## 1. Introduction

It is generally accepted that eddy diffusivity is crucial to the simulation of general circulation in the ocean, especially in coarse-resolution models that do not explicitly resolve eddy transport processes (e.g., Bauer et al. 1998; Stammer 1998). Such diffusive processes are usually described by the advection–diffusion model, in which a so-called diffusivity coefficient is adopted. The magnitude of diffusivity and how it varies in space and time are still open issues (Marshall et al. 2006). Taylor (1922) related the diffusivity coefficient to the Lagrangian integral scales by assuming a homogeneous and stationary eddy field. Davis (1991) took inhomogeneity into account and described diffusive processes in terms of a diffusivity tensor. To compute the components of the diffusivity tensor using the above two methods categorized as single particle statistics (LaCasce 2008), one requires quasi-Lagrangian current-following observations of oceans.

During the past decades, the increase of Lagrangian observations by satellite-tracked subsurface floats and surface drifters has facilitated research on Eulerian mean currents and Lagrangian statistics over oceans. Related studies have focused on different ocean basins, such as the Atlantic Ocean (Bograd et al. 1999; Krauss and Böning 1987; Lumpkin et al. 2002; Veneziani et al. 2004), the Pacific Ocean (Bauer et al. 2002, 1998; Lumpkin and Flament 2001; Zhurbas and Oh 2003, 2004), and the Southern Ocean (Marshall et al. 2006; Sallée et al. 2008), as well as on marginal seas including the Adriatic Sea (Poulain 2001), Black Sea (Poulain et al. 2005), East Sea (Oh et al. 2000), the Tyrrhenian Sea (Rinaldi et al. 2010), and the South China Sea (Centurioni et al. 2004, 2009).

Several methods concerning the computation of Lagrangian diffusivity have been developed and modified. The most classical method was proposed by Taylor (1922), which, however, is quite sensitive to the asymptotic behavior of the autocorrelation at large time lags (Lumpkin et al. 2002). Griffa et al. (1995) tried to fit the autocorrelation with a known-shape curve prescribed by some parameters, so that the problems with the asymptotic behavior of autocorrelation could be avoided by only estimating the parameters. Davis (1987, 1991) took into account the inhomogeneous velocity field, which is not applicable for Taylor’s theorem, and made the estimation of diffusivity hold true for the shear mean flows (Zhurbas and Oh 2003). To make the approach applicable to inhomogeneous mean flows, the removal of the mean flows is necessary and has also been elaborately designed. Taking spatial variation of the mean flows into account, Bauer et al. (1998) used a bicubic spline interpolation scheme to fit the mean flows and then removed them to minimize the mean shear-induced dispersion. Taking temporal variation of the mean flows into account, Lumpkin (2003) used the Gauss–Markov method to estimate the mean, seasonal, and semiannual cycles of flows and then removed them all to obtain the residual velocity.

The applicability of these theories to different oceans with various environmental conditions has increased our understanding of the dynamical characteristics of the oceans and hence facilitates regional ocean modeling. As the largest semienclosed marginal sea in Southeast Asia, the South China Sea (SCS) plays an important role in governing and modulating regional climate variability. Water mass exchanges with the neighboring Pacific and Indian Oceans may convey the signal of remote forcings to the basin and excite local responses (e.g., Qu et al. 2004). Through regional air–sea interactions, the SCS, as a part of the western Pacific warm pool, could also affect multiscale variabilities of both the atmosphere and ocean (e.g., Alexander et al. 2002; Wang et al. 2009).

Because of its special role in regional climate, the SCS has become the focus of worldwide attention and a number of Lagrangian drifter-based studies over this region have been carried out, targeted at revealing the characteristics of water mass exchanges, surface circulations, mesoscale eddies, etc. Centurioni et al. (2004) investigated the intrusion of Philippine Sea surface water through Luzon Strait using drifter data and pointed out that surface inflow occurs primarily between October and January. Their later work (Centurioni et al. 2009) proposed a new method to estimate the geostrophic, Ekman, and residual components of surface currents over the SCS during the winter monsoon. The Ekman flow estimated by their method is shown to be more downwind and slightly larger than previous estimation. Chow et al. (2008) illustrated the typical features of Dongsha cyclonic eddy based on drifter observations and found that ~90% of the Dongsha cyclonic eddy formed in winter or spring except one that was influenced by a tropical cyclone. Li et al. (2011) made a more systematic study of the eddy features in the northern SCS using drifter data up to 2010.

The above drifter-based studies shed light on the features of both the surface circulations and eddies in the SCS. However, there are still many issues that remain unsolved. For example, although Centurioni et al. (2004, 2009) had computed the Lagrangian time and length scales primarily outside Luzon Strait as well as the mid-to-north SCS, the Lagrangian diffusivity over these regions still remains unknown. Besides, the SCS is strongly influenced by the monsoon wind, showing significant low-frequency variabilities (e.g., annual cycle). Effects of these low-frequency variabilities on the calculation of diffusivity also remain unexplored. Since surface mixing processes are one of the crucial dynamics for regional climate simulation, an investigation of the surface mixing dynamics over the SCS is necessary. Results would provide useful information for optimizing mixing coefficients in regional numerical models and thus improve regional simulations. With the increasing surface observations over the SCS from the Global Drifter Program (GDP) accumulated up to the most recent year of 2011, a more systematic study of the Eulerian and Lagrangian statistics over the SCS is imperative and feasible.

The present study is a preliminary investigation on mixing processes via Eulerian and Lagrangian statistics of the surface flows over the SCS based on satellite-track quasi-Lagrangian surface drifter observations, with the consideration of both the complex geometry of the basin and the seasonal variation of the surface forcing by the monsoon winds (Wyrtki 1961). The rest of the paper is organized as follows. Section 2 describes the data and methods. Section 3 presents the characteristics of the dataset. Sections 4 and 5 show the Eulerian and Lagrangian statistics, respectively. The summary and discussion are given in section 6.

