their vertical resolution, and typically it is comparable to the mean depth (11 m) of the shallowest section across the strait—the Karimata “choke point”(Fig. 8b); as a result, the cross-sectional area of the choke point is overestimated, weakening the hydrodynamic drag to the flow and allowing stronger currents. This situation occurs in even very highly resolved models like OfES. The minimum depth in OfES is about 15 m (the separation of 3 grid points at the ocean surface), 4 m deeper than the mean depth at the choke point; as a result, the total cross-sectional area of the strait is 5.8 km^{2}, about 38% larger than that in the 0.016° General Bathymetric Chart of the Oceans (GEBCO) bathymetry (4.2 km^{2}). In addition, the bottom relief across the choke section is smoother and less obstructed by islands in the OfES representation (Fig. 8b). Of course, there is also substantial uncertainty in the Karimata Strait topography in existing bathymetric products, for example, with an 8.4-m rms difference between the GEBCO and 5-minute gridded elevations/bathymetry for the world (ETOPO5) datasets, significant in comparison to the mean depth (24 m) of the area shown in Fig. 8a.

Qualitatively, the lower transport in the optimized solution is reasonable because otherwise maximum current speeds through the strait are large, perhaps unrealistically so. For example, the maximum outflow through Karimata Strait in our initial solution is 3.9 Sv during January (Fig. 3). Given the choke point cross-sectional area of 4.2 km^{2}, this transport corresponds to a uniform speed of 93 cm s^{−1} everywhere across the section shown in Fig. 8b; more realistically, we might expect even larger speeds in the deep part of the choke section with smaller speeds over shallower parts. The maximum transport in the optimized solution is 1.3 Sv in November (Fig. 6), corresponding to a more reasonable, uniform speed of 30 cm s^{−1}.

#### 5) Taiwan Strait

The annual-mean transport through the Taiwan Strait in the optimized solution is 0.6 ± 0.5 Sv. The observed Taiwan Strait transport has never been regularly monitored. On the other hand, the recent observational estimate of 0.86 ± 0.20 Sv by Sen and Chao (2003) is consistent with our computation. Moreover, results from current measurements using bottom-mounted ADCPs across the central Taiwan Strait suggest that there is no persistent northward flow there during winter (Teague et al. 2003; Lin et al. 2005), also in agreement with our solution (Fig. 6). As expected, results from coarse-resolution models differ, but collectively they suggest an annual-mean value of about 0.6–0.8 Sv, similar to our value; for instance, Metzger and Hurlburt (2001) obtained a value of 0.7 Sv in a 0.125° regional layer model, Fang et al. (2005) obtained 0.45 Sv at 0.16° resolution, and Lebedev and Yaremchuk (2000) obtained 0.8 ± 0.4 Sv in their data-controlled diagnostic computation.

In the highly resolved OfES model, however, the annual-mean transport is 1.4 Sv. Again, inspection of the details in bottom topography may provide a possible clue for the larger transport in the OfES solution: The cross-sectional area across the choke point of Taiwan Strait is 60% larger in the OfES representation (9.2 km^{2}) than in the GEBCO bathymetry (5.8 km^{2}), reducing the hydrodynamic drag and allowing a larger transport.

### c. Sensitivities

To study the robustness of the optimized solution, we conducted an adjoint sensitivity analysis in the vicinity of the optimized state. Since the major reduction of the model − data misfit is due to the removal of biases in thermohaline structure, we choose target functionals to be the annual-mean basin-averaged temperature *σ _{qi}*.

Overall, Table 2 shows a remarkable control of the model’s annual-mean thermohaline structure by the port transports: The average of all the sensitivities for the port transports (lines 1–3) is 3.5, almost twice the value of 1.8 owing to surface fluxes (lines 4–6). The optimized Mindoro Strait transport is particularly robust in that *S**_{3} and *T**_{3}.

Among the surface fluxes, the freshwater and heat fluxes have a stronger control over the annual-mean fields than does the local wind stress. It is noteworthy that the variation of the layer 1 and layer 2 salinity fields are more sensitive to the heat flux than is temperature (line 5)—likely because * Q* impacts evaporation strongly. As might be expected, the layer 3 variables (columns 3 and 6) are largely determined by the port transports and to a much lesser extent by the surface fluxes.

Of the model parameters, the detrainment (*H _{d}*,

*t*) and mixed layer (

_{d}*m*,

*n*) parameters have the largest impacts, particularly on mixed layer thickness: Increases in

*H*and

_{d}*t*decrease detrainment into layer 3, allowing

_{d}*m*and convective overturning

*n*act to thicken the mixed layer. As noted above (section 2a),

*m*and

*n*doubled and

*t*halved during the assimilation, thereby allowing more freshwater to be pumped down into layer 3. This result is partially confirmed in Table 2, in which layer 3 salinity is freshened considerably with increases in

_{d}*m*and

*n*, but there is only a weak response to

*t*.

