Econometric modeling of ocean eddy kinetic energy and Atlantic Meridional Overturning Circulation transport with implications for Sustainable Development Goal 14

1 Introduction

The Sustainable Development Goals (SDGs) of the United Nations encompass several dimensions for improving the lives and wellbeing of populations globally (United Nations, 2017). Moreover, SDG 14 emphasizes conservation efforts for oceans, especially in the wake of climate change. While the interconnections between the 17 SDG and their subcomponents are important for formulation of evidence-based policies (Bhargava, 2019), it is important to provide rigorous empirical evidence on ocean dynamics for the achievement of SDG 14 (Zhang et al., 2021; Spenger et al., 2024). Increases in greenhouse gas emissions and atmospheric concentrations, heat emissions from land, carbon dioxide emission from shipping, and agricultural and industrial runoffs into rivers and oceans are exacerbating eutrophication of coastal waters, raising sea surface temperatures (SST), reducing sea ice concentrations (SIC), and lowering biodiversity (Rabalais et al., 2009; Intergovernmental Panel on Climate Change, 2014, 2019; Dasgupta, 2021; Bhargava, 2022; Yu et al., 2024).

From a methodological standpoint, there are complex interrelationships between baroclinic instability, SST, wind stress, and ocean eddy energy (EKE-complex conjugate of zonal and meridional ocean surface current velocities, Equation 2 below) that merit empirical investigations (Eady, 1949; Spall, 2007; Ezer and Dagendorf, 2021; Martinez-Moreno et al., 2021; Cai et al., 2022). The velocity changes in ocean currents such as the Atlantic Meridional Overturning Circulation (AMOC) are likely to determine the ultimate global climatic patterns (e.g. Weijer et al., 2019). It is therefore critical to analyze repeated time series (“longitudinal”) remote sensing data on ocean surface current velocities for understanding the role played by SST and SIC. It is also important to investigate the interrelationships between ocean EKE (Richardson, 1983) and in situ moorings data on AMOC transport for predicting climatic patterns and for enhancing ocean sustainability.

While the use of differential equations for tidal behavior and sea water movements was advocated by Laplace (Proudman, 1941), Stommel (1948, 1961) underscored the inherent turbulence in ocean currents and the difficulties in analyzing the effects of thermohaline convection for ocean circulation. For example, Stommel noted the “artificial nature of theoretical models” and that multiple equilibria may occur depending on whether “temperature differences dominate the density differences.” Moreover, partial differential equations employed in oceanographic research need to tackle discontinuity of higher order derivatives and boundary conditions (Kim et al., 2016). Thus, it is important to analyze remote sensing longitudinal data on ocean surface current velocities using econometric methods that invoke minimal assumptions on sea water movements and tackle the interdependence between climatic indicators.

Further, the seasonal patterns in Northern and Southern hemispheres affect quarterly SST and SIC, and influence zonal and meridional velocities of ocean surface currents. While increases in SST are likely to reduce SIC, remote sensing data on sea ice thicknesses are mainly available for the Northern hemisphere. Moreover, modeling of zonal and meridional velocities is complicated by Earth's rotation, reflected in the Coriolis and centrifugal forces, and by the nonlinear effects of salinity, temperatures, and pressure for sea water movements (Johnson et al., 2007; Lumpkin and Johnson, 2013; Wunsch and Ferrari, 2018; Loeb et al., 2021; Rohrs et al., 2021). For example, while higher SST may increase sea water movements, the impacts may not be apparent in zonal and meridional velocities owing to the turbulence and energy dissipation via viscosity. It is important to analyze quarterly data on zonal and meridional velocities in the hemispheres by initially estimating bivariate correlations between SST, SIC, zonal and meridional velocities, and ocean EKE. In addition, wind stress is utilized in construction of ocean surface current velocities (Bonjean and Lagerloef, 2002) so that it is useful to employ an econometric approach for analyzing the effects of previous wind stress magnitudes for ocean EKE, especially in the Southern hemisphere (Cai et al., 2022).

