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...he model and approximately 400 m at the surface. A sponge layer was placed above 14 km to dampen vertical motions at the top of the model and to prevent reflection of gravity waves off the top of the model domain. The model time step for dynamics was between 1.0 and 10.0 s and was chosen to be the largest time step that was computationally stable. MAY 2013 S CHM IDT AND GARRETT 1413 For radiative transfer, the UU LESM uses a planeparallel broadband approach, using a d-four stream scheme for parameterization of radiative transfer (Liou et al. 1988) based on the correlated-k distributionmethod (Fu and Liou 1992). Radiative transfer calculations were performed at a time step of 60 s. The dimensionless numbers presented in this study emphasized the effects of radiative flux divergence on cloud evolution. For all cases examined, the model was initialized with a standard tropical profile of temperature and atmospheric gases with a buoyancy frequency of approximately 0.01 s21 throughout the depth of the model domain. Relative humidity was set in two independent layers. In the bottom layer of the model, which extends from the surface to 7.8 km, the relative humidity was set to a constant 70% with respect t...

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...ther than laminar flows. A third adjustment mode relates to evaporation, which erodes cloudy air as it lofts, regardless of its optical density. The dominant mode is determined from two dimensionless numbers, whose predictive power is shown in comparisons with high-resolution numerical cloud simulations. The power and simplicity of the approach hints that fast, subgrid-scale radiative–dynamic atmospheric interactions might be efficiently parameterized within slower, coarse-grid climate models. 1. Introduction Cloud–climate feedbacks remain a primary source of uncertainty in climate forecasts (Dufresne and Bony 2008), mainly because clouds both drive and respond to the general circulation, the hydrological cycle, and the atmospheric radiation budget. Unlike fields of water vapor, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and dive...

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...tialized as homogeneous cylindrical ice clouds with neutral buoyancy, as shown in Fig. 3. Ice particles within the cloud were of uniform size with a fixed effective radius of 20 mm and an initially uniform mixing ratio as prescribed by the particular case. Cloud radius was prescribed according to the particular case, but in each case the thickness was set to 2500 m with the cloud base set at 8.8 km. Cloud base was chosen such that the cloud top would be placed at approximately 200 mb, in rough accordance with the average cirrus anvil height indicated by the fixed anvil temperature hypothesis (Hartmann and Larson 2002). Both the cloud and surrounding atmosphere were initialized to be at rest. No precipitation was allowed in any of the model simulations. Cloud particle fall speed was also neglected. A rough estimate for the importance of cloud particle fall speed is provided in the discussion of precipitation in section 4d. All cases were run for 1 h of model simulation time. The model was initialized with an idealized cloud in buoyant equilibrium with its lateral surroundings. With no vertical radiative contrasts allowing for the absorption of thermal radiation, the model cloud would continue to sit at rest...

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...mospheric radiation budget. Unlike fields of water vapor, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and divergence within cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et al. (2011) suggest th...

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... of isotropic radiation to vertical fluxes. Usually m; 0:6 (Herman 1980). The absorption optical depth is determined by the cloud icemixing ratio qi, as well as the ice crystal effective radius re, through tabs 5 k(re)qirDz where k is the mass specific absorption cross-section density, r is the density of air, and Dz is the vertical pathlength through which the radiation is absorbed. The characteristic depth is the e-folding pathlength for the attenuation such that tabs/m5 1: h5 m k(re)qir . (3) Assuming an effective radius of 20 mm, the value for k(re) in cirrus is approximately 0.045 m2 g21(Knollenberg et al. 1993). Taking, for example, qi values of 1 g kg 21 that have been observed in medium-sized cirrus anvils in Florida (Garrett et al. 2005), h would be about 30 m. As a contrasting example, a cloud with qi values of 0.01 g kg 21, similar to those observed in thin cirrus (Haladay and Stephens 2009), would have a radiative penetration depth h of about 3000 m. Thus, the deposition of radiative energy in this layer increases its potential temperature at rate H5 du dt 5 21 rP dF dz ’ DFnet rcph 5 4s ~T 3 c k(re)qi mcp D ~T , (4) where P is the Exner function [P 5 cp(T/u)]. Insofar as the dynamic adjustmen...

