Because aerosol-cloud interactions are the most uncertain climate forcing in the Earth system, it is important to better understand the aerosol-cloud interactions, especially how they will change with climate. This study carries out large eddy simulations (LES) of a 3-day stratocumulus-tocumulus transition (SCT) along an airmass-following trajectory in the Northeast Pacific Ocean. By perturbing aerosol concentrations within the marine boundary layer (MBL) in the SCT simulations, we evaluate aerosol-cloud interactions in both the present day as well as in a double-CO2 climate. We find that aerosol-induced cloud changes, including first (Twomey effect) and second adjustments of cloud fraction and liquid water path (LWP) indirect effects of aerosols, tend to be inhibited in a double-CO2 climate. The LWP adjustment is more sensitive to global warming than the Twomey effect. By decomposing the aerosol-induced cloud radiative effect change (CRE), we find that the aerosol-induced cloud fraction change (CF) shows the largest contribution to CRE in our simulations. Aerosol-induced droplet number concentration change (Nc) shows a cooling effect, while the LWP change shows a warming effect that is consistent with aerosol-induced cloud thinning. Overall, the cooling effect associated with increased aerosol concentrations will weaken in a double-CO2 climate. Our results can also help to understand and predict the cooling potential of marine cloud brightening (MCB) in a changing climate. Clouds are one of the most important components in our climate system and have significant influences on radiation, precipitation, temperature, etc [Bony et al., 2015;Ceppi et al., 2017;Rosenfeld et al., 2014]. Aerosols (both natural and anthropogenic) can serve as cloud condensation nuclei (CCN) to contribute to cloud formation [Williamson et al., 2019;Zheng et al., 2021], and then further influence cloud radiative effects (e.g., Twomey effects [Platnick, S., & Twomey 1994;Twomey. 1977]). A better understanding of aerosol-cloud interactions can improve the accuracy of weather and climate predictions. For example, recent studies show that reduced aerosols due to IMO 2020 regulation (IMO2020) could potentially increase global warming by dimming the marine low clouds [Gettleman et al., 2024;Jordan and Henry. 2024;Quaglia and Visioni. 2024;Yuan et al., 2024]. It is known that aerosols, clouds, and their interactions introduce huge uncertainties in global climate models (GCMs) [Fan et al., 2016;Morrison et al., 2020;Seinfeld et al., 2016]. The inability of GCMs to resolve subgrid processes (e.g., boundary layer turbulences and moist convection) makes them highly dependent on various parameterizations [McFarlane. 2011;Thayer-Calder et al., 2015]. On the other hand, large eddy simulation (LES) has a sufficiently high vertical and horizontal resolution to explicitly simulate much of the energy-containing turbulence and is a widely used tool to simulate cloud microphysics [e.g., Atlas et al., 2020;Shi et al., 2018] and cloud organization [e.g., Narenpitak et al, 2021;Seifert and Heus, 2013;Zhou and Bretherton, 2019]. There are many LES studies focusing on how climate change influences stratocumulus clouds, such as CFMIP/GASS Intercomparison of Large-Eddy and Single-Column Models (CGILS) [Blossey et al., 2013;Blossey et al., 2016]. However, limited LES studies evaluated how global warming influences aerosol-cloud interactions in the marine boundary layer (MBL). This study uses a Lagrangian LES method (Erfani et al., 2022;Sandu et al., 2011), simulating the stratocumulus-to-cumulus transition (SCT) along an airmass-following trajectory [Chun et al., 2025;Krueger et al., 1995;Sarkar et al., 2020;Wyant et al., 1997;Wood et al., 2018;Yamaguchi et al., 2015] in the SAM model [Khairoutdinov and Randall, 2003] with different MBL aerosols perturbations (control cases vs. perturbed cases) in a changing climate, with the present day (PD) to a warmer climate (P2CO2, with double-CO2 and sea surface temperature warmed by 2K) scenarios following the approach in a previous study [Bretherton and Blossey, 2014] named here as BB2014. More detailed information on model settings can be found in the Data and Methods. We use the LES simulations to evaluate aerosol-cloud interactions (Section 