1 Introduction

Clouds play a critical role in regulating the energy balance and hydrological cycles in atmosphere, but they are among the most uncertain processes represented in climate models (Bony and Dufresne 2005; Klein and Hartmann 1993; Webb et al. 2006; Zhou et al. 2016). In spite of promising improvement of cloud and turbulence parameterizations during the past several decades (Angevine et al. 2018; Bony and Emanuel 2001; DelGenio et al. 1996; Hourdin et al. 2019; Lock 2001; Suselj et al. 2019; Teixeira and Hogan 2002; Tompkins 2002), many important characteristics of clouds are still not accurately represented in modern weather and climate models (e.g., Brown et al. 2014; Collins et al. 2011; Duynkerke and Teixeira 2001; Emanuel 2020; Golaz et al. 2007; Siebesma et al. 2004), resulting in large uncertainties in quantifying the cloud-climate feedbacks and climate sensitivity using climate models (Bony et al. 2006; Bony and Dufresne 2005; Jackson et al. 2004; Loeb et al. 2009; Qian et al. 2016; Sherwood et al. 2014; Stainforth et al. 2005; Stephens 2005; Vial et al. 2013; Wan et al. 2014; Wyant et al. 2006; Zhang et al. 2018). Global climate models also exhibit difficulties in skillfully simulating all cloud regimes (i.e., shallow cumulus, stratocumulus, and deep convective clouds as well as transitions between them) simultaneously over different regions (Yang et al. 2021).

Moist convection is one of the primary mechanisms governing cloud formation and evolution. It produces precipitation, redistributes heat and moisture in the atmosphere, drives atmospheric circulation by releasing latent heating, and influences stratiform/cirrus cloud and large-scale environment through detrainment (Arakawa 2004; Collins et al. 2001; Harrop et al. 2018; Hartmann et al. 2001; Rio et al. 2019). Because the small-scale convection process cannot be resolved by contemporary global models, sub-grid parameterization is needed to represent its effects, which introduces many uncertain parameters (Hourdin et al. 2017; Jackson et al. 2008; Phillips et al. 2004; Yang et al. 2013, 2022; Zou et al. 2014). The values of these uncertain parameters are often empirically determined, so they can be optimized or tuned (Gong et al. 2015; Neelin et al. 2010; Ollinaho et al. 2013, 2014; Severijns and Hazeleger 2005; Wang et al. 2014; Williamson 2013). However, one of the biggest challenges often encountered in model tuning is that improvements in one evaluation metric or simulation region often led to degradations of performance for others (Yang et al. 2019). This implies the presence of structural uncertainties in climate models associated with limitations in our understanding and model representation of physical processes at unresolved scales (Leung et al. 2020; McNeall et al. 2016; Williamson et al. 2015).

The transect established by the GEWEX Cloud System Study (GCSS) Pacific Cross-section Intercomparison (GPCI) project (Bretherton et al. 2005; Teixeira et al. 2011; Wood and Bretherton 2004; Wood et al. 2011) provides a natural testbed to characterize clouds over multiple climatically critical cloud regimes in the real world and in climate models. The goal of the GPCI project is to evaluate turbulence, convection, clouds, and radiation in weather prediction and climate models in the tropics and sub-tropics, with the help of satellite observations (Marchand et al. 2008; Notz 2015; Xie and Arkin 1997). Along the GPCI transects, stratocumulus-topped marine boundary layer (MBL) off the west coast of continents first transitions to a cumulus-dominated MBL and then to deep cumulus convection regions in the intertropical convergence zone (ITCZ). The different cloud regimes along the GPCI transects play a key role in modulating atmospheric circulation patterns in the tropics and subtropics (Larson et al. 1999; Ma et al. 1996; Philander et al. 1996). However, global weather and climate models have large errors and spread in representing clouds along GPCI (Teixeira et al. 2011). For instance, there are overestimation of trade-wind cumuli and tropical deep convective clouds, underestimation of cloud amount in the marine stratocumulus regime, large multi-model spreads in the vertical cross section of cloud properties, vertical velocity, and relative humidity (Sun et al. 2010; Wang et al. 2015). These results motivate the need to identify the cause of the model errors and better understand the cloud and convection processes and their subgrid representations in global climate models (Guo et al. 2015).

