Constraints on regional projections of mean and extreme precipitation under warming

Significance

How the hydrological cycle will change in the warming future is one of the central questions in climate studies; however, current climate models still show large uncertainties in these projections. Some progress has been made to constrain the global mean precipitation changes; however, the global mean responses cannot represent the regional responses. Policymakers, stakeholders, and the public want to know the projections for the regions that concern them or where they live. Here, based on physical arguments, we used the observational past warming trend to constrain future projections of mean and extreme precipitation on both global and regional scales with the emergent constraint method. The results provide valuable information for the community and enlighten constraints on other regional climate projections.

Abstract

The projected changes in the hydrological cycle under global warming remain highly uncertain across current climate models. Here, we demonstrate that the observational past warming trend can be utilized to effectively co1nstrain future projections in mean and extreme precipitation on both global and regional scales. The physical basis for such constraints relies on the relatively constant climate sensitivity in individual models and the reasonable consistency of regional hydrological sensitivity among the models, which is dominated and regulated by the increases in atmospheric moisture. For the high-emission scenario, on the global average, the projected changes in mean precipitation are lowered from 6.9 to 5.2% and those in extreme precipitation from 24.5 to 18.1%, with the inter-model variances reduced by 31.0 and 22.7%, respectively. Moreover, the constraint can be applied to regions in middle-to-high latitudes, particularly over land. These constraints result in spatially resolved corrections that deviate substantially and inhomogeneously from the global mean corrections. This study provides regionally constrained hydrological responses over the globe, with direct implications for climate adaptation in specific areas.

Global warming intensifies the hydrological cycle on the global scale (1–3), posing significant threats to human society and the natural system. An accurate and confident projection of future changes in mean and extreme precipitation is critical for climate change adaptation, making it a central component in many multi-model climate simulation projects (4, 5). However, due to the deficiencies in and differences among the global climate models (GCMs), the projected precipitation changes showed large uncertainties (4–8). For example, under a high-emissions scenario, the projected increases in globally averaged mean precipitation range from 3.4 to 10.9% (5 to 95%) (5) and the increases in extreme precipitation (50-y annual maximum precipitation) range from 17.6 to 44.9% (10 to 90% ranges) (9) among models participating in the Coupled Model Intercomparison Project Phase 6 (CMIP6) (10). These large uncertainties seriously undercut the value of the GCM projections.

More importantly, the changes in mean and extreme precipitation under warming show distinctive regional patterns (SI Appendix, Fig. S1 A and B) (8, 9, 11–13). The regional responses substantially differ from the global mean response. The uncertainties among GCM projections also show large geographic heterogeneity (SI Appendix, Fig. S1 C and D). The regional hydrological responses to global warming are much more meaningful than their global means to policymakers and stakeholders. Thus, an accountable regional projection of mean and extreme precipitation with reduced uncertainty is of great value, which is the aim of this study.

An effective approach to reduce the uncertainties in multi-model results is the emergent constraint method (EC) (14–16). The EC framework first establishes an empirical relationship (the emergent relationship) between a simulated (but observable) current climate variable (predictor) and the future change (predictand) across GCMs. Then, the observed value of the predictor is used to correct the simulated predictand. A credible emergent relationship needs to not only meet the statistical requirements but also be built on reliable physical basis (17, 18). A recent progress is the use of the past warming trend in constraining the future global mean precipitation changes (19). Its physical basis is that the GCMs that overestimate the past warming trend tend to overestimate the future warming (20–23) and the associated intensification of precipitation (3). The study of ref. 19 inspires an idea of constraining the extreme precipitation changes using past warming. Such reasoning seems promising since extreme precipitation changes are largely controlled by changes in atmospheric moisture and thus temperature.

Some previous studies explored precipitation changes aggregated over a continental scale (19, 24–30). For example, ref. 27 constrained the extreme precipitation changes over the tropical belt by its interannual variability; ref. 30 constrained the changes of extreme precipitation aggregated over several regional boxes using present-day precipitation variability. The spatial aggregations in these previous studies are too coarse to capture the geographic patterns of the hydrological responses. Besides, using metrics of precipitation observations as the predictor may not be the best choice due to the limitation in data quality (31–34). The surface temperature record is arguably the most reliable observation, which is our choice of predictor here.

