# Coherence of Eddy Kinetic Energy Variation during Eddy Life Span to Low-Frequency Ageostrophic Energy

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Satellite Altimetry Data and Eddy Tracking Dataset

**U**

_{ga}used in this study, which were accessed from the Copernicus Marine Environment Monitoring Service (CMEMS) website. The datasets were merged products derived from Topography Experiment (TOPEX) Poseidon (T/P), Jason-1/2 (French–US altimeter satellites), ERS-1/2 (European Remote Sensing satellites), and Envisat (European Remote Sensing satellite) altimeters. The gridded data were geophysically/meteorologically corrected (tides, ionosphere, wet, and dry troposphere) and interpolated onto Mercator grids of 0.25° horizontal resolution with global ocean coverage except for high latitude oceans [36]. The temporal resolution of the data is one day, and the time coverage is from 1 January 1993 to 31 December 2019.

_{span}is calculated from the track day of eddies with the same tracking number. The data span the period from 1 January 1993 to 31 December 2019 as the geostrophic velocity anomaly

**U**

_{ga}.

#### 2.2. Reanalysis Dataset of Total Current Velocity

**U**, including geostrophic and ageostrophic components, is derived from the global ocean eddy-resolving reanalysis dataset (GLORYS12V1) provided by Copernicus Marine Environment Monitoring Service (CMEMS). The model component is the NEMO (Nucleus for European Modeling of the Ocean) platform, driven at the surface by ECMWF (European Centre for Medium-Range Weather Forecasts) ERA-Interim then ERA5 reanalyzes for recent years. A reduced-order Kalman filter was used to assimilate observations, including along track altimeter data, Satellite Sea Surface Temperature, Sea Ice Concentration, and in situ temperature and salinity (T/S) vertical Profiles. Moreover, a 3D-VAR scheme corrects for temperature and salinity biases that are slowly evolving on a large scale. The spatial and temporal resolutions of the dataset are (1/12)° and one day, respectively. The time coverage is from 1 January 1993 to 31 December 2019, the same as that of

**U**

_{ga}.

#### 2.3. Calculation of Geostrophic and Ageostrophic Kinetic Energy

**U**include all velocity components induced by different dynamic processes, of which the geostrophic velocity can be provided by the satellite observations. Here, we decompose the total velocity at each grid

**U**into three components: climatologic mean velocity

**U**

_{0}from time-averaged

**U**, geostrophic current velocity anomaly

**U**

_{ga}from satellite observations, and ageostrophic velocity

**U**

_{ag}:

_{ga}, and ageostrophic kinetic energy (AKE) E

_{ag}are calculated as follows:

**U**

_{alp}from the time series of

**U**

_{ag}at each data grid point; the remaining part of

**U**

_{ag}, subtracting

**U**

_{alp}, is the high-frequency ageostrophic velocity

**U**

_{ahp}, i.e.,

**U**

_{ahp}=

**U**

_{ag}−

**U**

_{alp}. Considering the periods of the submesoscale motions, a seven-day cutoff frequency is used, as in previous studies [14,38,39]. Since the temporal and spatial scales of ocean motions are generally coherent, the low-frequency ageostrophic motions are generally overlapped by the large-scale motions. Similar to the calculation of E

_{ag}, the high-frequency ageostrophic energy (HAE) E

_{ahp}and the low-frequency ageostrophic energy (LAE) E

_{alp}are calculated from

**U**

_{a}

_{hp}and

**U**

_{alp}, respectively.

^{2}/s

^{2}are mainly distributed in the western boundary current, the Antarctic circumpolar current (ACC), and the equatorial current regions, where maximum values are even higher than 1000 cm

^{2}/s

^{2}. Mesoscale eddies are frequently active in regions with strong currents due to barotropic and baroclinic instabilities of the background currents, while ageostrophic motions could be enhanced due to active nonlinear eddy–eddy interactions and eddy–current interactions. In addition, ageostrophic forces, such as buoyancy fluxes and wind forcing, are generally large in these areas and induce high ageostrophic energy [40,41,42]. In shallow shelf waters, bottom friction may also contribute to the ageostrophic motions [15,43]. Though high values of $\overline{{E}_{ag}}$ and $\overline{{E}_{ga}}$ are in close regions, they are dislocated in detail since high ageostrophic motions are prone to appear around mesoscale eddies. $\overline{{E}_{ga}}$ have more coverage of high values than $\overline{{E}_{ag}}$ indeed.

