Physically based equation representing the forcing-driven precipitation in climate models

This study aims to improve our understanding of the response of precipitation to forcings by proposing a physically-based equation that resolves simulated precipitation based on the atmospheric energy budget. The equation considers the balance between latent heat release by precipitation and the sum of the slow response by tropospheric temperature changes and the fast response by abrupt radiative forcing (RF) changes. The equation is tuned with three parameters for each climate model and then adequately reproduces time-varying precipitation. By decomposing the equation, we highlight the slow response as the largest contributor to forcing-driven responses and uncertainty sizes in simulations. The second largest one to uncertainty is the fast-RF response from aerosols or greenhouse gases (GHG), depending on the low or highest Coupled Model Intercomparison Projection 6 future scenarios. The likely range of precipitation change at specific warming levels under GHG removal (GGR) and solar radiation management (SRM) mitigation plans is evaluated by a simple model optimizing the relationship between temperature and decomposed contributions from multi-simulations under three scenarios. The results indicate that GGR has more severe effects from aerosols than GHG for a 1.5 K warming, resulting in 0.91%–1.62% increases in precipitation. In contrast, SRM pathways project much drier conditions than GGR results due to the tropospheric cooling and remaining anthropogenic radiative heating. Overall, the proposed physically-based equation, the decomposition analysis, and our simple model provide valuable insights into the uncertainties under different forcings and mitigation pathways, highlighting the importance of slow and fast responses to human-induced forcings in shaping future precipitation changes.


Introduction
Precipitation is a fundamental component of the Earth's climate system, responsible for regulating the distribution of water and energy across the atmosphere, oceans, and land (Bengtsson 2010, Gleeson et al 2020).Within the hydrological cycle, liquid water evaporates from the ground and remains in the atmosphere as water vapor.It forms clouds and eventually returns to the surface as precipitation.
This process involves the exchange of energy, with evaporation absorbing the surrounding energy from the surface and condensation releasing it into the atmosphere.Water vapor acts as both a forcing (Kondratev 1972) and a response to the external forcings due to its temperature-dependent relationship with the Clausius-Clapeyron relationship (Held and Soden 2006).Changes in this hydrological cycle have critical impacts on extreme climate risks (Paik et al 2020), water resources for human use (Kundzewicz and Döll 2009), agricultural production (Hirabayashi et al 2013), and ecosystems (Walther et al 2002).
Previous studies have examined the precipitation variation by analyzing the general circulation model (GCM) involved in the Coupled Model Intercomparison Projection (CMIP).Globally averaged outputs of GCMs are examined to generalize the precipitation response to the forcings.Without forcing variations, this precipitation can be controlled by the temperature, known as hydrological sensitivity (HS), showing 2-3%/K (Allen andIngram 2002, Pendergas 2020).Unlike the counterfactual conditions of HS, the world perpetually interacts with external forcings, and the atmospheric energy balance is perturbed and adjusted even on a short-term scale.The actual changes in precipitation are affected by HS and instantaneous radiative forcing (RF).The forcing-driven changes in precipitation are diverse depending on the types of GCMs and amount of external forcings (Fläschner et al 2016, Samset et al 2016).
After the Paris Agreement, global warming mitigation is promised.The mitigation plans must be determined based on the latest scientific understanding of forcing-driven impacts on possible approaches, like GHG removal (GGR) or solar radiation management (SRM).Some previous studies examined the precipitation change under GGR or SRM pathways.For examples, Sanderson et al (2017) used a single GCM to achieve a 1.5 and 2.0 K stable climate at the end of the 21st century under GGR methods, showing more increase in precipitation under warmer conditions.Laakso et al (2020) used two GCMs under GGR and SRM approaches, showing much drier conditions in SRM relative to GGR.However, high computational resources for simulations limit GCM numbers to explore the difference between GGR and SRM and hinder the robustness of their findings.
A reduced-complexity emulator for precipitation is the countermeasure to investigate possible changes under the given forcing conditions.Only a few studies examined the time-varying precipitation with a numerical equation.Shine et al (2015) proposed a conceptual model based on the impulse response functions but their parameters are theoretical values.Richardson et al (2018) considered the CO 2 and non-CO 2 together, but their parameters are estimated in averaged sense from limited numbers of GCMs.MacMartin et al (2018) developed an integrated equation but only considered the CO 2 and SRM.Laakso et al (2020) proposed an equation that captured the GGR and SRM in two GCMs, but its non-CO 2 part is empirical.Yeh et al (2021) introduced the integrated time-varying equation in a simplified form.However, the equation is designed for a single GCM and only applicable to CO 2 -forcing conditions.Given the wide range of uncertainty in precipitation among GCMs, there is a need to improve the timevarying numerical equation to capture the diverse characteristics of multiple GCMs and include physically missed fast responses driven by RF of non-CO 2 GHG and aerosols.
This study suggests an equation that resolves the global mean precipitation change at given forcing conditions.This equation physically explains the precipitation variation and quantifies the uncertainty sources.Finally, we evaluate reliable ranges of precipitation changes with a model based on the statistically optimized functions and compare the differences between GGR and SRM approaches.This paper is structured as follows.Section 2 describes model and analysis methods, including the definition of the equation resolving precipitation responses, three parameters indicating model sensitivity, and the uncertainty among GCMs.Results are provided in section 3 for evaluating our equation, quantifying the size of contributions to the precipitation variations and uncertainty among GCMs, and estimating the likely ranges of precipitation change under mitigation pathways.Section 4 gives a summary and discussion.

