Main

The ocean is constantly moving, driven by momentum, heat and freshwater fluxes at the air–sea interface, transporting heat and material from one place to another1,2,3,4,5. The rapid increase in greenhouse gas emissions over recent decades has caused notable changes in these fluxes and would probably keep doing so in the future6,7,8,9. Comprehending the responses of ocean currents to global warming is thus crucial to predict the ocean’s capacity for heat uptake and carbon sequestration, as well as its effects on future climate changes10,11,12.

Owing to the sparsity of ocean observations and coarse resolution of numerical climate simulations, existing knowledge of anthropogenic ocean current changes is mainly limited to its component at the largest spatio-temporal scale, that is, the large-scale ocean circulations, including the overturning circulations, basin-scale gyres, equatorial currents and boundary currents13,14,15,16,17,18. The responses of regional large-scale ocean circulations to historical anthropogenic forcing show large diversity and might be masked by natural variabilities15,16,17. Yet recent studies have consistently demonstrated an increased globally integrated kinetic energy (KE) of large-scale ocean circulations in the upper layer, despite ongoing debate regarding its underlying drivers14,18,19.

Overlooked in these analyses are the ocean currents at spatial scales from tens to hundreds of kilometres and temporal scales from days to months, loosely referred to as mesoscale eddies20,21,22. These mesoscale eddies pervade the entire ocean and form the major fraction of the KE spectrum of the ocean currents23,24,25. Satellite observations26 reveal an enhanced surface eddy activity, measured by eddy KE (EKE), over the past three decades, especially in eddy-rich regions such as the western boundary currents and their extensions (WBCEs) and the Antarctic Circumpolar Current. This finding is in broad agreement with the projected future EKE change based on a global climate simulation with regionally refined grids to resolve mesoscale eddies in the eddy-rich regions27. However, little is known about the EKE trend below the sea surface. It remains unclear whether the surface intensification of mesoscale eddies is a useful fingerprint of total EKE change in the global ocean under global warming. Neither do we know its contribution to the total KE change.

Observational records are probably too short to distinguish anthropogenic climate changes from natural multidecadal variabilities27,28,29. State-of-the-art climate models provide an alternative to evaluate the response of total ocean KE to global warming27,30,31. Here we assess the future changes in the total ocean KE contributed by large-scale ocean circulations and mesoscale eddies under a high carbon emission scenario, using an ensemble of high-resolution climate simulations based on the Community Earth System Model31 (CESM-H; see ‘CESM-H’ in Methods). Benefiting from resolving mesoscale eddies, the CESM-H shows much-improved performance in reproducing the KE in the ocean reanalysis (see ‘Reanalysis dataset’ in Methods), including its magnitude, vertical profile and geographic distribution, compared with its low-resolution counterpart from the CESM Large Ensemble project32 (CESM-L; Fig. 1 and Supplementary Fig. 1; see ‘Low-resolution climate models’ in Methods). The overly small KE in the CESM-L compared with the reanalysis and CESM-H is consistent with the dominant contribution of the EKE to the KE23,24,25 (Supplementary Fig. 1), giving us confidence in the CESM-H’s superiority of projecting the KE change under global warming over the standard low-resolution climate models in the Coupled Model Intercomparison Project Phase 6 (CMIP6)33.

Fig. 1: Dominant contribution of mesoscale eddies to ocean KE in the global ocean.
figure 1

a–c, Geographic distribution of time-mean (1993–2020) vertically integrated KE in the GLORYS12 reanalysis (a), CESM-H (b) and CESM-L (c). Mesoscale eddies are resolved in the GLORYS12 reanalysis and CESM-H, but not in the CESM-L.

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Response of KE to global warming in the CESM-H

The linear trend of vertically integrated KE (\(\widetilde{{\rm{KE}}}\)) over 2001–2100 is geographically heterogeneous (Fig. 2a). One striking feature is the widespread decrease of \(\widetilde{{\rm{KE}}}\) in the basin interior, where the climatological mean \(\widetilde{{\rm{KE}}}\) is at a low level (Fig. 1a,b). The \(\widetilde{{\rm{KE}}}\) trend in the high-\(\widetilde{{\rm{KE}}}\) region is more complicated. In the subtropical WBCEs of the Northern Hemisphere (that is, the Kuroshio and Gulf Stream), there is a significant decrease of \(\widetilde{{\rm{KE}}}\), whereas the opposite is true over their recirculation areas. In contrast, \(\widetilde{{\rm{KE}}}\) is significantly increased in the subtropical WBCEs of the Southern Hemisphere, especially in the East Australian Current, Malvinas Current and Brazil Current. Moreover, \(\widetilde{{\rm{KE}}}\) experiences a significant reduction over a large fraction of the equatorial region, except for the central Pacific Ocean. As for the Southern Ocean, the \(\widetilde{{\rm{KE}}}\) trend is characterized by alternating zonal bands of positive and negative values.

