The Predictability Limit of Oceanic Mesoscale Eddy Tracks in the South China Sea

Oceanic mesoscale eddies (OMEs) are active in the South China Sea (SCS), with a typical lifespan of several months and spatial scales of ~100 km (Wang et al., 2003; Chen et al., 2011; He et al., 2018). They play an essential role in the transport of mass, heat, salt, and nutrients around the ocean, thereby becoming critical for issues such as water mass distribution (e.g., Zhang et al., 2014), heat and salt transportation (e.g., Chen et al., 2012), variation of sea surface temperature and wind (e.g., Chow and Liu, 2012), general circulation (e.g., Wang et al., 2012), and chlorophyll distribution (e.g., He et al., 2016).

Predicting OMEs is important to understanding oceanic circulation but poses a great challenge to ocean forecasting systems because of the high nonlinearity and the complex dynamical processes inherent to an OME (Treguier et al., 2017; de Vos et al., 2018). Many studies have focused on predicting OMEs in the SCS. Based on an ocean model assimilated system, Xu et al. (2019) pointed out that when the observed amplitude of OMEs is higher than 8 cm in the northern SCS, the generation, development, decay, and movement of two anticyclonic eddies (AEs) can be well reproduced. The quality of the OME forecast largely depends on the quality of the initial conditions. Using the multiple linear regression method and five-year satellite altimeter data, Li et al. (2019) found that the forecasting skill has reached four weeks and the accuracy of the OME forecasting in the SCS is sensitive to eddy polarity and the forecast seasons. Xie et al. (2020) found that data assimilation of altimetry data improves the representation of OME properties, especially for the seasonal occurrence of CEs and AEs. All the above studies improved our understanding of forecasting OMEs and their propagation dynamics in the SCS.

Although the properties and prediction of OMEs in the SCS have received much attention in recent years, our understanding of the predictability limit of OME in the SCS is still hampered by their nonlinear nature and lacking a suitable method. In this study, we will adopt the nonlinear local Lyapunov exponent (NLLE) method (Chen et al., 2006; Ding and Li, 2007) to estimate the predictability limit of OME tracks for both the annual and seasonal means. The NLLE method mainly discusses the initial error growth of the nonlinear behavior by measuring the chaotic nature of the system using an NLLE, noting that the timescale obtained by NLLE can be considered as a proxy for predictability. Numerous studies have employed the NLLE method to assess the predictability of oceanic and atmospheric phenomena, including tracks of tropical cyclones (Zhong et al., 2018), precipitation (Liu et al., 2016), sea surface temperature (Li and Ding, 2013), and geopotential height (Ding and Li, 2009). Our recent study estimated the predictability limit of OME tracks within the Kuroshio Extension (KE) region by using the NLLE method and observed OME propagation data (Meng et al., 2021). The results show that the generation location and properties primarily influence the KE region’s predictability limit of OME tracks. In this study, we investigate the relationship between the predictability limit of OME tracks and their properties in the SCS and validate our findings if the previous conclusion is generalizable. Meanwhile, we also investigate whether the strong seasonal change of OMEs would have an additional effect on SCS eddy tracks.

In addition to their properties, the motion of OMEs would be affected by other factors, such as topography (Yang et al., 2017), strong currents (Zhai et al., 2010), splitting and merging of OMEs (Li et al., 2016), and eddy-eddy interaction (Early et al., 2011; Ni et al., 2020). For the SCS region, Chu et al. (2020) recently pointed out the annual occurrence of some OMEs in the SCS with a long lifetime and strong intensity, called “periodic OMEs”. The generation of these eddies is closely related to the topography and seasonality of the large-scale circulation. Therefore, the predictability limit of these specific eddies is also included in this study.

The remainder of this paper is organized as follows. Section 2 introduces the details of the datasets and methods. The results are shown in section 3. Finally, section 4 provides concluding remarks. Ultimately, this study provides a probable timescale of OME prediction and carries a baseline for predicting eddies for the eddy forecasting system in the SCS.

This study uses three datasets of global eddy propagation trajectories [Table S1 in the electronic supplementary material (ESM)]. One of the datasets was published by Chelton et al. (2011), which can be downloaded on the Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO) (https://www.aviso.altimetry.fr/en/data/products/value-added-products/global-mesoscale-eddy-trajectory-product.html) which is still being updated, referred to hereafter as Chelton. The Chelton dataset includes records for the OMEs derived from altimetry datasets, with daily location, speed, radius, and other associated metadata. 1699 OMEs with a lifetime over 28 days occur in the SCS region (5°–25°N, 108°–123°E) from January 1993 to December 2018.

The remaining two datasets, hereafter referred to as Faghmous and Dong, were from Faghmous et al. (2012, 2013) and Dong et al. (2011). The Faghmous and Dong datasets share identical eddy properties with the Chelton dataset but with a weekly interval. Predictability calculations were conducted using the native time resolution of the OME datasets. Notably, the Faghmous dataset has 2798 OMEs, while the Dong dataset comprises 1855 OMEs, spanning the period 1993 to 2010. These two OME datasets are used to cross-compare with Chelton and test the uncertainty among the three datasets.

