1. Introduction
Most climate models have long-lasting biases in the central United States (CUS); their simulated summer surface conditions are significantly warmer and drier than observations (Klein et al. 2006; Cheruy et al. 2014; Lin et al. 2017; Zhang et al. 2018; Ma et al. 2018; Morcrette et al. 2018; Van Weverberg et al. 2018). Given nonlinear interactions and feedbacks in the Earth system, these biases diminish the reliability of climate predictions and scenario projections (Liang et al. 2008; Lin et al. 2017; Palmer and Stevens 2019; Jiang et al. 2021). Consequently, the warm and dry biases disrupt the nonlinear relationships among surface heat fluxes and water storage and thus reduce the accuracy in predicting the critical feedback processes (Rupp et al. 2017). They decrease model ability in predicting extreme heat or rainfall events and severe weather outbreaks (Kunkel et al. 2010; Pendergrass et al. 2020; Sun and Liang 2020a), which have profound socioeconomic impacts (Smith and Matthews 2015; NOAA 2021; Kotz et al. 2022; Liang 2022), including extensive damages on agricultural productivity (Liang et al. 2017; Y. Li et al. 2019; Ortiz-Bobea et al. 2021). Therefore, they deteriorate the prediction reliability on climate–crop interactions in the CUS—the heartland of U.S. agriculture with summer as the primary growing season (Mueller et al. 2016). To boost the confident use of climate predictions or projections in decision-making, significant model improvements must be made to correct these essential biases.
The modeling community has made tremendous efforts to reduce these biases. Among these are various empirical bias-correction techniques (Bellprat et al. 2016; Ardilouze et al. 2019; Chang et al. 2019) to provide improved climate data necessary for impact assessments. Such posterior corrections, while useful, do not actually improve the modeling system representation of nonlinear physical relationships between atmosphere and land processes, rendering model predictions hard to explain, and thus may mislead decision-making that requires integrated knowledge (Ehret et al. 2012). Given that the failure to capture convective scale processes may cause the warm and dry biases (Liang et al. 2007; Lin et al. 2017; Sun and Liang 2020a), many efforts turn to high-resolution modeling to replace physics parameterizations. Liu et al. (2017) demonstrated that even using a grid spacing of 4 km (convection-permitting), severe biases (up to 6°C) in the CUS still exist. S. Li et al. (2019) tuned model parameters through a multiphase refinement approach, which reduced the warm biases to about 3°C. Cheruy et al. (2020) improved atmospheric and land surface physics, whereas more than 2°C warm biases remain. Barlage et al. (2021) incorporated groundwater processes in a convection-permitting model to reduce the warm biases from 5°–6°C to still 2°–3°C. Sun and Liang (2020b) attempted to infer the physical causes of the dry biases through the structural equation model analysis. Morcrette et al. (2018), Van Weverberg et al. (2018), Ma et al. (2018), and Zhang et al. (2018) established a special project named CAUSES (Clouds Above the United States and Errors at the Surface) to understand the physical mechanisms leading to CUS warm biases based on surface radiation budget decomposition analysis. They found that surface evaporative fraction underestimation and solar radiation overestimation were the two leading contributors to the warm biases. However, all previous studies have not fully disentangled the complex feedback mechanisms among radiation, convection, precipitation, and surface processes that result in the coupled warm and dry biases. These endeavors revealed the challenges in resolving the CUS warm and dry biases.
Our companion paper Sun and Liang (2022, hereafter S&L) took a solid step to understand and reduce CUS warm and dry summer biases in the regional Climate–Weather Research and Forecasting (CWRF) Model (Liang et al. 2012) by improving its ensemble cumulus parameterization (ECP; Qiao and Liang 2015, 2016, 2017). The improvements included a trigger for mesoscale convective systems to occur in unfavorable environmental conditions, a constraint to suppress convections from shallow boundary layers, and a cloud-to-rainwater conversion to favor anvil formation. In addition, the trigger defines the cloud base at the lifting condensation level (LCL) rather than the level of free convection (LFC), which lowered the cloud base and increased the cloud depth. Together, the new ECP produced more low- and high-level clouds to reduce surface shortwave radiation and increase outgoing longwave radiation, and hence eliminated the warm and dry biases in a physically consistent manner. S&L demonstrated the importance of identifying the physical processes and understanding the underlying mechanisms in order to actually reduce or eliminate such biases in a specific model.
The CAUSES project has been working on three competing hypotheses of the key causes for the CUS warm biases: underestimation of precipitation (Lin et al. 2017), cloud (Cheruy et al. 2014), and evapotranspiration (Mueller and Seneviratne 2014). To this end, S&L explained how improving cumulus parameterization can increase both low-level cloud and precipitation amounts to consistently eliminate the warm and dry biases with a realistic atmospheric energy balance. We did that from the perspective of atmospheric forcing since all changes started from altering cumulus parameterization that led to surface responses. As such, we did not address the specific role of evapotranspiration in the loop, although its interannual differences between the new and original ECP contained about 20%–70% (depending on regions) of total variance in surface air temperature and precipitation biases. Presumably, there exist strong land–atmosphere coupling mechanisms that link underestimations of low cloud, precipitation, and evapotranspiration to the warm and dry biases. However, such links are complex, strongly depending on climate regimes and model configurations, and it is particularly challenging to determine which mechanism dominates and whether the atmosphere or surface controls the pathways (Findell and Eltahir 2003a,b; Koster et al. 2004; Ferguson and Wood 2011; Taylor et al. 2012; Santanello et al. 2018; Wei and Dirmeyer 2012; Zhou et al. 2019).
Land–atmosphere interactions involve soil moisture and temperature, evapotranspiration and sensible heat, solar absorption and infrared emission, and cloud, convection, and precipitation through various positive and negative (local or nonlocal) feedbacks (Santanello et al. 2018). Numerous studies offered diagnostic methods to dissect soil moisture–precipitation coupling mechanisms containing all above variables and processes (Brubaker and Entekhabi 1995, 1996; Schär et al. 1999; Salvucci et al. 2002; Findell and Eltahir 2003a; Dirmeyer and Brubaker 2007; Wei and Dirmeyer 2012; Wulfmeyer et al. 2018; Hu et al. 2021). Most of these methods focused on analyzing the feedbacks in individual models, which may reach varying conclusions specific to land component deficiencies or different parameterizations used in the coupled land–atmosphere system (Williams et al. 2016; Wei et al. 2021). They may not be as effective for application to identify the key mechanisms that govern the spread of biases in all related variables among a large suite of models. They also require high-frequency data that are typically not available from all models in comparison.
