An idealized mesoscale eddy is initialized with an axisymmetric Gaussian-type profile based on long term moored observations (Zhang et al., 2013), Argo float data and the merged data products of satellite altimeters (Chen et al., 2010). Temperature profiles from observations are fitted into an equation of T(z)=Tb(z)+aze-x2+y22L2, where Tb(z) is background temperature; az is a function parameter varying with depth (z) and L is constant 1.5 × 104 m; x, y and z are position coordinates.

The eddy's initial velocity is calculated using the thermal wind balance with zero velocity at the ocean bottom. The density distribution is obtained from a state equation according to Jacket and Mcdougall (1995). Figure 1 shows the temperature and azimuthal velocity distribution on the cross section through the eddy centre. The initial eddy is 60 km in diameter and 500 m in depth with a total water depth of 2000 m. The maximum surface velocity is about 0.9 m s-1, and the maximum surface elevation is 0.5 m.

There are different methods to identify an eddy and here we use the Okubo–Weiss method (Okubo, 1970; Weiss, 1991) to identify the eddy that we constructed in the model and define the boundary of the eddy. The Okubo–Weiss parameter W is given by W=sn2+ss2-ω2,ω=∂v∂x-∂u∂y,sn=∂u∂x-∂v∂y,ss=∂v∂x+∂u∂y, where ω is the vertical component of relative vorticity; sn and ss represent the strain and shear deformation, respectively; and u and v are eastward and northward velocities, respectively.

Because the velocity field within an eddy is dominated by its rotation, ocean eddies are generally characterized by negative values of W. In this study, we use W < -0.2σw to define the core region of the eddy, where σw is the standard deviation of W in the study region. This way to identify an eddy has a tendency towards excess of eddy detection (Doglioli et al., 2007), so we combine the PVA distribution, velocity field and temperature anomaly to determine the main eddy that we focus on, and ignore the smaller circulations due to the eddy–topography interaction.

In our first set of numerical experiments, an eddy (warm or cold) is located at the centre (x = 250 km, y = 225 km) of the domain with open boundaries and a flat bottom (Fig. 2). When the eddy is a warm eddy (anticyclonic eddy in the northern hemisphere), it moves towards the southwest direction in a flat bottom ocean. At the beginning of the model integration, the eddy will adjust itself to a dynamic balance. As a result, the speed of the eddy movement is relatively small. After the model reaches its balance, the speed of the eddy increases, to a constant value of 2.4 km day-1 after 40 days. The speed of the warm eddy in the model is similar to that of the study of (Wei and Wang, 2009). The eddy propagation speed is influenced by the eddy size and the β effect, which is a function of the local latitude. Therefore at a different latitude, the eddy has different speed. Figure 3 shows that the average speed over 50 days of the warm eddy is 1.75 km day-1, which is smaller than the eddy speed in the natural conditions in the SCS region because of the adjustment in the early stage of the model run.

With a cold eddy (cyclonic eddy in the northern hemisphere) in the same situation, the movement direction is northwest under β and nonlinear effects. The speed increases from the beginning of the model integration which is the same as for the warm eddy. The speed reaches a constant value of 2.3 km day-1 after 40 days. From the trajectory and speed variation during the eddy movement, we can see that the warm and cool eddy have similar kinematic characteristics. However, the cold eddy moves to a higher latitude in the northern hemisphere while the warm eddy moves to a lower latitude because of their different spin directions.

When an isolated eddy propagates in open oceans with a flat bottom, while the β effect drifts the eddy westwards (Shi and Nof, 1994), nonlinearity provides the meridional component of movement (e.g. Chang et al., 2012; Hyun and Hogan, 2008). From the results of the model, the warm eddy generally moves in the southwest direction in the northern hemisphere, which agrees with previous studies. The trajectory of the cold eddy is mirror symmetric with the warm eddy (Fig. 2). Both eddies' propagation speed is about 2.4 km day-1, which is smaller than the value in previous numerical investigations which used mesoscale eddies with a 100 km horizontal scale (Wei and Wang, 2009; Sutyrin et al., 2003). The eddy propagation speed associated with the eddy size increases with increasing eddy size but will be limited by the maximum Rossby wave phase speed.