## 2. Data and methods

### a. Drifter dataset

Lagrangian drifting buoys of the Surface Velocity Program (SVP) provide an invaluable tool for investigating near-surface circulations over different ocean basins (e.g., Lumpkin and Pazos 2007; Niiler 2001). The modern dataset of the SVP drifters, including all drifters deployed since 1979 with a holey-sock drogue centered at 15-m depth, is known collectively as the Global Drifter Program (GDP). The raw fixes of drifter positions are inferred from the Doppler shift of their transmission, and then interpolated to uniform 6-h intervals using an optimal procedure known as kriging interpolation (Hansen and Herman 1989; Hansen and Poulain 1996).

The SVP drifter dataset over the world’s oceans is provided by the Atlantic Oceanographic and Meteorological Laboratory/National Oceanic and Atmospheric Administration (http://www.aoml.noaa.gov/phod/dac/dacdata.php), including quality-controlled positions and velocities that are interpolated to uniform 6-h intervals. A subset in which the drifters are located within the domain (0°–27°N, 98°–126°E) covering the entire SCS is extracted for the present study.

### b. Methods

*u*〉

_{i}_{E}, velocity covariance matrix

_{E}is the average in a given bin and in time,

*u*the two components of the drifter velocity, and

_{i}*i*and

*j*take the value 1 or 2 for the zonal and meridional directions, respectively. The MKE and EKE are defined as

*κ*

_{ij}(

*τ*), velocity covariance matrix components

*P*(

_{ij}*τ*), Lagrangian integral time scale

*T*(

_{i}*τ*), and length scale

*L*(

_{i}*τ*), which are defined aswhere 〈 〉

_{L}is the average over time and space computed at time lag

*τ*before/after the particles are located in a given domain, and

*κ*

_{ij}is estimated by

*K*is a constant of about 4 (Davis 1991).

## 3. Characteristics of the drifter dataset over the SCS

All the SVP drifter observations within the region (0°–27°N, 98°–126°E) are selected for studying the statistics of the surface currents over the SCS. Up to 2011, this subset contains a total number of 1215 drifters (identified by different IDs). After removing records that are not within the domain and those with undefined velocities, the subset spans a total of 157.2 drifter years, with the longest drifter spanning 1.1 drifter years.

Figure 1 shows the temporal distribution of the SVP drifter subset. Although the whole SVP dataset started as early as 1979, there are no drifters in the region until 1986 and the number of drifter observations remained below 6000 before 2003 (Fig. 1a). There is a remarkable increase of the drifter observations within the SCS region since 2003, with the largest number of observations (more than 41 000) occurring in 2005. However, no continuous increment is found in the following years and the number remains relatively steady around 15 000. The distribution in each month is also nonuniform (Fig. 1b). The numbers of observations from April to September, on an average of 13 000, are roughly half of those from other months. This may result in a sampling bias so that the Eulerian mean state of surface currents derived from this subset contains more characteristics belonging to autumn and winter seasons (section 4b provides a more detailed discussion).

The trajectories of the drifters in this subset are shown in Fig. 2a. It is found that most of the drifters entered the SCS from the western North Pacific via Luzon Strait. These drifters followed the northern SCS currents, moved westward to the east off Vietnam, and then turned southward along the coast of Vietnam. Some drifters got into the SCS through the Philippine Islands at about 10°N. However, this does not significantly contribute to the observations over the southeastern SCS (Fig. 1b).

Although the present study includes all the SVP drifter observations up to 2011, the spatial distribution of the data is inhomogeneous. From Fig. 2b we can see that regions with heavy drifter sampling are the Kuroshio path, the Luzon Strait, the northern SCS, and the east–southeast coast of Vietnam. The other parts of the SCS are rather poorly sampled. Such an inhomogeneous distribution is mainly due to the lack of targeted experiments in the SCS, which would deploy drifters at regular time and cover the entire basin. Although there were deployments of Lagrangian drifter in the SCS during an international project in the summer of 2005 (Centurioni et al. 2007), the deployment locations are primarily on the west of Luzon Strait and the drifters were mostly influenced by summer monsoon and could hardly reach west of 114°E (see their Fig. 3.4). This deployment cannot help to compensate the data sparseness over the southern and southeastern parts of the SCS. Therefore, the number of drifters deployed within the SCS is still very limited and the distribution of the drifter observations depends highly on the basin-scale circulations in the SCS.

## 4. Eulerian statistics

In this section, the Eulerian statistics over the SCS will be presented, including the mean state of the surface currents, as well as its seasonal variation. A classical way to address Eulerian mean currents is to bin all the drifter velocities into spatial and temporal means. To choose an appropriate bin size, one should take into account at least two concerns. First, a bin size should be large enough so that the mean obtained from the samples within each bin has statistical sense. Second, a bin size should not be too large so that the spatial variation of the mean field is not smoothed out. This is important in the region with sharp velocity gradient or narrow front or jet such as the Kuroshio. In this study, a bin size of 0.5° × 0.5° is chosen to maintain a balance between the above two concerns.

### a. Mean currents

Figure 3a is the pseudo-Eulerian map of the mean currents and MKE derived from all drifter observations from 1986 to 2011, using 0.5° bins. Neglecting those bins that contain no more than 15 observations for a robust estimation, the drifter observations within each bin span a minimum of 0.01 and a maximum of 0.67 drifter years. The energetic Kuroshio and Vietnam coastal current with high MKE values are clearly seen in the mean current map. The intrusion currents of Kuroshio at Luzon Strait, the northern SCS currents, and the Vietnam coastal currents form a semiclosed cyclonic circulation over the entire SCS basin. Another half of the circulation over the southeastern part of the SCS is not clearly revealed due to only a few drifter observations in this region. Such cyclonic circulation over the basin is similar to the wintertime SCS flow pattern (e.g., Hu et al. 2000; Hwang and Chen 2000). This seasonal sampling bias, as well as its spatial distribution, will be discussed in the next subsection.