Data
To check the universal application of our equation, we mainly focus on climate experiments shared by many GCMs.We use simulations participating in the CMIP Phase 6 (CMIP6) project on the condition that GCMs integrate all four experiments: piControl, abrupt-4xCO2, 1pctCO2, and historical.We have 55 GCMs and describe their details in table S1.We also examine 14 GCMs having hist-GHG, hist-nat and histaer simulations to assess the precipitation responses under each group of climate forcing.For future analysis, we select 40 GCMs having all simulations under the three Shared Socioeconomic Pathways (SSP) scenarios (table S1).For the reference values of concentration and effective RF of scenarios, we use the single protocol dataset of the Reduced Complexity Model Intercomparison Project (RCMIP, Nicholls et al 2020).Additionally, CMIP6 GCM having SSP1-1.9 or SRM simulations (G6solar) are examined to validate our findings (table S1).SSP1-1.9 is the ambitious mitigation scenario comparable to the 1.5 K warming levels (e.g.Kriegler et al 2018).G6solar scenario controls the solar constant to reduce the SSP5-8.5 scenarios to reach the global warming levels of SSP2-4.5 (Visioni et al 2021(Visioni et al , 2023)).Notably, we calculate the individual model results and then average them for the multi-model ensemble (MME) mean.To test any bias due to dependency among GCM members (Abramowitltz et al 2019), we quantify the model dependency effects (check details in figure S1).All possible MME candidates proved insignificant differences.Therefore, our results are globally averaged, D Lee et al including ocean and land areas, and MME covers all available GCMs.

