## 1. Introduction

Comprehensive climate models suggest that a global increase in absorbed solar radiation by 1 W m^{−2} would lead to a 0.6°–1.1°C increase in global-mean surface temperatures (Soden and Held 2006). The amount of solar radiation absorbed or reflected by Earth depends on the solar zenith angle *ζ* or the angle that the sun makes with a line perpendicular to the surface. When the sun is low in the sky (high *ζ*), much of the incident sunlight may be reflected, even for a clear sky; when the sun is high in the sky (low *ζ*), even thick clouds may not reflect most of the incident sunlight. The difference in average zenith angle between the equator and poles is an important reason why the albedo is typically higher at high latitudes.

*S*

_{0}and the cosine of the solar zenith angle,

*μ*≡ cos

*ζ*:

*I*〉 =

*S*

_{0}/4 ≈ 342 W m

^{−2}(in this paper, we will denote spatial averages with 〈

*x*〉 and time averages with

*μ** and an effective solar constant

The details of these additional assumptions are quite important to the simulated climate, because radiative transfer processes, most importantly cloud albedo, depend on *μ* (e.g., Hartmann 1994). For instance, the most straightforward choice for a planetary-average calculation might seem to be a simple average of *μ* over the whole planet, including the dark half, so that

*P*(

*μ*) is the probability distribution function of global surface area as a function of

*μ*over the illuminated hemisphere. For the purposes of a planetary average,

*P*(

*μ*) simply equals 1. This can be seen by rotating coordinates so that the North Pole is aligned with the subsolar point, where

*μ*= 1; then

*μ*is given by the sine of the latitude over the illuminated Northern Hemisphere, and since area is uniformly distributed in the sine of the latitude, it follows that area is uniformly distributed over all values of

*μ*between 0 and 1. Hereafter, when discussing planetary averages, it should be understood that integrals over

*μ*implicitly contain the probability distribution function

*P*(

*μ*) = 1. Evaluation of Eq. (3) gives

The daytime-average cosine zenith angle of 0.5 has been widely used. The early studies of radiative–convective equilibrium by Manabe and Strickler (1964), Manabe and Wetherald (1967), Ramanathan (1976), and the early review paper by Ramanathan and Coakley (1978) all took

To our knowledge, no studies of global-mean climate with radiative–convective equilibrium models have used an insolation-weighted cosine zenith angle of ⅔. The above considerations regarding the spatial averaging of insolation, however, also apply to the temporal averaging of insolation that is required to represent the diurnal cycle, or combined diurnal and annual cycles, with a zenith angle that is constant in time. In this context, Hartmann (1994) strongly argues for the use of insolation-weighted zenith angle and provides a figure with appropriate daily-mean insolation-weighted zenith angles as a function of latitude for the solstices and the equinoxes (see Hartmann 1994, his Fig. 2.8). Romps (2011) also uses an equatorial insolation-weighted zenith angle in a study of radiative–convective equilibrium with a cloud-resolving model, though other studies of tropical radiative–convective equilibrium with cloud-resolving models, such as the work by Tompkins and Craig (1998), have used a daytime-weighted zenith angle. In large-eddy simulations of marine low clouds, Bretherton et al. (2013) advocate for the greater accuracy of the insolation-weighted zenith angle, noting that the use of daytime-weighted zenith angle gives a 20 W m^{−2} stronger negative shortwave cloud radiative effect than the insolation-weighted zenith angle. Biases of such a magnitude would be especially disconcerting for situations where the surface temperature is interactive, as they could lead to dramatic biases in mean temperatures.

Whether averaging in space or time, an objective decision of whether to use daytime-weighted or insolation-weighted zenith angle requires some known and unbiased reference point. In section 2, we develop the idea of absorption-weighted zenith angle as such an unbiased reference point. We show that if albedo depends nearly linearly on the zenith angle, which is true if clouds play a dominant role in solar reflection, then the insolation-weighted zenith angle is likely to be less biased than the daytime-weighted zenith angle. We then calculate the planetary-average absorption-weighted zenith angle for the extremely idealized case of a purely conservative scattering atmosphere. In section 3, we perform calculations with a more detailed shortwave radiative transfer model and show that differences in planetary albedo between ^{−2}. In section 4 we show that the superiority of insolation-weighting also applies for diurnally or annually averaged insolation. Finally, in section 5, we discuss the implications of our findings for recent studies with global models.