Figure 3b displays the spatial distribution of the velocity variance ellipses and the EKE. The velocity variance, computed following Morrow et al. (1994), represents the variability caused by small-scale eddies, wind-driven current events, and the seasonal modulation of the surface circulations. Large variance is seen in the regions of energetic currents (e.g., Kuroshio and Vietnam coastal current) as well as Celebes Sea. Values of velocity variance at the coast of Vietnam reach 5000 cm^{2} s^{−2}, corresponding to a root-mean-square speed of ~70 cm s^{−1}. These velocity fluctuations can be as large as the local mean, and are typically oriented in the direction (north–south) of the mean currents (see ellipse orientation in Fig. 3b), indicating the velocity variations are closely related to seasonal reversals of the prevailing currents associated with the East Asia monsoon. Generally, the orientations of both the mean current vectors and the principal axes of velocity variance follow the coast or the bathymetric contour lines owing to topographic constraint. The eccentricity of the velocity variance ellipse ranges from 0.2 to 0.8 in the internal regions to values of larger than 0.9 in the coastal areas (figure not shown). The characteristics over these coastal areas (eccentricity > 0.9) show significant anisotropy.

The EKE is expected to be the largest around the Kuroshio path. However, Fig. 2b shows that the largest EKE values occur at the coastal region of Vietnam, the northeastern part of Celebes Sea, and the Luzon Strait. Although the pattern of EKE is similar to that estimated based on altimetry data (see Fig. 3 from Cheng and Qi 2010), the magnitude near the coast of Vietnam is much larger (~4000 cm^{−2} s^{−2}) than that estimated by altimetry data (~1400 cm^{−2} s^{−2}). The reason may be that the EKE estimated by drifters contains not only the mesoscale variability and turbulent energy, but also low-frequency variability such as the seasonal cycle. The altimetry data of a 7-day resolution may smooth out the high-frequency variabilities. Besides, the altimetry data over the coastal regions are problematic (Roblou et al. 2011).

### b. Seasonal variation

Figure 4 shows the mean amplitudes and phases in 0.5° bins. Small values of amplitudes (<0.4) are found in the regions east off the Philippines, the Luzon Strait, and northern SCS (Fig. 4a), indicating that these regions do not have a significant seasonal sampling bias. In contrast, the values of amplitudes are close to one in the regions around the western boundary of the SCS as well as the internal regions of the SCS, indicating a significant bias over these regions. The corresponding phases over these regions range from 330 to 360 and 0 to 60 (Fig. 4b), implying that the drifter observations are primarily distributed in winter.

Since the SCS region is significantly influenced by the monsoon with strong seasonal cycle (e.g., Wyrtki 1961; Shaw and Chao 1994), the sampling bias can be seen more clearly through averaging the drifter observations over different seasons. Each seasonal mean circulation with the number of observations is shown in Fig. 5. There is a striking contrast in the number of observations between summer (Fig. 5b) and winter (Fig. 5d) over the southeast coastal region of Vietnam. In winter (Fig. 5d), more drifters coming from the western North Pacific tend to intrude into the SCS through Luzon Strait, leading to a large number of observations over the northern SCS. Following the surface currents driven by the winter monsoon, these drifters are advected to the western boundary and then to the southwest of the SCS along the coastal region of Vietnam. In summer (Fig. 5b), however, because of the reversal of monsoon winds, the coastal currents east off Vietnam reverse their direction so that those drifters intruding from the Luzon Strait only reach to the east off Hainan Island. If the Luzon Strait is regarded as a source of deployed drifters, then the mean circulation over the entire SCS in winter (summer) acts as a transporter (blocker) to those drifters, leading to the seasonal sampling bias toward winter (Fig. 1b). Despite the low number of observations in summer, significant seasonal differences of the mean currents between winter and summer can be found over the northern SCS regions, with strong southwestward currents in winter (Fig. 5d) and weak northeastward currents in summer (Fig. 5b). For spring and autumn (i.e., the transitional seasons of monsoon), the numbers of the observations are slightly increasing over the regions south off the Hainan Island and east off Vietnam, and the differences of the mean currents regarding to the strength and the direction are also found over these regions (Figs. 5a,c). Nevertheless, although the seasonal differences of the mean currents are obvious due to the reversal of monsoon wind, the overall mean of all seasons (Fig. 3a) shows more characteristics of winter circulation, especially over the coastal regions of Vietnam, which could be attributed to both the strength of winter circulation and the sampling bias toward winter.

## 5. Lagrangian statistics

Lagrangian statistics include integral time and length scales, which are closely related to the important parameter, that is, diffusivity. Since diffusion is caused by the spreading of particles following flows, diffusivity is most efficiently described in Lagrangian framework (Hanna 1981). The Lagrangian integral time scale is a typical scale over which the Lagrangian velocities are correlated and represents a “memory” scale following the particles (LaCasce 2008). That is, within such a time scale, a particle “remembers” its past motion state by showing a strong autocorrelation. After that time scale, the particle “forgets” the past state, decorrelates with itself, and thus becomes statistically independent. Correspondingly, the Lagrangian length scale is defined as the distance a particle travels within the time scale at the characteristic speed. As the basic indicators of Lagrangian predictability, these scales are important parameters for identifying the dynamic nature of diffusive process. For example, in a short-time period (much shorter than the time scale), the diffusion is in a ballistic regime; in a long-time period (much longer than the time scale), the diffusion becomes a random-walk regime (e.g., Lumpkin and Flament 2001).

According to Taylor’s (1922) classical theory, diffusivity can be related to the mean square of velocity fluctuation and the Lagrangian integral time scale by assuming that the eddy field is statistically homogeneous and stationary. Davis’ (1987, 1991) theory assumes a less stringent condition compared to that of Taylor (1922), that is, the probability density function (PDF) of residual velocity should be approximately Gaussian. Therefore, it is necessary to explore the PDF of residual velocity observed by Lagrangian drifters first.