This study aims to identify the model deficiencies that may likely explain common CUS warm and dry biases among recent climate models and develop analytical modeling to quantify the major contributing factors and associated feedbacks. The identification and attribution were approached from the dominant behavior of a large multimodel ensemble rather than a single model like CWRF. However, the rigorous process understanding based on CWRF sensitivity experiments in S&L offered a unique guidance for the ensemble quantification. Section 2 describes the ensemble simulations from the Coupled Model Intercomparison Project phase 5 (CMIP5) and phase 6 (CMIP6) and the High-Resolution Model Intercomparison Project (HighResMIP, including only models with grid spacing < 30 km) as well as observational data used. Section 3 presents the warm and dry biases common to these climate models and conducts composite, decomposition, and regression analyses to distinguish their strong sensitivity to cumulus parameterization and statistical relationships among key variables. In particular, the analyses confirmed that defining the cloud base at LCL rather than LFC is a major contributor to reducing the biases. Section 4 develops the analytical bias model (ABM) to understand the responsible physical processes and quantify the key feedback mechanisms. Section 5 conducts counterfactual experiments with ABM to disentangle the feedback loops and identify the key mechanisms that dominate the systematic biases and spreads among the latest CMIP6 plus HighResMIP models. Especially, we found thresholds for varying evapotranspiration–precipitation feedbacks to occur and thus cause systematic warm and dry biases in CUS. Section 6 gives the conclusions.
2. CMIP simulations and observational data
This study analyzed all present-day simulations currently available from 61 CMIP6 and 11 HighResMIP models, where different resolutions of a same model are counted as different models (see Tables S1 and S2 in the online supplemental material for model resolution and data availability). Most models have multiple simulations (or variants) that were realized with small perturbations in initial states and physical parameters. In total, there are 592 simulations, with the actual number of variants available ranging from 592 to 402 among the 15 variables examined (Table 1). The analyses below were mainly based on the ensemble mean of all available variants for a specific variable in each model together with these individual variants depicting the uncertainty. To identify feedbacks and mechanisms in which the physical consistency is essential, we chose the first member (rather than ensemble mean) of each model with the variant label “r1i1p1f1” (with only a few exceptions due to naming differences), where the letters r, i, p, and f denote respectively the realization, initialization, physics, and forcing indices of the simulation. We also evaluated 49 CMIP5 models with a total of 212 variants (Table S3) to see if any reductions in the warm and dry biases were made over the years. For all these model simulations, our analyses focused on summer (June–August) averages during the common period from 1 January 1980 through 31 December 2014.
Observations, resolutions, available years, and their references.
Table 1 summarizes the observational data used in this study as the reference to calculate model biases. All data were available at varying temporal and spatial resolutions and mapped onto CWRF 30-km grid by conservative or linear interpolation.
3. Common climate model biases and sensitivity to cumulus parameterization
Figure 1 compares the geographic distributions of summer temperature and precipitation biases from observations over the contiguous United States among the models separately in the CMIP5, CMIP6, and HighResMIP-high (grid spacing < 30 km) ensembles. To avoid the result statistics being dominated by models with large realizations, we first calculated the mean of all the variants from each model and then averaged the mean biases over all models in an ensemble. The CMIP5 ensemble showed average warm biases of 1°–3°C in CUS. The CMIP6 ensemble produced similar warm biases but in more extensive areas, expanded from CUS into the central-southern Great Plains. The HighResMIP-high ensemble simulated even larger warm biases over broader areas, with centers of greater than 3°C. Correspondingly, all ensembles underestimated precipitation in CUS, with the magnitudes and coverages of the dry biases (over 0.5 mm day−1) both increased from CMIP5 to CMIP6 to HighResMIP-high.
Thus, CMIP6 still simulated large warm and dry biases in CUS over broader areas than CMIP5, albeit incorporating more advanced physical parameterizations and higher resolutions (Eyring et al. 2019). Even HighResMIP-high with grid sizes reaching 25 km could not resolve the warm and dry bias (see Fig. S1 in the supplemental material for a more balanced comparison between high- versus low-resolution simulations from the same models). Simply increasing model resolution cannot eliminate the biases, while refining physics representation such as for mesoscale convective systems is essential (S&L). CMIP5 had other biases, including wet biases in the northern states between the Rocky Mountains and Great Plains and along the Appalachian Mountains as well as dry biases in the coastal areas along the Gulf of Mexico. These precipitation biases exhibited no apparent link with those of temperature and were reduced in CMIP6 and more significantly diminished in HighResMIP-high. This reduction could partly result from model resolution increase, a topic warranting a separate study.
S&L clearly demonstrated the importance of cumulus parameterization to the CWRF summer warm and dry biases in CUS among other major regions. Given the corresponding physical mechanisms identified earlier, here we focused on the key characteristic differences in cumulus parameterization that may explain the warm and dry biases common to current global climate models. One of key findings in S&L was that the cloud base definition in cumulus schemes plays a critical role. Below we separated all 72 models from CMIP6 plus HighResMIP (hereafter referred to as CMIP6X for brevity) into nine groups by the cloud base definition in their cumulus scheme as summarized in Table S4, which includes the respective abbreviation, main features, and references. The models were first categorized by cumulus schemes, including BMJ (Betts–Miller–Janjić; Betts and Miller 1986), AS (Arakawa and Schubert 1974), GR (Gregory and Rowntree 1990), PCMT (Prognostic Condensates Microphysics and Transport; Lopez 2002; Piriou et al. 2007; Geleyn et al. 2008; Guérémy 2011), LMDZ (Hourdin et al. 2020), ZM (Zhang and McFarlane 1995), GY (Del Genio and Yao 1993), and TDK (Tiedtke 1989). Since BMJ does not use cloud mass flux explicitly, the first-order relationships among downdraft, cloud depth, and others are irrelevant. AS and GR define the cloud base at LFC, PCMT at LCL, and LMDZ at 40 hPa above LCL, while ZM and GY define it at the particle lifting level (PLL). In contrast, TDK adopts either the LCL or LFC definition and hence is subdivided into TDK_LCL or TDK_LFC.
