Climates of Warm Earth-like Planets. I. 3D Model Simulations

All simulations herein utilize ROCKE-3D (Way et al. 2017). ROCKE-3D is a generalized version of the Goddard Institute for Space Studies (GISS) ModelE2 (Schmidt et al. 2014) GCM. ROCKE-3D has been developed to allow a larger range of input variables than ModelE2 such as higher and lower atmospheric temperatures, rotation rates, atmospheric compositions, and pressures.

We use a baseline version of ROCKE-3D known as Planet_1.0, which is described in detail in Way et al. (2017). In summary, ROCKE-3D is a Cartesian gridpoint model run at 4° × 5° latitude×longitude resolution with 40 atmospheric layers and a top at 0.1 hPa. For this study the model is coupled to two different types of ocean with the same resolution as the atmosphere. The first is a "Q-flux" ocean (Russell et al. 1985), i.e., a thermodynamic mixed layer whose temperature is determined by radiative and turbulent energy exchange with the atmosphere. Lateral OHT is set to zero (Q-flux = 0). The second ocean is a fully coupled dynamic ocean with 13 vertical layers down to a possible depth of 5000 m (Russell et al. 1995). However, the ocean is limited to a depth of 1360 m for this study, which is the bottom of the 10th ocean layer in ROCKE-3D. This allows the model to come into radiative equilibrium faster than it would with the full-depth 5000 m ModelE2 default Earth ocean.

The experiments use a baseline model that is similar to that used by ModelE2 for modern Earth climate studies, but differs from that model in several ways. At model start the following properties were specified:

• 1.  
  Planetary obliquity and eccentricity are zero. This avoids the additional complications of unraveling effects related to seasonal cycles, as we see on Earth and Mars.
• 2.  
  Continental layout and land topography are roughly that of modern Earth. Some shallow ocean, sea, and lake regions were replaced with land at locations that tend to freeze to the bottom at lower insolations. For example, land replaces the Baltic, Mediterranean, Red, and Black Seas, and the Hudson Bay. The higher latitude seas freeze to the bottom at low insolations as a consequence of the zero-obliquity used. Currently ROCKE-3D is unable to interactively change ocean gridboxes that freeze to the bottom to land ice.⁵ The height of the replacement land is set to the average height of the neighboring land grid cells. As well, some land is replaced by ocean, where it may also cause problems in neighboring shallow continental shelf areas (e.g., between Australia and Indonesia). See Figure 8. A number of the default Earth boundary condition files were modified for these purposes (see Table 10), especially the river directions file, which now contains the river runoff direction for every grid cell. Runoff was also allowed for the Caspian Sea region, which otherwise does not have a runoff direction in the default Earth river directions file.
• 3.  
  The dynamic fully coupled ocean is a bathtub type ocean. Continental shelf regions are set to 591 m in depth (ocean model level 8 of 13), while the rest of the ocean is set to 1360 m (ocean model level 10 of 13). See Section 3.1 and Figure 8.
• 4.  
  Land albedo is set to 0.2 everywhere. The value is chosen to be the same as that used in the few non-aquaplanet simulations in Yang et al. (2014) to enable some inter-comparison. Those authors chose this value as a rough average albedo of a clay and sand mix. There is no glacial ice at model start, but snow is allowed to accumulate on land, which changes the albedo. Greenland and Antarctica maintain their present-day topographies though they have no land ice at model start. Note that the default land surface albedo in ROCKE-3D is spectrally flat.
• 5.  
  Lakes are allowed to shrink and expand, but excess runoff is transported downhill when the water exceeds the pre-defined lake height (also known as "sill depth").
• 6.  
  Soil texture was chosen to be comparable to a "clay and sand" mix as in Yang et al. (2014) for the majority of simulations herein. The fractions for each particle class were not explicitly defined in that study, so we set them to 0.5 each. However, this corresponds to a soil texture classified as a sandy clay on the soil texture triangle (Jury et al. 1991), an arbitrary choice with regard to soil physical properties of porosity (saturated capacity) and water holding capacity. The latter is also known as "available water" after soil has drained from saturation, excluding water too tightly bound to soil particles to be available for life. For plants, that lower threshold of unavailable water is generally considered to be the "wilting point," ∼−15 bar matric potential, though in nature it may be more extreme, with microbes also able to extract water at lower matric potentials. At the hygroscopic point water is bound to soil particles by molecular forces and cannot be evaporated. To set lower and upper bounds on the potential availability of soil water (see Paper III⁶ for details), we observed which simulations were very dry and very wet with the 50/50 clay/sand soil, and then we added to these end points more simulations with all sand soil (61D in Table 8) and all silt soil (22D and 72D in Table 8), which have, respectively, the lowest and highest water holding capacities. Texture-dependent soil physical properties are generally only estimated empirically, and representations differ among GCMs. Those used in ROCKE-3D's ground hydrology scheme for our chosen soil textures are listed in Table 1.
• 7.  
  No vegetation is included.
• 8.  
  Maximum land soil depth is 3.5 m uniformly around the globe, which is a restriction of the current ground hydrology scheme. Subsurface runoff is transported directly to the ocean. Solar radiative heating is allowed to penetrate to a soil depth of 3.5 m. This choice is appropriate for the length of day on modern Earth. However, for the slower rotation periods that we simulate, a variable penetration depth consistent with the length of the day would be appropriate to implement in future versions of ROCKE-3D.
• 9.  
  Since the canonical habitable zone is the region around a star where an orbiting planet with an Earth-like atmosphere (i.e., N₂, CO₂, CH₄, H₂O) could maintain surface liquid water (e.g., Hart 1979; Kasting et al. 1993), we prescribe present-day Earth surface atmospheric pressure (984 millibars), with N₂ as the major constituent and prescribed spatially uniform concentrations of the minor radiatively active constituents CO₂ = 400 ppmv⁷ and CH₄ = 1 ppmv and a modern Earth water vapor profile at model start. O₃ is set to zero in line with most previous exoplanet studies, with the exception of Godolt et al. (2015), who retained O₃ in most of their simulations. Aerosols (i.e., hazes from photochemical, volcanic, or dust-stirring or ocean-stirring processes) are also not explicitly included, although their presence as nuclei for cloud formation is implicit.
• 10.  
  H₂O is the only other radiatively active constituent besides CO₂ and CH₄. Its concentration varies in space and time in accordance with the parameterized convection, condensation, and evaporation physics and dynamical transport in the model. The effect of variable H₂O on atmospheric mass is neglected. The maximum monthly mean gridbox specific humidity (mass concentration of H₂O) in the lowest altitude model layer in the runs with the warmest surface temperatures were found to be ≲0.1 in any gridbox. Thus, our neglect of H₂O mass is a ∼10% or smaller effect in our simulations.