### a. The PDF of residual velocity

The residual velocities are calculated by the following procedure. First, all 6-h velocity observations are smoothed using 9-point (~2 day) smoothing approach. This procedure not only suppresses the energy aliasing of high-frequency inertial oscillation into low-frequency motions (Swenson and Niiler 1996), but also reduces the possible artificial errors in drifter positioning (Bracco et al. 2000). Second, these smoothed velocities are grouped into bins. Bins with fewer than 20 velocity observations are discarded. The mean and standard deviation of each bin are derived accordingly. At last, all velocities are demeaned by subtracting local means and then normalized by dividing corresponding standard deviations. This approach, as summarized by LaCasce (2008), could increase the accuracy of the PDFs. To investigate the influence of different bin sizes on the results, PDFs are computed using the bin size of 0.25°, 0.5°, and 1°, respectively.

*σ*and

*μ*are obtained by a maximum likelihood estimate of zonal or meridional residual velocities. At a first glance, the PDFs of both zonal (left column of Fig. 6) and meridional (right column of Fig. 6) residual velocities follow the Gaussian distribution better than the exponential distribution, regardless of the bin sizes (different rows of Fig. 6). The central cores of all PDFs are as smooth as the Gaussian distribution. The extended tails of zonal and meridional velocities (except for negative zonal velocities) deviate slightly from the Gaussian distribution to the exponential distribution. The kurtoses (the fourth-order moments) computed for all PDFs are larger than that of Gaussian distribution (i.e., a value of 3). In addition, all skewnesses are larger than zero, indicating a positive bias of all PDFs compared to the Gaussian distribution. The Kolmogorov–Smirnov (K–S) test (Press et al. 1992) is adopted here to find whether these PDFs, both zonal and meridional computed using three different bin sizes, are Gaussian. The degree of freedom used for the K–S test is computed as half of the total length of the samples divided by the typical time scale (Riser and Rossby 1983), which is taken constantly as 2 days here. The corresponding K–S probabilities are 0.04, 7.6 × 10

^{−4}, and 1.3 × 10

^{−4}for the zonal velocities in 0.25°, 0.5°, and 1° bins, and 0.13, 1.0 × 10

^{−3}, and 1.1 × 10

^{−4}for the meridional velocities in 0.25°, 0.5°, and 1° bins. These probabilities mean that the chance by which the K–S statistics would be at least as large as those observed is quite small. Such small values indicate that the PDFs are significantly different from Gaussian at a 95% confidence level.

It can be also seen from the K–S probabilities, kurtoses, and skewnesses that the PDFs of meridional velocities are closer to Gaussian distribution than those of the zonal components regardless of the bin sizes, although the negative tails of zonal velocities are much closer to Gaussian distribution than those of the meridional ones. This is different from other regions such as the surface over the California Current (Swenson and Niiler 1996) and subsurface in the Newfoundland Basin (Zhang et al. 2001).

Different bin sizes exert a certain influence on the results. For 0.25° bins, the K–S probabilities are much larger than those of larger bins. This is partially due to the relatively smaller degree of freedom derived from fewer samples (i.e., 212 138) for the 0.25° bin, since bins with fewer than 20 velocity observations are discarded. However, even using all the samples without discarding any bins so that different bin sizes have the same number of samples, smaller bins would have larger K–S probabilities (figure not shown). This result is similar to that of Swenson and Niiler (1996), which confirms their conclusion that the errors in accounting for the spatial variations of the mean fields (larger bins with larger errors) tend to deviate the PDFs from Gaussianity.

As noted earlier, the advective–diffusive formalism assumes that the Lagrangian velocities are normally distributed, or less stringently, quasi-normally distributed. Although the PDFs of residual velocities in the SCS are not exactly Gaussian, their kurtoses (~4) are only somewhat larger than that (3) of Gaussian distribution, and the differences are small enough so as not to warrant concern (LaCasce 2008). Therefore, the PDFs of residual velocities are considered to follow quasi-Gaussian distribution so that Davis’s diffusive theory (Davis 1987, 1991) is applicable to the SCS.

### b. Lagrangian scales and diffusivity

Lagrangian statistics are computed using Eqs. (3)–(6). It is interesting to investigate the characteristics of the diffusivity in different regions associated with different circulations or water mass in the SCS. Considering the number of drifter observations, we focus on the following six subregions (see black boxes in Fig. 2a): region 1 (19°–21.5°N, 121.5°–124°E), region 2 (21°–23°N, 117°–120°E), region 3 (18°–20.5°N, 117°–120°E), region 4 (17.5°–20.5°N, 113°–116°E), region 5 (16°–19°N, 109°–112.5°E), and region 6 (6.5°–10.5°N, 105°–109.5°E). These regions are associated with the Kuroshio (region 1), the intrusion of the Kuroshio through Luzon Strait (regions 2 and 3), the northern SCS currents (regions 4 and 5), and the fresh water plume from the Mekong River (region 6), whose characteristics are expected to be different. More importantly, the Lagrangian observations are relatively dense within these regions. Figure 7 shows the number of drifter observations as a function of time lag. Except for region 6, all other regions contain up to ~10 000 observations (approximately 2500 buoy days) at day 0. As the time lag increases negatively, the numbers of observations in each region only decrease slightly. Even at day −15, all the numbers of observations are above ~4000 (~1000 buoy days), which are sufficient to obtain statistical estimates.

For the estimations of Lagrangian statistics over these six regions, a maximum time lag of 15 days is chosen. According to previous studies (e.g., Lumpkin et al. 2002; Marshall et al. 2006; Zhurbas and Oh 2003), the typical time scale of surface flows over different oceans is usually below 6 days, especially over energetic regions such as western boundary currents. Therefore, a maximum time lag of 15 days would be long enough to find the asymptotic features of Lagrangian statistics.

*T*and

*T*are the total extent of observations and the Lagrangian integral time scale (in buoy days), respectively,

_{L}*N** the independent observations or degree of freedom (Emery and Thomson 2001), and

*σ*the standard deviation of the velocity. Results suggest that all estimated mean velocities over different regions are robust since the confidence intervals are relatively small compared to their means. This is primarily due to the large numbers of observations within these regions.

Mean velocities and the 95% confidence interval (cm s^{−1}) based on all drifter observations within the six subregions shown in Fig. 2a.