In this study, we show the robust emergent relationships between future responses in mean and extreme precipitation and the past warming trend across the CMIP6 simulations on the global scale and the regional scale over most regions. On the global scale, this constraint reduces model uncertainty in mean and extreme precipitation projections by 20 to 30%. On the regional scale, the correction and uncertainty reduction by EC are geographically inhomogeneous. The regional-scale constrained future projections on mean and extreme precipitation contain much more information than the global-scale constrained results do and provide great socio-economic values to climate change adaptation planning.

1. Results

We use daily data from 28 climate models in the CMIP6 (Methods and SI Appendix, Table S1) historical runs (1981 to 2014) and high-emission (SSP5-8.5) scenario runs by the end of the twenty-first century (2081 to 2100). The results of the medium-emission (SSP2-4.5) scenario are shown in supporting information. The historical trend is calculated for the period after 1981 to exclude the influence of aerosols whose global emissions have been nearly constant since 1981 (21, 35, 36). Mean and extreme precipitation refer to annual mean daily precipitation (P) and annual maximum daily precipitation (P_e), respectively. Their future projections are quantified by the fractional changes in the mean (dlnP) and extreme (dlnP_e) precipitation by the end of the century relative to the historical period. We use the global mean surface temperature (GMST) trend from 1981 to 2014 as the predictor for the EC. The trend in this paper is calculated using the ordinary least square regression. The simulated and observed historical GMST trends are denoted as $d{\overline{T}}_{\mathit{his}}$ and $d{\overline{T}}_{his, obs}$ (overbar denotes global averaging, subscript “his” denotes historical trend, and subscript “obs” denotes observation), respectively. The temperature observations (Methods) are from the HadCRUT4 (Hadley Centre/Climatic Research Unit Temperature version 4) (37) and GISTEMP4 (Goddard Institute for Space Studies Surface Temperature Analysis version 4) (38) datasets.

1.1. Global Scale Constraints.

First, we constrain the projection of globally averaged mean precipitation ($dln\overline{P}$). The results we obtain are very close to those in ref. 19; we keep those results here for validation of the method and integrity of presentation. By the end of this century, $dln\overline{P}$ from the multi-model mean of the CMIP6 unconstrained outputs is 6.9% with an inter-model spread of 3.0 to 10.9% (the 5 to 95% range assuming a Gaussian distribution). $dln\overline{P}$ may be expressed as the climatic sensitivity of $\overline{P}$ times the degree of warming, i.e., $dln\overline{P}=\frac{dln\overline{P}}{d\overline{T}}\times d\overline{T}$, where $d\overline{T}$ is the increase in GMST by the end of the century. The mean precipitation sensitivity (${\displaystyle \frac{dln\overline{P}}{d\overline{T}}}$) is well controlled by the global energy budget to be ~2.3%/K (SI Appendix, Fig. S2A) (3, 39); the inter-model spread in $dln\overline{P}$ is mainly due to the spread in $d\overline{T}$. Furthermore, given that the climate sensitivity is relatively constant in individual models, the models that have stronger past warming predict stronger future warming, which is confirmed by the strong inter-model correlation between $d\overline{T}$ and $d{\overline{T}}_{\mathit{his}}$ (SI Appendix, Fig. S2C, r = 0.67). Thus, there is a strong correlation between $dln\overline{P}$ and $d{\overline{T}}_{\mathit{his}}$ across the CMIP6 models (r = 0.63; Fig. 1A). Given this emergent relationship, one can use the observed warming trend ($d{\overline{T}}_{his,obs}$) to constrain $dln\overline{P}$ (Methods). The constrained $dln\overline{P}$ is reduced to 5.2% (the correction of EC is 6.9 to 5.2% = 1.7%), and the inter-model spread is narrowed to 1.9 to 8.5%, which corresponds to a relative reduction of variance (RRV) by nearly one-third (31.0%, SI Appendix, Table S3A).

Fig. 1.

Observational constraints on the future increases in globally averaged (A) mean precipitation ($dln\overline{P}$) and (B) extreme precipitation ($dln\overline{P_e}$). The y axes are the future changes (fractional changes of 2081 to 2100 to 1981 to 2014) in the CMIP6 models. The x axes are the past (1981 to 2014) warming trends (°C per 34 y). Each dot denotes a CMIP6 model result, and the dashed line is the fitting line of the emergent relationship according to the hierarchical EC (Methods). The vertical bars denote the multi-model means and spreads (17 to 83% range and 5 to 95% range) for the CMIP6 unconstrained results (black) and observationally constrained results (red for HadCRUT4, yellow for GISTEMP4, and blue for the mean of the two observations). The horizontal bar denotes the observed GMST trend from the mean of HadCRUT4 and GISTEMP4 and the 5 to 95% uncertainty range assuming a Gaussian distribution.

