Variability in modeled cloud feedback tied to differences in the climatological spatial pattern of clouds

Abstract

Despite the increasing sophistication of climate models, the amount of surface warming expected from a doubling of atmospheric CO$_2$ (equilibrium climate sensitivity) remains stubbornly uncertain, in part because of differences in how models simulate the change in global albedo due to clouds (the shortwave cloud feedback). Here, model differences in the shortwave cloud feedback are found to be closely related to the spatial pattern of the cloud contribution to albedo ($\alpha$) in simulations of the current climate: high-feedback models exhibit lower (higher) $\alpha$ in regions of warm (cool) sea-surface temperatures, and therefore predict a larger reduction in global-mean $\alpha$ as temperatures rise and warm regions expand. The spatial pattern of $\alpha$ is found to be strongly predictive ($r=0.84$) of a model’s global cloud feedback, with satellite observations indicating a most-likely value of $0.58\pm 0.31$ Wm$^{-2}$ K$^{-1}$ (90% confidence). This estimate is higher than the model-average cloud feedback of 0.43 Wm$^{-2}$ K$^{-1}$, with half the range of uncertainty. The observational constraint on climate sensitivity is weaker but still significant, suggesting a likely value of 3.68 ± 1.30 K (90% confidence), which also favors the upper range of model estimates. These results suggest that uncertainty in model estimates of the global cloud feedback may be substantially reduced by ensuring a realistic distribution of clouds between regions of warm and cool SSTs in simulations of the current climate.

1 Introduction

Clouds cool the planet by reflecting sunlight back to space. As surface temperatures rise, the fraction of incoming sunlight reflected by clouds (defined here as the cloud contribution to albedo, $\alpha$) may change, resulting in a shortwave cloud feedback. While climate models predict this feedback to be slightly positive on average (thus amplifying global warming), its magnitude and sign vary widely, accounting for much of the inter-model spread in equilibrium climate sensitivity (ECS) (e.g., Stocker et al. 2013; Soden and Vecchi 2011; Forster et al. 2013; Zelinka et al. 2013; Sherwood et al. 2014; Caldwell et al. 2016).

Several recent studies that have sought to constrain the global cloud feedback (GCF) have focused on regions of large-scale subsidence that extend over much of the tropical and subtropical oceans (e.g., Qu et al. 2014, 2015; Zhai et al. 2015; Brient et al. 2015; Brient and Schneider 2016; Myers and Norris 2016). These regions are often characterized by abundant clouds in the marine boundary layer, capped by a strong inversion that inhibits mixing with the drier free troposphere. With few clouds above to prevent the escape of outgoing longwave radiation, low clouds in these regions have a disproportionate impact on the global energy budget (Hartmann et al. 1992), and their response to warming therefore has a disproportionate impact on the GCF (e.g., Bony and Dufresne 2005; Webb et al. 2013; Zelinka et al. 2013). Recent studies have shown that, among climate models participating in the most recent Coupled Model Intercomparison Project (CMIP5), the response of such clouds to seasonal and interannual variations in sea-surface temperature (SST) closely mimics their response to global warming, allowing the cloud feedback (and to some degree, ECS) to be estimated from satellite observations of tropical low-cloud variability in the current climate (Qu et al. 2014, 2015; Zhai et al. 2015; Brient and Schneider 2016; Myers and Norris 2016). While the mechanisms responsible for such model differences continue to be refined, parameters governing the strength of low-level mixing and shallow convection are thought to play a significant role (Brient et al. 2015; Sherwood et al. 2014; Qu et al. 2014, 2015).

Meanwhile, a pioneering study by Volodin (2008) identified a separate way in which cloud-feedback variability relates to mean-state cloud differences among models. Analyzing an ensemble of global-warming simulations from an earlier generation of coupled climate models (CMIP3), Volodin found a significant correlation between the change in global cloud fraction with warming and the climatological gradient in cloud fraction between the tropics and the extratropical Southern Hemisphere. Based on this result and satellite-based estimates of cloud fraction, he calculated a most-likely ECS value of $3.6\pm 0.3$ K, though he cautioned against placing too much confidence in this estimate given little knowledge of the underlying mechanisms.

