1 Introduction

The Gulf of Mexico (GM) is a marginal sea of the Atlantic Ocean. It is connected to the Caribbean Sea by the Yucatan Channel (YC) in the south and to the Atlantic Ocean by the Straits of Florida in the east (Fig. 1). The Loop Current (LC), which flows into the GM through YC with transport of 23 ~ 28 Sv (Candela et al. 2002; Sheinbaum et al. 2002; Athié et al. 2020), is the dominant feature of the circulation in the eastern GM (e.g., Oey et al. 2005). As part of the western boundary current of subtropical Atlantic, the LC has been proved to be important in regulating the variability of the oceanic circulation (Enfield et al. 2001; Xu et al. 2013; Buckley and Marshall 2016) and overlying atmosphere (e.g. Hong et al. 2000; Shay 2009).

Fig. 1
figure 1

Bathymetry (colored shading; unit: m) based on Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) in the Gulf of Mexico, Caribbean Sea, and western Atlantic Ocean. The isobaths of 500, 1500, 2500, and 3500 m are indicated by gray lines

Being an intense oceanic current, LC is characterized by large meander and pinched-off eddies. Based on both observations and model simulations, it is found that the LC experiences an anticyclonic eddy (with diameters of 250 ~ 300 km) detachment every 3 ~ 18 months (Cooper et al. 1990; Forristall et al. 1992; Sturges and Leben 2000). Since the ideal numerical experiment by Hurlburt and Thompson (1980), researchers have explored the underlying dynamics of eddy shedding in the GM based on the accumulated observations and model data (e.g., Vukovich 1988; Sturges 1994; Oey 1996; Sturges and Leben 2000; Vukovich 2007; Chang and Oey 2011, 2012; Liu et al. 2016; Lugo-Fernández et al. 2016; Weisberg and Liu 2017; Chiri et al. 2019). By now, several mechanisms have been proposed and can be divided into the following three categories. The first group (Pichevin-Nof mechanism; Pichevin and Nof 1997; Nof 2005) attributes the eddy detachment to the competition between β effect and eddy growth rate. Fueled by the mass influx from the YC, the meander of the LC begins to develop. With the accumulation of water and negative potential vorticity, the meander keeps growing larger and thus forms an anticyclonic eddy. Finally, the eddy is pinched-off when the westward Rossby wave speed exceeding its growth rate. In this theory, LC transport through YC is the major factor that influences the eddy shedding. Rather than focusing on the meander, the second group treats the LC evolution and eddy shedding processes as the intrinsic oceanic variability of a nonlinear dynamical system. The characteristics of this system (shedding interval, eddy size, and meander growth etc.) are regulated by several parameters (lateral viscosity, topography and inflow transport, etc.). On the basis of the above two theories, a lot of attention has been paid on the processes associated with YC transport, such as upstream flow (e.g., Welsh and Inoue 2000; Bunge et al. 2002; Chang and Oey 2011) and local wind field (e.g., Murphy et al. 1999; Candela et al. 2002; Oey 2004; Chang and Oey 2013). With the development of high-resolution observational methods and numerical models, the third group pays attention to the eddy-mean flow interaction in the GM, especially the energetics. Drawn energy from the inertial current, BC (e.g., Yin and Oey 2007; Oey 2008; Xu et al. 2013; Donohue et al. 2016; Hamilton 2019) and BT (Yang et al. 2020) are found to affect the meander growth and eddy formation based on the in situ data and model simulations. Yang et al. (2020) investigate the relative contributions of BT and BC during the formation of the eddy shedding. Yet, a work that addresses the evolution of energetics during the cycle of eddy shedding based on abundant eddy shedding events is still absent. On the one hand, suffering from the span coverage of mooring arrays, the observations in this region fail to provide a full picture of the eddy-mean flow energy transfer. On the other hand, energetic analyses based on high resolution models are mainly focused on long-term effect or state-mean value and thus cannot give insightful information about the energy exchange during the shedding processes. Therefore, it is necessary to revisit the energetics in this region, especially during the eddy shedding processes.