ROCKE-3D was run first at Earth's present-day sidereal rotation period t_rot (X001) = 1 and insolation (S0) = 1360.67 W m⁻², and then in a series of experiments at higher insolation and longer sidereal days; see Tables 2 and 3. Simulations were generally conducted for surface temperatures that remain lower than 400 K. This is close to the upper temperature limit for accurate calculations with our radiation scheme SOCRATES (Edwards & Slingo 1996; Edwards 1996) which uses HITRAN 2012 (Rothman et al. 2013). See Goldblatt et al. 2013 for details on the accuracy of HITRAN2008 versus HITEMP2010, although our use of HITRAN 2012 may imply even smaller differences with the HITEMP database.

The suite of simulations is summarized in Tables 2 and 3.

Most previous exoplanet GCM studies have utilized a static thermodynamic ocean with no OHT, because of its computational efficiency (e.g., Merlis & Schneider 2010; Shields et al. 2013; Yang et al. 2014; Wolf & Toon 2015; Turbet et al. 2016). In this section we examine the effect that a dynamic fully coupled ocean with OHT has on planetary climate and impressions of liquid water habitable zone extent.

Figure 1(a) shows the global mean surface temperature of ROCKE-3D as a function of insolation for sidereal day lengths of 1 Earth day to 256 Earth days for a 100 m deep thermodynamic Q-flux = 0 ocean. Our results are qualitatively similar to those of Yang et al. (2014), who used the NCAR CAM3⁸ in an aquaplanet configuration (their Figure 1). Recall again that we use a modern Earth-like topography, not an aquaplanet. Mean surface temperature increases sharply with insolation at present-day Earth's day length (X001), but is less sensitive as the day length increases. There are several quantitative differences between ROCKE-3D and the CAM models used by Yang et al. (2014) as far as one can compare our Earth-like land/sea mask to their aquaplanet configuration. For the Q-flux = 0 runs, ROCKE-3D is more sensitive to increasing insolation at present-day Earth's day length (X001) and at 16 sidereal days (X016). The trend reverses for day lengths of 64 days (X064) and larger. Consequently, the ROCKE-3D simulations divide clearly into two classes: moderate to rapidly rotating planets with high sensitivity to insolation, and slowly rotating planets with low sensitivity, as opposed to the CAM simulations (Figure 1 of Yang et al. 2014), for which sensitivity is a more continuously decreasing function of rotation period. We return to this question below.