Fig. 2: Changes of ocean KE under global warming projected by CESM-H.
figure 2

a–c, Geographic distribution of linear trend of vertically integrated KE (a), MKE (b) and EKE (c) anomalies during 2001–2100 in the CESM-H. The anomalies of KE, MKE and EKE are obtained by subtracting their pre-industrial values. Regions with the trends insignificant at a 95% confidence level are filled in white.

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Both changes of EKE and KE of large-scale ocean circulations (mean KE and MKE for short) contribute to the trend of \(\widetilde{{\rm{KE}}}\) (Fig. 2b,c; see ‘Definition of large-scale ocean circulations and mesoscale eddies’ in Methods). In the high-\(\widetilde{{\rm{KE}}}\) regions, the trends of vertically integrated EKE (\(\widetilde{{\rm{EKE}}}\)) and MKE (\(\widetilde{{\rm{MKE}}}\)) are generally in phase with each other. Such consistency is expected as mesoscale eddies are generated primarily via the baroclinic and barotropic instabilities of large-scale ocean circulations34,35,36,37, so that stronger (weaker) large-scale ocean circulations cause more (less) energetic mesoscale eddies. In contrast, the trends of \(\widetilde{{\rm{EKE}}}\) and \(\widetilde{{\rm{MKE}}}\) differ evidently from each other in the low-\(\widetilde{{\rm{KE}}}\) regions. Although the trend of \(\widetilde{{\rm{MKE}}}\) in the low-\(\widetilde{{\rm{KE}}}\) regions shows patches of positive and negative values, the trend of \(\widetilde{{\rm{EKE}}}\) is generally negative and accounts primarily for the widespread decrease of \(\widetilde{{\rm{KE}}}\) in the basin interior.

These geographically heterogeneous trends of \(\widetilde{{\rm{KE}}}\) as well as \(\widetilde{{\rm{EKE}}}\) and \(\widetilde{{\rm{MKE}}}\) reflect the complicated dynamics governing the regional changes of ocean currents and may overshadow their global responses to the greenhouse gas forcing. For instance, the increased \(\widetilde{{\rm{KE}}}\) in the subtropical WBCEs of the Southern Hemisphere may result from the intensification and poleward shift of the surface westerlies associated with a positive trend in the Southern Annular Mode17,38,39. The slowdown of the Atlantic Meridional Overturning Circulation could contribute to the widespread decrease of \(\widetilde{{\rm{KE}}}\) in the Atlantic Ocean, especially along the western boundary region (for example, Gulf Stream and its extension)40,41. It may also play a role in the decreased \(\widetilde{{\rm{KE}}}\) in the Indian Ocean by weakening the Indonesian Throughflow42,43. Meridional shift of the Antarctic Circumpolar Current under global warming44,45 may cause alternating zonal bands of positive and negative \(\widetilde{{\rm{KE}}}\) trends. To uncover the total KE change across the global ocean, we compute the horizontally integrated KE (\(\langle {\rm{KE}}\rangle\)) and its linear trend during 2001–2100 (Fig. 3). There is a significant positive trend (6.34 ± 3.40 × 1016 J per century, or equivalently 3.3 ± 1.7% per century; mean ± 2 × s.e.m.) of \(\langle {\rm{KE}}\rangle\) integrated over the upper 200 m dominated by that (5.67 ± 2.40×1016 J per century) of the horizontally integrated MKE (\(\langle {\rm{MKE}}\rangle\)). This is consistent with the existing literature14,18 suggesting accelerated large-scale ocean circulations in the upper ocean under global warming. In addition, the surface horizontally integrated EKE (\(\langle {\rm{EKE}}\rangle\)) shows a significant increase (Supplementary Fig. 2), which is also in line with the previous estimates26,27. However, the trend of \(\langle {\rm{EKE}}\rangle\) attenuates rapidly with increasing depth and reverses its sign below 100 m (Supplementary Fig. 2), so that the \(\langle {\rm{EKE}}\rangle\) integrated over the upper 200 m is almost unchanged (Fig. 3f). It thus suggests that the surface imprint of the \(\langle {\rm{EKE}}\rangle\) trend is not representative of its vertical integral.