The nonlinear behavior of dynamic systems needs to be considered upon estimating the predictability limit of atmospheric and oceanic systems (Mu, 2000). The NLLE was recently proposed and applied to investigate the predictability limits for an n-dimensional chaotic system or a single variable (Chen et al., 2006; Ding and Li, 2007). Ding and Li (2007) suggested that the nonlinear Lyapunov exponent is superior in determining the predictability limits of chaotic systems compared with the traditional linear exponent (LE) method. It estimates the average growth rate of initial error based on a nonlinear dynamical model without linearizing the governing equations. In Fig. 1, the errors of linear and nonlinear growth are similar at the beginning. In contrast, the nonlinear error growth departs from the LE with increasing time and finally reaches saturation. When it comes to the constant saturation level, all the initial information of systems is lost, and the prediction at this time, $ {T}_{p} $, is meaningless. The linear error is constantly increasing monotonously with the increased time. When the system lasts long enough, the LE theory is not applicable anymore. Thus, the NLLE is more suitable and accurate for quantitatively estimating the predictability limit of a nonlinear chaotic system.

Figure 1.  A schematic diagram of (a) the procedure to calculate the NLLE from the observational eddy track data and (b) the mean error growth of a nonlinear dynamic system obtained by the NLLE (red) and traditional LE (blue) approaches. In panel (a), the trajectory of an OME analogous to the reference OME is termed the analogous trajectory. The NLLE is estimated by utilizing the growth rate of absolute distance errors between the reference OME and its analogous counterpart at each time step ($ {t}_{k} $, $ k $=0,1,2,3,…). In panel (b), the logarithmic scale on the y-axis enhances the discernibility of disparities in error evolution between linear and nonlinear trajectories in mean error growth. $ {T}_{p} $ is the point at which NLLE reaches saturation status, representing the predictability limit of NLLE.

For an n-dimensional chaotic system, the NLLE is defined as:

where $ \lambda \left({\bf{x}}\left({t}_{0}\right),{\boldsymbol{\delta }}\left({t}_{0}\right),\tau \right) $ is the function of the initial state $ {\bf{x}}\left({t}_{0}\right) $, initial error $ \boldsymbol{\delta }\left({t}_{0}\right) $, and time $ \tau $.

For a single variable, the NLLE is defined as:

The ensemble mean NLLE and the mean relative growth of the initial error (RGIE) can be obtained by:

and

where $ { < > }_{N} $ denotes the ensemble mean of samples of sufficient sample size $ N(N\to \infty ) $, $ {\xi }_{i} $ denotes the relative growth of one initial error, $ \stackrel{P}{\to } $ denotes the convergence in probability, $ {\bar{\mathrm{\Phi}}}_{i} $ represents the relative growth of the initial error of the ensemble mean of samples, and the constant c is considered to be the theoretical saturation value of $ {\bar{\mathrm{\Phi}}}_{i}$ when the sample size $ N $ tends toward infinity. According to the definition, the predictability limit is $ {T}_{p} $ as indicated in Fig. 1.

Given a time series of observational data, Ding and Li (2007) derived an effective algorithm for identifying local dynamical analogs (LDAs) to compute the NLLE. In chaotic systems, the trajectories of two states are inherently analogous over a brief time interval if they are similar from the initial stage (Lorenz, 1969; Wolf et al., 1985). The LDAs algorithm is able to seek analogous initial state and evolution from the time series. Identifying two analogous OME tracks relies on four key factors: (a) the initial distance, denoted as $ {d}_{0}=d\left({t}_{0}\right) $, representative of the separation between reference OME and its analogous at the initial state (Fig. S1); (b) the evolutionary distance $ {d}_{e} $; (c) the vorticity direction of OMEs, categorized into CEs and AEs; (d) the intensity of OMEs, characterized as the maximum anomaly in sea surface height. A minimal initial distance value ($ {d}_{0} $) ensures a close genesis location for two independent OMEs, while a small $ {d}_{e} $ ensures consistency in the trajectories of the OMEs and their analogous counterparts.

The initial distance between the reference OME and analogous OME is given by:

where $ {\bf{x}}_{{i}}\left({t}_{0}\right) $ and $ {\bf{x}}_{{j}}\left({t}_{0}\right) $ is the location of reference OME and analogous OME.

The evolutionary distance $ {d}_{e} $ which is defined by:

where $ {d}_{e} $ is the mean distance between two analogous OMEs within a period from $ {t}_{0} $ to $ {t}_{K} $, after OMEs have formed, and the value of K is subjected to the sensitivity test outlined in section 2.2.3.

Then the analogous OMEs based on the total distance$ {d}_{t} $ is determined by:

If $ {d}_{t} $ is small enough, the two OMEs are thought to be analogous at the initial time.