Precipitation equation
We build a physically based equation to emulate precipitation variation.Here, the emulation implies the computed precipitation according to the equation.Allen and Ingram (2002) introduced the straightforward concept of precipitation change with atmospheric energy variation.The hydrological response is balanced not by the thermodynamic responses of moisture but by the availability of atmospheric energy.The energy budget equation of the troposphere in the transient phase can be written as follows: (1) α is HS parameter, and ∆T is the changes in surface temperature.This first term (α•∆T) illustrates radiative cooling due to tropospheric warming, called HS effects or slow response (SLOW).R C is another radiative cooling from tropospheric RF, independent of temperature changes.F S is the net heat flux into the oceans in the transient, and this transient energy imbalance is scaled by model-dependent K. L is approximately 1 W•m −2 for a 1% precipitation increase.This second term represents the RF response in changing forcing conditions, known as the fast response (FAST).Yeh et al (2021) further simplify equation (1): (2) The term (∆R C + k•F S )/L in equation ( 1) is replaced by −(β/L)•∆F CO2 , implying the fraction (β) of CO 2 RF (∆F CO2 ), and this can be simplified by using equation (3) of Myhre et al (1998) explaining RF as the expression of CO 2 concentration (C) change relative to the base level of it (C 0 ).This number explains the fraction of CO 2 RF that reheats the atmosphere, also known as the fast CO 2 response (FAST CO2 ), eventually inhibiting latent heat releases.Although Allen and Ingram (2002) and Yeh et al (2021) explained well the precipitation changes under the transient forcing conditions, those equations only explain the precipitation changes driven by CO 2 .
First, our study expands equation (2) to reflect the GHG effects beyond Yeh et al (2021) approach.
(4) Equation ( 4) shares most of the terms in equation ( 2) except for the atmospheric RF amount (∆F).Although the previous study follows Myhre et al (1998), here we chose the approach of Etminan et al (2016) because they proved a better performance than Myhre et al (1998).The atmospheric RF equations for the first three major components of GHGs: CO 2 (∆F CO2 ), CH 4 (∆F CH4 ), and N 2 O (∆F N2O ) are ) . (7) Those expressions (equations ( 5)-( 7)) are adopted from Etminan et al (2016) and written in concise format.Please note that M and N indicate the concentration of CH 4 and N 2 O, and M 0 and N 0 show their base climate level.For the time-varying functions of concentration in the left term (f C * , f M * , and f N * ), we describe their full expression in table S2.
Consequently, the total sum of atmospheric RF from GHG components (∆F GHG ) is In equation ( 8), ∆F GHG is explained mainly by three dominant components.Since other well-mixed GHG components may have effects in the recent decade , we check their potential effects by adopting the Priestley-Center data (Smith 2020), which evaluated various effective RF of CMIP6 simulations.As the effective RF values are likely around RF values (Lee et al 2021), we test the effects of other well-mixed GHG by adding their effective RF values as the RF on equation ( 8).Eventually, we represent the GHG RF by simply equation ( 8) because emulations of hist-GHG perform better without their effects (figure S2(E)).
After reflecting on the fast response of GHG (FAST GHG ) to equation (4), we introduce a new term to reflect the effects of aerosols.Since aerosols-induced shortwave impacts change tropospheric temperature and they are reflected in SLOW by definition, we only consider the RF response to the precipitation independent of the temperature variation (FAST Aerosols ).FAST Aerosols is comparable to the net amount of longwave radiation in the atmosphere and sensible heat flux from the atmosphere to the surface (e.g.Zhou and Savijärvi 2014, Richardson et al 2016, Myhre et al 2018, Zhang et al 2021).Thus, equation ( 4) is now updated: The FAST Aerosols , (γ/L)•∆F Aerosols , is expressed by the scaled coefficient parameter (γ) multiplied by the RF in the third term (∆F Aerosols ).We adopt the evaluated effective RF values (Smith 2020) to account for ∆F Aerosols .Since model estimated longwave radiation amount caused by aerosols are highly uncertain for both aerosol-radiation interactions and aerosol-cloud interactions (Zelinka et al 2014, Smith et al 2020, 2021), here we introduce the modeldependent parameter γ indicating the residual atmospheric remained RF scaled from the effective RF values (e.g. Lee et al 2021).Positive γ implies a further decrease in precipitation under the rising aerosols concentration.

Three parameters of models
Three parameters illustrate the model sensitivity to the temperature and RF of GHG and aerosols.The parameter α is estimated by the regression slope between temperature and precipitation from the simulation of abrupt-4xCO2 (figure S2(A)), excluding the first 20 years to escape unresolved feedback effects from abrupt changing of forcing (Andrews et al 2015).The parameter β is measured from 1pctCO2 by the scale factor based on equation (2) (figure S2(B)).For the parameter γ, we first test the direct measurement from 14 GCMs with hist-aer simulations.After that, we estimate the γ indirectly from all forcing simulations by relocating equation ( 9) (FAST Aerosols ≈ ∆P − SLOW − FAST GHG ).Since forcing conditions are unlike abrupt-4xCO2, we use the entire periods of simulations to estimate parameters β and γ to acquire stability under the transient forcing conditions (Rugenstein et al 2020).We confirm the indistinctive differences between MME mean results of direct and indirect γ (figure S2(F)).Since we also find a significant correlation between them (r = 0.80), all of our main results use the indirect γ, having more GCM numbers.

