Introduction

Policymakers increasingly require accurate future climate projections to properly design climate change adaptation and mitigation strategies1. However, despite significant advances in Earth System Models (ESMs), including higher vertical and horizontal resolution of the ocean, and more components in models of the biosphere2, the latest Coupled Model Intercomparison Project (CMIP) phase 6 (CMIP6) ESMs have a wider range of uncertainty in the future global warming rate than CMIP phase 5 ESMs3,4. The ESMs included in the CMIP have a systematic uncertainty despite having the same emission scenarios5. Therefore, understanding uncertainty is essential to increase the confidence of future projections.

Uncertainty comes from three factors: internal variability uncertainty due to the natural variability in the climate system, model uncertainty from the different response of ESMs to the same external forcing due to different parameterizations and governing equations of physical and biogeochemical systems, and climate change scenario uncertainty from different future greenhouse gas emissions6. Quantifying and understanding the contribution of each uncertainty could potentially provide a narrower range of future climate projections. Model Uncertainty is particularly high in physically complex regions, such as the polar regions and those related to ocean circulation, and in variables with large uncertainties including precipitation. As a result, reducing model uncertainty is critical to improving scientific understanding of the climate system and making effective projections of future climate changes5,7,8.

The North Atlantic subpolar region has experienced a cooling trend, which is referred to as the North Atlantic warming hole (hereafter, the warming hole)9, despite global warming during the historical period. This phenomenon is associated with the reduced meridional transport of oceanic heat content in the North Atlantic10,11 or a weakening of the Atlantic Meridional Overturning Circulation (AMOC)12,13,14 due to increasing anthropogenic greenhouse gases15,16. Furthermore, a melting of Arctic sea ice17,18 and atmospheric processes including wind changes and cloud feedback9,11,19,20 contribute to this phenomenon. These complicated mechanisms lead to a large bias to simulate a realistic warming hole from one ESM to another, which in turn creates a large amount of future uncertainty. Indeed, salinity in the warming hole region determines the amount of anthropogenic carbon uptake in the North Atlantic21. The salinity in the warming hole region, which represents the AMOC intensity, determines the amount of anthropogenic carbon uptake in the North Atlantic. The higher the salinity (i.e., the stronger AMOC intensity), the more carbon and heat is transported from the surface to the deep ocean, resulting in more uptake of anthropogenic carbon dioxide from the atmosphere at the ocean surface22. Therefore, changes in the warming hole are closely linked to shifts in the Intertropical Convergence Zone (ITCZ), which has important implications for the global distributions of precipitation and temperatures23,24,25. Thus, reduction of the uncertainty of future projections of the warming hole is essential for accurate future climate projections.

Here, we apply emergent constraints to reduce uncertainties in future warming hole changes. This method reduces uncertainty by combining observations with a high correlation between present and future changes between ESMs (see Methods)26,27. By analyzing 32 ESMs from CMIP6 with different scenarios, we found that ESMs with a lower present-day surface density tend to simulate a weaker the warming hole intensity change (indicating more warming in the North Atlantic subpolar region) in the future than ESMs with a higher present-day surface density because of intensified stratification.

Results

Future North Atlantic warming hole

Historical simulations with different Shared Socioeconomic Pathway (SSP) experiments (SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5) from 32 ESMs in CMIP6 (see Methods and Supplementary Table 1) were used to project future (2070–2099) changes relative to the pre-industrial period (1850–1900)28,29.