## 2. Absorption-weighted zenith angle

*α*=

*f*

_{α}(

*μ*) =

*f*

_{α}(cos

*ζ*)], then we can choose a zenith angle

*P*(

*μ*)]:

*f*

_{α}is smooth and monotonic in the zenith angle—the likely (albeit not universal) case for planetary reflection—then

*f*

_{α}can be inverted, and the problem is well posed, with a unique solution:

*f*

_{α}. For the case of planetary-average solar absorption, the probability density function of

*μ*over the sunlit half of the globe is uniform [see discussion following Eq. (3)]. Taking

*P*(

*μ*) = 1, Eq. (5) simplifies to

*α*〉 is the planetary albedo or ratio of reflected to incident global shortwave radiation. Note that a bias in planetary albedo by 1% would lead to a bias in planetary-average absorbed shortwave radiation of 3.42 W m

^{−2}.

*α*

_{max}is the maximum albedo (for

*μ*= 0), and

*α*

_{Δ}is the drop in albedo in going from

*μ*= 0 to

*μ*= 1. In this case, we can show that the absorption-weighted zenith angle is exactly equal to the insolation-weighted zenith angle, regardless of the form of

*P*(

*μ*). From Eqs. (3), (4), and (8), it follows that

*μ*, then we expect the insolation-weighted zenith angle to closely match the absorption-weighted zenith angle.

*P*(

*μ*) allows us to perform an additional analytic calculation of the absorption-weighted zenith angle. Consider an albedo similar to Eq. (8), but which may now vary nonlinearly, as some power of the cosine of the zenith angle:

*b*is likely equal to or less than 1, so that the albedo is more sensitive to the zenith angle when the sun is low than when the sun is high. For a general value of

*b*, the planetary albedo and absorption-weighted zenith angle are given by

*μ*(

*b*= 1), then the absorption-weighted zenith angle has a cosine of ⅔, which is equal to the planetary-average value of

*b*< 1,

*e*

^{−1/2}≈ 0.607 and ⅔, suggesting that

*μ*, with significant weight at low

*μ*, in order to obtain values of

### Example: A pure scattering atmosphere

*μ*for a less idealized function

*f*

_{α}(

*μ*)? For a pure conservative scattering atmosphere, with optical thickness

*τ**, two-stream coefficient

*γ*[which we will take equal to ¾, corresponding to the Eddington approximation (Pierrehumbert 2010)], and scattering asymmetry parameter

*α*

_{g}, and a diffuse atmospheric albedo

*τ** ≈ 0.12,

*τ** = 3.92,

*μ*, in order to focus on the atmospheric contribution to planetary reflection. The particular surface albedo value of 0.12 is chosen following the observed global-mean surface reflectance from Fig. 5 of Donohoe and Battisti (2011) (average of the hemispheric values from observations). Of course, surface reflection also generally depends on

*μ*, with the direct-beam albedo increasing at lower

*μ*, but surface reflection plays a relatively minor role in planetary albedo, in part because so much of Earth is covered by clouds (Donohoe and Battisti 2011).

We can also use these results to calculate what bias would result from the use of the daytime-weighted zenith angle *μ** = ½ will overestimate the negative shortwave radiative effect of clouds, while choices of *μ** > ⅔ will underestimate the negative shortwave radiative effect of clouds. Our calculations here, however, are quite simplistic, and do not account for atmospheric absorption or wavelength dependence of optical properties. In the following section, we will use a more detailed model to support the assertion that the insolation-weighted zenith angle leads to smaller albedo biases than the daytime-weighted zenith angle.

Planetary albedos and biases.

## 3. Calculations with a full radiative transfer model

The above calculations provide a sense for the magnitude of planetary-albedo bias that may result from different choices of average solar zenith angle. In this section, we calculate albedos using version 3.8 of the shortwave portion of the Rapid Radiative Transfer Model for application to GCMs (RRTMG_SW, v3.8; Iacono et al. 2008; Clough et al. 2005); we refer to this model as simply “RRTM” for brevity. Calculations with RRTM allow for estimation of biases associated with different choices of *μ* when the atmosphere has more realistic scattering and absorption properties than we assumed in the pure scattering expressions above [Eqs. (12) and (13)]. RRTM is a broadband, two-stream, correlated-*k* distribution radiative transfer model, which has been tested against line-by-line radiative transfer models, and is used in several GCMs. For calculation of radiative fluxes in partly cloudy skies, the model uses the Monte Carlo independent column approximation (McICA; Pincus et al. 2003), which stochastically samples 200 profiles over the possible range of combinations of cloud overlap arising from prescribed clouds at different vertical levels and averages the fluxes that result.