For every reference OME, many analogous OMEs can be discerned when the total distance $ {d}_{t} $ is sufficiently small. Then we calculate the growth rate of the initial error at every time step, $ {t}_{k}(k=\mathrm{0,1},\mathrm{2,3},\dots ) $:

The above procedures are reiterated for every analogous OME pair. Subsequently, the approximation of the mean RGIE can be derived as follows:

where $ N $ is the total number of the analogous eddy pairs, and $ M $ is the total number in the time series.

The NLLE method involves three fundamental parameters: the distance $ {d}_{t} $ [Eq. (7)], the initial stage K [Eq. (6), the period from the start of eddy generation to day K], and the sample size. Prior to delving into the analysis of the results, the sensitivity of these parameters to the predictability limit has been meticulously examined. Figure 2 shows the estimated predictability limit of OME tracks by testing the sensitivity to $ {d}_{t} $, K, and sample size.

Figure 2.  The sensitivity tests of the OME’s predictability limits as a function of (a) $ {d}_{t} $ (units: km), (b) K (units: d), and (c) the minimum sample size. The blue points are for CE, and the red points are for AE. The boxplot on the right side reflects the variation of the predictability limit within the sensitivity test. The “+” represents the abnormal values.

The application of the NLLE to estimate the predictability limit necessitates a sufficiently large number of samples (Liu et al., 2016). The satellite observation period limits the OMEs’ track datasets to only about 20–25 years, hindering the need to obtain enough observational OMEs within a limited region and period. To get enough samples, one reference OME typically corresponds to multiple analogous OMEs within the specified limits of $ {d}_{t} $. Conversely, an increase in the criterion of $ {d}_{t} $ also may lead to the discovery of additional analogous OMEs.

The predictability limit of CE tracks shows a significant fluctuation in Fig. 2a, but the predictability limit of AEs shows a relatively stable value with a different value of $ {d}_{t} $. The CE lifetime decreases from about 63 days at 100 km of $ {d}_{t} $ to 48 days at 80 km of $ {d}_{t} $. According to Eq. (1), as the total error or distance ($ {d}_{t} $) increases, the NLLE value becomes smaller, leading to a decrease in the predictability limit, as illustrated in Fig. 2a. It is important to note that $ {d}_{t} $ represents the total error or distance, not solely the initial error. Here, we choose 80 km as the $ {d}_{t} $ value in the stable range. which ensures the distance between OMEs is small enough as the mean radius of OMEs is 109 km. Accordingly, the identified number of analogous pairs has reached 2864, demonstrating adequate sampling for the subsequent analysis. It is noteworthy that all errors mentioned herein refer to the distance between the center points of two independent OMEs.

In accordance with NLLE theory, two states are considered analogous at the initial time if their evolutions in phase space exhibit similarity over a brief time. The total days of the initial state in Eq. (6), denoted by K, aid in effectively excluding local non-analogous samples. Figure 2b shows a gradual decrease in the predictability limit as K increases. A larger K value mitigates the erroneous growth rate during the initial stage. Notably, there are prominent increments observed from 7–8 days to 11–12 days. In this context, we determined the K as ten days.

The sample size is another pivotal parameter. Figure 2c presents an examination of the minimum sample size for calculating the mean value of RGIE at a given time $ {t}_{k} $. The value of the predictability limit stabilizes within the range of 300 to 450, with an increment of 50 for each step. Within this context, we established the sample threshold at 300. The sensitive parameter values are the same as those in our previous study in the KE region (Meng et al., 2021).

Since many studies have investigated the properties of OMEs in the SCS (e.g., Wang et al., 2003; Chen et al., 2011), we compare some critical features of OMEs of three datasets and show the spatial distribution of OME properties. The period-mean values of OME properties for three datasets are shown in Table 1. The probability density functions (PDFs) for lifetime, radius, and amplitude in the SCS for all the datasets are shown in Fig. 3. The eddy numbers represent the difference among the three datasets. Chelton identified the smallest eddy number, about 918 for CEs and 781 for AEs from 1993–2018. It also seems that the OME radii of the Dong datasets are often smaller than the other two datasets. We find the radius PDF for Dong exhibits a significant shift to the side of the smaller values (Figs. 3c, d). That may be caused by a different definition of radius used in Dong. Faghmous has the shortest lifetime but with differences from others of several days. It is also clear that the PDF of Faghmous has a larger proportion in the short lifetime ranges (Figs. 3a, b). However, these discrepancies among datasets are not so significant as to affect the predictability limit of OME tracks, as shown in the results below. Consequently, we will only show the Chelton results for further analysis of the properties.

Figure 3.  The probability density distribution (PDFs) of (a) lifetime (units: d), (c) radius (units: km), and (e) amplitude (units: cm) of CEs for Chelton (blue), Faghmous (green), and Dong (black) in the SCS spanning 1993–2018 in Chelton and 1993–2019 in Faghmous and Dong. Panels (b, d, f) are the same as (a, c, e), but for AEs.

In all the datasets, AEs are fewer but bigger, stronger, and of greater longevity than CEs. On average, there are five more CEs per year than AEs within the whole SCS. Figures 4a and 4b show the number of CE and AE in a 1° × 1° grid, respectively, and reveal that both CEs and AEs are frequently generated in the western regions of Luzon Strait and the western part of the Philippine Archipelago. Furthermore, Figs. 4j and 4l indicate that the occurrence of AEs generated to the west of the Luzon Strait surpasses that of CEs, particularly for long-lived OMEs (>112 days).