Next, we analyze the changes in globally averaged extreme precipitation ($dln\overline{P_e}$). $dln\overline{P_e}$ from the CMIP6 unconstrained outputs is 24.5%, more than three times of $dln\overline{P}$, and the inter-model spread is quite large (7.0 to 42.1%). We also find a statistically significant inter-model correlation between $dln\overline{P_e}$ and $d{\overline{T}}_{\mathit{his}}$ (r = 0.54; Fig. 1B) albeit slightly lower than that of mean precipitation. The reasoning behind the emergent relationship of $dln\overline{P_e}$ is similar to that of $dln\overline{P}$. We may express $dln\overline{P_e}$ by $dln\overline{P_e}=\frac{dln\overline{P_e}}{d\overline{T}}\times d\overline{T}$. The climatic sensitivity of extreme precipitation ($\frac{dln\overline{P_e}}{d\overline{T}}$) approximately follows the increases of atmospheric water vapor by the Clausius–Clapeyron (CC) scaling (9, 40, 41). There is a strong correlation between $dln\overline{P_e}$ and $d\overline{T}$ across the CMIP6 models (SI Appendix, Fig. S2b), suggesting a relatively constant $\frac{dln\overline{P_e}}{d\overline{T}}$. Again, given the strong correlation between $d\overline{T}$ and $d{\overline{T}}_{\mathit{his}}$ (SI Appendix, Fig. S2C), we have a significant positive correlation between $dln\overline{P_e}$ and $d{\overline{T}}_{\mathit{his}}$ (Fig. 1B). After applying the EC, $dln{P}_e$ is reduced to 18.1% (the correction of EC is 6.4%), with a reduced range of 2.7 to 33.5% (Fig. 1B). The RRV of $dln\overline{P_e}$ is nearly one-fourth (22.7%, SI Appendix, Table S3B).

1.2. Regional Scale Constraints.

The global means do not reflect the large geographic inhomogeneities in the hydrological changes under warming. How about the regional responses? In the following, we explore the idea of EC on regional precipitation changes.

We first examine the validity of the above emergent relationship on the regional scale. The inter-model correlation between $\mathit{dlnP}$ and $d{\overline{T}}_{\mathit{his}}$ is statistically significant over many regions, particularly northern hemisphere land (Fig. 2A). Similar to the argument of the globally averaged EC, a regional EC of $\mathit{dlnP}$ using past warming requires that the regional precipitation sensitivity (${\displaystyle \frac{dlnP}{d\overline{T}}}$) to be relatively consistent among the models. This is confirmed by the high regional correlations between $\mathit{dlnP}$ and $d\overline{T}$ (SI Appendix, Fig. S3A) and similarity of ${\displaystyle \frac{dlnP}{d\overline{T}}}$ across the models (SI Appendix, Fig. S4). The correlation between $\mathit{dlnP}$ and $d\overline{T}$ (or $d{\overline{T}}_{\mathit{his}}$) is weak and fails to pass the statistical significance mostly over the regions where the inter-model spread of regional precipitation sensitivity is large (SI Appendix, Fig. S6A).

Fig. 2.

The emergent relationship for regional precipitation changes. (Upper) Inter-model correlations between future precipitation changes and recent GMST trends for (A) mean and (B) extreme precipitation. (Lower) the slope of EC for (C) mean and (D) extreme precipitation. Dots denote regions where the correlations are statistically significant at the 95% confidence level based on the Student’s t test.

The inter-model consistency of ${\displaystyle \frac{dlnP}{d\overline{T}}}$ reflects that most models can simulate the large-scale patterns of mean precipitation responses to warming based on simple physical argument. Under global warming, almost all the models project increases in middle- and high-latitude mean precipitation due to the dominant effects of increased atmospheric moisture (11). The increases in near-equator precipitation and broad decreases in subtropical precipitation are also consistent among most models, corresponding to the robust circulation response of deep-tropical narrowing and subtropical expansion (42–44). The inconsistency of ${\displaystyle \frac{dlnP}{d\overline{T}}}$ (SI Appendix, Fig. S6A) is mainly over subtropical oceans and Sahara (45) where precipitation is sensitive to the patterns of sea surface temperature (SST) changes (46) among the models. The fact that the emergent relationship is valid over most land regions where large population lives is a beneficial coincidence that increases the value of the following regional EC.