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Our results for Earth's rotation period (X001) can also be compared with those in Figure 1 of Wolf & Toon (2015), who plotted climate sensitivity for their CAM3 & CAM4 results, but also for the LMDG⁹ results of Leconte et al. (2013a). ROCKE-3D is much more sensitive than the CAM3 and CAM4 results to a 10% increase in insolation (S0X = 1.1), but much less sensitive than the LMDG GCM, which has a large warming for insolation increases of 5%–10%. For a 20% increase (S0X = 1.2, the only other simulation comparable in Wolf & Toon 2015) ROCKE-3D is less sensitive than CAM4 because CAM4 warms dramatically at a 11%–12% insolation increase, while ROCKE-3D does not.

Figure 1(b) is identical to Figure 1(a) but for a dynamic fully coupled ocean. The results for day lengths of 64 d (X064) and longer are very similar to those for the thermodynamic ocean case. However, for the X001 day lengths the dynamic ocean model is significantly warmer (see Figure 2), and considerably less sensitive to insolation increases.

The climate sensitivity to changes in sunlight is usually expressed as the response of surface temperature to the absorbed sunlight rather than the insolation itself (see, e.g., Equation (2) of Wolf & Toon 2015). This quantity is shown in Figure 3 for both types of ocean. Climate sensitivity is highest for the fastest rotators and most weakly irradiated planets and is very small for the slow rotators, regardless of absorbed sunlight. The dynamic ocean simulations have a markedly lower climate sensitivity than the Q-flux = 0 simulations for fast rotation periods.

To explain these differences among models, Figures 5 and 6 shows the extent to which the two different ocean types affect aspects of the planetary (Bond) albedo, ocean ice fractional extent, and cloudiness at different levels of the model. The upper panels of Figure 5 show that at the lower insolation values, the rapidly and slowly rotating planets behave in opposite fashions: planetary albedo initially decreases with insolation in the shorter day length simulations (more so for the Q-flux = 0 ocean), but increases with insolation for longer day lengths. At higher insolations, all simulations exhibit increasing planetary albedo as insolation increases. This is partly explained by the sensitivity of the ocean ice fraction to insolation, rotation, and ocean dynamics. The zero-obliquity of this Earth-like world allows ocean ice to grow considerably at higher latitudes when the insolation is low, but the ice melts (and thus the surface albedo decreases) when either insolation increases or the length of day increases. The latter dependence occurs because heat is more efficiently transported poleward on slowly rotating planets with broader Hadley cells (Del Genio & Suozzo 1987). The higher ocean ice fractions for the Q-flux = 0 versus dynamic ocean at lower insolations clearly point to the role that a dynamic ocean plays in equator-to-pole heat transport.

The increase in planetary albedo with insolation for the longer day lengths is due to the now well-documented substellar cloud bank, as shown in previous 3D model simulations (e.g., Yang et al. 2014). This occurs because moist convection at low latitudes increasingly transports water vapor and cloud particles upward as insolation increases. The convection itself occupies a small area but produces more extensive anvil clouds at the levels where water vapor and cloud particles detrain into the environment (Fu et al. 1990). On Earth, the anvil clouds are primarily a feature of the upper troposphere. This is also true in the current experiments at low insolation. Interestingly, though, as insolation increases, convection depth seems to decrease, because high cloud cover decreases as middle level cloud cover increases in Figure 6. The latter clouds apparently dominate the optical thickness and thus the dependence of planetary albedo on insolation. Low-level clouds decrease with increasing insolation at low insolation values, similar to how 3D Earth climate models respond to 21st century increases in anthropogenic greenhouse gas concentrations (Klein et al. 2017). However, at higher insolation low cloud cover tends to increase with insolation instead. For example, in Figure 4 the decrease in low clouds from insolation values of 1.0 to 1.2 and the increase from 1.2 to 1.4, are both related to relative humidity, mostly at high latitudes. This make sense because there is less shallow convection at high latitudes.

