## 1. Introduction

Phenomena associated with the spreading of passive tracers in the ocean, such as plankton, marine debris, and chemical pollutants (e.g., oil), are of practical importance. These problems are usually described by an advection–diffusion model, in which a diffusivity coefficient is employed. Finding a proper diffusivity coefficient for the advection–diffusion model is the pursuit of many studies since the magnitude of diffusivity and how it varies in space and time still remain unclear.

Theoretical studies have tried to seek mathematical expressions of diffusivity. Under the assumptions of a homogeneous and stationary eddy field, Taylor (1922) related the diffusivity to the product of eddy velocity variance and Lagrangian integral time scale. Davis (1987, 1991) regarded the diffusivity as an observable quantity and generalized the diffusivity into a second-order tensor to describe quasi-2D diffusion in an inhomogeneous eddy field. These methods are categorized as single-particle statistics (LaCasce 2008) that require only a large number of current-following (Lagrangian) observations from oceans. The increasing Lagrangian observations by satellite-tracked subsurface floats and surface drifters have facilitated researches on the estimates of diffusivity over various ocean basins (LaCasce 2008; Lumpkin and Pazos 2007).

One important assumption upon which those diffusion theories are based is the scale-separation assumption. However, this assumption is not satisfied because no spectral gap between the large- and small-scale motions is found in the real ocean. Fortunately, Davis’s (1991) theory relaxes this hypothesis by introducing a “history term” that allows interaction between large- and small-scale motions, and the scale-separation assumption could thus be made for practical purposes. In this assumption, velocity

Inhomogeneity of the mean flow implies that

Nonstationarity of the mean flow implies that

Although Qian et al. (2013) as well as Zhurbas et al. (2014) have shown that seasonal variation would bias the diffusivity if not removed from

The rest of paper is organized as follows to address the above issues. Section 2 describes the methods. Results are presented in sections 3 and 4. Conclusions and discussion are given in section 5.

## 2. Methods

Generating synthetic drifters in idealized or model output flow has been widely used in diffusion-related studies (e.g., De Dominicis et al. 2012; Griesel et al. 2010; Koszalka and LaCasce 2010; McClean et al. 2002; Oh et al. 2000; Veneziani et al. 2005). This method has several obvious advantages. First, the background mean flow

### a. Stochastic model

To simulated synthetic drifters in a prescribed background mean flow, stochastic (random flight/walk) models are usually adopted. Zero-order, first-order, and second-order stochastic models are most commonly used (Griffa 1996; LaCasce 2008), in which displacement, velocity, and acceleration are noised variables, respectively. The simplest zeroth-order model cannot be applied to inhomogeneous turbulent field because it does not meet the “well-mixed” criterion proposed by Thomson (1987). Moreover, it cannot be applied to motions of time scale smaller than the Lagrangian integral velocity time scale, since its autocorrelation function is a delta function (Griffa 1996). The first-order model becomes a little more complex and could resolve finite velocity time scale motion. Although it has an unrealistic delta autocorrelation function of acceleration as compared to that of the second-order model and may also fail the well-mixed criterion, it is sufficient for the present study in which diffusivity is specified constantly and will be used here.

### b. Estimation of lateral eddy diffusivity

### c. Gauss–Markov estimator

*T*stands for matrix transpose. After obtaining

## 3. Results from idealized scenarios

Idealized tests are carried out first to show the effects of nonstationary ^{7} cm^{2} s^{−1} and

Summary of the zonal mean advective flow in four idealized tests (Test I–IV). The mean flow is the sum of a constant mean flow, harmonic oscillations, and a mean flow shear. The variables ^{−1} per degree latitude in Test IV.

As a first step of the test, a constant ^{−1} is used (Test I). The mean flow is completely homogeneous and stationary so that the simple binning technique is sufficient to provide an accurate estimate of ^{7} cm^{2} s^{−1} after 20 days. The asymmetric components (

^{−1}, period

^{−1}is the constant background mean as in Test I. For simplicity, only the zonal component of the flow is considered (Fig. 2a) and discussed hereafter.