Figure 1 shows the future multi-model ensemble mean (MMEM) sea surface temperature (SST) changes in the Northern Hemisphere (NH) under different emission scenarios from low (SSP1-2.6) to high (SSP5-8.5) using 32 different ESMs. Future NH SSTs tend to increase more at higher latitudes than at lower latitudes under the low emissions scenario (Fig. 1a). In particular, the Norwegian Sea and Barents Sea show strong warming, which is mainly due to the Arctic Ocean amplification30,31. The strength of Arctic Ocean amplification is strongly correlated with the amount of sea ice reduction in the low-emission scenarios, while this relationship weakens in the high-emission scenarios32. As a result, the pattern of SST change in the NH, with higher SST increases at higher latitudes than at lower latitudes, is stronger in the low-emission scenario than in the high-emission scenario (Supplementary Fig. 1). However, a relatively weak warming trend is projected for the North Atlantic subpolar region (green boxes in Fig. 1, 45°N–65°N and 50°W–15°W) along with the largest inter-model spread for future SST projections (one standard deviation of the future SST) compared to other regions of the NH (contour in Fig. 1). It is noteworthy that the ESMs tend to disagree on the sign of future SST projections, with some ESMs projecting cooling in the future and others projecting the opposite (stipples in Fig. 1, model agreement <80%). Although the magnitude of SST change in the NH is large under different emission scenarios, the pattern of future NH SST projections, characterized by strong warming at high latitudes, is similar under different emission scenarios including SSP2-4.5, SSPP3-7.0, and SSP5-8.5 (Fig. 1b–d). However, there is still a discrepancy in the North Atlantic subpolar SST projection along with relatively weak warming among the ESMs. These projections indicate that the warming hole is evident in all future climate scenarios along with a large inter-model uncertainty. Therefore, emergent constraints using the combination of observations and the correlation between the current and future climate in the ESMs can be useful to reduce the uncertainty in future climate projections27.

Fig. 1: Future changes in sea surface temperature (SST).
figure 1

SST changes of the multi-model ensemble mean (MMEM) in the future (2070–2099) relative to the pre-industrial period (1850–1900) based on 32 CMIP6 Earth System Models (ESMs) under (a) SSP1-2.6, (b) SSP2-4.5, (c) SSP3-7.0, and (d) SSP5-8.5 scenarios. The stipples indicate the area where the ESMs have less than 80% agreement in sign (25 ESMs out of 32 in total). The blue contours indicate the 2 °C line of inter-model spread, defined as 1 standard deviation of future SST change among ESMs. The green boxes indicate the subpolar North Atlantic (45°N–65°N, 50°W–15°W).

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Source of warming hole uncertainty

The warming hole area, defined as green boxes (45°N–65°N and 50°W–15°W) in Fig. 1, is nearly the same in four future scenarios. We have defined “warming hole intensity” to identify warming hole changes, hereafter, if it is strong (weak), that implies that the warming in the North Atlantic subpolar region is weak (strong). To identify the source of uncertainty in the future warming hole intensity, we first compared the decadal-mean time series of the ESMs with observations of the warming hole intensity (Fig. 2a). The warming hole intensity in the observations shows an insignificant change based on the Mann-Kendall test, while the CMIP6 MMEM underestimates the observations (indicating a warming trend in the subpolar North Atlantic region) in the early 21st century. However, the trend discrepancy between the observations and the MMEM is not large, and the observations are within the range simulated in the ESMs. Therefore, the CMIP6 ESMs tend to accurately reproduce the observed warming hole intensity. By the end of the 21st century, the warming hole intensity is projected to weak under all scenarios, with greater weakening in the high emissions scenario (SSP5-8.5) than in the low emissions scenario (SSP1-2.6), resulting in stronger warming in the subpolar North Atlantic under SSP5-8.5 than under SSP1-2.6. The spread of the projected future warming hole intensity, defined as one standard deviation among the ESMs, also tends to increase across all scenarios. Indeed, the high-emission scenarios tend to have a large spread (right bars in Fig. 2a). This is associated with differences in climate sensitivity to external forcings, including cloud feedback. Because the ESMs have different climate sensitivities to external forcing, the spread of the projected future warming hole intensity is greater in high-emission scenarios in which greenhouse gas forcing is stronger33,34.

Fig. 2: Time series of the warming hole intensity and the source of uncertainty.
figure 2

a Decadal running means of the warming hole intensity from MMEM with observations (HadISST, purple dashed line), all relative to 1850–1900. The thick lines illustrate the MMEM of each scenario and the thin lines illustrate each ESM. The bars illustrate the 90% range (5–95%) of the future change in the warming hole intensity among CMIP6 ESMs with MMEM (circles). b Source of uncertainty in the multi-model, multi-scenario mean projection of the warming hole intensity. c Fractional contribution of each source to the total uncertainty of the warming hole intensity.