In this paper, we seek to shed further light on Volodin’s essential result by performing a related analysis of CMIP5 climate models, which in general have more sophisticated cloud parameterizations than those used in Volodin’s study. Consistent with Volodin, we find that inter-model variability in the GCF in the latest models is closely related to variability in the spatial distribution of $\alpha$ in the mean state. However, we also find that the correlation between the GCF and $\alpha$ is not governed by latitude per se, but rather by SST, with positive (negative) correlations in cool (warm) regions, and a large gradient in correlation where the annual-mean SST is between about 14 and 27 $^\circ$C. This correlation pattern persists as SSTs increase, explaining the large variability in shortwave cloud feedback in regions where the correlation is most sensitive to warming. Because the climatological pattern of $\alpha$ is tightly constrained by satellite observations, this relationship allows us to estimate the GCF in nature with higher confidence than previously possible (Dessler 2010; Zhou et al. 2013).

Unlike other observational constraints that have been proposed, we find that the relationship between the GCF and $\alpha$ in the current climate is not restricted to low clouds or to regions of large-scale subsidence, but includes contributions from all vertical levels and varied dynamical regimes, suggesting the influence of mechanisms operating beyond just the marine boundary layer. In particular, our results suggest the possibility that GCF variability may be largely due to mechanisms whose influence on $\alpha$ is confined to regions where SSTs are either warm or cool. In order to achieve global-mean radiative balance at the top of the atmosphere, cool-SST clouds must be tuned to compensate for variability in warm-SST mechanisms, and vice versa. This compensation between warm and cool regions explains why, despite large variability in $\alpha$ regionally, most climate models are able to simulate realistic global-mean surface temperatures in the current climate. However, as SSTs rise and warm regions expand, the balance between warm and cool regions is disrupted, leading to higher GCFs in models that exhibit lower $\alpha$ at warm SSTs. While other mechanisms undoubtedly also contribute to inter-model variability in the GCF (and even more so for ECS), these results point to concrete steps that can be taken to improve future generations of climate models and reduce uncertainty in their projections of global warming.

2 Data and methods

In order to eliminate potential differences in $\alpha$ caused by biases or variability in SSTs, we focus on atmosphere-only (AMIP) simulations from 20 distinct CMIP5 models run with observed SSTs (Taylor et al. 2011) (Table 1). AMIP simulations were available for 22 models, but two models (MPI-ESM-MR and ACCESS1-3) were excluded because their GCF and ECS estimates were very similar to those of related models (MPI-ESM-LR and ACCESS1-0). Although each simulation spanned 1979–2008, we limit our analysis to the last 6 years, during which well-calibrated satellite observations of $\alpha$ can be compared directly with model output. While the choice of time period was found to have a minor impact on the spatial pattern of $\alpha$ in each model, it had no discernible impact on the pattern of inter-model variability, which is the focus of this paper.

Table 1 AMIP model information

The cloud contribution to albedo was calculated for each model as the ratio between the annual-mean upwelling shortwave radiation due to clouds and the annual-mean downwelling shortwave radiation at the top of the atmosphere. An observational map of $\alpha$ was calculated using estimates of both these quantities taken from NASA’s Clouds and Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) monthly top-of-atmosphere flux product (Ed2.8). While CERES data begins in 2001, the data prior to 2003 was ignored because it is based on observations from just one satellite, which precludes an objective error estimate (see Appendix). To facilitate direct comparison across models and between models and observations, all data (modeled and observed) were regridded to a lowest-common horizontal resolution of 96 longitude points by 64 latitude points, using a flux-conserving algorithm to ensure no change in the global mean.

GCF and ECS values were taken from a previous study of coupled atmosphere-ocean models forced by an instantaneous quadrupling of CO$_2$ (Caldwell et al. 2016), and represent the change in the global cloud radiative effect per degree of global surface warming, adjusted to account for cloud masking of the water vapor, lapse rate, and surface-albedo feedbacks.

3 Statistical analysis

Figure 1a shows the correlation coefficient between the GCF and $\alpha$ at each grid point across the 20-member AMIP ensemble. Land masses are ignored because they lack a common surface boundary condition and because they contribute little to GCF variability across models (Soden and Vecchi 2011; Webb et al. 2013). Consistent with what Volodin (2008) found in CMIP3 models, correlations are significantly negative throughout much of the tropics equatorward of 30$^{\circ }$, indicating that models with a higher GCF tend to exhibit less tropical cloud cover. In contrast, broad positive correlations are found poleward of 30$^{\circ }$, as well as in subtropical regions off the west coasts of continents known for extensive marine stratocumulus clouds.