It is useful to model the effects of SST and SIC for ocean EKE using quarterly data on 1° × 1° grids for 2000–2019 in the hemispheres. The remote sensing measurements for zonal and meridional velocities may be differentially affected by satellite tracks (Johnson et al., 2007). While these velocities are estimated from satellite signals based on geostrophy, Ekman dynamics, and thermal winds (Bonjean and Lagerloef, 2002; Rayner et al., 2003; Stroeve and Notz, 2018), the interpolation methods also utilize changes in SST for their construction. Thus, measurement issues surrounding zonal and meridional velocities are intertwined with the use of appropriate statistical methods (Bhargava and Echenique, 2022). It is important to avoid spurious associations when investigating the effects of higher SST and lower SIC for sea water movements. For example, while ocean EKE for grids in a quarter might be influenced by measurements in the previous quarter, “dynamic” models may not provide robust estimates due to interpolations invoked (Wiener, 1949). Instead, “static” models that exclude lagged ocean EKE as an explanatory variable but take in account unobserved heterogeneity across the grids are useful, especially in non-linear settings.

This article modeled the effects of quarterly SST and SIC for ocean EKE in the hemispheres, taking in account non-linearities in the relationships using 3-year quarterly averages for 1° × 1° grids. The separate analyses tackled issues such as directional changes in ocean currents and seasonal factors affecting SST and SIC. The analysis also addressed the bidirectional relationships between wind stress and ocean EKE using an econometric approach. The framework was useful for investigating the changes over time in AMOC transport (Little et al., 2020; Le Bras et al., 2023; McCarthy and Caesar, 2023; Johns et al., 2023; Frajka-Williams et al., 2023; Ditlevsen and Ditlevsen, 2023). The analysis of AMOC transport data around 35°N showed significantly negative effects of ocean EKE but not of SST. The interrelationships between AMOC transport and ocean EKE were analyzed using dynamic and static models, and the empirical results provide insights for the formulation of evidence-based policies for achieving the SDG 14.

2 Materials and methods

2.1 The remote sensing and moorings data sets

The longitudinal remote sensing data for SST, SIC and ocean surface current velocities were previously processed (Bhargava and Echenique, 2022). Briefly, the data on SST for 0.05° × 0.05° grids were from “Level 4 analyses” (Copernicus Climate Change Service, 2021a,b; Merchant et al., 2019). The data were processed for 64,800 1° × 1° grids and 3-yearly averages were created for the quarters for reducing internal variation.

The radar altimetry data for SIC, expressed as percentages of the area of the grid covered by sea ice, were obtained via satellite signals [Meier and the National Center for Atmospheric Research (NCAR) Staff, 2014; Ricker et al., 2015]. The “Level 3” monthly data were produced for 64,800 1° × 1° grids, and merged with the data for SST. The monthly data on velocities of ocean surface currents were available from the Ocean Surface Current Analysis Real-time (OSCAR; Jet Propulsion, Laboratory, 2021) and merged with the SST and SIC data. Note that the near-surface zonal and meridional velocities were derived as nonlinear functions of sea surface heights, wind stress, sea surface displacement, depth, eddy viscosity, vertical sheer, and buoyancy force (Bonjean and Lagerloef, 2002; Johnson et al., 2007; Kang and Curchitser, 2017). The wind stress magnitudes data from ERA5 reanalysis (Cai et al., 2022) were also merged with the monthly database.

Lastly, monthly data on AMOC transport during 2005–2013 were obtained via moorings for the coordinates {40.3°N, 70.2°W} to {35°N, 66.4°W} and then eastward along 35°N (Le Bras et al., 2023). The latitude 35°N was selected for covering the crossing of the Gulf Stream extension. The monthly data for SST and ocean EKE for the 2° × 55° grid encompassing this AMOC region were merged with the time series transport data. The importance of analyzing the data from “RAPID” array at 26°N for the period 2005–2019 (Petit et al., 2025) in future research is noted in the Discussion.

2.2 The analytical framework, methodological aspects, and empirical models

It is common in the oceanographic literature to employ geostrophic equations for studying wind-driven circulation in an enclosed and midaltitude basin. The “stream-function” formulation is defined on a plane domain Θ with boundary Γ and given by

1ReΔ2ψ+J(ψ,Δψ)-1Ro∂ψ∂x=1RoF(1)

where ψ is the velocity stream-function, F is the forcing, and J is the Jacobian operator. The parameters Re and Ro denote Reynolds and Rossby numbers, respectively; the model is supplemented by boundary conditions and requires continuity corrections for higher-order derivatives (Kim et al., 2016). However, in view of “chaotic” nature of sea water movements (Stommel, 1948, 1961) and the emphasis on the interrelationships between SST, SIC and sea water movements, remote sensing data for 1° × 1° grids were analyzed on ocean EKE, that is the complex conjugate (or squared modulus) of zonal (u) and meridional velocities (v) and is given for grid i in time period t by:

(Ocean eddy kinetic energy)it=0.5(uit2+vit2) (i=1,…,N;t=1,…,T)(2)

