Abstract

The mechanisms that govern the response of shallow cumulus, such as found in the trade wind regions, to a warming of the atmosphere in which large-scale atmospheric processes act to keep relative humidity constant are explored. Two robust effects are identified. First, and as is well known, the liquid water lapse rate increases with temperature and tends to increase the amount of water in clouds, making clouds more reflective of solar radiation. Second, and less well appreciated, the surface fluxes increase with the saturation specific humidity, which itself is a strong function of temperature. Using large-eddy simulations it is shown that the liquid water lapse rate acts as a negative feedback: a positive temperature increase driven by radiative forcing is reduced by the increase in cloud water and hence cloud albedo. However, this effect is more than compensated by a reduction of cloudiness associated with the deepening and relative drying of the boundary layer, driven by larger surface moisture fluxes. Because they are so robust, these effects are thought to underlie changes in the structure of the marine boundary layer as a result of global warming.

1. Introduction

The idea that the atmospheric relative humidity does not change with changing surface temperatures has long been a staple of climate change studies (Held and Soden 2000). This idea is the basis for the water vapor feedback, wherein the absolute humidity must rise with rising temperatures if the relative humidity is to remain constant. Because an atmosphere that maintains a constant distribution of relative humidity could be expected to maintain a constant distribution of clouds, it is not surprising that cloud feedbacks emerged quite late as a theme in climate change research, and then largely qualitatively (e.g., Plass 1956; Möller 1963). The first quantitative studies to formally incorporate cloud feedbacks in a representation of the climate system began to appear in the early 1970s (Schneider 1972), partly motivated by interest in aerosol–cloud interactions (Twomey 1971). A further decade passed before clear ideas of why and how cloudiness might depend on surface temperature began to emerge. Paltridge (1980), for instance, argued that even in an atmosphere with constant relative humidity, clouds should consist of more liquid water, and be brighter, in a warmer climate. The physical basis for his argument is the recognition that the adiabatic lapse rate of liquid water, which accompanies the expansional cooling of saturated air, increases with temperature. Such an effect can be thought of as a negative feedback, as with warming temperatures clouds should be comprised of more water and reflect more sunlight to space, increasing Earth’s albedo and causing a reduction in the radiative forcing that initially gave rise to warmer temperatures. This feedback is sometimes called the optical depth feedback or the liquid water lapse-rate feedback. The latter terminology identifies it as the counterpart to the more common idea of a temperature lapse-rate feedback; both would act to reduce the dependence of surface temperature on the radiative forcing.

Marine boundary layer clouds have emerged as a focal point in the quest to understand how cloud feedbacks contribute to climate change. The focus on these particular cloud systems arises largely because diagnostic studies show that the radiative budget at the top of the atmosphere is especially sensitive to changes in tropical and subtropical marine boundary layer clouds (Hartmann and Short 1980) and because model-based estimates of the climate sensitivity are sensitive to the representation of such clouds (Bony and Dufresne 2006). Increasingly comprehensive frameworks, developed around a hierarchy of models, have begun to emerge for studying how such clouds might change with a changing climate. For instance, Zhang and Bretherton (2008) propose a conceptual framework that links the background structure of subtropical marine boundary layers to changes in the tropical sea surface temperature. In this conceptual framework, like in many others, a central feature is that the large-scale processes act to maintain a constant background relative humidity, in which context ideas such as the liquid water lapse-rate feedback gain relevance.

The rates at which temperature decreases and liquid water is produced in the adiabatic expansion of rising air are not the only processes that are temperature dependent. Held and Soden (2000) point out that in a constant relative humidity atmosphere, surface moisture fluxes should increase in proportion to the saturation specific humidity at the surface. Changes in the surface moisture fluxes can in turn be expected to influence the development of the cloud-topped marine boundary layer. These considerations motivate us to more generally consider the question of how clouds would be different in an atmosphere that is uniformly warmed, and over a surface that is warmed by the same amount, but the relative humidity, winds, and large-scale forcings are otherwise assumed not to change, and clouds are assumed shallow enough that precipitation is not a factor. The question lends itself to the use of large-eddy simulation (LES) wherein a time-scale separation argument justifies the exploration of how the cloud-topped marine boundary layer develops given an imposed large-scale environment. We take such an approach to investigate the relative strength of the liquid water lapse-rate feedback, as well as possible effects of surface-flux changes that accompany a change in temperature with constant relative humidity.

We study the evolution of transient simulations using a case based on observations of shallow cumulus layers as observed during the recent Rain in Cumulus over the Ocean (RICO) field study (Rauber et al. 2007). The widespread occurrence of shallow cumulus within the trades and the sensitivity of the climate system to their representation (Bony and Dufresne 2006) make shallow cumulus trade wind layers a natural starting point. The assumption of a constant relative humidity surely is not a complete representation of how the large-scale environment is expected to change with a warming climate, as it neglects possible contributions from changing large-scale pressure gradients, changes in the atmospheric stability, and the magnitude of the large-scale vertical motion, among other factors. The failure to allow these other processes to compete with those that we study, so as to determine the equilibrium structure of the trade wind layer, means that our results cannot be interpreted as indicative of the net change in boundary layer cloudiness in a climate that warms while large-scale processes act to maintain a constant relative humidity. Rather, the simulations are designed to help isolate and articulate the role of specific processes, in particular the thermodynamic tendency toward increasing liquid water lapse rates within clouds and moisture fluxes at the surface in a warming climate; these are issues that we believe are central to a more comprehensive understanding of how the marine boundary layer depends on the mean temperature of the working fluid.

The remainder of the manuscript is organized as follows. In section 2 we present the lapse-rate feedback ideas of Paltridge (1980) and discuss more fully how and why surface fluxes should be allowed to increase with temperature in an atmosphere whose relative humidity does not change. The design of the large-eddy simulations and a description of the code used to conduct them is presented in section 3. Results of the simulations are presented in sections 4 and 5. In section 6 we restate our major findings and reflect on how limitations in our formulation of the problem might influence the more general applicability of our results. Conclusions are presented in section 7.

2. Temperature-dependent processes in a constant relative humidity atmosphere

In this section we explore some of the boundary layer and cloud-related processes that one expects to depend on temperature T in an atmosphere where large-scale processes act to keep relative humidity constant. Where appropriate this includes a review of earlier literature on the subject.

a. The liquid water lapse-rate feedback

Paltridge’s argument follows from the thermodynamic properties of saturated air. For saturated adiabatic ascent, the adiabatic liquid water lapse rate can, to a good degree of approximation, be written as follows:

 
formula
 
formula

Here qs and qt are the saturation and total water vapor specific humidities, respectively; p is the total pressure; pd is the partial pressure of “dry” air; Lυ is the enthalpy of vaporization, Rυ and Rd are the gas constant for vapor and dry air, respectively; and g is the magnitude of the gravitational acceleration. At a temperature of 290.5 K the liquid water lapse rate is 2.18 g kg−1 km−1, increasing to 2.25 g kg−1 km−1 as temperature is increased by 2 K (Fig. 1).