Fig. 1

a The correlation coefficient between the cloud contribution to albedo ($\alpha$) at each grid point and the net global cloud feedback (GCF) across 20 AMIP simulations. Stippling indicates that the correlations lack statistical significance at the 95% confidence level. The black contour represents the 23.5 $^\circ$C SST isotherm, which approximately corresponds to where the correlations change sign. b Annual mean SST between 2003 and 2008. c The correlation coefficient from a versus SST from b at each grid point in the Northern Hemisphere (red) and Southern Hemisphere (blue). The average correlation as a function of SST is shown in green, indicating that models with high GCF tend to exhibit lower $\alpha$ where SSTs are warm ($>23.5$ °C), and higher $\alpha$ where SSTs are cool

While the correlation pattern in Fig. 1a appears to depend significantly on latitude, it bears an even stronger resemblance to the pattern of annual-mean SSTs shown in Fig. 1b. Comparing the two maps, we find that high-GCF models tend to exhibit lower $\alpha$ wherever SSTs exceed about 23.5 $^\circ$C (black contour, Fig. 1a), and higher $\alpha$ otherwise. The robustness of this relationship is further demonstrated by plotting the correlation coefficient versus SST at each grid point (Fig. 1c). While the average correlation (green line) changes little with SST at the warmest and coldest temperatures, it decreases sharply between 14 and 27 $^\circ$C, from about 0.65 to −0.40.

Fig. 2

Climatological mean $\alpha$ in a the IPSL-CM5A-LR model, and b the GISS-E2-R model, which exhibit the highest and lowest GCFs, respectively, among all the models listed in Table 1. c The difference in $\alpha$ between the IPSL-CM5A-LR and GISS-E2-R models. The black contour represents the 23.5 $^\circ$C SST isotherm

In order to interpret Fig. 1 correctly, two points should be emphasized. First, the correlation pattern in Fig. 1a—and its relationship to mean-state SST—does not imply that a similar relationship should exist between $\alpha$ and SST in any particular model. On the contrary, the spatial pattern of $\alpha$ bears little resemblance to SST in every model. To illustrate this point, Fig. 2 shows the mean-state patterns of $\alpha$ for the IPSL-CM5A-LR and GISS-E2-R models, which exhibit the highest and lowest GCFs, respectively, among the 20 models in Table 1. Comparing each figure to the pattern of mean-state SSTs in Fig. 1b, we do not find a strong similarity in either case. However, if we instead look at the difference in $\alpha$ between the two models (Fig. 2c), we find a much more coherent pattern, with the high-GCF model (IPSL-CM5A-LR) exhibiting lower (higher) $\alpha$ at warm (cool) SSTs, regardless of whether a particular region in either model is relatively cloudy or clear.

Second, the correlations in Fig. 1a are due almost entirely to variability in the shortwave component of the GCF (SWGCF), with very little contribution from the longwave component (LWGCF). The reason for this is that the GCF is highly correlated with the SWGCF at $r=0.86$, but essentially uncorrelated with the LWGCF ($r=0.01)$ among the models included in our analysis. Moreover, Fig. 1a is very similar to the correlation map between $\alpha$ and the SWGCF alone (not shown), with a spatial correlation of $r=0.98$ between the two maps.

Fig. 3

a The leading empirical orthogonal function (EOF) of the shortwave cloud radiative effect (CRE) over the oceans among the 20 AMIP models, which explains 33% of the total inter-model variance. CRE is by definition positive downward, so positive (negative) values indicate lower (higher) $\alpha$. b The GCF versus the leading principal component (PC) of shortwave CRE, which are significantly correlated at $r=0.75$

In light of the two facts above, we propose the following explanation for the correlation pattern between $\alpha$ and the GCF observed in Fig. 1a. In the current climate, incident sunlight at the top of the atmosphere is equally partitioned between ocean regions that are warmer and cooler than 23.5 $^\circ$C (51 vs. 49%). Therefore, in order for each model to achieve a reasonable global albedo, models with lower $\alpha$ in regions where SSTs are warmer than 23.5 $^\circ$C should tend to exhibit higher $\alpha$ in regions where SSTs are cooler than 23.5 $^\circ$C. This trade-off is hinted at in Fig. 2c, and it is further reinforced in Fig. 3, which shows the leading empirical orthogonal function (EOF) of shortwave cloud radiative effect among models (accounting for 33% of the total inter-model variance). While the EOF pattern exhibits large magnitudes locally (e.g., $>10$ Wm$^{-2}$ in much of the tropics), its global-mean value is close to zero (1.16 Wm$^{2}),$ implying almost total cancellation between tropical and extratropical differences. As SSTs rise, however, the 23.5$^\circ$ isotherm expands poleward, tipping the scales in favor of warm-SST regions. Thus, if inter-model differences in $\alpha$ depend on SST in the same way in a future climate as they do in the current climate (i.e., Fig. 1c), this would explain why models that exhibit lower $\alpha$ at warm SSTs tend to exhibit a lower global-mean $\alpha$ (and thus a higher GCF) as the climate warms.