The ocean dynamics are strongly tied to the rotation rate of the planet. This can most clearly be shown by looking at a range of rotation rates for a fixed insolation, as shown in Figure 7. At the faster rotations (X001–X004) the ocean currents are very Earth-like, with gyres in the northern hemisphere in both ocean basins and an Antarctic circumpolar current in the southern mid-latitudes. The gyres clearly transport heat poleward, as the real Gulf Stream does on Earth, but because there is more sea ice on this planet due to the zero-obliquity, the transport is very important at low S0X and fast rotation. The dynamic ocean runs have less sea ice than their Q-flux = 0 equivalents and thus the dynamic ocean climates are generally warmer, as described earlier and shown in Figure 2. The trend goes up until about X016, by which time the Hadley cell now stretches to the pole and the OHT is directed equatorward with no gyres. By this rotation the sea ice is already gone, so the addition of OHT does not do very much, hence the insensitivity of the results to OHT at slower rotations.

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Despite the advantages of a dynamic ocean, limitations exist as well. The ROCKE-3D ocean model utilizes the Gent & McWilliams (1990) Earth-specific parameterization for mesoscale mixing by unresolved eddies, which is appropriate for the quasi-geostrophic ocean flow on a rapidly rotating planet. Little information exists about the behavior of mesoscale ocean eddies in the slowly rotating dynamical regime, although several studies have explored the rotation dependence of overall OHT (Farneti & Vallis 2011; Cullum et al. 2014). However, in a Proxima Centauri b simulation with ROCKE-3D, it was found that changing the mesoscale diffusivity had little effect (Del Genio et al. 2018). As mentioned previously, the fully coupled ocean simulations have only two depths: 591 m near continent boundaries and 1360 m elsewhere (see Figure 8). This allows the model to come into thermodynamic equilibrium faster than it would if we used all 13 ocean model layers to 5000 m, and hence lowers the total computational time per simulation. The results of Russell et al. (2013) and Hu & Yang (2014) suggest that OHT increases with ocean depth, with concomitant effects on climate, although this is likely to be partly compensated by increased atmospheric transport (Stone 1978). Ocean bottom topography, the location of continents, and the steepness of continental boundaries all influence ocean circulation and thus heat transports. Likewise, salinity is specified as that of Earth's ocean, which need not be the case on another planet. This has implications for the density-driven component of the ocean circulation and especially the freezing temperature (see Del Genio et al. 2018), which in turn affects albedo. Finally, the impact of tides, which are strongly correlated with the size/mass and distance of the Earth's moon, is neglected in ROCKE-3D. These are not strictly limitations, since exoplanets can be expected to have a large variety of ocean depths and either no moons or moons of varying size and distance, and these will likely be unconstrained by observations for the foreseeable future. It may, however, be a consideration for deep paleo-Earth studies simulating time periods when the moon was closer to Earth and the tides were higher.

Clouds are considered the most uncertain aspect of GCMs, accounting for much of the inter-model spread in estimates of the global climate sensitivity of Earth to increases in greenhouse gas concentrations (Vial et al. 2013). A coupled ocean-atmosphere climate model must be in global top-of-atmosphere radiation balance at something close to Earth's observed surface temperature in order to be used for climate change applications. Given the uncertainties in GCM cloud parameterizations, free parameters are usually adjusted within reasonable limits to achieve balance while not overly compromising other aspects of the simulated climate (Mauritsen et al. 2012). For exoplanets, no effective observational constraints exist, nor is there any guarantee that a model developed for modern Earth will reach radiative balance on its own. Thus, adjustments of the free parameters may be necessary. Here, we consider two cloud uncertainties to illustrate the limitations of 3D studies of exoplanets and the conclusions that can be drawn despite the limitations.