Figure 4.  The mean geographic distribution of the (a) generation number, (c) lifetime, (e) amplitude, and (g) radius in a 1° × 1° grid. Panels (i) and (j) show the trajectories for CE within 56 days and over 112 days during 1993–2018 of CEs for the Chelton dataset. Panels (b, d, f, h, k, l) are the same as (a, c, f, h, i, j) but for AEs.

In Table 1, the area-mean lifetime, amplitude, and radius of AEs are about 2–6 days longer, 0.3–0.8 cm, and 3–9 km larger than that of CE, respectively. These are also obvious in Figs. 4c–h. The lifetime distribution of CEs and AEs is obviously different, as seen in Figs. 4c and 4d. CEs with long lifetimes mainly distributed to the east of Vietnam and the central region of the SCS. Long-lived AEs are located in the northern SCS near the continental slope, the central SCS, and to the southeast of Vietnam. The amplitudes of CEs and AEs show a similar northeast-southeast distribution, but the large amplitudes of CEs and AEs are near Vietnam and the Luzon Strait, respectively (Figs. 4e, f). The radii of CEs and AEs exhibit a spatially similar distribution.

Figures 4i and 4j (4k and 4l) show the CE (AE) lifetime trajectories of less than 56 days and more than 112 days. CEs and AEs in the northern SCS (north of 18°N) move southwest roughly along the slope and eventually dissipate at the western boundary of the SCS. The mean propagation distance (distance between generation location and termination location) of short-lived CEs (AEs) is 183 km (189 km), but 451 km (466 km) for long-lived CEs (AEs). To the south of 18°N, the long trajectories of CE and AE are mainly concentrated in the southwest region of the SCS and close to Luzon Island. The OME tracks in the southeast area of the SCS are scarce because the intricate seabed topography poses challenges to the generation and propagation of OMEs.

There is a strong correlation between CE and AE activities and the monsoon in this region, exhibiting significant seasonal variations (Fig. 5). For spring and summer, CEs with enormous amplitude are located north of the SCS (Figs. 5a, b), and for autumn and winter, CEs with enormous amplitude are located to the south of the SCS (Figs. 5c, d). While AEs tend to have a large amplitude along the north slope and the west of Luzon strait throughout the years. AEs with large amplitudes are normally observed offshore of Vietnam in the summer (Fig. 5f). In terms of lifetime, seasonality is not a very significant factor but also shows some effects. The long-lived CEs and AEs mainly occur in the center of the SCS during all seasons. However, long-lived AEs have a location preference along the north slope in spring and winter (Figs. 5m, p). Basically, the radius of an OME is related to the Rossby deformation radius, which means it varies according to latitude, larger near the equator and smaller away from the equator. The seasonal features of OME radii are less evident than those of their amplitude and lifetime. CEs (AEs) have larger radii in autumn (Fig. 5s) and winter (Fig. 5t) (spring and summer, Figs. 5u, v). Seasonal variations exist in the northern continental shelf and southeast of Vietnam, which aligns with seasonal sea surface height variability in the SCS (Zhuang et al., 2010). Moreover, the mean atmospheric seasonal circulations trigger a chain of air-sea interaction that shapes the eddy statistics over the SCS (Xie et al., 2003; Liu et al., 2004).

Figure 5.  The mean geographic distribution of (a–d) amplitude, (i–l) lifetime, (q–t) radius, and (y–B) generation number of CEs in spring (MAM), summer (JJA), autumn (SON), and winter (DJF). Panels (e–h, m–p, u–x, C–F) are the same as (a–d, i–l, q–t, y–B) except for AEs.

The normalized trajectories of OME tracks in four seasons are shown in Fig. 6. The majority of CEs and AEs move toward the southwest. In assessing the complexity of OME tracks, we introduce an Eddy Complexity Index ($ \mathrm{C}\mathrm{I} $) for OME trajectories, defined as follows:

Figure 6.  Normalized trajectories of OME tracks for CEs (blue) and AEs (red) in (a, b) spring (MAM), (c, d) summer (JJA), (e, f) autumn (SON), and (g, h) winter (DJF). The value of the mean Complexity Index (CI) of trajectories is also shown in each figure.

where $ D $ represents the straight-line distance from the generation point to the disappearance point, and$ L $ represents the trajectory distance during either a portion or the entire lifespan of the eddy, noting the constraint ( $ L\geqslant D $). The simplest trajectory is a straight line from the generation point to the extinction point (L = D). The most complex route can be understood as the eddy starting from the generation point and then returning to that point to die out ($ L\gg D $). Here, $ \mathrm{C}\mathrm{I} $ is a positive number between 0 and 1 corresponding to L=D and $L\gg D $, respectively. A smaller $ \mathrm{C}\mathrm{I} $ corresponds to a more complex trajectory. This index has been applied to Meng et al. (2021) with reasonable results.