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Uncertainty in the future warming hole intensity projections can be quantified by the three contributions defined above: internal variability uncertainty, model uncertainty and scenario uncertainty (see Methods)5,6. Figure 2b illustrates the decadal mean time series of the warming hole intensity since 1905 relative to 1850, which is averaged over all four scenarios from the CMIP6 MMEM, along with the quantified sources of uncertainty. Towards the end of the 21st century, the uncertainty in future projections of the warming hole intensity increases, with a particularly large contribution from model and scenario uncertainty. Scenario uncertainty increases over time due to the different forcings applied toward the end of the 21st century35, while internal variability uncertainty remains relatively constant. There are differences in the internal variability among the ESMs (a minimum of 0.02 °C for CAS-ESM2-0 and 0.30 °C for EC-Earth3) although the multi-model mean of the internal variability is 0.09°C (Supplementary Fig. 2). Despite these differences in internal variability, its contribution to the total uncertainty of the future warming hole intensity projection is small, as the scenario uncertainty is 0.67 °C and the model uncertainty is 4.88 °C at the end of the 21st century.

The fractional contribution to the total uncertainty offers an efficient way to show which uncertainties are important at a given time (Fig. 2c). In the early 21st century, the contribution of scenario uncertainty is negligible because the differences in forcing between the scenarios are small, and internal variability uncertainty and model uncertainty largely account for the total uncertainty of the warming hole intensity projection. In particular, model uncertainty accounts for more than 80% of total uncertainty. Moving forward into the 21st century, the contribution of scenario uncertainty increases, while the effect of model and internal variability uncertainty decrease. The decrease of the contribution of internal variability uncertainty is particularly notable, while model uncertainty increases until the mid-21st century and then decreases slightly toward the end of the century. However, even at the end of the 21st century, the contribution of model uncertainty to the total uncertainty is >80%. This high contribution indicates that model uncertainty is the most important source for future warming projections, unlike uncertainty in temperature-related variables, particularly hemispheric surface temperature, where scenario uncertainty is the dominant source of uncertainty5 (see also Supplementary Fig. 3). It is noteworthy that no other region shows such a high contribution of model uncertainty to future projections as does the warming hole region (Supplementary Fig. 4). Therefore, reducing the model uncertainty is essential to reducing the uncertainty of future projections of the warming hole intensity.

Emergent relationship between present-day surface density and future warming hole

The North Atlantic subpolar region, where the warming hole occurs, is a hotspot for oceanic deep convection and the formation of thermohaline circulation36. We emphasize that stratification plays a key role in oceanic deep convection, leading to strong oceanic mixing between surface and deep waters in this region. Therefore, understanding the influence of stratification can reduce the model uncertainty37.

Because stratification is closely associated with density, the vertical profile of density in the North Atlantic subpolar region simulated by the CMIP6 ESMs in the present-day climate is important for stratification and density increases from the surface to the deep ocean (Fig. 3a). The inter-model density spread is larger at the surface than in the deep ocean. We calculate the stratification index (SI), which is defined as the sum of the differences between the density in each layer of the water column and the surface density (see Methods)38. ESMs with higher surface densities (i.e., stronger oceanic deep convection) tend to have a lower SI in the present-day than ESMs with lower surface densities. By the end of the 21st century, all ESMs tend to simulate decreasing density at all ocean depths with greater density loss near the surface under the high emissions scenario SSP5-8.5 (Fig. 3b). ESMs with low surface density in the present-day that have a high SI tend to have a greater decrease in near-surface density in the future than the others. This implies that ESMs with strong stratification in the present day will also have strong stratification in the future, resulting in a statistically significant inter-model correlation between the present-day stratification and future stratification, with a correlation coefficient (r) of 0.81 (Fig. 3c). Therefore, ESMs with stronger stratification produce a larger decrease in the warming hole intensity (same as stronger warming in there) because oceanic convection is weaker and the transport of warm water from the surface to the deep ocean is weaker than in ESMs with weaker stratification (shading in Fig. 3c). In addition, it is found that ESMs with a deep mixed layer depth (MLD) in the present-day (indicating strong oceanic convection) also tend to simulate a deep MLD in the future, with a correlation coefficient of 0.89 (Fig. 3d and see Methods). In summary, MLD and stratification, indicative of oceanic convection, are highly correlated. ESMs with a deep present-day MLD tend to simulate a weak stratification in the present-day, and this relationship holds into the future. Note that the correlation coefficient between the present-day MLD and the present-day stratification in the subpolar North Atlantic region is −0.78 in 28 ESMs. This correlation also holds for the future, where the correlation between future MLD and future stratification is also −0.78. Similar conclusions are also obtained when other future emission scenarios are analyzed (Supplementary Fig. 5). These results indicate that the present-day surface density has an emergent relationship with future changes in the warming hole intensity.

