State-of-the-art climate models reduce dominant dynamical uncertainty in projections of extreme precipitation

Extreme precipitation can lead to severe environmental and economic impacts. Thus, future changes in extreme precipitation and their uncertainties are of major interest. Changes in extreme precipitation can be decomposed into thermodynamic (temperature-related) and dynamic (vertical velocity related) contributions with a scaling approach for extreme precipitation. Applying this approach to the global climate model ensembles CMIP5 and CMIP6, we decompose projection uncertainties of extremes in daily precipitation into uncertainties of thermodynamic and dynamic changes. We analyze regional patterns of the total uncertainties in extreme precipitation projections, as well as the thermodynamic and dynamic contributions to these uncertainties. Total uncertainties relative to the projected multi model mean are dominated by the dynamical contributions, and are large over the tropics and subtropics, but smaller over the high and mid-latitudes. Uncertainties in the thermodynamic contribution are generally small. From CMIP5 to CMIP6, uncertainties in thermodynamic and dynamic changes are slightly reduced in the high and mid-latitudes, while there is a substantial reduction of the uncertainties of the dynamic changes in the tropics and subtropics.


Introduction
Future changes in extreme precipitation cause severe environmental and economic impacts (Ranasinghe et al 2021).Projections by global climate models (GCMs) show an intensification of extreme daily precipitation with global warming (O'Gorman 2015) for global and zonal averages.However, at the regional scale and for individual seasons, extreme precipitation is expected to decrease across many subtropical regions (e.g.Pfahl et al 2017, Gutiérrez et al 2021, Seneviratne et al 2021, suppl. information).Uncertainties about these changes, however, are substantial: in many regions, not only the magnitude of precipitation trends, but even their sign is uncertain (Gutiérrez et al 2021).These uncertainties limit the possibility for efficient adaptation.It is therefore important to better understand the sources of these uncertainties, and to assess whether they have been reduced across recent generations of climate models.
Extreme precipitation events are shaped by thermodynamic and dynamical processes.The intensification of precipitation intensities is closely linked to an increase of atmospheric saturation water vapor pressure by about 7% per Kelvin warming as given by the Clausius-Clapeyron relation (Allen andIngram 2002, O'Gorman 2015).But significant differences can be detected at the regional scale (O'Gorman 2015, Pfahl et al 2017, John et al 2022).Changes in the atmospheric circulation modulate vertical velocities, moisture convergence and in turn extreme precipitation.Consequently, changes in the atmospheric dynamics need to be considered to understand spatial and seasonal variations and changes in extreme precipitation.Uncertainties in changes of dynamical processes are substantial, and a key source of uncertainties about regional climate change (Shepherd 2014).
O' Gorman and Schneider (2009) developed a diagnostic that expresses the amount of extreme precipitation events as a function of thermodynamic and dynamic processes, namely the column-integrated saturation vapour pressure and vertical velocities.Pfahl et al (2017) applied this approach to the Coupled Model Intercomparison Project Phase 5 (CMIP5) to understand the contribution of thermodynamic and dynamic changes to the overall changes of extremes in daily precipitation.Consistent with earlier findings, the thermodynamic changes are globally homogeneous and follow the Clausius-Clapeyron scaling, while changes in vertical velocities regionally reduce or amplify the uniform signal.In a brief analysis, Pfahl et al (2017) also found that projection uncertainties in CMIP5 are mainly caused by dynamical uncertainties and are largest in the tropics.
Over recent years, the new generation of CMIP6 models has been published.Wehner et al (2020) compared the performance of both ensembles at reproducing observed 20 year return values of daily precipitation, but found no significant differences.Model performance in present climate, however, cannot necessarily be related to projection uncertainties (Flato et al 2014).However, Ritzhaupt and Maraun (2023) found a high agreement in seasonal 20 year return value changes of daily precipitation over Europe between CMIP5 and CMIP6.
Here, we therefore extend the uncertainty analysis of Pfahl et al (2017).We consider seasonally-resolved projections for the IPCC AR6 regions (Gutiérrez et al 2021) and compare the CMIP5 and CMIP6 ensembles.We use the approach by Pfahl et al (2017) and decompose projection uncertainties of extremes in daily precipitation into uncertainties of thermodynamic and dynamic changes.Specifically, we assess (1) how well regional patterns of uncertainties of changes in extreme precipitation are explained by the chosen diagnostic, (2) how much uncertainties in thermodynamic and dynamic changes contribute to regional projection uncertainties, and (3) whether and to what extent these uncertainties have been reduced from CMIP5 to CMIP6 models.