2.1) and their radiative effects (Section 2.2) in a changing climate, which may help to constrain the uncertainties in weather and climate predictions. In the control cases without aerosol perturbation (PD c and P2CO2 c ), dry aerosol concentration stabilizes around 30 cm -3 (blue and red solid lines in Figure 1b), creating a clean marine boundary layer (MBL). The source of MBL dry aerosols includes surface emissions and entrainment, and the dominant sink of dry aerosols is activation (Figure S2). The dry aerosol concentration is slightly higher in the P2CO2 than in the PD, which may result from less interstitial scavenging (due to clouds and rain) in a double-CO2 climate (Figure S2.b). Due to the lack of cloud condensation nuclei (CCN) (solid lines in Figure 1b) in the clean MBL, clouds in the control cases (PD c and P2CO2 c ) will quickly drizzle out (solid lines in Figures 1c and 1f) and result in a low cloud fraction (solid lines in Figure 1h). Note that the overcast stratocumulus cloud breaks up much earlier in the PD c simulation than in BB2014, likely because BB2014's prescribed cloud droplet number concentration of 100 cm -3 leads to much weaker precipitation. In the perturbed cases (PD p and P2CO2 p ), a shorter during, spatially-uniform increase in the surface aerosol flux at Day196.6 (see Data and Methods) leads to perturbed dry aerosol concentrations of approximately 120 cm -3 (dashed lines in Figure 1b, PD p and P2CO2 p ), about 4 times larger than in the unperturbed simulations (solid lines, PD b and P2CO2 b ). Observations have shown that accumulation-mode aerosol concentrations can reach up to 150 cm -3 [Bretherton et al., 2019], suggesting that the elevated perturbed aerosol levels in our study are within a realistic range. Increased aerosols in the perturbed cases (PD p and P2CO2 p ) can serve as CCN, forming more smaller cloud droplets through activation and condensation processes, which is known as the first indirect effect of aerosols (or Twomey effect [Quaas et al., 2009;Twomey. 1974]). Compared to the control cases (solid lines in Figure 1), larger dry aerosol concentration (dashed lines in Figure 1d) in the perturbed cases can increase cloud droplet number concentration (Nc) to up to 150 cm -3 .
This increased Nc also inhibits precipitation formation and, in this case, increases the lifetime of overcast stratocumulus clouds [e.g., Albrecht, 1989]. Subsequently, Nc gradually decreases in the 3-day simulation corresponding to the decrease of Nd, as the activation is the dominant source of Nc (Figures 1b and S2d). By comparing aerosol-induced cloud changes between the present day and a double-CO2 climate (e.g., blue vs. red lines in Figure 1), we can evaluate aerosol-cloud interactions in a changing climate. The aerosol-induced increase of Nc in a double-CO2 climate is almost identical to that in the present day, especially on Day 1 (mainly stratocumulus clouds). Compared to the P2CO2, the PD shows a slightly larger scavenging sink (Figure S3.b) on Day 3, resulting in a smaller Nc concentration . But Nc caused by the activation process (Figure S2.d) doesn't change much in a changing climate, indicating that the first indirect effect of aerosols wouldn't change much in a changing climate. Since rain formation relies on larger cloud droplets, the more numerous and smaller cloud droplets in the perturbed cases (induced by increased aerosols on Day 1) can inhibit autoconversion and accretion when compared to the control cases (Figure S3c), which further depresses the precipitation (Figures 1c) and contributes to an increase in the liquid water path and cloud cover (or cloud fraction) . These changes (or adjustments) in cloud cover and LWP represent the second indirect effect of aerosols [Albrecht. 