Fig. 3: Oceanic density and the future warming hole.
figure 3

a Oceanic density in the North Atlantic subpolar region (boxed area) in the present-day (1981–2010) from 32 CMIP6 ESMs. Purple line indicates the observation. b Future changes (2070–2099) relative to pre-industrial period (1850–1900) of the oceanic density in the North Atlantic subpolar region. The line colors indicate the present-day stratification index. Statistically significant (P value < 0.001) relationship between future climate and present-day of the climate in the North Atlantic subpolar region for stratification (c) and mixed depth layer (d). Shading indicates corresponding changes in the future warming hole intensity. The solid vertical purple line (shading) indicates the climatology (one standard error) of the WOA18. The dashed ellipse indicates the 5–95% range derived from the multivariate normal distribution. The correlation coefficient (r), slope, and P value obtained by a two-tailed Student’s t test are also provided with a regression line (solid black). Mixed layer depth is only available in 28 ESMs.

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Reducing uncertainty in warming hole projections

The present-day surface density in the North Atlantic subpolar region has a strong negative correlation with the future warming hole intensity projection under the low emissions SSP1-2.6 scenario (r = −0.62). This relationship is stronger in higher emission scenarios (r values of −0.75, −0.78, and −0.82 in SSP2-4.5, SSP3-7.0, and SSP5-8.5, respectively). The higher correlation in the high-emission scenario is associated with the stronger impact of greenhouse gases compared to the low-emission scenario4. Based on this relationship, we apply an emergent constraint to reduce the uncertainty in the future warming hole intensity based on the robust emergent relationship between the present-day surface density and the future warming hole intensity projection. Consequently, both the MMEM and inter-model uncertainty of the future warming hole intensity projection decrease in all scenarios after application of the constraint (Table 1, Fig. 4). Although some ESMs share key physical components and codes, we assume that all ESMs are independent by using only one ensemble member from each ESM39. The observed surface density and its uncertainty over the period of 1981–2010, as obtained from the World Ocean Atlas 2018 (WOA18)40, is estimated to be 26.83  ±  0.09 kg m−3. This value is similar to the mean value of 26.69  ±  0.35 kg m−3 obtained from the CMIP6 ESM, but the ESM underestimated the observations. By combining observations with the statistically significant emergent relationship from CMIP6, we can reduce the uncertainty of future warming hole intensity changes (Table 1 and Fig. 4). Furthermore, because future changes in the warming hole intensity are highly correlated with future NH warming rates in all scenarios (Supplementary Fig. 6), this reduction of the uncertainty based on the emergent relationship can be applied to the reduction of uncertainty in future NH warming rates (shading in Fig. 4).

Table 1 Unconstrained (before constraint) and constrained (after constraint) warming hole intensity projection in CMIP6
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Fig. 4: Emergent constraint on the future warming hole intensity change.
figure 4

In the top panel, projected changes in the warming hole intensity under (a) SSP1-2.6, (b) SSP2-4.5, (c) SSP3-7.0, and (d) SSP5-8.5 scenarios versus surface density in the warming hole region in the present-day climate (1981–2010). The solid black line follows the linear regression of 32 CMIP6 models, while the dashed black lines indicate prediction errors with one standard deviation, indicating 68% confidence intervals. The solid vertical purple line (shading) indicates the climatology (one standard error) of the WOA18. The slope and P value are obtained by a two-sided Student’s t test. The shading indicates the corresponding changes of the surface temperature in the northern hemisphere. The bottom panel illustrates probability density functions for the projected warming hole intensity changes before (“CMIP6 prior”, transparent) and after (“after constraint”, opaque) the emergent constraint is applied.