Data
We compare the two GCM ensembles CMIP5 (Taylor et al 2012) and CMIP6 (Eyring et al 2016).The mean resolution of these GCMs are 150 km and 100 km, respectively.For this study, the necessary variables were available for 25 models of CMIP5 and 17 models of CMIP6 (see supplementary information for a list).In case of CMIP6, we excluded three models (AS-RCEC, CCCR-IITM and NIMS-KMA) where the chosen approach highly overestimates extreme precipitation at many gridpoints.The GCM ensembles are forced with the strongest available emission scenario for both ensembles to minimize the influence of internal variability.We use the RCP8.5 scenario for CMIP5 and the SSP5-8.5 scenario for CMIP6, respectively.The results are expressed per degree global warming to allow a direct comparison of the two ensembles.

Methods
We use the approach by O' Gorman and Schneider (2009) to decompose extreme daily precipitation into its thermodynamic and dynamic contributions.The intensity of an extreme precipitation event (P e ) scales with the net column integrated condensation rate and can be expressed as follows: Here, {.} indicates a mass-weighted integral over the troposphere, ω e is the pressure vertical velocity, and dqs dp | θ * is the vertical derivative of the saturation specific humidity assuming a moist adiabatic lapse rate.To calculate dqs dp | θ * , the vertical temperature T e is used on the days of the extreme events.ω e is also evaluated on the days of the extreme events.
The scaling relation can be derived either by assuming a moist-adiabatic, saturated ascent of air (O'Gorman and Schneider 2009) or by using a dry static energy budget approach which does not need the precondition of saturated ascent (Muller et al 2011).The latter is relevant for the tropics.
Changes in precipitation extremes can be decomposed into a thermodynamic contribution due to changes in T e and a dynamic contribution due to changes in ω e .To obtain the thermodynamic contribution, ω e is held constant over the entire time period.Following (Pfahl et al 2017), we calculate the dynamic contribution as the difference between the full scaling and the thermodynamic contribution.As 'full scaling' , we refer to the precipitation calculated by equation (1).
Climate change signals (CCS) of the full scaling as well as the thermodynamic and dynamic contributions are calculated between two 30 year periods for seasonal maxima of daily precipitation as CCS = P sim e − P baseline e P baseline e 1 ∆T • 100%.
P sim e and P baseline e are the mean intensities of the precipitation events in the simulation (2070-2100) and baseline  periods.T e and ω e are taken on the day and location of each daily maximum precipitation event to calculate the full scaling.The calculations are carried out on the native grids and the change signals are re-gridded on a 1 • × 1 • grid before applying ensemble statistics.Very low seasonal maximum precipitation leads to unrealistically large extreme precipitation sensitivities at some gridpoints (Pfahl et al 2017).Following (Williams and O'Gorman 2022), gridpoints, where the average seasonal maximum precipitation amount in the baseline period is less than 0.5 mm d −1 , are masked prior the calculation of ensemble statistics.
We express the CCS per degree Kelvin global warming (∆T).This rescaling has two advantages: first, it allows us to better compare the RCP8.5 and SSP5-8.5 scenarios; and second it removes uncertainties to the differing global climate sensitivities, which are on average higher in CMIP6 than in CMIP5 (Forster et al 2020, Zelinka et al 2020).
We interpret the model spread within each ensemble as a measure for the projection uncertainty of the corresponding ensemble.More precisely, we always consider the within-ensemble standard deviation of climate change signals, relative to the multi model mean (MMM) of the climate change signal of the full scaling (or precipitation).We believe that this relative spread is more useful than the absolute spread considered by Pfahl et al (2017) as the expected change defines the scale relative to which uncertainties are relevant.To assess the significance of differences in model spread between CMIP5 and CMIP6, we employ an F-Test.Using this test, we implicitly assume that the change signals in each ensemble are sampled from a normal distribution.We provide results separately for the four meteorological seasons.