In addition to the 100-km-scale anti-cyclonic eddy, observations also capture cyclonic eddies in the vicinity of the LC meander (e.g., Cochrane 1972; Paluszkiewicz et al. 1983; Hamilton 1992; Oey and Lee 2002). Cochrance (1972) is the first scholar to find two cyclonic eddies locating at the west of the LC near the Campeche Bank and the east of the LC near the West Florida Shelf, respectively. Subsequent studies further indicate that these cyclonic eddies are generated at any time of the LC cycle with an irregular period of 75 days (Hurlburt 1986), which contributes to the detachment of eddy from LC (Cochrane 1972; Vukovich and Maul 1985; Fratantoni et al. 1998; Chérubin et al. 2006; Oey 2008; Le Hénaff et al. 2012; Huang 2013; Androulidakis et al. 2014; Rudnick et al. 2015). The generation of cyclonic eddy is investigated based on observation and model simulations. Oey (2008) shows that BC of the LC can generate the cyclonic eddy using a high-resolution numerical model. In an eddy-shedding case study based on observation and model, Xu et al. (2013) also find that small cyclones are caused by BC. On the other hand, Chérubin et al. (2006) suggest the contribution of BT in the generation of cyclonic eddy in the surface layers. By now, it remains unsolved that whether BT or BC dominates the generation of cyclonic eddy.

The abovementioned issues motivate us to investigate the energetics during the process of eddy shedding in the GM region. Here, we will focus on the energy exchange between mean flow and eddy during the shedding processes based on model product. The rest of paper is organized as follows: Sect. 2 and Sect. 3 briefly describe the data and analysis method, respectively. In Sect. 4, a detailed exploration of energetics associated with the eddy shedding in GM is presented. The discussion is in Sect. 5. This paper ends with a summary and conclusions in Sect. 6.

2 Data

2.1 Model data

In this study, Estimating the Circulation and Climate of the Ocean (ECCO) Phase II product (ECCO2, http://apdrc.soest.hawaii.edu/data/; cube92 version) is used. Based on the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997), the global model solves the primitive equations on the cube-sphere grid with a horizontal resolution of 0.25° (Adcroft et al. 2004). It has 50 levels in vertical direction with thicknesses varying from 10 m near the surface to 456 m near the bottom. Quadratic drag law and biharmonic friction are used to parameterize the bottom stress and horizontal viscosity, respectively. The K-profile parameterization (KPP) vertical mixing scheme from Large et al. (1994) is employed to parameterize subgrid-scale vertical mixing processes. The eddy-permitting ECCO2 ocean state estimate is obtained by a least square fit of the MITgcm to available observations. Using the Green function approach (Menemenlis et al. 2005), the least squares fit is applied for number of model control parameters, initial conditions, and boundary conditions rather than modifying the data directly with artificial technics. With these optimized control parameters, the model is run forward freely without unphysical sources/sinks and artificial jumps. Thus, the solution is dynamically consistent and appropriate for process and budget analyses (Wunsch et al. 2009). The ECCO2 state estimate has been used to diagnose the eddy energetics both in the global ocean (Chen et al. 2014) and the Kuroshio Extension region (Yang et al. 2018). In this study, the 3-day-averaged dataset in the GM region (100°W–75°W, 15°N–35°N) from 1992 to 2016 is used.

2.2 Satellite observation

Satellite data can capture the characteristics of eddy shedding from LC (e.g., Zavala-Hidalgo et al. 2003; Athie et al. 2012). To validate the ECCO2 data in the GM region, the merged Sea Surface Height (SSH) product derived from measurements of several satellites (e.g., JASON-1 and 2, ENVISAT, ERS-1 and 2) is utilized in this study. This dataset is provided by Copernicus Marine Environment Monitoring Service (CMEMS, http://marine.copernicus.eu/) and is more efficient than the datasets using a single altimeter in resolving the mesoscale spatial and temporal variability of the ocean circulation (Ducet et al. 2000). The horizontal resolution of the absolute dynamic height (ADT) dataset is 0.25° at daily intervals. Similar to the ECCO2 data, the concurrent SSH fields during 1993–2016 within the region (100°W–75°W, 15°N–35°N) are analyzed in the study.