We now use the observed GMST trends ($d{\overline{T}}_{his,obs}$) to constrain local dlnP over the regions where the emergent relationship holds. A key parameter here is the slope of the local emergent relationship (Fig. 2C), which is proportional to the correlation coefficient and SD in predictor and predictand (Methods). The correction of EC and reduction of uncertainties are proportional to the slope. In the main text, we present the regional EC results (Fig. 3) of an aggregated version (Methods) following the IPCC AR6 WGI recommendation (47), while the grid-scale results are shown in SI Appendix. The emergent relationship is valid for 30 regions out of the 58 IPCC AR6 WGI reference regions (Fig. 3A and SI Appendix, Table S4A), covering over 60.8% of the global area and 54.9% of the current global population [SI Appendix, Table S5, the population data is from the Gridded Population of the World Version 4 (48) for the year 2020]. Qualitatively consistent with the global mean results, the constrained dlnP (Fig. 3A and SI Appendix, Fig. S7A and Table S4A) shows a weaker response than that from the unconstrained outputs. However, the regional correction by EC (Fig. 3A and SI Appendix, Fig. S7B) is substantially different from the global mean correction (1.7%). For example, dlnP is reduced from 26.6% in unconstrained outputs to 20.6% after constraint over South Asia (SAS), and from 67.3 to 47.1% over the Arabian Peninsula (ARP). On the other hand, the drying over Southwest US, East Australia, and the Southern Indian Ocean becomes severer after EC correction (SI Appendix, Fig. S7A). The reduction of uncertainties also varies geographically (Fig. 3A and SI Appendix, Fig. S7E). Over high-latitude regions, the reduction in the inter-model spread (RRV) is quite substantial. For example, the RRV over Arctic Ocean is 39.5% and over Russian-Arctic is 36.4%.

Fig. 3.

Constrained future (A) mean and (B) extreme precipitation changes in the 58 IPCC AR6 reference regions. Within each hexagon, the filled color of each hexagon denotes the constrained projection, the Left and Right dots denote the EC-induced correction and RRV, respectively. The red (blue) dot denotes a positive (negative) value after subtracting the constrained from unconstrained value and the size of dot is proportional to the absolute value. The hexagons without the diagonal lines indicate that the emergent relationship is statistically significant at the 95% confidence level. The hexagons for ocean regions are highlighted with bold borders. The values for making the plot are listed in SI Appendix, Table S4.

Lastly, we constrain the regional projections of extreme precipitation (dlnP_e). A similar emergent relationship between dlnP_e and $d{\overline{T}}_{\mathit{his}}$ is also valid over many regions (Fig. 2B). Previous studies (12, 13, 49) decomposed the climatic sensitivity of extreme precipitation ($\frac{dln{P}_e}{d\overline{T}}$) into a component associated with the increased water vapor (the thermodynamic component) and a component associated with the changes of vertical motion during extreme precipitation (the dynamic component). The thermodynamic component follows the CC scaling and is geographically homogeneous and quite consistent among the models (12). The dynamic component accounts for the geographical inhomogeneity and has larger inter-model differences. However, the inter-model differences of the dynamic component are mainly confined in the low-latitude ocean. In the middle and high latitudes, the dynamic component plays a less important role than the thermodynamic component (12, 13). Besides, due to model development, the inter-model differences of ($\frac{dln{P}_e}{d\overline{T}}$) are notably reduced in CMIP6 comparing to those in CMIP5 (50). Although there is still significant uncertainty in $\frac{dln{P}_e}{d\overline{T}}$ in CMIP6, the regions where the inter-model differences of $\frac{dln{P}_e}{d\overline{T}}$ are large (and thus the emergent relationship breaks) are mostly confined to the subtropical and tropical oceans (SI Appendix, Figs. S5 and S6B). The emergent relationship is valid for 45 regions out of the 58 IPCC AR6 WGI reference regions (Fig. 3B and SI Appendix, Table S4B), covering 84.0% of the global area and ~80% of current global population (SI Appendix, Table S5). It is interesting to note that the regions where the emergent relationship holds for P_e occupy a larger area than those for dlnP; this reflects the robust and dominant effects of the thermodynamic component in $\frac{dln{P}_e}{d\overline{T}}$ (Section 1.3).