Fig. 3: Response of mesoscale eddies in the deep ocean to global warming dominates the long-term change of globally integrated KE.
figure 3

a,c,e, Time–depth plot of horizontally integrated KE (a), MKE (c) and EKE (e) anomalies during 1920–2100 in the CESM-H. b,d,f, Linear trends of KE (b), MKE (d) and EKE (f) anomalies integrated over the upper ocean (≤200 m), deep ocean (>200 m) and entire water column during 2001–2100, with the error bars denoting their 95% confidence intervals. The anomalies of KE, MKE and EKE are obtained by subtracting their pre-industrial values.

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Below 200 m, \(\langle {\rm{KE}}\rangle\) shows a negative trend (−21.79 ± 5.75 × 1016 J per century or equivalently −6.5 ± 1.7% per century) mainly contributed by that (−15.22 ± 3.69 × 1016 J per century) of \(\langle {\rm{EKE}}\rangle\) (Fig. 3). This deep-ocean decrease of \(\langle {\rm{KE}}\rangle\) overwhelms the increase of \(\langle {\rm{KE}}\rangle\) in the upper 200 m, causing the globally integrated KE to decline at a rate of −15.45 ± 7.98 × 1016 J per century or equivalently −2.9 ± 1.5% per century. This highlights the fundamental role of deep-ocean mesoscale eddies in determining the response of globally integrated KE to global warming. The globally integrated EKE decreases at a rate of −14.55 ± 4.81 × 1016 J per century, accounting for more than 94% of the decrease of globally integrated KE. In particular, most of the decrease of the globally integrated EKE occurs below 200 m (Fig. 3f).

Underlying dynamics of the reduced EKE in CESM-H

To shed light on the underlying processes accounting for the weakened mesoscale eddies under global warming, we perform an EKE budget analysis (see ‘EKE budget’ in Methods) using diagnostic outputs from the CESM-H. The EKE budget decomposes the EKE change into components driven by different dynamics, including barotropic energy conversion (BT) from MKE to EKE, baroclinic energy conversion (BC) from eddy available potential energy (EAPE) to EKE, surface wind power (WP), interior dissipation (ID) and bottom drag (BD). For the climatological mean state, the EKE budget reveals a delicate balance between energy sources and sinks (Fig. 4a). In terms of the global integral, the generation of EKE is primarily attributed to the BC term, resulting in an EKE source of 1.08 TW. The BT and WP terms have minor effects, contributing jointly to an EKE source of 0.11 TW. These sources of EKE are largely balanced by the ID term (−1.09 TW) and to a much lesser extent by the BD term (−0.12 TW). Under global warming, the BC term shows a 39% decrease (−0.42 TW) and is primarily responsible for the decreased EKE (Fig. 4b). By weakening the EKE, the reduced BC term leads to a decrease of the ID term (0.43 TW) so that a new quasi-equilibrium state can be established. Changes in the BT, WP and BD terms play a minor or negligible role.

Fig. 4: Dynamical processes responsible for the reduced EKE under global warming.
figure 4

a, Time-mean (1920–1934) globally integrated EKE budget with TD representing the EKE tendency, BT the barotropic energy conversion from MKE to EKE, BC the baroclinic energy conversion from EAPE to EKE, WP the surface wind power, ID the interior dissipation and BD the bottom drag. b, Same as a, but for the difference of EKE budgets between 2086–2100 and 1920–1934. c, Time series of globally integrated MAPE anomaly in the CESM-H (blue) and ensemble mean of 28 low-resolution climate models in the CMIP6 (red) with the shading representing the 95% confidence interval of the CMIP6 ensemble mean. The MAPE anomaly is obtained by subtracting its pre-industrial values.

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In the BC pathway, mesoscale eddies are essentially fuelled by the mean available potential energy stored in the large-scale ocean circulations (MAPE)34,35,46. In particular, the BC term is maintained by the conversion from MAPE to EAPE. Under global warming, the MAPE-to-EAPE conversion rate is changed by an amount of −0.49 TW, fairly close to a −0.42 TW change of the BC term. The MAPE-to-EAPE conversion rate is directly proportional to the horizontal density gradient associated with large-scale ocean circulations, but inversely proportional to the vertical stratification (see ‘EKE budget’ in Methods). Under global warming, the increased (decreased) MKE in the upper (deep) ocean enhances the vertical shear of large-scale ocean circulations. This corresponds to a stronger horizontal density gradient through the thermal wind balance47 (Supplementary Fig. 3a). However, such an effect on the MAPE-to-EAPE conversion rate is overwhelmed by the intensified vertical stratification, especially in the deep ocean, so that the MAPE-to-EAPE conversion rate is reduced under global warming (Supplementary Fig. 3b,c). As the intensified vertical stratification under global warming is a very robust feature consistently obtained from the observations, reanalysis datasets and climate simulations48,49,50, this lends support to the fidelity of the reduced MAPE-to-EAPE conversion rate simulated by the CESM-H.