Quantification of uncertainty sources
Since precipitation change uncertainty over the globe and the majority of sub-regions is dominantly driven by the model uncertainty rather than internal variability or scenario uncertainty (e.g.Hawkins and Sutton 2011, Nguyen et al 2018, Lehner et al 2020), this study focuses on the model uncertainty which is measured by the variance (Var) utilizing the mean function (E in equation ( 10)).
Arithmetically, the uncertainty size can be decomposed by three different terms having four variables.Please note that we use the single timeseries of RF (Etminan et al 2016), and effective RF (Smith 2020) for all GCMs.First, we simplified equation ( 9) as below: The (12) Equation ( 12) explains the model uncertainty by the sum of variance (SV) from a single variable (main effect) and covariance (Cov) from multiple variables (Interactions).The first term Var(ab) can be decomposed.
In equation ( 13), the size of Var(a)•Var(b) has a negligible contribution (figure S3).When we exclude the Cov, all the contributors to the uncertainty are always positive and additive.Therefore, we can calculate the percentage uncertainty from these main variables (e.g.Hawkins and Sutton 2009, 2011, Yip et al 2011, Lehner et al 2020, Lee et al 2023).
Eventually, equation ( 15) quantifies the contributions from various components.From left to right, the uncertainties in HS parameter (SLOW α), in temperature (SLOW ∆T), FAST GHG (sum of FAST CO2 , FAST CH4 , and FAST N2O ,), and FAST Aerosols , respectively.Check figure S3 for the quantified size details.

Evaluation for physically based equation
Forcing-driven changes in precipitation are illustrated in figure 1.Given the proliferated CO 2 conditions, as the temperature rises, precipitation increases (figures 1(A) and (B)).The atmospheric radiative cooling caused by tropospheric warming in equation ( 9) explains the precipitation variation under equilibrium responses after quadrupling CO 2 concentration (figure 1(A)).This SLOW response also explains volcanic eruption responses injecting sulfate aerosols to the atmosphere (figure S2(D)).However, SLOW represents the overestimation relative to the simulation of 1pctCO2 and historical (figures 1(B) and (C)).The gap between simulation and SLOW is mostly transient RF effects (FAST), inhibiting a room for latent heat release resulting decrease in precipitation.It is noteworthy that equation ( 9) can capture both MME mean responses and the spread of ensemble members, proving the validity of our equation to capture precipitation variation (figures 1 and S2).To check the likely ranges of computation error between emulation and simulation, we draw the one-to-one scatter plot and reaffirm that error size is mainly within ±0.5% when all components are considered (figure 1(D)).We confirm emulations usually have the size of error less than 0.1% in climate sense (figure S4).