1989;Glassmeier et al., 2021]. The stratocumulus clouds, which cannot be maintained in the control cases, are formed and maintained in the perturbed cases (dashed lines in Figure 1h). Figures 1f and 1h show that the aerosol-induced increase of LWP and CF becomes smaller in a double-CO2 climate (P2CO2 p vs. P2CO2 c ) than in the present day (PD p vs. PD c ). If we shift our perspective and consider the impact of a warmed climate by comparing the two cases with aerosol perturbations (PD p vs. P2CO2 p ), we find cloud changes with warming that are consistent with other studies [e.g., Blossey et al., 2016;Schneider et al., 2019], indicating that stratocumulus cloud formation favors the present day over a double-CO2 climate as global warming tends to break up stratocumulus clouds by several mechanisms, including the entrainment liquid-flux adjustment [Bretherton and Blossey, 2014;Bretherton, 2015]. From Day 1 to Day 3, stratocumulus tends to break up and transit to cumulus clouds, mainly driven by the increase of SST [Sandu et al., 2011;Wyant et al., 1997]. As previous studies indicate that the most salient feature of SCT is the structural change in the boundary layer circulation [Krueger et al., 1995;Sandu et al., 2011], Figure 2 shows the daytime-mean cloud fraction (shaded) and boundary-layer structure (black, orange, and red bars) [Blossey et al., 2016] in the three days. The black bar in Figure 2 is the inversion height Zi, where the relative humidity crosses 50%. The red bar is the lifting condensation level (LCL) of air with the properties at 0.1 x Zi, representing the lowest cloud base. The orange bar is the lifting condensation level (LCL) of air with properties at 0.9 x Zi, representing the stratocumulus cloud base. There is no stratocumulus cloud layer if the orange bar is near or above the black bar. The distance between the orange and red bars can be defined as a decoupling index: LCL 10 90 = LCL(z = 0.9Zi)  LCL(z = 0.1Zi). The cloudy boundary layer is well-mixed if the orange and red bars are close, indicating a less decoupled boundary layer. On Day 1, the perturbed cases show a stratocumulus cloud layer (between black and orange bars in Figure 2a), while there are almost no stratocumulus clouds in the control cases because the low aerosol conditions lead to persistent precipitation and cloud breakup [Yamaguchi et al., 2017]. The perturbed cases show a stratocumulus-topped boundary layer with more mixing (compared to the control cases) in the upper boundary layer (smaller distance between red and orange bars in Figure 2a). As shown in Figure S4, there are two TKE peaks (produced by buoyancy and wind shear) in the MBL: one in the cloud layer (dominated by buoyancy production) and another near the surface (dominated by shear production). The perturbed cases have more TKE in the cloud layer while the control cases have more TKE near the surface. Cloud-top entrainment , modulated by the evaporation cooling and LW radiative cooling at the top of the stratocumulus clouds, is much larger in the perturbed cases than in the control cases. From Day 1 to Day 3, clouds transit from stratocumulus to cumulus, with the deepening (Zi tends to increase) and decoupling (LCL 10 90 tends to increase) of the MBL . Compared to the present day (PD p ), global warming (P2CO2 p ) decreases the thickness of stratocumulus clouds (i.e., the distance between black and orange bars in Figure 2) in the perturbed cases but doesn't change the boundary layer structure (e.g., decoupling index LCL 10 90 is similar in PD p vs. in P2CO2 p ). As pointed out by previous studies [Blossey et al., 2016;Bretherton 2015], global warming can cause shrinkage of the cloud and subcloud layer depths, but cause little change in the buoyancy fluxes within either region. To further explore how increased aerosols change the cloud radiative effect, we first decompose the aerosol-induced CRE into two components (See Section 5.4.1), representing contributions from the aerosol-induced CF (change of cloud fraction) and A (change of cloud albedo): CRE = CRE_CF + CRE_A. In both PD and P2CO2 cases, CF has much larger contributions to the CRE than the contribution from A (i.e., CRE_CF vs. CRE_A). Aerosol-induced CF and A are smaller in the double-CO2 climate than in the present day. Furthermore, CRE_A is mainly contributed from aerosol-induced Nc (change of cloud droplet number concentration) and LWP (change of liquid water path): CRE_A = CRE_ANc + CRE_ALWP + CRE_ARes (See Section 5.4.2), where CRE_ARes is a negligible residual term.