The constrained dlnP_e also shows weakened increases over most regions (Fig. 3B and SI Appendix, Fig. S7B and Table S4B). Consistent with the geographic distribution of the slope of the emergent relationship (Fig. 2D), the EC correction shows a geographic distribution deviating from that in the globally averaged EC (6.4%). For example, the EC correction over SAS (13.7%), ARP (17.9%), and Northwest South America (12.1%) is much greater than the global-mean value; while that over West North America (3.7%), West and Central-Europe (2.2%), and Mediterranean (2.7%) is much smaller. The regional EC effectively reduces the uncertainty in extreme precipitation changes in Eurasia, mid-latitude oceans and high latitudes (Fig. 3B and SI Appendix, Fig. S7F). Notably, the RRV in 13 IPCC AR6 WGI reference regions, including East Asia (EAS, 29.6%), SAS (24.3%), north Pacific Ocean (30.3%) and some high-latitude regions (up to 37.4%), is even larger than the global mean value of 22.7%. It indicates that the EC is rather effective in constraining regional extreme precipitation changes.

1.3. Physical Basis of the Regional EC.

A solid physical basis of the emergent relationship is vital for one to grant his/her confidence to the EC results. To reveal the underlying physical basis for the regional EC, we decompose mean and extreme precipitation changes into the thermodynamic (which represents the increases of atmospheric moisture with warming) and dynamic components (Methods) (11, 12). For the mean precipitation changes (SI Appendix, Fig. S8), the spatial pattern of its thermodynamic component resembles that of mean precipitation change (SI Appendix, Fig. S1A), while the dynamic component indicates weakening of vertical ascent during precipitation over most regions (3, 11). Importantly, the thermodynamic component is strongly correlated with $d{\overline{T}}_{\mathit{his}}$ over most regions (SI Appendix, Fig. S8C). The exception is only limited in equatorial central-east Pacific and subtropical Atlantic. For the extreme precipitation changes (SI Appendix, Fig. S9), the thermodynamic component exhibits a spatially uniform increase (~7%/K) while the dynamic component leads to the geographic pattern of dlnP_e (12). Comparing their contributions, the thermodynamic component dominates dlnP_e in extratropical regions. Moreover, the thermodynamic component is again strongly correlated with $d{\overline{T}}_{\mathit{his}}$ over almost every region (SI Appendix, Fig. S9C). Thus, the emergent relationship reflects the dependence of the thermodynamic component in both mean and extreme precipitation on global-mean warming. Since the increase of atmospheric moisture (the thermodynamic component) is strongly tied to warming itself and is a robust climate response (3), the emergent relationship here (not the uncertainties in the observed and projected GMST trend) may have similar level of confidence to that in the climate responses of atmospheric moisture.

The EC in this study is insensitive to the ensemble selection of CMIP6 models (SI Appendix, Figs. S10–S12). We performed a bootstrap analysis (50), in which we randomly take 18 out of 28 CMIP6 models to calculate the emergent relationship for 10⁴ times. The probability distribution of the correlation coefficients of the emergent relationship for the global mean hydrological responses (SI Appendix, Fig. S10) shows a low dispersion across different ensemble compositions. The bootstrap analysis on regional scale emergent relationship (SI Appendix, Figs. S11–S12) also indicates that the results in Fig. 2 are not sensitive to reasonable perturbations in model selection. The emergent relationship is not sensitive to perturbations in the selection of historical time periods. For example, we have repeated the analyses, but choose the historical period as 1970 to 2014 (instead of 1981 to 2014). The emergent relationship for both the global and regional precipitation changes remain almost unchanged (SI Appendix, Figs. S13 and S14).

The ECs are also valid when applied to the CMIP6 outputs of the SSP2-4.5 scenario (SI Appendix, Figs. S15–S18). The globally averaged $dln\overline{P}$ is reduced from 4.7% (2.0 to 7.4%) to 3.6% (1.3 to 6.0%), and globally averaged $dln\overline{P_e}$ is reduced from 13.2% (3.1 to 23.2%) to 10.2% (0.9 to 19.5%). The RRV is 26.9% and 14.8%, respectively. The regions where the local emergent relationship holds (SI Appendix, Fig. S16) are smaller than those in the SSP5-8.5 results due to weaker signal-to-noise ratio. The regional aggregated results (SI Appendix, Fig. S18) are still valid over many land regions with large populations. Again, these results provide locally constrained regional projections beyond the global means.