In this study, we use the U.S. Department of Energy’s Energy Exascale Earth System Model (E3SM) (Golaz et al. 2019) Atmosphere Model version 1 (EAMv1) (Rasch et al. 2019) to investigate the model sensitivity to uncertain parameters in cloud and convective parameterizations. EAMv1 deploys many new features and various configurations with a high level of complexity (Rasch et al. 2019; Xie et al. 2018). To quantify the parametric sensitivity and better understand the model physics and behaviors, Qian et al. (2018) conducted short-term perturbed parameter ensemble (PPE) and performed a systematic parametric sensitivity analysis for EAMv1, focusing on global mean features and quantities. While the values of tunable parameters in most global climate models are fixed temporally and spatially, model responses to perturbations of parameters may vary over different regions and cloud regimes (Bellprat et al. 2012; Zhang et al. 2012). It is therefore important to understand how cloud properties in different regions or regimes respond differently to the imposed parameter perturbation and how they interact with each other in the global atmospheric system (S. Liu et al. 2022). In this study, using the same set of PPE simulations as described in Qian et al. (2018) and a similar sensitivity analysis framework Murphy (Mulholland et al. 2017; Murphy et al. 2014), we identify the most sensitive parameters in different regions, including the GPCI transect, and compare model responses to the parameters for several important fidelity metrics related to clouds in different regimes. A brief introduction of the approach and analysis strategy is presented in Sect. 2 and analysis results are shown in Sect. 3. Our focus in this paper is to characterize the fast responses of clouds and convection to parameter perturbations. Such fast responses are known to have long-term impacts that can be evaluated with reasonable accuracy using short (few-day) simulations. Results from this study provide useful information of parametric sensitivity over different climate regimes in EAMv1, which improves understanding of the treatment of model physics in terms of tunable parameters and their interactions and helps guide future parametrization development for climate models.

2 Model and methodology

In this study, EAMv1 (Rasch et al. 2019) is run on a ne30np4 cubed-sphere grid (approximately 100 km horizontal spacing) with 72 vertical layers. As described in Xie et al. (2018), the vertical grid was designed to have higher vertical resolution in the lower troposphere to better represent low clouds. The Fig. 1 of Xie et al. (2018) illustrates the vertical grid and model top was set at 0.1 hPa to capture features in the upper troposphere and stratosphere. EAMv1 generally produces better cloud characteristics compared to CFMIP1 and CFMIP2 models (Zhang et al 2019). Boundary layer clouds show only marginal sensitivity to vertical resolution unless the vertical resolution is increased to what is typically adopted in LES (Bogenschutz et al. 2021).

Fig. 1
figure 1

(Left) Location of six regions of analysis focus. Three regions over land are (4) Amazon (297.5–300 E, – 4.24 to – 2.36S), (5) Southern Great Plain (SGP, 261.25–263.75 E, 35.34–37.22N), and (6) Arctic North Slope of Alaska (Arctic, 201.25–203.75 E, 70.2–72.09 N), respectively, together with the E3SM simulated high-level cloud cover. Three regions over ocean are along (Right) a cross-section from the equator, across the trade cumulus regions, to the stratocumulus regions off the west coast of California over eastern Pacific based on the GEWEX Cloud System Study (GCSS) Pacific Cross-section Intercomparison (GPCI) project, together with the ISCCP low cloud cover (%) climatology for the June–July–August. (1) Ocean_deep (197.5–200 E, 7.068–8.95 N, (2) Ocean_transition (213.75–216.25 E, 19.32–21.20N), (3) Ocean_stratoCu (230–232.5 E, 30.63–32.51 N)

Full size image

2.1 Parameters selection

Major physics parameterizations related to clouds in EAMv1 include: (1) the Cloud Layers Unified By Binormals (CLUBB) scheme, an incomplete third-order turbulence closure parameterization coupled with assumed probability density functions (PDFs) to provide a unified treatment of boundary layer turbulence, shallow convection, and stratiform cloud macrophysics (Golaz et al. 2002; Larson 2017; Larson and Golaz 2005; Larson et al. 2012), (2) the Morrison and Gettelman (MG) microphysical scheme version 2 (MG2) that is compatible with CLUBB (Gettelman and Morrison 2015; Gettelman et al. 2015), and (3) an improved Zhang‐McFarlane Deep Convection Scheme (Zhang and McFarlane 1995) with a diluted CAPE (Convective Available Potential Energy) modification by Neale et al. (2008), which is based on a plume ensemble concept assuming the ensemble of plumes acts to relax the atmosphere to a neutral buoyancy state and uses the quasi‐equilibrium assumption to construct a bulk representation of cumulus ensemble. These parameterization schemes are used to represent the unresolved physical processes related to turbulence, shallow convection, deep convection, and cloud microphysics.