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...to a maximum grid spacing of approximately 300 m at the top of the model and approximately 400 m at the surface. A sponge layer was placed above 14 km to dampen vertical motions at the top of the model and to prevent reflection of gravity waves off the top of the model domain. The model time step for dynamics was between 1.0 and 10.0 s and was chosen to be the largest time step that was computationally stable. MAY 2013 S CHM IDT AND GARRETT 1413 For radiative transfer, the UU LESM uses a planeparallel broadband approach, using a d-four stream scheme for parameterization of radiative transfer (Liou et al. 1988) based on the correlated-k distributionmethod (Fu and Liou 1992). Radiative transfer calculations were performed at a time step of 60 s. The dimensionless numbers presented in this study emphasized the effects of radiative flux divergence on cloud evolution. For all cases examined, the model was initialized with a standard tropical profile of temperature and atmospheric gases with a buoyancy frequency of approximately 0.01 s21 throughout the depth of the model domain. Relative humidity was set in two independent layers. In the bottom layer of the model, which extends from the surface to 7.8 km...

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... S. Despite these variations in the decay of S over time, the trend is for large values of S to decay rapidly and undergo a transition into the lofting regime with values of S, 1. This decay roughly follows an anticipated t23 power law because of the negative feedback associated with the increasing dilution of radiative heating frommixedlayer deepening. As a contrasting example, contrail formations are typically optically thin and horizontally narrow. In some cases they can evolve into broad swaths of cirrus that persist for up to 17 h after initial formation and radiatively warm the surface (Burkhardt and Kaercher 2011). We did not specifically model contrails in this study. While noting that there are differences in heat, moisture, and turbulence when compared to the clouds modeled in this study, all three become rapidly diluted since the width of a contrail is much greater than the width of a jet engine. The theoretical principles that we discuss in this work can provide guidance for how contrails might be expected to evolve over longer time scales of minutes to hours. Immediately following ejection from a jet engine, the contrail air has water contents of a few tenths of a gram per meter cubed (Spinhirne ...

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...ycle, and the atmospheric radiation budget. Unlike fields of water vapor, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and divergence within cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et...

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...xing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et al. (2011) suggest that radiative cooling can help to initiate thin tropopause cirrus formation, while subsequent radiative heating in an environment of weak stability can induce the small-scale convection currents that are required to maintain the cloud against gravitational sedimentation and vertical wind shear. Where these recent studies directly simulated the highly interactive and complex nature of cloud processes, an alternative and perhaps more general approach is to start with simple, analytical, and highly idealized models that emphasize specific aspects of the relevant ph...

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...cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et al. (2011) suggest that radiative cooling can help to initiate thin tropopause cirrus formation, while subsequent radiative heating in an environment of weak stability can induce the small-scale convection currents that are required to maintain the cloud against gravitational sedimentation and vertical wind shear. Where these recent studies directly simulated the highly interactive and complex nature of cloud processes, an alternative and perhaps more general approach is to start with simple, analytical, and highly idea...

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...0.68 1418 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70 5. Discussion We have separated the evolutionary response of clouds to local diabatic heating into distinct modes of cross-isentropic lifting, along-isentropic spreading, and evaporation of cloud condensate. A straightforward method has been described for determining how a cloud will evolve based on ratios of the associated rates. The dominant modes of evolution are outlined in Table 5. For example, cirrus anvils begin their life cycle as dense cloud from convective towers that have reached their level of neutral buoyancy (Scorer 1963; Jones et al. 1986; Toon et al. 2010). Such broad optically thick clouds are associated with high values of S owing to their large horizontal extent and high concentrations of cloud ice. Radiative flux convergence is confined to a thin layer at cloud base. Heating is so intense, and the cloud is so broad, that the cloudy heated air cannot easily escape by spreading into surrounding clear air. Instead, large values of S favor the development of a deepening mixed layer. The mixed layer still spreads, but in the form of turbulent density currents rather than laminar motions. However, as the cloud spreads and thins...