3 Method

In this study, eddy energetics of the eddy shedding in the GM region are analyzed using multiscale energy and vorticity analysis method (MS-EVA; Liang 2016). Based on wavelet analyses (Meyer wavelet is used in this study), MS-EVA decomposes time series into serval time-scale windows orthogonally without changing the total energy. Here, we decompose the variables into two windows where ~ 0 and ~ 1 represent mean flow and oceanic perturbations respectively:

$$A={A}^{\sim 0}+{A}^{\sim 1}$$

The cutoff period should be divided by integer power of 2 according to the length of data (details refer to Liang 2016). The LC is found to shed eddy with an interval of 9 months, which will be discussed in Sect. 4.2. Therefore, the cutoff period which is calculated from the length and temporal resolution of the dataset is designed as 285 (T/2 = 3044 × 3/26 = 142.6875) days for variables. Cyclonic eddies are generated at any time of the LC cycle with an irregular period of around 75 days (Hurlburt 1986). The cutoff period is designed as 72 (T/2 = 3044 × 3/28 = 35.6719) days for variables in Sect. 4.3. The budget equation for kinetic energy perturbation (EKE) is as follows:

$$\begin{array}{c}\underbrace{\frac{\partial EKE}{\partial t}}_{Tend}=\frac{\partial }{\partial t}{\rho }_{0}\frac{{\left({u}^{\sim 1}\right)}^{2}+{\left({v}^{\sim 1}\right)}^{2}}{2}=\underbrace{-{\rho }^{\sim 1}{w}^{\sim 1}g}_{BF}\underbrace{-\frac{1}{2}{\rho }_{0}\left\{{\left({\mathrm{vv}}_{H}\right)}^{\sim 1}:\nabla {\mathrm{v}}_{H}^{\sim 1}-\left[\nabla \bullet {\left({\mathrm{vv}}_{H}\right)}^{\sim 1}\right]\bullet {\mathrm{v}}_{H}^{\sim 1}\right\}}_{BT}\underbrace{-\nabla \bullet \left[\frac{1}{2}{\rho }_{0}{\left({\mathrm{vv}}_{H}\right)}^{\sim 1}\bullet {\mathrm{v}}_{H}^{\sim 1}+{p}^{\sim 1}{\mathrm{v}}^{\sim 1}\right]}_{Q}\\ \underbrace{+{\mathrm{v}}_{H}^{\sim 1}\bullet \left({A}_{M}{\nabla }^{2}{\mathrm{v}}_{H}^{\sim 1}+\frac{\partial }{\partial z}\mu \frac{{\partial \mathrm{v}}_{H}^{\sim 1}}{\partial z}\right)}_{{D}_{K}}\end{array}$$

Here, v = (u,v,w) and vH = (u,v) represent the full and horizontal velocity vectors, respectively. ρ is the density with reference value ρ0, and p indicates pressure. AM denotes the coefficients for horizontal viscosity, and µ is the corresponding vertical mixing coefficients which depend on the local state and mixing parameterization. The operator ∇ represents the three-dimensional gradient operator, and symbol “:” is defined as (AB):(CD) = (A·B)(C·D). The term on the right side of the Eq. (2) describe conversion of energy potential perturbation (EPE) to energy kinetic perturbation through buoyancy forcing (BF, associated with BC), energy transfer from mean flow through BT, nonlocal processes of energy flux divergence Q through advection and pressure work, and energy dissipation DK through friction, wind stress, and bottom drag. As certain variables are not available in the ECCO2 output, DK is not explicitly diagnosed and treated as residual in the budget.