Many uncertain parameters used in the parameterization schemes cannot be derived theoretically or obtained directly from observations, so the individual parameters can be tuned separately within their own reasonable ranges or optimized as a group to improve the overall model fidelity, as measured by certain evaluation metrics. Based on consultation with the developers of the original parameterization schemes and previous works (Guo et al. 2014, 2015; Qian et al. 2015; Yang et al. 2013), we select 18 tunable and potentially sensitive parameters from the parameterizations of turbulence and shallow convection, deep convection, cloud microphysics, and gravity wave drag as summarized in Table 1. More detailed descriptions of those parameters were introduced in Qian et al. (2018).

Table 1 The 18 selected parameters for perturbation and their ranges of values
Full size table

2.2 Parameters sampling and sensitivity analysis (SA)

After the parameters are selected and their ranges of values are determined (assuming a uniform distribution in the range for each parameter), we use a Latin hypercube sampling (LHS) approach (Helton and Davis 2003; Stein 1987) to generate large sets of parameter values for conducting the PPE simulations. The sensitivity analysis (SA) is performed for a few objective (interested) variables to quantify the parameters responses after all short simulations for each realization (sampled parameter set) are completed (Di et al. 2015; Gleckler et al. 2016; Hou et al. 2012; Johannesson et al. 2014; Rougier et al. 2009; Williamson et al. 2017; Yang et al. 2012; Zhao et al. 2013). We apply the generalized linear model (GLM) (Mackinnon and Puterman 1989) to the results from the PPE simulations to account for both linear and nonlinear interaction effects of the parameters. By analyzing the interpreted variances, goodness of fit (R2), and statistical t and F tests, GLM can effectively quantify the proportion of the original variance attributed to each input parameter for every object variable. Additionally, GLM offers valuable guidance into whether a factor/predictor holds statistical significance or not. More details regarding the GLM can be found in McCullagh and Nelder (1989). This SA framework has gained widely adoption within the statistical community and recently found applications in the atmospheric and climate modeling community (Zhao et al. 2013; Qian et al. 2015; Yan et al. 2015).

In practice, it is generally recommended to have a number of points in the LHS that is at least 10 times the number of parameters being investigated (Loeppky et al. 2009). For this study, we employed the LHS approach to generate 256 parameter sets, which allowed us to achieve adequate coverage of the parameter space from the low to high bounds. The selection of this number of points was made as a tradeoff between computational time and numerical error, considering that E3SM simulations are computationally demanding. Running thousands of simulations was not practical within the scope of this study.

2.3 Short simulations design

We generate 256 combinations of parameters values, corresponding to the 256 sample points obtained from the LHS approach (Queipo et al. 2005). For each set of parameters, we run 12 simulations each initialized at the beginning of each month of the year to account for seasonal variability. We run a total of 256 × 12 PPE simulations with a length of 5 days each, following the approach of Cloud Associated Parameterization Testbed (CAPT) and transpose Atmosphere Model Intercomparison Project (AMIP) paradigm (Kanamitsu et al. 2002; Liu et al. 2016; Phillips et al. 2004). CAPT is a model framework that utilizes the numerical weather prediction (NWP) technique in climate models so that climate models can be initialized with date from NWP analyses or re-analyses and evaluated with detailed process observations such as field campaign data. In this study, the initial conditions are from a multi-year simulation previously conducted with the default parameter values (Qian et al. 2018). As demonstrated in Xie et al. (2012) and Ma et al. (2014), many well-known climate errors in cloud, precipitation, and radiation often develop in the early periods of model forecasts and steadily grow with forecast lead time. This approach of generating PPE using short simulations is more relevant to the “fast processes” associated with clouds and convection (Ma et al. 2014; Xie et al. 2012), which is the focus of this study to improve the process-level understanding of the climate model physics parameterizations. Our analysis in this paper focuses on the physical quantities associated with convections, clouds, radiation and precipitation, including variables like cloud fraction, radiative fluxes, precipitable water, surface fluxes and precipitation. More details of our simulation strategy and computational and analysis method can be found in Qian et al. (2018). This PPE has been used for guiding the recalibration of the EAMv1 (Ma et al. 2022), which was largely adopted in E3SMv2 resulting in reduced model biases in quantities such as global mean shortwave cloud forcing and precipitation (Golaz et al. 2022).

2.4 Selection of analysis regions

We select six regions to represent different climate and cloud regimes for our analysis. Three regions over land are Amazon, Southern Great Plains (SGP) and North Slope of Alaska (NSA), representing the tropical, mid-latitude and high-latitude land environments, respectively. Each region encompasses 3 × 3 grid points (See Fig. 1 Caption for more details). Three regions over ocean are along the GPCI cross-section, transitioning from the equator deep convection to trade cumulus and to stratocumulus regions off the west coast of California over eastern Pacific, as shown in Fig. 1.