The reduced MAPE-to-EAPE conversion rate under global warming can also be understood based on the decrease of globally integrated MAPE (Fig. 4c; see ‘Computation of MAPE’ in Methods). Again, the latter is primarily attributed to the enhanced vertical stratification. Recomputing the MAPE with the vertical stratification replaced as its pre-industrial value reverses the MAPE trend (Supplementary Fig. 4), as the intensified horizontal density gradient of large-scale ocean circulations alone would increase the MAPE under global warming. In summary, the enhanced vertical stratification under global warming diminishes the MAPE, causing the EAPE and further EKE to reduce by inhibiting the BC pathway (Extended Data Fig. 1). It should be noted that the decrease of globally integrated MAPE under global warming in the CESM-H agrees well with that in the ensemble mean of low-resolution climate models in the CMIP6 (Fig. 4c; see ‘Low-resolution climate models’ in Methods), lending further support to the robustness of the results derived from the CESM-H.

Finally, we remark that the above analysis, focusing on the globally integrated property, only provides a first-order understanding of EKE change under global warming and its underlying dynamics. Locally, the EKE trend (Figs. 2c and 3e) can differ substantially from its global mean value, suggesting complicated dynamics governing the local EKE response to global warming. We perform the EKE budget analyses over some selected layers and regions (Extended Data Figs. 2 and 3). Although there is a universal reduction of the BC term under global warming, the change of the WP term, BT term and energy convergence though the advection and pressure flux could locally be important for the response of EKE to global warming. Furthermore, the reduced BC term locally is not necessarily attributed to the enhanced vertical stratification. The change of wind- and buoyancy-driven large-scale ocean circulations such as the WBCEs and Atlantic Meridional Overturning Circulation could also play a role40,41,42,43,44,45. A comprehensive understanding of dynamics governing the spatial variability of EKE trend deserves further investigation in future studies.

Validation based on idealized simulations

A set of idealized simulations of wind-driven double gyres is conducted to further demonstrate the effects of enhanced vertical stratification on the EKE change and its underlying dynamics (Fig. 5; see ‘Idealized numerical simulations’ in Methods). In these simulations, mesoscale eddies are generated primarily through the baroclinic instabilities of wind-driven large-scale ocean circulations, mimicking the situation in the CESM-H. The surface wind forcings are identical among the simulations, whereas the vertical stratifications differ as the sea surface temperature (SST) is resorted to different values. Consistent with the CESM-H’s results, there is a monotonic decrease of domain-integrated EKE with the intensified vertical stratification that diminishes the BC term despite the enhanced horizontal density gradient of large-scale ocean circulations (Fig. 5c–f).

Fig. 5: Effects of global warming on EKE change in idealized simulations.
figure 5

a, Simulated time-mean sea surface height (contour) and SST (shading) during the last 40 years in the CTRL run. b, Snapshot of sea surface height (SSH) anomaly on some day in the CTRL run. c, Differences of time-mean domain-averaged vertical stratification during the last 40 years between the ΔSST-Forc runs and CTRL run. d–f, Same as c, but for the KE (d), BC term (e) and squared horizontal density gradient of large-scale ocean circulations (f).

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The EKE dissipation in climate models is parameterized31 and thus has large uncertainties. We utilize the idealized numerical simulations to test whether the reduced EKE under global warming is robust regardless of the parameterized EKE dissipation. Sensitivity experiments are conducted by varying the parameterizations of viscous mixing and grid sizes (see ‘Idealized numerical simulations’ in Methods). These experiments show a universal EKE reduction in response to the intensified vertical stratification (Supplementary Fig. 5), despite some quantitative differences. This lends further support to the robustness of the reduced EKE under global warming.

Discussion

Although previous analyses have focused mainly on the anthropogenic change of global ocean currents in the upper ocean14,15,16,17,18, this study uncovers a more quiescent deep ocean under global warming. In particular, the weakened mesoscale eddies in the deep ocean, overlooked in the existing literature, dominate the global KE change. The current generation of global climate models is generally too coarse to resolve mesoscale eddies33. Accordingly, the simulated KE change in these low-resolution climate models is dominated by the MKE change, which may misrepresent the total KE change under global warming (Supplementary Fig. 6). In fact, the changes of globally integrated MKE under global warming show large inter-model differences in the low-resolution climate models in the CMIP6 with their ensemble mean MKE trend insignificant (Supplementary Fig. 7), despite the consistent MAPE reduction across these models (Fig. 4c). It should be noted that the BC pathway is parameterized rather than resolved in low-resolution climate models51,52. Accordingly, the reduced MAPE under global warming in these models (Fig. 4c) cannot affect the EKE change as in reality and CESM-H.