Two techniques are used to estimate

The estimated zonal velocity autocovariance from the different methods are shown in Fig. 2b. The results from the different methods are indistinguishable. The estimated autocovariance decreases within ~22 days and then oscillates around the analytic result. This is not surprising for the classical binning method because it completely overlooks the sinusoid signal contained in

In Test III, two more realistic sinusoids are added, one with ^{−1}, ^{−1},

In Test IV, the situation becomes a little more complex. Spatial shear equal to ^{7} cm^{2} s^{−1}) after 20 days. Note that the shear dispersion is partially resolved even without ST because of the spatial binning technique and the unresolved part of shear is inside bins. If no binning is applied, the estimated diffusivity will not asymptote to a constant at all (e.g., Bauer et al. 1998) since its velocity spectrum is red (see blue solid line in Fig. 5a). With the ST included, the GM method is able to resolve both the seasonal variation and spatial shear of

Figure 5b shows the variance-preserving spectra (spectral density multiplied by

Note that the GM method performs near perfectly in these cases. This is because the mean flow is designed consistently with Eq. (6), that is, a linear combination of temporal and spatial variations by assuming stationary spatial variation and homogeneous temporal variation. These mean flows are intentionally designed so that we can clearly see different performances of methods caused only by the nonstationarity of the mean flow rather than other factors. Actually, there are also many cases in which the GM method performs not so well or even fails. For example, the spatial variation of the mean flow follows higher-order (third order) polynomial variation rather than the second-order form used in Eq. (6) or even nonpolynomial variations (e.g., exponential variation). Besides, if the shear strength changes with time (nonstationary spatial variation) or amplitude of temporal variation changes with space (inhomogeneous temporal variation), that is, the prescribed form in Eq. (6) does not exactly apply, the GM method is likely to fail in fitting such mean flows. Fortunately, if the GM method is used in conjugation with the spatial binning technique (domain is divided into smaller bins), the performance of the GM would improve because homogeneity is approximately valid inside a smaller bin. Thus, the second-order polynomial fit in Eq. (6) is good enough to resolve any kind of nonsharp spatial variation inside a smaller bin, and the amplitude of temporal variation will also remain roughly a constant. The GM performance in these cases will be shown in the next section through more realistic tests in which the mean flows are derived from satellite-based observations without prescribed forms.

Conclusions can now be drawn from the above idealized tests. If

## 4. Results from the OSCAR scenario

The above tests may be too idealized because the velocity spectra in real oceans may be quite different. Therefore, in this section, a more realistic scenario regarding the background flows in the Indian Ocean (IO) is considered. Synthetic drifters are deployed in currents depicted by the global Ocean Surface Currents Analyses–Real time (OSCAR) data obtained online (from www.oscar.noaa.gov). The velocity of OSCAR is directly derived from sea surface height, surface wind, and sea surface temperature collected from both satellites and in situ instruments, including geostrophic, Ekman, and Stommel shear dynamics and a term from the surface buoyancy gradient (Bonjean and Lagerloef 2002). The 5 yr of data from 2006 to 2010 are used, with 1/72-yr (5 days) temporal and ⅓° spatial resolutions. Only the Indian Ocean is examined here (50°S–30°N, 20°–120°E; Fig. 5) because of strong seasonal cycle within this region. Notice that the Rossby radius and thus the scale of eddies gets small at 50°S, and so the satellite altimetry (and the OSCAR data) would tend to underestimate the in situ eddy kinetic energy at the southern edge of our domain. However, as this only affects the southernmost part of the domain, results shown here are not sensitive to this.

This OSCAR scenario is different from the previous idealized scenario in several aspects. First of all, the velocity spectra of the OSCAR data range from mesoscale to interannual variabilities that are expected to be closer to reality than in the previous scenario. The second aspect is that not only the amplitude and phase of the seasonal cycle vary with space (inhomogeneous temporal variation), but also the shear strength would change with time (nonstationary spatial variation). Therefore, no prescribed mathematical form of the mean flow is assumed. Third, only segments of Lagrangian observations contain the seasonal signal when drifters traveled over the region of strong seasonal cycle, which is different from the earlier scenario.

The deployment strategy is as follows. Synthetic drifters are released in four patches each year (one patch every 3 months). Drifters in each patch are deployed simultaneously at a uniform 1.5° latitude/longitude interval over the whole basin and then tracked for half a year using Eq. (1) with ^{3} cm^{2} s^{−1} and ^{7} cm^{2} s^{−1} in idealized tests) is only used to introduce a random movement of drifters without affecting the velocity spectra of the OSCAR data, so that drifters follow different pathways even if they are deployed at exactly the same initial location. OSCAR data are first linearly interpolated into daily data and drifter velocities are recorded every day. Tracking is stopped if a drifter moves onto land.