Fig. 1.

Liquid water lapse rate Γ as a function of temperature. The dashed line shows the cloud-base temperature taken from the RICO intercomparison case.

Fig. 1.

Liquid water lapse rate Γ as a function of temperature. The dashed line shows the cloud-base temperature taken from the RICO intercomparison case.

Following the notation of Somerville and Remer (1984), we define a liquid water sensitivity parameter f that measures the relative change in the liquid water q accompanying a temperature change δT such that

 
formula

If we assume that q ∝ Γh, where h denotes the height above cloud base (assuming h is constant), then it follows that

 
formula

At a temperature of 290.5 K and a pressure of 930 hPa, f = 1.6% K−1 Calculating f in this manner is appropriate provided that the cloud liquid water scales linearly with the adiabatic lapse rate of liquid water. As discussed by Rieck (2011), such an assumption is implicit in many models of shallow convection (e.g., Betts 1982) and borne out by observations of boundary layer clouds (Brenguier et al. 2000). The liquid water sensitivity parameter calculated with Eq. (4) is smaller than the sensitivity of saturation specific humidity to temperature, which in a similar temperature range is near 6% K−1 However, like δ(lnqs)/δT, f is a decreasing function of temperature, so the liquid water lapse-rate feedback is most potent at colder temperatures (Betts and Harshvardhan 1987).

A simple set of calculations shows that even for small values of f the temperature dependence of the lapse rate can lead to changes in the albedo that would substantially modify the radiative forcing associated with the doubling of atmospheric CO2. If we take the temperature response to a doubling of CO2 to be 3 K, we would expect cloud liquid water to increase by 4.5%, all other things being equal. The albedo dependence on cloud optical thickness τ can be roughly expressed as

 
formula

where 6.8 ≈ 1/(1 − g), where g = 0.15 is the asymmetry parameter, β ≈ 2 is a constant factor related to the efficiency of light scattering by water droplets and the subadiabaticity of the cloud, re is the effective radius, h is the depth of the cloud layer, and ρw is the density of liquid water (Stephens 1978; Brenguier et al. 2000). Assuming clouds with an initial optical thickness of about 10, so that their albedo is about 0.6 (which is likely an exaggeration of their reflectiveness, given the simplified nature of the above expression), it is straightforward to show that , which as a rough estimate shows that (all other factors being equal) changes in the liquid water lapse rate would increase the mean cloud albedo from 0.60 to 0.612, or by about 2%. Between 75 and 80 W m−2 of the reflected incoming solar radiation is attributed to clouds (Stevens and Schwartz 2012). If all clouds were affected by such an albedo increase, this would increase the reflected shortwave radiation by about 1.5 W m−2. However, not all clouds can be expected to be equally sensitive to the temperature dependence of the lapse rate. For instance, in some clouds liquid water is limited by precipitation, and in deeper clouds the albedo saturates at high optical depths and becomes insensitive to perturbations in the liquid water amount. If we assume that roughly one-third of the clouds are susceptible to such a change, this implies an effect whose magnitude is about −0.5 W m−2, which is about 15% of the initial radiative forcing of 3.7 W m−2 that is normally ascribed to a doubling of CO2. This radiative forcing is commensurate with the changes in the direct radiative effects thought to arise from anthropogenic contributions to atmospheric aerosols (Forster et al. 2007).

Initial studies on cloud liquid water lapse-rate feedbacks used similar considerations as the one above and suggested that through its effect on cloud albedo the liquid water lapse rate constitutes a significant negative feedback of surface temperature on the radiative forcing (Paltridge 1980; Charlock 1982; Somerville and Remer 1984). The more recent literature on the topic, using remote sensing data and cloud-resolving simulations, has been less conclusive, even with respect to the sign of the feedback (Tselioudis et al. 1992; Del Genio and Wolf 2000; Xu et al. 2010). An analysis of one year of International Satellite Cloud Climatology Project (ISCCP) data by Tselioudis et al. (1992) shows that while for cold clouds f is consistent with the hypotheses arising from simple models, for warm clouds it changes sign. Del Genio and Wolf (2000) and Xu et al. (2010) highlight the role of complicating factors: as surface temperatures increase, cloud liquid water lapse-rate changes are modulated by other changes, such as the height of cloud base, cloud fraction, the thickness of the cloud layer, and surface fluxes, a topic that we address next.

b. Surface-flux feedbacks

What are possible effects of surface-flux changes that accompany a change in temperature when relative humidity is constant? To a good approximation the fluxes of moisture at the air–sea interface can be parameterized by the bulk aerodynamic formulas (Fairall et al. 2003),

 
formula

where V is a positive definite exchange velocity that is linearly related to the surface wind speed through an exchange coefficient that depends on the stability. Here we denote surface values by subscript 0 so that T0 is the sea surface temperature and q2m denotes the specific humidity at a height of 2 m above the surface. It follows that Q can be approximated as

 
formula
 
formula

Differences between T and T0 [hence in qs(T) and qs(T0)] are small over the tropical oceans, and this approximation is a good one. Equation (8) shows that in a warming atmosphere with a constant relative humidity, if V remains constant, surface moisture fluxes will increase substantially—to a first approximation proportionally to qs(T0).

It is important to understand what increase in surface moisture flux is necessary to maintain a constant relative humidity. Figure 2 illustrates two layers (boxes) that have the same relative humidity initially, but different temperatures (t = 0, T2 > T1, left top and bottom). The term Q1 denotes the moisture flux into a layer at temperature T1 with a relative humidity . The change in humidity of the (cold) layer during a period δt of constant moisture supply Q1 follows from , where α measures the depth of the layer being moistened multiplied by the time interval over which the moistening flux is applied. Likewise, . If the two layers are to humidify (in the relative sense) by an equal amount , this requires that . In other words, if the moisture flux scales with the saturation humidity at the working temperature of the layer, then the moisture flux will increase by precisely the amount required to increase the relative humidity of the warmer layer by the same amount as a layer with initially the same relative humidity with a different working temperature.

Fig. 2.