Fig. 4

a The correlation coefficient between $\alpha$ at each grid point and the GCF across a subset of 11 AMIP simulations for which AMIP4K simulations were also available. The pattern is similar to that calculated using the full 20-member ensemble (Fig. 1a). The black contour represents the 23.5 $^\circ$C SST isotherm, which separates positive and negative correlations in the full ensemble. b As in a, but for the 11 AMIP4K simulations, with the 23.5 $^\circ$C SST isotherm shifted poleward as a result of uniform 4 K warming. c The difference between the AMIP4K (b) and AMIP (a) correlation patterns. d As in c, but with the change in correlation plotted at each grid point as a function of mean-state SST. The solid green line shows the average change, while the dashed green line shows the predicted change based on a uniform 4 K shift in the mean-state correlation (Fig. 1c). As predicted, the largest decrease in correlation is observed between 14 and 27 $^\circ$C

If this hypothesis is correct, we should expect the region of negative correlation in Fig. 1a to expand with warming as the 23.5 $^\circ$C isotherm shifts poleward. To test this, we calculate the correlation between $\alpha$ and the GCF as before, but this time within an 11-member ensemble of “AMIP4K” simulations (Table 1), which are identical to the AMIP simulations but with SSTs increased by a uniform 4 $^\circ$C (Taylor et al. 2011). The correlations for the subset of 11 AMIP models are shown in Fig. 4a, alongside those for the same subset of AMIP4K simulations (Fig. 4b). Comparing the two maps, we find that the region of negative correlations is indeed larger in the AMIP4K simulations, and as expected, its expansion approximately tracks the 23.5$^\circ$ isotherm (black contour). Moreover, when we examine the differences between the two maps (Fig. 4c), we find that the decrease in correlation with warming is largely confined to regions where mean-state SST is between 14 and 27 °C (green lines), which is precisely where the correlation between $\alpha$ and the GCF is most sensitive to temperature in the current climate (i.e., where the slope of the green line in Fig. 1c is negative).

If the correlation between $\alpha$ and the GCF were governed by SST alone, then the change in correlation between the AMIP and AMIP4K simulations (Fig. 4c) should be well predicted by a uniform 4 K shift in the mean-state correlation shown in Fig. 1c. To test this, we plot the grid-point-wise change in correlation as a function of mean-state SST in Fig. 4d. The solid green line represents the average change over both hemispheres, while the dashed green line represents the predicted change implied by a 4 K shift in the mean-state correlation pattern. Consistent with Fig. 4c, the largest decrease in correlation occurs between 14 and 27 $^\circ$C. However, the magnitude of the decrease is overpredicted by as much as 0.2 above 18 °C, and slightly underpredicted below 13 $^\circ$C. This discrepancy implies that the changes in correlation in Fig. 4c are not driven exclusively by SST warming, suggesting the possibility that other factors like inversion strength may also play a role (Qu et al. 2014, 2015).

On the whole, however, the evidence presented in Fig. 4 is broadly consistent with our hypothesis for why GCF variability among models is related so strongly to differences in the climatological distribution of clouds: because high-GCF models exhibit lower $\alpha$ at warm SSTs, they exhibit lower $\alpha$ globally as the region of warm SSTs expands.

4 Mechanistic interpretation

The above results suggest that existing theories may not fully explain the causes of inter-model variability in the GCF. Consider, for example, the notion that differences in low-level mixing account for much of the variability in the feedback from low clouds, as recently put forth by Sherwood et al. (2014). While their metric of low-level mixing ($S+D$) is significantly correlated with ECS ($r=0.54$ among the 20 models in Table 1), we find a much weaker correlation with the GCF ($r=0.26$). This suggests that the physical mechanism proposed by Sherwood et al. (2014) is not primarily responsible for our results.