The influence of an island on the eddy deformation is explored in this study. According to the eddy-splitting at Dongsha island in the SCS (Chang et al., 2012), we set an island on the path of the warm eddy based on the first case we have examined. The diameter of the island is 20 km. At the beginning of the model integration, the eddy is not influenced by the island because the distance between the eddy and the island is not sufficiently close. As the eddy moves towards the island along its trajectory, the eddy eventually interacts with the island.

It is evident from Fig. 4 when the eddy collides with the island, that there is another weak warm eddy formed on the other side of the island. The two eddies have similar diameters but the secondary eddy is weaker than the main one, which can be seen from sea surface height (SSH), temperature, potential vorticity anomaly (PVA), and O–W field (Fig. 5). In eddy-splitting, the temperature and PVA can be seen as a tracer. From the temperature distribution we can find that the water of the secondary eddy is derived from the original eddy, so we believe that the secondary eddy comes from eddy-splitting rather than being formed independently. After the eddy-splitting, the two eddies move away from the island along their own trajectories as independent eddies. When a cold eddy encounters an island with 20 km diameter in its trajectory, the eddy split in the same way with the warm eddy (Fig. 6). Therefore, only the warm eddies are used to study the influence of an island/seamount on the eddy-splitting.

As mentioned previously, when an eddy collides with an island, the eddy can split into two eddies with similar rotation characters. Here we examine the evolution of the eddy-splitting process. Figure 7 shows the temperature field evolution of an anticyclonic eddy colliding with an island with a diameter of 20 km. The eddy is initially located at 40 km northeast of the island. Then the eddy moves towards the island at 0.023 m s-1. At t = 20 days, the eddy gradually collides with the island and the isolated anticyclone is cut by the island. The fluid at the edge leaks to the right (looking off-shore) due to the presence of the solid boundary of the island. The eddy loses mass along the edge of the island, creating a jet moving away from the eddy.

As the inertia and β effect push the eddy continually closer to the island, more and more warm water leaks to form a jet with higher velocity. From t = 26 days, because of the curved edge of the island, the jet moves forward off the boundary. The jet trajectory curves to the right side under the influence of the earth's rotation. Until t = 32 days, the water leaking as a jet becomes weaker as the eddy stops squeezing onto the island. At the same time, the warm water trapped by the jet gathers at the downstream and merges into the newly formed anticyclone eddy.

The radius of the newly formed anticyclone is about 25 km, which is similar with the parent eddy, but its strength is weaker. Under the boundary effect, both eddies move away from the island. As a result, the parent anticyclonic eddy splits into two anticyclonic eddies during the interaction with the island.

For a better understanding of the mechanism of the eddy-splitting process, the PVA field is analysed, which is shown in Fig. 8. The eddy is composed by two parts: one is the inner part with negative PVA and the other is the outer annulus with positive PVA. As shown in the figure, from t = 22 days, at the start stage, the water leaked out is outer annulus water with positive PVA and forms the original jet. When the jet flows off the boundary from t = 24 days, there is an anticyclonic eddy formed due to the flow shear effect at the corner, which is the separation point of the jet and the boundary. At t = 32 days, as the eddy pushes closer to the island, more warm water with lower vorticity flows into the newly formed anticyclone under the influence of the Coriolis force. When the warm water merges into the anticyclonic eddy, the new anticyclone matures gradually similar to the parent eddy by geostrophic adjustment and moves off the island. As shown in Fig. 8, the newly formed anticyclonic eddy is weaker than its parent eddy counterpart.

The position of the newly formed anticyclone is controlled by the separation point of the jet and the island boundary and therefore is influenced by the boundary curvature, which is a function of the island scale. As the island scale increases, the azimuthal angle (clockwise is positive) of the new anticyclonic eddy to the parent eddy decreases. The relationship between the positions of the eddies and the island will be discussed in Sect. 4.3.