The diagonal (symmetric) components of the diffusivity tensor (*κ*_{11} and *κ*_{22}) are related to the dispersive effects of the unresolved flows. We first compute these two components using Eq. (3), in which the residual velocity is obtained by removing the 0.5°-bin mean, and the results are shown in Fig. 9. At smaller time lags (<3 days), diffusivity tends to grow with time in a quadratic manner. At larger time lags, however, the zonal diffusivities in regions 2–6 and the meridional ones in regions 5–6 increase quasi linearly without approaching any steady values. As documented by previous studies (e.g., Lumpkin and Flament 2001; Oh et al. 2000), such behavior could be attributed to the shear of the mean flow (i.e., shear dispersion). It is thus speculated that 0.5°-bin mean does not resolve sufficiently the shear of the mean flow so that the shear effect retains in the residual velocity and leads to the increase of the diffusivities. However, very similar results are shown when finer bin size of 0.25° is used (figure not shown), although a finer bin slightly reduces the slope of diffusivity at larger time lags. Therefore, the shear dispersion may not be the major factor that causes the quasi-linear increases of the diffusivities shown in Fig. 9.

Another factor which may affect the estimation of diffusivity is the strong seasonal variation over the SCS (Fig. 5) owing to the influence of the monsoon winds. Although the mean of each bin is subtracted to obtain the residual velocity, low-frequency temporal variabilities such as strong seasonal cycle are still retained in the residual velocity, and thus affect the estimation of diffusivity. Taking this into account, we compute the residual velocity by employing the Gauss–Markov (GM) decomposition (Lumpkin 2003; Lumpkin and Garraffo 2005). A time-invariant mean, an annual cycle, and a semiannual cycle are estimated in 0.5° bins by the GM method and then removed to obtain the residual velocity.

Figure 10 shows the diffusivities computed by incorporating the GM decomposition. Compared to the results shown in Fig. 9, the new estimations of diffusivities are much smoother at larger time lags (asymptotic behaviors). It is noteworthy that the curves of region 1 obtained by the two methods do not show much difference in shape. This is because the amplitude of the seasonal variation in region 1 is not as large as those in other regions (Fig. 5) so that the removal of the annual/semiannual cycles takes little effect on the shape of the curves. Changes in regions 2–6 are obvious since the seasonal variability of the surface currents over these regions is significant.

According to Davis (1991), in regions where diffusivity reaches a stable value quickly, a traditional advection–diffusion model with diffusive coefficient (i.e., flux versus gradient relationship) will be adequate to describe the evolution of large-scale passive tracer fields. Strictly speaking, such behavior is only found in the regions 4 and 5 (Fig. 10). Behaviors in regions 1, 2, and 6 are very similar to the asymptotic behavior but also show meandering features. According to Swenson and Niiler (1996), regions where diffusivity levels out for short lags and then meanders for larger lags could be attributed to the presence of the coast that introduces spatial inhomogeneity. Thus diffusivities in regions 1, 2, and 6 may be strongly affected by the coast. The zonal and meridional components of diffusivity in all regions except region 5 are distinct from each other, showing that significant anisotropy and inhomogeneity of diffusivity are the major features over the SCS. The curves of zonal and meridional components of diffusivity in region 3 overlap on each other at smaller time lags but bifurcate after 3 days. The zonal component increases linearly while the meridional one decreases linearly. It is speculated that this behavior could be attributed to the long-living “terminal” eddy (Swenson and Niiler 1996), that is, the Luzon cold eddy over this region. Studies (e.g., Qu 2000) have shown that the Luzon cold eddy first appears in November, reaches maximum strength in January–February, and decays after May–June. Although other regions (e.g., region 2) may also be eddy-active areas, the lifetime of eddies in these regions is much shorter (e.g., about ~40–150 days in region 2; Xiu et al. 2010) than that of the Luzon cold eddy (~8 months). Therefore, the diffusivity in region 3 may be affected by the long-living Luzon cold eddy and apparently different from those in other regions.

Tables 2–4 present the Lagrangian statistics over the six regions. The asymptotic diffusivities, as well as the Lagrangian scales, are estimated as the maximum values within 15 days. The diffusivities in all regions except for region 5, which is located southeast off Hainan Island, show significant anisotropy. The zonal diffusivities in regions 3 and 4 are much larger than the meridional ones while those in regions 1 and 6 are opposite. This feature may be attributed to the constraint of coastal boundary, which prohibits particles from diffusing in the direction perpendicular to the coastal line. The study by Oh et al. (2000) shows that the typical Lagrangian time scale and length scale over the western North Pacific are ~1.7–3.7 days and 18–62 km. Compared to their results, the time scales (Table 3) and length scales (Table 4) estimated over the selected regions in SCS are slightly smaller. The estimated maximum time scale is just above 2 days whereas the maximum length scale is 44.4 km.

Estimates of asymptotic diffusivity (10^{7} cm^{2} s^{−1}) for the six regions shown in Fig. 2a.

As in Table 2, but for time scale (day).

As in Table 2, but for length scale (km).

The asymmetric part of the diffusivity tensor (*κ*_{12} and *κ*_{21}), which describes the tendency of particles to veer (Davis 1991), is usually investigated in terms of mean angular momentum *M*(*τ*) in regions 3–5 are negative at small time lags but become positive at large time lags, indicating that these regions are dominated by anticyclonic eddies at small scales but by cyclonic circulations (e.g., mean flow) at large scales. These results are consistent with previous eddy-related studies over the SCS (e.g., Li et al. 2011; Wang et al. 2003; Xiu et al. 2010). Using satellite altimeter data, Wang et al. (2003) showed that the southwest of Taiwan (roughly region 2) is dominated by anticyclonic eddies (see their Fig. 1), whereas many cyclonic eddies clustered at the northwest of Luzon (roughly region 3). Xiu et al. (2010) found similar results in their modeled SCS environment (see their Fig. 11). Li et al. (2011) illustrated statistical features of eddies also using Lagrangian drifter data. Their results show (see their Fig. 6), in a more consistent sense that anticyclonic eddies appear to be small while cyclonic eddies are much larger in the northwest of Luzon (region 3) as well as to the south of the exit region of the Pearl River (roughly region 4). The southeast off Hainan (roughly region 5) is dominated by large cyclonic eddies. The consistency with the previous studied concentrated on the SCS eddies implies that the computation for the asymmetric components of diffusivity tensor, as an independent approach from those eddy-identification-based methods, provides useful insights of eddy features.