3 Results

3.1 Variation in the relative importance of parameters among different regions

Figure 2 demonstrates the relative importance, represented by the ratio of variance contribution to the total explained variances calculated by GLM, of the perturbed parameters (18 of them) and their interactions with shortwave cloud forcing (SWCF), longwave cloud forcing (LWCF), total precipitable water (TMQ) and precipitation (PRECT), over the six regions. The analysis in this figure and a few others that follow are based on the results of day 5, and Sect. 3.4 will discuss how model sensitivity evolves with the forecast lead time. The relative importance of parameters to the SWCF varies across regions, while c1 in the CLUBB scheme is one of the most influential parameters in all six regions, especially over ocean because c1 effectively affects the turbulent mixing and then the low-level clouds. SWCF is also very sensitive to the timescale of consumption of convective available potential energy (CAPE) tau in the ZM deep convection scheme for the deep convective clouds over land and tropical ocean. Other important parameters include dp1 and c0_ocn for deep convection over ocean and c8 and gamma over middle-latitude ocean, and dmpdz over SGP. Overall, the contribution of individual parameters accounts for over 80% of the total variance, except over the Arctic and marine stratocumulus regimes where the interaction terms account for more than 20% of the variance. Comparing to previous studies (Qian et al. 2018, 2015), the contribution of interaction terms to the total variance is larger at the regional scales than the global mean quantities.

Fig. 2
figure 2

Fractional contributions of the 18 perturbed parameters to the total variance estimated by GLM fit for the annual mean Shortwave Cloud Forcing (a, SWCF), Longwave Cloud Forcing (b, LWCF), vertically integrated precipitable water (c, TMQ) and total precipitation (d, PRECT) over the six regions shown in Fig. 1. The white portion of each bar (“Int” term) denotes the summation of contributions from all interactions between parameters. Contributions of ZM and CLUBB scheme parameters are marked with hatches and dots, respectively

Full size image

Most of the important parameters for SWCF are also sensitive parameters for LWCF (Fig. 2b) and FLNT (Net Longwave Flux at the Top of the Atmosphere, not shown), though the summation of contributions from all interactions between parameters is larger for LWCF than for SWCF. The most influential parameter for total precipitable water (TMQ) is c_k10 because of its impact on surface wind through the turbulent diffusion that affects evaporation. The parameters tau and dmpdz of the ZM convection scheme dominate the contribution to the variance of total precipitation (PRECT). A non-negligible role is also played by c_k10. The role of interaction terms is more important for the variance of PRECT than for other variables, and the contribution to the total variance of PRECT from all interaction terms is more than 50% over ocean. This suggests that precipitation is not only regulated by multiple relatively fast physical processes such as turbulence, cloud macro- and micro-physics, and convective mixing, but also by the nonlinear effects from feedbacks (e.g., dynamical feedback through circulation change) that usually evolve at longer time scale.

The different cloud regimes along the GPCI transect play a key role in modulating the atmospheric circulation in the tropics and subtropics. However, global weather and climate models often have large errors and spreads in the representation of clouds along the GPCI. Quantification of parametric sensitivities along the GPCI transect may help us better understand EAMv1 model bias and behavior over the different cloud regimes associated with cloud parameterizations. Figure 3a shows the parametric sensitivity of SWCF along the GPCI transect from deep convection (left) to cumulus and further transition to marine stratocumulus clouds (right). Over tropical ocean with active deep convective activities, the ZM deep convection parameters such as tau, dmdpz, dp1 and c0_ocn are all important, as expected, and they together contribute to around 60% of the total SWCF variance. Over the stratocumulus-to-cumulus transitional area, the sensitivity to c1, c8 and gamma in CLUBB scheme gradually increases. Over the stratocumulus region, c1, c8 and gamma become the dominant contributors, together accounting for more than 60% of the total variance. The contribution of interaction term accounts for approximately 10–20% across all cloud regimes. The major parameters and their contributions to LWCF variance also vary along the GPCI transect (Fig. 3b), and the most important parameters are c1, dmpdz, tau and c0_ocn. The contribution of parameters interaction term accounts for 20–50% for LWCF, much larger than for SWCF.