From a methodological standpoint, note that the EKE viscosity parameter inversely affects the Reynolds number in Equation 1 thereby contributing to discontinuities in higher-order derivatives. Moreover, eddy viscosity is utilized in the construction of zonal and meridional velocities so that it is useful to employ an econometric approach. Second, while overall means of velocities in Equation 2 are often subtracted in aggregate time series analyses (Richardson, 1983), differences in the time means at the grid level are accounted for in the longitudinal data analyses via random effects. Moreover, a constant term is included in the model for ocean EKE (Equation 3 below) which converts all variables into deviations from their respective overall means. Third, analysis of ocean eddy EKE (or its variants) is appealing because the empirical fourth-order moments of the data were invariably finite and did not exhibit discontinuities in the quarters; robustness of the estimated model parameters were appraised. Fourth, higher SST and lower SIC can increase ocean EKE so that it is important for empirical models to allow for interactive effects and non-linearities with respect to these variables. The quarterly data for consecutive years were averaged for reducing internal variation, and 3-yearly averages at seven time points {2001, 2004, 2007, 2010, 2013, 2016, 2019} were analyzed. Lastly, SST and ocean EKE can in turn affect AMOC transport assessed via moorings in contiguous regions (Le Bras et al., 2023), so that it is important to investigate the empirical interrelationships between AMOC transport and ocean EKE.

Next, it would be helpful to appraise the internal consistency of remote sensing data by computing bivariate correlations between SST, SIC, zonal and meridional velocities, and ocean EKE using quarterly data for 1° × 1° grids for the hemispheres. Subsequently, ocean EKE for the quarters can be explained by latitudes and longitudes, indicator variables for 5 years {2007, 2010, 2013, 2016, 2019} for capturing the underlying trends, and a second-degree polynomial in SST and SIC. Note that while dynamic models (Bhargava and Sargan, 1983) that include previous levels of ocean EKE were also estimated, the results are reported for static models (Bhargava, 1991) that exclude the lagged dependent variables; the processing of remote sensing signals for zonal and meridional velocities apparently complicated the dynamic properties of ocean EKE. The static models took in account the unobserved heterogeneity across the grids and serial correlations in the errors. The model for 3-year averages of quarterly ocean EKE for ith grid in time period t was given by:

(Ocean eddy kinetic energy)it=a0+a1ln(Latitude)i + a2ln(Longitude)i+a3(Year 2007)t + a4(Year 2010)t +a5(Year 2013)t + a6(Year 2016)t+ a7(Year 2019)t + a8ln(Sea surface temperatures)it + a9ln(Sea ice concentrations)it + a10[ln(Sea surface temperatures)it × ln(Sea ice concentrations)it] + a11[ln(Sea surface temperatures)it]2 + a12[ln(Sea ice concentrations)it]2 + wit (i=,1,2,…N;t=1,2,…T)(3)

Note that ln represents natural logarithms, and SST were measured in degrees Celsius, with 2 °C added for circumventing zero values; the freezing point of sea water is −2 °C. The model in Equation 3 was separately estimated for Northern and Southern hemispheres, and the geodesic variables accounted for regional characteristics. In addition, interactions between wind stress and ocean EKE are important due to circumpolar flow in the Southern Ocean (Cai et al., 2022). Thus, enlarged models for ocean EKE, that included lagged wind stress magnitudes as explanatory variables, were estimated for the Southern hemisphere for assessing the robustness of the results. Lastly, spatial correlation patterns were accounted for in the model via the inclusion of explanatory variables and the EKE time series were stationary. Note that the analyses annual data were reported by Bhargava and Echenique (2022) and the analyses of 3-year averages provide insights into the interactions between the explanatory variables.

The errors w_it's affecting Equation 3 can be decomposed in a simple random effects fashion as (Equation 4):

wit=δi+rit (i=1,2,…,N;t=1,2,…,T)(4)

where the δ_*i*'s were grid-specific random effects that were assumed to be distributed with zero mean and constant variance, and r_it were distributed with zero mean and constant variance. However, a more general error structure was employed where the w_it's were a drawing from a multivariate distribution with an unrestricted T × in T dispersion matrix Ω (see below). Note that the coefficients a₁₁ and a₁₂ of the squared variables SST and SIC, respectively, were not statistically significant in the models for Northern hemispheres and were dropped.