Contributions to forcing-driven precipitation change
Our equation can decompose the precipitation variation into two components: SLOW and FAST.FAST is divided into RF types (CO 2 , CH 4 , N 2 O, and aerosols).We investigate their contributions to the MME mean and uncertainty in rainfall change (figure 2).Changes in the precipitation of simulations are reproduced reasonably by emulations (figures 1(C) and 2(A)).Our decomposition reveals that a tremendous amount of precipitation increases due to global warming is offset by RF in the atmosphere coupled with the anthropogenic components (figure 2(A)).The size of uncertainty among GCMs increases with time (figures 1(C) and 2(B)) as the increase in the variance of simulated temperature (figure S3(A)).In the last 20 years of historical periods (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014), MME mean of simulations expect a 0.45% increase, and MME variance is around 0.95% 2 .As MME mean is about half size of MME standard deviation, about 36% GCMs show negative changes (figure S4), suggesting a noticeable disagreement in precipitation trend under the historical experiment.The total variance is comparable to SV (figures 2(B) and S3(A)).We decompose SV with equation (15) (section 2.4) and manifest the quantified contributions from each component (figure 2(C)).The driving contributor is the temperature (SLOW ∆T), taking more than half of SV.The second largest contributor is FAST Aerosols , reflecting the largest spread among parameters (table S3).
Similarly, we apply this equation to understand the precipitation changes under SSP scenarios (figure 3).These future results validate the reliability in our equaiton because most SSP emulations have an error size of less than 1% without updating any parameters (figure S5).Precipitation change are mainly driven by SLOW and offset by FAST CO2 following the results of the historical period (figure 2(A)).Contrary to the historical case, the FAST Aerosols increase precipitation due to the decrease in aerosols concentration in all future scenarios (figures 3(A)-(C)).The size of SV in the late 21st Century (2081-2100) is 1.76, 2.47, and 6.99% 2 for SSP1-2.6,SSP2-4.5, and SSP5-8.5, emphasizing the larger uncertainty for the more  severe warming scenarios.Accordingly, SV primarily originates from SLOW (at leasts 80% for all SSP scenarios), highlighting noticeable model disagreement on the effects of global warming.While the size of uncertainty in FAST Aerosols is noticeable for the low-emission scenario, SSP1-2.6 (figure 3(D)), it is not vital for the high-emission scenario, SSP5-8.5, because the FAST CO2 is much larger than FAST Aerosols (figure 3(F)).This characteristic is also reflected in the fraction of uncertainty (figures 3(G)-(I)).

Precipitation change at global warming mitigation pathways
Since the temperature change is the vital contributor to the mean response and uncertainty (figures 2 and 3), here we assess the statistically estimated ranges of precipitation changes conditional on the different global warming levels (figure 4).First, we built a function (y = slope × ∆T exponent ) finding the best relationship between decomposed equation ( 9) and global warming levels.We found the optimized parameters (slope and exponent) fitting with 120 samples (40 GCMs × 3 SSP scenarios) for SLOW, FAST GHG , and FAST Aerosols components at the late 21st century.The optimized functions are 2.53×∆T 1.0 , −0.69×∆T 1.1 , and 0.26 ×∆T −0.2 for SLOW, FAST GHG , and FAST Aerosols , respectively.The ranges of the likely mean of emulations at specific warming levels can be measured statistically from a simple model, which is the sum of these functions.We obtain a 99% confidence interval (99% CI) from bootstrapping samples of alternative slope in each function with a half size of GCM members.
First, the estimated changes in precipitation caused by temperature, SLOW (figure 4(A)), are similar to the averaged parameter of 55 GCMs, 2.55%/K (table S3).Different GHG amount in three scenarios and small spread in parameter β (table S3) makes a statically sturdy relationship (r = −0.80) between global warming levels and FAST GHG (figure 4(B)), illustrating the possible relationship between GHG RF amount and global warming levels.While, for the RF effects of aerosols, the estimated function is nearly constant due to the negligible differences in aerosols' effective RF of three scenarios and the sizeable intermodel spread in parameter γ (table S3).As a result, the function for the radiative effect of aerosols is much more sensitive to the member of models rather than global warming levels (r = −0.10).
Finally, this simple model predicts changes in precipitation under specific warming conditions under GGR and SRM (figure 4(D)).We reflect the observed anthropogenic warming level of 1.0 K in 2001-2020 (https://globalwarmingindex.org,Haustein et al 2017) to the relative warming levels in late 21st Century.All 120 simulations are positioned around the simple model.For validation, we compare 14 simulations of SSP1-1.9 which are not involved in optimizing functions but are located around the prediction of the simple model.About GGR, the reversed D Lee et al contribution sign between FAST GHG and FAST Aerosols implies that aerosols would affect more than GHG in the specific range of temperatures (figures 4(B) and (C)).We compute this particular global warming level with the negative FAST GHG offsetting the positive FAST Aerosols , resulting in no combined FAST contribution, which is about 1.5 K. Thus, the mitigation pathways below 1.5 K are more likely to be influenced by RF of aerosols than GHG.We also compare the estimated results of GGR and SRM together.Here, to calculate SRM prediction, we first select six GCMs having SRM experiments (table S1) and find their MME mean changes in temperature and precipitation under SSP5-8.5.Then we draw the lines crossing this MME mean only driven by the SLOW.This simple prediction is well around the SRM simulations, affirming drier conditions in SRM than GGR due to remaining atmospheric GHG radiative effects (MacMartin et al 2018, Laakso et al 2020, Visioni et al 2021, 2023).
Under the global warming mitigation plans of GGR according to the Paris Agreement (1.5 K or 2.0 K), our simple model predicts a precipitation increase of about 0.91 ∼ 1.62%, or 1.80 ∼ 2.39% (99% CI).All predictions from 1.1 to 4.0 K are summarized in table S4.Our simple model with optimized functions highlights that SRM effectively cools global warming but accompanies severe drought than GGR approaches.To check the robustness of our optimized functions, we examined the effects of different based periods on optimizing functions (table S5).We confirmed that each term of function is mostly invariant.However, watchfulness is required for the base periods after 2020 because aerosols and GHG emissions are heterogeneous in SSP scenarios (Lund et al 2019, Meinshausen et al 2020, Smith et al 2021), affecting statistical optimization.