The Yang et al. (2014) baseline model has a global surface temperature of 287 K for Earth's rotation period (their Figure 1(c)), remarkably close to Earth's observed surface temperature. Given that the Yang et al. (2014) runs are aquaplanets, while those herein are Earth-like, it should not be surprising that ROCKE-3D's Q-flux = 0 run (see Table 9) is cooler (265 K), while our dynamic ocean simulation (Table 8) has a temperature of 284 K. The reason for the difference between these temperatures and that of modern Earth is that the planet being simulated is clearly not Earth (e.g., it has no aerosols or vegetation). Most importantly, the planet has zero-obliquity and eccentricity. Earth's present-day obliquity warms the poles and cools the tropics. We expect the first of these to be more important because it reduces sea ice and snow cover and decreases the Bond albedo, at least for planets in the quasi-geostrophic dynamical regime that have large equator-pole temperature gradients (Del Genio 2013; Madeleine et al. 2014). Thus, although the global temperature of our idealized planet cannot be constrained, if such a zero-obliquity planet existed, it would likely be colder than Earth. The warmer temperature of the dynamic ocean model reflects the role that ocean thermal inertia and heat transport play in limiting sea ice extent.

To test the sensitivity of the Q-flux = 0 model temperature to the choice of model tuning, we created an alternate version of our baseline planet (see the legend for "t" in Table 2 and X001B in Figure 1(a)) with free parameters in the cloud parameterization chosen differently but still within the range of uncertainty for modern Earth to produce radiation balance at a different climatic state. Specifically, we changed the values of two parameters that determine the threshold relative humidity at which stratiform clouds begin to form (one for free troposphere clouds, one for boundary layer clouds) and thus the cloud fraction. We also changed a parameter that affects how quickly small cloud ice crystals grow into large snowflakes by gravitational coalescence and precipitate out of the cloud, thus altering the cloud's ice water content and optical thickness.

Figure 1(a) (X001B, dashed black line) shows the global mean surface temperature of this alternate model as a function of insolation for Earth's rotation period, for comparison with the baseline model (X001, solid black line). The alternate model has a global mean surface temperature of 259.5 K for Earth's insolation (S0X = 1.0) compared to the baseline model's 265.4 K (a difference of ∼6 K), but its sensitivity to insolation change is fairly similar to that of the baseline model (see Figure 3(a)). The result is that this planet can sustain a 20% increase in insolation and remain only slightly warmer (11K) than the baseline model does for a 10% increase in insolation. This implies that model-based estimates of the edges of the habitable zone, whether 1D or 3D, are only weak constraints on where habitable planets may be found, since model clouds can be tuned to give a variety of climates.

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The more important question is whether any definitive statements can be made at all about the effects of clouds on exoplanet climates. The primary finding of Yang et al. (2014) is that a planet can remain habitable at much higher values of insolation if it is slowly rotating than if it is rapidly rotating. This occurs because at slow enough rotation rates (long day lengths) for significant day–night temperature differences to arise, convergence on the dayside leads to rising motion and moist convection that produces a shield of high, optically thick clouds that limit warming there and stabilize the planet's climate. The fact that ROCKE-3D produces qualitatively similar behavior (Figure 1) is an encouraging sign that this might be a robust feature of planetary climates. In our own solar system, this process may be relevant to the question of whether ancient Venus was habitable (Way et al. 2016).

On the other hand, moist convection is one of the most challenging and uncertain aspects of terrestrial climate models (Del Genio 2015). The dayside cloud that stabilizes the climate at slow rotation is the end result of a series of parameterized cloud and convection processes: a decision to initiate convection, an assumption about the updraft mass flux and vertical velocity, and an assumption about how much of the condensate that forms in the updraft is transported upward and detrained along with saturated updraft vapor. ROCKE-3D does this by diagnosing an updraft speed profile, assuming particle size distributions and size-dependent fall speeds, and interactively calculating the fraction of the condensate that precipitates versus being transported upward to form a thick anvil cloud (Del Genio et al. 2005, 2007). Each of these steps introduces uncertainty, e.g., the parameterization in the baseline model overestimates the ice carried upward by convective events (Elsaesser et al. 2017). Since ∼80% of the cloud ice that results from deep convection in the terrestrial GCM is the result of upward transport of particles rather than in situ ice formation from supersaturated air, it is reasonable to ask whether the rotation dependence of surface temperature is parameterization-dependent.

To address this, we performed another sensitivity test in which we assume that all condensate in the convective updraft precipitates. We denote such runs "convective condensate precipitates" in Tables 2, 3, 8 and 9. Thus, middle and upper level clouds can only form from supersaturated vapor detrained by convective updrafts or created by large-scale resolved upward motion. We conducted one simulation for four rotation periods at different insolations (open circles in Figure 1). The simulations without upward transport of condensate are predictably warmer, but only by a couple of degrees, than the simulations with the full convective physics, independent of rotation period. That is, cloud that forms in situ from the vapor detrained by the updraft or the resolved upward motion is primarily what stabilizes the temperature. Insensitivity of the results to this type of structural modification, rather than simply a parameter change, strengthens the case for the conclusions of Yang et al. (2014).