Several other studies have focused on the mechanisms governing the low-cloud feedback in subtropical regions of large-scale subsidence (Bony and Dufresne 2005), where the marine boundary layer contains extensive stratocumulus cloud decks capped by a strong temperature inversion. In these regions, inter-model variability in the low-cloud feedback has been connected to the magnitude of shallow convection and boundary-layer turbulence (Brient et al. 2015), the strength of the capping inversion and boundary-layer moistening (Qu et al. 2014, 2015; Myers and Norris 2016), and the covariance between SST and low-cloud amount on seasonal-to-annual timescales (Zhai et al. 2015; Brient and Schneider 2016; Myers and Norris 2016).

Fig. 5

a The ensemble-mean pattern of 500-hPa vertical velocity in pressure coordinates ($\omega$), with positive (negative) values indicating subsidence (ascent). Green contours represent the 14 and 27 $^\circ$C mean-state SST isotherms. b The slope of the regression of the shortwave cloud radiative effect feedback (i.e., the change in SWCRE divided by 4 K) onto the GCF, indicating the shortwave contribution to GCF variability at each grid point. The two maps are spatially correlated at $r=0.55$

While we do not question the validity of the studies cited above, however, two pieces of evidence suggest that they, too, cannot fully account for our results. First, significant shortwave contributions to GCF variability are not confined to regions of large-scale subsidence. This point was made in a previous study of CMIP5 models by Vial et al. (2013), and it is also evident in our Fig. 5, which shows the mean-state vertical velocity ($\omega = dp/dt$) at 500 hPa (a) alongside a regression map of the shortwave cloud radiative effect feedback onto the GCF, which can be interpreted as the shortwave contribution to GCF variability at each grid point (b). The two maps exhibit a spatial correlation of $r=0.55$, indicating a modest tendency for regions of large-scale subsidence to contribute more to GCF variability. However, some regions of large-scale ascent, such as the South Pacific Convergence Zone, also contribute significantly to GCF variability, while some regions of large-scale subsidence do not. By comparison, the connection to mean-state SST is much cleaner, with the largest contributions to GCF variability almost invariably found between the 14 and 27 $^\circ$C isotherms (Fig. 5b)—a result which is closely related, both physically and mathematically, to the warming-induced change in correlations between $\alpha$ and the GCF observed in Fig. 4c.¹

Fig. 6

The correlation coefficient between the GCF and the average cloud water mixing ratio (liquid + ice) in the a upper (< 440 hPa), b middle (440–680 hPa), and c lower (> 680 hPa) troposphere across a subset of 11 AMIP simulations for which AMIP4K simulations were also available. d–f As in a–c, but for the AMIP4K simulations. g–i The gridpoint-wise change in correlation coefficient versus mean-state SST, analogous to Fig. 4e. The solid green line shows the average change, while the dashed green line shows the predicted change based on a uniform 4 K shift in the mean-state relationship between the correlation coefficient and SST

Second, variability in the shortwave cloud feedback is not due exclusively to low-level clouds. This is illustrated in Fig. 6, which shows the correlation between the GCF and the total cloud-water mixing ratio (liquid + ice) in the lower, middle, and upper troposphere across both the AMIP and AMIP4K ensembles, along with the difference in correlation between the two ensembles as a function of mean-state SST (analogous to Fig. 4d). Comparing the two ensembles, we find that not only do the correlations decrease throughout the troposphere in response to warming, but the magnitude and spatial pattern of the correlation changes are remarkably well predicted by a uniform 4 K shift in the mean-state correlation pattern at all levels.² Thus, variability in the cloud feedback appears to be closely related to differences in mean-state cloud amount not only at low levels but throughout the troposphere, suggesting that the underlying mechanisms are likewise not confined to the marine boundary layer.

The inability of existing theories to explain the connection between differences in mean-state $\alpha$ and the GCF would seem to implicate other mechanisms that have not yet been identified. One possible explanation, which we refer to as the “tuning hypothesis”, is that inter-model differences in the spatial pattern of $\alpha$ are the result of multiple mechanisms operating in different models together with model tuning (e.g., of a threshold relative humidity for cloud formation) to achieve radiation balance at the top of the atmosphere. These mechanisms might primarily influence $\alpha$ either at warm SSTs or at cool SSTs, with the tuning process forcing compensating differences in $\alpha$ in the opposite regime. To be consistent with our results, the mechanisms must favor a larger positive shortwave cloud feedback for GCMs in which either cool-SST $\alpha$ is large or warm-SST $\alpha$ is small.