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Meanwhile, the results of the emergent constraint using the present-day stratification are almost identical to those using the present-day surface density, indicating that both stratification and surface density represent oceanic convection (Supplementary Fig. 7). In particular, the inter-model correlation between the present-day stratification and the future warming hole intensity change is almost as strong as when the surface density is used, with ESMs with stronger stratification in the present-day tending to have a greater weakening of the warming hole intensity. However, the amount of reduced uncertainty after the emergent constraint using the present-day stratification is less than that using surface density (Supplementary Table 2). This is because the stratification uncertainty in the observations is larger than that of the surface density (see Methods).

In addition, we applied an emergent constraint on the present-day MLD to reduce the uncertainty of future warming hole intensity changes (Supplementary Fig. 8). ESMs with a deeper present-day MLD, indicating stronger oceanic convection, tend to project stronger future warming hole intensity than the ESMs with a shallower present-day MLD, consistent with the stratification and surface density results. However, the correlation coefficients between the present-day MLD and the future warming hole intensity change in the ESMs are relatively lower than when using stratification and surface density. As a result, the uncertainty reduction from the emergent constraint using MLD is smaller compared to the results using stratification and surface density (Supplementary Table 2). The results using MLD, which represents oceanic convection, are similar to those using stratification and surface density, but with a smaller reduction in the uncertainty of the future warming hole intensity change, which is reduced by the relatively low inter-model correlation and large observational uncertainty (Supplementary Fig. 8 and Supplementary Table 2).

To test the robustness of the emergent constraint results obtained from CMIP6, we applied an out-of-sample test using CMIP527. The results of the emergent constraint between the present-day surface density in the North Atlantic subpolar region and future warming hole intensity change using CMIP5 was also statistically significant (Supplementary Fig. 9). The emergent relationship in CMIP5 between future warming hole intensity change and present-day surface density is lower than in CMIP6, which could be associated with improved simulation performance in the North Atlantic subpolar SST in CMIP641.

Discussion

We quantified the uncertainty in the future projections of the warming hole intensity, one of the representatives in climate tipping points42, by dividing the total uncertainty into internal variability uncertainty, model uncertainty, and scenario uncertainty S6. Temperature-related variables generally have a large contribution of scenario uncertainty5, but the model uncertainty was the largest contributor to the warming hole intensity projections. By the end of the 21st century, the model uncertainty accounted for more than 80% of the total uncertainty. Changes in the warming hole alter the position of the NH jet23, shift the position of the ITCZ14, and are closely associated with Arctic sea ice43. The warming hole is also one of the hotspots for the global carbon cycle and climate tipping, which is closely linked to irreversible climate change through its relationship with the AMOC21,42,44. Therefore, reducing the uncertainty in projections of the future warming hole is a very efficient way to improve the accuracy of future climate projections.

We demonstrate that the present-day surface density in the North Atlantic subpolar region is closely associated with uncertainty in the future warming hole intensity, which allows for the reduction of uncertainty by the emergent constraint. The strong correlation between density and stratification suggests that future ocean warming is highly correlated with current climate conditions, and lower density with strong stratification in the present-day ESMs tends to project weaker future warming hole intensity, indicating stronger warming in there (Fig. 3). By combining the emergent relationship with long-term observations (WOA18), we are able to reduce the uncertainty of the future warming hole intensity projections by up to 39% in high-emission scenarios. The emergent relationship between the present-day surface density and the future warm hole intensity change is high in 15 ESMs including a biogeochemical scheme compared to that of 32 ESMs (Supplementary Fig. 10). The biogeochemical scheme may allow for a better simulation of the North Atlantic climate, as the subpolar North Atlantic region is a major carbon sink and important for the global carbon cycle45

Uncertainty in the future NH warming rate could be reduced by using the emergent relationship between surface density and future warming hole intensity changes, because of a high correlation between warming hole intensity changes and the NH warming rate (Fig. 4). The warming hole intensity is strongly associated to the AMOC intensity, which influences the global warming rate through uptake of carbon and heat. In particular, ESMs with a strong AMOC intensity in the present-day climate can absorb more carbon and heat in the future climate21, reducing the NH warming rate. However, the correlation between changes in the AMOC intensity and the global warming rate is not significant among ESMs14, therefore, further studies on the oceanic and atmospheric processes associated with warming hole intensity change are needed to reduce the uncertainty in the future NH warming rate9. Furthermore, reduced uncertainty of the future warming hole intensity can also be applied to estimates of the climate sensitivity with respect to the AMOC14, the Arctic climate43, and ITCZ46.