## 6. Summary

Eulerian and Lagrangian statistics of the surface flows over the SCS basin are investigated based on the observations of satellite-tracked quasi-Lagrangian drifters of Surface Velocity Program. Although the dataset used in this study spans a time period from 1986 to 2011, the spatial and temporal distributions of the data are quite inhomogeneous. With the lack of specific Lagrangian experiments in the SCS, drifters over the SCS come primarily from the western North Pacific through Luzon Strait, which makes Luzon Strait act like a deploying location of drifters. Regions around Luzon Strait as well as the northern and western parts of the SCS are therefore heavily sampled by drifter observations whereas the internal SCS is less sampled.

Eulerian mean currents derived from all available drifters clearly show a semiclosed cyclonic circulation over the SCS basin, which consists of the intrusion of Kuroshio at Luzon Strait, the east–west-oriented currents at the northern SCS, and the western boundary currents down to Karimata Strait. Another half of the circulation over the southeast part of the basin is not clear because of the lack of drifter observations. Velocity variance ellipses orient in the direction of the mean flow and show obvious anisotropy over the entire SCS basin where the intrusion region of the Luzon Strait and the coastal regions of Vietnam are most energetic.

The western parts of the SCS are not well sampled in summer because drifters coming from Luzon Strait usually do not reach the western parts of the SCS because of the local northeast-flowing currents driven by the southwest monsoon winds in summer. In contrast, drifters could be easily advected down to the southwest of the SCS by wintertime currents driven by the northeast monsoon winds. The reversal of the monsoon winds leads to a bias of observation density toward the winter season, and therefore makes the derived Eulerian mean circulation resemble the one in the wintertime.

Although the probability density functions of the residual velocities derived from the surface drifters over the SCS follow the Gaussian distribution better than the exponential distribution, they are not Gaussian in terms of the stringent K–S test. Considering that their kurtoses are not too large, however, they can still be treated as quasi Gaussian, and thus Davis’ (1991) theory can be adopted to compute the Lagrangian statistics over different regions in the SCS.

Using the residual velocity obtained by removing the bin mean, the diffusivities tend to grow with time and do not approach any steady values. It is found out that the low-frequency variabilities such as annual/semiannual cycles may lead to significant bias in the estimation of diffusivity. Therefore, it is necessary to remove these signals before calculating the diffusivity. By incorporating the GM method for the removal of these low-frequency variabilities, the typical values of diffusivity, time scale, and length scale in the regions around the Luzon Strait (i.e., regions 1–3) are 3.7–11.9 × 10^{7} cm^{2} s^{−1}, 0.8–1.9 days, and 16.4–44.4 km, respectively (Tables 2–4). In the regions around Hainan Island and the western boundary of the SCS, the typical estimated diffusivity and length scale are slightly smaller, that is, 2.3–6.9 × 10^{7} cm^{2} s^{−1} and 12.8–33.7 km, while the time scale (0.8–2.0 days) are approximately the same as that in the regions around the Luzon Strait. The “flux versus gradient” relationship is likely to be held in these regions, except region 3, which is located at southwest of the Luzon Strait. Whether the relationship is applicable to the internal regions of the SCS still remains open owing to the sparse drifter observations there.

The angular momentum analysis for the asymmetric parts of the diffusivity tensor (*κ*_{12} and *κ*_{21}) indicates that, in some regions (including the east and the northwest of the Luzon Strait and the exit region of Mekong River), the flows are dominated by anticyclonic eddies, while in other regions, such as the northern SCS, the flows are characterized by anticyclonic eddies at small scales but by cyclonic circulations at large scales. These results are consistent with previous eddy studies (e.g., Li et al. 2011; Wang et al. 2003; Xiu et al. 2010), indicating that the computation of asymmetric parts of diffusivity tensor could be a valuable tool in accessing eddy features.

It is important to note that the estimation of the statistics in the present study, especially the diffusivity, cannot be directly applied to numerical models. The diffusivity estimated here is based on Lagrangian observations (following the fluid particles freely) and it is thus called Lagrangian diffusivity. The ocean models, usually constructed on fixed grids, are based on the Eulerian reference frame (at fixed positions). Different reference frameworks lead to different concepts of these statistics as well as their magnitudes. As summarized by LaCasce (2008), a number of studies (e.g., Corrsin 1959; Davis 1982, 1983; Hanna 1981; Middleton 1985) have already focused on the Eulerian–Lagrangian relationship and their mutual transformations. Therefore, for the purpose of practice one needs to seek the relationship through which the Lagrangian statistics could be changed into Eulerian statistics suitable for model parameterizations. Middleton (1985) derived simple relations under some assumptions, which may be a good start but still needs to be checked within the SCS.

The present study discusses the annual cycle in detail. Other low-frequency variabilities such as interannual signal may also exert an influence on the statistics. However, it is not easy to define the interannual signal and remove it from the time series (based on drifter observations) within a bin. This is because the time series of a bin are not equally sampled in each year, that is, some years are heavily sampled while some are not. Figure 1a shows a significant interannual sampling bias that would certainly distort the interannual signal and thus make the removal of interannual signal difficult. That is probably why Bauer et al. (2002) ignored the interannual variability even though their target was the tropical Pacific region. A large number of drifter observations without significant seasonal/interannual sampling bias would facilitate a future study concerning this topic.

To make a complete understanding of the dispersion statistics of the SCS, including the internal deep sea, specific Lagrangian experiments should be designed and carried out. Drifters should be deployed 1) during summer to reduce the seasonal sampling bias and 2) regularly from year to year to reduce the interannual sampling bias. The locations of deployment should be also carefully chosen so that drifters will follow the southwest summer monsoon and sample the summertime surface currents from south to north, such as deploying at Karimata Strait. In addition, the internal regions as well as the southeast part of the basin also need deployment owing to the sparse observations in all seasons.