#### 2.4. Matching Gridded Data and Normalization

_{ga}(eddy,t) for a certain eddy can be captured along the eddy track on each day t of the eddy life span. In fact, the variation of E

_{ga}(eddy,t) here within 0.5–1.5 R is consistent with that derived from the maximum speed of eddies (not shown), although the mean E

_{ga}within 0.5–1.5 R is smaller than that from the maximum speed.

_{span}, i.e., T

_{e}= t/T

_{span}. Averaging all energy curves along the eddy track in the normalized life span, we could gain the direct mean energy curve E

_{xx}(T

_{e}) from all eddies, i.e.,

_{xx}(eddy, T

_{e}) by its life-mean EKE ${E}_{ga}^{ave}(eddy)$:

_{xxn}(eddy, T

_{e}) along the track in the normalized life span for all mesoscale eddies are averaged to obtain the mean curve of E

_{xxn}(T

_{e}) throughout the eddy life cycle, i.e., ${E}_{xxn}({T}_{e})={\displaystyle \sum _{i=1}^{N}{E}_{xxn}(eddy,{T}_{e})}/N$. Therefore, the variation curves of EKE E

_{agn}, AKE E

_{agn}, LKE E

_{alpn}, and HKE E

_{ahpn}are derived. The mean standard error, std/N

^{1/2}, is used to measure the deviation of the mean curves, where std is the standard deviation of the data and N is the number of averaged eddy samples.

## 3. Results

#### 3.1. Eddy Kinetic Energy Variation during Eddy Life Span

_{g}

_{a}for all eddies (black curve), AEs (red curve), and CEs (blue curve) along their tracks during their normalized life span. One can see that the three curves show a similar pattern, i.e., increasing at the beginning, maintaining in the middle, and decreasing at the end. Specifically, E

_{ga}for all eddies increases from 190 cm

^{2}/s

^{2}at the beginning of the life span to 237 cm

^{2}/s

^{2}at 0.3 T

_{e}. Subsequently, E

_{ga}maintains around 240 cm

^{2}/s

^{2}until 0.7 T

_{e}and decreases rapidly to 200 cm

^{2}/s

^{2}at the end of the life span (black line). The mean E

_{ga}over the life span is 229 cm

^{2}/s

^{2}, lower than 245 cm

^{2}/s

^{2}for CEs, and higher than 213 cm

^{2}/s

^{2}for AEs.

_{ga}(T

_{e}) for total eddies with time, we divide the whole eddy life span into three stages with a critical slope of 0.35 cm

^{2}/s

^{2}per 0.01 T

_{e:}the growing stage from 0 to 0.3 T

_{e}, the mature stage from 0.3 to 0.7 T

_{e}, and the decaying stage from 0.7 to 1 T

_{e}, as shown in Figure 3a. The energy curves of AEs (red line) and CEs (blue line) have similar patterns to the total one (black line), while the EKE values of CEs are systematically higher than those of AEs, as also observed by Chen and Han [29]. Note that the curves of E

_{ga}(T

_{e}) are not symmetric. The energy values at the beginning are 5–15 cm

^{2}/s

^{2}smaller than that at the end of the eddy life span. The differences between the curves of CEs and AEs in the growing stage are also smaller than that in the decaying stage.

_{gan}(T

_{e}) curve is shown in Figure 3b. One can see that the mean E

_{gan}(T

_{e}) was 1.02 in the mature stage, while the minimum values are 0.90 and 0.94 at the beginning and the end of the eddy life span, respectively. In contrast to the higher E

_{ga}(T

_{e}) for CEs in Figure 3a, the normalized E

_{gan}(T

_{e}) of AEs is generally close to that of CEs, and the E

_{gan}(T

_{e}) of AEs has larger values in the growing stage. The difference may be attributed to more strong CEs than AEs. The statistics reveal 25.7% of CEs have a ${E}_{ga}^{ave}(eddy)$ larger than 240 cm

^{2}/s

^{2}, while the ratio is only 23.2% for AEs. In the direct mean of E

_{ga}(T

_{e}), strong eddies contribute more to the mean value, while the normalized mean curve E

_{gan}(T

_{e}) reveals the real portion of energy variation over the life span for each eddy. On the other hand, the asymmetry of the curve in the beginning and ending are enhanced in the normalized curve since the eddies are weaker in the growing and decaying stages.

