Energetics of Multiscale Interactions in the Agulhas Retroflection Current System

Mengmeng Li; Chongguang Pang; Xiaomei Yan; Linlin Zhang; Zhiliang Liu

doi:10.1175/JPO-D-21-0275.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Numerical simulation
- a. Numerical simulation settings
- b. Evaluation of the ECCO2 reanalysis against data
3. Methodology
- a. Multiscale window transform
- b. Localized multiscale energy and vorticity analysis
4. Results
- a. Characterization of multiscale energy reservoirs
- b. Characterization of local multiscale interactions between reservoirs and nonlocal energy transport within the reservoirs
5. Summary and discussion
Acknowledgments.
Data availability statement.
APPENDIX
- Spatial Correlations between the Model and Observational Results
REFERENCES

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Fig. 1.

Altimeter-based mean sea surface height (white contours; contour interval: 0.1 m) and bathymetry (shaded areas; m) in the study region. The black lines are the bathymetry contours of 0, 300, and 1000 m. Three red arrows mark the northern Agulhas Current (NAC), SAC, and BC. Two green arrows highlight the ARC and the ACC.

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Fig. 2.

Long-term mean and vertically averaged (upper 1500 m) EAPE^* (colored areas; J m⁻³) obtained from (a) Argo data and (b) ECCO2 outputs. Long-term mean surface EKE^* (colored areas; J m⁻³) and SSH (white contours; contour interval: 0.2 m) derived from (c) ADT and (d) ECCO2 outputs. Four labeled boxes mark the four subdomains examined in this work: the retroflection (S1), rings drift (S2), meanders (S3), and stable (S4) regions. In (b), the letters A, B, C, and D denote the four vertices of the rings drift region (S2), the black dotted line connecting the midpoints of BC and AD represents the horizontal axis of the rings drift subdomain (S2) part in Fig. 7, the black solid line connecting the midpoints of AB and CD represents the horizontal axis in Fig. 14, the white solid and dashed lines indicate the sections examined in Figs. 5a,b and Figs. 5c,d, respectively. In (c) and (d), the black solid lines are the 0.5- and 0.1-m MSSH contours in the ADT and ECCO2, respectively, and the magenta pentagrams mark the westernmost retroflection points.

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Fig. 3.

The power spectra for area-mean SSH averaged over the four subdomains indicated in Fig. 2. The blue and red solid lines show results from the ECCO2 and ADT, respectively, with the blue and red dashed lines denoting the 95% confidence level. The black dashed lines indicate the 192- and 3072-day cutoff periods for the separation of the three scale windows marked in (a).

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Fig. 4.

Schematic of the energy flow for the three-window decomposition. Orange arrows represent energy transfers within different scale windows, green arrows indicate buoyancy conversions, blue solid and dotted arrows illustrate nonlocal transports due to advection and pressure work, respectively, and gray dotted arrows denote the forcing and dissipation processes.

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Fig. 5.

Vertical distributions of the time-mean (a) meridional velocity (m s⁻¹) along 38.0°S (the white solid line in Fig. 2b) and (c) zonal velocity (m s⁻¹) along 21.0°E (the white dashed line in Fig. 2b) and the corresponding velocity variance (m² s⁻²) along (b) 38.0°S and (d) 21.0°E, respectively. The superimposed black solid lines are isotherms with an interval of 2°C. In (a) and (c), the dashed and dotted lines are ±0.1 m s⁻¹ and 0 m s⁻¹ velocity contours, respectively. In (b) and (d), the dashed lines are the 0.02 m² s⁻² velocity variance contours.

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Fig. 6.

Horizontal structures of the vertically (upper 1500 m) averaged (a) MKE, (b) LKE, (c) EKE, (d) MAPE, (e) LAPE, and (f) EAPE (J m⁻³). White solid lines are the 0-, 0.2-, and 0.4-m MSSH contours. The four boxes marked with red dashed lines are the four subdomains indicated in Fig. 2: the retroflection region (S1), the rings drift region (S2), the meanders region (S3), and the stable region (S4).

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Fig. 7.

Vertical distributions of the short side-averaged^² multiscale energy components (J m⁻³). The superimposed black lines are contours of the (a)–(c) corresponding meridional velocity (solid lines for northward and dashed lines for southward; contour interval: 6 cm s⁻¹) or (d)–(f) perturbation density (contour interval: 0.1 kg m⁻³). The four boxes marked with gray lines are the four subdomains indicated in Fig. 2: the retroflection region (S1), the rings drift region (S2), the meanders region (S3), and the stable region (S4). Notice that the y axis of 0–50 m is elongated to show the near-surface structures more clearly.

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Fig. 8.

Meridional–vertical distributions of the zonally (14.0°–24.0°E) averaged multiscale energy components (J m⁻³) in the retroflection region (S1). The superimposed black lines are contours of the (a)–(c) zonally averaged velocity (solid lines for eastward and dashed lines for westward; contour interval: 3 cm s⁻¹) or (d)–(f) perturbation density (contour interval: 0.08 kg m⁻³). Notice that the y axis of 0–50 m is elongated to show the near-surface structures more clearly.

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Fig. 9.

Horizontal structures of the depth-averaged (0–1500 m) multiscale energetics (10⁻⁴ W m⁻³). Superimposed black lines are the 0-, 0.2-, and 0.4-m MSSH contours. The four boxes marked with red dashed lines are the four subdomains indicated in Fig. 2: the retroflection region (S1), the rings drift region (S2), the meanders region (S3), and the stable region (S4).

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Fig. 10.

As in Fig. 9, but for the nonlocal energy transports.

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Fig. 11.

Vertical distributions of the area-mean multiscale energetics (10⁻⁴ W m⁻³) averaged over the four subdomains indicated in Fig. 2.

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Fig. 12.

Meridional–vertical distributions of the zonally (14.0°–24.0°E) averaged multiscale energetics (10⁻⁴ W m⁻³) in the retroflection region (S1). Black lines are the contours of the (a)–(c),(j)–(l) zonal mean velocity (solid lines for eastward and dashed lines for westward; contour interval: 3 cm s⁻¹) or (d)–(i) perturbation density (contour interval: 0.08 kg m⁻³). Notice that the y axis of 0–50 m is elongated to show the near-surface structures more clearly.

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Fig. 13.

Schematics of the volume-averaged multiscale energy budget for the four subdomains indicated in Fig. 2. The multiscale energy components are in units of J m⁻³, and the energy transfers, conversions, and transports are in units of 10⁻⁶ W m⁻³. Notice that the most important terms are highlighted with thick arrows and bold red texts.

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Fig. 14.

As in Fig. 12, but for the long side-averaged multiscale energetics in the rings drift region (S2). The superimposed black lines in (a)–(c) and (j)–(l) are the contours of velocities parallel to the line AB. Solid and dashed lines denote northwestward and southeastward velocities, respectively.

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Fig. 15.

As in Fig. 12, but for the zonally averaged (24.0°–36.0°E) multiscale energetics in the meanders region (S3). The red and blue boxes mark the positions of the meander crests and trough, respectively.

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Fig. 16.

As in Fig. 12, but for the zonally averaged (36.0°–46.0°E) multiscale energetics in the stable region (S4).

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Fig. A1.

Horizontal distributions of the spatial correlation coefficients for (a) surface EKE^* and (c) MSSH between the ECCO2 and ADT results, and for (b) depth-averaged (0–1500 m) EAPE^* between the ECCO2 and the Argo-based results. White lines indicate the 0.4 and 0.9 correlation coefficients. Gray lines are the 0-, 0.2-, and 0.4-m ECCO2 MSSH contours. The four boxes marked with red dashed lines are the four subdomains indicated in Fig. 2: the retroflection region (S1), the rings drift region (S2), the meanders region (S3), and the stable region (S4).