This work was jointly supported by the MOST of China (Grant 2011CB403505), Chinese Academy of Sciences through the Project KZCX2-EW-208, the National Natural Science Foundation of China (Grant 41076009), and the Hundred Talent Program of the Chinese Academy of Sciences. Thanks to Rick Lumpkin from NOAA for providing his code of GM decomposition. Constructive comments and suggestions from the two anonymous reviewers are also gratefully acknowledged.

## REFERENCES

Alexander, M. A., , I. Bladé, , M. Newman, , J. R. Lanzante, , N.-C. Lau, , and J. D. Scott, 2002: The atmospheric bridge: The influence of ENSO teleconnections on air–sea interaction over the global oceans.

,*J. Climate***15**, 2205–2231.Bauer, S., , M. S. Swenson, , A. Griffa, , A. J. Mariano, , and K. Owens, 1998: Eddy-mean float decomposition and eddy-diffusivity estimates in the tropical Pacific Ocean. 1. Methodology.

,*J. Geophys. Res.***103**(C13), 30 855–30 871.Bauer, S., , M. S. Swenson, , and A. Griffa, 2002: Eddy-mean float decomposition and eddy-diffusivity estimates in the tropical Pacific Ocean. 2. Results.

,*J. Geophys. Res.***107**, 3154, doi:10.1029/2000JC000613.Bograd, S. J., , R. E. Thomson, , A. B. Rabinovich, , and P. H. LeBlond, 1999: Near-surface circulation of the northeast Pacific Ocean derived from WOCE-SVP satellite-tracked drifters.

,*Deep-Sea Res. II***46**, 2371–2403.Bracco, A., , J. H. LaCasce, , and A. Provenzale, 2000: Velocity probability density functions for oceanic floats.

,*J. Phys. Oceanogr.***30**, 461–474.Centurioni, L. R., , P. P. Niiler, , and D.-K. Lee, 2004: Observations of inflow of Philippine Sea surface water into the South China Sea through the Luzon Strait.

,*J. Phys. Oceanogr.***34**, 113–121.Centurioni, L. R., , P. P. Niiler, , Y. Y. Kim, , D.-K. Lee, , and V. A. Sheremet, 2007: Near-surface dispersion of particles in the South China Sea.

*Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics,*A. Griffa et al., Eds., Cambridge University Press, 73–75.Centurioni, L. R., , P. P. Niiler, , and D.-K. Lee, 2009: Near-surface circulation in the South China Sea during the winter monsoon.

,*Geophys. Res. Lett.***36**, L06605, 10.1029/2008GL037076.Cheng, X., , and Y. Qi, 2010: Variations of eddy kinetic energy in the South China Sea.

,*J. Oceanogr.***66**, 85–94.Chow, C.-H., , J.-H. Hu, , L. R. Centurioni, , and P. P. Niiler, 2008: Mesoscale Dongsha cyclonic eddy in the northern South China Sea by drifter and satellite observations.

,*J. Geophys. Res.***113**, C04018, doi:10.1029/2007JC004542.Corrsin, S., 1959: Progress report on some turbulent diffusion research.

Vol. 6, Academic Press, 161–164.*Advances in Geophysics,*Davis, R. E., 1982: On relating Eulerian and Lagrangian velocity statistics: Single particles in homogeneous flows.

,*J. Fluid Mech.***114**, 1–26.Davis, R. E., 1983: Oceanic property transport, Lagrangian particle statistics, and their prediction.

,*J. Mar. Res.***41**, 163–194.Davis, R. E., 1987: Modeling eddy transport of passive tracers.

,*J. Mar. Res.***45**, 635–666.Davis, R. E., 1991: Observing the general circulation with floats.

,*Deep-Sea Res. I***38**, S531–S571.Emery, W. J., , and R. E. Thomson, 2001:

*Data Analysis Methods in Physical Oceanography.*2nd ed. Elsevier, 654 pp.Griffa, A., , K. Owens, , L. Piterbarg, , and B. Rozovskii, 1995: Estimates of turbulence parameters from Lagrangian data using a stochastic particle model.

,*J. Mar. Res.***53**, 371–401.Hanna, S. R., 1981: Lagrangian and Eulerian time-scale relations in the daytime boundary layer.

,*J. Appl. Meteor.***20**, 242–249.Hansen, D. V., , and A. Herman, 1989: Temporal sampling requirements for surface drifting buoys in the tropical Pacific.

,*J. Atmos. Oceanic Technol.***6**, 599–607.Hansen, D. V., , and P.-M. Poulain, 1996: Quality control and interpolations of WOCE-TOGA drifter data.

,*J. Atmos. Oceanic Technol.***13**, 900–909.Hu, J., , H. Kawamura, , H. Hong, , and Y. Qi, 2000: A review on the currents in the South China Sea: Seasonal circulation, South China Sea warm current and Kuroshio intrusion.

,*J. Oceanogr.***56**, 607–624.Hwang, C., , and S.-A. Chen, 2000: Circulations and eddies over the South China Sea derived from TOPEX/Poseidon altimetry.

,*J. Geophys. Res.***105**(C10) 23 943–23 965.Krauss, W., , and C. W. Böning, 1987: Lagrangian properties of eddy fields in the northern North Atlantic as deduced from satellite-tracked buoys.

,*J. Mar. Res.***45**, 259–291.LaCasce, J. H., 2008: Statistics from Lagrangian observations.

,*Prog. Oceanogr.***77**, 1–29.Li, J., , R. Zhang, , and B. Jin, 2011: Eddy characteristics in the northern South China Sea as inferred from Lagrangian drifter data.

,*Ocean Sci.***7**, 661–669.Lumpkin, R., 2003: Decomposition of surface drifter observations in the Atlantic Ocean.

,*Geophys. Res. Lett.***30**, 1753.Lumpkin, R., , and P. Flament, 2001: Lagrangian statistics in the central North Pacific.

,*J. Mar. Syst.***29**, 141–155.Lumpkin, R., , and Z. D. Garraffo, 2005: Evaluating the decomposition of tropical Atlantic drifter observations.