CRE_ANc in Table 1 shows a cooling radiative effect due to the Twomey effect (aerosols induce more smaller cloud droplets to reflect more shortwave radiation), which is almost identical between the present day to the double-CO2 climate. This result, that the Twomey effect is not significantly influenced by global warming, is consistent with the nearly identical changes in Nc found in Section 2.1. Aerosol-induced LWP has two ways to contribute to the CRE. On the one hand, CRE_ALWP in Table 1 shows a slightly warming radiative effect due to the aerosol-induced decrease of cloud thickness (Figures 2 and S5), which is a warming contribution to CRE_A. On the other hand, overall there is an increase in aerosol-induced LWP , consistent with the increase of CF , which has a cooling contribution to CRE_CF. Through a temporary increase in the surface dry aerosol flux, we study how increased aerosols influence clouds in the NE Pacific marine boundary layer in a changing climate (i.e., PD vs. P2CO2). Figure 3a summarizes the aerosol-cloud interactions discussed in this study, which employs high-resolution large eddy simulations of the stratocumulus-to-cumulus transition along quasi-Lagrangian (airmass-following) trajectories. The effects of changes in aerosols and climate on such simulations have been studied before [Bretherton and Blossey, 2014;Chun et al., 2025;Prabhakaran et al., 2024;Sandu and Stevens, 2011;Yamaguchi et al., 2017;], but, to our knowledge, this is the first study to characterize the interactions of aerosol and climate change in such simulations. Compared to the present day, the double-CO2 climate decreases the aerosol-induced Nc by less than 10% but decreases the aerosol-induced LWP by more than 40% (red vs. blue circles in Figure 3b). Thus, global warming slightly inhibits the first indirect effects of aerosols (aerosol-induced Nc), but notably inhibits the second indirect effects of aerosols (aerosol-induced LWP), which results in an inhibition of aerosol-cloud interactions in a double-CO2 climate. To explore how aerosol-induced cloud changes influence the cloud radiative effect, we use a decomposing method to decompose the aerosol-induced changes of cloud radiation effects (CRE) into contributions from aerosol-induced CF, LWP, and Nc . From the control case (clean MBL) to the perturbed case (increased aerosols shown in Figure 1b), aerosolinduced CF shows the largest contribution (cooling effects) to the CRE. Aerosol-induced Nc causes a cooling effect by increasing cloud albedo due to the Twomey effect. Aerosolinduced changes to cloud macrophysical properties have a cooling contribution to CRE (related to the increase of cloud fraction) and a warming contribution to CRE (related to the decrease of cloud thickness). All analysis in this paper focuses on daytime (shortwave radiative effects), a comparison between daytime and nighttime (diurnal evolution of aerosol-cloud interactions in a changing climate) will be our next step. This study focuses on how aerosols influence clouds in a changing climate, but from a different perspective, we may evaluate climate sensitivity by comparing the shortwave cloud radiative effects (SW CRE) in the present day (PD) to those in a double-or quadruple-CO2 climate (P2CO2 or P4CO2) under different aerosol perturbations (e.g., P2CO2 p -PD p ), as shown in Figure 3c. The asterisks in Figure 3c represent the SW CRE in the present day (PD) and a 4CO2 climate (P4CO2) from BB2014 using simulations with fixed Nc of 100 cm -3 . The slight difference in climate sensitivity between our results and BB2014 is expected due to the discrepancy of model settings: our simulations predict rather than prescribe aerosol and cloud droplet concentrations, and they have a much larger horizontal domain of 51.2 km  51.2 km than BB2014 4.48 km  4.48 km. SW CRE changes with doubling CO2 (P2CO2 b -PD b ) are weak when Nc is small (for Nc ~ 20 cm -3 ) but SW CRE weakens by ~30 W/m2 with a doubling of CO2 when Nc is larger (for Nc ~110 cm -3 ). The change of SW CRE caused by quadrupling CO2 (~40 W m -2 ) is about twice as large as that by doubling CO2, especially when Nc>100 cm -3 . The main analyses in this study are based on the differences between the present day (PD) and a 2CO2 scenario (P2CO2). To test the sensitivity of our results, we also carried out another set of simulations in a 4CO2 scenario (P4CO2) to evaluate aerosol-induced cloud changes (magenta points in Figure 3b). In the P4CO2 scenario (Figure S1), the CO2 concentration is increased to 4 times (4CO2), SST is