Further, static models for monthly data on AMOC transport around 35°N were given by:

(Transport)t=b0+b1(Trend)t+b2(Trend)2t + b3(Sea surface temperatures)t + b4(Ocean eddy kinetic energy)t + wt(t=1,2,…,T)(5)

Because ocean EKE may itself be affected by AMOC transport, a similar model was estimated for ocean EKE. Moreover, in view of possible interdependence between these variables, dynamic models with nonlinear interactions between lagged explanatory variables were estimated. For example, the dynamic model for AMOC transport was:

(Transport)t=c0+c1(Sea surface temperatures)t−1 + c2[(Sea surface temperatures)t−1 ]2 + c3(Ocean eddy kinetic energy)t−1 + c4[(Ocean eddy kinetic energy)t−1]2 + c5(Transport)t−1+ wt (t=2,3,…,T)(6)

A similar dynamic model was estimated for the ocean EKE energy that was explained by lagged SST and transport, and their respective squared values (Table 7). Note that it is difficult to assert a priori “causality” in such interrelationships (Cox, 1992); it is useful to include lagged values of certain explanatory variables for circumventing the problems of “endogeneity” of variables (see below).

2.3 Statistical and econometric methods

The static random effects models for ocean eddy EKE in Equation 3 were estimated by stepwise methods, assuming that the number of grids (N) was large but number of time periods (T) was fixed (Bhargava, 1991). The errors affecting the models were assumed to be independent drawings from a multivariate distribution with a T × T symmetric positive definite dispersion matrix Ω. The criterion function minimized using a FORTRAN program was the trace of a matrix:

tr {Ω-1(A R A′)}(7)

where the matrix A contained the coefficients in Equation 3, and R was a T × T matrix containing the second moments of “predicted” parts of variables that may be correlated with the error terms. Note that “exogeneity” hypotheses of zero correlation between certain explanatory variables and the errors were tested using statistics that were distributed for large numbers of grids as Chi-squared variables. The normality assumptions were not necessary for the errors affecting static random effects models. Note that the parameters from random effects models can be estimated using less general formulations in StataCorp (2022). While one could estimate models with “fixed effects” using data for large number of grids over a few time periods, the increase in “dummy” variables for grids with sample size leads to the “incidental parameters” problem (Neyman and Scott, 1948). The models in Equations 5, 6 were estimated using Ordinary Least Squares and errors affecting the static models were tested for serial independence against first order positive autocorrelation using the Durbin–Watson statistics (StataCorp, 2022).

3 Results

3.1 Sample means for sea surface temperatures, sea ice concentrations and ocean eddy kinetic energy

Table 1 reports the quarterly sample means and standard deviations of SST, SIC, zonal and meridional velocities, and ocean EKE for 1° × 1° grids for 3-year averages corresponding to years 2001, 2004, 2007, 2010, 2013, 2016, and 2019 in the Northern hemisphere. The corresponding sample means for the Southern hemisphere are in Table 2. Significant changes (P < 0.05) in the variables between 2001 and 2019 from the applications of paired t-test statistics are marked in Tables 1, 2. First, in Table 1, the sample means of SST increased from 2001 to 2019 in all quarters and paired t-tests showed that the increases were statistically significant. Moreover, SIC were significantly lower in 2019 than in 2001 in all quarters. While the changes between 2001 and 2019 in zonal and meridional velocities were ambiguous, ocean EKE was significantly higher except for the second quarter. For illustrating the trends, the quarterly sample means for the subset of years 2001, 2007, 2013, and 2019 are displayed in Figure 1 for the Northern hemisphere.

Table 1. Sample means and standard deviations of 3-yearly averages of quarterly sea surface temperatures, sea ice concentrations, zonal and meridional ocean current velocities, and ocean eddy kinetic energy for 1° × 1° grids in the Northern hemisphere during 2000–2019.^{a, b}

^*Paired t-tests were significant (P < 0.05) for changes in the 3-yearly averages between 2001 and 2019.

^aQuarterly data for seven 3-yearly averages were used for up to 32,400 1° × 1° grids in the Northern hemisphere for computing sample means and standard deviations.

^bQuarterly data were available in year 2001 for sea surface temperatures, sea ice concentrations, and zonal and meridional velocities for, respectively, 17,896, 10,281, 12,452, and 12,461 grids; twice the values of ocean EKE are reoprted and analyzed.

Table 2. Sample means and standard deviations of 3-yearly averages of quarterly sea surface temperatures, sea ice concentrations, zonal and meridional ocean current velocities, and ocean eddy kinetic energy for 1° × 1° grids in the Southern hemisphere during 2000–2019.^{a, b}

^*Paired t-tests were significant (P < 0.05) for changes in the 3-yearly averages between 2001 and 2019.