A necessary step to estimate the eddy energy budget is to identify the eddy shedding processes. Our method for detecting the eddy shedding consists of two steps. First, the path of the LC jet axis within the GM is determined based on a contour line of a fixed SSH (Sasaki and Minobe 2015). To exclude the influence of seasonal evolution and long-term trend of background flow, the sea level anomaly is used in this study. The anomaly donates as sSLA. \(sSLA(x,y,t)=SSH(x,y,t)-\overline{SSH }(t)\), where \(\overline{SSH }(t)\) is spatial mean SSH for each time step in the domain region (100°W–75°W, 15°N–35°N). Then, we identify a path of a fixed continuous sSLA contour in the GM in the range from – 2 to 10 cm at 0.5-cm intervals and averaged surface absolute velocity along each continuous contour of 25 years. The results show that absolute velocity reaches maximum along 2.5-cm sSLA isoline, indicative of the main current axis (Fig. 2a). The second step involves calculating length of the LC at each time step and investigating its temporal fluctuation. If the length of the jet decreases by more than 320 km in 3 days (corresponding to diameter of 100 km), we consider this decreasing as an eddy detachment.

Fig. 2
figure 2

a sSLA (spatial mean SSH within 100°W-75°W,15°N-35°N is excluded) field (contour; unit: cm) and its standard deviation (colored shading; unit: cm) in the Gulf of Mexico from ECCO2. The thick black line denotes the axis of Loop Current. The black box indicates the area for eddy shedding energy budget analysis; b the same as (a) but based on satellite data. The thick black line denotes the axis of Loop Current. c Bathymetry of the GM (shading; m) and eddy centers at the time of detachment (circles). Red and yellow circles indicate category 1 and category 2 (last detachment), respectively. d The same as (c) but based on satellite data. The black stars in (c) and (d) are the averaged positions of eddy centers at the time of detachment

4 Results

4.1 The evaluation of ECCO2

Before exploring the eddy generation mechanism in the GM region, it is necessary to quantify whether ECCO2 can capture the characteristics of the LC and its eddy shedding as observed by satellite. Figure 2a and b show the mean and standard deviation of sSLA fields derived from ECCO2 and satellite observation, respectively. The pattern of the ECCO2-simulated LC resembles that from satellite observation. It enters the GM from the Caribbean Sea through the YC and then turns to northeast of the Campeche Bank, forming an intense anticyclonic flow. Finally, the LC enters the Atlantic Ocean through the Florida Strait and joins the Gulf Stream. The transport of the LC at YC is 23.6 Sv, which is also close to previous observations (e.g., Candela et al. 2002, 2019; Sheinbaum et al. 2002). In addition to the mean circulation, sSLA standard deviation reveals that both oceanic variability reaches maximum in the vicinity of the large meander. According to previous study, this maximum reflects the meander and eddy shedding processes (Yang et al. 2020). Moreover, eddies are found to be shed at 87.3°W ± 0.5°, 25.3°N ± 0.3° (eddy center when it is pinched-off, Fig. 2a,c) in ECCO2, which is also close to 88.0°W ± 2.1°, 26.2°N ± 1.9° (Fig. 2b,d) satellite observation. This suggests that ECCO2 reasonably simulates both the mean state and variability in the GM region.

Based on ECCO2, eddy shedding of the LC can be divided into two categories: (1) After an eddy is shed, it migrates southwestward directly and eventually decays in the western GM, and (2) the shed eddy rejoins the LC and detaches from the LC again within the following 3 months, and then it moves southwestward and decays. To make the statistical analysis more confident, here we define the occurrence as an eddy shedding event when the westward-moving eddy crosses the 90°W line. According to this definition, 34 events are identified in 25 years based on the ECCO2, indicative of a frequency about 8.8 month–1 on average. This result is close to satellite data (8.5 month–1) and previous studies (Sturges and Leben 2000; Chang and Oey 2011). It should be noted that more shed eddies move westward directly in ECCO2 (21 events in category 1 and 13 events in category 2, Fig. 2c), whereas they tend to rejoin the LC in observation (14 events in category 1 and 20 events in category 2, Fig. 2d). This discrepancy indicates that ECCO2 is not appropriate to investigate the process of eddy reattaching to the LC. For events belonging to category 2, we will focus on the time before the first detachment and after the last detachment.