Results
As an initial check, we compare the seasonal MMM changes in the full scaling with the MMM changes in actual extreme daily precipitation (supplementary figures S1 for CMIP5 and S2 for CMIP6).The changes in the full scaling reproduce very well the regional patterns of extreme precipitation changes where the results for CMIP5 and CMIP6 are very similar.The thermodynamic contribution (supplementary figure S3) is globally homogeneous and consistent with the Clausius-Clapeyron scaling (∼7% K −1 ), almost identical for CMIP5 and CMIP6.The dynamic contribution (supplementary figure S4) varies seasonally and regionally, thereby modulating the thermodynamic response.Differences in the MMM between CMIP5 and CMIP6 are in general small while they are strongest in the tropics, where the mean change itself is strongest.All CMIP5 results are in agreement with the findings of Pfahl et al (2017).However, minor differences can be found in JJA over middle Africa, where our results show stronger increases in the changes of the dynamical and full scaling, and in DJF and JJA over the tropical east Pacific where our results show some decreases embedded in a region of strong increases in the dynamical and full scaling.
In the following, we address uncertainties about these MMM changes, quantified by the within-ensemble spread.First, we assess whether the spread in the full scaling changes represents the spread in the actual extreme daily precipitation changes.Figure 1 shows the standard deviation across the CMIP6 models, relative to the CMIP6 MMM, for actual precipitation (left) and the full scaling (right), separately for each season.The corresponding results for CMIP5 are similar (supplementary figure S5).Negative values correspond to regions where the MMM is negative.The relative spread in the full scaling very well represents the relative spread in the actual extreme precipitation changes.While the relative spread is lowest in the high latitudes, the largest relative spread is found in the tropics and subtropics, in particular in the transition regions between positive and negative changes of extreme precipitation (where the MMM changes are close to zero).These results are in qualitative agreement with the annual results of Pfahl et al (2017).
In the following, we present an in-depth analysis of the thermodynamic and dynamic contributions to the overall uncertainty in extreme precipitation projections.To analyze the regionally-varying model spread beyond a single statistic such as the standard deviation, we consider regional-average changes for all IPCC regions (Gutiérrez et al 2021).
Figure 2 shows scatter plots of the thermodynamic (red) and dynamic (blue) change contributions (y-axis) versus the change in the full scaling (x-axis), for two selected IPCC regions.We selected one region representative of the tropics/subtropics (south-east Asia) and one region representative of the mid-to high latitudes (Northern Europe).The corresponding results for all other regions can be found in the supplementary information.In south-east Asia (two left columns) the spread in the thermodynamic scaling is small and explains only very little of the spread in the full scaling.The dynamic scaling, instead, is almost perfectly correlated with the full scaling.Both results hold irrespective of the season.In contrast, in Northern Europe, the spread of the thermodynamic and dynamic scaling are of similar magnitude, and explain less of the spread in the full scaling (as both relationships form a cloud rather than a line).
We further assess the regional patterns of relative spread in thermodynamic and dynamic changes by summarizing the relative spread as the corresponding standard deviation, relative to the MMM of the change in the full scaling, separately for each grid-box.Figure 3 shows the relative spread in the thermodynamic changes (left), the relative spread in the dynamic changes (center), and the contribution of the relative spread in the dynamic changes to the relative spread in the full scaling changes (right) for CMIP6, separately for each season (rows).The results for CMIP5 are similar (supplementary figure S7).The spread in dynamic changes is almost everywhere larger than the spread in the thermodynamic changes and contributes more than 65% to the total spread over most of the globe.In the tropics and subtropics, the contribution is often higher than 80%.Uncertainties in thermodynamics only play a notable role in the high latitudes.Because of the normalization, the relative spread in both contributions is highest in the subtropical transition regions, where MMM trends in the full scaling are close to zero.For the thermodynamic changes, this might be considered an artefact of the normalization (also given the narrow bands of high spread).This is different for the spread in the dynamic changes, where the high values in these regions are embedded into generally high values across the subtropics and tropics.
In the following, we compare how these results differ between CMIP5 and CMIP6.We start by getting back to figure 2. For both regions considered in this figure, the spread in the dynamic scaling (along the y-axis) is smaller in CMIP6 than in CMIP5, hinting at reduced uncertainties in the latest model generation.The reductions in south-east Asia are 3-5 %K −1 depending on the season, in Northern Europe less than 0.7 %K −1 .A notable fraction of the spread in the dynamic contribution (and consequently the full scaling) in south-east Asia seems to stem from 3-4 models with high change signals of 10-20 %K −1 .These models are from IPSL (IPSL-CM5A-LR, IPSL-CM5A-MR) and NOAA-GFDL (GFDL-EMS2G, GFDL-ESM2M).The climatology of the IPSL-CM6A-LR model of CMIP6 improved much over the previous versions of CMIP5 due to model improvements like more physically based parameterizations or more realistic implementation of some forcings (Boucher et al 2020).To investigate whether the reduction in model spread is related to these models only, we consider all subsequent results for the full ensembles and sub-ensembles, where all models from IPSL and NOAA-GFDL are excluded both in CMIP5 and CMIP6.The sub-ensemble plot corresponding to figure 2 can be found in supplementary figure S6: here the results for CMIP6 barely change, whereas the spread in CMIP5 decreases markedly.But whereas the spread in the CMIP5 and CMIP6 sub-ensembles are comparable for Northern Europe, it is still larger in CMIP5 for south-east Asia.Overall, we find that CMIP6 models have a smaller spread than CMIP5 models, but a considerable fraction of the reduction in spread can be attributed to improvements of a few models.
To further investigate these changes between CMIP5 and CMIP6, we summarize the spread in scaling in box-whisker plots, separately for all IPCC regions (figure 4 for DJF, figure 5 for JJA).While these plots do not explicitly show relationships between the spread in full scaling and its thermodynamic and dynamic contributions, they indicate how the mean and spread of the former depend on the mean and spread of the latter.These plots show that a small reduction in the spread of both thermodynamic and dynamic scaling changes can be found in many mid-latitude regions.But a large reduction of the spread in the dynamic scaling change can be found in many sub-tropical regions such as Northern Australia, northern South-America, the Arabian Peninsula and south-east Asia (see also supplementary material).These results remain in principle valid when removing the IPSL and NOAA-GFDL models, albeit with in general much weaker reductions from CMIP5 to CMIP6.
We again summarize these regional comparisons of CMIP5 and CMIP6 into global maps.Figure 6 shows the difference in the standard deviation of the thermodynamic (left), dynamic (center) and the full scaling changes (right) between CMIP6 and CMIP5, relative to the overall MMM change across both ensembles in the full scaling.We define the latter as ( std(CMIP5)−std(CMIP6) 1 2 (MMM(CMIP5)+MMM(CMIP6)) ).In weighting CMIP5 and CMIP6 equally, we disregard that the number of models in the two ensembles differs.The spread in thermodynamic changes is reduced from CMIP5 to CMIP6 in most regions of the extratropics, although the reductions are in general small.The reductions in the spread of the dynamic changes are modest over the mid-and high latitudes, but substantial in the subtropics and tropics.Here, the reduction in the spread is in the order of the MMM climate change.Only in some tropical regions, the spread of dynamical changes increase from CMIP5 to CMIP6, such as over the east Atlantic and Africa.The change from CMIP5 to CMIP6 in the spread of the full scaling can almost perfectly be explained by the patterns for the dynamic scaling.Overall it is evident that the spread in full scaling changes is reduced from CMIP5 to CMIP6 over much of the globe, apart from some tropical regions, and in particular over the tropics and subtropics.These findings are statistically significant (blue shading in figure 6, see methods for details) in areas with strong reductions in CMIP6 compared to CMIP5, especially in the dynamical and full scaling.Excluding again the IPSL and NOAA-GFDL models, these results remain qualitatively the same, but with smaller absolute values and less statistically significant areas.