The trajectories in the autumn and winter (Figs. 6e–h) are slightly less complicated, with a $ \mathrm{C}\mathrm{I} $ of more than 0.4. The highest $ \mathrm{C}\mathrm{I} $ value for AE tracks, 0.48, occurs in autumn, which means that the trajectories are extended and smooth. Some AEs exhibit northeast trajectories with lower values in spring and summer.

Figure S1 in the ESM shows the eddy trajectories for CEs and AEs in all four seasons. The north continental shelf of the SCS is a good place for forming the routes of OME tracks. Other locations include the center of the SCS during all seasons and the southwestern part along the boundary in the autumn and winter for CEs and spring and summer for AEs.

Based on the NLLE method and LDAs algorithm, the mean error growth of OME tracks can be obtained by Eq. (9) using three OME propagation datasets (Fig. 7). The mean errors of the three datasets all increase quickly and show linear growth in the initial stage. After 15–20 days, mean error growth reaches a nonlinear increasing stage and finally enters a saturation level. The average relative error growth defines the saturation value after 50 days for the Chelton dataset. To reduce the effects of sampling fluctuations, we define the predictability limit as the time at which the mean error growth reaches 95% of the saturation value (the dashed line in Fig. 7) following previous studies (e.g., Ding and Li, 2009; Zhong et al., 2018; Meng et al., 2021). Ding and Li (2007) stated that the saturation level depends on the dynamics of the system. Based on a threshold defined by 95% level of the saturation value, the predictability limit of OME tracks is 44 days for AEs and 39 days for CEs in the Chelton (Fig. 7a). AEs tend to have a slightly longer predictability limit than CEs in the SCS.

Figure 7.  The average error growth of OME tracks from (a) Chelton, (b) Faghmous, and (c) Dong in the SCS. The blue curves represent CEs, while the red curves represent AEs. The dashed line corresponds to the 95% level of the saturation value as derived from the mean after 50 days for Chelton and 45 days for Faghmous and Dong.

We also calculate the predictability limit of OME tracks from Faghmous and Dong in Figs. 7b and 9c to estimate the uncertainty of different datasets. The predictability limit of AE (CE) tracks is 44 (35) days for Faghmous and 46 (35) days for Dong. The value for AEs shows a strong agreement across three datasets with discrepancies of under four days. The predictability limit of CE tracks from Chelton is four days larger than those from Faghmous and Dong, but only by ~10%. Thus, alternative datasets do not significantly affect the capacity to the predictability limit. This result could also be seen in the KE region (Meng et al., 2021). Hereafter, we mainly use Chelton to further analyze the effects of seasons and eddy properties.

We further discuss the predictability limit of OME tracks over four seasons. Figure 8 shows the mean error growth, the predictability limit, and analogous eddy numbers for all four seasons. All analogous eddy numbers are over 68 (Fig. 8f), sufficient to estimate the seasonal predictability. In Fig. 8e, AEs and CEs have the highest predictability limit in autumn (52 days) and winter (53 days), respectively. The predictability limit of AE tracks in winter is 45 days, but only about 40 days in spring and summer. CE has the lowest predictability limit in summer (30 days). Seasonal variations affect the predictability limit of OME tracks in the SCS.

Figure 8.  The predictability limit for different seasons of OME tracks in the SCS, including (a) spring, (b) summer, (c) autumn, and (d) winter. Panels (e) and (f) present the value of the predictability limit and analogous eddy number in four seasons, respectively.

In the SCS, the spatial distribution of AE and CE predictability limits also changes with the seasons. In spring and summer, the predictability limit of CE tracks is higher to the south of 18°N (Figs. 9a, b, i, j), particularly just to the east of Vietnam and west of Luzon Island in spring and in the southern sea area in summer. However, in autumn and winter, the predictability limit of CE tracks in the northern part of the SCS increased, specifically in the area west of the Luzon Strait. It can also be seen from Figs. 9d and 9l that the pixels with a higher predictability limit of CE tracks in winter are located near the exit region of the Luzon Strait, east of Vietnam, and to the southwest of Luzon Island.

Figure 9.  The spatial distribution of area means the predictability limit (days) of CE tracks in a 2° × 2° box in four seasons, including (a) spring (MAM), (b) summer (JJA), (c) autumn (SON), and (d) winter (DJF). Panels (e–h) are the same as (a–d), but for AE. Panels (i–p) are the predictability limit anomaly of CE and AE tracks in four seasons.

As with CEs, the predictability limit of AE tracks also varies with the season. In summer, the higher predictability limit of AE tracks obviously coincides with the seabed slope, indicating that the sloped boundary in the seabed topography is an essential path for OME movement. In autumn, the AEs (Figs. 9g, o) have the highest predictability limit, and the high-value areas of the predictability limit are mainly concentrated to the north of 16°N. In winter, the high value of the predictability limit of AE tracks is concentrated in the vicinity of the Luzon Strait. Notably, there is a strong correlation between the high-value region of the predictability limit of OME tracks and the spatial distribution of OME properties (Fig. 5). These phenomena prove again that the OMEs with a long lifetime and strong intensity are more predictable.