4.2 Energetics of the eddy shedding

Figure 3 shows the ensemble sSLA maps for the 34 eddy shedding events with time indicating the days passed since the date of eddy detachment (for events with more than one detachment, time indicates the days before the first detachment and after the last detachment). In 150 ~ 45 days before shedding, the meander of the LC begins to form at the north of the YC and expands northward (Fig. 3a-c). Accompanied by its northward penetration, the LC strengthens, and its length continually elongates (Fig. 3i). In Fig. 3c, a prominent negative sSLA center begins to appear at the southeastern side of the LC (83.4°W, 24.13°N), and then it quickly strengthens in the following period and squeezes the east part of the LC (Fig. 3d). Finally, the neck of the meander is pinched off (Fig. 3e), resulting in the detachment of the anticyclonic eddy (Fig. 3f). The LC sharply shortens at the time of eddy detached from the LC (Fig. 3i). Meanwhile, the LC axis retreats to the south and flows along the north coast of Cuba (Fig. 3g). The westward speed of the pinched-off eddy is about 4.5 cm/s, consistent with observations (Vukovich 2007). Seventy-five days after the detachment, it enters the western Gulf and eventually decays (Fig. 3h,a and b). Accompanied by the above processes, the LC begins to reform (Fig. 3i), and the cycle repeats. One thing should be noted that these 34 cases present high regularity, which is idealized but still an advantage for understanding the processes and underlying dynamics.

Fig. 3
figure 3

The ensemble maps for the 34 events with time indicating the days relative to the date of eddy detachment (a)–(h). Ensemble of sSLA is shaded with color (unit: cm). White contour in the range from – 3 to 6 cm at 1.5-cm intervals. The thick black line in (a) ~ (h) denotes the ensemble axis of loop current (2.5 cm sSLA isoline). The time of panel in (a) ~ (h) denotes days before (negative value) or after (positive value) the detachment. i The ensemble length of the LC (black thick line) with time indicating the days relative to the date of eddy detachment. The gray shading is the standard deviation of 34 events

To clarify the dominant energy sources and sinks during this process, we examine the energy budget based on Eq. (2). Figure 4 presents the ensemble energy terms based on 34 eddy shedding events integrated in the upper 1500 m layer within selected region (Fig. 2). BT prominently dominates the energy budget in the process of eddy shedding (Fig. 4). Energy transfers from mean flow to eddy. To get a better understanding of the dynamics, spatial distribution of 1500-m-integrated BT (Fig. 5a-c) and BF (Fig. 5d-f) are also provided. In stage1 (– 150 to – 45 days), the LC intrudes and penetrates northward (Fig. 5a). Significant barotropic energy conversion is found along the LC, especially near its western part (Fig. 5a). Meanwhile, the net contribution of BF is negative (Fig. 4), indicating that energy flux from EKE backs to EPE. The LC system moves westward, resulting in the intrusion to Campeche Bank. The returning flow of LC comes across the topography, leading to the upwelling of denser water. During stage 1, the EKE tendency is positive (0.06 W/m2) but pretty small, and most of the generated energy is balanced by DK (Fig. 4). After the LC is well developed, its neck is squeezed and finally pinched off (– 45 to + 15 days, stage 2). An anticyclonic eddy is shed from the LC. A remarkable positive EKE tendency (0.3 W/m2) consists with the development of the eddy. The EKE tendency integrated over the up 1500 m reaches 0.7 × 109 W (Fig. 4). The growth of EKE is inhomogeneous. The EKE reaches the maximum around the day of eddy detachment. Compared to stage 1, BT experiences a significant increase caused by the much stronger shear and strain of the current. In particular, strong released barotropic energy is found at both the western part of the LC and the neck of the meander (Fig. 5b). By contrast, BF presents negative values in the vicinity of the LC, indicating more energy stored as EPE. BF makes minor contribution to the generation of eddy shedding. In stage2, parts of generated energy are diverged through BF and dissipated by DK (Fig. 4). After the detachment (+ 15 to + 90 days, stage 3), the anticyclonic eddy moves westward, and the energy transfer terms shrink (Figs. 4 and 5c,f). The YC “directly” leaves the GM through the Florida Strait (Fig. 5c). Two things should also be noted. First, the magnitude of BT is four times greater than BF (Fig. 5; note the different scales used for upper and lower panels in Fig. 5), proving that the released barotropic energy is the energy source of eddy shedding process, which is obtained by Yang et al. (2020) as well, and the contribution of BF can be neglected. Second, the nonlocal processes (Q) play the energy sink during the cycle of eddy shedding (Fig. 4). The value of Q cannot be ignored compared to the others in each stage. The details of pressure work’s contribution will be discussed in Sect. 5.