Fig. 3
figure 3

Parametric sensitivity of SWCF (a, top) and LWCF (b, bottom) along the GPCI transect from deep convection on the left to cumulus and further transition to marine stratocumulus on the right. Values (0–60) on the x-axis go from Southwest to Northeast, as shown in Fig. 1b for the GPCI location

Full size image

Figure 4 shows the spatial distribution of contributions to the total variance of SWCF by four CLUBB parameters (gamma, c1, c8 and beta). In general, there is a large spatial variability in the subtropical eastern Pacific, with large contributions of gamma, c1 and c8 over the stratocumulus regimes near the coast (e.g., near California and Peru), and also the region of western boundary currents (e.g., Kuroshio Current and Gulf Stream where sea surface temperatures are relatively warm, inducing relatively strong MBL turbulence.). Figure 4 suggests that the near-coastal focus of gamma and c8 could be combined with the more widespread influence of c1 to improve the simulated spatial pattern of low clouds in E3SM. Indeed, this tuning strategy was successfully used by Ma et al. (2022). That tuning exercise combining physical intuition with a large ensemble of parameter perturbation experiments might have been facilitated by more detailed prior information about the sensitivity. Overall, this suggests that the processes governing the dynamic and thermodynamic turbulent structures are the most influential ones for simulating clouds that show a clear cloud-regime dependence. This behavior agrees very well with previous studies that perturbed CLUBB and ZM parameters in the Community Atmosphere Model (Guo et al. 2015).

Fig. 4
figure 4

Spatial distribution of individual contributions to the total variance of SWCF by four CLUBB parameters (gamma, c1, c8 and beta)

Full size image

3.2 Response to parameter perturbation over different regions

Climate models often suffer from the problem that improvement of performance in one region may result in degradation in other regions when parameters are tuned for specific target variables. Qian et al. (2018, 2015) identified the most sensitive parameters for several selected variables and showed the response curve of simulated fields to parameter perturbations. While the information based on global mean quantities is useful for better understanding the overall model behavior at the global scale, we need to know how the model responds to parameter perturbations in different regions in order to understand the region/regime dependence of model physics and better tune the model at the regional scale.

Figure 5 shows the responses of SWCF to nine key parameters over the six selected representative regions. Here we highlight two parameters, dmpdz and tau, respectively, associated with entrainment rate and CAPE consumption rate. SWCF responds to the same parameter perturbations differently over different regions. In some cases, the responses are even in opposite directions. For example, with increasing dmpdz, SWCF decreases over both tropical ocean (black) and land (Amazon, purple) but increases over mid-latitude land (SGP, green), although all these regions are deep convection regimes. Similarly, with an increasing tau, SWCF increases over both tropical ocean and land but decreases over mid-latitude land (SGP) and Arctic region. In Sect. 3.3 below, we will provide more in-depth discussion on the spatial dependence of parametric sensitivity.

Fig. 5
figure 5

Responses of SWCF to nine key parameters over six regions (see Fig. 1a, black, red and blue are three regions from deep convection to stratocumulus-to-cumulus transition and further to marine stratocumulus cloud along the GPCI transect. Purple: Amazon; Green: Southern Great Plains; Light blue: Arctic/North Slope of Alaska). For each parameter over a specific region, the 256 sets of simulations are evenly assigned to eight bins according to parameter values. The mean value and range of SWCF anomaly in each bin are denoted by solid square and vertical bar, respectively. Numbers in red (indicating the 95% statistically significant contribution) or black above each parameter panel show the fractional contribution (%) of individual parameter, calculated by the GLM (with the interaction terms excluded)

Full size image

3.3 Interpretation of spatial dependence of parametric sensitivity

What are the underlying physical mechanisms for the different responses of SWCF to model parameters (dmpdz and tau) over different regions? Convective parameters influence clouds over tropics and mid-latitude SGP differently through their associated thermodynamical process. Figure 6 shows the vertical profiles of detrained ice from deep convection for eight groups of simulations (based on tau values from small to large) over the three regions. A larger tau (e.g., bin8 in purple color) means weaker convection for each plume, producing a lower amount of detrained ice clouds and resulting in less high and mid-level clouds (not shown). Note that high-cloud variation is more important for SWCF and LWCF over regions with intense deep convection. That’s why a longer consumption time of CAPE (i.e., larger value of tau) tends to increase SWCF over the tropical ocean and Amazon.

Fig. 6
figure 6

Vertical profiles of detrained ice from deep convection for eight bins grouped based on the values of parameter tau over tropical ocean, Amazon and SGP, respectively. Note that different scales are used in the x-axis for the three regions

Full size image

In contrast, a larger value of tau over SGP results in a lower SWCF. With a smaller tau (i.e., a stronger convection), convective drying is very strong in the lower troposphere but weak in the upper troposphere (see Fig. 7). This contrasting response of convective drying in the lower and upper troposphere over SGP is different from the similar response throughout the troposphere over tropical ocean and Amazon. Since moisture over SGP is not as abundant as over the tropics, larger convective drying in the lower troposphere with smaller tau means faster consumption of moisture and suppression of the upward transport of water vapor by shallow convection or resolved vertical motion. Furthermore, in SGP with a relatively drier troposphere compared to the tropics, there are much less high-level clouds, as indicated by the much lower detrained ice amount in SGP relative to the tropics (Fig. 6). A very large tau (weak convection) is not favorable for the formation of high cloud because of the very weak detrainment but this impact in SGP is not as large as in the tropics. This can be further confirmed by the humidity profiles (not shown).