The advent of satellite altimeters advances the knowledge of mesoscale eddies at the sea surface, including their response to global warming26. However, we remark that such knowledge is prone to conceal the total EKE change over the entire water column and could even yield misleading conclusions, as the enhanced vertical stratification may have opposite effects on the surface and vertically integrated EKE. On the one hand, the enhanced vertical stratification weakens the BC pathway, reducing the vertically integrated EKE. On the other hand, it tends to make the EKE more concentrated in the upper ocean21,53 and may partially account for the surface-intensified EKE under global warming.

Finally, there is a caveat that the CESM-H does not resolve all the oceanic motions and their contributions to the total KE change under global warming. In the deep ocean, these unresolved motions are mainly composed of submesoscale eddies, internal waves, and turbulence54,55. Their responses to global warming remain unknown, except that a recent study56 suggests weakened internal lee wave generation in a warming climate. Nevertheless, as these smaller-scale motions possess much less KE compared with the mesoscale eddies and large-scale ocean circulations54,55, their KE changes under global warming are unlikely to dominate those of mesoscale eddies and large-scale ocean circulations. Nowadays, ocean currents below the sea surface are still poorly observed, especially in regions deeper than 2,000 m (ref. 57). New in situ observation platforms working efficiently in deep waters will be necessary to enhance our capability of monitoring anthropogenic changes of ocean currents in the deep ocean and developing adaptive strategies to better cope with climate change.

Methods

CESM-H

High-resolution climate simulations resolving ocean mesoscale eddies are performed based on the CESM1.3 (ref. 31). The CESM-H includes the atmospheric component of the Community Atmosphere Model version 5 (CAM5), the oceanic component of the Parallel Ocean Program version 2 (POP2), the sea-ice component of the Community Ice Code version 4 (CICE4), and the land component of the Community Land Model version 4 (CLM4). The POP2 and CICE4 use the same nominal horizontal resolution of ~0.1°, and the CAM5 and CLM4 use the same nominal horizontal resolution of ~0.25°. There are 62 vertical levels in the ocean model, with increasing grid space from 10 m near the surface to 250 m near the maximum depth of 6,000 m. More details on model configurations can be found in ref. 31.

The CESM-H consists of a 500-year-long pre-industrial control (PI-CTRL) climate simulation and three members of historical-and-future transient (HF-TNST) climate simulation using time-varying historical forcing from 1850 to 2005 and representative concentration pathway 8.5 (RCP8.5) forcing during 2006–2100 following the protocol for the CMIP5 experiments58. The first member (CESM-H1) is initialized from the PI-CTRL simulation at year 250, whereas the second and third members (CESM-H2 and CESM-H3) are branched from the CESM-H1 simulation in 1920 with slightly perturbed initial atmospheric conditions. The three members are integrated to 2100.

All the simulations save monthly mean three-dimensional velocity, temperature, salinity and KE from 1920 to 2100. Furthermore, daily mean three-dimensional velocity, temperature, salinity, and diagnostic outputs for the momentum equations are stored during 1920–1934 and 2086–2100 in the CESM-H1 to investigate the underlying dynamics of EKE changes under global warming. The time series of KE, as well as MKE, MAPE and EKE, is computed as the ensemble mean over the three members, whereas the EKE budget is computed based on the CESM-H1 alone, as there are no daily mean outputs for the CESM-H2 and CESM-H3 due to the storage limitation. Although there are inter-member differences in the globally integrated KE at interannual and decadal scales, their long-term trends are qualitatively consistent with each other (Supplementary Fig. 8), giving us confidence that the change of EKE budget under global warming in the CESM-H1 should be representative of its ensemble mean.

After a spinup of 250 years, the model drift in the deep ocean becomes small but still noticeable, especially for the temperature and salinity. This may bias the responses of large-scale ocean circulations and mesoscale eddies to the greenhouse gas forcing. To minimize such bias, we subtract the time series of KE, MKE, MAPE and EKE during the years 250–500 in the PI-CTRL simulation (Supplementary Fig. 9) from their counterparts during 1850–2100 in the HF-TNST simulation (for example, 2100 in the HF-TNST simulation corresponds to the year 500 in the PI-CTRL simulation), and refer to the resultant values as the KE, MKE, MAPE and EKE anomalies.