Fig. 4
figure 4

The ensemble EKE budget integrated over the up 1500 m in the selected region (Fig. 2a). Errors bars indicate the standard deviation of 34 cases

Fig. 5
figure 5

Ensemble horizontal distribution (colored shading; unit: W/m2) of BT (a) ~ (c) and BF (d) ~ (f). Contours are coinstantaneous mean sSLA isolines. The black box indicates the area for energy budget analysis

4.3 Energetics of cyclonic eddy

Ensemble analysis of 34 events in the last subsection indicates the existence of the cyclonic eddy at the southeastern side of the LC before eddy shedding (Fig. 3c,d). Besides the ECCO2, the cyclonic eddy is detected in observation as well. Thirty-two of those thirty-four eddy shedding events with observing cyclones are found in the southeastern side of the LC from satellite data. Previous research found cyclonic eddies could be generated around the LC as consequence of BT and BC (e.g., Vukovich et al. 1979; Elliot, 1979; Hurlburt 1986; Chérubin et al. 2006) or by the advection of YC (Candela et al. 2002; Athie et al. 2012). Therefore, we select the region (90°W-84°W, 23°N-27.5°N, black box in Fig. 7a) to investigate the cyclonic eddy. Next, we will explore its generation mechanism through energetic analysis based on Eq. (2). Considering the fast-growing nature of this cyclonic eddy, the cutoff period between mean flow and perturbation is modified to 72 days. Then, the terms of Eq. (2) are calculated for budget analysis.

Through case-by-case investigation, cyclonic eddy can be captured before anticyclonic eddy detachment based on the simulation of ECCO2. To clarify the dominant energy source for the cyclonic eddy, Fig. 6 presents the energy budget in stage 1 and stage 2, which are used for the anticyclonic eddy. The significant energy convergences are BFC and BTC (Fig. 6, subscript C means the energy flux of cyclonic eddy). The mean flow (period over 72 days) transfers energy to cyclonic eddy. To further understand the contribution of each term, the spatial distribution of main energy source is shown in Fig. 7 (different scales are used for upper and lower panels in Fig. 7). During stage 1, BFC dominates the energy budget. The denser water from the north sinks in the east part of the LC, indicative of the potential energy release. Remarkable positive values occur in the eastern part of the LC (Fig. 7c). Meanwhile, the alternatively positive and negative values of BTC along the axis make no effectivity contribution to the generation of cyclonic eddy (Fig. 7a). The generated energy is balanced by the nonlocal divergence term QC and energy dissipation term DKC. With the development of the meander, the value of spatially integrated BFC has doubled compared with that in stage 1 (Fig. 6). EKE gets more energy from EPE due to BC. In this stage, BTC makes great contribution to the generation of cyclonic eddy, which accounts for 28% of the energy source. Significant barotropic energy conversion is found along the LC, which can reach 0.06 W/m2, especially near its eastern part (Fig. 7b). The local value of BTC is twice than that of BFC (Fig. 7b,d). Similar to the GM scenario, energetics of the cyclonic eddy generated from the Kuroshio Loop in the Luzon Strait are also studied. Zhang et al. (2017) point out that the energy for eddy pairs southwest of Taiwan is primarily from BT of the mean flow. The net energy of BTC integrated in selected region is smaller than that of BFC resulting in the energy divergence in the eastern part of the LC. The negative value of BTC suggests energy transfer from EKE back to mean flow. To sum up, BF is the main energy source for the generation of cyclonic eddy based on the composed results; meanwhile, BT also plays an indispensable role.