This supports the idea that the fundamental behavior operating in all these models is not the details of the cloud physics but the interaction between the dynamics and radiation. The transition from a very sensitive to a weakly sensitive climate as day length increases depends on two transitions in dynamical regime that determine how heat is transported. The first is from the quasi-geostrophic regime characteristic of rapidly rotating planets to the quasi-barotropic regime of more slowly rotating planets. This transition has been well-studied for rocky planets (Williams & Holloway 1982; Del Genio & Suozzo 1987; Del Genio et al. 1993; Allison et al. 1995; Navarra & Boccaletti 2002; Showman et al. 2013). Both regimes are dominated by poleward heat transport. In the quasi-geostrophic regime this is accomplished by a Hadley circulation at low latitudes, with mean rising motion near the equator and sinking motion in the subtropics, and by baroclinically unstable eddies that produce low and high pressure centers and fronts at higher latitudes. In the quasi-barotropic regime, the Hadley cell spans most latitudes and dominates the global heat transport. This transition occurs approximately when the Rossby radius of deformation (the spatial scale of rapid baroclinic eddy growth) approaches the size of the planet (Del Genio et al. 1993; Edson et al. 2011; Showman et al. 2013). It differentiates the atmospheric circulations of Earth and Mars from those of Venus and Titan. Our simulations with 1 day rotation period are in the quasi-geostrophic regime, while the 16 day period simulations are in the quasi-barotropic regime. In Figure 9 we show how the Hadley cell changes in size when going between these regimes and how it clearly affects the surface temperatures and ocean ice fraction at high latitudes (see Figure 5).

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The second transition is from a circulation that transports heat poleward to a diurnally driven circulation that transports heat from a dayside region of rising motion to a nightside region of sinking motion, with day–night transport both over the poles and across the terminators. The day–night circulation has been explored by Joshi et al. (1997), Joshi (2003), and Yang et al. (2013) for synchronously rotating planets and by Yang et al. (2014) for slowly rotating but asynchronous planets. In our simulations the transition appears to occur between 32 and 64 day rotation periods. This can be understood by considering the radiative relaxation timescale t_rad = pc_pT/(gF), where p is pressure, c_p is the specific heat at constant pressure, T is the temperature, g is the acceleration of gravity, and F is the emitted thermal flux to space. For Earth, heated at 1 au by the Sun, t_rad ∼ 1–2 months, depending on the pressure level one chooses for the calculation. When the length of the solar day t_sol ≪ t_rad, day–night temperature differences are small, and equator-pole temperature gradients drive the circulation. When t_sol ≳ t_rad, day–night temperature contrasts become important and the circulation develops a strong diurnally driven component. This regime transition occurs in the 32–64 day rotation interval (for which t_sol is only slightly longer than the rotation period) for the planets we simulate. Showman et al. (2015) invoked a similar argument to define the boundary between circulations characteristic of weakly irradiated Jovian planets and strongly irradiated hot Jupiters.

These changing transport patterns among the three dynamical regimes matter for habitability because they determine where optically thick clouds form and thus where sunlight is more strongly reflected. They also matter for the water cycle because of their effect on precipitation patterns, as we will explore in future papers in this series. Despite uncertainties in parameterized cloud physics, the only requirement of the cloud/convection physics is that thick clouds form where large-scale upward motion is prevalent. This basic behavior should be model-independent for any mass flux cumulus parameterization, since it requires only that rising air adiabatically cool and eventually saturate.

Figure 10 shows the global mean short wave (SW) and long wave (LW) cloud radiative forcing (CRF), as well as the column-integrated cloud condensed water (liquid + ice; CLDW) as a function of incident solar flux S_o and rotation period. CRF measures the effect of clouds on a planet's top-of-atmosphere radiation budget (e.g., Cess et al. 1990). We use the convention that downward/upward fluxes (i.e., heating/cooling of the planet) are positive/negative and define the absorbed SW flux as

$$\begin{eqnarray}&&Q={S}_{{\rm{o}}}(1-A)/4,\end{eqnarray} \tag{ 1 }$$

where A is the planetary (Bond) albedo, and the emitted LW flux

$$\begin{eqnarray}&&F=\sigma {T}_{{\rm{e}}}^{4},\end{eqnarray} \tag{ 2 }$$

where T_e is the equilibrium temperature.