_{gan}increases by 12.1% of ${E}_{ga}^{ave}(eddy)$ in the growing stage and decreases by 8% of ${E}_{ga}^{ave}(eddy)$ in the decaying stage.

#### 3.2. Ageostrophic Kinetic Energy Variation during Eddy Life Span

_{agn}(T

_{e}) during the eddy life span. One can see that the E

_{agn}of total eddies has a maximum value of 1.16 in the growing stage, a minimum value of 1.09 in the mature stage, and 1.15 in the decaying stage. It has an upside-down variation trend compared to the curve of E

_{gan}shown in Figure 3b. As listed in Table 2, E

_{agn}decreased by 0.07 in the eddy growing stage and increased by 0.06 in the decaying stage, which accounted for 57% and 75% of |∆E

_{gan}| in the two stages, respectively. CEs and AEs both have similar variation trends in the normalized AKE, while the values of AEs are higher than that of CEs, probably due to stronger geostrophic strain in AEs [14].

_{alpn}(>7 d) and the normalized HAE E

_{a}

_{hpn}(<7 d) during the eddy life span are further shown in Figure 4b,c. One can see that the energy curves of both E

_{alpn}and E

_{a}

_{hpn}are similar to that of the total ageostrophic energy E

_{agn}shown in Figure 4a. The maximum and minimum values of E

_{alpn}are 0.84, 0.80, and 0.83 in the three stages, accounting for 62% of the values of total ageostrophic energy E

_{agn}. On the other hand, the extreme values of E

_{a}

_{hpn}for all eddies are only 0.22, 0.21, and 0.22 in the three stages. They are four times lower than those of E

_{a}

_{lpn}, indicating that E

_{alpn}is the dominant component in E

_{an}.

#### 3.3. Analysis of Coherence of EKE Variation to Low-Frequency Ageostrophic Motions

_{alpn}and E

_{ahpn}are highly coherent with that of E

_{gan}during the eddy life span. The correlation coefficient reaches −0.94 and −0.89 with a confidence level of 99% for E

_{alpn}and E

_{ahpn}, respectively. This indicates that both low-frequency and high-frequency ageostrophic motions contribute to EKE variation during the eddy life span, while the low-frequency ageostrophic motions are more closely correlated to the EKE variation.

_{alpn}decreases by 0.05 from 0.84 to 0.79, contributing 38% of $\Delta {E}_{gan}^{G}$. On the other hand, E

_{alpn}increases by 0.03 (42% of $\Delta {E}_{gan}^{D}$) in the decaying stage. On average, the energy provided by the low-frequency ageostrophic motions contributes 40% of the total variation of E

_{gan}. For comparison, the contributions of high-frequency ageostrophic motions to the variation of E

_{gan}are also listed in Table 2. The ratios are only 8% and 15% in the growing and decaying stages, respectively, 3–5 times lower than the ratio of E

_{alpn}to E

_{gan}. The low-frequency ageostrophic motions are a major contributor.

#### 3.4. Barotropic Conversion Rate

_{ga}curve (not shown). We calculate the time derivative dE

_{gan}/dt of the mean normalized iEKE curve in the eddy life span. The resulted dE

_{gan}/dt is shown by the blue curve in Figure 5.

_{gan}/dt, blue line in Figure 5), one can see that the two rates have the same decreasing trend during the eddy life span and change the sign of variation rate in the mature stage. Their correlation coefficient reaches 0.73 with 99% significance, indicating that the BT rate is the dominant term in the eddy energy variation. The BT rate is generally larger than the EKE variation rate dE

_{gan}/dt. In the eddy growing stage, EKE increases with a positive dE/dt as LAE transfers to EKE with a positive BT rate; EKE varies little with |dE

_{gan}/dt| less than 1.5 × 10

^{−3}as the BT rate decrease rapidly in the mature stage. Then, EKE decreases with a negative rate in the decaying stage, as the BT rate becomes negative. It is clear that low-frequency ageostrophic motions serve as the energy sources for eddy growth and the energy sink for eddy decay. Furthermore, the BT rate was positive between 0 and 0.62 T

_{e}, while dE

_{gan}/dt becomes negative after 0.47 T

_{e}, implying EKE loss possibly probably due to high-frequency ageostrophic motions, viscous dissipation, wind killing, and energy transfer to potential energy [32,45,46]. The positive anomaly after 0.95 T

_{e}may be caused by eddy shape deforming or eddy–mean flow interaction at the western boundary [47,48].