*U.S. Standard Atmosphere, 1976*; and the subarctic winter atmosphere, and we perform calculations with clear skies, as well as two cloud-profile assumptions (Table 2). One cloud profile is a mixed sky, intended to mirror Earth’s climatological cloud distribution, with four cloud layers having fractional coverage, water path, and altitudes based on RS99; we call this the RS99 case. The other cloud profile is simply fully overcast with a low-level “stratocumulus” cloud deck, having a water path of 100 g m

^{−2}. Table 2 gives the values for assumed cloud fractions, altitudes, and in-cloud-average liquid and ice water in clouds at each level. Cloud fractions have been modified from Table 4 of RS99 because satellites see clouds from above and will underestimate the true low-cloud fraction that is overlain by higher clouds. If multiple cloud layers are randomly overlapping and seen from above, then, indexing cloud layers as (1, 2, …) from the top down, we denote

*i*, and

*σ*

_{i}as the true cloud fraction in layer

*i*. The true cloud fraction in layer

*i*is

*i*, so the term in parentheses gives the fraction of clear sky above level

*i*, which is equal to the ratio of observed cloud fraction to true cloud fraction in layer

*i*(again assuming random cloud overlap). Applying this correction to observed cloud fractions

Cloud profiles used in calculations with RRTM. The multiple cloud layers of RS99 are used together and are assumed to overlap randomly. Cloud fractions are based on Table 4 of RS99 but adjusted for random overlap and observation from above (see text). Cloud-top altitudes are based on top pressures from RS99 and the pressure-height profile from the *U.S. Standard Atmosphere, 1976*. Cloud water/ice allocation uses 260 K as a threshold temperature.

To isolate the contributions from changing atmospheric (and especially cloud) albedo as a function of *μ*, the surface albedo is set to a value of 0.12 for all calculations, independent of the solar zenith angle. Using RRTM calculations of albedo at 22 roughly evenly spaced values of *μ*, we interpolate *f*_{α}(*μ*) to a grid in *μ* with spacing 0.001, calculate the planetary albedo 〈*α*〉 from Eq. (6), and find the value of *α*〉. The dependence of albedo on *μ* is shown in Fig. 3; atmospheric absorption results in generally lower values of albedo than in the pure scattering cases above as well as lower sensitivity of the albedo to zenith angle. For partly or fully cloudy skies, the albedo is approximately linear in the zenith angle. Note that *f*_{α}(*μ*) here is not necessarily monotonic, as it decreases for very small *μ*. This implies that the inverse problem can return two solutions for

For clear skies, biases in 〈*α*〉 are nearly equal in magnitude for *α*〉 are much larger for ^{−2}. While we have only tabulated biases for the *U.S. Standard Atmosphere, 1976*, results are similar across other reference atmospheric profiles.

## 4. Diurnal and annual averaging

Thus far, we have presented examples of albedo biases only for the case of planetary-mean calculations. The absorption-weighted zenith angle can also be calculated and compared to daytime-weighted and insolation-weighted zenith angles for the case of diurnal- or annual-average solar radiation at a single point on Earth’s surface, using Eq. (5). The latitude and temporal-averaging period both enter into the calculation of the probability density function *P*(*μ*) as well as the bounds of the integrals in Eq. (5). We will look at how *f*_{α}(*μ*) as calculated by RRTM, for the *U.S. Standard Atmosphere, 1976*, and the mixed-sky cloud profile of RS99.