When the eddy encounters an obstacle, the trajectories and speed are usually drastically altered. The results show that the speed of the eddy decreases significantly when the eddy interacts with the island. Shi and Nof (1994) pointed out that the image effect and the rocket effect (caused by the jet) usually dominate when colliding with a solid obstacle, and the effect would change the original movement trend combined with the boundary effect. At the same time, the generation of a weak cyclonic eddy during the interaction of warm eddies with an island/seamount adds a significant effect to the eddy propagation.

Actually, an anticyclonic eddy can never split on its own. Nof (1990) demonstrated this by applying the conservation law of integrated angular momentum (IAM). As a result, when a warm eddy splits, the IAM has to increase as the newly formed eddies move away from their original centre. When a warm eddy is forced by the solid boundary of an island or the lower layer of a seamount, there has to be a transfer of IAM from the surrounding fluid to the core region of the eddy (Drijfhout, 2003). In order to show the change of IAM before and after the eddy interacts with the island, the PVA at the surface layer (depth = 100 m) and deep layer (depth = 1000 m) at t = 30 days and t = 50 days are shown in Fig. 9. Compared with the PVA field at t = 30 days, the maximum upper anticyclonic PVA decreases at t = 50 days because of the splitting while maximum lower cyclonic PVA increases.

Observational data, including satellite images and in situ measurements, indicate that when an eddy collides with a continental slope or a small island there is no eddy-splitting, only changes to its trajectory (Jacob et al., 2002; Nan et al., 2011; Wei and Wang, 2009). In order to find out the parameter ranges of eddy-splitting, we use a series of islands with different diameters at the same location in the model. Before that, interactions of different sized islands and eddies were investigated. Take, for example, the eddies with 90 km (Eddy90) and 60 km (Eddy60) in diameter, the eddy-splitting pattern of Eddy90 interacting with an island of 120 km diameter is similar to that of Eddy60 interacting with an island of 90 km (Fig. 10). Although the islands and eddies are all different in size in comparison, they have approximately the same ratio of the island radius to the eddy radius in each experiment.

We, therefore, define two dimensionless parameters R and S to represent the size and submergence depth of an obstacle, namely, R=RobRed,S=DsbDed, where Rob is the radius of an obstacle; Red is the radius of an eddy; Dsb is the seamount submergence depth and Ded is the vertical extent of an eddy.

The eddy collides with the islands in 20 days and interacts with them as we have described previously. Figure 11 shows when the island is small enough, namely R < 1/4, the eddy does not split. Instead, the eddy will move through the obstacle, although the eddy structure deforms during the interaction process, and then recovers back after the interaction. As the island diameter increases, the “passing through” eddy gradually turns to splitting as a result of the eddy–island interaction. The eddy-splitting happens in the parameter range of 1/4 < R < 2. As the island diameter increases to R > 2, a filament splits-off from the eddy. This phenomenon is not considered as eddy-splitting in this study. In the last example, when the eddy collides with a solid wall (which can be seen as an island with an infinite diameter), the eddy propagates to the higher latitude along the boundary, which agrees with previous studies (Wei and Wang, 2009).

From the eddy-splitting processes with different sizes of islands, we can find that the locations of the secondary eddy split-off are related to the island size. Figure 12 shows the position relationship of the two eddies and the island. The angle (θ) between the secondary eddy and the position of collisional origin varies with the different island sizes (R). The distribution of the angle (θ) and the island size (R) are shown in Fig. 13. The fitting curve demonstrates that the empirical relation between the angle and the island size can be written by θ∼f(R)=2.6R-0.663, where f(R) is the angle (rad) between the two split eddies related to the island.

In natural oceans, islands are just part of the topography and there are more seamounts which are submerged under the sea surface. The effect of seamounts on ocean dynamics is different from that of islands. The submergence depth and the size of a seamount are key factors in the eddy-splitting. During the interaction between an eddy and a seamount, the lower part of the eddy is affected directly by the solid seamount while the upper part is not, then the vertical structure of the eddy is deformed significantly. As a result, its trajectory and splitting process is different from that of the interaction between an eddy and an island.