## 4. Discussion

_{gn}and E

_{alpn,}marked as r

_{,}with different numbers of random eddy samples in the global ocean. One can see that r oscillates around 0 as the number of eddy samples N is less than 2000, increases gradually from about −0.73 at N = 2000 to −0.90 as N reaches 7000, and keeps to a little more than −0.92 with N > 10,000. It indicates that the coherence between the curves of EKE and LAE could be established if there were more than 2000 eddies. We further calculate r between E

_{gn}and E

_{alpn}of the eddies in each year. As shown in Figure 6b, r has the largest value of −0.95 in 1998 and then varies between −0.7 and −0.94 with a 2–3 year period in the recent 20 years, which seems to be related to the interannual variability of eddies and background currents influenced by ENSO [23,49,50]. For spatial variability of the coherence shown in Figure 6c, the correlation r between E

_{gn}and E

_{alpn}is larger than −0.8 in most oceans except in the Atlantic, probably due to the narrow and asymmetric ocean basin. Mechanisms about the coherence variation need further pursuing.

## 5. Conclusions

_{gan}curve and the E

_{alpn}curve throughout the eddy life span have a correlation coefficient of −0.94. Low-frequency ageostrophic motion contributes about 38% of the EKE variation in the eddy generation phase and 42% in the decaying phase. Thus, it is the low-frequency ageostrophic motions that are closely coherent with the EKE variation during the eddy life span. In addition, the BT conversion rate analysis confirmed the relationship between the energy transfer between LAE and EKE. Moreover, the EKE decrease in the decaying stage may also be attributed to other processes such as wind work, submesoscale processes, and topography effect.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Roemmich, D.; Gilson, J. Eddy Transport of Heat and Thermocline Waters in the North Pacific: A Key to Interannual/Decadal Climate Variability? J. Phys. Oceanogr.
**2001**, 31, 675–687. [Google Scholar] [CrossRef] - Chelton, D.B.; Schlax, M.G.; Samelson, R.M. Global observations of nonlinear mesoscale eddies. Prog. Oceanogr.
**2011**, 91, 167–216. [Google Scholar] [CrossRef] - Dong, C.; McWilliams, J.C.; Liu, Y.; Chen, D. Global heat and salt transports by eddy movement. Nat. Commun.
**2014**, 5, 3294. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, Z.; Wang, W.; Qiu, B. Oceanic mass transport by mesoscale eddies. Science
**2014**, 345, 322–324. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Qiu, B.; Klein, P.; Travis, S. The influence of geostrophic strain on oceanic ageostrophic motion and surface chlorophyll. Nat. Commun.
**2019**, 10, 2838. [Google Scholar] [CrossRef] [PubMed][Green Version] - Adams, D.K.; McGillicuddy, D.J., Jr.; Zamudio, L.; Thurnherr, A.M.; Liang, X.; Rouxel, O.; German, C.R.; Mullineaux, L.S. Surface-generated mesoscale eddies transport deep-sea products from hydrothermal vents. Science
**2011**, 332, 580–583. [Google Scholar] [CrossRef][Green Version] - Hausmann, U.; Czaja, A. The observed signature of mesoscale eddies in sea surface temperature and the associated heat transport. Deep Sea Res. Part I Oceanogr. Res. Pap.
**2012**, 70, 60–72. [Google Scholar] [CrossRef] - Kamenkovich, I.; Berloff, P.; Haigh, M.; Sun, L.; Lu, Y. Complexity of Mesoscale Eddy Diffusivity in the Ocean. Geophys. Res. Lett.