Discussion and conclusions
In this study, we have extended the analysis of Pfahl et al (2017) on uncertainties in the projections of extremes in daily precipitation.We measure these uncertainties by the model spread of the CMIP5 and CMIP6 projections.
We show that a scaling diagnostic developed by O'Gorman and Schneider ( 2009) can be used to successfully explain uncertainties in these projections by uncertainties in changes in the condensation rate in vertical motion at the seasonal scale, and to decompose these uncertainties into uncertainties about thermodynamic changes (changes in the saturation vapour pressure with temperature) and dynamic changes (changes in the vertical velocities).
The projection uncertainties, even relative to the expected climate change (measured by the MMM) are highest in the tropics and subtropics.They are dominated by uncertainties in dynamic changes, in particular in the subtropics and tropics.Only in some regions of the mid-to high latitudes, uncertainties in the thermodynamics play an equally important role.
From CMIP5 to CMIP6, we find a slight reduction in the uncertainties of the thermodynamic changes and dynamic changes in the mid-to high latitudes, but a substantial reduction in the uncertainties of the dynamic changes in the tropics and subtropics.This reduction is dominated by the improvement of some individual models, but holds also for the full ensemble irrespective of these models.A question we cannot answer with our study is whether this reduction of uncertainty is actually a real improvement by better representing relevant processes, or caused by modeling centers tuning their models towards the MMM.Yet many other studies have shown that, in present climate, many aspects of atmospheric dynamics are better represented by CMIP6 models compared to CMIP5.This holds, for instance, for the jet stream and storm tracks (Harvey et al 2020), atmospheric blocking (Davini andd'Andrea 2020, Schiemann et al 2020), or circulation types (Cannon 2020) in the mid-to high latitudes.For the subtropics and tropics, the simulation of the Indian summer monsoon has improved from CMIP5 to CMIP6 (Gusain et al 2020) as well as the (tropical) precipitation-sea surface temperature feedback (Yang and Huang 2023) which is important for the representation of the effects of the Indian Ocean dipole, the monsoon and ENSO (McKenna et al 2020).Furthermore, the representation of different modes of variability, especially of ENSO and the PDO, has improved from CMIP5 to CMIP6 (Fasullo et al 2020).These findings hint at some real reduction of uncertainties, but further studies, e.g. based on emergent constraints (Hall et al 2019, Brient 2020) are required to substantiate this reasoning.
Our study contributes to the process-based understanding of projections and may, thereby, increase trust in the projections of extreme precipitation (Doblas-Reyes et al 2021).Similar to Pfahl et al (2017), we find that the identified dynamical changes, their uncertainties, and the differences between CMIP5 and CMIP6 can essentially be explained by changes, uncertainties, and differences in column-averaged vertical velocities (see supplementary figure S74).But further studies are needed, investigating the large-scale drivers of especially the dynamical changes, and their uncertainties.Such research links to the challenge of understanding regional patterns of global circulation changes (Woollings 2010, Zappa 2019).It may be based on storylines (Zappa andShepherd 2017, Shepherd et al 2018) and may address the role of Arctic vs. tropical amplification (Zappa and Shepherd 2017), the expansion of the Hadley cell (Vallis et al 2015), or the slow down of the Atlantic meridional overturning circulation (Jackson et al 2015).Against this background, our results, that CMIP6 seems to be superior to CMIP5, are encouraging.