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Editorial Type:: Article

Article Type:: Research Article

Energetics of Multiscale Interactions in the Agulhas Retroflection Current System

Mengmeng Li

Mengmeng Li ^aKey Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China
^cUniversity of Chinese Academy of Sciences, Beijing, China

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Chongguang Pang

Chongguang Pang ^aKey Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China
^bPilot National Laboratory for Marine Science and Technology (Qingdao), Qingdao, China
^dCenter for Ocean Mega-Science, Chinese Academy of Science, Qingdao, China

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Xiaomei Yan

Xiaomei Yan ^aKey Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China
^bPilot National Laboratory for Marine Science and Technology (Qingdao), Qingdao, China
^dCenter for Ocean Mega-Science, Chinese Academy of Science, Qingdao, China

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Linlin Zhang

Linlin Zhang ^aKey Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China
^bPilot National Laboratory for Marine Science and Technology (Qingdao), Qingdao, China
^dCenter for Ocean Mega-Science, Chinese Academy of Science, Qingdao, China

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Zhiliang Liu

Zhiliang Liu ^eResearch Center for Marine Science, Hebei Normal University of Science and Technology, Qinhuangdao, China
^fHebei Key Laboratory of Ocean Dynamics, Resources and Environments, Hebei Normal University of Science and Technology, Qinhuangdao, China

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Online Publication:: 19 Jan 2023

Print Publication:: 01 Feb 2023

DOI:: https://doi.org/10.1175/JPO-D-21-0275.1

Page(s):: 457–476

Received:: 23 Nov 2021

Accepted:: 01 Oct 2022

Published Online:: 19 Jan 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Using the recently developed multiscale window transform and multiscale energy and vorticity analysis methods, this study diagnoses the climatological characteristics of the nonlinear mutual interactions among mesoscale eddies, low-frequency (seasonal to interannual) fluctuations, and the decadally modulating mean flow in the Agulhas Retroflection Current System (ARCS). It is found that mesoscale eddies are generated primarily in the retroflection region by mixed barotropic and baroclinic instabilities. The barotropic instability dominates the generation of eddy kinetic energy (EKE) here, contributing power roughly 10 times larger than the baroclinic one. These locally generated eddies are transported away. In the rings drift and meanders regions, the nonlocal transport serves as an important energy source for the eddy field, making a contribution comparable to that of the baroclinic instability for the EKE production. Contrarily, in the stable region, the EKE is generated mainly due to the baroclinic instability. In most of the ARCS area, the kinetic energy (KE) is further transferred inversely from mesoscale eddies to other lower-frequency motions. In particular, in the retroflection, rings drift, and stable regions, the inverse KE cascade plays a leading role in generating seasonal–interannual fluctuations, providing roughly 3–5 times as much power as the forward KE cascade from the mean flow and the advection effect. In the meanders region, however, the forward cascade contributes 4 times more KE to the low-frequency variabilities than the inverse one. All the results provide a model-based benchmark for future studies on physical processes and dynamics at different scales in the ARCS.

Corresponding author: Xiaomei Yan, yanxiaomei@qdio.ac.cn

Abstract

Corresponding author: Xiaomei Yan, yanxiaomei@qdio.ac.cn

Keywords: Indian Ocean; Eddies; Instability; Mesoscale processes

1. Introduction

As the western boundary current of the south Indian Ocean, the Agulhas Current (AC) is one of the most energetic regions in the world (Lutjeharms 2006; Beal et al. 2015). Particularly, it links the Indian and Atlantic Oceans together and plays a fundamental role in the global ocean–atmosphere system (Lutjeharms 2006). The Agulhas Retroflection Current System (ARCS) is the southern extension of the AC, consisting of the southern Agulhas Current (SAC) and the Agulhas Return Current (ARC) (Wells et al. 2000; Fig. 1). The SAC shows sideways-meandering and shear edge eddies influenced by the gentle and wide continental shelf below it (Lutjeharms et al. 1989, 2003b; Tedesco et al. 2019; Schubert et al. 2021). It is bounded by Port Elizabeth (∼33.5°S) to the north and moves southwestward as a free jet until ∼37.0°S (Beal and Bryden 1997). Then the current turns tightly eastward, as the ARC, along which there exist two stable meander crests near 26.0° and 33.0°E with a trough between them (Belkin and Gordon 1996). The AC also leaks from the Indian Ocean into the Atlantic Ocean and facilitate the interbasin exchanges. This Agulhas leakage plays a significant role in the Atlantic meridional overturning circulation and even the global oceanic circulation as well as the climate (Peeters et al. 2004; Biastoch et al. 2008; Beal et al. 2011).

Fig. 1.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

The ARCS is characterized by vigorous intraseasonal to interannual variabilities (de Ruijter et al. 1999a; Lutjeharms 2006; Biastoch et al. 2009; Putrasahan et al. 2016; Elipot and Beal 2018; Zhu et al. 2018). Large-amplitude rings and other active mesoscale eddies have been identified from satellite altimeter observations (Gordon and Haxby 1990; van Ballegooyen et al. 1994; Goñi et al. 1997; Chelton et al. 2011) and hydrographic measurements (Schmitz 1996; Braby et al. 2016; Souza et al. 2011) and are usually analyzed by numerical models (Chassignet and Boudra 1988; Raith et al. 2017; Holton et al. 2017; Tedesco et al. 2022). In particular, in the Agulhas retroflection region, anticyclonic Agulhas rings (ARs) are shed and propagate westward, acting as vehicles transporting warm, salty water and energy from the southwestern Indian Ocean to the southern Atlantic Ocean (Feron et al. 1992; Schouten et al. 2000). Compared with eddies and rings in other western boundary currents, the ARs are more linear and have mechanical energies of approximately an order of magnitude higher (Olson and Evans 1986). In recent decades, the formation, propagation trajectories, and volume transports of the ARs have been studied extensively (Schouten et al. 2000; Weijer et al. 2013; Beron-Vera et al. 2013; Raith et al. 2017; Laxenaire et al. 2018). Based on energetics analyses, it is revealed that mesoscale eddies in the retroflection region, including the ARs, are generated due to mixed barotropic–baroclinic instabilities (Chassignet and Boudra 1988; Wells et al. 2000; Dijkstra and de Ruijter 2001; Weijer et al. 2013; Loveday et al. 2014; Tedesco et al. 2022). Using a high-resolution numerical simulation (∼1 km), Schubert et al. (2020) further suggested that mesoscale eddies here could be strengthened by submesoscale eddies. Previous studies also indicated that the ARs mainly propagate from the Agulhas retroflection region into the southern Atlantic Ocean (Schouten et al. 2000; Beron-Vera et al. 2013) and some of them might propagate back to the Indian Ocean and reabsorbed by the ARC (Gordon et al. 1987; Lutjeharms and van Ballegooyen 1988; Laxenaire et al. 2018).

On the whole, there are several factors involved in the variation of eddy kinetic energy (EKE) within a fixed ocean domain: the local canonical transfer resulting from nonlinear multiscale interactions, the nonlocal EKE transport via advection by the eddies and mean flow and due to eddy pressure work, the external wind and buoyancy forcings, and dissipation processes through mixing and bottom drag. Here we mainly focus on the former two factors. Regarding the local one, energy transfers between different scales occur in the ARCS. In the Agulhas retroflection region, as mentioned above, previous studies have shown that the barotropic and baroclinic instabilities of the background currents are two important mechanisms responsible for the formation of mesoscale eddies with the barotropic instability being dominant (Wells et al. 2000; Dijkstra and de Ruijter 2001; Weijer et al. 2013). These studies have examined the interaction between mesoscale eddies and the AC/ARC. However, in addition to mesoscale eddies and the strong mean flow, other low-frequency variabilities on seasonal and interannual time scales are also prominent in the ARCS (Lutjeharms 2006; Biastoch et al. 1999; Bryden et al. 2005; Krug and Tournadre 2012). Sérazin et al. (2018) and Yang and Liang (2019c) have qualitatively pointed out that the inverse energy transfer from mesoscale eddies could contribute to low-frequency fluctuations here. Therefore, the interactions among processes in these three scale windows, especially those associated with the low-frequency variabilities, still need a systematic quantitative analysis. In terms of the nonlocal factor, using a high-resolution (∼2.5 km) numerical simulation, Tedesco et al. (2022) confirmed that the locally generated EKE is significantly transported away from the SAC region. Contributions of these exported EKE to the downstream eddy fields, corresponding to the westward (i.e., toward the Atlantic Ocean) and eastward (i.e., toward the Indian Ocean) propagation of mesoscale eddies like the ARs, also remain to be quantitatively examined.

Motivated by the aforementioned issues, this study uses a newly developed methodology, the multiscale window transform (MWT) and the MWT-based localized multiscale energy and vorticity analysis (MS-EVA) (Liang and Robinson 2005, 2007; Liang and Anderson 2007), to diagnose the multiscale energetics in the ARCS. Associated fields are separated into three parts: mesoscale eddies, seasonal to interannual fluctuations, and the decadally modulating mean flow. The three-dimensional structures of multiscale energy reservoirs, local energy transfers, and nonlocal energy transports are investigated in detail.