In cool regions, an example of one such mechanism is the temperature threshold for converting liquid water to ice. Because liquid cloud water is more reflective than ice, models with a lower temperature threshold (and thus a higher liquid-to-ice ratio) have been found to exhibit higher $\alpha$ in the extratropics (Kay et al. 2016; McCoy et al. 2015). This in turn has been linked to a higher GCF and ECS (Kay et al. 2016; McCoy et al. 2015), consistent with the correlations presented in Fig. 1a. Among the models in Table 1, we find that the measure of cloud glaciation temperature introduced by McCoy et al. (2015) is correlated with the GCF at $r=0.47$. This is significant, but not as large as the correlation of our $\alpha$ pattern with the GCF, suggesting that another mechanism—possibly operating at warm SSTs—may also be important (e.g., see discussion of convective precipitation efficiency below).

The study by Kay et al. (2016) is also important as a practical illustration of how model tuning can involve a trade-off between warm and cool regions. In order to fix the low-$\alpha$ bias over the Southern Ocean in their model (CESM), the authors increased the temperature threshold for cloud glaciation, which led to an increase in the liquid-to-ice ratio in extratropical clouds. However, in order to avoid significant global cooling, this change had to be accompanied by a decrease in $\alpha$ elsewhere, which they accomplished by increasing the relative humidity threshold for cloud formation. While the effect of the relative humidity adjustment was not limited to a particular region, the combined result of both adjustments was a mean-state climate with higher $\alpha$ in the extratropics and lower $\alpha$ in the tropics, much like the pattern of inter-model $\alpha$ variability evident in Figs. 1a and 3a.

While Kay et al. (2016) began by altering a cool-SST mechanism and then tuned their model to maintain a reasonable global-mean $\alpha$, in general the tuning process can also be driven by variations in warm-SST mechanisms. In particular, one mechanism that appears to play an important role in warm regions is the convective precipitation efficiency (CPE), which we approximate here as

$$\begin{aligned} \mathrm {CPE}\approx \frac{P_c}{Mq^*}, \end{aligned}$$

(1)

where $P_c$ is precipitation due to parameterized convection, M is the convective mass flux just above the atmospheric boundary layer (700 hPa), and $q^*$ is the saturated specific humidity at the same level. By this definition, the CPE represents the fraction of the convective moisture flux out of the boundary layer that ends up as precipitation at the surface, and could therefore depend on a number of convective and microphysical parameters within a given model, some of which (e.g., those governing autoconversion of cloud into rain) are commonly used to tune cloud radiative properties. Because the convective mass flux was not a required output variable for CMIP5, we were only able to calculate the CPE for 13 models, which are indicated in the fourth column of Table 1.

Fig. 7

a The correlation between the convective precipitation efficiency (CPE) and the leading principle component of CPE across models, considering only ocean grid points where convective precipitation exceeds 0.5 m/year on average (excluded grid points appear white). The correlation pattern is nearly uniform, which indicates that models with high CPE in one location tend to exhibit high CPE throughout the tropics. b The leading PC of CPE versus the leading PC of the shortwave CRE from Fig. 3. The two PCs are significantly correlated at $r=0.79$, indicating that models with high CPE also tend to exhibit high shortwave CRE (i.e., low $\alpha$) in the tropics. c The leading PC of CPE versus the GCF, which are significantly correlated at $r=0.70$

In previous simulations performed with variations of a single model, high CPE was found to be associated with low rates of detrainment from convective plumes, resulting in less overall cloudiness (Zhao 2014; Zhao et al. 2016). To illustrate the potential influence of CPE on our earlier results, Fig. 7 shows the leading EOF of CPE across models (a), along with scatter plots showing the leading principal component (PC) of CPE versus (b) the leading PC of SWCRE and (c) the GCF. In Fig. 7a, the leading EOF (which explains 90% of the total variance) has been normalized everywhere by the standard deviation of CPE, and therefore represents the correlation between the leading PC and CPE at each grid point. The near-uniformity of the EOF pattern indicates that models with high CPE in one location tend to exhibit high CPE everywhere. Plotting the leading PC of CPE versus the leading PC of SWCRE from Fig. 3, we find a strong correlation of $r=0.79$ (Fig. 7b). This indicates that AMIP models with lower $\alpha$ (higher SWCRE) in the tropics tend to exhibit higher CPE in convecting regions (Fig. 7a), consistent with the previous findings of Zhao (2014). Perhaps more importantly, we also find a significant correlation between the CPE index and the GCF ($r=0.70,$ Fig. 7c), echoing previous studies that have found the GCF in specific models to be sensitive to various parameters known to influence CPE, including the ice fall speed, the threshold for cloud-to-rain autoconversion, and the rate of mixing between a convective plume and its environment (Sanderson et al. 2008; Tomassini et al. 2015; Mauritsen et al. 2012; Zhao 2014; Zhao et al. 2016).