Reanalysis dataset

To assess the realism of the simulated KE over the global ocean by the CESM-H, we use the Global Ocean Physics Reanalysis products (GLORYS12)59 as a benchmark. The GLORYS12 has a horizontal resolution of 1/12° and 50 vertical levels, with 22 levels in the upper 100 m. It provides daily mean horizontal velocity, temperature and salinity from 1993 to 2020.

Low-resolution climate models

The output from the CESM-L32 consisting of 40 members is used to compare the KE changes under global warming between low-resolution and high-resolution climate models. The experimental setup of the CESM-L is similar to that of the CESM-H1, but its oceanic resolution (~1°) is insufficient to resolve mesoscale eddies. To test the robustness of the projected MAPE change by the CESM-H, we use another 28 climate model simulations in the CMIP633 under the Shared Socioeconomic Pathways (SSP585; Supplementary Fig. 7), in addition to the CESM-L. Unlike the CESM-L, these 28 climate models do not output the monthly mean KE, so they cannot be used to estimate the response of KE to global warming.

Definition of large-scale ocean circulations and mesoscale eddies

The large-scale ocean circulations and mesoscale eddies are distinct from each other both in their spatial and temporal scales20. Here, we isolate them based on their different temporal scales due to the data availability. Specifically, large-scale ocean circulations are obtained from a three-month average of ocean currents, while mesoscale eddies are computed as anomalies from the three-month average. In this case, the EKE during 1920–2100 can be computed as:

$${\mathrm{EKE}}=\frac{1}{2}{\rho}_{0}\left(\overline{{{{{{u}}}}{\prime} }^{2}+{{{{{v}}}}{\prime} }^{2}}\,\right)=\frac{1}{2}{\rho }_{0}\overline{\left({{{{{u}}}}}^{2}+{{{{{v}}}}}^{2}\right)}-\frac{1}{2}{\rho }_{0}\left({\overline{{{{{u}}}}}}^{\,2}+{\overline{{{{v}}}}}^{\,2}\right)$$
(1)

where the overbar denotes the three-month average, ρ0 is a reference density, and u and v are the zonal and meridional velocity, respectively. The first term on the rightmost side of equation (1) is computed based on the monthly mean KE output by the CESM-H, and the second term is computed based on the monthly mean velocity output by the CESM-H.

To examine whether the response of EKE to global warming depends on the way mesoscale eddies are defined, we compare the EKE changes before and after global warming with mesoscale eddies defined as anomalies from a 3-month average, a 91-day low-pass filter, a 3° × 3° horizontal Gaussian kernel filter and a 4° × 4° horizontal running mean. As the latter three definitions require daily mean velocity to compute EKE, the computation is conducted during 1920–1934 and 2086–2100 when the CESM-H1 outputs daily mean velocity. As shown in Supplementary Fig. 10, using different definitions of mesoscale eddies leads to qualitatively consistent EKE changes from 1920–1934 to 2086–2100. All reveal reduced deep-ocean EKE under global warming.

EKE budget

To reveal the underlying dynamics for the EKE change under global warming, an EKE budget analysis60,61 is performed:

$$\begin{array}{lll} \displaystyle\frac{\partial }{\partial t}\iiint {\rm{EKE}}{\mathrm{d}}V_{\mathrm{TD}} &=& \displaystyle -\iiint g\overline{\rho {\prime} w{\prime} }{\mathrm{d}}V_{\mathrm{BC}} \\ && \displaystyle -\, \iiint g{\rho }_{0}\left(\overline{u{\prime} {\bf{u}}{{{\prime} }}}\cdot \nabla \bar{u}+\overline{v{\prime} {\bf{u}}{{{\prime} }}}\cdot \nabla \bar{v}\right){\mathrm{d}}V_{\mathrm{BT}}\\ && \displaystyle +\, \iint \overline{{{\mathbf{\uptau }}{{{\prime} }}}_{{\mathrm{w}}}\cdot {{\bf{u}}{{{\prime} }}}_{{\mathrm{s}}}}{\mathrm{d}}x{\mathrm{d}}y_{\mathrm{WP}} - \iint \overline{{{\mathbf{\uptau }}{{{\prime} }}}_{{\mathrm{b}}}\cdot {{\bf{u}}{{{\prime} }}}_{{\mathrm{b}}}}{\mathrm{d}}x{\mathrm{d}}y_{\mathrm{BD}}\\ && \displaystyle +\, \iiint {\rho }_{0}\left(\overline{u{\prime} {D{\prime} }_{{\mathrm{u}}}+v{\prime} {D{\prime} }_{{\mathrm{v}}}} \right.\\&&\displaystyle- \left.\overline{\left(\left({A}_{{\mathrm{v}}}\frac{\partial u}{\partial z}\right){\prime} \frac{\partial u{\prime} }{\partial z}+\left({A}_{{\mathrm{v}}}\frac{\partial v}{\partial z}\right){\prime} \frac{\partial v{\prime} }{\partial z}\right)}\,\right){\mathrm{d}}V_{\mathrm{ID}}\end{array}$$
(2)