Fig. 7
figure 7

Same as Fig. 6 but for moisture tendency in the ZM convection parameterization

Full size image

In contrast to deep convective regimes, cloud features over the stratocumulus regimes (e.g., Peru and California) are mostly determined by CLUBB parameters, especially those related to the skewness of sub-grid vertical velocity, w \(^{\prime}\). Figure 4 shows that gamma ( \(\upgamma \)) and c8 have a higher contribution to the sensitivity near the coast, But, on the other hand, c1 strongly contributes to the sensitivity both near the coast and offshore. Why is c1 more important offshore than \(\upgamma \) and c8? This is a complex question that we cannot answer, but we mention some preliminary considerations here, based on inspection of CLUBB’s equations (Larson 2022).

First, the parameters enter the equations through different pathways. γ directly influences the width of PDFs in w-coordinate (\({\widetilde{\sigma }}_{w}^{2}\)), defined in CLUBB as

$$ \tilde{\sigma }_{w}^{2} = \gamma ~\left[ {1 - \max \left( {\frac{{\overline{{w^{\prime}q_{t}^{{\text{'}}} }} ^{2} }}{{\overline{{w^{{'2}} }} \cdot \overline{{q_{t}^{{{\text{'}}2}} }} }},\frac{{\overline{{w^{\prime}\theta _{l}^{{\text{'}}} }} ^{2} }}{{\overline{{w^{{'2}} }} \cdot \overline{{\theta _{l}^{{{\text{'}}2}} }} }}} \right)} \right] $$
(1)

Because γ is a PDF parameter, it directly influences all terms that are closed by the CLUBB’s PDF, including CLUBB’s higher-order buoyancy terms and its third-order turbulent advection terms. However, we speculate that γ’s biggest indirect impact in these simulations comes through the influence of \({\widetilde{\sigma }}_{w}^{2}\) in modifying the skewness of vertical velocity, \(\widehat{Skw}=\frac{Skw}{{\left(1-{\widetilde{\sigma }}_{w}^{2}\right)}^{1.5}}=\frac{1}{{\left(1-{\widetilde{\sigma }}_{w}^{2}\right)}^{1.5}}\frac{\overline{{w }^{\mathrm{^{\prime}}3}}}{{\left(\overline{{w }^{\mathrm{^{\prime}}2}}\right)}^{1.5}}\). When \(\upgamma \) increases, keeping all else unchanged, the PDF becomes more skewed (Guo et al. 2014, 2015). Then CLUBB’s third-order moments and consequently its vertical turbulent advection terms are amplified [see Appendix B of Larson and Golaz (2005)]. The amplified turbulent advection deepens the boundary layer, making it more cumulus-like, with decreased cloud fraction.

The pathway of influence of c8 is simpler. This parameter appears in only one term in CLUBB. Namely, c8 is the coefficient of a Newtonian damping term on \(\overline{{w }^{\mathrm{^{\prime}}3}}\) (Larson 2022). When c8 is increased, \(\overline{{w }^{\mathrm{^{\prime}}3}}\) tends to decrease. Therefore c8 has similar but inverse effects on skewness as the effects of \(\upgamma \), and the rest of CLUBB’s solutions.

Finally, c1 is the coefficient of a Newtonian damping term on \(\overline{{w }^{\mathrm{^{\prime}}2}}\) (V. E. Larson 2022). Therefore, c1 affects CLUBB’s solutions through the denominator of the skewness. Through the skewness, c1 has similar effects as \(\upgamma \) and c8. However, unlike \(\overline{{w }^{\mathrm{^{\prime}}3}}\), \(\overline{{w }^{\mathrm{^{\prime}}2}}\) appears in the turbulent production term, \(\overline{{w }^{\mathrm{^{\prime}}2}}\frac{\partial \overline{x}}{\partial z }\), of the prognostic equation for a scalar turbulent flux, \(\overline{{w }^{\prime}x^{\prime}}\). Here x denotes either total water mixing ratio or liquid water potential temperature. Through the production terms, c1 indirectly influences the scalar fluxes through an important pathway that c8 and \(\upgamma \) do not.