2. Conclusions and Discussion

We constrained the future projections of mean precipitation and extreme precipitation on global and regional scales in CMIP6 using the past global warming trend. On the global scale, the projected changes in mean precipitation are lowered from 6.9% (3.0 to 10.9%) to 5.2% (1.9 to 8.5%) and those in extreme precipitation from 24.5% (7.0 to 42.1%) to 18.1% (2.7 to 33.5%), with the inter-model variances reduced by 31.0% and 22.7%, respectively, under the high-emission scenario by the end of the twenty-first century. The local emergent relationship between future precipitation changes and $d{\overline{T}}_{\mathit{his}}$ is also valid over many regions, particularly over land. The physical basis for the strong emergent relationship is that the changes in the thermodynamic processes (moisture), which are robustly correlated with temperature changes, are a major contributor to the future precipitation changes in most regions around the globe, especially for dlnP_e. The regional EC corrections are highly heterogeneous geographically. The subtropical continents and EAS show reductions several times of the global means. The regional constrained projections of the hydrological cycle provided here are of great value for future impact assessment, natural risk management, and climate adaptation in specific areas.

There are some limitations of this study, which may be remedied in future work. There are discrepancies between the surface warming pattern in the historical simulations and observations. This so-called pattern effect leads to an underestimation of model’s equilibrium climate sensitivity in ECs (5, 51). Although its impact on the estimation of transient climate response is much weaker (5, 20), more work is needed to constrain the future warming. Moreover, this study highlights the inter-model consistency of the precipitation sensitivity ($\frac{\mathit{dlnP}}{d\overline{T}}$ and $\frac{dln{P}_e}{d\overline{T}}$), which facilitate the emergent relationship. However, the uncertainty in the precipitation sensitivity cannot be ignored, particularly in the tropical and subtropical regions (SI Appendix, Figs S4–S6). There are still substantial inter-model spreads in future precipitation changes after the ECs using the past GMST trends. Future work shall explore ways to constrain the precipitation sensitivity (50) and combine the results with the method here to further reduce the uncertainties in projected precipitation responses.

The present study reveals that regional EC works better in middle and high latitudes. In the low latitudes, the regional precipitation projections are mainly determined by the projected SST (46). Thus, future work needs to be conducted in correcting model biases in the low latitudes, such as the East Pacific cold tongues (52), to improve the precipitation projections there. Another perspective for improvement is to add more predictors in the emergent relationship. This is along the same line of the recent development of using machine-learning-based methods to constrain future projections (53). Nevertheless, this study suggested that it is time to move forward to the regional constraints of weather and environmental changes under warming. For weather or environmental changes that are geographically inhomogeneous, such as precipitation changes demonstrated here, wildfire, sand storm, and drought, their regional changes are more relevant to policy makers or stockholders than the global means are, and thus there are high demands for regional constraints in future research.

3. Methods

3.1. CMIP6 Simulations.

The 28 CMIP6 models are listed in SI Appendix, Table S1. Given that the spatial resolutions are different among the CMIP6 models, each model outputs are first re-gridded to a uniform horizontal resolution of 2.5° × 2.5°. For calculating the globally averaged changes, we first calculate the (area-weighted) global means and then the fractional changes, i.e., $dln\overline{P}=\left({\overline{P}}_{\mathit{fut}}-{\overline{P}}_{\mathit{his}}\right)/{\overline{P}}_{\mathit{his}}$. $dln\overline{P_e}$ is calculated similarly. For calculating the regional EC over the IPCC AR6 WGI reference regions (47), we first calculate the regional means of mean or extreme precipitation and then the fractional changes.