^aQuarterly data for seven 3-yearly averages were used for up to 32,400 1° × 1° grids in the Southern hemisphere for computing sample means and standard deviations.

^bQuarterly data were available in year 2001 for sea surface temperatures, sea ice concentrations, and zonal and meridional velocities for, respectively, 21,810, 12,931, 21,243, and 21,243 grids; twice the values of ocean EKE are reoprted and analyzed.

The sample means in Table 2 for the Southern hemisphere were similar to the corresponding results in Table 1. SST significantly increased from 2001 to 2019, whereas SIC significantly declined during this period in all quarters. Moreover, ocean EKE was higher in 2019 except for the last quarter. Figure 2 showed similar patterns for the hemispheres though the scales on the y-axis for SST and SIC were slightly different; ocean EKE was generally lower in the Southern hemisphere.

Further, the bivariate correlations between SST, SIC, zonal and meridional velocities, and ocean EKE in the first quarter for the Northern hemisphere are reported in the upper panel of Table 3. The corresponding results for the third quarter covering the winter months in the Southern hemisphere are presented in the lower panel. Note that in Table 3, correlations are also presented for the same variables in 2001 and 2019, partly to appraise the dynamic aspects of climate indicators. Similar results were obtained for remaining quarters in both hemispheres but are not reported for economizing on space.

Table 3. Bivariate correlations among climate indicators using 3-yearly averages for 1° × 1° grids in winter quarters 1 and 3, respectively, in Northern and Southern hemispheres.^{a, b}

^*P < 0.05.

^aThe correlations were calculated using data on 12,452 and 19,048 1° × 1° grids for Zonal and Meridional velocities and Eddy kinetic energy (EKE) in 2001 in the Northern and Southern hemispheres, respectively.

^bNumber of grids for Sea Surface Temperatures and Sea Ice Concentration in 2001 were 6,621 and 2,693 in Northern hemisphere, and 16,216 and 9,626 in Southern hemisphere, respectively.

The salient features of the results for Northern hemisphere were that the zonal velocities were negatively correlated with SST, whereas meridional velocities were positively correlated. This was also true for the correlations between zonal and meridional velocities and SIC. By contrast, ocean EKE was positively and negatively correlated, respectively, with SST and SIC in both hemispheres. Thus, from an internal data consistency standpoint, bivariate correlations indicated a preference for analyzing the data on ocean EKE rather than on zonal and meridional velocities in the empirical models. The internal consistency of remote sensing data was further corroborated by positive correlations between the same variables in 2001 and 2019, and by negative correlations between SST and SIC for all quarters in the hemispheres. Note that the correlations between ocean EKE and wind stress magnitudes in the first quarters of 2001 and 2019 in the Southern hemisphere were 0.022 and 0.037, respectively; such issues are addressed in the analyses below.

3.2 Efficient estimates of model parameters for ocean eddy kinetic energy in the Northern hemisphere

Table 4 presents the estimated coefficients from models explaining ocean EKE in the quarters in the Northern hemisphere by geodesic variables, indicators for time periods, SST, SIC, and interactions between SST and SIC. The coefficients of the squared SST and SIC were generally insignificant, presumably because the data were available for relatively small numbers of grids; such coefficients are reported for the Sothern hemispheres in Table 5. First, the coefficients of latitudes were not statistically significant in the models for all quarters. By contrast, the coefficients of longitudes were negative and statistically significant in all quarters. Second, the indicator variables for the years were seldom statistically significant, though they were estimated with positive coefficients for 2019, indicating slight increases in the ocean EKE over time.

Table 4. Efficient estimates of static random effects models for ocean eddy kinetic energy using 3-year quarterly averages during 2000–2019 for 1° × 1° grids in the Northern hemisphere explained by geodesic variables, sea surface temperatures, sea ice concentrations and time indicators.^{a, b}

^*P < 0.05.

^aSlope coefficients and standard errors are reported using seven 3-year quarterly observations on 1° × 1° degree grids for eddy kinetic energy in the Northern hemisphere.

^bQuadratic terms in Sea Surface Temperatures and Sea Ice Concentration were not statistically significant for the Northern hemisphere.

Table 5. Efficient estimates of static random effects models for ocean eddy kinetic energy using 3-year quarterly averages during 2000–2019 for 1° × 1° grids in the Southern hemisphere explained by geodesic variables, sea surface temperatures, sea ice concentrations and time indicators.^{a, b}

^*P < 0.05.

