To identify oceanic mononuclear mesoscale eddies, a threshold-free splitting
method was developed based on the watershed. Because oceanic eddies are
similar to plateaus and basins in the map of the sea level anomaly (SLA)
data, the natural divisions of the basins are the watersheds between them.
The splitting algorithm is based on identifying these watersheds by finding
the path of steepest descent. Compared to previous splitting methods, the
proposed splitting algorithm has some advantages. First, there are no
artificial parameters. Second, the algorithm is robust; the splitting
strategy is independent of the algorithm and procedure and automatically
guarantees that the split mononuclear eddies are simply connected pixel
sets. Third, the new method is very fast, and the time complexity is

To investigate the dynamics and roles of oceanic eddies in the environment, these eddies must first be automatically identified and tracked, especially when they are close to each other. In general, the automated eddy detection algorithms are categorized into three types: (1) physical parameter-based algorithms, e.g. Okubo–Weiss (Isern-Fontanet et al., 2003; Chaigneau et al., 2008); (2) flow geometry-based algorithms (Fang and Morrow, 2003; Chaigneau et al., 2011; Petersen et al., 2013; Chelton et al., 2011; Xu et al., 2011; Wang et al., 2015); and (3) hybrid methods, which involve physical parameters and flow geometry characteristics (Nencioli et al., 2010, Xiu et al., 2010; Dong et al., 2011; Yi et al., 2014). However, each identification method poses a multinuclear eddy identification problem, e.g. multiple sea level anomaly (SLA) extremes (Chelton et al., 2011). This problem can occur when multiple eddies are physically close together. Note that such multiple eddies are very common in SLA data (Li et al., 2014; Wang et al., 2015).

A simple method to avoid the problem is to reduce the number of contours of the SLA until there is only one extreme in the contour (Chaigneau et al., 2011). Thus, only one extreme is located in the eddy, as shown in Fig. 1a. However, reducing this contour will lead to reductions in both the area and the amplitude of the eddy. The identified eddies are much smaller and weaker. For example, the amplitudes of the identified eddies were only approximately 2–3 cm (Chaigneau et al., 2008), whereas they could be in the range of 20 to 30 cm in other eddy identifications (Chelton et al., 2011; Xiu et al., 2010).

The best approach to solve the multinuclear eddy identification problem is by directly splitting multinuclear eddies, as shown in Fig. 1b. This splitting is not easily achieved. Chelton et al. (2011) attempted to split multinuclear eddies using various methods. However, their splitting process often resulted in some track problems, and it was finally abandoned. Subsequently, Yi et al. (2014) applied a hybrid detection approach by integrating the ideas of the Okubo–Weiss method and the SLA-based method. Li et al. (2014), following the approach proposed by Chelton et al. (2011), attempted to split multiple eddies according to SLA with two simple strategies and a threshold for strategy choice.

Note that Yi's hybrid method does not include any splitting strategy or method. As a result, Yi's hybrid method simply identifies the boundary of the multinuclear eddy using one parameter and identifies the centres of multinuclear eddies using another parameter but cannot actually split multinuclear eddies into single ones. Li's method, which includes the splitting method, requires an additional threshold. In addition, these splitting methods have difficulty in identifying very close multinuclear eddies.

The goal of this study was to establish a splitting strategy that could separate multinuclear eddies into mononuclear eddies. The idea is based on the fact that the values of eddy parameters (e.g. SLA) are similar to plateaus (anti-cyclonic eddies) and basins (cyclonic eddies) in a map and that the vortex is similar to a funnel like a black hole (Haller and Beron-Vera, 2013). The natural divisions of the basins are the watersheds between them. For basins, the “watershed” is a ridge between them, while it is a valley for plateaus.

In this paper, we do not try to find the exact location of the watersheds, but rather we only use the property of watershed (ridge): a particle cannot roll across the ridge from one basin to another one. We use the valley (ridge) to split the anti-cyclonic (cyclonic) multi-nuclear eddy into mononuclear ones. To simplify the descriptions, we use only cyclonic eddies as examples. The anti-cyclonic eddies can be split in a similar way.

The SLA data used in this study were from the MSLA (maps of sea level
anomalies), a merged and gridded satellite product, which is produced and
distributed by AVISO (archiving, validation, and interpretation of satellite
oceanographic data at

To identify eddies, a physical definition of an eddy is required. In
general, an eddy is considered a coherent structure characterized by
water rotating around a common centre (Chelton et al., 2011; Faghmous et
al., 2013) and a structure that retains all its initial mass as it
propagates (Haller and Beron-Vera, 2013). Because this study focuses mainly
on the splitting strategy, the choice of parameters is not of concern, and
we simply use SLA as an example. The following mononuclear eddy definition
is from previous studies (Li et al., 2014). Each pixel has eight nearby
neighbours. A point within the region is a local extremum if it has an SLA
greater or less than all of its nearest neighbours. We also use such
definition of extremum in our following studies, in which the extrema are
identified by checking each pixel in the map and the eight pixels around them.
An eddy is defined as a simply connected set of pixels that satisfies the
following criteria:

Only one SLA extremum exists in the set.