The rest of this paper is organized in the following manner. First, the model configuration is briefly introduced and the model outputs are validated in section 2, and the MWT-based MS-EVA method is described in section 3. Then the major results are presented in section 4. Finally, section 5 summarizes the study.

2. Numerical simulation

a. Numerical simulation settings

We use the reanalysis outputs from the NASA project Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) to conduct the energetics analysis. The ECCO2 state estimate is based on the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997), which solves nonlinear primitive equations with the hydrostatic and Boussinesq approximations in the global ocean by employing a cube-sphere grid projection (cube92 version). It has an eddy-permitting horizontal resolution of 1/4° × 1/4° and 50 vertical levels with thicknesses ranging from 10 m near the surface to 456 m near the bottom (Menemenlis et al. 2005). Note that the model is essentially a free forward run based on a number of control parameters (e.g., forcing and viscosity) that are optimized by observations using the Green’s function approach (Menemenlis et al. 2005). Therefore, although constrained by observations, the ECCO2 state estimate is dynamically and kinematically consistent (Wunsch et al. 2009). It has been used extensively to examine mesoscale eddy dynamics and ocean energetics (Fu 2009; R. Chen et al. 2014; Z. Chen et al. 2014; Zemskova et al. 2015; Chen et al. 2016; Qiu et al. 2017; Yang et al. 2017; Zhu et al. 2018). In this study, the 3-day-averaged sea surface height (SSH), temperature, salinity, and velocity fields during 1993–2009 are used.

b. Evaluation of the ECCO2 reanalysis against data

To validate the ECCO2 reanalysis in the study region, the delayed gridded absolute dynamic topography (ADT) product operated by the Data Unification and Altimeter Combination System and distributed by the Copernicus Marine Environment Monitoring Service is used. This altimetric data has a spatial resolution of 1/4° × 1/4° and an interval of 1 day. To synchronize with the ECCO2 reanalysis, the ADT data from 1993 to 2009 is adopted. It is used to validate the modeled surface circulation and EKE. For the eddy available potential energy (EAPE) computed with ECCO2 outputs, it is compared with that derived from Argo floats by Roullet (2020). Here the EKE and EAPE fields are estimated as follows:

{EKE}^{*} = \frac{g^{2}}{2 f^{2}} \bar{[{(\frac{\partial h_{a}}{\partial x})}^{2} + {(\frac{\partial h_{a}}{\partial y})}^{2}]}, and

(1)

{EAPE}^{*} = \frac{g^{2}}{2 ρ_{0} N^{2}} \bar{ρ_{a}^{*}^{2}},

(2)

where g is the acceleration due to gravity, f is the Coriolis parameter, ρ₀ = 1025 kg m⁻³ is the reference density, h_a and

ρ_{a}^{*}

are the deviations from the long-term (1993–2009) mean SSH (MSSH) and potential density (ρ^*), respectively, and

N = \sqrt{- (g / ρ_{0}) (\partial \bar{ρ^{*}} / \partial z)}

is the buoyancy frequency. The overbar denotes the time average during 1993–2009. Note that to be consistent with the method used by Roullet et al. (2014) and Roullet (2020) in the EAPE estimation with the global Argo profiles, here the potential density ρ^* and density anomalies

ρ_{a}^{*}

are computed with respect to the local vertical position following Luecke et al. (2017). Besides, to differentiate from the EKE and EAPE in the multiscale energetics framework as illustrated later in this work, in this section, they are denoted as EKE^* and EAPE^*.

As demonstrated in Fig. 2, the mean oceanic circulations as well as the EAPE^* and EKE^* estimated using the ECCO2 outputs generally show similar patterns with the observations. Specifically, the Agulhas retroflection point, that is, the westernmost part of the 0.1-m (0.5-m) ECCO2 (ADT) MSSH contour marked with the magenta pentagram in Figs. 2c and 2d, is located at the longitude of ∼14.4°E (∼14.5°E) in the model (observation). Moreover, both the modeled and observed ARC meanders have one trough and two crests located at ∼30.0°, ∼26.0°, and ∼33.0°E, respectively. For the EAPE^*, EKE^*, and MSSH, the spatial correlation coefficients between the simulated and observational results in the entire study area are 0.64, 0.70, and 0.98, respectively, all of which are far beyond the 95% significance level. The detailed horizontal distributions of the correlations are presented in the appendix. It is worth mentioning that there is a discrepancy in the southwest Indian Ocean Subgyre where the EKE^* and EAPE^* from ECCO2 are larger than the observations (circled with a magenta ellipse in Figs. 2b,d). Previous studies have reported that eddies in this subgyre may come from the upstream of the AC and eddy shedding from the ARC meanders (Gründlingh 1978; Lutjeharms and Valentine 1988; de Ruijter et al. 1999b; Gründlingh 1995). A closer examination suggests that in the ECCO2, such eddy events occur more frequently and are individually more intense (figures not shown), resulting into higher EKE^* and EAPE^* then. One of the possible causes is that the ECCO2 is run forward freely with no observational data taken in directly. This needs a further investigation, which, however, is beyond the scope of this study. Here we mainly focus on the multiscale energetics in the ARCS (the four boxes labeled in Fig. 2) where the overall good correspondence between the model and observations lends support to the use of the ECCO2 outputs.

Fig. 2.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

According to the horizontal structures of the EKE^*, EAPE^*, and background currents, we divided the ARCS into four subdomains: the retroflection (S1), rings drift (S2), meanders (S3), and stable (S4) regions (Fig. 2). In each subdomain, the power spectrum of the area-mean SSH obtained from ECCO2 is compared with that from ADT. As illustrated in Fig. 3, the retroflection region (S1) shows particularly vigorous variabilities on intraseasonal (∼150 days) to interannual (4 yr) time scales, the rings drift (S2) and stable (S4) regions are dominated by the seasonal cycle (1 yr) and weak interannual variabilities (∼2–3 yr), and the meanders region (S3) is characterized by a broadband of peaks ranging from 60 days to 1 yr as well as interannual variabilities (>2 yr). In general, the spectra in the four subdomains have similar structures, showing peaks in the bands of 2–6 yr, 1 yr, and <200 days. Overall, the ECCO2 successfully reproduces the variabilities at these time scales. Large differences exist mainly at periods longer than 3000 days (∼8.5 yr). Further comparisons also show that the 3000-day low-pass filtered ECCO2 SSH is less consistent with the satellite observation (figure not shown), which can also be attributed to that the model is a free forward run as mentioned in section 2a. Nevertheless, the decadal (>8.5 yr) variability is insignificant and is much weaker than variabilities on intraseasonal to interannual time scales in the ARCS. In this work, the slowly modulating flow at periods longer than 8.5 yr is treated as the mean flow and we mainly focus on mesoscale (<200 days) and seasonal-to-interannual (1–6 yr) variabilities. Thus, the ECCO2 state estimate is appropriate for the analysis here.

Fig. 3.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

3. Methodology

Energetics analysis is a powerful approach to diagnose multiscale interactions and flow instabilities. This study adopts the novel multiscale energetics analysis tools known as the MWT and MS-EVA. The methods were developed by Liang and Robinson (2005, 2007) and Liang and Anderson (2007) and have been applied successfully in energetic analyses of the Kuroshio system (Yang and Liang 2016; Yang et al. 2017; Yang and Liang 2018, 2019a,b,c), the Gulf of Mexico (Yang et al. 2020), and the Iceland–Faeroe front system (Liang and Robinson 2004). In this section, a brief introduction of these methods is given.

a. Multiscale window transform

Liang and Robinson (2005) suggested that oceanic processes tend to occur on an exclusive range of scales (hereinafter referred to as windows) and developed a localized orthogonal complementary subspace decomposition apparatus, that is, MWT. The MWT is better than traditional filters as it can ensure a faithful representation of localized energy processes (Liang and Robinson 2005; Liang and Anderson 2007). Within the MWT framework, the kinetic energy (KE) and available potential energy (APE) are, respectively, calculated as follows:

K_{^{n}}^{ϖ} = \frac{1}{2} ρ_{0} {\hat{U}}_{n}^{\sim ϖ} \cdot {\hat{U}}_{n}^{\sim ϖ}, and

(3)

A_{^{n}}^{ϖ} = \frac{g^{2}}{2 ρ_{0} N^{2}} {({\hat{ρ}}_{n}^{\sim ϖ})}^{2},

(4)

where U = (u, υ, w) is the velocity vector, ρ is the density perturbation from the background profile

\bar{ρ} (z)

N = \sqrt{- (g / ρ_{0}) (\partial \bar{ρ} / \partial z)}

is the buoyancy frequency, and the operator

{\hat{(\cdot)}}_{n}^{\sim ϖ}

denotes MWT on window ϖ at time step n. For more details, please refer to Liang (2016).