(left) Schematic diagram illustrating two boxes with identical and depth α/δt but different temperatures (T2 > T1, so that q2 > q1). (right) After a period δt of moistening during which the moisture flux is held constant for each layer, differences in the relative humidity arise that depend on the respective surface fluxes.

Fig. 2.

(left) Schematic diagram illustrating two boxes with identical and depth α/δt but different temperatures (T2 > T1, so that q2 > q1). (right) After a period δt of moistening during which the moisture flux is held constant for each layer, differences in the relative humidity arise that depend on the respective surface fluxes.

When asking how clouds will change in an atmosphere where large-scale processes act to keep relative humidity constant, it is thus appropriate to allow surface moisture fluxes to increase following Eq. (6). Failing to do so would, in the absence of other sources of moisture, cause the warmer atmosphere to dry in a relative sense, as compared to the cooler atmosphere. However, the surface moisture flux is important to the evolution of the turbulence and cloud dynamics, with the rate of deepening of the cloud layer depending directly on this quantity (Stevens 2007). Changes in the surface fluxes hence drive changes in the structure of the cloud layer. In our methodology, outlined in the next section, we use simulations that can separate surface-flux effects from effects associated with the changing liquid water lapse rates, by choosing to use either variable or constant surface fluxes.

We note that aside from the liquid water lapse-rate and surface-flux feedback, other temperature-dependent processes are at play in a constant relative humidity atmosphere. For instance, we can consider the mixing of air masses that have different constant relative humidity (and whose mixtures will have a relative humidity that depends on the working temperature) and a change in the static stability (due to a change in the lapse rate of specific humidity), both of which are explained in more detail in the  appendix. Similar considerations apply on the microscale, where an increased liquid water lapse rate may affect the precipitation efficiency, which in turn helps regulate the distribution of liquid water in the atmosphere, and wherein changes in surface temperature may be coupled to changes in the aerosol loading, which may also affect cloud processes. Finally, we note that the surface energy budget will also change because the downward flux of longwave radiation also changes as the absolute humidity of the layer changes; this change is not consistent with the enhancement of evaporation implied by the surface-flux formulation (assuming and V are constant). How this inconsistency is resolved is an interesting, and open, question.

3. Methodology

We explore our questions using large-eddy simulations developed around observations of trade wind cumulus during the RICO field study. RICO sampled conditions typical of the winter trade winds from late November 2004 until nearly the end of January 2005. Observations were centered around a study area in the Atlantic Ocean upwind (northeast) of Antigua and Barbuda. More details about the measurement systems deployed, the weather regimes sampled, and the motivation for doing so are provided by Rauber et al. (2007).

The specific setup of our simulations is based on the case developed by vanZanten et al. (2011). Our decision to focus on this case, as opposed to other trade wind cases in the literature (e.g., Stevens et al. 2001; Siebesma et al. 2003; Stevens 2007; Bellon and Stevens 2012) is motivated by several factors. Not only is the RICO case the best observed trade wind cumulus case, but the behavior of large-eddy simulation has been extensively studied for this case (vanZanten et al. 2011; Matheou et al. 2011; Stevens and Seifert 2008) and it lends itself well to extensions that consider the role of precipitation, which for reasons of simplicity we neglect in the present study.

a. Experimental design and protocol

The RICO case as defined by vanZanten et al. (2011) serves as our control experiment (CTL). This case explores the 24-h evolution of the trade wind layer given a free-tropospheric profile that is in balance with large-scale forcings. The forcings comprise surface fluxes, which are determined using a fixed sea surface temperature and a bulk aerodynamic formula; a fixed rate of large-scale subsidence, which given the thermodynamic profiles results in a drying and warming of the layer; and constant forcing terms calculated to mimic the effects of radiative cooling and large-scale advective forcing. This formulation, which includes advective forcing with a fixed sea surface temperature, is representative of the Eulerian budget, as sampled during the RICO field study.

In addition to the control case, two additional experimental configurations are explored by increasing the temperature profile of the initial sounding of the control experiment by either 2 K (+2 K) or 8 K (+8 K) and adjusting the specific humidity to keep the relative humidity constant. The +2 K and +8 K simulations are collectively referred to as the perturbed simulations. In the perturbed simulations, all additional model specific parameters remain unchanged compared with the control experiment, but the temperature dependence of the surface fluxes implies that they will be different across the simulations as discussed above. Because surface fluxes are important driving forces for convective boundary layers, in a second set of three simulations we replace the surface fluxes calculated by the model with the average values derived from the last hours of the control experiment so as to maintain a constant surface flux across the control and perturbed simulations. This second set of three simulations helps separate the effect of lapse-rate feedbacks from surface-flux feedbacks.

Initial profiles from the control and perturbed simulations differ by a constant temperature offset and in their absolute humidity, but their relative humidity structure is, by design, the same (Fig. 3). An important feature of the RICO case is the balanced initial conditions such that the free-tropospheric moisture gradients are balanced by the imposed subsidence, and the temperature profiles in the free troposphere are balanced by the combination of a prescribed cooling and subsidence. Between the control and perturbed cases the horizontal large-scale temperature and moisture tendencies are held constant. Because the subsidence drying is applied by specifying a subsidence velocity, this increases the absolute rate of drying from subsidence, in a way that can be shown to be consistent with what would be expected if large-scale processes acted to maintain a constant relative humidity. In reality one would expect that as temperature increases, horizontal moisture gradients would scale with qs(T0) and this large-scale drying, associated with the advection of upstream air into the domain, should also decrease. Here, we leave out such effects, which would, if anything, strengthen the signals we identify.

Fig. 3.

Initial profiles for liquid water potential temperature θ, total water specific humidity qt, and relative humidity , showing the control (black), +2 K perturbed (dark gray), and +8 K perturbed (light gray) simulations.

Fig. 3.

Initial profiles for liquid water potential temperature θ, total water specific humidity qt, and relative humidity , showing the control (black), +2 K perturbed (dark gray), and +8 K perturbed (light gray) simulations.

b. The large-eddy simulation model

Simulations were performed with the University of California, Los Angeles (UCLA) LES code (Savic-Jovcic and Stevens 2008). The model has been extensively used to study the cloudy boundary layer, and its solutions have been routinely compared with those using other numerical solvers. The model solves the Navier–Stokes equations in the anelastic limit over a three-dimensional grid using the Runge–Kutta third-order method for advancing quantities in time. Additionally, a conservation law for scalar quantities, an equation of state, and a model for the subgrid fluxes/stresses are used to close the system of equations. The model operates with constant horizontal and stretchable vertical grid spacing. Subgrid fluxes for scalars and momentum are modeled using the Smagorinsky–Lilly approach Lilly (1967). Sea surface temperatures are held constant and surface fluxes are formulated using a bulk aerodynamic law (Fairall et al. 2003), such as Eq. (6), with V given by CdU‖, where Cd is a bulk aerodynamic coefficient determined from similarity theory and ‖U‖ denotes the surface wind speed at each grid cell. This surface-flux parameterization was found to approximate the fluxes well that were measured during the RICO campaign (Nuijens et al. 2009). Here we use a simple saturation adjustment for the microphysics, and a specified cooling rate in lieu of radiation. Using the saturation adjustment scheme as a microphysical representation limits our considerations to nonprecipitating clouds.