increased by 4 K, subsidence is reduced by 10%, and specific humidity is increased to keep the relative humidity identical to that in PD and P2CO2. From the P4CO2 scenario, we find that:  Because global warming tends to decrease clouds (e.g., leading to thinner stratocumulus clouds that break up sooner), generally the aerosol-induced cloud changes (see LWP and Nc in Figure 3b) are smaller in a warmer climate. From P2CO2 to P4CO2 (red vs. magenta circles in Figure 3b), aerosol-induced increases in LWP weaken by ~80%, while aerosolinduced Nc decreases by only ~7%. This indicates that the second indirect effect of aerosols is more sensitive to global warming than the first indirect effect of aerosols when the unperturbed MBL is clean. Figure S5 shows the vertical profile of cloud fraction and TKE. In the control case without perturbed aerosols, global warming tends to lift the cloud layer upward (in our model settings, subsidence is decreased in a 2CO2 climate as is expected with a weaker overturning tropical circulation). In the perturbed case with increased aerosols, global warming tends to lower the cloud layer due to the decrease of TKE (Figures 5d-5f), which is because (1) more CO2 and water vapor in the FT inhibit cloud-top longwave cooling and (2) stronger inversion as the FT warms faster than the surface. Convective clouds (on Day 3) are more sensitive to global warming than the stratocumulus clouds (on Day 1), as shown by the larger difference between P4CO2 and P2CO2 on Day 3 than that on Day 1 in Figure S5.
To highlight aerosols' interactions with clouds in a case with strong potential for cloud thickening in response to natural or anthropogenic aerosol perturbations, the control case in this study has an initially clean MBL with a dry aerosol concentration of approximately 30 cm -3 . To test the sensitivity of our results to different initial (i.e., unperturbed) MBL aerosol concentrations, we carried out a set of 1-day simulations using a more polluted MBL with a larger initial aerosol concentration of approximately 75 cm-3 (red lines in Figures 3b and S6b). These short simulations focus on how stratocumulus early in the Sc-to-Cu transition respond to aerosol perturbations when the background aerosol is larger than in the control simulations above. With the same increase of surface dry aerosol flux in this new set of perturbed simulations, we find that:  Unlike the clean MBL in which the increased aerosols (on Day 196.5) take over most interactions with clouds, the increased aerosols in the polluted MBL have much weaker aerosol-cloud interactions because the higher initial aerosol concentration in the polluted MBL leads to more widespread cloud cover. Figure S6 shows that, when aerosol perturbations are introduced into an MBL with higher background Na, most aerosol-induced cloud changes (e.g., CF, LWP, Nc) are much smaller (solid vs. dashed red lines) than in the clean MBL (solid vs. dashed yellow lines) on the first day. Unlike the clean background that cannot maintain stratocumulus clouds (solid yellow line in Figure S6h) due to the high precipitation (solid yellow line in Figure S6c), the higher background aerosol concentrations can inhibit the precipitation (solid red line in Figure S6c) and maintain stratocumulus clouds (solid red line in Figure S6h) at the beginning of the simulation even before the aerosol perturbation is introduced. Unlike the clean MBL where the aerosol perturbation inhibits precipitation (dashed yellow line in Figure S6c) and increase stratocumulus cloud (dashed yellow line in Figure S6h), the aerosol perturbation has negligible influences on precipitation (dashed red line in Figure S6c) when the background aerosol concentration is larger and may even slightly decrease the cloud fraction and LWP of stratocumulus clouds (dashed red lines in Figures S6f and S6h) due to the increase of entrainment and evaporation at the top of the clouds. The opposite LWP responses to aerosol perturbations in a clean MBL (increased LWP due to aerosolinduced precipitation inhibition) versus in an MBL with higher background aerosol concentrations (decreased LWP due to aerosol-induced increase of entrainment and evaporation) indicate high uncertainties in aerosol-induced LWP adjustment [Glassmeier et al., 2021;Florent