*ϕ*, the cosine of the zenith angle is given by

*μ*(

*h*) = cos

*ϕ*cos[

*π*(

*h*− 12)/12], where

*h*is the local solar time in hours. Since time

*h*is uniformly distributed, this can be analytically transformed to obtain the probability density function

*μ*< cos

*ϕ*. For the equinoctial diurnal cycle, daytime weighting gives

^{−2}, respectively. For clear-sky calculations (not shown), results are also similar to what we found for planetary-average calculations: the two choices are almost equally biased, with albedo underestimated by about 0.5% when using

For the full annual and diurnal cycles of insolation, *P*(*μ*) must be numerically tabulated. For each latitude band, we calculate *μ* every minute over a year, and construct *P*(*μ*) histograms with bin width 0.001 in *μ*, then we calculate the insolation-weighted, daytime-weighted, and absorption-weighted cosine zenith angles and corresponding albedos (Fig. 5). For partly cloudy skies, the insolation-weighted zenith angle is a good match to the absorption-weighted zenith angle, with biases in albedo of less than 0.2%. Albedo biases for the daytime-weighted zenith angle are generally about 2%–3%, with a maximum of over 3% around 60° latitude. The solar absorption biases at the equator are similar to those found in the equinoctial diurnal average, though slightly smaller. Overall, these findings indicate that insolation weighting is generally a better approach than daytime weighting for representing annual or diurnal variations in insolation.

## 5. Discussion

The work presented here addresses potential climate biases in two major lines of inquiry in climate science. One is the use of radiative–convective equilibrium, either in single-column or small-domain cloud-resolving models, as a framework to simulate and understand important aspects of planetary-mean climate, such as surface temperature and precipitation. The second is the increasing use of idealized three-dimensional general circulation models (GCMs) for understanding large-scale atmospheric dynamics. Both of these categories span a broad range of topics, from understanding the limits of the circumstellar habitable zone and the scaling of global-mean precipitation with temperature in the case of radiative–convective models to the location of midlatitude storm tracks and the strength of the Hadley circulation in the case of idealized GCMs. Both categories of model often sensibly choose to ignore diurnal and annual variations in insolation so as to reduce simulation times and avoid unnecessary complexity. Our work suggests that spatial or temporal averaging of solar radiation, however, can lead to biases in total absorbed solar radiation on the order of 10 W m^{−2}, especially if the models used have a large cloud-area fraction.

The extent to which a radiative–convective equilibrium model forced by global-average insolation accurately captures the global-mean surface temperature of both the real Earth and more complex three-dimensional GCMs is a key test of the magnitude of nonlinearities in the climate system. For instance, variability in tropospheric relative humidity, as induced by large-scale vertical motions in the tropics, can give rise to dry-atmosphere “radiator fin” regions that emit longwave radiation to space more effectively than would a horizontally uniform atmosphere, resulting in a cooling of global-mean temperatures relative to a reference atmosphere with homogeneous relative humidity (Pierrehumbert 1995). This radiator fin nonlinearity can appear in radiative–convective equilibrium simulations with cloud-resolving models as a result of self-aggregation of convection with a large change in domain-average properties such as relative humidity and outgoing longwave radiation (Muller and Held 2012; Wing and Emanuel 2014). But many other potentially important climate nonlinearities—such as the influence of ice on planetary albedo, interactions between clouds and large-scale dynamics (including midlatitude baroclinic eddies and the clouds that they generate), and rectification of spatiotemporal variability in lapse rates—would be quite difficult to plausibly incorporate into a radiative–convective model. Thus, despite its simplicity, the question of how important these and other climate nonlinearities are—in the sense of how much they alter Earth’s mean temperature as compared to a hypothetical radiative–convective model of Earth—remains a fundamental and unanswered question in climate science.

The recent work of Popke et al. (2013) is possibly the first credible stab at setting up an answer to this broader question of the significance of climate nonlinearities. Popke et al. (2013) use a global model (ECHAM6) with uniform insolation and no rotation to simulate planetary radiative–convective equilibrium with column physics over a slab ocean, thus allowing for interactions between convection and circulations up to planetary scales. One could imagine a set of simulations with this modeling framework in which various climate nonlinearities were slowly dialed in. For example, simulations could be conducted across a range of planetary rotation rates as well as with a range of equator-to-pole insolation contrasts; progressively stronger midlatitude eddies would emerge from the interaction between increasing rotation rate and increasing insolation gradients, and the influence of midlatitude dynamics on the mean temperature of the Earth could be diagnosed. But the study of Popke et al. (2013) does not focus on comparing the mean state of their simulations to the mean climate of Earth; they find surface temperatures of approximately 28°C, which are much warmer than the observed global-mean surface temperature of approximately 14°C. The combination of warm temperatures and nonrotating dynamics prompts comparison of their simulated cloud and relative humidity distributions to Earth’s tropics, where they find good agreement with the regime-sorted cloud radiative effects in the observed tropical atmosphere.