The model domain is fixed over a central latitude of 18°N. Simulations are performed using the same grid structure chosen by vanZanten et al. (2011), a doubly periodic square horizontal grid measuring 12.5 km on a side with an isotropic horizontal grid spacing of 100 m. The vertical is spanned by a uniform grid with 40-m grid spacing over 5-km depth. All simulations performed are 24 h long with a maximum time step of 4 s.

The statistics that we present are domain-averaged profiles of cloud variables (averaged over the last 6 h of 24-h simulations) or time series of domain-averaged variables at a temporal resolution of 30 s. At the end of 24 h the simulations have not reached equilibrium, although during the analysis period they are evolving only slowly with time. Longer simulations using the RICO forcing suggest that the cloud layer will continue to slowly deepen in time. In reality this tendency toward an ever deeper layer should increasingly be countered by enhanced advective drying and warming at cloud top. These drying and warming tendencies are expected to result from an increased along-wind gradient in the depth of the boundary layer that must accompany a deepening layer.

For the above-stated reasons, and because some of the distinctions we make among simulations are rather fine, whereas the model grid is relatively coarse, we have rerun individual simulations with slightly different initial conditions so as to obtain another realization, or compared different time periods (e.g., integrated the simulations for 48 h in time). In all cases the features we identify as meaningful are robust to such changes. Based on the results of Matheou et al. (2011), we expect that our results will be quantitatively affected by resolution, but not qualitatively. A further check on the qualitative robustness of the features we identify is provided through the consistency of the qualitative response between the +2 K and +8 K simulations.

4. Cloud changes at constant humidity

Before discussing the differences between the control and perturbed simulations, it is useful to note some basic and common features to all the simulations. All of the simulations produce a well-developed cloud layer, with cloud base near 700 m, and clouds extending to a depth of around 3 km (Fig. 4). The cloud layer has a thin cloud-base layer where cloud fraction reaches a maximum value of near 0.05 very near cloud base. Cloud fraction then decreases sharply with height over a distance of a few hundred meters, thereafter maintaining a relatively constant value of about 0.02 through what we call the cumulus layer, until a height of about 2 km, where the domain averaged liquid water reaches a maximum. Above this height temperature increases and humidity decreases (not shown), resulting in a sharp decrease in cloudiness and defining the top of the trade wind layer. Above a height of about 3 km no further cloud is present.

Fig. 4.

Profiles of domain-averaged (left) liquid water and (right) cloud fraction for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. Each profile is based on the final 4 h of a 24-h simulation.

Fig. 4.

Profiles of domain-averaged (left) liquid water and (right) cloud fraction for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. Each profile is based on the final 4 h of a 24-h simulation.

As temperature increases the domain-averaged liquid water specific humidity decreases within the bulk of the cumulus layer, mostly as a result of a smaller cloud fraction (Fig. 4). The changes in cloud water are less evident in the vertically integrated quantities, which are examined as a function of time in Fig. 5. The cloud cover, averaged over the last 6 h of the simulation, decreases from 12.2 in the CTL to 10.5 in the +8 K simulation (11.2 in the +2 K simulation). Overall the general tendency of the +2 K and +8 K simulations is similar, but more marked in the +8 K simulation.

Fig. 5.

Temporal evolution of the (top) liquid water path and (bottom) cloud cover for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. The ordinate displays an average value for the final 4 h of each simulation.

Fig. 5.

Temporal evolution of the (top) liquid water path and (bottom) cloud cover for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. The ordinate displays an average value for the final 4 h of each simulation.

From the mean profile of liquid water there is little evidence of a liquid water lapse-rate effect manifesting itself. Because the mean profile is influenced by the cloud fraction, which tends to decrease as temperature increases, it proves useful to conditionally sample the clouds to see if the in-cloud liquid water profile is influenced by temperature.

Through the bulk of the cloud layer, there is a slight increase in the liquid water lapse rate, as measured by the vertical gradient of for the simulations at higher temperatures (Fig. 6). For reference the adiabatic lapse rates are also plotted. The picture is complicated because behaves differently within the cumulus layer as compared to the inversion layer where the stability increases markedly and the relative humidity and cloud amount drop sharply. In the upper (inversion) layer the liquid water is much more adiabatic, reflecting the greater fraction of mixtures that are subsaturated and thus are not able to bias the sampling toward drier values. To quantify the change in the lapse rate in the conditionally sampled profile, we average the lapse rate of over the lower 1 km (of clouds) of each simulation and denote changes in the lapse rates defined in this manner by . Comparing the +2 K and +8 K simulations to the control simulation yields values of = 3.6% and 2.1% K−1, respectively. Recall that theoretical considerations predicted f ≈ 1.5% K−1 for the range of temperature changes considered here. Thus the simulated values of are of the expected order of magnitude; lack of more quantitative agreement with the theory is perhaps not surprising given the other changes in the cloud layer.

Fig. 6.

Profiles of conditionally and time-averaged (left) cloud water and (right) relative humidity for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. Dashed lines in the left panel correspond to the adiabatic lapse rates; dotted line in right panel indicates initial relative humidity profile.

Fig. 6.

Profiles of conditionally and time-averaged (left) cloud water and (right) relative humidity for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. Dashed lines in the left panel correspond to the adiabatic lapse rates; dotted line in right panel indicates initial relative humidity profile.

Although by design the CTL, +2 K, and +8 K simulations have the same initial relative humidity profile, the ongoing dynamics affect the distribution of relative humidity, leading to the differences in its structure later on (Fig. 6). The relative humidity shows a systematic tendency toward lower values in a warmer atmosphere, up to the inversion layer (where the deeper clouds at higher temperatures are associated with increased relative humidity). The reduced humidity through the bulk of the cloud layer is consistent with the slight reduction in cloud fraction, while the enhanced inversion layer humidity is consistent with the somewhat slower decline of cloud fraction with height in the perturbed simulations, especially so in experiment +8 K (e.g., Figs. 4 and 6). The reduced humidity in the subcloud layer in the perturbed experiment is also consistent with the tendency of cloud base to rise slightly in those same simulations, increasingly so for experiment +8 K. This suggests that at warmer temperatures the changes in liquid water lapse rates are a rather minor effect compared with the macroscopic changes evident within the cloud layer. At warmer temperatures the boundary layer is deeper, the inversion layer is more distributed, cloud base is higher, and the boundary layer is drier.