In CLUBB’s equations, c8 and \(\upgamma \) influence CLUBB’s variables (mainly) through the skewness. In CLUBB’s solutions, the parameters c8 and \(\upgamma \) have their biggest effect on near-coastal stratocumulus (Fig. 4). This may be surprising, given that stratocumulus have lower skewness than cumulus. However, in these simulations, CLUBB overpredicts the skewness in stratocumulus [compare Fig. 8 with Stevens et al. (2005) and Hogan et al. (2009)]. It appears possible that increases to CLUBB’s skewness can cause the simulated stratocumuli to transition toward cumulus-like behavior, including decreased cloudiness. We speculate, however, that the pre-existing cumuli farther offshore may be more robust to perturbations in skewness.

Fig. 8
figure 8

Vertical profiles of \(\overline{{w }^{\mathrm{^{\prime}}2}}\), \(\overline{{w }^{\mathrm{^{\prime}}3}}\) and skewness of vertical velocity for the GPCI transition region (top row) and marine stratocumulus region (bottom row)

Full size image

It is unclear why c8 and \(\upgamma \) have less influence on shallow cumulus than stratocumulus but c1 influences both cloud types equally (Fig. 4). However, the cumulus may be influenced by c1’s additional effect on turbulent production of scalar fluxes, as mentioned earlier. We speculate that an increase in moisture flux may loft more moisture into the cloud layer, increasing the cloud fraction.

As compared to \(\upgamma \), the beta ( \(\upbeta \)) PDF parameter, which helps determine the skewness of \(\overline{{\mathrm{q} }_{\mathrm{t}}}\), \(\overline{{\uptheta }_{\mathrm{l}}}\) and the turbulence advection terms, has a smaller impact on the global low-cloud fraction and SWCF. The smaller impact on those fields may be related to the fact that changes in \(\upbeta \) have zero direct impact on two important higher-order moments, namely, \(\overline{{w }^{{\prime}4}}\) and \(\overline{{w }^{{\prime}2}{x}{\prime}}\) [Appendix B of Larson and Golaz (2005)].

The parameter c_k10 for the eddy diffusivity of momentum is an influential parameter over storm tracks and near-coastal regions. This is partly because wind shear, as the main source term for generating turbulence, is strong there. Momentum fluxes, i.e., \( \mathop {u^{\prime}w^{\prime}}\limits^{ - } \) and \( \mathop {v^{\prime}w^{\prime}}\limits^{ - } \) in CLUBB, are closed by a downgradient scheme, as shown in Eqs. 2 and 3 (Golaz et al. 2002).

$$ \overline{{u^{\prime}w^{\prime}}} = - Ck_{{10}} \cdot Ck~L~\sqrt {\bar{e}} ~\frac{{\partial \bar{u}}}{{\partial z}} $$
(2)
$$ \overline{{v^{\prime}w^{\prime}}} = - Ck_{{10}} \cdot Ck{\text{~}}L{\text{~}}\sqrt {\bar{e}} {\text{~}}\frac{{\partial \bar{v}}}{{\partial z}} $$
(3)

where, \(\overline{u }\) and \(\overline{v }\) are the horizontal winds, z is the altitude, \(e=\frac{1}{2}(\overline{{u }^{\mathrm{^{\prime}}2}}+\overline{{v }^{\mathrm{^{\prime}}2}}+\overline{{w }^{\mathrm{^{\prime}}2}})\) is the TKE and L is the turbulent mixing length scale. \(Ck\) and \(C{k}_{10}\) (i.e., c_k10 in EAMv1) are the coefficients of eddy diffusivities.

The momentum fluxes rely on the vertical gradient of horizontal winds (\(\frac{\partial \overline{v}}{\partial z }\) and \(\frac{\partial \overline{u}}{\partial z }\)) as well as the turbulent mixing length scale (or the time scale). Over the storm track regions, strong westerlies increase with height and naturally produce negative momentum flux (i.e., flux from the free atmosphere to the PBL). Meanwhile, the turbulent mixing scales are usually small at the PBL top and the surface, generating strong negative vertical gradients in momentum fluxes and subsequently stronger surface wind at the grid-scale. Moreover, the stronger negative momentum flux causes stronger surface TKE through the turbulence productions of \(\overline{{u }^{\mathrm{^{\prime}}2}} \, {\rm and} \,\overline{{v }^{\mathrm{^{\prime}}2}}\) that can exceed \(\overline{{w }^{\mathrm{^{\prime}}2}}\) near the surface. Thus, along the storm track cloud amount and SWCF increase with \(C{k}_{10}\).

3.4 Parameter sensitivities evolve with forecast lead time

The analyses in the previous subsections are entirely based on the day-5 mean of the PPE simulations. One may still wonder how the responses evolve with time given the dependence on cloud regimes and lower boundary conditions. Figure 9 shows the evolution of the parameter sensitivities with prediction length from day 1 to 5 for SWCF, LWCF, high-cloud fraction and low-cloud fraction over MBL cloud regimes near California coast. Parameters c1, c8 and gamma are three dominant ones for SWCF, LWCF and low cloud, and their relative importance are very stable with time from day 1 to 5, especially for SWCF and low cloud. For high cloud, the parameter accounting for maximum relative humidity threshold for ice cloud is the dominant one contributing the most to the total variance, especially during the first few days, but its contribution is unstable and decays quickly. High cloud amount is very small over the stratocumulus cloud regime.