Figure 2 compares the ensemble mean temperature biases among the nine groups. Each group contains all the models adopting a same cumulus scheme, and each model is also shown with its own multirealization mean biases averaged in CUS. In general, the groups defining the cloud base at LCL (TDK, PCMT) simulated smaller warm biases than LFC (TDK, AS, GR). The most direct comparison was made with TDK, in which using LCL rather than LFC produced systematically cooler air in most of the eastern United States, essentially eliminating the large warm biases in CUS. This TDK_LCL resembled our new ECP in CWRF improved by S&L as they used the same cumulus trigger. In contrast, LMDZ still had warm biases in CUS and the Northeast, likely because it elevated the cloud base by 40 hPa above LCL. On the other hand, ZM estimated PLL as the maximum moist static energy level. Since this PLL is usually lower than LCL (Wu 2012), a parcel rising from PLL to LCL may experience negative buoyancy (Zhang and McFarlane 1991), reducing the total cloud-work function. The net effect of defining the cloud base at PLL is equivalent to reducing the effective cloud depth. This may partially explain why ZM using PLL generated large warm biases like other schemes using LFC. A seeming contradiction appeared with GY, which also used PLL but yielded cold biases (see below for further discussion).
Figure 3 compares the ensemble mean precipitation biases among the nine groups. Clearly, TDK using LCL (rather than LFC) largely reduced the dry biases in CUS. Meanwhile, the cumulus schemes using LFC (TDK, GR) and PLL (ZM) simulated notable dry biases. These were consistent with the reduction of the warm biases from LCL versus LFC. On the other hand, the dry biases in CUS (excluding the southern region) were not evident in AS using LFC but obvious in PCMT and LMDZ using LCL. In addition, GY even overestimated precipitation, accompanying a cold bias. This could result from the precipitable water increase by allowing convective rain evaporation above the cloud base (Kim et al. 2012), which was deactivated in the latest model (Rind et al. 2020). It is difficult to interpret these mixed results since no clean experiment was available for these cumulus schemes directly separating LCL versus LFC as in TDK. Nevertheless, the choice of the cumulus parameterization scheme plays a significant, if not dominant, role in the contrast in precipitation biases among the models. While parameterized (convective) and resolved (large-scale) precipitations affect cloud–radiation interactions very differently (Lin et al. 2013), their relative contributions to model biases or even extreme events are not fully understood (Liang et al. 2019; Sun and Liang 2020b). However, our analysis of convective precipitation (Fig. S2) and its ratio to total amount (Fig. S3) showed no significant correspondence with CUS warm and dry biases (Fig. S4).
Figure 4 compares CUS regional mean temperature, precipitation, total cloud cover, and precipitable water biases among 72 CMIP6X models as grouped by cumulus schemes and cloud base definitions. In general, the models using the cumulus schemes defining the cloud base at LCL (PMCT, TDK, LMDZ) simulated the least dry and warm biases associated with more cloud cover and less precipitable water, while those defining at LFC (AS, TDK, GR) or PLL (GY, ZM) produced larger biases with less cloud and more precipitable water (see Figs. S5 and S6 for their spatial distributions). Since high- and low-level clouds have generally opposite radiative warm and cooling effects (partially explained LMDZ’s total cloud underestimate), a more physically consistent picture would be drawn if cloud data at different levels were available for direct comparison. As discussed earlier, GY_PLL is an exception, having cold and wet biases. On the other hand, BMJ behaved differently from all others, producing cold but dry biases, indicating inconsistent physics representation (see below for further discussion). TDK_LFC had the largest bias spread; a closer scrutiny (Table S5) showed that the outliers (with largest high warm and dry biases) were from two high-resolution simulations (MPI-ESM1-2-XR and INM-CM-5).
Figure 5 compares CUS regional mean biases in temperature, precipitation, cloud albedo [shortwave reflection; see the supplemental material for calculation following Betts (2007)], and surface energy fluxes among 72 CMIP6X models as grouped by cumulus cloud base definitions, which include LCL (15), LFC (26), PLL (27), and BMJ (4). Apparently, BMJ is an outlier with large cold but relatively small dry biases (−0.6°C, −0.3 mm day−1). It substantially overestimated both shortwave surface reflection and cloud attenuation (17.6 and 5.7 W m−2), causing large underestimates of surface net shortwave and hence total (shortwave plus longwave) radiation fluxes (−10.6 and −7.0 W m−2). Corresponding to its cold bias, BMJ simulated surface net (shortwave, longwave, total) radiation and sensible heat biases totally opposite to those by other cloud base definitions, indicating inconsistent treatment for surface reflection and cloud effect.
Clearly, LCL had the smallest temperature and precipitation biases (0.2°C, −0.3 mm day−1), whereas LFC yielded much larger biases (2.3°C, −0.6 mm day−1). Consistently, LCL simulated the most realistic surface shortwave cloud radiative effect, net shortwave, longwave, and total radiation, and sensible heat with small biases (−0.4, 7.4, −3.0, 4.4, and 4.0 W m−2). As demonstrated in S&L, LCL reduced excessive convective mass flux and thus cumulus drying and warming so to generate sufficient low-level clouds and attenuate more downwelling shortwave radiation. On the other hand, LFC largely underestimated cloud amount so to systematically overestimate cloud radiative effect and surface fluxes (17.3, 16.4, −8.3, 8.1, and 16.9 W m−2), all of which were much larger in magnitude than the corresponding values of LCL. PLL resembled LFC with not only close temperature and precipitation biases (2.2°C, −0.7 mm day−1) but also similar biases in respective cloud radiative effect and surface fluxes (6.1, 13.8, −7.3, 6.5, and 15.2 W m−2). The main PLL difference from LFC was in reducing the cloud radiative effect by 65%. LCL also simulated a much smaller surface latent heat bias than LFC and PLL (−10.4 vs −18.8 and −19.6 W m−2). This was well reflected in the contrast among their precipitation biases.
In summary, the CUS summer warm and dry biases and their spread among the latest CMIP6X models strongly depend on cumulus parameterization, in which the cloud base definition is a critical factor separating systematic model differences. In particular, defining the cumulus cloud base at LCL results in the least warm and dry biases with more realistic surface energy partitioning among sensible and latent heat, and radiation components.