The SLA values of the eddy are above (below) a given SLA threshold
associated with data error, e.g. 3 cm for anti-cyclonic
(e.g.

The amplitude of the eddy is larger than the data error (e.g. 3 cm).

The eddy is identified by the following procedures. First, we find a simply connected region with a given a threshold. Second, we check whether there is at least one extremum in the region. Then we check whether the region satisfies the eddy conditions (2) and (3). Finally, we check whether the eddy is multinuclear. As both conditions (2) and (3) allow the eddy to be multinuclear, we explicitly add condition (1) as a constraint. However, we need a splitting method to implement this.

In this study, an eddy is split based on the fact that the negative gradient vector of the SLA points toward the eddy centre of an ideal circular-shaped eddy (Li et al., 2014) and the fact that the vortex is similar to a funnel (Haller and Beron-Vera, 2013). Because oceanic cyclonic eddies are similar to basins in the map of the SLA data, the natural divisions of the basins are the watersheds between them.

Figure 2 illustrates this eddy splitting strategy. Figure 2a shows two individual but close eddies. The pixels between the two dashed lines are naturally divided by the watershed. As shown in Fig. 2b, the cross section of the eddy clearly shows that two closely located particles on the left and right sides of watershed slide along their ways to different eddy centres. The shape of SLA can provide sufficient information to split the multinuclear eddy into mononuclear ones.

To make the strategy more effective, we assume that all of the particles fall only along the path of steepest descent. This assumption ensures that the particle at each pixel has only one path to the eddy centre. As the path to the centre is mathematically well defined, it is obvious that such a path does not depend on the search method or procedure.

A simple example of the splitting procedure for cyclonic multinuclear eddy is
illustrated in Fig. 2c. The procedure for the anti-cyclonic one is similar
but with a little bit of difference in Sect. 3.4. At first, the extremes with
the definition in Sect. 2.2 are labelled as

Label the extremes as cyclonic eddies of

Mark the pixels in the multinuclear eddy as 1, 2, 3, …,

Let the index

Take the

Is it marked as part of any eddy? If yes, go to (8). If no, go to (6).

Find the path and eddy label “Cx” for the

Mark all of the pixels in the path as cyclonic eddy “Cx”.

Let the index

Stop.

Second, the algorithm is linear and very fast. Each pixel is scanned only
once; thus, the time complexity is

In the splitting procedure, we need to find the path of steepest descent.
Noting that each pixel is surrounded by eight discrete neighbours, the paths
are only the connections of the nearest pixels with approximation, when the
particles roll straight downhill (in continuous field). A simple example of
such a path is illustrated in Fig. 2c. The arrows indicate the path of
steepest descents from pixel

Let

Take pixel “

Find the pixel “

Check whether “

Let

Return along the path of

Stop.

We apply this method to some examples. Figure 3a shows four cyclonic eddies
that are difficult to split because they are very close to each other. Li et
al. (2014) suggested re-identifying a multinuclear eddy if too many extremes
exist (

However, this new method can also avoid another problem in many SLA-based
identification methods. As shown in Fig. 4a, the colour contours show a
simply connected region above a critical value. Part of an eddy

When the eddies are anti-cyclonic like plateaus, the above method cannot be
directly used. One may transform the SLA values into negative ones by
multiplying by

In general, the splitting strategy should meet the following requirements. First, the strategy should be threshold-free. Any artificial threshold might be unphysical and controversial. Second, the strategy should be robust: the splitting strategy should be independent of the numbers of extremes and independent of the algorithm and procedure. Third, the strategy should be independent of the parameter(s) usable. Because there are many eddy parameters (e.g. SLA, geostrophic potential vorticity, Okubo–Weiss parameter), the best parameter for the physical definition of an eddy remains unknown. The present algorithm satisfies all of these requirements.

In this study, a watershed splitting strategy was used for mononuclear eddy identification. The splitting strategy has the following advantages. First, the strategy is threshold-free. No artificial threshold was required in the proposed procedure. Second, the strategy is robust and independent of the algorithm and procedure used. Third, the strategy is very fast, regardless of how many extremes there are. Fourth, the strategy is independent of the parameter used (e.g. SLA, geostrophic potential vorticity, Okubo–Weiss parameter). In addition, the present strategy can be applied to automatic identification of troughs and ridges from weather charts. Due to the potential general applications of eddy splitting, we denoted it the Universal Splitting Technology for Circulations (USTC) method.

We thank D. G. Bowers and another referee for comments and suggestions. This work was supported by the National Basic Research Program of China (no. 2013CB430303) and the National Foundation of Natural Science (no. 41376017). We thank AVISO for providing the SLA data. Edited by: O. Zielinski