Fig. 9
figure 9

Parameter sensitivities evolve with prediction length over the stratocumulus cloud regime near west coast of California

Full size image

Figure 10 shows the response curves of SWCF to a few selected key parameters over the six regions. The response of SWCF evolves steadily with prediction length, indicating a stable and solid impact of those parameters on SWCF. Parametric sensitivity evolving with prediction length varies in different cloud regimes and, in general, it’s more stable in the stratocumulus than the cumulus cloud regime.

Fig. 10
figure 10

Response of SWCF to parameters evolving with time over different regions, see Fig. 5

Full size image

4 Summary and discussion

In this study, using a parametric sensitivity analysis approach applied to an EAMv1 PPE, we identify parameters that cause largest sensitivities for different regions or cloud regimes and compare the model response to the perturbation of parameters for several important fidelity metrics of clouds over different regimes. Here is a brief summary of our findings: (1) Parametric sensitivity and relative importance vary with region, underlying surface type and cloud regime. (2) Cloud forcing has opposite response to some convective parameters over mid-latitude vs. tropical land through competing thermodynamic processes. (3) The sensitivity of cloud forcing to key parameters of the CLUBB shallow convection and turbulence scheme shows a large spatial variability in the subtropical eastern Pacific, in association with the second and third moments of vertical velocity. (4) Parametric sensitivity evolving with prediction length varies in different cloud regimes (e.g., more stable in stratocumulus than cumulus cloud regime). These findings improve our process-level understanding of the subgrid parameterizations for cloud, turbulence, and convection used in EAM, and provides insights into developing regime-aware parameterizations.

This study mainly focuses on the effects of atmospheric fast processes on cloud simulations with short-term PPE experiments. Note that SST is prescribed while land surface properties are allowed to interact with atmosphere in our simulations, which might lead to different responses of model behaviors over land and ocean. We believe that this impact should be small for our analysis because the evolution of SST is slow, but it can be important in long-term simulations.

Our results regarding the spatial variations of parametric sensitivity suggest that when tuning to improve global model performance, it is important to make sure that the tuning not only improves global or overall statistics but also maintains internal consistency in terms of model physics for each parameterization module in each cloud regime. In other words, over-tuning the model for improving overall statistics might produce unrealistic parameterization behavior. For example, the stronger w skewness in the cloud-topped boundary layer in the marine stratocumulus region than that in the transitional region further offshore shown in Fig. 8 hints at an over-tuning aimed at maintaining low cloud cover offshore. This kind of over-tuning sometimes betrays conceptual design of the parameterization.

Our results show that parameter perturbations affect cloud simulations via thermodynamical interactions among multiple processes, such as shallow convection, deep convection, and stratiform cloud microphysics, suggesting that it is important to develop a unified cloud parameterization or at least to include the physical interactions among these processes in separate cloud parameterizations for better cloud simulations in global climate models (Yang et al. 2021, 2022). Meanwhile, cloud systems over different regimes can interact with each other via atmospheric circulation, suggesting that at longer time scales, cloud response in one region might be affected by parameters in other regions (Liu et al. 2022), which warrants further investigations in the future.

Our analysis presented in this paper reveals that, over the 5-day period of the PPE simulations, the model responses related to low-level clouds and SWCF are rather stable (i.e., change little with simulation time), while the model responses related to high-level clouds and LWCF change more significantly after day 3 (Fig. 9), possibly due to the gradual loss of predictability of deep convection activities and the spin-up of slower processes. Ideally, it would have been useful to extend the length of our simulations or increase the number of 5-day simulations for each of the 256 parameter sets, and then further evaluate the parameter sensitivities related to deep convection and the dependence of the conclusions on the simulation and analysis strategy. It is worth noting that our 256 × 12 PPE simulations were conducted during the development of EAMv1. Despite each simulation being only 5 days in length, the PPE experiment was expensive in terms of both computing time and storage needs. The high cost and the status of the v1 model development made it impractical to extend each run or conduct more simulations at the time. More recently, however, the climate modeling community has been showing a new surge of interest in PPE simulations and the available computing resources have increased substantially. It will be useful to conduct some follow-up studies to further assess the robustness of parameter sensitivities related to cumulus convection and explore computationally efficient ways to quantify such sensitivities.