Fig. 6
figure 6

The ensemble EKE budget integrated over the up 1500 m in the selected region (90°W-84°W, 23°N-27.5°N). Errors bars indicate the standard deviation of 34 cases. Subscript of each term represents cyclonic eddy

Fig. 7
figure 7

Ensemble horizontal distribution (colored shading; unit: W/m2) of BTC (a) ~ (b) and BFC (c) ~ (d). Contours are coinstantaneous mean sSLA isolines. The black box in (a) indicates the area for energy budget analysis

5 Discussion

Vertical velocity is hard to observe. The BF cannot be diagnosed from observation data. BC is used in energy analysis. Based on the MS-EVA (Liang 2016), \(BC=-\frac{{g}^{2}}{{2\rho }_{0}{N}^{2}}\left\{{\left(\mathrm{v}\rho \right)}^{\sim 1}\bullet {\nabla \rho }^{\sim 1}-\left[\nabla \bullet {\left(\mathrm{v}\rho \right)}^{\sim 1}\right]\bullet {\nabla \rho }^{\sim 1}\right\}\) BCC (subscript C means the energy flux of cyclonic eddy) is calculated. The mean state of BCC is combined from stage 1 and stage 2. Ensemble horizontal distribution of (Fig. 8a) shares the similar horizontal distribution with BFC. The spatial-integrated BCC and BFC vary consistently (Fig. 8b). BFC accounts for 32% of the value of BCC based on ECCO2. One thing should be noted that ECCO2 cannot find cyclonic eddy in west part of the LC. This is discrepancy with satellite observation, and reasons are unclear.

Fig. 8
figure 8

a The mean state of ensemble horizontal distribution (colored shading; unit: W/m.2) of BC. b The integrated BC (red) and BF (blue) in black box of (a)

During the cycle of eddy shedding process, the nonlocal processes of energy flux divergence Q obviously is negative to the generated energy (Fig. 4). This term includes the process of advection and pressure work. Maslo et al. (2020) find that 75% of the energy maintaining deep kinetic energy is transferred from upper layer to deep layer via PW. Therefore, we pay attention to the effect of PW during the cycle of eddy shedding. The term of Q in Eq. (2) is split into the flux of PW and other nonlocal process of energy flux divergence (QK) components:

$$Q=\underbrace{-\nabla\cdot\left[\frac12\rho_0\left({\mathrm{vv}}_H\right)^{\sim1}\cdot\mathrm v_H^{\sim1}\right]-\nabla_H\cdot\left(p^{\sim1}\mathrm v_H^{\sim1}\right)}_{Q_K}\underbrace{-\frac\partial{\partial z}\left(p^{\sim1}w^{\sim1}\right)}_{PW}$$

where \(w\) is the vertical velocity and. \(\nabla_H=\frac\partial{\partial x}\vec{i}+\frac\partial{\partial y}\vec{j}\) Taking Eq. (3) into Eq. (2), the budget equation can be written as follows:

$$\begin{array}{c}\underbrace{\frac{\partial EKE}{\partial t}}_{Tend}=\underbrace{-{\rho }^{\sim 1}{w}^{\sim 1}g}_{BF}\underbrace{-\frac{1}{2}{\rho }_{0}\left\{{\left({\mathrm{vv}}_{H}\right)}^{\sim 1}:\nabla {\mathrm{v}}_{H}^{\sim 1}-\left[\nabla \cdot {\left({\mathrm{vv}}_{H}\right)}^{\sim 1}\right]\cdot {\mathrm{v}}_{H}^{\sim 1}\right\}}_{BT}\underbrace{-\nabla \cdot \left[\frac{1}{2}{\rho }_{0}{\left({\mathrm{vv}}_{H}\right)}^{\sim 1}\cdot {\mathrm{v}}_{H}^{\sim 1}\right]-{\nabla }_{H}\cdot \left({p}^{\sim 1}{\mathrm{v}}_{H}^{\sim 1}\right)}_{{Q}_{K}}\\ \underbrace{-\frac{\partial }{\partial z}\left({p}^{\sim 1}{w}^{\sim 1}\right)}_{PW}\underbrace{+{\mathrm{v}}_{H}^{\sim 1}\cdot \left({A}_{M}{\nabla }^{2}{\mathrm{v}}_{H}^{\sim 1}+\frac{\partial }{\partial c}\mu \frac{{\partial \mathrm{v}}_{H}^{\sim 1}}{\partial z}\right)}_{{D}_{K}}\end{array}$$