**2021**, 48, e2020GL091719. [Google Scholar] [CrossRef] - Ferrari, R.; Wunsch, C. Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks. Annu. Rev. Fluid Mech.
**2009**, 41, 253–282. [Google Scholar] [CrossRef][Green Version] - Huang, R.X. Ocean Circulation: Wind-Driven and Thermohaline Processes; Cambridge University Press: Cambridge, UK, 2010; p. 195. [Google Scholar]
- Chen, R.; Flierl, G.R.; Wunsch, C. A Description of Local and Nonlocal Eddy–Mean Flow Interaction in a Global Eddy-Permitting State Estimate. J. Phys. Oceanogr.
**2014**, 44, 2336–2352. [Google Scholar] [CrossRef][Green Version] - Vallis, G.K. Atmospheric and Oceanic Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Kang, D.; Curchitser, E.N. Energetics of Eddy–Mean Flow Interactions in the Gulf Stream Region. J. Phys. Oceanogr.
**2015**, 45, 1103–1120. [Google Scholar] [CrossRef] - Zhang, Z.; Qiu, B. Evolution of Submesoscale Ageostrophic Motions Through the Life Cycle of Oceanic Mesoscale Eddies. Geophys. Res. Lett.
**2018**, 45, 11847–11855. [Google Scholar] [CrossRef][Green Version] - McWilliams, J.C. Submesoscale currents in the ocean. Proc. Math. Phys. Eng. Sci.
**2016**, 472, 20160117. [Google Scholar] [CrossRef] [PubMed] - Xu, C.; Shang, X.-D.; Huang, R.X. Estimate of eddy energy generation/dissipation rate in the world ocean from altimetry data. Ocean Dyn.
**2011**, 61, 525–541. [Google Scholar] [CrossRef] - Yang, Z.; Zhai, X.; Marshall, D.P.; Wang, G. An Idealized Model Study of Eddy Energetics in the Western Boundary “Graveyard”. J. Phys. Oceanogr.
**2021**, 51, 1265–1282. [Google Scholar] [CrossRef] - Xu, C.; Zhai, X.; Shang, X.-D. Work done by atmospheric winds on mesoscale ocean eddies. Geophys. Res. Lett.
**2016**, 43, 12174–112180. [Google Scholar] [CrossRef][Green Version] - Chelton, D.B.; Gaube, P.; Schlax, M.G.; Early, J.J.; Samelson, R.M. The influence of nonlinear mesoscale eddies on near-surface oceanic chlorophyll. Science
**2011**, 334, 328–332. [Google Scholar] [CrossRef] - Chen, G.; Wang, D.; Hou, Y. The features and interannual variability mechanism of mesoscale eddies in the Bay of Bengal. Cont. Shelf Res.
**2012**, 47, 178–185. [Google Scholar] [CrossRef] - Zhai, X.; Marshall, D.P. Vertical Eddy Energy Fluxes in the North Atlantic Subtropical and Subpolar Gyres. J. Phys. Oceanogr.
**2013**, 43, 95–103. [Google Scholar] [CrossRef] - Trott, C.B.; Subrahmanyam, B.; Chaigneau, A.; Delcroix, T. Eddy Tracking in the Northwestern Indian Ocean During Southwest Monsoon Regimes. Geophys. Res. Lett.
**2018**, 45, 6594–6603. [Google Scholar] [CrossRef] - Zheng, S.; Feng, M.; Du, Y.; Meng, X.; Yu, W. Interannual Variability of Eddy Kinetic Energy in the Subtropical Southeast Indian Ocean Associated With the El Niño-Southern Oscillation. J. Geophys. Res. Ocean.
**2018**, 123, 1048–1061. [Google Scholar] [CrossRef] - Li, J.; Roughan, M.; Kerry, C. Dynamics of Interannual Eddy Kinetic Energy Modulations in a Western Boundary Current. Geophys. Res. Lett.
**2021**, 48, e2021GL094115. [Google Scholar] [CrossRef] - Chen, S.; Qiu, B. Variability of the Kuroshio Extension Jet, Recirculation Gyre, and Mesoscale Eddies on Decadal Time Scales. J. Phys. Oceanogr.