All of these changes can be explained as a consequence of the larger surface moisture fluxes, which of themselves do not moisten the layer any more than the proportionally smaller surface fluxes found in the colder layer, but which drive a deeper boundary layer and hence mix more dry and warm air to the surface. Figure 7 presents the time series of surface fluxes and boundary layer depth for the control and perturbed simulations over the course of the 24-h simulation. In the +2 K and +8 K simulations there is an immediate change in the surface moisture flux, as predicted in section 2. Increased surface moisture fluxes increase the surface buoyancy flux, which in turn is accompanied by an increase in the liquid water flux carried by the clouds, into the inversion layer. Through the injection-deepening mechanism formalized by Stevens (2007), one would expect the simulations with larger moisture fluxes to deepen more rapidly in time. This mechanism states that the liquid water flux into the inversion scales with the surface buoyancy flux and the depth of the cloud layer, and deepens the boundary layer through the moistening and cooling that accompanies the evaporation of liquid water detrained in the inversion layer. Accompanying this deepening will be a progressive warming and drying of the cloud and subcloud layers, which will amplify the differences in surface moisture fluxes and reduce the surface fluxes of sensible heat [Fig. 7; see also Nuijens and Stevens (2012) for a discussion of this point]. Both the warming and drying contribute to a reduction of the relative humidity through the cloud and subcloud layers as observed in Fig. 6.

Fig. 7.

Temporal evolution of the (top) boundary layer depth, (middle) latent heat flux, and (bottom) sensible heat flux for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. The ordinate displays an average value for the final 4 h of each simulation.

Fig. 7.

Temporal evolution of the (top) boundary layer depth, (middle) latent heat flux, and (bottom) sensible heat flux for the control (black), +2 K (dark gray), and +8 K (light gray) simulations. The ordinate displays an average value for the final 4 h of each simulation.

As discussed in the  appendix, increasing the temperature while keeping the relative humidity constant also modifies the static stability. For the RICO case one expects a reduction in the static stability, and some of the additional growth in the layer may be a result of these changes in stability. These two effects, increased surface driving accompanied with reduced atmospheric stability, are difficult to separate in our simulations; however, given the 50% increase in surface moisture fluxes for the +8 K simulations, and noting that the reduction in static stability between the surface and 2 km is roughly 10% for this case, we believe that the surface fluxes play a fundamental role.

From the calculations in section 2 we know that the increased surface moisture flux in the warmer case (which because it depends linearly with qs increases by precisely the desired amount) is just sufficient to maintain the relative humidity of the layer (assuming constant depth) in the absence of other sources of moisture. Hence the deepening of the layer and the incorporation of drier air from aloft leads to these layers being drier than they otherwise would have been.

5. Cloud changes at constant humidity with fixed surface fluxes

To separate the influence of the surface fluxes from that of the changing liquid water lapse rates, we carried out simulations in which the surface fluxes are identical in the CTL and +2 K and +8 K simulations.

When the surface fluxes are not allowed to change with increasing temperature, there is relatively little change in the depth of the boundary layer among the simulations. Initially the simulations deepen more rapidly, consistent with the decreased static stability and a constant forcing. But over longer times there is an increasing tendency toward a shallower boundary layer with increasing temperature. There is also a progressive tendency for the cloud fraction and cloud cover to decrease with increasing temperature (Figs. 8 and 9), which suggests that the drying of the cloud layer (in terms of relative humidity) reduces the efficiency with which liquid water is transported into the inversion, thereby helping to deepen the cloud layer.

Fig. 8.

Profiles of clouds domain-averaged (a) liquid water and (b) cloud fraction for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with surface fluxes held constant.

Fig. 8.

Profiles of clouds domain-averaged (a) liquid water and (b) cloud fraction for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with surface fluxes held constant.

Fig. 9.

Temporal evolution of the (top) liquid water path and (bottom) cloud cover for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with surface fluxes held constant. The ordinate displays an average value for the final 4 h of each simulation.

Fig. 9.

Temporal evolution of the (top) liquid water path and (bottom) cloud cover for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with surface fluxes held constant. The ordinate displays an average value for the final 4 h of each simulation.

In contrast to the simulations with interactive surface fluxes, the dynamics of the layer here lead to a reduction in relative humidity almost everywhere (Fig. 10). This is to be expected because the effect of additional moisture input by a higher surface moisture flux (at warmer temperatures) is absent when the surface fluxes are held constant.1 The reduction of the relative humidity is most pronounced at the surface and at the top of the cloud layer. The value at cloud base, because it plays such a role in defining the cloud-base level, is relatively unchanged, although cloud base is higher in the warmer simulations because of the drying of the subcloud layer. Hence in contrast to the simulations with interactive surface fluxes, the boundary layer is drier throughout, which further reduces cloud amount. It is interesting that the drying of the subcloud layer is similar in all perturbed simulations, irrespective of whether surface fluxes are interactive or not. This suggests that the additional moisture that enters the boundary layer when surface fluxes are allowed to freely evolve is disproportionately used to deepen the boundary layer.

Fig. 10.

Profiles of conditionally and time-averaged (left) cloud water and (right) relative humidity for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with fixed surface fluxes. Dotted line in right panel indicates initial relative humidity profile.

Fig. 10.

Profiles of conditionally and time-averaged (left) cloud water and (right) relative humidity for the control (black), +2 K (dark gray), and +8 K (light gray) simulations with fixed surface fluxes. Dotted line in right panel indicates initial relative humidity profile.

In these fixed-flux simulations the increase in the lapse rate of in the +2 K and +8 K simulations is somewhat less than in the simulations where surface fluxes are computed interactively. For both the +2 K and +8 K simulations with fixed surface fluxes, ≈1.2% K−1. The difference between these simulations and those with interactive surface fluxes, where was a factor of 2 larger, probably reflects the different humidities in the cloud layer. In a drier cloud layer clouds become progressively more dilute with height. This suggests that , as constructed from the profiles of liquid water averaged only over the clouds themselves, depends on both temperature and the relative humidity of the layer in which the clouds form. The presence of a positive liquid water lapse-rate response to increasing temperature is robust; but even in the case where additional energetic input is not allowed to deepen the boundary layer,2 the desiccation of the boundary layer associated with the relative surface-flux deficit dominates, and the overall change in cloudiness is negative.