To understand the trajectory movement with a shorter and longer predictability limit, we show the distribution of the four seasons with a predictability limit of fewer than 20 days and greater than 40 days. $ {\mathrm{CI}} $ mainly discusses the complexity of the trajectories and the movement at each season. The $ {\mathrm{CI}} $ shows that CE and AE tracks with higher predictability tend to have simpler trajectories (larger $ {\mathrm{CI}} $ values). It should be noted here that the long-lived eddy $ {\mathrm{CI}} $ values of Figs. 10g and 10m are too small. Possible explanations affecting the predictability include an insufficient eddy number, insufficient accuracy of the statistical results, and/or other factors.

Figure 10.  Trajectories of CEs (blue) with a predictability limit (PL) of less than 20 and greater than 40 days in four seasons, including (a, e) spring (MAM), (b, f) summer (JJA), (c, g) autumn (SON), and (d, h) winter (DJF). Panels (i–p) are the same as (a–h), but for AE (red). The value of the mean Complexity Index (CI) of trajectories is also shown in each figure. Solid dots and hollow circles represent the locations of eddy generation and extinction.

The major trajectories and generation locations of the CEs vary with the seasons (Figs. 10a–h). In spring, the CEs generated near the west of Luzon Island have a relatively high predictability limit. According to Fig. 10e, these CEs moved westward across the submarine plain and died out along the western boundary of the SCS. The remaining CEs with high predictability limits were generated on the northern slope and moved southwest along the sloped boundary. In summer, CEs with greater predictability limits are mainly located east of Vietnam. Such CEs appear as long-lived steady OMEs without sudden turning, making them easier to predict (Fig. 10f). After comparison, we found that the CEs east of Vietnam during the summer are likely to be periodic CEs, usually generated around August (Chu et al., 2020). In autumn, the predictability limit of CE tracks with a higher value is significantly reduced, and the distribution is not concentrated. Still, most of them are located in the northern region. In winter, the predictability limit of CE tracks with higher values has three routes. First, CEs that generate near the mouth of the Luzon Strait, move along the slope and eventually die at the western border. Second, CEs that generate in the southwestern region of Luzon Island before moving into the central basin and dying out. Third, CEs that generate in the southwest corner of the SCS continue to move to the southwest before dying out.

The same is true for AE tracks (Figs. 10i–p). The AE tracks in autumn are very neat and regular. There are three main routes: (1) AEs that generate to the west of the Luzon Strait and move along the land slope before finally dying out at its western border; (2) AEs that generate to the east of Luzon Island and move westward to the western border before dying out; (3) a small number of AEs that generate near the southwestern area of the SCS move a short distance to the north. In spring, the predictability limit of AE tracks is lowest, and only a few AEs with higher predictability gather in the northwestern region of Luzon, move westward, and slowly die out. The AE tracks in summer are more predictable in the central part of the SCS, mainly moving westward. The AE tracks in winter are similar to those in autumn. The predictability limit of AE tracks is generated near the east coast of Vietnam, moving a short distance to the west and gradually disappearing after staying in the area for a long time. The above predictable AE is consistent with the generation time and location of periodic OMEs such as Vietnam cold and warm eddies, Vietnam eddy pairs, Luzon cold and warm eddies, Taiwan cold and warm eddies, and Huangyan Island eddies (Chu et al., 2020).

The eddy properties could be important factors that determine the predictability limit. The previous study suggested that when the amplitude is greater than 8 cm, two AEs can be well predicted in the SCS (Xu et al., 2019). We have investigated the relationship between the predictability limit and eddy properties in the Kuroshio Extension (KE) region (Meng et al., 2021). Eddies that are long-lived, of large amplitude, and have large radii tend to have a higher predictability limit in the KE region.

We classified all OMEs according to their lifetime, amplitude, and radius: short-lived (<57 days) and long-lived (>57 days), weak (<6 cm) and strong (>6 cm), and small (<106 km) and big (>106 km). The classification criteria are based on the mean value of their properties in the SCS. Figure 11 shows the mean error growth for different lifetimes, amplitude, and radius. There are 1118 (581) OMEs that have lifetimes shorter (greater) than 57 days, and we found 1066 (452) analogous OME pairs based on LDAs criteria. According to the 95% saturation level, the predictability limit of short-lived OME tracks is 33 days for AE and 36 days for CE, while for the long-lived OME tracks the predictability limit is 72 days for AE and 60 days for CE (Figs. 11a, b). This indicates a trend wherein long-lived OMEs exhibit a heightened predictability limit in the SCS.

Figure 11.  The same as in Fig. 9, but for different OME properties, including (a, b) lifetime, (c, d) amplitude, and (e, f) radius. The dashed lines represent the 95% level of the saturation value obtained by the mean value after (a) 40 days, (c) 50 days, and (b, d, e, f) 60 days.