Figure 6 shows the statistical relationships among CUS summer biases in cloud albedo, surface radiation fluxes, and surface heat fluxes simulated by 72 CMIP6X models. For each model and each field, all its variants were used to calculate the mean and standard deviation, with the latter depicting uncertainty. Linear regressions were then made on these means averaged in CUS among these fields, with the respective one deviation depicting their uncertainty range. For the surface energy budget averaged over all models, biases (in W m−2) in radiation components were the largest for SWd (23.9), reduced by about half for SWu (11.8) and LWu (10.5), and the smallest for LWd (4.3). The net total radiation (Rn) was overestimated by 5.9 W m−2, which was compensated by smaller LH (−16.6) and larger SH (12.1). This resulted in a net energy surplus of 10.4 W m−2, which was balanced by larger heat flux into the ground.
For the two radiation components directly affected by the atmosphere and as the energy inputs to the surface, the variance explained is 89% for SWd by cloud albedo alone and 64% for LWd by precipitable water. While the two inputs are independent (with 0 covariance), their partitioning into other surface energy components is our main concern. The variance percentage explained for SWu, LWu, SH, and LH was respectively 22, 37, 27, and 4 by SWd and 2, 46, 16, and 8 by LWd. It is surprising to notice that biases in LH (hence evapotranspiration or ET) and SWd had almost no correspondence among the models. Little correspondence was seen between SWu and LWd biases. The model spread in SWd biases was distributed mostly into those in LWu, SH, and SWu, while the LWd spread was partitioned mainly among those in LWu, SH, and LH. It is important to note that the spread in net radiation gain (Rn) explained 95% variance of moist enthalpy flux (SH + LH), whereas the residual (ground heat flux) barely contributed any to temperature (0.07) or precipitation bias (0.05). The sum of LWd and SH explained 83% variance of LWu, much more than each individually (46% and 69%; not shown).
Figure 7 illustrates the statistical relationships among CUS summer biases in precipitation and its decomposed diabatic cooling components simulated by the 72 models. The model spread in precipitation biases was mainly determined by those in SH and LWu, which explained respectively 64% and 55% variance. The remaining radiation components, LWd, SWu, SWd, TOA shortwave reflection (
4. Developing the analytical bias model (ABM) to explain CMIP ensemble errors
The results presented in section 3 with Figs. 1–7 were all based on statistical analyses of CMIP model biases, showing a strong sensitivity to cumulus parameterization and significant relationships among key variables. These statistics cannot reveal nonlinear feedbacks essential to the coupled climate system nor reflect the causality of the actual physical system (Pearl 2009). Therefore, below we developed the ABM that captures the physical mechanisms underlying the CUS warm and dry biases. The LCL-based schemes simulated more low-level cloud and less total precipitable water, which reduced net shortwave and longwave downwelling at the surface (Figs. 4–6; S&L). Thus, the purpose of the ABM is to link the coupled (T2, PR) biases to these surface radiation (LWd, SWn) differences. Our focus is on the key mechanisms and feedbacks that dominate the systematic biases and spreads among the latest CMIP6X models. Table S4 summarizes the important notations used in this study. For radiation fluxes, unless specifically denoted with a superscript TOA for the top of atmosphere, they are all at the surface, where the subscripts d, u, and n represent downwelling, upwelling, and net (d minus u) fluxes, respectively. Table 2 lists the key parameters used in the ABM, where the subscript 0 depicts a prescribed value, and the overbar denoted CUS summer averaging over all analysis years (35) and across all CMIP6X models (72).
Model parameters and their reference values.
5. ABM simulations and stability analyses for CMIP6X bias mechanisms
Figure 9 compares the (T2, PR) biases simulated by the ABM and CMIP6X models. The ABM can explain 82% temperature and 81% precipitation variance of the biases among all CMIP6X models. These percentages or coefficients of determination (R2) were based on the one-to-one relationship between the ABM and CMIP6X biases, rather than the linear regression fitting them. This avoids overlooking the confidence of the result with a high correlation but incorrect amplitude (Legates and McCabe 1999). The regression (slope, intercept) of CMIP6X with ABM are (0.81° ± 0.04°, 0.32° ± 0.10°C) for temperature and (1.19 ± 0.04, −0.12 ± 0.02 mm day−1 for precipitation. As compared with the existing CAUSES outcome (R2 = 0.61), our result represents a significant improvement for both temperature and precipitation, having higher R2 scores, slopes closer to 1, and intercepts closer to 0. The first two measures indicate that the ABM well captures the spread of biases among all CMIP6X models, while the third measure depicts a high certainty in estimating the system sensitivity as derived in Eq. (S8). Therefore, the ABM can represent the principal relationships among surface climate variables in summer CUS, rendering a significant confidence for application to study the physical mechanisms of biases.
Below we conducted ABM counterfactual experiments to identify the key mechanisms for the warm and dry bias. In each experiment, for a specific input factor (LWd, SWn, PRs), we replaced its values from all individual CMIP6X models with the mean of the subset models using the LCL-based cumulus parameterizations to quantify its contribution to the bias (see the supplemental material for calculation details). This experimental approach enables us to consider interactions among feedback processes in the entire system, which is critical but often overlooked in previous studies on bias mechanisms (Stephens 2005). Additional experiments were conducted to examine the ABM result sensitivity to the prescribed key parameters (γeb0, ωap0). This approach may not test uncertainties associated with all physical assumptions and numerical approximations used in building the ABM. However, our focus is to seek a useful alternative physical proxy that enables a better understanding of the complex nonlinear feedbacks underlying the fully coupled CMIP6X simulations and hence offers an explicit explanation for the spread of biases among models. This alternative approach is recommended for feedback analysis (Stephens 2005).
Figure 10 compares the attributions to the CUS summer warm and dry bias. When adopting a common LWd value as the mean of the subset models using the LCL-based parameterization, other cumulus schemes (LFC, PLL, BMJ) would significantly reduce the warm bias by (1.7°, 2.1°, 0.7°C) and increase precipitation by (0.6, 0.6, 0.4 mm day−1). The BMJ models contain significantly larger uncertainty than other groups. Correspondingly, when adopting a common SWn value from the mean of the LCL models, other cumulus schemes would reduce the warm bias by 0.7°, 0.7°, and 0.0°C and increase precipitation by 0.3, 0.3, and 0.4 mm day−1, respectively. These warm and dry bias reductions induced by SWn are much smaller than those by LWd, with the former accounting for only 20%–50% of the latter. See appendix for the explanation why CUS summer is more sensitivity to LWd. The uncertainties in estimating the contributions decrease from LWd to SWn. In addition, when adopting a common PRs value from the mean of the LCL models, other cumulus schemes would reduce the warm bias by 0.4°, 0.5°, and 0.1°C and increase precipitation by 0.4, 0.4, and 0.1 mm day−1, respectively. These bias reductions are comparable with those induced by SWn, although the uncertainties of the estimation increase.