6. Discussion

Our results suggest that, in the absence of other processes, shallow cumulus layers will tend to have fewer clouds if they develop in a warmer climate where large-scale tendencies act to maintain a constant relative humidity. This response is illustrated by a conceptual diagram in Fig. 11. The diagram illustrates how, in a large-scale environment that acts to maintain a constant relative humidity (and with other factors remaining constant), surface moisture fluxes will increase. Increased moisture fluxes result in cumulus clouds that carry more liquid and deepen more rapidly. The deepening of the layer leads to a redistribution of the available moisture over a larger vertical extent and brings increased amounts of warm and dry air into the boundary layer, which further reduces the relative humidity, leading to a reduction of clouds overall. Although our simulations suggest that the basic concept of the liquid water lapse-rate feedback is operative, this effect is more than compensated by the reduction in cloud amount that results from the drying of the boundary layer with a deepening. These dynamics are not that different from the arguments generally used to explain the climatological reduction of cloudiness as the marine boundary layer advects over warmer waters (Bretherton and Wyant 1997). Perhaps the key additional insight is the recognition of the need for surface moisture fluxes to increase in a warming climate, assuming that large-scale processes act to maintain a constant relative humidity.

Fig. 11.

Schematic diagram showing the main response of the cloud-topped boundary layer to a change in temperature, assuming large-scale processes act to keep the humidity constant.

Fig. 11.

Schematic diagram showing the main response of the cloud-topped boundary layer to a change in temperature, assuming large-scale processes act to keep the humidity constant.

The simulations performed by Xu et al. (2010) share some important features with the present study, in that both explore the response of clouds to temperature in a constant relative humidity atmosphere, but the temperature sensitivity of the clouds in our simulations is less (in magnitude) and different in sign than what is found in Xu et al. (2010). Their simulations, however, explore the equilibrium behavior of the simulations and allow additional processes (other than surface fluxes) to change with increasing temperatures. For instance, in their simulations the background potential temperature profile and the subsidence also change [following the framework developed by Zhang and Bretherton (2008)] with changing surface temperature. And while the approach employed by Xu et al. (2010) could be argued to give a more complete picture of how marine boundary layers will change as a function of temperature, the equilibrium solutions they find are likely sensitive to some unsatisfactory aspects of the Zhang and Bretherton (2008) formulation. Most notable is the tendency for the prescribed large-scale drying at the top of the boundary layer to turn to moistening at a height that depends on the depth of the control simulation. This formulation was designed to maintain stationarity in the free-troposphere humidity in the control simulation, but the failure to relate the advective forcings to the depth of the cloud layer distorts the humidity budget at cloud top. We suspect these differences are decisive in explaining the differences in the sensitivity of the cloud development between the two studies. Because our purpose is to understand how some processes work in a simplified setting, rather than to provide a final quantification of cloud feedbacks, much more interesting is to point out that both studies show a small change in the liquid water lapse rate in the lower part of the cloud layer, and both studies suggest that the cloud and subcloud layers dry, while surface latent heat fluxes increase, and the boundary layer deepens as temperature increases.

This leads to the point that the interplay of processes controlling the structure of the cloud-topped boundary layer is wide ranging and involves a multitude of physical processes operating on a wide range of space and time scales. The ultimate response of shallow convection in the trades may depend on subtle tradeoffs between competing effects. As such it cautions against drawing definitive conclusions from the present study as to the fate of shallow cumulus layers in a warmer climate. Nonetheless, the present framework identifies an important and robust response and helps explain why a deeper and drier cloud and subcloud layer would be expected in a warmer climate. In so doing it also raises an interesting question. Given that expected changes in the surface heat budget cannot be expected to support the increased rate of evaporation demanded by a warmer atmosphere with constant relative humidity (Stephens and Hu 2010), and given that the boundary layer dynamics in response to an increase in surface moisture fluxes appears to demand further increases in the surface moisture fluxes, it remains unclear how these competing demands are reconciled. One possibility is through an increased stabilization of the surface layer (a process not considered here, but to be expected as a result of the enhanced warming that accompanies the deepening of the boundary layer). Such a stabilization is consistent with both theoretical constraints on radiative transfer (Stephens and Hu 2010) and general circulation modeling (Richter and Xie 2008).

7. Conclusions

Large-eddy simulation is used to explore the basic mechanisms that govern how marine shallow cumuli will react if large-scale atmospheric processes act to maintain a constant relative humidity in the atmosphere, even as temperatures rise, say in response to elevated concentrations of greenhouse gases. The question is motivated by a desire to identify and understand cloud feedback processes in the climate change context.

Early work on this topic argued that because the lapse rate of liquid water accompanying saturated adiabatic ascent increases with temperature, in a warmer climate clouds should consist of more water, and thus have a larger optical depth. Such behavior would constitute a negative feedback process, thereby reducing the sensitivity of surface temperatures to a radiative forcing. We identify an additional process that accompanies warming in a constant relative humidity atmosphere, namely an increase in surface moisture fluxes, which in turn affects the dynamics of the cloud-topped boundary layer.

These effects are explored using large-eddy simulations of shallow cumulus clouds based on initial data and forcing derived from data taken from the recent RICO field study. In addition to a number of sensitivity studies to explore the robustness of our results to arbitrary choices, such as the random perturbations that initialize the simulations, we conduct two sets of three simulations. One set of simulations is based on a dynamic and hence temporally evolving representation of the surface fluxes, and the other fixes the fluxes using values taken from the end of the control simulation from the first set. The three simulations for each set consist of a control simulation based on the RICO case and two perturbation simulations, with the temperature everywhere increased by 2 and 8 K, respectively.

The simulations confirm the presence of a negative liquid water lapse-rate feedback and show that the strength of this effect scales reasonably well with the liquid water lapse-rate changes. However, the net effect of the increase in liquid water is wholly compensated by macroscopic changes in the structure of the clouds that results from increasing surface fluxes. Larger surface fluxes in a warmer climate drive a deeper boundary layer, which is necessarily drier (in a relative sense) and thus characterized by reduced cloud fraction. This surface-flux desiccation feedback is a positive feedback as it tends to reduce cloud cover as the temperature increases, assuming that large-scale processes act to keep the relative humidity constant.