Summary and discussion
This study introduces a physically based equation to examine changes in global mean precipitation under various scenarios having different mitigation pathways.The latent heat release by precipitation is balanced by SLOW and FAST, representing the temperature-dependent and temperatureindependent change, respectively.After finding the three parameters of each GCM, the equation can capture both the mean and the spread of simulations.The decomposition of equation reveals that a tremendous amount of precipitation increase due to global warming is offset by RF in the atmosphere coupled with anthropogenic forcings.Since most uncertainty is derived from the disagreement in global warming levels, this study proposes a simple model predicting precipitation change based on the relationship between each term of our equation and global warming levels.This simple model reveals that the size of aerosols' RF is critical for precipitation change around 1.5 K. Based on the physically based characteristics of GGR and SRM, we compare the results from different mitigation approaches, and our simple model physically proves the drier conditions in SRM than the GGR.
Our results are broadly consistent with previous studies analyzing the global mean precipitation changes under future scenarios of CMIP Phase 5 (Caesar et al 2013, Hegerl et al 2015), CMIP6 (Tebaldi et al 2021), and SRM (MacMartin et al 2018, Laakso et al 2020, Visioni et al 2021, 2023).A simple model uncovers the nonlinearity in precipitation increase to global warming levels originating from FAST GHG corresponding to the results of various GCM experiments (e.g.Mitchel et al 2016, Salzmann 2016, Li et al 2019).This simple equation can explain the higher slope of total precipitation response conditional on total temperature change for low-emission than highemission scenarios (Hegerl et al 2015).The most considerable fraction of precipitation uncertainty is from the disagreement in global warming levels, supporting the findings of Fläschner et al (2016).HS in CMIP6 has less bias than it in CMIP5, possessing bias in cloud interaction (Watanabe et al 2018), according to the anti-correlated relationship between equilibrium climate sensitivity and HS (Watanabe et al 2018, Pendergrass 2020, Ribes et al 2021).We expect a simple climate model (e.g.Smith et al 2018) can employ our equation, and this employment enables the prevision of precipitation variations under various future scenarios, much more effectively than complex climate models.These theoretical foresights can impress policymakers, such as the importance of redeeming a pledge to implement mitigation pathways aimed at the Paris Agreement.
Nonetheless, caveats remain in our study.First, our emulations cannot reproduce the precipitation variation on annual and decadal scales (figure S3).Reflection of the natural variability to the emulations will diminish the errors (e.g.Lenssen et al 2020).Second, our parameters are concisely defined but have further uncertainty sources that should be explored, such as ocean warming patterns effects (e. letters a, b, c, and d indicate each model's variables of HS parameter α, temperature changes (∆T), FAST GHG , and FAST Aerosols .With this equation (11), model uncertainty can be written by Var (∆P) = Var (ab) + Var (c) + Var (d) + 2 • Cov (ab, c) + 2 • Cov (ab, d) + 2 • Cov (c, d) .