^aSlope coefficients and standard errors are reported using seven 3-year quarterly observations on 1° × 1° grids for eddy kinetic energy in the Southern hemisphere.

^bQuadratic terms in sea surface temperatures and sea ice concentration are included in the models for the Southern hemisphere.

Third, the coefficients of SST and SIC were estimated with positive signs and were statistically significant in all quarters. Moreover, the interactive terms for these variables were estimated with negative coefficients that were significant. For illustrating the interactive effects, the partial derivates of ocean EKE with respect to SST and SIC in the first quarter are given, respectively, in Equations 8, 9:

{∂EKE/∂ln(SST+2)}=0.00325-0.00053{ln(SIC)}(8)

{∂EKE/∂ln(SIC)}=0.0104-0.00053{ln(SST+2)}(9)

Thus, the points of inflection of ocean EKE with respect to SST and SIC were 5.116 °C and 460%, respectively. More importantly, the partial derivatives of ocean EKE with respect to SST were positive for all admissible values of SIC, i.e., below 100%. In contrast, the partial derivatives of ocean EKE with respect to SIC were negative insofar as SST were below 5.116 °C which was seldom the case except in extremely cold regions. Taken together, these results reveal asymmetries in the effects on ocean EKE of SST and SIC. For example, ocean EKE was likely to increase with SST, whereas it would be difficult to lower ocean EKE via increases in SIC because SST were above 5.116 °C for most grids. However, SIC can only be indirectly influenced by climate policies (see the Discussion). Note that these computations are mainly illustrative since confidence intervals for the coefficients were likely to be wide due to the nonlinear relationships and relatively modest number of grids in the Northern hemisphere.

3.3 Efficient estimates of model parameters for ocean eddy kinetic energy in the Southern hemisphere

Table 5 presents the results for ocean EKE explained by geodesic variables, time indicators, and a second-degree polynomial in SST and SIC for the quarters in the Southern hemisphere. First, the coefficients of latitudes were estimated with positive signs and were statistically significant for all quarters. The data for larger number of grids facilitated precise estimation of the coefficients. The longitudes were estimated with negative coefficients and were statistically significant in all quarters, as for the Northern hemisphere.

Second, the coefficients of SST, SIC, and their respective squared terms were generally significant in the models; interactive terms between SST and SIC were invariably significant in all quarters. The interactive effects for first quarter using the model with a second-degree polynomial in SST and SIC, respectively, are given in Equations 10, 11:

{∂EKE/∂ln(SST+2)}=0.0035−0.0077{ln(SST+2)} − 0.0015{ln(SIC)}(10)

{∂EKE/∂ln(SIC}=0.0008−0.0002{ln(SIC)} − 0.00053{ln(SST+2)}(11)

The points of inflection with respect to SST can be computed by setting the partial derivative to zero at the sample mean of SIC. From an analytical standpoint, Equation 10 implied that lower SIC were likely to lead to larger increases in ocean EKE with higher SST. By contrast, Equation 11 implied that ocean EKE will decrease more slowly with increases in SIC if SST were higher. However, it would be complicated to compute the confidence intervals for the net effects of SST and SIC for ocean EKE from these estimates. Instead, in the special cases where the models excluded squared SST and SIC, the empirical results showed that the coefficients of the interaction terms were invariably significant for all quarters in the Southern hemisphere. These issues are revisited in the Discussion.

Lastly, Table 6 reports the efficient estimates of models for ocean EKE in the Southern hemisphere where lagged wind stress magnitudes were included as explanatory variables for avoiding spurious associations (Bonjean and Lagerloef, 2002). Note that the squared of SST and SIC were omitted from these models. The Chi-squared tests for zero correlations (“exogeneity”) between lagged wind stress and the errors affecting the models were statistically significant for all quarters. However, the empirical results in Table 6 were robust to changes in the exogeneity assumptions. The main findings in Table 6 were that the coefficients of lagged wind stress magnitudes were positive and significant for all quarters in the Southern hemisphere. As noted above, wind stress variables were employed in constructing the zonal and meridional velocities so that the use of lagged wind stress magnitudes in the empirical models was likely to ameliorate the problems of spurious associations.

Table 6. Efficient estimates of static random effects models for ocean eddy kinetic energy using 3-year quarterly averages during 2000–2019 for 1° × 1° grids in the Southern hemisphere explained by geodesic variables, sea surface temperatures, sea ice concentrations, time indicators, and lagged wind stress magnitude.^{a, b}

^*P < 0.05.