**2005**, 35, 2090–2103. [Google Scholar] - Liu, Y.; Dong, C.; Guan, Y.; Chen, D.; McWilliams, J.; Nencioli, F. Eddy analysis in the subtropical zonal band of the North Pacific Ocean. Deep Sea Res. Part I Oceanogr. Res. Pap.
**2012**, 68, 54–67. [Google Scholar] [CrossRef] - Chelton, D.B.; Schlax, M.G.; Samelson, R.M. Randomness, Symmetry, and Scaling of Mesoscale Eddy Life Cycles. J. Phys. Oceanogr.
**2014**, 44, 1012–1029. [Google Scholar] - Ji, J.; Dong, C.; Zhang, B.; Liu, Y.; Zou, B.; King, G.P.; Xu, G.; Chen, D. Oceanic Eddy Characteristics and Generation Mechanisms in the Kuroshio Extension Region. J. Geophys. Res. Ocean.
**2018**, 123, 8548–8567. [Google Scholar] [CrossRef] - Chen, G.; Han, G. Contrasting Short-Lived With Long-Lived Mesoscale Eddies in the Global Ocean. J. Geophys. Res. Ocean.
**2019**, 124, 3149–3167. [Google Scholar] [CrossRef] - Huang, R.; Xie, L.; Zheng, Q.; Li, M.; Bai, P.; Tan, K. Statistical analysis of mesoscale eddy propagation velocity in the South China Sea deep basin. Acta Oceanol. Sin.
**2021**, 39, 91–102. [Google Scholar] [CrossRef] - Zhai, X.; Johnson, H.L.; Marshall, D.P. Significant sink of ocean-eddy energy near western boundaries. Nat. Geosci.
**2010**, 3, 608–612. [Google Scholar] [CrossRef] - Kang, D.; Curchitser, E.N.; Rosati, A. Seasonal Variability of the Gulf Stream Kinetic Energy. J. Phys. Oceanogr.
**2016**, 46, 1189–1207. [Google Scholar] [CrossRef] - Macdonald, H.S.; Roughan, M.; Baird, M.E.; Wilkin, J. The formation of a cold-core eddy in the East Australian Current. Cont. Shelf Res.
**2016**, 114, 72–84. [Google Scholar] [CrossRef] - Geng, W.; Xie, Q.; Chen, G.; Zu, T.; Wang, D. Numerical study on the eddy–mean flow interaction between a cyclonic eddy and Kuroshio. J. Oceanogr.
**2016**, 72, 727–745. [Google Scholar] [CrossRef] - Yang, H.; Wu, L.; Liu, H.; Yu, Y. Eddy energy sources and sinks in the South China Sea. J. Geophys. Res. Ocean.
**2013**, 118, 4716–4726. [Google Scholar] [CrossRef] - Pujol, M.-I.; Faugère, Y.; Taburet, G.; Dupuy, S.; Pelloquin, C.; Ablain, M.; Picot, N. DUACS DT2014: The new multi-mission altimeter data set reprocessed over 20 years. Ocean Sci.
**2016**, 12, 1067–1090. [Google Scholar] [CrossRef][Green Version] - Schlax, M.G.; Chelton, D.B. The “Growing Method” of Eddy Identification and Tracking in Two and Three Dimensions; College of Earth, Ocean and Atmospheric Sciences, Oregon State University: Corvallis, OR, USA, 2016; pp. 1–8. Available online: https://www.aviso.altimetry.fr/fileadmin/documents/data/products/value-added/Schlax_Chelton_2016.pdf (accessed on 14 December 2021).
- Thomas, L.N.; Tandon, A.; Mahadevan, A. Submesoscale processes and dynamics. In Ocean Modeling in an Eddying Regime; Geophysical Monograph Series; Wiley-Blackwell Publishing: Hoboken, NJ, USA, 2008; pp. 17–38. [Google Scholar]
- Lévy, M.; Ferrari, R.; Franks, P.J.S.; Martin, A.P.; Rivière, P. Bringing physics to life at the submesoscale. Geophys. Res. Lett.