The above ABM experiments allow us to attribute the major cause for the warm and dry bias to the excessive radiative energy reaching the surface together with insufficient rainfall. An important question is why most of the excessive energy flows into the sensible rather than latent heat and what mechanisms amplify initial perturbations toward significant biases—a potential runaway problem. To address this question, we conducted a feedback analysis following Roe (2009) and showed that the system bias behavior depends on the fb range (see appendix for derivations). The fb range can be divided by a threshold at 0.6, which corresponds approximately to a mean precipitation amount of 8.8 mm day−1. Since summer CUS mean precipitation is only 3.3 mm day−1 in observations and less than 3.9 mm day−1 in all CMIP6X models, the combined radiative forcing
6. Summary and conclusions
This study showed that substantial warm and dry biases persisted in summer CUS among most climate models despite remarkable improvements in the latest CMIP6 (Eyring et al. 2019) and even in high-resolution simulations. Meanwhile, S&L demonstrated that CWRF defining the cloud base at LCL in cumulus parameterization can significantly reduce the biases. Following this idea, we grouped all 72 CMIP6X models by four cloud base definitions in their cumulus schemes. Our composite analysis indicated that the models using the LCL-based schemes simulated systematically smaller warm and dry biases than others. A more comprehensive analysis of precipitation decompositions, radiative fluxes, and surface budgets confirmed that the cloud base definition is the dominant factor determining the spread of the biases among the CMIP6X models and those using the LCL definition performed the best.
The statistical analyses cannot determine the causality or physical mechanisms underlying the relationships among model biases. To identify these feedback mechanisms and quantify the key contributions to the spread of the biases among the CMIP6X models, we developed a physically based ABM that captures the principal energy balances of the coupled surface–atmosphere system. The ABM consists of three major physical parameterizations that link 1) precipitation with surface net radiation minus sensible heat flux and ground temperature departure from a reference state; 2) surface latent heat flux (and hence evapotranspiration) with precipitation in proportion to a generalized Budyko curve of an effective aridity index; and 3) surface net longwave radiation with a negative proportion of sensible heat flux. With these new parameterizations, we introduced two key dimensionless numbers, γeb and ωpe, to respectively characterize the land–atmosphere coupling strength through energy and water exchanges. The ABM solves recursively the coupled (T2, PR) responses to the driving biases and spreads of (LWd, SWn, PRs) among the CMIP6X models. This is analogous to the surface forcing and response framework (Andrews et al. 2009), the ABM enables us to gain physical insight into the bias causes. Using the prescribed parameters from observational data or multimodel ensemble means, the ABM has significant explanatory power for CUS summer warm and dry biases, capturing 80% variance of temperature and precipitation biases among all CMIP6X models.
We conducted ABM counterfactual experiments to attribute the key factors for the warm and dry bias and found that the LCL cumulus parameterization reduces the warm and dry bias through two principal mechanisms. First, it produces more low clouds and smaller total precipitable water, which reduce respectively the downwelling (and net) shortwave and longwave radiation reaching the surface and thus provide less energy available for surface heating and evapotranspiration, causing a cooler and wetter soil. Second, it produces more rainfall and wetter soil conditions, which prevent the land surface runaway toward the hot-drought state as driven by the strong positive ET–PR feedback and hence eliminate the warm and dry bias. Further analysis revealed that, under relative dry conditions in most CMIP6X models, primary surface downwelling longwave errors combined with secondary net shortwave flux errors drive the climate system into the runaway stage, causing a lock in a dry and warm bias loop.
This study offered a systematic approach to determine common deficiencies, identify sensitive factors, quantify relative contributions, and understand dominant mechanisms for CUS summer warm and dry biases and their spreads across the CMIP climate models. The developed ABM, with large explanatory power for CUS summer biases, can be applied to other regions with similar climate characteristics. This analytical modeling and physical understanding laid a solid foundation for not only climate model improvement but also more interpretable and reliable climate prediction in CUS. Our ABM analysis supports the projected coexistence of severe heat stress with more devastating drought (Zhou et al. 2019; Ting et al. 2021) and the observational interpretation of the potential feedback processes in CUS summer (Taylor et al. 2012). It also explains how underestimation of precipitation (Lin et al. 2017), cloud (Cheruy et al. 2014), and evapotranspiration (Mueller and Seneviratne 2014) can jointly cause the warm and dry bias and quantifies their relative contributions. Our ABM analysis indicated that CUS summer warm and dry biases among CMIP6X models are attributed mostly by surface downwelling longwave radiation errors and second by surface net shortwave radiation errors, with the former 2–5 times larger. These two errors as weighted by their relative contributions form an effective radiative forcing to induce runaway temperature and precipitation feedbacks, which collaborate to cause CUS summer warm and dry biases. Note that the ABM circumvents explicit representation of the large uncertainty associated with clouds by taking input of surface radiation fluxes from CMIP6X models. This implicit treatment could underestimate the buffering effect from cloud-induced negative feedbacks (Stephens and Webster 1981). The consequence of this limitation warrants further investigation.
Acknowledgments.
The research was supported by the U.S. National Science Foundation Innovations at the Nexus of Food, Energy and Water Systems under Grant EAR1639327 and the U.S. Department of Agriculture–National Institute of Food and Agriculture under Grant 20206801231674 for developing the Dashboard for Agricultural Water use and Nutrient management at the University of Maryland and Grant 2015-34263-24070 for the UV-B Monitoring and Research Program at Colorado State University. Additional support came from the U.S. Department of Commerce Educational Partnership Program–National Oceanic and Atmospheric Administration Center for Atmosphere and Meteorology (NCAS-M) under Grant NA16SEC4810006. The simulations and analyses were conducted on supercomputers supported by the National Center for Atmospheric Research Computational and Information Systems Lab, the Maryland Advanced Research Computing Center, and the Atmospheric River Program funded by the California Department of Water Resources and the Forecast Informed Reservoir Operations Program funded by the U.S. Army Corps of Engineers Engineer Research and Development Center. We acknowledge the World Climate Research Programme with its Working Group on Coupled Modeling for coordinating the CMIP5/6 efforts. We thank all the climate modeling groups for producing and making available their CMIP5/6 outputs, the Earth System Grid Federation (ESGF) for archiving this data and providing access, and the multiple funding agencies who support CMIP5/6 and ESGF.