_{d}## 4. Summary and discussion

Our previous numerical solutions showed that the thermohaline structure of the upper SCS is sensitive to the SCSTF (Yu et al. 2007, 2008). In this study, we take advantage of this sensitivity, using an inverse modeling approach to estimate quantitatively the SCSTF transport and structure. Specifically, we use 4dVar to obtain an optimized solution to the 4½ layer model of Yu et al. (2008) that best fits climatological, temperature, salinity, and mixed layer thickness fields from the *WOA01* dataset. Control variables are the transports through the basin inflow and outflow ports, surface heat and freshwater fluxes, and model parameters.

The annual-mean SCSTF (equal to the annual-mean Luzon Strait transport) in our optimized solution is 2.4 ± 0.6 Sv, a low value in comparison to other numerical estimates. It results from the solution’s weak Karimata Strait transport of 0.3 ± 0.5 Sv, in contrast to other estimates of 1–2 Sv. Our result is robust, however, owing to high sensitivity of the model temperature and salinity fields to *could be* overestimated in many OGCMs. The Mindoro Strait provides the only deep (layer 3) passage from the SCS, and half of the model SCSTF exits the basin via this deep passage (1.2 ± 0.3 Sv). This result is robust, required in order to keep layer 3 salinity sufficiently salty (line 1 of Table 2); it also agrees with solutions to OfES and HYCOM. The annual-mean Taiwan Strait transport in our optimized solution is 0.6 ± 0.5 Sv, consistent with available measurements and numerical simulations except for OfES (1.4 Sv), again possibly due to inaccurate bottom topography and drag. The vertical structure of the Kuroshio inflow is sensitive to the data within the SCS: its layer 4 transport is almost eliminated and the transport in layers 1–3 is strengthened. These changes appear to result from the strong dependence of the SCSTF heat and salt content on the baroclinic structure (model layering) of the Kuroshio at the inflow port.

A sensitivity analysis allows us to quantify the impacts of variations in the control variables in the neighborhood of the optimized solution. The analysis confirms that the port transports are at least as important as surface fluxes in shaping the thermohaline structure of the upper SCS. Significant sensitivity is also found with respect to the parameterization of the vertical exchange processes between layer 3 (the NPTW layer) and overlying layers; these processes convey the impacts of surface heat and freshwater fluxes to the subsurface ocean.

To conclude, we obtained a first-order, observationally consistent estimate of the SCSTF transport and structure, which are not well known observationally owing to the lack of long-term direct measurements in the Mindoro, Karimata, and Taiwan Straits. Our analysis also provides the first quantitative appraisal of the relative importance of the processes responsible for shaping the upper-thermohaline structure in the SCS and points toward the importance of the correct representation of the shallow bottom topography and bottom drag in OGCMs. A potential limitation of our present model is that it lacks shallow shelves, which could underestimate mixing processes and freshwater fluxes that impact circulations in the interior basin (see, however, the discussion in section 2a). Another limitation is that our model does not allow for circulations deeper than layer 4 (∼700 m), and so, for example, cannot consider the impacts of the deep overflow of Pacific water through Luzon Strait (Qu et al. 2006b). Finally, the model is not eddy resolving, which will affect the advection of salinity and temperature throughout the basin. It will be interesting to carry out a similar inverse modeling study using an ocean model that overcomes these limitations.

## Acknowledgments

This study was supported by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), by NASA through Grant NNX07AG53G, and by NOAA through Grant NA17RJ1230 through their sponsorship of the research activities at the International Pacific Research Center.

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Values of model parameters adjusted in obtaining the first-guess solution. Column *q*_{in} lists values of parameter *q* used to obtain the initial solution, columns *q*_{min} and *q*_{max} list the minimum and maximum values allowed during the optimization of *q*, and the last column lists the first-guess value. Parameters are collected into three blocks, according to whether they are (top) vertical-mixing parameters, (middle) adjustments to the Kuroshio inflow temperatures and salinities, or (bottom) initial layer thicknesses. During the main assimilation run (section 2c), parameters in the top and middle blocks are kept fixed at their values; those in the bottom block are not used.

Sensitivities of * Q*,

**,and**

τ *); and model parameters (*P

*H*,

_{d}*t*,

_{d}*m*, and

*n*). Sensitivity of each functional

*′*q

_{i}=

q

_{i}/

*σ*in response to 5% perturbations of the transports, surface fluxes, and model parameters. Sensitivities exceeding 5% are boldfaced.

_{qi}