where u = (u, v, w) is the three-dimensional velocity, z and t are the vertical and temporal coordinates, ρ is the density, g is the gravity acceleration, \({{\mathbf{\uptau }}}_{{\mathrm{w}}}\) (\({{\mathbf{\uptau }}}_{{\mathrm{b}}}\)) is the surface wind stress (bottom stress), us (ub) is the horizontal current at the sea surface (bottom), Av is the turbulent vertical viscosity parameterized by a K-profile parameterization62, Du (Dv) is the horizontal mixing for zonal (meridional) momentum, the overbar (prime) represents the three-month average (anomaly), EKE is defined as \(1/2{\rho }_{0}(\overline{{{{{{u}}}}{\prime} }^{2}+{{{{{v}}}}{\prime} }^{2}})\), \({\iiint} {\cdot} {\mathrm{d}}{V}\) denotes the volume integral over the global ocean, and \({\iint} {\cdot} {\mathrm{d}}x{\mathrm{d}}y\) denotes the area integral over the global sea surface or bottom.

The term on the left-hand side of equation (2) is the EKE tendency (denoted as TD). The terms on the right-hand side (including the sign) in sequence are the baroclinic energy conversion rate (BC) from EAPE to EKE, barotropic energy conversion rate (BT) from MKE to EKE, surface wind power (WP), bottom drag (BD) and interior dissipation (ID). All these terms in equation (2) can be explicitly computed based on the CESM-H’s daily outputs during 1920–1934 and 2086–2100.

It should be noted that the WP term is dominated by the wind power on Ekman currents that is almost dissipated within the surface Ekman layer via the ID term and unable to fuel mesoscale eddies61,63. To extract the usable wind power on mesoscale eddies that are largely in geostrophic balance, we re-define the WP term as the wind power on surface geostrophic currents61:

$${\rm{WP}}={\iint} \overline{{{\mathbf{\uptau }}{{{\prime} }}}_{{\mathrm{w}}}\cdot {{\bf{u}}{{{\prime} }}}_{{\mathrm{s}},{\mathrm{g}}}}{\mathrm{d}}x{\mathrm{d}}y$$
(3)

where us,g is the surface geostrophic current computed from the sea surface height using the geostrophic balance. As the geostrophic balance breaks down around the Equator, the integral excludes the region between 2° S–2° N where the EKE change contributes to less than 1% of the globally integrated EKE trend. To remove the ID of Ekman currents from the EKE budget, the ID term is recomputed by subtracting the difference between the WP terms computed according to equations (2) and (3) from the ID term in equation (2) based on the premise that the wind power on Ekman currents is balanced by the ID.

Supplementary Fig. 11 compares the EKE budgets computed by defining the WP and ID terms in two different ways. As expected, defining the WP and ID terms according to equation (2) causes them to be the leading terms in the time-mean EKE budget during 1920–1934, making the EKE budget dynamically less relevant to mesoscale eddies. Nevertheless, the decrease of the BC term under global warming accounts primarily for the reduced EKE in the global ocean, regardless of the definitions of the WP and ID terms.

Computation of MAPE

The globally integrated MAPE in this study is defined following refs. 46,64:

$$\left\langle \widetilde{{\rm{MAPE}}}\,\right \rangle =\iiint g\left[z-z^{\star}\left(\overline{{\;\rho }_{\uptheta }}\;\right)\right]\overline{{\rho }_{\uptheta }}{\mathrm{d}}V$$
(4)

where \(\overline{{\rho }_{\uptheta }}\) is the potential density of large-scale ocean circulations, and \(z^{\star} (\;\overline{{\rho }_{\uptheta }}\;)\) is the height of a fluid parcel in the background state \(\overline{{\rho }_{\uptheta }}^{\star} (z)\) associated with the minimum potential energy attainable through an adiabatic reordering of \(\overline{{\rho }_{\uptheta }}\). The \(\langle \widetilde{{\rm{MAPE}}}\rangle\) defined according to equation (4) is free from the assumption of a small perturbation of \(\overline{{\rho }_{\uptheta }}\) from \(\overline{{\rho }_{\uptheta }}^{\star} (z)\), and thus more desirable for a global scale analysis46 compared with the MAPE definition used in the Lorenz energy cycle60,61.