We also discuss different amplitudes and radii in the same way. It should be noted that the amplitude and radius of OMEs change over the course of their entire life, so we used the average value during their entire lifespan. Each group contains over 321 OMEs and 200 analogous OME pairs. In Figs. 11c–f, the predictability limit of OME tracks is 43.5 days for those with small amplitudes (35 for AE and 52 for CE) and 58.5 days for those with large amplitudes (55 days for AE and 62 days for CE). This result is consistent with Xu et al. (2019) in that larger amplitude OMEs in the SCS are much more predictable. The mean predictability limit of OME tracks is 43 days for those with small radii (46 days for AE and 40 days for CE) and 55.5 days for those with large radii (54 days for AE and 47 days for CE). All of these results indicate that long-lived, stronger, and bigger OMEs have longer predictability limits, or are more predictable.

The high predictability limit often is linked with simple OME tracks. Figure 12 shows the trajectory distribution for different properties. The numbers in the upper-left corner of the panels represent the average CI of the trajectories. The long-lived OMEs mainly move along the slope in the northern part of the SCS, generated from the eastern boundary, and move westward until they disappear (Figs. 12b, d). At the same time, their CI-value is larger than the short-lived OMEs, so the trajectories of long-lived OMEs are simpler. When classified according to amplitude (Figs. 12f, h), OME tracks that have an amplitude greater than 6 cm are also simpler. It is also evident that most tracks are distributed in the northern part of the SCS, are generated from the Luzon Strait, and moved to the southwest boundary. The comparison of CI for CE does yield a statistically significant difference. However, the CI of AE is 0.4 and 0.47 for amplitudes less than 6 cm and greater than 6 cm, respectively, which are primarily located in the southeastern SCS with complex topography, making it difficult to predict the OME tracks.

Figure 12.  The trajectories of all CEs and AEs for various categories: (a–d) lifetime, (e–h) amplitude, and (i–l) radius. Additionally, the mean Complexity Index (CI) values for the trajectories are displayed in each respective figure.

There are small-radii OMEs concentrated in the eastern region of the SCS, indicating that in addition to OMEs with long lifetimes and large amplitudes, there are also OMEs with short predictabilities in the east (near the Luzon Strait). In particular, the CI-values of larger-radii AEs are smaller than smaller-radii AEs. This indicates that other factors (such as latitude) have the potential to affect the predictability limit in AEs with a larger radius.

By comparing the positions of periodic OMEs in different seasons of the SCS (Chu et al., 2020), we found that the positions with higher predictability limits are highly coincident with the positions of periodic OMEs, e.g., the Huangyan Island CE (located at 118.9°E, 13.4°N), Western SCS AE in summer (located at 111.2°E, 13.0°N), and three AEs streets located on the northern slope often generated in summer, etc.

Periodic OMEs are usually generated at an almost fixed time and place, forming the current SCS eddy field. The reasons for periodic OMEs are related to the monsoon dominance of the SCS, its circulation field, and complex terrain. According to the definition of a periodic OME in Chu et al. (2020), periodic OMEs need to be defined and extracted with some of the criteria that have been modified in our study. Four criteria have been proposed: (1) the lifetime should be greater than 35 days, which corresponds to the shortest lifetime of periodic OME in Chu et al. (2020); (2) the nonlinear parameter (rotation velocity/moving velocity) must be greater than 1, the rotation characteristics of the OMEs can be more obvious; (3) the occurrence rate of OMEs should be greater than 80%, that is, during the period 1993–2018, periodic OMEs must occur for more than 20 years; (4) an OME should be generated in a certain position. We obtained 548 periodic OMEs, accounting for 30.8% of the total. The average lifetime, amplitude, and radii of periodic OMEs are 63 days, 7.5 cm, and 105.7 km, respectively. Compared with the average properties of all OMEs in the SCS, periodic OMEs are characterized by longer lifetimes and larger amplitudes.

Figure 13 shows the mean error growth of periodic OME tracks, and the predictability limit is determined based on the 95% saturation level. The predictability limit for AE and CE is around 49 days, more than five days longer than the average result in section 3.3. In addition, it should be noted that the above selection criteria cannot exclude all non-periodic OMEs or extract all periodic OMEs, consequently resulting in a low estimate of the predictability limit of periodic OME tracks.

Figure 13.  Normalized trajectories of OME tracks for (a) CEs (blue) and (b) AEs (red). The value of the mean Complexity Index (CI) of trajectories is also shown. Trajectories of (c) CE (blue) and (d) AE (red) are shown with schematic periodic OME locations. The solid and empty circles represent the generation and termination locations of OME. The abbreviations in the legend are clarified as follows: TCE—Taiwan CE, TAE—Taiwan AE, LCE—Luzon CE, LAE—Luzon AE, HCE—Huangyan Island CE, HAE—Huangyan Island AE, CCE—Center CE, TA—Three AEs, VEP(CE)—Vienan Eddy Pair (CE), VEP(AE)—Vienan Eddy Pair (AE), VCE—Vietan wintertime CE, WAE—Western SCS AE, and WanCE—Wan’an Tan CE. (e) The mean error growth of periodic OME tracks in the SCS.