**2012**, 39. [Google Scholar] [CrossRef][Green Version] - Colas, F.; Capet, X.; McWilliams, J.C.; Li, Z. Mesoscale Eddy Buoyancy Flux and Eddy-Induced Circulation in Eastern Boundary Currents. J. Phys. Oceanogr.
**2013**, 43, 1073–1095. [Google Scholar] [CrossRef] - Stammer, D.; Böning, C.; Dieterich, C. The role of variable wind forcing in generating eddy energy in the North Atlantic. Prog. Oceanogr.
**2001**, 48, 289–311. [Google Scholar] [CrossRef][Green Version] - Hogg, A.M.; Meredith, M.P.; Chambers, D.P.; Abrahamsen, E.P.; Hughes, C.W.; Morrison, A.K. Recent trends in the Southern Ocean eddy field. J. Geophys. Res. Ocean.
**2015**, 120, 257–267. [Google Scholar] [CrossRef][Green Version] - Semtner, A.J.; Mintz, Y. Numerical Simulation of the Gulf Stream and Mid-Ocean Eddies. J. Phys. Oceanogr.
**1977**, 7, 208–230. [Google Scholar] [CrossRef][Green Version] - Kundu, P.K.; Cohen, I.M. Fluid Mechanics; Elsevier Science: Amsterdam, The Netherlands, 2010. [Google Scholar]
- Rai, S.; Hecht, M.; Maltrud, M.; Aluie, H. Scale of oceanic eddy killing by wind from global satellite observations. Sci. Adv.
**2021**, 7, eabf4920. [Google Scholar] [CrossRef] - Yang, Q.; Nikurashin, M.; Sasaki, H.; Sun, H.; Tian, J. Dissipation of mesoscale eddies and its contribution to mixing in the northern South China Sea. Sci. Rep.
**2019**, 9, 556. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jan, S.; Mensah, V.; Andres, M.; Chang, M.H.; Yang, Y.J. Eddy-Kuroshio Interactions: Local and Remote Effects. J. Geophys. Res. Ocean.
**2017**, 122, 9744–9764. [Google Scholar] [CrossRef][Green Version] - Chern, C.-S.; Wang, J. Interactions of Mesoscale Eddy and Western Boundary Current: A Reduced-Gravity Numerical Model Study. J. Oceanogr.
**2005**, 61, 271–282. [Google Scholar] [CrossRef] - Kuo, Y.-C.; Tseng, Y.-H. Influence of anomalous low-level circulation on the Kuroshio in the Luzon Strait during ENSO. Ocean Model.
**2021**, 159, 101759. [Google Scholar] [CrossRef] - Langlais, C.E.; Rintoul, S.R.; Zika, J.D. Sensitivity of Antarctic Circumpolar Current Transport and Eddy Activity to Wind Patterns in the Southern Ocean. J. Phys. Oceanogr.
**2015**, 45, 1051–1067. [Google Scholar] [CrossRef][Green Version] - Zhang, N.; Liu, G.; Liu, Q.; Zheng, S.; Perrie, W. Spatiotemporal Variations of Mesoscale Eddies in the Southeast Indian Ocean. J. Geophys. Res. Ocean.
**2020**, 125, e2019JC015712. [Google Scholar] [CrossRef] - Ayouche, A.; De Marez, C.; Morvan, M.; L’Hegaret, P.; Carton, X.; Le Vu, B.; Stegner, A. Structure and Dynamics of the Ras al Hadd Oceanic Dipole in the Arabian Sea. Oceans
**2021**, 2, 105–125. [Google Scholar] [CrossRef] - Gnevyshev, V.G.; Malysheva, A.A.; Belonenko, T.V.; Koldunov, A.V. On Agulhas eddies and Rossby waves travelling by forcing effects. Russ. J. Earth Sci.
**2021**, 21, 3. [Google Scholar] [CrossRef] - Laxenaire, R.; Speich, S.; Blanke, B.; Chaigneau, A.; Pegliasco, C.; Stegner, A. Anticyclonic Eddies Connecting the Western Boundaries of Indian and Atlantic Oceans. J. Geophys. Res. Ocean.
**2018**, 123, 7651–7677. [Google Scholar] [CrossRef] - Keppler, L.; Cravatte, S.; Chaigneau, A.; Pegliasco, C.; Gourdeau, L.; Singh, A. Observed Characteristics and Vertical Structure of Mesoscale Eddies in the Southwest Tropical Pacific. J. Geophys. Res. Ocean.
**2018**, 123, 2731–2756. [Google Scholar] [CrossRef][Green Version] - Errico, R.M. The Strong Effects of Non-Quasigeostrophic Dynamic Processes on Atmospheric Energy Spectra. J. Atmos. Sci.
**1982**, 39, 961–968. [Google Scholar] - Straub, D.N.; Bartello, P.; Ngan, K. Dissipation of Synoptic-Scale Flow by Small-Scale Turbulence. J. Atmos. Sci.
**2008**, 65, 766–791. [Google Scholar]