To remove the effect of vertical stratification change on the MAPE change under global warming, the globally integrated MAPE is recomputed as:

$${\left\langle \widetilde{{\rm{MAPE}}}\,\right\rangle }^{N}=\iiint g\left[z-z^{\star} \left({\overline{{\;\rho }_{\uptheta }}}^{\,N}\right)\right]{\overline{{\rho }_{\uptheta }}}^{\,N}{\mathrm{d}}V$$

where \({\overline{{\rho }_{\uptheta }}}^{\,N}=\overline{{\rho }_{\uptheta }}-{\overline{{\rho }_{\uptheta }}}_{H}+{\overline{{\rho }_{\uptheta }}}_{H,{\rm{PI}}\mbox{-}{\rm{CTRL}}}\) with the subscript H representing the horizontal average and PI-CTRL representing the counterpart in the PI-CTRL simulation (for example, 2100 in the HF-TNST simulation corresponds to the year 500 in the PI-CTRL simulation).

Idealized numerical simulations

To demonstrate the effect of enhanced vertical stratification on the EKE change, we conducted an idealized baroclinic double-gyre ocean circulation experiment using the Massachusetts Institute of Technology General Circulation Model (MITgcm)65. The model is configured as an enclosed fluid sector on a sphere, spanning from 12° N to 72° N and being 60° long in the zonal direction. There are 49 vertical layers with increasing space from 5 m near the sea surface to 150 m in the deep ocean. The model adopts the Leith viscosity scheme66. In addition, a K-profile parameterization scheme62 is adopted to parameterize the turbulent vertical mixing. The equation of state is simplified as a linear dependence of potential density on potential temperature with the salinity’s effect neglected. The model is forced by a zonal wind stress that is constant in time and varies sinusoidally in the meridional direction. In addition, the SST is restored to a prescribed climatological profile (SSTClim) ranging from 30 °C at the southern boundary to 0 °C at the northern boundary with a nudging time scale of 30 days, to simulate the buoyancy forcing at the sea surface.

The ocean is initialized as a rest warm pool with the potential temperature decreasing from 30 °C at the sea surface to 2 °C at the bottom layer. The model is spun up for 200 years at a 0.25° horizontal resolution, followed by another 60-year spinup at a 0.1° resolution, after which the large-scale ocean circulations and mesoscale eddies have reached their equilibrium states. Then, the simulation is split into five branches differing only in their SST restoring. The SST in the first branch (denoted as CTRL run) is still resorted to SSTClim, whereas the SST in the other four branches (ΔSST-Forc runs) is restored to SSTClim + ΔSST with ΔSST set as 0.2 °C, 0.6 °C, 1.0 °C and 2.0 °C over the entire sea surface, respectively. Each branch is integrated for 80 years, with the last 40 years used for analysis. As the value of ΔSST increases, the domain-averaged vertical stratification in the ΔSST-Forc runs becomes progressively stronger, mimicking the enhanced vertical stratification due to the ocean’s heat uptake under global warming18. The differences in large-scale ocean circulations and mesoscale eddies between the CTRL and ΔSST-Forc runs (Fig. 5) are qualitatively similar to those in the CESM-H before and after global warming. Specifically, the MKE in the ΔSST-Forc runs is increased (reduced) in the upper (deep) ocean compared with that in the CTRL run, as evidenced by the enhanced horizontal density gradient of large-scale ocean circulations. Such changes in large-scale ocean circulations become more evident with the larger ΔSST. As to the domain-integrated EKE and BC term, their values in the ΔSST-Forc runs are lower than their counterparts in the CTRL run and decrease monotonically as ΔSST increases.

To test whether the reduced EKE under global warming is robust regardless of the parameterized EKE dissipation in numerical models, we re-conduct the CTRL run and ΔSST-Forc (2 °C warming) run, using a biharmonic viscous mixing scheme with viscosity values set as 3.2 × 108 m4 s−1 and 80 × 108 m4 s−1. Although the experiments using different parameterizations of viscous mixing differ considerably in their simulated EKE levels in the CTRL run, they show similar percentage differences of EKE between the CTRL run and ΔSST-Forc (2 °C warming) run (Supplementary Fig. 5). In addition, we re-conduct the CTRL run and ΔSST-Forc (2 °C warming) run by refining the model grid size from 0.1° to 0.033°. It is found that changing the model grid size does not qualitatively alter the result, either (Supplementary Fig. 5).