F. Malavelle., 2017]. Our study suggests that cloud fraction should be considered in addition to LWP when evaluating the response of clouds to aerosols: increased aerosols tend to increase LWP when CF is smaller (e.g., circles in Figure 3b for a clean MBL) but may decrease LWP when CF is near 1 (e.g., triangles in Figure 3b for a polluted MBL). As an application of the aerosol-cloud interactions, marine cloud brightening (MCB) proposes the idea of injecting aerosols into the stratocumulus clouds, the most common cloud type covering more than 20% of the ocean surface [Warren et al., 1988;Wood. 2012]. These additional aerosols would lead to more numerous and smaller cloud droplets, which can increase cloud albedo and reflect more incoming solar radiation [Diamond et al., 2022;Feingold et al., 2024;Latham et al., 2012;Rasch et al., 2024]. Our results indicate that global warming can weaken the cooling effects of MCB (CRE in Table 1) mainly by inhibiting the second indirect effect of aerosols . This study also indicates that the cooling effect of MCB is weaker in an MBL with higher background aerosol concentrations. In some cases, deploying MCB in a polluted MBL may lead to a decrease in cloud thickness (e.g., negative LWP shown by triangles in Figure 3b) due to aerosol-induced entrainment and evaporation, which will have a negative contribution to MCB's cooling effects. These results can help design MCB injection strategies to achieve a better cooling effect (e.g., applying MCB preferentially to clean rather than polluted MBLs, or by adjusting the MCB injection rate based on background aerosol concentration). Figure 3c shows that aerosol perturbations lead to much stronger cloud brightening (i.e., more negative SW CRE changes) when the background MBL aerosol concentration is ~30 cm -3 than when the background aerosol concentration is larger (~75 cm -3 ). This may indicate a saturation (ceiling) of MCB's cooling effect as the background aerosol concentration increases. This MCB's saturation (ceiling) cooling effect is approximately -100 W/m2 (averaged on the 1 st day) in PD, then decreases to -80 W/m2 in P2CO2 and -60 W/m2 in P4CO2. We used the SAM model to carry out the large eddy simulations along a Lagrangian trajectory following a previous study named here as BB2014 (Bretherton and Blossey, 2014) with the domain center starting at (25N, 125W) in the NE Pacific. The SAM model has 288 vertical levels with a model top at 4.2 km. The horizontal domain size is 51.2 km  51.2 km, with a horizontal resolution of 100 m. The whole simulating time is 3 days in summertime (196.1-199.1). The first half day (196.1-106.6) is considered the spin-up time. The transient nudging of temperature (T), water vapor (q), and aerosols is applied in the marine boundary layer (MBL) in the first 6 hours. T and q are nudged in the free troposphere (FT) in the whole 3-day simulation. SAM model includes an interactive aerosol scheme [Berner et al., 2013] with representation of the PBL aerosol budget for a single accumulation mode, with sources and sinks from the surface and FT. Before the aerosol perturbation, the background aerosol concentration is 70 cm -3 in FT [Bretherton et al., 2019;Bretherton and Blossey. 2014] and 30 cm -3 in MBL, which represents a clean boundary layer (i.e., PD b and P2CO2 c ). Compared to PD, P2CO2 represents a double-CO2 climate [Bretherton and Blossey, 2014] due to doubling CO2.
For P2CO2, the CO2 concentration is increased to 2 times (2CO2), SST and absolute temperature are increased by 2 K, subsidence is reduced by 5%, and specific humidity is increased to keep the relative humidity identical to that in PD. More details can be found in Figure S1. This paper focuses on how aerosol-cloud interactions respond to climate change including all warming perturbations. In the next step, we will evaluate how each component of composite warming perturbations (e.g., the increase of SST, the decrease of subsidence) influences ACI, which can help us better understand the performance of cloud controlling factors (CCFs) in a changing climate. In the clean case, the increase of aerosols (due to less precipitation caused by global warming) is the dominant CCF that facilitates cloud formation, while in the polluted case with enough aerosols, the