,*J. Atmos. Oceanic Technol.***22**, 1403–1415.Lumpkin, R., , and M. Pazos, 2007: Measuring surface currents with Surface Velocity Program drifters: The instrument, its data, and some recent results.

*Lagrangian Analysis and Predication of Coastal and Ocean Dynamics,*A. Griffa et al., Eds., Cambridge University Press, 39–67.Lumpkin, R., , A.-M. Treguier, , and K. Speer, 2002: Lagrangian eddy scales in the northern Atlantic Ocean.

,*J. Phys. Oceanogr.***32**, 2425–2440.Marshall, J., , E. Shuckburgh, , H. Jones, , and C. Hill, 2006: Estimates and implications of surface eddy diffusivity in the Southern Ocean derived from tracer transport.

,*J. Phys. Oceanogr.***36**, 1806–1821.Middleton, J. F., 1985: Drifter spectra and diffusivities.

,*J. Mar. Res.***43**, 37–55.Morrow, R., , R. Coleman, , J. Church, , and D. B. Chelton, 1994: Surface eddy momentum flux and velocity variances in the Southern Ocean from

*Geosat*altimetry.,*J. Phys. Oceanogr.***24**, 2050–2071.Niiler, P. P., 2001: The world ocean surface circulation.

*Ocean Circulation and Climate: Observing and Modelling the Global Ocean,*G. Siedler, J. Church, and J. Gould, Eds., Academic Press, 193–204.Oh, I. S., , V. Zhurbas, , and W. Park, 2000: Estimating horizontal diffusivity in the East Sea (Sea of Japan) and the northwest Pacific from satellite-tracked drifter data.

,*J. Geophys. Res.***105**(C3), 6483–6492.Poulain, P.-M., 2001: Adriatic sea surface circulation as derived from drifter data between 1990 and 1999.

,*J. Mar. Syst.***29**, 3–32.Poulain, P.-M., , R. Barbanti, , S. Motyzhev, , and A. Zatsepin, 2005: Statistical description of the Black Sea near-surface circulation using drifters in 1999-2003.

,*Deep-Sea Res. I***52**, 2250–2274.Press, W. H., , S. A. Teukolsky, , W. T. Vetterling, , and B. P. Flannery, 1992:

*Numerical Recipes in Fortran 77: The Art of Scientific Computing.*2nd ed. Cambridge University Press, 933 pp.Qu, T., 2000: Upper-layer circulation in the South China Sea.

,*J. Phys. Oceanogr.***30**, 1450–1460.Qu, T., , Y. Y. Kim, , M. Yaremchuk, , and T. Tozuka, 2004: Can Luzon Strait transport play a role in conveying the impact of ENSO to the South China Sea?

,*J. Climate***17**, 3644–3657.Rinaldi, E., , B. B. Nardelli, , E. Zambianchi, , R. Santoleri, , and P.-M. Poulain, 2010: Lagrangian and Eulerian observations of the surface circulation in the Tyrrhenian Sea.

,*J. Geophys. Res.***115**, C04024, doi:10.1029/2009JC005535.Riser, S. C., , and H. T. Rossby, 1983: Quasi-Lagrangian structure and variability of the subtropical western North Atlantic circulation.

,*J. Mar. Res.***41**, 127–162.Roblou, L., and Coauthors, 2011: Post-processing altimeter data towards coastal applications and integration into coastal models.

*Coastal Altimetry,*S. Vignudelli et al., Eds., Springer-Verlag, 217–246.Sallée, J. B., , K. Speer, , R. Morrow, , and R. Lumpkin, 2008: An estimate of Lagrangian eddy statistics and diffusion in the mixed layer of the Southern Ocean.

,*J. Mar. Res.***66**, 441–463.Shaw, P.-T., , and S.-Y. Chao, 1994: Surface circulation in the South China Sea.

,*Deep-Sea Res. I***41**, 1663–1683.Stammer, D., 1998: On eddy characteristics, eddy transports, and mean float properties.

,*J. Phys. Oceanogr.***28**, 727–739.Swenson, M. S., , and P. P. Niiler, 1996: Statistical analysis of the surface circulation of the California Current.

,*J. Geophys. Res.***101**, 22 631–22 645.Taylor, G. I., 1922: Diffusion by continuous movements.

,*Proc. London Math. Soc.***s2-20**, 196–212.Veneziani, M., , A. Griffa, , A. M. Reynolds, , and A. J. Mariano, 2004: Oceanic turbulence and stochastic models from subsurface Lagrangian data for the northwest Atlantic Ocean.

,*J. Phys. Oceanogr.***34**, 1884–1906.Wang, B., , F. Huang, , Z. Wu, , J. Yang, , X. Fu, , and K. Kikuchi, 2009: Multiscale climate variability of the South China Sea monsoon: A review.

,*Dyn. Atmos. Oceans***47**, 15–37.Wang, G., , J. Su, , and P. C. Chu, 2003: Mesoscale eddies in the South China Sea observed with altimeter data.

,*Geophys. Res. Lett.***30**, 2121, doi:10.1029/2003GL018532.Wyrtki, K., 1961:

*Physical Oceanography of the Southeast Asian Waters.*Scripps Institution of Oceanography, 195 pp.Xiu, P., , F. Chai, , L. Shi, , H. Xue, , and Y. Chao, 2010: A census of eddy activities in the South China Sea during 1993-2007.

,*J. Geophys. Res.***115**, C03012, doi:10.1029/2009JC005657.Zhang, H.-M., , M. D. Prater, , and T. Rossby, 2001: Isopycnal Lagrangian statistics from the North Atlantic current RAFOS float observations.

,*J. Geophys. Res.***106**(C7), 13 817–13 836.Zhurbas, V., , and I. S. Oh, 2003: Lateral diffusivity and Lagrangian scales in the Pacific Ocean as derived from drifter data.

,*J. Geophys. Res.***108**, 3141, doi:10.1029/2002JC001596.Zhurbas, V., , and I. S. Oh, 2004: Drifter-derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns.

,*J. Geophys. Res.***109**, C05015, 10.1029/2003JC002241.