Because the increase in surface fluxes is in fact necessary to maintain the rate of humidification of the boundary layer as the climate warms, simulations with fixed surface fluxes are drier and less cloudy. Increasing the temperature while holding surface fluxes fixed does not produce a deepening of the boundary layer, confirming that the deepening desiccation feedback is a result of the tendency for a warmer climate to have larger surface fluxes.

Although the positive surface-flux desiccation feedback dominates the behavior of our simulations, work by others (Xu et al. 2010) suggests that the sufficient deepening of the cloud layer may offset the drying of that layer so as to increase cloud cover and liquid water path overall, thus changing once more the sign of the feedback. This interplay of complex and competing processes recalls the discussion of the tendency of cloud-scale processes to buffer the system against external changes as a result of aerosol perturbations (Stevens and Feingold 2009). While the net change in cloudiness, or shortwave cloud forcing, is difficult to unambiguously assess based on our simulations, our work does suggest that the key process determining the character of cloud feedbacks in boundary layer clouds is the degree to which increasing surface fluxes drive a deepening of the cloud layer, and the effect of the deepening on the structure of the cloud fields both directly by changing cloud depth and indirectly by additionally drying the boundary layer.

Acknowledgments

The simulations were performed using the facilities of the Deutsches Klimarechenzentrum. M. Rieck thanks Thijs Heus and Irina Sandu for their technical help with the large-eddy simulation. B. Stevens thanks Sandrine Bony for the initial suggestion to look into these questions. Cathy Hohenegger and Robert Pincus are thanked for their critical comments on an earlier version of this manuscript, and C. Bretherton’s insightful comments motivated us to reconsider the role of humidity in regulating the static stability. The presentation also benefited from constructive comments by two anonymous reviewers. The NCL group at NCAR is thanked for their support of a high-quality open source graphics language designed explicitly for use with climate and atmospheric data. This research was made possible through the support of the Max Planck Society for the Advancement of Science.

APPENDIX

Further Temperature-Dependent Processes in a Constant Relative Humidity Atmosphere

Aside from the surface-flux and liquid water lapse-rate feedback, other temperature-dependent processes are at play in a constant relative humidity atmosphere. For instance, given that relative humidity is a nonlinear function of temperature, mixing two reservoirs whose relative humidity is constant can produce mixtures whose relative humidity depends on the working temperature. This effect, however, is small. To appreciate this, consider two reservoirs (cases 1 and 2), each with a temperature Ta and relative humidity in the upper layer and with temperature Tb and relative humidity in the lower layer (Fig. A1). Furthermore, assume that and are the same in cases 1 and 2, with temperatures offset by a constant factor, so that T2a − T1a = T2b − T1b = T2T1. If the layers in each reservoir are mixed, it is straightforward to show that the difference in the resultant humidities follows as

 
formula

which is small. For , T2T1 = 5 K, and T1aT1b = 5 K, or about 1%.

Fig. A1.

Schematic diagram illustrating two cases, each initially with a two-layer structure of lower humidity at one temperature overlying higher humidity at another temperature and then transformed to a well-mixed layer with a single temperature and humidity structure. The only difference between the two cases is an offset between the temperatures so that T2 > T1.

Fig. A1.

Schematic diagram illustrating two cases, each initially with a two-layer structure of lower humidity at one temperature overlying higher humidity at another temperature and then transformed to a well-mixed layer with a single temperature and humidity structure. The only difference between the two cases is an offset between the temperatures so that T2 > T1.

Changing the temperature of the atmosphere but keeping the relative humidity constant also implies a difference in the static stability with respect to saturated and unsaturated ascent. For the unsaturated case, the static stability depends, in part, on the gradient of the specific humidity, which is not invariant for a change of temperature. Consider the static stability defined as

 
formula

where θυ = θ(1 + εqυ) with ε = Rυ/Rd−1 ≈ 0.608. This expression can be formulated in terms of the ambient humidity, by recognizing that . With the help of the Clausius–Clapeyron equation, we can write the static stability in terms of the gradient of θ and the gradient in relative humidity, such that

 
formula

This equation shows that an increase in absolute humidity that accompanies a warming at fixed relative humidity tends to increase the atmospheric stability, most notably through the term proportional to in Eq. (A3), but the tendency for the humidity to decrease in the atmosphere leads to a destabilizing effect associated with the term, which in this case would be negative. The two terms are of similar order. The destabilizing effect dominates when

 
formula

For the RICO case, where /dz ≈ 6 K km−1 and ≈ 0.3 km−1, this suggests that N2 will be reduced as temperature increases and relative humidity is kept constant, which may contribute to the more rapid deepening simulated in the warmer case.

In addition, the change of the moist adiabatic lapse rate with temperature is expected to be reflected in the temperature profiles in the free troposphere (e.g., Zhang and Bretherton 2008) so that /dz increases in a warmer atmosphere, stabilizing the large-scale environment. Changes in the absolute humidity of the free atmosphere (which affect the radiative cooling) and changes in the lapse rate (which affect the rate of adiabatic warming) are expected to change the subsidence rate.