decrease of turbulence in a double-CO2 climate (due to reduced longwave radiative cooling) thins and lowers clouds. In this study, cloud-top entrainment ( e ) is calculated as: Where is the rate of change of inversion height (Z inv ) over time,  ls,inv is the large-scale vertical velocity at the inversion height. For two simulating cases named as control (solid line in Figure 1) and perturbed (dashed line in Figure 1), the change of cloud radiative effects between the two cases (dashed vs. solid lines in Figure 1g) can be calculated as: CRE p means the cloud radiative effect (CRE) from the perturbed case (e.g., PD p and P2CO2 p in this study), CRE c mean the CRE from the control case (PD c and P2CO2 c in this study), CRE is the change of CRE between the two cases (e.g., PD p vs. PD c ). In this study, we ensure that positive values of CRE always mean warming effects. CRE can be calculated as: S is the incoming shortwave (SW) flux at the top of the atmosphere (TOA). A is the total overcast albedo (i.e., the albedo as seen from space, also known as planetary albedo or TOA albedo) [Diamond et al., 2020;Qu et al., 2005;Donohoe et al., 2011], which is calculated by: CF is the cloud fraction, A cld is the overcast albedo over cloudy areas, A clr is the overcast albedo over the clear-sky areas. Based on Eq. (3), we can calculate the total overcast albedo for the control and perturbed cases: For perturbed case: By combining Eqs. ( 1) and ( 2), we get: Substituting Eqs. ( 4) and ( 5) into Eq. ( 6): So far, we decompose the change of CRE between two cases into two parts: the first part represents the contribution from the change of cloud albedo (A cld = A cld p  A cld c ), the second part represents the contribution from the change of cloud fraction (CF p  CF c ). In the next step, we will analyze how cloud number concentration (Nc) and liquid water path (LWP) inside clouds influence the change of cloud albedo. Based on previous studies [Chun et al., 2023;Diamond et al., 2020;Ehsan et al., 2022], the changes of cloud albedo (A cld ) can be considered as a function of changes of Nc and LWP. As mentioned before, the albedo (A) we have discussed so far is the overcast albedo (i.e., the albedo as seen from space, also known as planetary albedo or TOA albedo) [Diamond et al., 2020;Qu et al., 2005;Donohoe et al., 2011]. By using  as the albedo of the individual constituent, we can get: A cld is the overcast albedo over the cloudy areas (seen from the space), while  cld is the cloud albedo. FT and T FT are the albedo and transmissivity of the free troposphere, respectively. A clr is the overcast albedo over clear-sky areas. atm and T atm are the albedo and transmissivity of the atmosphere, respectively. sfc is the surface albedo (i.e., the ratio of surface upward SW radiation to the surface downward SW radiation). is derived from Eq. ( 9). To calculate A cld,Nc and A cld,LWP based on Eqs. ( 11) and ( 12), we need to calculate  cld,Nc ,  cld,LWP ,  cld , T FT , and  FT , which is shown in the following. The change of cloud albedo between two cases due to the change of Nc can be calculated as [Wood et al., 2021]: 13) and ( 14) is the cloud albedo from the control case, while  cld used in Eqs. ( 11) and ( 12) is set as the average of cloud albedos from control and perturbed cases (i.e.,  cld =  cld c + cld p 2 ,  cld c and  cld p can be calculated based on Eq. ( 15)). Note: the value assigned to  cld can introduce bias for the final results, but the bias is negligible. The transmissivity of the free troposphere (T FT ) is calculated as the ratio of downward SW radiation near the inversion height to the solar irradiation (S). The albedo of the free troposphere can be calculated by:  atm is the albedo of the atmosphere, which can be estimated from Eq. ( 10). T atm is the transmissivity of the atmosphere (i.e., the ratio of downward SW radiation reaching the surface to the solar irradiation S under the clear-sky condition). This research has been NA22OAR4310474 and NA20OAR4320271). This work used Bridges-2 at Pittsburgh Supercomputing Center through allocation EES210037 from the Advanced This research has been NA22OAR4310474 and NA20OAR4320271). This work used Bridges-2 at Pittsburgh Supercomputing Center through allocation EES210037 from the Advanced