^aSlope coefficients and standard errors are reported using six 3-year quarterly observations on 1° × 1° grids for eddy kinetic energy in the Southern hemisphere.

^bLagged wind stress magnitude was an explanatory variable reducing the number of time observations to 6.

3.4 Results for Atlantic Meridional Overturning Circulation transport at 35°N

Table 7 presents the results from static models for monthly AMOC transport and ocean EKE in columns 1 and 3, respectively, that were explained by a quadratic trend variable, SST, and ocean EKE (or AMOC transport) for the 2° × 55° grid covering the flow region. Moreover, columns 2 and 4 report the results from dynamic models for AMOC transport and ocean EKE, respectively, with nonlinearities in lagged explanatory variables. The results for static model for AMOC transport in column 1 showed that the coefficients of the trend variable and its squared were statistically significant. Thus, AMOC transport was initially increasing until the 39th month, i.e., until January, 2008. Subsequently, there was a decline in transport until 2013. Ocean EKE was significantly negatively associated with AMOC transport though SST were not. By contrast, in column 3, the trend variables were not significantly associated with ocean EKE, presumably due to small sample sizes. Thus, it seems important to formulate non-linearities in AMOC transport (cf. Le Bras et al., 2023; Ditlevsen and Ditlevsen, 2023). Also, SST and AMOC transport were significantly negatively associated with ocean EKE in column 3. Note that the Durbin–Watson statistics indicated significant positive autocorrelation in the errors affecting static models for AMOC transport and ocean EKE. By contrast, the errors on dynamic models were generally serially uncorrelated using tests statistics that were distributed in large samples as Chi-squared variables with one degree of freedom. Moreover, the adjusted R² increased from 0.435 for the static model to 0.484 for the dynamic model thereby indicating a preference for the dynamic model.

Table 7. Ordinary least squares estimates from static and dynamic models for monthly Atlantic Meridional Overturning Circulation (AMOC) transport and eddy kinetic energy explained by a quadratic trend, sea surface temperatures, and lagged explanatory variables in the 2° × 55° grid during 2005–2013.^{a, b}

^*P < 0.05.

^aSlope coefficients and standard errors are reported for Atlantic Meridional Overturning Circulation transport (Sv) using monthly data from moorings during 2005–2013.

^bData on sea surface temperatures and eddy kinetic energy were computed for 2° × 55° grid covering the AMOC region.

^cIndicates significant positive correlation in the errors.

^dIndicates no serial correlation in the errors affecting dynamic models.

Lastly, the dynamic models for AMOC transport in column 2 showed that lagged ocean EKE was significantly associated with transport but not its squared value. The lagged SST and its squared were estimated with insignificant coefficients. Further, the lagged values of AMOC transport were estimated with a large positive and significant coefficient 0.89 indicating gradual evolution of AMOC transport. By contrast, in column 4, lagged ocean EKE was estimated with a coefficient that was not significantly different from zero. The coefficients of lagged SST and their squared values were also not significant in the model for ocean EKE in column 4. Note that Lagrange multiplier type tests (StataCorp, 2022) for the serial independence of the errors affecting dynamic models for AMOC transport and Ocean EKE were 0.209 and 0.636, respectively. Thus, the lagged dependent variables accounted for the serial correlation in the errors and the null hypotheses of serial independence could not be rejected. The adjusted R² for the dynamic models for AMOC transport and ocean EKE in columns 2 and 4, respectively, were 0.48 and 0.13. Thus, the empirical results in Table 2 support the formulation that ocean EKE was negatively associated with AMOC transport, and that AMOC transport was generally not a significant predictor of ocean EKE (see below).

4 Discussion

This article modeled quarterly remote sensing longitudinal data for ocean EKE for 1° × 1° grids using the explanatory variables latitudes and longitudes, time indicators, SST, and SIC in the Northern and Southern hemispheres. The interactions between SST and SIC were underscored, and the consequences of higher ocean eddy EKE for AMOC transport at 35°N were investigated using moorings data. The analytical framework postulated bidirectional relationships between ocean EKE, wind stress, and AMOC transport, and econometric methods afforded systematic longitudinal data analyses for formulating evidence-based policies that can facilitate the achievement of SDG 14.

First, processing of remote sensing data for the zonal and meridional velocities of ocean surface currents invokes several algorithms so that it is necessary to appraise the internal consistency of the variables. The analysis of quarterly data at the grid level using 3-year averages reduced internal variation in the data. The application of paired t-tests in Tables 1, 2 showed that quarterly mean sea SST and ocean EKE significantly increased between 2001 and 2019, whereas there was a significant decline in SIC in both hemispheres. In addition, the bivariate correlations in Table 5 indicated ambiguous correlation patterns for quarterly SST and SIC with zonal and meridional velocities in the hemispheres. By contrast, ocean EKE was invariably positively and negatively correlated, respectively, with SST and SIC.