As have been demonstrated in Fig. 3, fluctuations in the ARCS are dominated by a 2–6-yr interannual cycle, an annual cycle, and a broadband of peaks with periods shorter than 200 days that correspond to transient mesoscale eddies. Besides, the time scale defined in MWT needs to be a power of 2 (Liang 2016), and the time step of the ECCO2 data is 3 days. Therefore, using the MWT method, the original variables are decomposed into three orthogonal windows with cutoff periods of 192 days (= 3 × 2⁶) and 3072 days (= 3 × 2¹⁰; 8.5 yr), respectively. The three orthogonal windows are the decadally modulating mean flow window (>8.5 yr), mesoscale eddy window (<192 days), and low-frequency variability window between them. They are denoted as windows 0, 2, and 1, respectively, hereinafter, for easy reference.

b. Localized multiscale energy and vorticity analysis

Based on the MWT, the MS-EVA was developed to investigate complex dynamics that are nonlinear, spatiotemporally intermittent, and involve multiscale interactions. In the MS-EVA, the governing equations for the multiscale KE and APE are (Liang and Robinson 2005):

\begin{array}{l} \frac{\partial K_{n}^{ϖ}}{\partial t} = \underset{\nabla \cdot Q_{K, n}^{ϖ}}{\underset{︸}{- \nabla \cdot [\frac{1}{2} ρ_{0} {\hat{U}}_{n}^{\sim ϖ} {(\hat{U U_{H}})}_{n}^{\sim ϖ}]}} \underset{\nabla \cdot Q_{P, n}^{ϖ}}{\underset{︸}{- \nabla \cdot ({\hat{U}}_{n}^{\sim ϖ} {\hat{p}}_{n}^{\sim ϖ})}} \\ \underset{T_{K, n}^{ϖ}}{\underset{︸}{- [\frac{1}{2} ρ_{0} {({\hat{u}}_{n}^{\sim ϖ})}^{2} \nabla \cdot U_{u}^{ϖ} + \frac{1}{2} ρ_{0} {({\hat{υ}}_{n}^{\sim ϖ})}^{2} \nabla \cdot U_{υ}^{ϖ}]}} \underset{b_{n}^{ϖ}}{\underset{︸}{- g {\hat{ρ}}_{n}^{\sim ϖ} {\hat{w}}_{n}^{\sim ϖ}}} + F_{K, n}^{ϖ}, and \end{array}

(5)

\frac{\partial A_{n}^{ϖ}}{\partial t} = \underset{\nabla \cdot Q_{A, n}^{ϖ}}{\underset{︸}{- \nabla \cdot [\frac{g^{2}}{2 ρ_{0} N^{2}} {\hat{ρ}}_{n}^{\sim ϖ} {(\hat{ρ U})}_{n}^{\sim ϖ}]}} \underset{T_{A, n}^{ϖ}}{\underset{︸}{- A_{n}^{ϖ} \nabla \cdot U_{ρ}^{ϖ}}} + \underset{- b_{n}^{ϖ}}{\underset{︸}{g {\hat{ρ}}_{n}^{\sim ϖ} {\hat{w}}_{n}^{\sim ϖ}}} + F_{A, n}^{ϖ},

(6)

respectively, where U_H = (u, υ) is the horizontal velocity vector, p is the dynamic pressure,

U_{S}^{ϖ} = {(\hat{U S})}_{n}^{\sim ϖ} / {\hat{S}}_{n}^{\sim ϖ}

is the S-coupled velocity when S = u, υ, or ρ,

K_{n}^{ϖ}

and

A_{n}^{ϖ}

are the KE and APE, respectively, on window ϖ at time step n. Then

K_{n}^{0}

K_{n}^{1}

, and

K_{n}^{2}

represent the KE of the mean flow (MKE), low-frequency variability (LKE), and eddy field (i.e., EKE), respectively, while

A_{n}^{0}

A_{n}^{1}

, and

A_{n}^{2}

represent the APE of the mean flow (MAPE), low-frequency variability (LAPE), and eddy field (i.e., EAPE), respectively. The

F_{K, n}^{ϖ}

and

F_{A, n}^{ϖ}

terms stand for the external forcing and internal dissipation processes. In this study, we focus on the multiscale interaction terms listed in Table 1. Note that the term T^ϖ represents the cross-scale energy transfer that results from interwindow interactions. It satisfies the following property:

\sum_{ϖ} \sum_{n} T_{n}^{ϖ} = 0,

(7)

where

\sum_{ϖ}

and

\sum_{n}

are the summation over all of the scale windows ϖ and sampling time steps n, respectively (Liang 2016). We use Tⁱ^→^j to represent the energy transfer from window ϖ = i to window ϖ = j. All the terms listed in Table 1 are four-dimensional and retain all the localized information. The energy diagram for the three-window decomposition is summarized in Fig. 4.

Fig. 4.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Table 1

Symbols and meanings for multiscale energetics terms (ϖ denotes the scale window, i.e., ϖ = 0, 1, 2).

4. Results

a. Characterization of multiscale energy reservoirs

Before examining the spatial structures of the multiscale energy components, we first investigate the distribution of the background currents. Figure 5a shows the vertical distribution of the meridional velocity along 38.0°S. The meanders across the ARC are surface-intensified with the maximum velocity of approximately 0.3 m s⁻¹. The velocity core extends to 1200 m where the meridional velocity drops to 0.1 m s⁻¹. The two meander crests are located at ∼26.0° and ∼33.0°E, respectively, consistent with the results observed by Belkin and Gordon (1996). Figure 5c presents the vertical section of the zonal velocity along 21.0°E. The SAC and ARC are centered at 39.0° and 37.5°S, respectively, with the surface-intensified cores in the upper 1500 m. The extreme zonal velocities are approximately ±0.5 m s⁻¹. The Agulhas retroflection is centered at 38.2°S, 21.0°E (Figs. 5a,c), and the variance of the horizontal velocity is largely confined to the upper 1500-m layer (Figs. 5b,d). The eddy activity here is more intense than that along the NAC, where the eddy signals are generally confined to the upper 0–500 m (Zhu et al. 2018). Based on these results, only the energetics above 1500 m are analyzed in this study.

Fig. 5.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Figures 6a–c show the horizontal structure of the vertically averaged (0–1500 m) KE in each window. In general, the EKE is the strongest, implying the presence of energetic mesoscale eddies. Moreover, it is significantly elevated (>100 J m⁻³) in the retroflection region (S1; Fig. 6c). Using the traditional filtering method, Zhu et al. (2018) obtained a similar horizontal structure for the EKE in the study area. The MKE is maximized along the AC axis and decays rapidly as the ARC flows eastward (Fig. 6a). Enhanced LKE (>20 J m⁻³) appears in the retroflection region (S1) and also along the ARC east of 28.0°E (Fig. 6b). With such a noticeable variability, the LKE is believed to be nonnegligible in the multiscale energetics analysis in the ARCS.

Fig. 6.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

The horizontal distributions of the APE in each window are illustrated in Figs. 6d–f. The LAPE and EAPE exhibit similar spatial patterns as the LKE and EKE, respectively. The EAPE, in particular, is about 50% weaker than the EKE, indicating that the eddy energy is mainly stored in the form of KE. In contrast, the MAPE is approximately one order of magnitude stronger than the MKE (Figs. 6a,d), which is a generic feature in the global ocean (Gill et al. 1974; Ferrari and Wunsch 2009). Moreover, compared with other energy components, the MAPE shows a unique horizontal structure with high values spreading widely east of the AC and north of the ARC. The horizontal distribution of the MAPE is analogous to that of the MSSH. This is not surprising, since that for the wind-driven upper-ocean circulation at midlatitudes that is in geostrophic and hydrostatic balances to leading order, the MAPE is stored in the tilting isopycnals, which are generally opposite to the sea level slopes (Gill et al. 1974).