Methods

Earth system models and observational dataset

We used 32 coupled general circulation models (CGCMs) in CMIP6 and 20 CGCMs in CMIP5. The model names and modeling groups are given in Supplementary Table 1. The surface air temperature, SST, surface salinity, ocean potential temperature, and salinity were used, and the criterion for including a given model was the inclusion of full values for historical and four future scenarios (SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5) experiments at the time of data download (September 2023). In addition, the mixed layer depth (MLD), defined as an ocean depth where the potential density differs by more than 0.125 kg m−3 relative to 10 m of ocean47, is used to analyze oceanic convection, although it is only available in 28 ESMs.

Only CGCMs that included all three Representative Concentration Pathways (RCP) scenarios (RCP2.6, RCP4.5, and RCP8.5) were used for CMIP5. All model simulations cover 1850–2099 (1861–2099 for GFDL-CM3 and GFDL-ESM2M, and 1860–2099 for HadGEM2-AO and HadGEM2-ES) and follow historical changes of greenhouse gases, aerosols, and natural forces for 1850–2005 (CMIP5) and 1850–2014 (CMIP6). For CMIP5, the simulations follow RCP for 2006–2100 and for CMIP6, they follow SSP scenarios for 2015–2100. We used the 1850–1900 mean in historical simulations for the pre-industrial climate and the 2070–2099 mean as the future climate. The differences represent future changes under anthropogenic warming. For observations, we used SSTs from the Hadley Center Sea Ice and Sea Surface Temperature dataset v1.148 and global mean surface temperature from the GISTEMP v449. The surface density and MLD data were averaged over 1981–2010 from the World Ocean Atlas 201840,50.

A common 1° × 1° horizontal grid interpolation was applied to all the model outputs and observations.

Uncertainty partitioning

There are three sources for the future climate change total uncertainty (T): the model uncertainty (M), the internal variability uncertainty (I), and the scenario uncertainty (S)6. T is calculated as the sum of the three uncertainty components and each of the uncertainties can be estimated as variance for a given time, t, and location, l, as follows:

$$T\left(t,l\right)=M\left(t,l\right)+I\left(t,l\right)+S\left(t,l\right)$$
(1)

with the fractional uncertainty from a given source calculated as \(\frac{M\left(t,l\right)}{T\left(t,l\right)}\), \(\frac{I\left(t,l\right)}{T\left(t,l\right)}\), and \(\frac{S\left(t,l\right)}{T\left(t,l\right)}\). This formulation assumes that the sources of uncertainty are additive, which strictly speaking is not valid due the terms not being orthogonal (e.g., M and S). In practice, an ANOVA formulation with interaction terms yields similar results and conclusions51.

M and I are calculated by the following pathways. The forced response is estimated as a fourth-order polynomial fit to the first ensemble member of each model. The reason of why we use one ensemble per model is intended to reduce the impact of bias in a particular model by giving each model the same weight. M is then calculated as the variance across the estimated forced responses. I is defined as the variance of the residuals from the fits (forced response) for 1850–2099, estimated independently of scenario and each ESMs. The multi-model mean of these variances is taken as I:

$$I=\frac{\mathop{\sum}\limits_{m}{{{{{\mathrm{var}}}}}}_{s,t}({\varepsilon }_{m,s,t})}{{{number}\, {of}\, {m}}}$$
(2)

where, \({{{{{\mathrm{var}}}}}}_{s,t}\) denotes the variance across scenarios and time, \(m\) denotes each model, \(s\) indicates scenarios and \(\varepsilon\) is the residual. I is constant in time. Prior to this calculation, time series are smoothed with the running mean corresponding to the target metric. Historical volcanic eruptions can thus affect I in CMIP. Estimating S relies on the availability of an equal set of models that were run under divergent emissions scenarios. We calculate S as the variance across the multi-model means calculated for four different emissions scenarios (SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5), using a consistent set of available models (32 models from CMIP6).