Data availability statement.
Data analyzed in this study were a re-analysis of existing data, which are openly available at locations cited in the reference section. Further documentation about data processing is available at the supplemental material.
APPENDIX
For precipitation it is a positive feedback when fb0 < n−1 ≈ 0.6. Given
REFERENCES
Andrews, T., P. M. Forster, and J. M. Gregory, 2009: A surface energy perspective on climate change. J. Climate, 22, 2557–2570, https://doi.org/10.1175/2008JCLI2759.1.
Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.
Ardilouze, C., L. Batté, B. Decharme, and M. Déqué, 2019: On the link between summer dry bias over the U.S. Great Plains and seasonal temperature prediction skill in a dynamical forecast system. Wea. Forecasting, 34, 1161–1172, https://doi.org/10.1175/WAF-D-19-0023.1.
Barlage, M., F. Chen, R. Rasmussen, Z. Zhang, and G. Miguez-Macho, 2021: The importance of scale-dependent groundwater processes in land–atmosphere interactions over the central United States. Geophys. Res. Lett., 48, e2020GL092171, https://doi.org/10.1029/2020GL092171.
Bellprat, O., S. Kotlarski, D. Lüthi, R. De Elía, A. Frigon, R. Laprise, and C. Schär, 2016: Objective calibration of regional climate models: Application over Europe and North America. J. Climate, 29, 819–838, https://doi.org/10.1175/JCLI-D-15-0302.1.
Betts, A. K., 2000: Idealized model for equilibrium boundary layer over land. J. Hydrometeor., 1, 507–523, https://doi.org/10.1175/1525-7541(2000)001<0507:IMFEBL>2.0.CO;2.
Betts, A. K., and M. J. Miller, 1986: A new convective adjustment scheme. Part II: Single column tests using GATE wave, BOMEX, ATEX and Arctic air-mass data sets. Quart. J. Roy. Meteor. Soc., 112, 693–709, https://doi.org/10.1002/qj.49711247308.
Betts, A. K., 2007: Coupling of water vapor convergence, clouds, precipitation, and land–surface processes. J. Geophys. Res., 112, D10108, https://doi.org/10.1029/2006JD008191.
Bohren, C. F., and B. A. Albrecht, 2000: Atmospheric thermodynamics. Amer. J. Phys., 68, 1159, https://doi.org/10.1119/1.1313524.
Brubaker, K. L., and D. Entekhabi, 1995: An analytic approach to modeling land–atmosphere interaction: 1. Construct and equilibrium behavior. Water Resour. Res., 31, 619–632, https://doi.org/10.1029/94WR01772.
Brubaker, K. L., and D. Entekhabi, 1996: Analysis of feedback mechanisms in land–atmosphere interaction. Water Resour. Res., 32, 1343–1357, https://doi.org/10.1029/96WR00005.
Budyko, M. I., 1974: Climate and Life. Vol. 18. Academic Press, 508 pp.
Chang, Y., S. D. Schubert, R. D. Koster, A. M. Molod, and H. Wang, 2019: Tendency bias correction in coupled and uncoupled global climate models with a focus on impacts over North America. J. Climate, 32, 639–661, https://doi.org/10.1175/JCLI-D-18-0598.1.
Cheruy, F., J. L. Dufresne, F. Hourdin, and A. Ducharne, 2014: Role of clouds and land–atmosphere coupling in midlatitude continental summer warm biases and climate change amplification in CMIP5 simulations. Geophys. Res. Lett., 41, 6493–6500, https://doi.org/10.1002/2014GL061145.
Cheruy, F., and Coauthors, 2020: Improved near-surface continental climate in IPSL–CM6A–LR by combined evolutions of atmospheric and land surface physics. J. Adv. Model. Earth Syst. ,12, e2019MS002005, https://doi.org/10.1029/2019MS002005.
Coakley, J. A., Jr., and P. Yang, 2014: Atmospheric Radiation: A Primer with Illustrative Solutions. John Wiley & Sons, 256 pp.
DelGenio, A. D., and M. S. Yao, 1993: Efficient cumulus parameterization for long-term climate studies: The GISS scheme. The Representation of Cumulus Convection in Numerical Models, Meteor. Monogr., No. 46, Amer. Meteor. Soc., 181–184.
Dirmeyer, P. A., and K. L. Brubaker, 2007: Characterization of the global hydrologic cycle from a back-trajectory analysis of atmospheric water vapor. J. Hydrometeor., 8, 20–37, https://doi.org/10.1175/JHM557.1.
Dominguez, F., P. Kumar, X.-Z. Liang, and M. Ting, 2006: Impact of atmospheric moisture storage on precipitation recycling. J. Climate, 19, 1513–1530, https://doi.org/10.1175/JCLI3691.1.
Ehret, U., E. Zehe, V. Wulfmeyer, K. Warrach-Sagi, and J. Liebert, 2012: HESS opinions “Should we apply bias correction to global and regional climate model data?” Hydrol. Earth Syst. Sci., 16, 3391–3404, https://doi.org/10.5194/hess-16-3391-2012.
Eyring, V., and Coauthors, 2019: Taking climate model evaluation to the next level. Nat. Climate Change, 9, 102–110, https://doi.org/10.1038/s41558-018-0355-y.
Ferguson, C. R., and E. F. Wood, 2011: Observed land–atmosphere coupling from satellite remote sensing and reanalysis. J. Hydrometeor., 12, 1221–1254, https://doi.org/10.1175/2011JHM1380.1.
Findell, K. L., and E. A. B. Eltahir, 2003a: Atmospheric controls on soil moisture–boundary layer interactions. Part I: Framework development. J. Hydrometeor., 4, 552–569, https://doi.org/10.1175/1525-7541(2003)004<0552:ACOSML>2.0.CO;2.
Findell, K. L., and E. A. B. Eltahir, 2003b: Atmospheric controls on soil moisture–boundary layer interactions. Part II: Feedbacks within the continental United States. J. Hydrometeor., 4, 570–583, https://doi.org/10.1175/1525-7541(2003)004<0570:ACOSML>2.0.CO;2.
Geleyn, J.-F., B. Catry, Y. Bouteloup, and R. Brozkova, 2008: A statistical approach for sedimentation inside a microphysical precipitation scheme. Tellus, 60, 649–662, https://doi.org/10.1111/j.1600-0870.2007.00323.x.