Figure 7 illustrates the vertical distributions of the multiscale energy components. It is clear that the KE components are surface-trapped and drop rapidly with depth in the upper 1000 m (Figs. 7a–c), whereas MAPE and EAPE exhibit interior maxima at depths of ∼200–300 m, which are comparable to the peaks at the surface. For the LAPE, high values (>100 J m⁻³) are limited primarily to the upper 100 m without obvious variations in the zonal direction. Such features of KE and APE are widely distributed in the global ocean (Wells et al. 2000; Roullet et al. 2014; Kang and Curchitser 2015; Yang and Liang 2016; Yan et al. 2019, 2022).

Fig. 7.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Notably, the retroflection region (S1), where the ARs are frequently observed to generate (Lutjeharms 2006; Souza et al. 2011; Raith et al. 2017; Holton et al. 2017), has the strongest variabilities. Hence the meridional–vertical distributions of the multiscale energy components in this subdomain are further examined in Fig. 8. The MKE is found to accurately depict the two velocity cores of the SAC and ARC, with the former being more energetic than the latter (Fig. 8a). Both the LKE and EKE are enhanced between the two jets (Figs. 8b,c). Using numerical simulations, Wells et al. (2000) found similar vertical structures of the MKE and EKE in this subdomain (see their Figs. 2a,b and 3a,b). For the MAPE near the surface, it decreases markedly southward as the density declines (Fig. 8d). Below the 100-m depth, the three APE components are increased around the center of the retroflection (∼38.2°S), also coincident with the horizontal structure of the perturbation density (Figs. 8d–f).

Fig. 8.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

b. Characterization of local multiscale interactions between reservoirs and nonlocal energy transport within the reservoirs

The horizontal distributions of the local multiscale interaction terms in each window are demonstrated in Fig. 9. In general, the eddy-mean flow energy transfers ( $T_{K}^{0 \to 2}$ and $T_{A}^{0 \to 2}$ ) and buoyancy conversions in the mean flow (b⁰) and mesoscale eddy (b²) windows are approximately 2–5 times greater than the other terms. Both the barotropic transfer $T_{K}^{0 \to 2}$ and the baroclinic transfer $T_{A}^{0 \to 2}$ are dominantly positive and negative along the SAC and ARC, respectively (Figs. 9a,d). Regarding the buoyancy conversions, b⁰ is the strongest and has positive and negative peak values stronger than 1.0 × 10⁻⁴ W m⁻³ appearing alternately along the SAC and ARC (Fig. 9g), and b² shows strong positive and negative values of about ±0.5 × 10⁻⁴ W m⁻³ along the SAC and the upstream of ARC (west of 32.0°E), respectively (Fig. 9i). The horizontal patterns of the eddy-mean flow energy transfers ( $T_{K}^{0 \to 2}$ and $T_{A}^{0 \to 2}$ ) as well as the eddy buoyancy conversion (b²) are consistent with those reported by Wells et al. (2000), R. Chen et al. (2014), and Zhu et al. (2018), all of which indicate the importance of barotropic and baroclinic instabilities for the generation of mesoscale eddies in the ARCS. Associated with the low-frequency variabilities, the barotropic transfer $T_{K}^{1 \to 2}$ is the strongest and is dominated by negative values (<−0.5 × 10⁻⁴ W m⁻³) in the ARCS (Fig. 9b), suggesting that the inverse KE cascade from eddies contributes to the generation of low-frequency fluctuations here. Based on spectral KE flux analyses and nonlinear multiscale interaction examinations, respectively, Sérazin et al. (2018; see their Fig. 3) and Yang and Liang (2019c; see their Fig. 5) obtained similar results in the Agulhas Current region.

Fig. 9.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Figure 10 shows the horizontal structures of nonlocal energy transports, including the pressure work ( $\nabla \cdot Q_{P}^{ϖ}$ ) and advection effect ( $\nabla \cdot Q_{K}^{ϖ}$ and $\nabla \cdot Q_{A}^{ϖ}$ ), in each scale window. These terms play important roles in redistributing the energies. They generally exhibit alternating positive and negative peaks in the study area. The pressure work on MKE ( $\nabla \cdot Q_{P}^{0}$ ) and EKE ( $\nabla \cdot Q_{P}^{2}$ ) as well as the advection of MKE ( $\nabla \cdot Q_{K}^{0}$ ) and EKE ( $\nabla \cdot Q_{K}^{2}$ ) are roughly 2–4 times stronger than the other terms. In particular, the EKE advection term ( $\nabla \cdot Q_{K}^{2}$ ) shows strong negative values (∼−1.0 × 10⁻⁴ W m⁻³) along the SAC and ARC and positive values (∼1.0 × 10⁻⁴ W m⁻³) in between (Fig. 10h). It implies that the EKE are advected from the two jets to the center of retroflection. Downstream of the ARC (east of 24.0°E), however, both the EKE advection ( $\nabla \cdot Q_{K}^{2}$ ) and EAPE advection ( $\nabla \cdot Q_{A}^{2}$ ) terms are weakly positive (<0.6 × 10⁻⁴ W m⁻³), suggesting the import of EKE and EAPE, respectively. Clearly, the nonlocal transport terms also contribute to the variation of eddy energies in the ARCS.

Fig. 10.

As in Fig. 9, but for the nonlocal energy transports.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Overall, in the horizontal direction, all the multiscale energetics terms are distributed inhomogeneously with enhanced values along the mainstreams of the SAC and ARC. Next, the vertical structures of these terms are further examined.

In Fig. 11, the vertical distributions of the area-mean energy transfer terms averaged over the four subdomains are depicted. In general, the barotropic ( $T_{K}^{0 \to 2}$ , $T_{K}^{1 \to 2}$ , and $T_{K}^{0 \to 1}$ ) and baroclinic ( $T_{A}^{0 \to 2}$ , $T_{A}^{1 \to 2}$ , and $T_{A}^{0 \to 1}$ ) terms reach their maximum values in the upper 100 m, whereas buoyancy conversions (b⁰, b¹, and b²) exhibit interior maxima at deeper depths where the APE components are also elevated (Fig. 7). Such vertical structures were also found in the Gulf Stream (Kang and Curchitser 2015), the Kuroshio (Yan et al. 2019; Yang and Liang 2019a), and the Kuroshio Extension (Yang and Liang 2018, 2019b) regions. The barotropic transfer $T_{K}^{0 \to 2}$ is strongly positive in the retroflection region (S1) with a maximum value of 0.9 × 10⁻⁴ W m⁻³ at the surface and weakly negative (>−0.2 × 10⁻⁴ W m⁻³) in the other three subdomains, implying that the eddy generation by the barotropic instability mainly takes place in the retroflection region (S1). In contrast, the area-mean buoyancy conversion b² and barotropic transfer $T_{K}^{1 \to 2}$ are positive and negative, respectively, throughout the upper 1500 m in all the four subdomains, suggesting that the EKE production due to the baroclinic instability and the inverse KE cascade from mesoscale eddies to low-frequency variabilities are two common phenomena in the upper ocean of the ARCS.

Fig. 11.

Vertical distributions of the area-mean multiscale energetics (10⁻⁴ W m⁻³) averaged over the four subdomains indicated in Fig. 2.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

1) The retroflection region (S1)

As shown in Figs. 9–11, the multiscale interactions in the retroflection region (S1) are the most pronounced. Figure 12 further demonstrates the meridional–vertical structure of the zonally averaged multiscale energetics in this region.

Fig. 12.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

It is found that the barotropic transfer $T_{K}^{0 \to 2}$ exhibits a large positive pool with values higher than 2.0 × 10⁻⁴ W m⁻³ in the SAC and a weak interior negative peak of approximately −0.5 × 10⁻⁴ W m⁻³ at the core of the ARC. It indicates that the EKE is generated due to the barotropic instability of the SAC, and a small portion of the EKE is transferred back to the ARC (Fig. 12a). With magnitudes weaker than 0.4 × 10⁻⁴ W m⁻³, the barotropic transfer $T_{K}^{0 \to 1}$ also shows positive and negative values in the SAC and ARC, respectively (Fig. 12c). Meanwhile, the barotropic transfer $T_{K}^{1 \to 2}$ is predominantly negative (<−0.5 × 10⁻⁴ W m⁻³) in the upper 500 m where the LKE is significantly enhanced (Fig. 12b). These results suggest that the LKE is generated both by the barotropic instability of the SAC and by the inverse KE cascade from mesoscale eddies. Regarding the baroclinic transfers, $T_{A}^{0 \to 2}$ is the strongest, with its magnitude comparable to the barotropic term $T_{K}^{0 \to 2}$ (Fig. 12d). Moreover, in the upper 500 m, it is dominantly positive (negative) in the SAC (ARC), indicative of the generation (dissipation) of EAPE there. Strong buoyancy conversions occur mainly below 100 m where the APE terms are elevated (Figs. 12g–i). In particular, the eddy buoyancy conversion (b²) below 100 m exhibits a similar structure as the baroclinic transfer $T_{A}^{0 \to 2}$ , also showing positive (negative) values along the SAC (ARC) with the strength weaker than 0.5 × 10⁻⁴ W m⁻³. All these results indicate that eddies are generated in the SAC through mixed barotropic ( $T_{K}^{0 \to 2} > 0$ ; MKE → EKE) and baroclinic ( $T_{A}^{0 \to 2} > 0$ and b² > 0; MAPE → EAPE → EKE) pathways, whereas in the ARC, eddies are dissipated via the inverse routes.