Letting subscript c indicate the fluxes that would exist in clear skies (calculated by a separate offline call to the radiation parameterization that ignores the clouds), and subscript o denote the fluxes that exist in overcast skies, we define^{10 ,11}

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{SWCRF} & = & Q-{Q}_{{\rm{c}}}=[{{fQ}}_{{\rm{o}}}+(1-f){Q}_{{\rm{c}}}]\\ & & -\,{Q}_{{\rm{c}}}={{fS}}_{{\rm{o}}}({A}_{{\rm{c}}}-{A}_{{\rm{o}}})/4,\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{LWCRF} & = & {F}_{c}-F=f({F}_{{\rm{c}}}-{F}_{{\rm{o}}})\\ & = & \sigma f({T}_{\mathrm{ec}}^{4}-{T}_{\mathrm{eo}}^{4}),\end{array}\end{eqnarray} \tag{ 4 }$$

where f is the cloud fractional area.

Positive/negative values of CRF indicate heating/warming by clouds, respectively. SWCRF < 0 because clouds are more reflective than the clear sky and most surfaces and thus make a planet cooler than it would be without clouds, while LWCRF > 0 because clouds absorb upwelling thermal radiation and re-radiate it at a lower temperature, producing greenhouse warming. Note that CRF depends not only on the coverage and properties of clouds but also on the properties of the clear sky atmosphere and planet surface through the terms Q_c and F_c. Thus, changes in CRF in response to a climate forcing such as an increase in incident solar flux capture the cloud feedback only in the context of the accompanying clear sky changes rather than isolating the cloud feedback itself (Soden & Held 2006).

Figure 10 shows that SWCRF increases monotonically in magnitude with incident solar flux and is somewhat stronger for the slowly rotating planets, as expected. Much of the insolation dependence reflects the increase in S_o itself, but for a doubling of S_o, SWCRF increases by about a factor of 3, indicating that cloud fraction and/or the albedo difference between overcast and clear skies increases with insolation as well. Cloud vertically integrated water (CLDW) actually increases more slowly with insolation for the slowly rotating planets, but this is misleading because the clouds on the slow rotators are much denser on the dayside, where they can contribute to SWCRF. LWCRF is smaller than SWCRF and increases slowly as incident solar flux increases. This does not appear to be due to increasing cloud height, given that high cloud cover decreases and middle level cloud cover increases with insolation, at least for the lower insolation values (Figure 6). Furthermore, the high cloud decrease exceeds the middle cloud increase, so cloud fraction does not appear to be the cause. Instead, the similarity between the LWCRF and cloud water dependences on temperature and insolation suggest that cloud opacity, and its implications for the radiating temperature contrast between cloud and clear skies, controls the LWCRF behavior. The net result of the SWCRF and LWCRF behavior is that the clouds exert a net cooling effect on the climate that becomes stronger as the incident flux increases, i.e., by this simple measure the net cloud feedback is negative.

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Rotation period does not directly affect temperature, so the cooling of the planet as rotation slows in Figure 1 must be the result of climate feedbacks that occur as the dynamical regime changes in response to the changing rotation. Figure 11 shows the LWCRF, SWCRF, and column-integrated water vapor as a function of rotation period for different values of S0X. From 1 to 16 day periods, SW cloud cooling generally weakens, though this is partly offset by an increase in LW cloud warming. Furthermore, sea ice decreases with increasing rotation period as the Hadley cell expands (Figure 9). Thus, neither cloud nor sea ice feedbacks can explain the generally cooler temperatures and lower climate sensitivity as rotation slows. Instead, the dramatic decrease in water vapor with increasing rotation period (>50% for most values of S0X) reduces the greenhouse effect of the clear sky atmosphere. Water vapor continues to decrease with longer rotation periods for periods up to 128 days, but SWCRF becomes increasingly negative and LWCRF becomes slightly less positive. All three feedbacks are thus negative, consistent with the large separation in temperature in Figure 1 between the more rapidly and more slowly rotating planets.

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