Gentine, P., G.-J. Steeneveld, B. G. Heusinkveld, and A. A. M. Holtslag, 2018: Coupling between radiative flux divergence and turbulence near the surface. Quart. J. Roy. Meteor. Soc., 144, 2491–2507, https://doi.org/10.1002/qj.3333.
Gerrits, A. M. J., H. H. G. Savenije, E. J. M. Veling, and L. Pfister, 2009: Analytical derivation of the Budyko curve based on rainfall characteristics and a simple evaporation model. Water Resour. Res., 45, W04403, https://doi.org/10.1029/2008WR007308.
Gimeno, L., and Coauthors, 2016: Major mechanisms of atmospheric moisture transport and their role in extreme precipitation events. Annu. Rev. Environ. Resour., 41, 117–141, https://doi.org/10.1146/annurev-environ-110615-085558.
Gregory, D., and P. R. Rowntree, 1990: A mass flux convection scheme with representation of cloud ensemble characteristics and stability-dependent closure. Mon. Wea. Rev., 118, 1483–1506, https://doi.org/10.1175/1520-0493(1990)118%3c1483:AMFCSW%3e2.0.CO;2.
Guérémy, J. F., 2011: A continuous buoyancy based convection scheme: One- and three-dimensional validation. Tellus, 63A, 687–706, https://doi.org/10.1111/j.1600-0870.2011.00521.x.
Hartmann, D. L., 2015: Global Physical Climatology. 2nd ed. Elsevier, 481 pp.
Hohenegger, C., P. Brockhaus, C. S. Bretherton, and C. Schär, 2009: The soil moisture–precipitation feedback in simulations with explicit and parameterized convection. J. Climate, 22, 5003–5020, https://doi.org/10.1175/2009JCLI2604.1.
Hourdin, F., and Coauthors, 2020: LMDZ6A: The atmospheric component of the IPSL climate model with improved and better tuned physics. J. Adv. Model. Earth Syst., 12, e2019MS001892, https://doi.org/10.1029/2019MS001892.
Hu, H., L. R. Leung, and Z. Feng, 2021: Early warm-season mesoscale convective systems dominate soil moisture–precipitation feedback for summer rainfall in central United States. Proc. Natl. Acad. Sci. USA, 118, e2105260118, https://doi.org/10.1073/pnas.2105260118.
Jiang, R., L. Sun, C. Sun, and X.-Z. Liang, 2021: CWRF downscaling and understanding of China precipitation projections. Climate Dyn., 57, 1079–1096, https://doi.org/10.1007/s00382-021-05759-z.
Jung, M., and Coauthors, 2019: The FLUXCOM ensemble of global land–atmosphere energy fluxes. Sci. Data, 6, 74, https://doi.org/10.1038/s41597-019-0076-8.
Kim, D., A. H. Sobel, A. D. Del Genio, Y. Chen, S. J. Camargo, M. S. Yao, M. Kelley, and L. Nazarenko, 2012: The tropical subseasonal variability simulated in the NASA GISS general circulation model. J. Climate, 25, 4641–4659, https://doi.org/10.1175/JCLI-D-11-00447.1.
Klein, S. A., X. Jiang, J. Boyle, S. Malyshev, and S. Xie, 2006: Diagnosis of the summertime warm and dry bias over the US southern Great Plains in the GFDL climate model using a weather forecasting approach. Geophys. Res. Lett., 33, L18805, https://doi.org/10.1029/2006GL027567.
Koster, R. D., and Coauthors, 2004: Regions of strong coupling between soil moisture and precipitation. Science, 305, 1138–1140, https://doi.org/10.1126/science.1100217.
Kotz, M., A. Levermann, and L. Wenz, 2022: The effect of rainfall changes on economic production. Nature, 601, 223–227, https://doi.org/10.1038/s41586-021-04283-8.
Kunkel, K. E., X.-Z. Liang, and J. Zhu, 2010: Regional climate model projections and uncertainties of U.S. summer heat waves. J. Climate, 23, 4447–4458, https://doi.org/10.1175/2010JCLI3349.1.
Legates, D. R., and G. J. McCabe Jr., 1999: Evaluating the use of “goodness‐of‐fit” measures in hydrologic and hydroclimatic model validation. Water Resour. Res., 35, 233–241, https://doi.org/10.1029/1998WR900018.
Li, S., and Coauthors, 2019: Reducing climate model biases by exploring parameter space with large ensembles of climate model simulations and statistical emulation. Geosci. Model Dev., 12, 3017–3043, https://doi.org/10.5194/gmd-12-3017-2019.
Li, W., L. Li, R. Fu, Y. Deng, and H. Wang, 2011: Changes to the North Atlantic subtropical high and its role in the intensification of summer rainfall variability in the southeastern United States. J. Climate, 24, 1499–1506, https://doi.org/10.1175/2010JCLI3829.1.
Li, Y., K. Guan, G. D. Schnitkey, E. DeLucia, and B. Peng, 2019: Excessive rainfall leads to maize yield loss of a comparable magnitude to extreme drought in the United States. Global Change Biol., 25, 2325–2337, https://doi.org/10.1111/gcb.14628.
Liang, X.-Z., 2022: Extreme rainfall slows the global economy. Nature, 601, 193–194, https://doi.org/10.1038/d41586-021-03783-x.
Liang, X.-Z., L. Li, A. Dai, and K. E. Kunkel, 2004: Regional climate model simulation of summer precipitation diurnal cycle over the United States. Geophys. Res. Lett., 31, L24208, https://doi.org/10.1029/2004GL021054.
Liang, X.-Z., M. Xu, K. E. Kunkel, G. A. Grell, and J. S. Kain, 2007: Regional climate model simulation of U.S.–Mexico summer precipitation using the optimal ensemble of two cumulus parameterizations. J. Climate, 20, 5201–5207, https://doi.org/10.1175/JCLI4306.1.
Liang, X.-Z., K. E. Kunkel, G. A. Meehl, R. G. Jones, and J. X. L. Wang, 2008: Regional climate models downscaling analysis of general circulation models present climate biases propagation into future change projections. Geophys. Res. Lett., 35, L08709, https://doi.org/10.1029/2007GL032849.