For the nonlocal terms, the eddy pressure work ( $\nabla \cdot Q_{P}^{2}$ ) is positive along the ARC and SAC jets and negative between them (Fig. 12j), while the EKE advection term ( $\nabla \cdot Q_{K}^{2}$ ) exhibits an opposite cross-stream structure (Fig. 12k). Both terms are dominated by negative values, implying that the EKE is mainly transported out of the subdomain. The EAPE advection term ( $\nabla \cdot Q_{A}^{2}$ ) in the upper 500 m is positive and negative along the ARC and SAC, respectively, indicating that the EAPE generated along the SAC through the baroclinic pathway ( $T_{A}^{0 \to 2} > 0$ ) would be advected southward to the ARC (Fig. 12l).

Figure 13a further shows the volume-averaged multiscale energy budget in the retroflection region (S1). Here the eddy-mean flow interactions ( $T_{K}^{0 \to 2}$ and $T_{A}^{0 \to 2}$ ) and nonlocal eddy energy transports ( $\nabla \cdot Q_{P}^{2}$ , $\nabla \cdot Q_{K}^{2}$ , and $\nabla \cdot Q_{A}^{2}$ ) are remarkable. On average, the energy is drained from the mean flow to the eddy field with the barotropic (baroclinic) term $T_{K}^{0 \to 2}$ ( $T_{A}^{0 \to 2}$ ) transferring 25.30 × 10⁻⁶ W m⁻³ (5.56 × 10⁻⁶ W m⁻³) of KE (APE). The EKE generation is dominant in this region, for which the barotropic ( $T_{K}^{0 \to 2} > 0$ , i.e., MKE → EKE) and baroclinic ( $T_{A}^{0 \to 2} > 0$ and b² > 0, i.e., MAPE → EAPE → EKE) instabilities contribute 25.30 × 10⁻⁶ and 2.28 × 10⁻⁶ W m⁻³, respectively. Clearly, it is the barotropic instability that dominates the production of EKE here. This phenomenon has been noticed in previous numerical studies (e.g., Wells et al. 2000; Dijkstra and de Ruijter 2001) and also theoretical analysis (e.g., Weijer et al. 2013). Regarding the nonlocal eddy energy transports, both the pressure work ( $\nabla \cdot Q_{P}^{2}$ ) and advection effect ( $\nabla \cdot Q_{K}^{2}$ and $\nabla \cdot Q_{A}^{2}$ ) are negative. The three terms transport 12.67 × 10⁻⁶, 5.87 × 10⁻⁶, and 2.84 × 10⁻⁶ W m⁻³ of EKE/EAPE out of this region, respectively. Another interesting feature is that for the low-frequency variabilities, the inverse KE cascade from mesoscale eddies ( $T_{K}^{2 \to 1}$ ) is dominant, which provides 3.81 × 10⁻⁶ W m⁻³ of power to the LKE, about 3 times as much as the contribution of the barotropic transfer $T_{K}^{0 \to 1}$ .

Fig. 13.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

Overall, the SAC and ARC serve as the energy source and sink for the eddy field, respectively. In the SAC, mesoscale eddies are generated due to mixed barotropic–baroclinic instabilities, with the barotropic instability being the dominant mechanism. In the ARC, contrarily, the eddy energies are dissipated via the inverse barotropic and baroclinic pathways. On average, the locally generated eddies provide energy for the localized low-frequency fluctuations via inverse KE cascades and for the nonlocal eddy field through the pressure work and horizontal advections. The pressure work acts as the largest sink of the eddy energy in this region.

2) The rings drift region (S2)

After being generated in the retroflection region (S1), most of the ARs propagate into the southern Atlantic Ocean and interact with the Benguela Current (BC) and the Antarctic Circumpolar Current (ACC) (Fig. 1; McDonagh et al. 1999; Schouten et al. 2000; Casanova-Masjoan et al. 2017). In this rings drift region (S2), the mean flow is quite weak and most (∼80%) of the KE is stored in the mesoscale eddy field (Figs. 7 and 13b). Figure 14 shows the vertical distributions of the long side-averaged^¹ multiscale energetics in this region. In general, all the terms except the eddy pressure work ( $\nabla \cdot Q_{P}^{2}$ ) have magnitudes of approximately one-fifth of those in the retroflection region (S1). For the barotropic terms, $T_{K}^{0 \to 2}$ is weakly negative (>−0.2 × 10⁻⁴ W m⁻³) in the cores of ACC and BC and weakly positive (<0.1 × 10⁻⁴ W m⁻³) in between, while $T_{K}^{1 \to 2}$ is negative (∼−0.2 × 10⁻⁴ W m⁻³) in most of the section (Figs. 14a,b). This implies that the EKE is transferred inversely both to the mean flow and low-frequency variabilities in this region. The baroclinic terms ( $T_{A}^{0 \to 2}$ , $T_{A}^{1 \to 2}$ , and $T_{A}^{0 \to 1}$ ) are almost positive with magnitudes weaker than 0.2 × 10⁻⁴ W m⁻³, indicating forward APE cascades from the mean flow to low-frequency variabilities and mesoscale eddies (Figs. 14d–f). Regarding the eddy buoyancy conversion (b²), it exhibits positive and negative peaks (±0.2 × 10⁻⁴ W m⁻³) along the BC and ACC, respectively (Fig. 14i). Such a cross-stream structure also appears in the EAPE advection term ( $\nabla \cdot Q_{A}^{2}$ ) (Fig. 14l). Then it could be concluded that the EAPE generated through the baroclinic pathways ( $T_{A}^{0 \to 2} > 0$ , $T_{A}^{1 \to 2} > 0$ , and b² < 0) is advected away from the ACC, whereas along the BC, the locally generated and nonlocally imported EAPE, probably from the retroflection region (S1), is primarily converted to EKE. The pressure work on EKE ( $\nabla \cdot Q_{P}^{2}$ ) and the advection of EKE ( $\nabla \cdot Q_{K}^{2}$ ) show more complex cross-stream variations (Figs. 14j,k). The former is negative along the ACC and BC jets and positive in between with the peaks stronger than 0.8 × 10⁻⁴ W m⁻³, while the latter has opposite signs and is relatively weaker (<0.6 × 10⁻⁴ W m⁻³). Being the strongest term, the pressure work $\nabla \cdot Q_{P}^{2}$ determines the cross-stream distribution of EKE, which is also enhanced between the ACC and BC as illustrated in Fig. 7c.

Fig. 14.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

In the entire region, the volume-averaged results confirm that the EKE is generated mainly due to the baroclinic instability ( $T_{A}^{0 \to 2} > 0$ and b² > 0) and the pressure work ( $\nabla \cdot Q_{P}^{2} > 0$ ) that amount to 3.04 × 10⁻⁶ and 2.40 × 10⁻⁶ W m⁻³, respectively (Fig. 13b). The advection effect ( $\nabla \cdot Q_{K}^{2}$ and $\nabla \cdot Q_{A}^{2}$ ) is much weaker, transporting 0.38 × 10⁻⁶ W m⁻³ (0.08 × 10⁻⁶ W m⁻³) of EKE (EAPE) into (away from) this subdomain. At the same time, there is 0.60 × 10⁻⁶ and 1.58 × 10⁻⁶ W m⁻³ of KE transferred inversely from eddies to the mean flow and low-frequency motions, respectively. Particularly, for the LKE, the inverse KE cascade ( $T_{K}^{2 \to 1}$ ) contributes 4 times more power than the advection effect ( $\nabla \cdot Q_{K}^{1}$ ). All the results suggest that the mesoscale eddies could help to accelerate the BC and ACC as well as the low-frequency motions here.