Figure 1 .
Figure 1.Uncertainty in relation to the expected change for CMIP6.Standard deviation across the CMIP6 models relative to the CMIP6 multi model mean for the actual extreme precipitation changes (left) and the full scaling changes (right).Each row represents one season.Negative values correspond to regions with a negative multi model mean.The results for CMIP5 can be found in supplementary figure S5.

Figure 2 .
Figure 2. Distributions of the thermodynamic and dynamic contributions.Scatter plots where the thermodynamic (red dots) and dynamic (blue triangles) change contributions (y-axis) are shown in relation to the full scaling changes (x-axis) for the IPCC regions south-east Asia (two left columns) and Northern Europe (two right columns).The columns represent CMIP5 and CMIP6 while the rows represent the four seasons.The standard deviations for the thermodynamic (std_t) and dynamic (std_d) contributions are given in each panel.The results without the IPSL and NOAA-GFDL models can be found in supplementary figure S6.

Figure 3 .
Figure 3. Relative uncertainty of the thermodynamic and dynamic changes, and contribution of dynamic uncertainty at total uncertainty for CMIP6.The left and middle columns show the standard deviations of the thermodynamic or dynamic changes in relation to the multi model mean of the full scaling changes.Negative values correspond to regions with a negative multi model mean.The right column shows the contribution of the relative uncertainty of the dynamic changes to the relative uncertainty of the full scaling changes.The total uncertainty is mainly dominated by the dynamic contribution (blue).Each row represents one season.The results for CMIP5 can be found in supplementary figure S7.

Figure 4 .
Figure 4. Box-whisker plots of the spread of the full scaling changes, thermodynamic changes and dynamic changes for CMIP5 (blue) and CMIP6 (orange).The y-axis represents the change signals in %K −1 .For DJF and all IPCC regions.The results without the IPSL and NOAA-GFDL models can be found in supplementary figure S8.

Figure 5 .
Figure 5. Same as figure 4 but for JJA.The results without the IPSL and NOAA-GFDL models can be found in supplementary figure S11.

Figure 6 .
Figure 6.Difference in the standard deviation of the thermodynamic (left), dynamic (center) and full scaling (right) changes between CMIP5 and CMIP6, relative to the overall multi model mean change across both ensembles in the full scaling ( std(CMIP5)−std(CMIP6) (MMM(CMIP5)+MMM(CMIP6))/2 ).Notice, we disregard the differing number of models in the ensembles when weighting CMIP5 and CMIP6 equally.The rows represent the four seasons.Blue crosses indicate regions where the results are statistically significant.The results without the IPSL and NOAA-GFDL models can be found in supplementary figure S14.