The trajectory distribution of periodic OME is shown in Fig. 13. According to CI, the trajectory complexity of CEs and AEs is 0.38 and 0.39, respectively, close to the CI of the total eddies (Fig. 6). We also found that periodic OMEs are mainly located in the eastern boundary of the SCS, including the Luzon Strait and the west coast area of Luzon Island (Figs. 13c, d). The periodic OMEs in this eastern border region are the Luzon CE and AE, Taiwan CE and AE, Huangyan Island CE, and AE (Chu et al., 2020). These eastern periodic eddies are highly correlated with the local wind stress curl, with oceanic processes that include the Kuroshio intrusion and the SCS outflow via the Luzon Strait or the Mindoro Strait. The western border also has periodic OMEs: the Vietnam eddy pair, Vietnam CE, Wan’an Tan CE, Center CE, Three AEs, and a Western SCS AE. These western periodic eddies are mainly related to the boundary of the west current in the SCS, which is primarily affected directly or indirectly by the seasonal wind (Chen and Xue, 2014).

In this study, we have adopted the nonlinear local Lyapunov exponent (NLLE) method (Chen et al., 2006; Ding and Li, 2007) to estimate the predictability limit of OME tracks qualitatively for both the annual and seasonal mean. We also investigated the relationship between the predictability limit of OME tracks and their properties in the SCS. In addition to their properties, the predictability limit of periodic OMEs is also investigated in the present study. The primary conclusions are as follows:

(1) The mean predictability limit of CE (AE) tracks is 44 (39) days for Chelton, 44 (35) days for Faghmous, and 46 (35) days for Dong in the SCS. In general, the predictability limit of AE tracks is about 5–11 days higher than that of CE tracks.

(2) Long-lived, large-amplitude, and large-radius OMEs exhibit a higher predictability, surpassing 60 days. In contrast, short-lived, small-amplitude, and small-radius OMEs tend to have lower predictability, with values around 40 days.

(3) We also introduced the complexity index (CI) to further explain the complexity of OME tracks. The long-lived and large amplitude OMEs mainly gathered in the northern slope of the SCS and to the west of Luzon Strait with a large CI. The tracks are simpler and smoother for the long-lived and large-amplitude OMEs. However, the CI value of AEs with larger radii is smaller, indicating that the CI cannot completely represent the trajectory complexity. In future studies, we plan to analyze further factors such as the shape, steering angle, and rapid turning. We will explore the use of weight assignments that combine these factors to generate a more comprehensive and integrated path complexity metric that more accurately reflects the characteristics of eddy trajectories.

(4) The predictability limit of CE and AE tracks varies according to season. The predictability limit of AE (CE) tracks is highest in autumn (winter) with 52 (53) days in Chelton, respectively. The predictability limit of AE tracks is 45 days for winter and 40 days for spring and summer. The predictability limit of CE is the lowest in summer with 30 days, 42 days for spring, and 40 days for autumn. The positions and trajectories of the larger predictability limit of OME tracks are often consistent with the periodic OMEs.

(5) The predictability limit of periodic OME tracks is 49 days for both CE and AE tracks. The periodic OMEs tend to be long-lived and stronger and are mainly located near the Luzon Strait and near the west coast of Luzon Island.

It has become clear that mesoscale eddy motion is the mainstay of ocean surface motion. Individual eddies carry their core water and transport it over long distances. As a result, ocean eddies play a crucial role in the transport and redistribution of heat, salt, nutrients, and other materials and energy in the global ocean, which in turn can regulate global climate change and influence the distribution of ocean resources (Zhang et al., 2014). A great deal of research investigates the issue of eddy prediction based on ocean models. However, there is still a lack of research on the predictability limit of OME tracks through observational data. This study provides insight into eddy predictability and will help ocean modelers realize the limits of predictability or chaos that the ocean models can depict.

Here, we estimate the predictability limit of OME tracks and select factors that could affect the predictability limit in the SCS. Conclusion (2) is consistent with our previous result in the KE (Meng et al., 2021). Compared with the predictability limit in the KE, the value in the SCS is shorter due to the different eddy behaviors and external environments, such as seasonal circulations and the monsoon of the SCS. Therefore, local specific influences need to be considered when discussing the predictability limit in different regions.

Only the measurable parameters are considered in this study. We cannot consider the more complex processes that affect the OMEs because little is known about how eddies interact with the continental shelf and seabed in the region and eddy-eddy interaction. These processes are likely to be important for regulating the routes of OMEs. In the future, high-resolution numerical models are needed to analyze the physical processes and determine the effects of other external factors (like surface wind, temperature, topography, etc.). At the same time, we could also validate the model simulations by calculating the predictability limit of modeled OME tracks.

Acknowledgements. We are grateful for the mesoscale eddy datasets provided by Chelton et al. (2011), Faghmous et al. (2013), and Dong et al. (2011). This study was supported by the National Key R&D Program for Developing Basic Sciences (2022YFC3104802).

Electronic supplementary material: Supplementary material is available in the online version of this article at https://doi.org/10.1007/s00376-024-3250-7.