3) The meanders region (S3)

The meridional–vertical distributions of the multiscale energetics in the meanders region (S3) are presented in Fig. 15. All the terms are about one-third of the magnitudes of those in the retroflection region (S1). South of the meander crests (∼37.7°E), the KE is transferred inversely from eddies to the mean flow and low-frequency motions ( $T_{K}^{0 \to 2} < 0$ , $T_{K}^{1 \to 2} < 0$ , and $T_{K}^{0 \to 1} > 0$ , i.e., EKE → MKE → LKE and EKE → LKE), while around the crests, it forms a closed energy transfer loop ( $T_{K}^{0 \to 2} < 0$ , $T_{K}^{1 \to 2} > 0$ , and $T_{K}^{0 \to 1} > 0$ , i.e., EKE → MKE → LKE → EKE) (Figs. 15a–c). For the APE, it is also transferred across the three scale windows through a closed route ( $T_{A}^{0 \to 2} < 0$ , $T_{A}^{1 \to 2} > 0$ and $T_{A}^{0 \to 1} > 0$ , i.e., EAPE → MAPE → LAPE → EAPE) (Figs. 15d–f). Regarding the buoyancy conversions, b⁰ is alternated between positive and negative values in the horizontal direction, while b¹ and b² are dominated by negative and positive values, respectively (Figs. 15g–i). As in the rings drift region (S2), the nonlocal transport terms ( $\nabla \cdot Q_{P}^{2}$ , $\nabla \cdot Q_{K}^{2}$ , and $\nabla \cdot Q_{A}^{2}$ ) are the strongest with magnitudes about 2–3 times larger than the other terms. The pressure work on EKE ( $\nabla \cdot Q_{P}^{2}$ ) is generally positive (>0.8 × 10⁻⁴ W m⁻³) in the upper 100 m and negative (∼−0.6 × 10⁻⁴ W m⁻³) below, whereas both the advection terms $\nabla \cdot Q_{K}^{2}$ and $\nabla \cdot Q_{A}^{2}$ are positive (∼0.8 × 10⁻⁴ W m⁻³) throughout the upper 1500 m (Figs. 15j–l). It means that a large portion of EKE and EAPE is advected into this region, mainly by the ARC from the retroflection region (S1; Figs. 10h–i).

Fig. 15.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

On average, in the meanders region (S3), the influences of the pressure work ( $\nabla \cdot Q_{P}^{2}$ ) and advection effect ( $\nabla \cdot Q_{K}^{2}$ and $\nabla \cdot Q_{A}^{2}$ ) on the eddy field are opposite (Fig. 13c). The former transports 4.92 × 10⁻⁶ W m⁻³ of EKE out of this region, whereas the latter advects 7.97 × 10⁻⁶ W m⁻³ (3.50 × 10⁻⁶ W m⁻³) of EKE (EAPE) into this region. The eddy energies, including 3.16 × 10⁻⁶ W m⁻³ of EKE and 1.98 × 10⁻⁶ W m⁻³ of EAPE, are further transferred back to the ARC. It suggests that the mesoscale eddies intensify the mean flow here. For the low-frequency fluctuations, 3.55 × 10⁻⁶ W m⁻³ of LKE and 2.97 × 10⁻⁶ W m⁻³ of LAPE are generated by the forward cascade from the mean flow, via the barotropic ( $T_{K}^{0 \to 1} > 0$ , i.e., MKE → LKE) and baroclinic ( $T_{A}^{0 \to 1} > 0$ , i.e., MAPE → LAPE) pathways, respectively. In addition, the inverse KE cascade from eddies also contributes 0.70 × 10⁻⁶ W m⁻³ of power to the LKE. Therefore, in this region, the generation of LKE is dominated by the barotropic instability of the mean flow.

In short, the mesoscale eddies in the meanders region (S3) are primarily generated by the nonlocal energy advections ( $\nabla \cdot Q_{K}^{2}$ and $\nabla \cdot Q_{A}^{2}$ ). Among the three scale windows, energy is mainly transferred from the eddies to the mean flow, and then to the low-frequency variabilities. In this way, the low-frequency fluctuations are enhanced as found in Figs. 6 and 7.

4) The stable region (S4)

Unlike the aforementioned three subdomains, the stable region (S4) is far away from the topography and the flow field is weakly affected by lateral boundaries (Fig. 1). The axis of the ARC here is located at about 39.8°S (Fig. 16). The three barotropic terms ( $T_{K}^{0 \to 2}$ , $T_{K}^{1 \to 2}$ , and $T_{K}^{0 \to 1}$ ) are dominated by negative values, implying that the KE is inversely transferred from mesoscale eddies to low-frequency motions and the mean flow (Figs. 16a–c). In particular, the barotropic transfer $T_{K}^{1 \to 2}$ is the strongest, with a magnitude of ∼−0.2 × 10⁻⁴ W m⁻³ (Fig. 16b). Contrarily, the baroclinic transfers ( $T_{A}^{0 \to 2}$ , $T_{A}^{1 \to 2}$ , and $T_{A}^{0 \to 1}$ ) are generally positive but are much weaker (<0.1 × 10⁻⁴ W m⁻³) (Figs. 16d–f). The buoyancy conversion(s) b² (b¹ and b⁰) is (are) predominately positive (negative), with the peak strengths weaker than 0.2 × 10⁻⁴ W m⁻³ (Figs. 16g–i). As in the meanders region (S3), the pressure work on EKE ( $\nabla \cdot Q_{P}^{2}$ ) here is strongly positive (>0.3 × 10⁻⁴ W m⁻³) in the upper 100 m and negative (∼−0.1 × 10⁻⁴ W m⁻³) below (Fig. 16j). The advections of EKE ( $\nabla \cdot Q_{K}^{2}$ ) and EAPE ( $\nabla \cdot Q_{A}^{2}$ ) are relatively weak with mixed positive and negative values weaker than 0.2 × 10⁻⁴ W m⁻³ (Figs. 16k,l).

Fig. 16.

As in Fig. 12, but for the zonally averaged (36.0°–46.0°E) multiscale energetics in the stable region (S4).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

On the whole, similar to the rings drift region (S2), the stable region (S4) is characterized by inverse KE cascade and forward APE cascade among the three scale windows (Fig. 13d). The EKE is generated mainly through the eddy buoyancy conversion (b²), which converts 4.18 × 10⁻⁶ W m⁻³ of EAPE to the EKE. Contributions of the pressure work ( $\nabla \cdot Q_{P}^{2}$ ) and the advection effect ( $\nabla \cdot Q_{K}^{2}$ ) to the EKE are opposite and much weaker, transporting 0.45 × 10⁻⁶ and 0.49 × 10⁻⁶ W m⁻³ of EKE out of and into this region, respectively. Besides, the barotropic term $T_{K}^{1 \to 2}$ transfers 2.63 × 10⁻⁶ W m⁻³ of EKE to the low-frequency variabilities, which is about 4 times higher than the advection term $\nabla \cdot Q_{K}^{1}$ . Therefore, for the enhanced low-frequency fluctuations in this region as shown in Figs. 6 and 7, it is also primarily due to the inverse KE cascade from mesoscale eddies.

5. Summary and discussion

The ARCS is rich in multiscale variabilities. Using the novel multiscale energetics analysis tools MWT and MS-EVA introduced by Liang and Robinson (2005, 2007) and Liang and Anderson (2007), we investigated the climatological characteristics of the multiscale energy interactions among three scale windows, that is, the mean flow, low-frequency fluctuations, and transient mesoscale eddies, based on the 1/4° ECCO2 reanalysis. In our analysis, we emphasized four subdomains that represent the retroflection (S1), rings drift (S2), meanders (S3), and stable (S4) regions of the ARCS:

The retroflection region (S1) is the key region where the most vigorous variabilities and the most pronounced multiscale interactions occur. The mean flow here can be considered as an energy source on average. Energies stored in the mean flow, mainly the SAC, are released to mesoscale eddies and low-frequency fluctuations through mixed barotropic–baroclinic instabilities. Particularly, for the EKE production that dominates this subdomain, the barotropic instability plays a leading role, contributing power about 10 times larger than the baroclinic one. The locally generated EKE is then transported out of this region, for which the contribution of the pressure work is about twice as large as that of the advection effect.
In the rings drift region (S2), the eddy window contains the largest portion (∼80%) of kinetic energy. The EKE is generated mainly due to the eddy pressure work and the eddy buoyancy conversion with their contributions comparable and is dissipated primarily via the inverse KE cascade toward the low-frequency variabilities.
The meanders region (S3) acts as a significant sink for remote EKE and EAPE, on average, especially for that advected from the retroflection region (S1). The imported eddy energies are further transferred inversely to the mean flow and then to the low-frequency variabilities.
In the stable region (S4), the eddy field is the weakest. Mesoscale eddies are generated (dissipated) by the baroclinic (barotropic) and forward (inverse) cascade from (toward) the mean flow and low-frequency fluctuations. The multiscale energy transfers are generally similar to those in the rings drift region (S2). The major difference is that the EKE here is generated primarily via the buoyancy conversion.