REFERENCES

REFERENCES
Bellon
,
G.
, and
B.
Stevens
,
2012
:
Using the sensitivity of large simulations to evaluate atmospheric boundary layer models
.
J. Atmos. Sci.
,
69
,
1582
1601
.
Betts
,
A. K.
,
1982
:
Saturation point analysis of moist convective overturning
.
J. Atmos. Sci.
,
39
,
1484
1505
.
Betts
,
A. K.
, and
Harshvardhan
,
1987
:
Thermodynamic constraint on the cloud liquid water feedback in climate models
.
J. Geophys. Res.
,
92
(
D7
),
8483
8485
.
Bony
,
S.
, and
J. L.
Dufresne
,
2006
:
Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models
.
Geophys. Res. Lett.
,
32
,
L20806
,
doi:10.1029/2005GL023851
.
Brenguier
,
J.-L.
,
H.
Pawlowska
,
L.
Schüller
,
R.
Preusker
,
J.
Fischer
, and
Y.
Fouquart
,
2000
:
Radiative properties of boundary layer clouds: Droplet effective radius versus number concentration
.
J. Atmos. Sci.
,
57
,
803
821
.
Bretherton
,
C. S.
, and
M. C.
Wyant
,
1997
:
Moisture transport, lower tropospheric stability, and decoupling of cloud-topped boundary layers
.
J. Atmos. Sci.
,
54
,
148
167
.
Charlock
,
T. P.
,
1982
:
Cloud optical feedback and climate stability in a radiative-convective model
.
Tellus
,
34
,
245
254
.
Del Genio
,
A. D.
, and
A. B.
Wolf
,
2000
:
The temperature dependence of the liquid water path of low clouds in the southern Great Plains
.
J. Climate
,
13
,
3465
3486
.
Fairall
,
C. W.
,
E. F.
Bradley
,
J. E.
Hare
,
A. A.
Grachev
, and
J. B.
Edson
,
2003
:
Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm
.
J. Climate
,
16
,
571
591
.
Forster
,
P.
, and
Coauthors
,
2007
:
Changes in atmospheric constituents and in radiative forcing. Climate Change 2007: The Physical Science Basis, S. Solomon et al., Eds., Cambridge University Press, 129–234
.
Hartmann
,
D. L.
, and
D. A.
Short
,
1980
:
On the use of Earth radiation budget statistics for studies of clouds and climate
.
J. Atmos. Sci.
,
37
,
1233
1250
.
Held
,
I. M.
, and
B. J.
Soden
,
2000
:
Water vapor feedback and global warming
.
Annu. Rev. Energy Environ.
,
25
,
441
475
.
Lilly
,
D. K.
,
1967
:
The representation of small-scale turbulence in numerical simulation experiments. Proc. IBM Scientific Computing Symp. on Environmental Sciences, Yorktown Heights, NY, IBM, 195–210
.
Matheou
,
G.
,
D.
Chung
,
L.
Nuijens
,
B.
Stevens
, and
J.
Teixeira
,
2011
:
On the fidelity of large-eddy simulation of shallow precipitating cumulus convection
.
Mon. Wea. Rev.
,
139
,
2918
2939
.
Möller
,
F.
,
1963
:
On the influence of changes in the CO2 concentration in air on the radiation balance of the Earth’s surface and on the climate
.
J. Geophys. Res.
,
68
,
3877
3886
.
Nuijens
,
L.
, and
B.
Stevens
,
2012
:
The influence of wind speed on shallow marine cumulus convection
.
J. Atmos. Sci.
,
69
,
168
184
.
Nuijens
,
L.
,
B.
Stevens
, and
A. P.
Siebesma
,
2009
:
The environment of precipitating shallow cumulus convection
.
J. Atmos. Sci.
,
66
,
1962
1979
.
Paltridge
,
G. W.
,
1980
:
Cloud-radiation feedback to climate
.
Quart. J. Roy. Meteor. Soc.
,
106
,
895
899
.
Plass
,
G. N.
,
1956
:
The carbon dioxide theory of climatic change
.
Tellus
,
8
,
140
154
.
Rauber
,
R. M.
, and
Coauthors
,
2007
:
Rain in shallow cumulus over the ocean—The RICO campaign
.
Bull. Amer. Meteor. Soc.
,
88
,
1912
1928
.
Richter
,
I.
, and
S.-P.
Xie
,
2008
:
Muted precipitation increase in global warming simulations: A surface evaporation perspective
.
J. Geophys. Res.
,
113
,
D24118
,
doi:10.1029/2008JD010561
.
Rieck
,
M.
,
2011
:
Testing the liquid water lapse rate feedback in shallow convection using large eddy simulations. M.S. thesis, University of Hamburg, 65 pp
.
Savic-Jovcic
,
V.
, and
B.
Stevens
,
2008
:
The structure and mesoscale organization of precipitating stratocumulus
.
J. Atmos. Sci.
,
65
,
1587
1605
.
Schneider
,
S. H.
,
1972
:
Cloudiness as a global climatic feedback mechanism: The effects on the radiation balance and surface temperature of variations in cloudiness
.
J. Atmos. Sci.
,
29
,
1413
1422
.
Siebesma
,
A. P.
, and
Coauthors
,
2003
:
A large eddy simulation intercomparison study of shallow cumulus convection
.
J. Atmos. Sci.
,
60
,
1201
1219
.
Somerville
,
R. C. J.
, and
L. A.
Remer
,
1984
:
Cloud optical thickness feedbacks in the CO2 climate problem
.
J. Geophys. Res.
,
89
(
D6
),
9668
9672
.
Stephens
,
G. L.
,
1978
:
Radiation profiles in extended water clouds. I: Theory
.
J. Atmos. Sci.
,
35
,
2111
2122
.
Stephens
,
G. L.
, and
Y.
Hu
,
2010
:
Are climate-related changes to the character of global-mean precipitation predictable?
Environ. Res. Lett.
,
5
,
025209
,
doi:10.1088/1748-9326/5/2/025209
.
Stevens
,
B.
,
2007
:
On the growth of layers of nonprecipitating cumulus convection
.
J. Atmos. Sci.
,
64
,
2916
2931
.
Stevens
,
B.
, and
A.
Seifert
,
2008
:
Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection
.
J. Meteor. Soc. Japan
,
86A
,
143
162
.
Stevens
,
B.
, and
G.
Feingold
,
2009
:
Untangling aerosol effects on clouds and precipitation in a buffered system
.
Nature
,
461
,
607
613
.
Stevens
,
B.
, and
S. E.
Schwartz
,
2012
:
Observing and modeling Earth’s energy flows
.
Surv. Geophys.
,
in press
.
Stevens
,
B.
, and
Coauthors
,
2001
:
Simulations of trade wind cumuli under a strong inversion
.
J. Atmos. Sci.
,
58
,
1870
1891
.
Tselioudis
,
G.
,
W. B.
Rossow
, and
D.
Rind
,
1992
:
Global patterns of cloud optical thickness variation with temperature
.
J. Climate
,
5
,
1484
1495
.
Twomey
,
S.
,
1971
:
The influence of atmospheric particulates on cloud and planetary albedo
.
Bull. Amer. Meteor. Soc.
,
52
,
265
266
.
vanZanten
,
M. C.
,
B.
Stevens
, and
L.
Nuijens
,
2011
:
Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during RICO
.
J. Adv. Model. Earth Syst.
,
3
,
M06001
,
doi:10.1029/2011MS000056
.
Xu
,
K.-M.
,
A.
Cheng
, and
M.
Zhang
,
2010
:
Cloud-resolving simulation of low-cloud feedback to an increase in sea surface temperature
.
J. Atmos. Sci.
,
67
,
730
748
.
Zhang
,
M.
, and
C. S.
Bretherton
,
2008
:
Mechanisms of low cloud–climate feedback in idealized single-column simulations with the Community Atmospheric Model, version 3 (CAM3)
.
J. Climate
,
21
,
4859
4878
.

Footnotes

1

Recall from section 2 that a larger input of moisture is in fact required to maintain a constant relative humidity if the temperature is larger.

2

Recall that if the surface flux is held constant, the relative humidity in the warmer layer will increase less rapidly (see, e.g., section 2).