Liang, X.-Z., and Coauthors, 2012: Regional climate–weather research and forecasting model. Bull. Amer. Meteor. Soc., 93, 1363–1387, https://doi.org/10.1175/BAMS-D-11-00180.1.
Liang, X.-Z., and Coauthors, 2017: Determining climate effects on US total agricultural productivity. Proc. Natl. Acad. Sci. USA, 114, E2285–E2292, https://doi.org/10.1073/pnas.1615922114.
Liang, X.-Z., Q. Li, H. Mei, and M. Zeng, 2019: Multi‐grid nesting ability to represent convections across the gray zone. J. Adv. Model. Earth Syst., 11, 4352–4376, https://doi.org/10.1029/2019MS001741.
Lin, Y., M. Zhao, Y. Ming, J. C. Golaz, L. J. Donner, S. A. Klein, V. Ramaswamy, and S. Xie, 2013: Precipitation partitioning, tropical clouds, and intraseasonal variability in GFDL AM2. J. Climate, 26, 5453–5466, https://doi.org/10.1175/JCLI-D-12-00442.1.
Lin, Y., W. Dong, M. Zhang, Y. Xie, W. Xue, J. Huang, and Y. Luo, 2017: Causes of model dry and warm bias over central U.S. and impact on climate projections. Nat. Commun., 8, 881, https://doi.org/10.1038/s41467-017-01040-2.
Linnet, K., 2000: Nonparametric estimation of reference intervals by simple and bootstrap-based procedures. Clin. Chem., 46, 867–869, https://doi.org/10.1093/clinchem/46.6.867.
Liu, C., and Coauthors, 2017: Continental-scale convection-permitting modeling of the current and future climate of North America. Climate Dyn., 49, 71–95, https://doi.org/10.1007/s00382-016-3327-9.
Loeb, N. G., and Coauthors, 2018: Clouds and the Earth’s Radiant Energy System (CERES) energy balanced and filled (EBAF) top-of-atmosphere (TOA) edition-4.0 data product. J. Climate, 31, 895–918, https://doi.org/10.1175/JCLI-D-17-0208.1.
Lopez, P., 2002: Implementation and validation of a new prognostic large-scale cloud and precipitation scheme for climate and data-assimilation purposes. Quart. J. Roy. Meteor. Soc., 128, 229–257, https://doi.org/10.1256/00359000260498879.
Ma, H.-Y., and Coauthors, 2018: CAUSES: On the role of surface energy budget errors to the warm surface air temperature error over the central United States. J. Geophys. Res. Atmos., 123, 2888–2909, https://doi.org/10.1002/2017JD027194.
McGuffie, K., and A. Henderson-Sellers, 2014: The Climate Modelling Primer. 4th ed. John Wiley and Sons, 456 pp.
Mesinger, F., and Coauthors, 2006: North American Regional Reanalysis. Bull. Amer. Meteor. Soc., 87, 343–360, https://doi.org/10.1175/BAMS-87-3-343.
Morcrette, C. J., and Coauthors, 2018: Introduction to CAUSES: Description of weather and climate models and their near‐surface temperature errors in 5 day hindcasts near the southern Great Plains. J. Geophys. Res. Atmos., 123, 2655–2683, https://doi.org/10.1002/2017JD027199.
Mueller, B., and S. I. Seneviratne, 2014: Systematic land climate and evapotranspiration biases in CMIP5 simulations. Geophys. Res. Lett., 41, 128–134, https://doi.org/10.1002/2013GL058055.
Mueller, N. D., E. E. Butler, K. A. McKinnon, A. Rhines, M. Tingley, N. M. Holbrook, and P. Huybers, 2016: Cooling of US Midwest summer temperature extremes from cropland intensification. Nat. Climate Change, 6, 317–322, https://doi.org/10.1038/nclimate2825.
Muller, C. J., and P. A. O’Gorman, 2011: An energetic perspective on the regional response of precipitation to climate change. Nat. Climate Change, 1, 266–271, https://doi.org/10.1038/nclimate1169.
Nicholls, S., and F. B. Smith, 1982: On the definition of the flux of sensible heat. Bound.-Layer Meteor., 24, 121–127, https://doi.org/10.1007/BF00121804.
NOAA, 2021: NOAA National Centers for Environmental Information (NCEI) U.S. billion-dollar weather and climate disasters. Accessed December 2021, https://www.ncdc.noaa.gov/billions/.
Ortiz-Bobea, A., T. R. Ault, C. M. Carrillo, R. G. Chambers, and D. B. Lobell, 2021: Anthropogenic climate change has slowed global agricultural productivity growth. Nat. Climate Change, 11, 306–312, https://doi.org/10.1038/s41558-021-01000-1.
Otterman, J., 1990: A simple two-system-parameter model for surface-effected warming of the planetary boundary layer. Bound.-Layer Meteor., 51, 213–227, https://doi.org/10.1007/BF00122138.
Palmer, T., and B. Stevens, 2019: The scientific challenge of understanding and estimating climate change. Proc. Natl. Acad. Sci. USA, 116, 24 390–24 395, https://doi.org/10.1073/pnas.1906691116.
Pearl, J., 2009: Causality. 2nd ed. Cambridge University Press, 464 pp.
Pendergrass, A. G., and Coauthors, 2020: Flash droughts present a new challenge for subseasonal-to-seasonal prediction. Nat. Climate Change, 10, 191–199, https://doi.org/10.1038/s41558-020-0709-0.
Piriou, J.-M., J.-L. Redelsperger, J.-F. Geleyn, J.-P. Lafore, and F. Guichard, 2007: An approach for convective parameterization with memory: Separating microphysics and transport in grid-scale equations. J. Atmos. Sci., 64, 4127–4139, https://doi.org/10.1175/2007JAS2144.1.
Platnick, S., and Coauthors, 2015: MODIS cloud optical properties: User guide for the collection 6 level-2 MOD06/MYD06 product and associated level-3 datasets. Version 1, 145 pp., https://modis-images.gsfc.nasa.gov/_docs/C6MOD06OPUserGuide.pdf.
Qiao, F., and X.-Z. Liang, 2015: Effects of cumulus parameterizations on predictions of summer flood in the central United States. Climate Dyn., 45, 727–744, https://doi.org/10.1007/s00382-014-2301-7.
Qiao, F., and X.-Z. Liang, 2016: Effects of cumulus parameterization closures on simulations of summer precipitation over the United States coastal oceans. J. Adv. Model. Earth Syst., 8, 764–785