Second, while partial differential equations for ocean circulation have been employed since Laplace, as noted by Stommel (1961), ocean currents are chaotic and are likely to entail multiple equilibria even in the simplest cases. It is therefore important to analyze the effects of policy instruments such as SST for ocean EKE. The results for ocean EKE in the Northern hemisphere in Table 4 from estimating static random effects models, that accounted for unobserved heterogeneity across grids and serial correlation in the errors, showed significant increases due to higher SST and lower SIC. Moreover, the interactions between these variables were significant for all quarters. The computation of the nonlinear effects of SST and SIC in Equations 8, 9, respectively, indicated that it is important to lower SST for reducing ocean EKE for any level of SIC. By contrast, the evaluation of partial derivatives in Equation 9 implied that it may not be fruitful to increase SIC- which cannot be directly achieved by climate policies. It is interesting that such aspects were evident in the analyses of remote sensing data on ocean EKE where non-linear interactions between the climatic indicators were taken in account. While the use of ocean EKE has been previously advocated (Richardson, 1983; Bhargava and Echenique, 2022), the present analyses provide insights into the nonlinear effects of SST and SIC for ocean EKE.

Third, the results for the Southern hemisphere in Table 5 for ocean EKE showed not only significant interactions between SST and SIC but also the importance of squared of these variables. The greater numbers of grids facilitated the estimation of models, and the results indicated certain difficulties in making future predictions regarding ocean EKE. For example, while it is important to reduce SST via lowering heat emissions, the benefits of such policies will accrue at declining rates and depend on SIC that are adversely affected by increases in ambient temperatures and by ocean acidification which lowers freezing point of sea water. Moreover, lagged wind stress magnitudes were positively associated with ocean EKE in the Southern hemisphere, presumably due to topography of the Southern Ocean and the circumpolar mean flow (Cai et al., 2022). In contrast, lagged wind stress magnitudes were generally not significantly associated with ocean EKE in the Northern hemisphere. Note that owing to the interpolations invoked in constructing zonal and meridional ocean surface current velocities, it is preferable to estimate static models for 3-year averages for ocean EKE rather than analyzing annual observations. In addition, the analyses of AMOC transport data utilized monthly observations on the variables. Overall, evidence-based policies should seek to simultaneously reduce greenhouse gas emissions (World Data Centre for Greenhouse Gases, 2021), heat emissions, and industrial and agricultural runoffs into the oceans (National Centers for Environmental Information, 2019) for achieving SDG 14. For example, reducing ocean acidification is also important for enhancing marine life health and for sustainable fisheries (Yu et al., 2024; Sun et al., 2025).

Finally, it is important to analyze the effects of higher SST and ocean EKE for AMOC transport. While SST have been suggested as proxies for transport (Ditlevsen and Ditlevsen, 2023), this practice is likely to be misleading (Little et al., 2020). In fact, the results in Table 7 showed the lack of significant associations between SST and AMOC transport at 35°N in static and dynamic models. By contrast, ocean EKE was a negative and significant predictor in the models for AMOC transport. Similarly, dynamic models for AMOC transport showed significant effects of lagged ocean EKE but not of lagged SST. Because ocean EKE reflects to some degree the effects of SST and SIC, it is a useful indicator for assessing the impact of global warming for ocean circulation. Furthermore, AMOC may be a system of interlinked currents (Weijer et al., 2019; Frajka-Williams et al., 2023) so that it is simplistic to make predictions about its future course without covering many flow regions. Because ocean EKE was negatively associated with transport around 35°N, it is important to investigate whether other regions exhibit similar patterns. It is critical to compile data for AMOC transport in different flow regions (Petit et al., 2025) and also measure variables such as the sea surface temperatures in situ. Because future climatic patterns in the continents are likely to be differentially affected by instabilities in AMOC, elaborate data analyses can provide useful insights for policy formulation and for the achievement of the SDG 14.

Correction note

This article has been corrected with minor changes. These changes do not impact the scientific content of the article.

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Footnotes

Edited by: Phoebe Koundouri, Athens University of Economics and Business, Greece

Reviewed by: Hongchu Yu, Wuhan University of Technology, China

Ling Sun, Shanghai Maritime University, China