The most obvious cause of the warmth of their simulations is that Popke et al. (2013) also find an anomalously low planetary albedo of about 0.2, much lower than Earth’s observed value of 0.3 (e.g., Hartmann 1994). Although part of the reason for this low albedo can be readily explained by the low surface albedo of 0.07 in Popke et al. (2013), the remaining discrepancy is large, in excess of 5% of planetary albedo. It is possible that this remaining discrepancy arises principally because of the lack of bright clouds from midlatitude storms. But our study indicates that their use of a uniform equatorial equinox diurnal cycle of insolation, with ^{−2} owing to zenith-angle considerations alone.

Simulations by Kirtman and Schneider (2000) and Barsugli et al. (2005) also find very warm global-mean temperatures when insolation contrasts are removed; both studies retain planetary rotation. Kirtman and Schneider (2000) obtain a global-mean surface temperature of approximately 26°C with a reduced global-mean insolation of only 315 W m^{−2}; realistic global-mean insolation leads to too-warm temperatures and numerical instability. Kirtman and Schneider (2000) offer little explanation for the extreme warmth of their simulations but apparently also choose to homogenize insolation by using an equatorial equinox diurnal cycle, with ^{−2}. Similar to Popke et al. (2013), Barsugli et al. (2005) also invoke a low planetary albedo of 0.21 as a plausible reason for their global warmth and explain their low albedo as a consequence of a dark all-ocean surface. This work, however, suggests that their unphysical use of constant *μ* = 1 may lead to a large albedo bias on its own. For RS99 clouds, we estimate an albedo of 28.8% for *μ* = 1, as compared to 34.6% for *μ* = ⅔, so their albedo bias may be as large as −5.8%, with a resulting shortwave absorption bias of +19.8 W m^{−2}. Use of these three studies (Kirtman and Schneider 2000; Barsugli et al. 2005; Popke et al. 2013) as a starting point for questions about the importance of climate nonlinearities may thus be impeded by biases in planetary albedo and temperature due to a sun that is too high in the sky. While it was not the primary focus of these studies to query the importance of climate nonlinearities, these studies nonetheless serve as a reminder that care is required when using idealized solar geometry in models.

Because global-mean temperatures are quite sensitive to planetary albedo, we have focused in this work on matching the top-of-atmosphere shortwave absorption. For either a radiative–convective model or a GCM, we expect biases in mean solar absorption to translate cleanly to biases in mean temperature. The bias in mean temperature *T*′, should scale with the bias in solar absorption ^{−2}), divided by the total feedback parameter of the model *λ* (W m^{−2} K^{−1}): *μ* ~ 0.58 will give both the correct planetary albedo and the correct partitioning of absorbed shortwave radiation for clear skies; however, for partly cloudy or overcast skies, a single value of *μ* cannot simultaneously match both the planetary albedo and the partitioning of absorbed shortwave radiation. Together with the correspondence between global precipitation and free-tropospheric radiative cooling (e.g., Takahashi 2009), the dependence of atmospheric solar absorption on zenith angle suggests that idealized simulations could obtain different relationships between temperature and precipitation owing solely to differences in solar zenith angle.

Finally, we note that the use of an appropriately averaged solar zenith angle still has obvious limitations. Any choice of insolation that is constant in time cannot hope to capture any covariance between albedo and insolation, which might exist because of diurnal or annual cycles of cloud fraction, height, or optical thickness. Furthermore, use of an absorption-weighted zenith angle will do nothing to remedy model biases in cloud fraction or water content that arise from the model’s convection or cloud parameterizations. We hope that the methodology and results introduced in this paper will mean that future studies make better choices with regard to solar-zenith-angle averaging and thus will not convolute real biases in cloud properties with artificial biases in cloud radiative effects that are solely related to zenith-angle averaging.

## Acknowledgments

Thanks to Kerry Emanuel, Martin Singh, Aaron Donohoe, Peter Molnar, and Paul O’Gorman for helpful conversations and to Dennis Hartmann, Aiko Voigt, and one anonymous reviewer for useful comments. This work was funded by NSF Grant 1136480: The Effect of Near-Equatorial Islands on Climate.

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