Figure 6 shows the number of daily observations in 2° bins after 6-yr simulations. The southern IO and the eastern equatorial IO are heavily sampled, whereas the western equatorial IO and the coastal regions are sampled less. For most parts of the IO, the number of daily observations is above 2000. As the drifters are deployed regularly (one patch every 3 months), drifter observations in each season would be roughly equal, and no significant seasonal sampling bias exists. On the other hand, drifters are deployed uniformly in space, and the array bias caused by nonuniform drifter observations should be partially reduced. However, Fig. 6 shows that drifter numbers also vary from bin to bin and such distribution is probably attributed to the divergence/convergence of OSCAR currents. Therefore, following Poulain (2001), the array bias is computed by assuming constant zonal (20 × 10^{7} cm^{2} s^{−1}) and meridional (10 × 10^{7} cm^{2} s^{−1}) diffusivities. Results (figure not shown) show that in the ocean interior, array biases are below 1 cm s^{−1}, except for the western equatorial regions where bias remains below 4 cm s^{−1}. This is relatively small as compared to the mean flow shown in Fig. 7 and would not much affect the results.

Figure 7 shows the spatial mean

The above comparison proves that the GM method with ST is capable of reproducing the true

For the OSCAR scenario, there is no analytic solution for comparison as in the idealized scenarios. We could nevertheless define a true diffusivity by taking the advantage of the exactly known Eulerian field. The inhomogeneous and nonstationary mean flow

The true and estimated diffusivities from the different methods are shown in Fig. 10. In view of the asymptotic nature, the true diffusivities of both zonal and meridional components over the three regions all approach some steady values at larger lags and thus could be used as references for the estimates from different techniques. At first glance, the classical binning method gives the largest estimates (in magnitude), followed by the two-season binning method. The four-season binning and GM with ST methods provide estimates quite close to the true diffusivity, especially in the EQ and WB regions where the seasonal signal is significant (Figs. 10a–d). In the SIO region where the seasonal cycle is negligible, the four methods produce similar estimates to the true diffusivity. Conclusions from previous idealized tests can also be validated here. In the EQ region, for example, where the seasonal cycle is strong, the estimate from the classical binning method seems to have a complete wave trough/ridge within half a year, indicating an oscillation with a period of 1 yr (see negative time lags in Fig. 10a and positive time lags in Fig. 10c). The two-season binning method also provides a similar estimate but the amplitude is reduced. The meridional estimates in the EQ and WB regions (Figs. 10b,d) show less oscillating characteristics, but the asymptotic values from the classical binning method (or two-season binning method) are also much larger than those from the GM method with ST (or true). Increasing the temporal bins to four seasons further reduces the oscillating features and makes the corresponding estimate much closer to the true diffusivity. However, apparent differences between the estimate by the four-season binning method and the true diffusivity can also be found in both the EQ and WB regions, especially in the meridional component (Figs. 10b, 10d). The GM method with ST included performs best and provides an estimate that never deviates far from the true diffusivity.

This conclusion also applies to the SIO region, but the differences are much smaller since the seasonal signal in this region is quite weak (Figs. 8, 9). Thus, the simple binning method would yield a reasonable result. To our surprise, for the zonal component (Fig. 10e), all methods (even the true diffusivity) provide results that only marginally asymptote to constant values after 90 days. In addition, the meridional estimates overshoot at 15 days and then level off to a constant after 50 days (Fig. 10f), indicating a significant negative lobe in the autocovariance function that is usually observed in cross-stream estimates (e.g., Bauer et al. 2002; Griesel et al. 2010; Klocker et al. 2012). These features are likely attributed to the low-frequency (Rossby) waves resolved by OSCAR data, which is in line with the description by Klocker et al. (2012).

## 5. Conclusions and discussion

To demonstrate the effects of a nonstationary mean flow on the estimation of Lagrangian diffusivity, two scenarios are designed in the present study. In the first scenario, synthetic drifters are deployed in a completely idealized mean flow that consists of a time-invariant part and a sinusoidal oscillated part. When decomposing the velocities sampled by drifters into eddy and mean components using the classical binning method, the oscillating part of the mean flow leaks into the eddy component, resulting in an oscillating estimate of diffusivity. Using a temporal binning method such as the seasonal binning method can partially resolve the nonstationary part of the mean flow and thus reduce the amplitude of oscillation in the estimated diffusivity. The new GM method (with spatial terms included) proposed by Lumpkin and Johnson (2013), fitting the mean flow in a continuous sense, gives the best estimate of diffusivity even when the mean flow is both nonstationary and inhomogeneous. Although the seasonal variability does not have a net contribution to the dispersion, it may cause large bias in the estimate of diffusivity within a time lag of ~2 months, especially when the classical binning method is used.