Fig. 8

a The average convective precipitation fraction (CPF, defined as the ratio of convective to total precipitation) among the 11-member subset of AMIP simulations for which AMIP4K simulations are also available. b As in a, but for AMIP4K simulations. c The change with warming (AMIP4K−AMIP) in the ensemble-mean CPF. d The inter-model standard deviation of the change in CPF with warming. Increases in CPF are widespread at the margin of convecting regions, and the pattern of $\Delta$CPF is significantly correlated with the regression pattern in Fig. 5b at $r=0.34$, suggesting an increased role for convection in regions where variability in the shortwave cloud feedback is large. However, the magnitude of $\Delta$CPF is too small in most places to distinguish between the AMIP and AMIP4K patterns

If the tuning hypothesis is correct, inter-model variability in the GCF should arise from some combination of (1) an expansion of regions where $\alpha$ variability is governed primarily by warm-SST mechanism(s), and/or (2) a contraction of regions where $\alpha$ variability is governed primarily by cool-SST mechanism(s). If the former is most important, we might expect the AMIP4K simulations to exhibit an increase in the ratio of convective-to-total precipitation (convective precipitation fraction) near the tropical margin, where the shortwave contributions to GCF variability are largest (Fig. 5b). To test this, Figs. 8a, b show the convective precipitation fraction averaged over the AMIP and AMIP4K ensembles, with their difference shown in Fig. 8c. As expected, the difference map reveals a widespread increase in convective precipitation fraction with warming at the margin of convecting regions. However, the magnitude of this change is quite modest relative to the mean-state pattern (Figs. 8a, b), and in most regions it does not stand out above the noise of inter-model variability (Fig. 8d), suggesting that convective expansion by itself is insufficient to account for GCF variability among models. Without a more complete understanding of all the mechanisms involved, we raise the tuning hypothesis with some reservation, but believe that it is an intriguing possibility that should be investigated further.

5 Use of $\alpha$ as an emergent constraint on GCF

Regardless of the validity of the tuning hypothesis, Fig. 1 demonstrates a clear relationship between the GCF and the distribution of $\alpha$ in simulations of the current climate. If we assume that uncertainty in the true cloud feedback is well represented by inter-model variability, then we can use observations of $\alpha$ from the Clouds and the Earth’s Radiant Energy Systems (CERES) satellites to estimate the GCF in nature, using a statistical method known as partial least-squares regression (Smoliak et al. 2010; Wallace et al. 2012; Christian et al. 2016). For each model, we calculate an albedo index, A, that measures the degree to which its pattern of $\alpha$ projects onto the correlation map in Fig. 1a. Put another way, if one thinks of the correlation map in Fig. 1a as a kind of normalized EOF, then A would represent the corresponding PC. This means that models with a higher A tend to exhibit lower $\alpha$ at warm SSTs and higher $\alpha$ at cool SSTs. Applying the same algorithm to satellite observations of $\alpha$ yields an observed albedo index that we then use to estimate the GCF in nature based on the regression of A onto the GCF in models (see Appendix). The result, shown in Fig. 9a, indicates a most-likely GCF of 0.58 W m$^{-2}$ K$^{-1},$with 90% confidence that it lies between 0.27 and 0.89 Wm$^{-2}$ K$^{-1}$. This estimate is higher than the model average (0.43 W m$^{-2}$ K$^{-1}$), and suggests that the likelihood of a negative GCF is exceedingly small.

Fig. 9

a GCF versus the albedo index, A, which was calculated by regressing each model’s $\alpha$ pattern onto the correlation map in Fig. 1a, and then standardizing the result to have zero mean and unit standard deviation. Satellite observations indicate a real-world A value of $0.48\pm 0.03$, which corresponds to a GCF of $0.58\pm 0.31$ (90% confidence). b As in a, but for ECS, based on the correlation pattern between $\alpha$ and ECS (not shown). Based on satellite observations ($A=0.36\pm 0.08$), we calculate a most-likely ECS of $3.68\pm 1.30$ K, or about 0.3 K higher than the model average (90% confidence)