3.2. Temperature Observations.

The observational temperature trends during 1981 to 2014 are calculated using the HadCRUT4 (37) and GISTEMP4 (38) dataset. The GMST in the observed data is a blend of surface air temperature over land and ice and SST over ocean, different from the GMST in climate models. To modify this blending bias, the temperature trend from HadCRUT4 is inflated by 1.074 (23) and the temperature trend from GISTEMP4 is added with 0.014 °C per decade (21). To estimate the internal climate variability, we use 9 CMIP6 historical simulations with large initial condition ensembles (SI Appendix, Table S2), following ref. 29. The mean SD is 0.12 °C for the 34-y GMST trends, similar to the value of 0.13 °C per 35 y during 1980 to 2014 estimated by ref. 19 and 0.035 °C per decade (corresponding to 0.12 °C per 35 y) in ref. 21.

3.3. Constraint on Future Precipitation Changes.

The constraining method is the hierarchical EC following ref. 54. Let y be the predictand and x be the predictor, the equation for a constrained predictand (y_c) is

$$y_c=y+k\left(x_o-x\right),$$ [1]

in which the slope (k) is

$$k=\frac{r(x,y)s\left(y\right)s\left(x\right)}{s^2\left(x\right)+s^2\left(x_o\right)}.$$ [2]

s(*) and s²(*) denote the SD and variance of *, respectively. r(x,y) denotes the correlation between x and y and is used to measure the validity of the emergent relationship. The significance of the correlation is tested based on the Student’s t test assuming that the models are independent of each other. $x_o$ is the observed predictor. $s^2\left(x_o\right)$ denotes the internal variability, estimated as the mean variance from 9 CMIP6 historical simulations with large initial condition ensembles following ref. 29. The reduction of the multi-model mean of precipitation changes by EC refers to $k\left(x_o-\overline{x}\right)$, where the overbar denotes the multi-model average. The constrained SD of predictand (i.e., the uncertainty) follows

$$s\left(y_c\right)=s\left(y\right)\sqrt{1-\frac{r^2\left(x,y\right)}{1+\left(\frac{s^2\left(x_o\right)}{s^2\left(x\right)}\right)}}.$$ [3]

The unconstrained and constrained ranges of the future changes are estimated by assuming a Gaussian distribution. The RRV is calculated with $RRV=\left(1-\frac{s^2\left(y_c\right)}{s^2\left(y\right)}\right)\times 100\%.$

Here, the difference between y and y_c is referred to the correction of EC. From the above equations, the correction of EC is proportional to the slope (k), which is further proportional to r(x,y). Thus, in the regions where the correlation coefficient of the emergent relationship is insignificant or small, the EC correction has little effect.

3.4. Thermodynamic and Dynamic Decomposition.

The decomposition of mean precipitation changes ($dln\overline{P}$) follows ref. 11. The mean precipitation can be expressed as $P=\int P_{\omega }{\mathit{Pr}}_{\omega }d\omega$, where P_ω is the expected values of daily precipitation given the vertical pressure velocity ω and Pr_ω is the probability density function of ω. Here, ω is divided into bins with a uniform width of 50 hPa/d. We have verified this expression for both historical and warmer climate simulations in CMIP6. For dlnP, its thermodynamic component is calculated using the above expression with fixed Pr_ω and changing P_ω, while its dynamic component is calculated with fixed P_ω and changing Pr_ω (11).

The decomposition of extreme precipitation changes ($dln\overline{P_e}$) follows ref. 12. Extreme precipitation can be approximated by the scaling $P_e=-\frac{1}{g}\int \omega {\left.\frac{d{q}_s}{\mathit{dp}}\right|}_{{\theta }^{\ast }}dp$, where ${\left.\frac{d{q}_s}{\mathit{dp}}\right|}_{{\theta }^{\ast }}$ is the vertical derivative of the saturation specific humidity ($q_s$) at constant saturation equivalent potential temperature (${\theta }^{\ast }$) (49). We have verified this approximation for both historical and warmer climate simulations in CMIP6. Then, the thermodynamic component of $dln{P}_e$ is calculated with fixed ω and changing ${\left.\frac{d{q}_s}{\mathit{dp}}\right|}_{{\theta }^{\ast }}$, while the dynamic component is the difference between approximation of dlnP_e using the full scaling and the thermodynamic component (12).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

The CMIP6 archive are available at (https://esgf-node.llnl.gov/search/cmip6/) (10). The observational data are available at (https://www.metoffice.gov.uk/hadobs/hadcrut4/) for HadCRUT4 (37) and (https://data.giss.nasa.gov/gistemp/) for GISTEMP4 (38). The analysis and codes are available at (https://zenodo.org/records/10404897) (55).