To clarify the effect of PW during eddy shedding process, we examine the energy budget based on Eq. (4). The cutoff between eddy and mean flow is designed as 285 days. Figure 8a presents the budget below 1500 m in the whole region of GM, which is obviously dominated by PW. This indicates that PW is the main energy source of abyssal ocean in selected frequency. Positive value means that energy flux transfers from upper layer to the deep layer. Other nonlocal process of energy flux divergence can be ignored during eddy shedding (Fig. 9a). Below 1500 m, energy is transferred though PW and dissipated by DK. Before the eddy detachment from LC, the abyssal ocean is inactive. When eddy shed from LC, the energy transfers to the abyssal ocean through PW (Fig. 9b). The value of integrated EKE below 1500 m sharply increases at eddy shed. With the eddy moving westward in the upper ocean, the energy flux of PW and spatial-integrated EKE consistently decrease. Lagrange tracking is employed to investigate the work of PW on eddy in the upper 1500 m. A box (2.5° × 2.5°) tracks the eddy after detachment. The center of box is the position of eddy center. After eddy shed, EKE decreases with detachment day from LC. Only less than 40% of the EKE has been left in 120 days since detachment day. Tracking the eddy, the integrated PW can reach nearly 3 × 108 J in the coinstantaneous time. Approximately 50% of the dissipation of EKE is transferred from the upper layer to the deep layer by PW.

Fig. 9
figure 9

a The ensemble mean stage of EKE budget integrated over the below 1500 m in the Gulf of Mexico (unit: 108 W). Errors bars indicate the standard deviation of 34 cases. b The ensemble energy flux of PW (red, unit: 108 W) and EKE (blue, unit: 1014 J) with time indicating the days relative to the date of eddy detachment

6 Conclusions

Based on the ECCO2 data from 1992 to 2016 through the MS-EVA, the energetics during eddy shedding process in the GM are explored in this study. ECCO2 can capture the main characteristics of GM. The climatological Yucatan Transport is 23.6 Sv derived from ECCO2. During 1992 to 2016, 34 eddy shedding events are captured by ECCO2, out of which 21 eddies are moving directly westward after shedding from LC, and 13 eddies rejoin the LC and detach again. The ensemble analysis reveals that BT is the main energy source of eddy shedding; meanwhile, the BF is not contributing to the energy source during eddy shedding period. During the stage of eddy shedding, energy transfers from upper layer to the deep layer by PW. The PW is the main source of abyssal EKE, in accordance with the previous study (Maslo et al. 2020).

ECCO2 can also capture the cyclonic eddy before eddy shedding in the eastern side of the LC. In previous research (e.g., Oey 2008; Xu et al. 2013), BC is what generates the process. Based on ECCO2 results, the integrated BF dominates the energy budget. However, BT also plays significant role on generation of cyclonic eddy. The value of BT is as twice as that of BF in the region of eastern part of the LC. In previous research based on site observations, BC, instead of BF, is usually diagnosed because of the lack of vertical velocity. Even though BC changes in pace with BF, the latter only accounts for 30% of the value of BC. This leads to the underestimating of barotropic instability’s effect on generation of cyclonic eddy. Based on the results, BT is comparable to BC for the contribution of eddy generation.

Several problems remain. In ECCO2 simulation, the period of eddy shedding case is more uniform than observations. Meanwhile, in observation, the process of eddy shedding is more complex, and more eddy shedding events belong to category 2. Answers may be found by improving the spatial and temporal resolution of model. The simulation of cyclonic eddy is not perfect; the generation of cyclonic eddy needs further study. Other energetics methods (e.g., Chen et al. 2014; Kang and Curchitser 2015) should be analyzed. As a part of AMOC, the transport of the LC may vary in low frequency, which may modulate the eddy shedding process. Therefore, the variation of eddy shedding in low frequency is expected to be investigated in future studies using high-resolution couple models.