To test the sensitivity of this estimate to our methodology, we repeat the regression using two other observable metrics as predictors in place of A: (1) the meridional gradient in zonal-mean $\alpha$ between 20$^\circ$ and 40$^\circ$ latitude (averaged over both hemispheres), and (2) the average difference in $\alpha$ between ocean grid points warmer and cooler than 23.5 $^\circ$C. Compared with A, each of these metrics exhibits a similarly strong correlation with the GCF across models ($r=0.84$ and 0.82, respectively). Moreover, satellite observations of each metric yield a similar estimate of the GCF in nature ($0.57 \pm 0.30$ and $0.62 \pm 0.41$ Wm$^{-2}$ K$^{-1}$), giving further credibility to our result found through partial least-squares.

The same partial-least-squares method can also be used to estimate the real-world ECS based on the correlation map between $\alpha$ and ECS among AMIP models (not shown), yielding a 90% confidence range of 3.68 ± 1.30 K (Fig. 9b). Although the range in this estimate is as large as the model spread, the mean is about 0.3$^\circ$ higher, implying that high-ECS models tend to exhibit more realistic cloud distributions than do low-ECS models.

6 Discussion

Our results highlight and expand upon an important but underappreciated aspect of inter-model variability in global cloud feedback that was first pointed out by Volodin (2008). In the past, model tuning has focused on how clouds affect the global energy budget, with an eye towards producing top-of-atmosphere radiative balance and realistic global-mean surface temperatures (Watanabe et al. 2010; Gent et al. 2011; Mauritsen et al. 2012; Hourdin et al. 2016). However, our analysis has demonstrated that the spatial distribution of clouds also matters, particularly for how models simulate the response of clouds to global warming. Using atmospheric models with common sea surface temperatures, we find that variability in the spatial pattern of $\alpha$ explains 71% of the variance in the global cloud feedback and substantially constrains it to the upper range of model estimates ($0.58 \pm 0.31$ W m$^{-2}$ K$^{-1}$). The constraint on ECS is weaker ($3.68 \pm 1.30$ K), but also indicates that higher-sensitivity models are more realistic.

Inter-model differences in the spatial pattern of $\alpha$ may reflect the combined effect of mechanisms operating in regions of warm and cool SSTs, whereby variability in one region is tuned to compensate for variability in the other. In cool regions, previous studies have tied $\alpha$ variability to the temperature at which cloud water freezes (Kay et al. 2016; McCoy et al. 2015). In warm regions, we find that differences in $\alpha$ are influenced in part by the efficiency with which moist-convective updrafts are converted to precipitation. If the tuning hypothesis is correct, inter-model variability in the GCF may be due to some combination of an expansion of regions where warm-SST mechanisms dominate, and/or a contraction of regions where cool-SST mechanisms dominate. Both of these scenarios would explain why models that exhibit lower (higher) $\alpha$ in regions of warm (cool) SST tend to predict a larger reduction in $\alpha$ (and thus a higher GCF) as warm regions expand and cool regions contract. While much remains unknown about the specific mechanisms involved, the tuning hypothesis is in our view the most straightforward interpretation of our results, and therefore warrants further investigation. At the very least, the large variability in the mean-state pattern of $\alpha$ and its significant correlation with the GCF suggests that modeling groups should pay more attention to tuning models to match the observed distribution of $\alpha$ between regions of warm and cool SSTs in simulations of the current climate.

Appendix: Estimating the GCF and ECS

Regressions in Fig. 9 were performed using orthogonal (i.e., Deming) regression after normalizing each variable to unit variance. Unlike ordinary least-squares regression, orthogonal regression minimizes the sum of the squared perpendicular distances from the (normalized) data points to the regression line. This is appropriate when using the partial least-squares method, for which uncertainty exists in both the predictor (A) and the predictand (GCF or ECS). Error estimates were calculated using standard algorithms and the t-statistic for 90% confidence (two-tailed).

The range of uncertainty in our estimates of the GCF and ECS was calculated as follows. First, the difference between the diurnally-corrected top-of-atmosphere fluxes measured by the two CERES satellites (Aqua and Terra) was used as an approximation of the error in $\alpha$. This difference pattern was multiplied by a random number selected from a standard normal distribution and added to $\alpha$, which was then projected onto the correlation map in Fig. 1a to estimate the observed albedo index, A. This process was repeated 5000 times using a different random number each time. The standard deviation of A was then scaled by the t-statistic for 90% confidence to approximate the uncertainty.