In the second scenario, a large number of synthetic drifters are released in the ocean currents in the IO basin prescribed by the 2006–10 OSCAR product. This scenario is more complex than the idealized one, and the velocity spectra from the former are expected to be closer to real oceanic conditions. In this case, the inhomogeneous and nonstationary Eulerian mean flow is exactly defined by OSCAR regular-gridded data. Hence, a true diffusivity could be computed using residual velocities with respect to the true Eulerian mean. When estimating diffusivities over regions where the seasonal cycle is strong (e.g., equatorial and western boundary regions), the classical binning method gives the largest estimates that increase within 2 months, showing an oscillating behavior similar to that in the idealized scenario. The two-season binning method reduces the magnitude of oscillation, while the four-season binning method provides acceptable results. The new GM method with ST included successfully captures the temporal variation as well as spatial variation of the mean flow, resulting in a quick convergence of estimates to the true diffusivity, especially in the eastern coast of Somalia and the equatorial IO where a strong signal of seasonal cycle exists in the surface currents influenced by the monsoon winds. For the region over the southern IO where seasonal variation is weak, the simple binning method is sufficient to give a result close to the true diffusivity.

The seasonal cycle focused on here is not the only signal that contributes to the nonstationary mean flow. At a shorter time scale, as shown in Fig. 11, there are also triannual (~120 days) peaks in the zonal velocity over the EQ and WB regions besides the annual and semiannual spikes. Actually, if trapped by a strong rotational eddy field, the autocovariance of a drifter would also show the oscillation feature (e.g., Veneziani et al. 2005). Therefore, at a longer time scale (cannot be shown in Fig. 11), interannual variability might also exert some influence on diffusivity estimation.

On the other hand, the present study only discusses the nonstationarity of

It is worth mentioning the strategy used by several studies (e.g., Krauss and Böning 1987; Lumpkin et al. 2002; Rupolo 2007) in which drifters are divided into equal length segments and then the Lagrangian mean and linear trend of segments are removed to obtain the residual velocity for diffusivity estimation. This approach cuts off the velocity spectrum at a specific period, and variabilities longer than that period are excluded naturally. Rupolo (2007) used 64-day segments that filter out the impact of seasonal and interannual variabilities much longer than 64 days. If the spectral plateau can be reached at 64 days (become a white spectrum), the true diffusivity could converge. If the spectrum was still red, no diffusivity would exist (LaCasce 2008). Velocities over some regions (e.g., the equatorial region) might be red at the cutoff period (Fig. 11). Increasing the cutoff period, such as in Lumpkin et al. (2002), who used 120-day segments, however, would face the potential problems of fewer segments.

Recently, discrepancies between diffusivity estimates from different methods are reported, especially those from drifter-based and tracer-based methods. While many studies have obtained larger values from drifter-based methods than those from tracer-based methods (e.g., Chiswell 2013; Lumpkin and Pazos 2007; Sallée et al. 2008; Sundermeyer and Price 1998), Klocker et al. (2012) has shown that the overestimated diffusivities from drifter-based methods would be reduced to agree with those from tracer-based methods when using sufficiently long times. The present study demonstrated that if the seasonal variability is removed from residual velocity, the magnitude of estimated diffusivity would also be reduced significantly for monsoon-dominated regions (also see Zhurbas et al. 2014). Another benefit of isolating the seasonal variability is that the estimates quickly converge to the true diffusivity. This may help to obtain the asymptotic value of diffusivity estimate in a relatively short time using real drifter data, since long time assessment (e.g., several months) is usually unpractical for the real drifter due to error increments (Davis 1991) and nonlocality (drifters would move outside the domain of interest). But one should be aware of other factors affecting the short time estimates of diffusivity such as the presence of Rossby waves. These waves tend to inhibit dispersion across the meridional mean vorticity gradient and are not resolved in the mean flow we remove, potentially complicating the diffusivity for small lags.

As already shown in the present study, nonstationary, as well as inhomogeneous, mean flows exert some influence on diffusivity estimates. Scale-separation methods should take them into account if they cannot be ignored. The new GM method with ST, as one of the choices but not the only, is very efficient in simultaneously removing both the seasonal variability and spatial shear of mean currents. Therefore, it is particularly appropriate for diffusivity estimation using real drifter data in those regions where significant nonstationarity and inhomogeneity exist, such as the Indian Ocean. This will be presented in our next publication.

This work is jointly supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA11010304), the MOST of China (2011CB403505, 2010CB950302), the Knowledge Innovation Program of the Chinese Academy of Sciences (SQ201305), and National Natural Science Foundation of China (41376021, 41306013), the Hundred Talent Program of the Chinese Academy of Sciences. The authors gratefully acknowledge the use of the HPCC at the South China Sea Institute of Oceanology, Chinese Academy of Sciences. R. Lumpkin was supported by NOAA’s Climate Program Office and the Atlantic Oceanographic and Meteorological Laboratory.

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