Density calculations and stratification index (SI)

We calculated the ocean density (⍴) from the potential temperature and practical salinity of CMIP6 ESMs following TEOS-10 standards52. Three-dimensional ⍴ fields were area-weighted and averaged along horizontal surfaces. Ocean levels were linearly interpolated to the 33 standard levels of Leviticus observations: 0, 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1750, 2000, 2500, 3000, 3500, 4000, 4500, 5000, and 5500 m53.

We calculated SI with37,38

$${SI}=\mathop{\sum }\limits_{i=1}^{25}{\rho }^{{z}_{i}}-{\rho }^{{z}_{0}}$$
(3)

where z0 is the sea surface and zi is each ocean depth for i  =  10, 20, …, 2000 m. Since WOA18 only provides standard errors at each depth, the uncertainty in the observed stratification (\({{SI}}_{u}\)) is calculated as:

$${{SI}}_{u}=\mathop{\sum }\limits_{i=1}^{25}{{\rho }_{{SE}}}^{{z}_{i}}+{{\rho }_{{SE}}}^{{z}_{0}}$$
(4)

where \({z}_{0}\) is the sea surface and \({z}_{i}\) is each ocean depth for i  =  10, 20, …, 2000 m. \({\rho }_{{SE}}\) is standard error at each depth. This method assumes that the uncertainty is maximized.

Emergent constraint

Probability density functions (PDFs) of the projected warming hole intensity were calculated following a previously established methodology54. The emergent relationship in this study was a linear regression of CMIP6 models between the simulated present-day state, \({x}_{n}\) (surface density), and projected changes, \({y}_{n}\) (future warming hole intensity), in the future (2070–2099). We used an ordinary least-squares regression for linear regression and only one ensemble from each model for equal model weighting. The prior PDF assumed that models were equally likely and followed a Gaussian distribution. The PDFs of observational constraints were defined as follows:

$$P\left(x\right)=\frac{1}{\sqrt{2\pi {{\sigma }_{x}}^{2}}}\exp \left\{-\frac{{(x-\bar{x})}^{2}}{2{\sigma }_{x}^{2}}\right\}$$
(5)

where \(\bar{x}\) is the averaged strength of the observed surface density in the North Atlantic subpolar region for 1981–2010 (present-day) and \({\sigma }_{x}\) is the corresponding standard error among these periods. The “prediction error” of the emergent multi-model linear regression (\({\sigma }_{f}(x)\)) defines contours of equal probability density around the multi-model linear regression, which represent the probability density of y given x.

$$P\left\{y{{{{{\rm{|}}}}}}x\right\}=\frac{1}{\sqrt{2\pi {\sigma }_{f}^{2}}}\exp \left\{-\frac{{(y-f(x))}^{2}}{2{\sigma }_{f}^{2}}\right\}$$
(6)

Here, \(f(x)\) is the fitted value of the linear regression and \({\sigma }_{f}\) is the prediction error of the linear regression. The emergent relationship was combined with the observational PDF by calculating the product of their PDFs and then integrating across the x-axis variable to derive a constrained PDF.

$$P\left(y\right)={\int }_{{\!\!\!}-{\infty }}^{{\infty }}P\left\{y|x\right\}P\left(x\right){dx}$$
(7)

Out-of-sampling test using CMIP5

CMIP5 with 20 ESMs was used to perform the same analysis that was applied to CMIP6. The CMIP5 MMEM also reproduced the observed warming hole intensity during the historical period well and projected a relatively weak warming at the end of the 21st century (Supplementary Fig. 11a). The magnitude of the absolute uncertainty also increased toward the end of the 21st century (Supplementary Fig. 11b). The dominant source of T in the warming hole intensity is also M, with the fraction of S increasing towards the end of the 21st century. However, the fraction of M at the end of the 21st century in CMIP6 is >80%, compared to ≤70% in CMIP5 (Supplementary Fig. 11c). These results are consistent with a broader range of climate sensitivities and transient responses for CMIP6 compared to CMIP53,4,5.

The emergent relationship between the present-day surface density in the North Atlantic subpolar region and future warming hole intensity change is also statistically significant in CMIP5 (Supplementary Fig. 9). After application of the emergent constraint, the uncertainty in future warming diagnosis is reduced to 14% in RCP2.6 and 16% in RCP8.5. The lower r values and uncertainty for CMIP5 compared to CMIP6 are expected because the spread of CMIP5 is smaller than that of CMIP6.