In addition to mesoscale eddies, seasonal to interannual variabilities are prominent in the ARCS. The LKE is significantly enhanced in the ARC and even dominates the KE in the stable region (S4). In all the four subdomains, the KE is inversely cascaded from mesoscale eddies to low-frequency variabilities. Such a phenomenon has been reported by Sérazin et al. (2018) and Yang and Liang (2019c). In this work, it is further shown that the LKE in the retroflection (S1) and meanders (S3) regions is also increased due to the forward KE cascade from the mean flow, while that in the rings drift (S2) and stable (S4) regions is enhanced by the advection effect. On average, the inverse KE transfer dominates the LKE generation in most of the ARCS, including the retroflection (S1), rings drift (S2), and stable (S4) regions, contributing power approximately 3–5 times as large as the forward KE cascade and the LKE advection. Only in the meanders region (S3), the forward KE cascade exceeds the inverse KE transfer with a ratio of 5 between them.

It is worth noting that the retroflection region (S1) is characterized by the generation and outgoing transportation of eddy energies. For the eddy generation, it is due to the mixed barotropic and baroclinic instabilities, which has been reported previously (e.g., Chassignet and Boudra 1988; Wells et al. 2000; Lutjeharms et al. 2003a; Tedesco et al. 2022). Moreover, the ratio of barotropic to baroclinic contributions to the EKE production here is approximately 11. In the Agulhas Current region of Tedesco et al. (2022), this ratio is ∼1. However, in the other three subdomains far away from the shelf, the baroclinic one plays a more important role in the eddy generation. Other studies found such ratio is ∼3 in the Kuroshio (Yang and Liang 2019a), ∼3/2–3 in the Gulf Stream (Gula et al. 2015; Kang and Curchitser 2015), ∼1/4 in the Kuroshio Extension (Yang and Liang 2016), ∼1/14 in the Southern Ocean (R. Chen et al. 2014), ∼1/80 in the North Atlantic Ocean (Beckmann et al. 1994), and ∼1/7 for the global ocean (von Storch et al. 2012). The comparisons suggest that for the EKE production, the barotropic instability is more important along the intense boundary currents than in the vast open ocean. Regarding the strong net outgoing of eddy energy from this region, it was also noticed by Tedesco et al. (2022) from a high-resolution (∼2.5 km) numerical simulation. Corresponding to the eddy energy exported here, the nonlocal transport acts as an important energy source for the mesoscale eddy field in the adjacent rings drift (S2) and meanders (S3) regions.

This paper presents a three-dimensional picture of the multiscale energies and interactions in the ARCS. The results provide a model-based benchmark for future studies on physical processes and dynamics at different scales in the ARCS. Nevertheless, it should be mentioned that, limited by the spatial resolution of the ECCO2, this study does not resolve the submesoscale variabilities, which could also impact the mesoscale processes via inverse energy cascades (Schubert et al. 2019, 2020, 2021; Tedesco et al. 2019, 2022). Thus, the results presented here might be sensitive to smaller scale variabilities and further studies are needed to clarify the multiscale interactions involved with submesoscale processes based on higher-resolution observations and numerical models.

We took the average of all the values on the lines parallel to the line AB indicated in Fig. 2b. For clarity, we used the line connecting the midpoints of AB and CD, i.e., the black solid line in Fig. 2b, to represent the horizontal axis in Fig. 14.

We took the average of all the values on the lines parallel to the line BC (indicated in Fig. 2) for the rings drift region (S2), and took the average meridionally for the other three subdomains. For clarity, we used the line connecting the midpoints of AD and BC, i.e., the black dotted line in Fig. 2, to represent the horizontal axis for the rings drift region (S2) part in Fig. 7.

Acknowledgments.

This work was supported by the Natural Science Foundation of Shandong Province (Grant ZR2021MD092), the National Natural Science Foundation of China (Grants 41606016, 41776021), the National Key R&D Program of China (Grant 2016YFC0301203), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant XDB42010102).

Data availability statement.

The ECCO2 outputs are available at http://apdrc.soest.hawaii.edu/index.php. The satellite altimeter data were downloaded from http://www.aviso.altimetry.fr. The EAPE dataset estimated with ARGO floats by Roullet (2020) was obtained from https://doi.org/10.17882/72432.

APPENDIX

Spatial Correlations between the Model and Observational Results

To quantitatively assess the performance of the ECCO2 reanalysis in the study area, the horizontal distributions of the spatial correlations between the modeled and observed EKE^*, EAPE^*, and MSSH are examined. The correlation coefficients are calculated in each 12° × 12° box (note that the results are insensitive to a slight shift of the box). As can be seen from Fig. A1, the EKE^*, EAPE^*, and MSSH derived from the model and observations are well correlated. In most of the ARCS, the correlation coefficients (r) for EKE^* and EAPE^* are higher than 0.4, while that for MSSH is larger than 0.7. The relatively weak correlations for EKE* with r < 0.4 appear in the southwest Indian Ocean Subgyre. This is discussed in section 2b. All the correlation coefficients mentioned above exceed the 95% significance level. Overall, the ECCO2 successfully reproduces the horizontal structure of the EKE^*, EAPE^*, and MSSH in the ARCS area and is appropriate for the analysis in this study.

Fig. A1.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0275.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Numerical simulation
- a. Numerical simulation settings
- b. Evaluation of the ECCO2 reanalysis against data
3. Methodology
- a. Multiscale window transform
- b. Localized multiscale energy and vorticity analysis
4. Results
- a. Characterization of multiscale energy reservoirs
- b. Characterization of local multiscale interactions between reservoirs and nonlocal energy transport within the reservoirs
5. Summary and discussion
Acknowledgments.
Data availability statement.
APPENDIX
- Spatial Correlations between the Model and Observational Results
REFERENCES

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Fig. 9.

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Fig. 10.

As in Fig. 9, but for the nonlocal energy transports.

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Fig. 11.

Vertical distributions of the area-mean multiscale energetics (10⁻⁴ W m⁻³) averaged over the four subdomains indicated in Fig. 2.

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Fig. 12.

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Fig. 13.

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Fig. 14.

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Fig. 15.

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Fig. 16.

As in Fig. 12, but for the zonally averaged (36.0°–46.0°E) multiscale energetics in the stable region (S4).

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Fig. A1.

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Energetics of Multiscale Interactions in the Agulhas Retroflection Current System

Abstract

Abstract

1. Introduction

2. Numerical simulation

a. Numerical simulation settings

b. Evaluation of the ECCO2 reanalysis against data

3. Methodology

a. Multiscale window transform

b. Localized multiscale energy and vorticity analysis

4. Results

a. Characterization of multiscale energy reservoirs

b. Characterization of local multiscale interactions between reservoirs and nonlocal energy transport within the reservoirs

1) The retroflection region (S1)

2) The rings drift region (S2)

3) The meanders region (S3)

4) The stable region (S4)

5. Summary and discussion

Acknowledgments.

Data availability statement.

APPENDIX

Spatial Correlations between the Model and Observational Results

REFERENCES

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Energetics of Multiscale Interactions in the Agulhas Retroflection Current System

Abstract

Abstract

1. Introduction

2. Numerical simulation

a. Numerical simulation settings

b. Evaluation of the ECCO2 reanalysis against data

3. Methodology

a. Multiscale window transform

b. Localized multiscale energy and vorticity analysis

4. Results

a. Characterization of multiscale energy reservoirs

b. Characterization of local multiscale interactions between reservoirs and nonlocal energy transport within the reservoirs

1) The retroflection region (S1)

2) The rings drift region (S2)

3) The meanders region (S3)

4) The stable region (S4)

5. Summary and discussion

Acknowledgments.

Data availability statement.

APPENDIX

Spatial Correlations between the Model and Observational Results

REFERENCES

Share Link

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References

Figures

Cited By

Metrics

Related Content

AMS Publications

Get Involved with AMS

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