**Figure 1.**Global distributions of climatological kinetic energy. (

**a**) Eddy geostrophic kinetic energy $\overline{{E}_{ga}}$, (

**b**) ageostrophic kinetic energy $\overline{{E}_{ag}}$, (

**c**) differences between $\overline{{E}_{ga}}$ and $\overline{{E}_{ag}}$.

**Figure 2.**Matching gridded data along eddy track. The background image shows the EKE field, the white curve represents the eddy trajectory, the black circles give the range of the eddy 0.5–1.5 R, and the scatters between the two concentric circles represent the data points used in the energy calculation for each eddy.

**Figure 3.**Curves of global mean eddy kinetic energy along the eddy track throughout the eddy life span. (

**a**) EKE E

_{ga}, (

**b**) Normalized EKE E

_{ga}

_{n}. Black curves represent the energy curves for total eddies. Red and blue curves represent the energy curves for AEs and CEs, respectively. Shadows around the energy curves represent the mean standard error.

**Figure 4.**Variation curves of global mean ageostrophic kinetic energy along the eddy track throughout the eddy life span. (

**a**) Normalized AKE E

_{agn}, (

**b**) normalized LAE E

_{alpn}(with period > 7 d), and (

**c**) normalized HAE E

_{a}

_{hpn}(with period < 7 d). Black curves represent the energy curves for total eddies. Red and blue curves represent the energy curves for AEs and CEs, respectively. Shadows around the energy curves represent the mean standard error.

**Figure 5.**Time derivative of normalized iEKE (blue line) and normalized barotropic conversion rate (BTn) (red line) throughout the eddy life span. Shadows around the BTn curves represent the mean standard error.

**Figure 6.**Variation of correlation coefficient between E

_{gan}and E

_{alpn}curves with (

**a**) eddy samples, (

**b**) eddy occurrence years, and (

**c**) ocean basins.

All | AEs | CEs | |
---|---|---|---|

Eddy number | 378,513 | 185,937 | 192,577 |

Eddy radius (km) median (mean ± std) | 74.20 (82.33 ± 35.93) | 74.95 (83.13 ± 36.23) | 73.45 (81.53 ± 35.60) |

Eddy lifespan (day) median (mean ± std) | 51.00 (73.38 ± 66.40) | 51.00 (74.43 ± 69.62) | 51.00 (72.60 ± 63.33) |

**Table 2.**Increments of E

_{gn}, E

_{an}, E

_{a}

_{lpn}, and E

_{a}

_{hpn}and contributions to E

_{gn}in the growing and decaying stages.

E_{gn} | E_{an} | E_{alpn} | E_{ahpn} | |
---|---|---|---|---|

Growing stage $\Delta {E}_{xx}^{G}(\Delta {E}_{xx}^{G}$$/\Delta {E}_{gn}^{G})$ | 0.12 (100%) | −0.07 (57%) | −0.05 (38%) | −0.01 (8%) |

Decaying stage $\Delta {E}_{xx}^{D}(\Delta {E}_{xx}^{D}$$/\Delta {E}_{gn}^{D})$ | −0.08 (100%) | 0.06 (75%) | 0.03 (42%) | 0.01 (15%) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Z.; Xie, L.; Zheng, Q.; Li, M.; Li, J.; Li, M.
Coherence of Eddy Kinetic Energy Variation during Eddy Life Span to Low-Frequency Ageostrophic Energy. *Remote Sens.* **2022**, *14*, 3793.
https://doi.org/10.3390/rs14153793

**AMA Style**

Zhang Z, Xie L, Zheng Q, Li M, Li J, Li M.
Coherence of Eddy Kinetic Energy Variation during Eddy Life Span to Low-Frequency Ageostrophic Energy. *Remote Sensing*. 2022; 14(15):3793.
https://doi.org/10.3390/rs14153793

**Chicago/Turabian Style**

Zhang, Zhisheng, Lingling Xie, Quanan Zheng, Mingming Li, Junyi Li, and Min Li.
2022. "Coherence of Eddy Kinetic Energy Variation during Eddy Life Span to Low-Frequency Ageostrophic Energy" *Remote Sensing* 14, no. 15: 3793.
https://doi.org/10.3390/rs14153793