D
Figure 1.Global averaged annual mean precipitation changes relative to the first twenty years (1-20 years of abrupt-4xCO2 and 1pctCO2, and 1850-1869 years of historical).(A)-(C) The ensemble distribution of simulations (solid lines) and emulations (dashed lines) are shown vertically together after applying smoothing by kernel density estimation for every twenty years of chunks.Horizontally drawn solid and dashed lines represent the ensemble-averaged results of simulations and emulations.(D) Scatter plot between simulations and emulations of 55 GCMs for 165 historical years (1850-2014).Black and two grey diagonal lines indicate one-to-one reference and ±0.5% error ranges in emulations, respectively.The results of emulations are categorized as three: (A)-(D) only temperature effects (SLOW, red), (B)-(D) only atmospheric radiative effects (FASTCO 2 ), or FASTGHG + FAST Aerosols , orange), and (B)-(D) the sum of these effects (SLOW + FASTCO 2 or SLOW + FASTGHG + FAST Aerosols , blue).

Figure 2 .
Figure 2. (A) Time-varying precipitation amount (%) relative to 1850-1869 years and its decomposed contribution from SLOW, FASTGHG (sum of FASTCO2, FASTCH4, and FASTN2O), and FAST Aerosols with colors.The MME mean results of 55 simulations (open circles) and emulations (solid black line: sum of SLOW, FASTGHG, and FAST Aerosols ) are shown together.(B) Time-varying size of uncertainty (% 2 ) among emulations, decomposed by various components.The difference between the variance of precipitation and the sum of single variable effects (Sum of Variance, SV) are shown by arrows for every five years.This size represents the uncertainty from interactions between multiple variables (Cov, see equation (12) in section 2.4).(C) The relative uncertainty contribution size from components to SV is illustrated (Single variable effects/SV * 100).

D
Figure 4. (A)-(C) Scatter plot between temperature changes and decomposed precipitation change (circle for each emulation) in 20 years (2081-2100) under three SSP scenarios (blue: SSP1-2.6,green: SSP2-4.5, and red: SSP5-8.5)relative to 20 years (2001-2020).Three different contributions and statistically fitted functions for them are shown for (A) temperature-driven radiative cooling, slow response (∆T effect), and instantaneous radiative forcing effects driven by (B) GHG (GHG Radiative Effect) and (C) aerosols (Aerosols Radiative Effect).The optimized temperature-dependent functions are established from 120 samples (40 GCMs × 3 SSPs).The likely ranges of parameter slope (dotted lines, 0.5 and 99.5th percentile values) are estimated by the bootstrapping method.(D) The prediction for precipitation change from a simple model, combining three functions and slopes in (A)-(C).For the global warming level in 2081-2100, we reflect the observed warming level in the recent 20 years, 1.0 K (https://globalwarmingindex.org/), to the relative changes in temperature shown in (A)-(C).An asterisk denotes each result of the GCM simulation.The reversed triangles represent the MME mean of simulations under the corresponding scenarios.Solid lines and shading area indicate the best estimation and 99% confidence interval ranges, respectively, conditional on the specific warming levels under the GGR.As a validation of this simple model, SSP1-1.9 and SRM simulations are shown together (cyan: SSP1-1.9, 14 GCMs and brown: SRM, 6 GCMs), and their simulations are remarked with black outlines.The diagonal dotted line represents the best estimation conditional on the specific warming levels under SRM by considering their GCM members.For SSP5-8.5, the reversed triangle with outline indicates MME mean under SSP5-8.5 after selecting GCM members of SRM.(A)-(D) The relative warming levels in 20 years of the late 21st Century are illustrated by vertical lines with colors (+0.5, +1.0, +1.5, +2.0, +2.5, and +3.0 K, respectively).
g. Zhang et al 2023), different characteristics in aerosols' sub-component (Richardson et al 2018, Zhang et al 2021), and model structure or component effects (e.g.Qian et al 2018, Tett et al 2022).Third, the simple prediction model for GGR is statistically designed with assumptions on linear-like forcing conditions between SSP scenarios.Our following study will examine dynamical climate model simulations under the adaptive emission scenarios (Terhaar et al 2022) to check the validity of the statistical assumption.The next goal of our equation is to resolve the mathematization of different responses over land and ocean.