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...ly unity within time t ’ 10t0. While the value of t0 is not explicitly defined, assuming that it is one buoyancy period 2p/N, then the time scale for the cirrus anvil to shift from turbulent mixing to isentropic adjustment is of order 1 h. Because this time scale is much less than tmax, the anvil never manages to enter a regime of runaway mixed-layer deepening where Eq. (26) is positive. What is interesting is that this time scale for a convecting anvil to move into a laminar flow regime is comparable to the few hours’ lifetime of tropical cirrus associated with deep-convective cloud systems (Mace et al. 2006). A transition to laminar behavior seems inevitable. Figure 10 shows numerical simulations for the time evolution of S within the cloud base domain. These reproduce the theoretically anticipated decay at a rate close to the anticipated t23 power law. The decay in S is dominated by mixed-layer deepening, which roughly follows the anticipated t1/2 power law, confirming that TABLE 5. Dominant modes of evolution observed in the simulations. Cases where S and E are much greater than 1 when evaluated at t 5 0 are indicated in boldface and italics, respectively. qi (g kg 21) L 100 m 1 km 10 km 0.01 E...

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...is the density of air, and Dz is the vertical pathlength through which the radiation is absorbed. The characteristic depth is the e-folding pathlength for the attenuation such that tabs/m5 1: h5 m k(re)qir . (3) Assuming an effective radius of 20 mm, the value for k(re) in cirrus is approximately 0.045 m2 g21(Knollenberg et al. 1993). Taking, for example, qi values of 1 g kg 21 that have been observed in medium-sized cirrus anvils in Florida (Garrett et al. 2005), h would be about 30 m. As a contrasting example, a cloud with qi values of 0.01 g kg 21, similar to those observed in thin cirrus (Haladay and Stephens 2009), would have a radiative penetration depth h of about 3000 m. Thus, the deposition of radiative energy in this layer increases its potential temperature at rate H5 du dt 5 21 rP dF dz ’ DFnet rcph 5 4s ~T 3 c k(re)qi mcp D ~T , (4) where P is the Exner function [P 5 cp(T/u)]. Insofar as the dynamic adjustment to radiative heating is concerned, it is not the potential temperature u that is relevant, but rather the virtual potential temperature, since this accounts for the density differences of vapor and condensate. However, to first order, du/dt; duy /dt: perturbations in the potential tempera...

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...or, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and divergence within cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et al. (2011) suggest that radiative cooling can help to initiate thin tropopaus...

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... effective brightness temperature difference between the lower-tropospheric air and cloud base. Provided the cloud is sufficiently opaque to act as a blackbody, radiative energy is deposited within a layer of characteristic depth h at the base of the cloud that is smaller than the depth of the cloud itself. Themagnitude of h can be obtained by considering that the thermal emissivity is given by « ’ 12 exp(2tabs/m) , (2) where tabs is the absorption optical depth and m is the quadrature cosine for estimating the integrated contribution of isotropic radiation to vertical fluxes. Usually m; 0:6 (Herman 1980). The absorption optical depth is determined by the cloud icemixing ratio qi, as well as the ice crystal effective radius re, through tabs 5 k(re)qirDz where k is the mass specific absorption cross-section density, r is the density of air, and Dz is the vertical pathlength through which the radiation is absorbed. The characteristic depth is the e-folding pathlength for the attenuation such that tabs/m5 1: h5 m k(re)qir . (3) Assuming an effective radius of 20 mm, the value for k(re) in cirrus is approximately 0.045 m2 g21(Knollenberg et al. 1993). Taking, for example, qi values of 1 g kg 21 th...

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...er than unity. If S. 1, the relevance of evaporation is less clear because a convective mixed layer develops, in which case one might expect continual reformation and evaporation of cloud condensate as part of localized circulations within the mixed layer. The more relevant comparison might be to rates of turbulent entrainment and mixing. 3. Numerical model To test the suitability of S [Eq. (17)] and E [Eq. (22)] for determining the cloud evolutionary response to local diabatic heating, we made comparisons to cloud simulations from the University of Utah Large-Eddy Simulation Model (UU LESM) (Zulauf 2001). An LES model is used because the resolved scales are sufficiently small to represent turbulent motions, convection, entrainment and mixing, and laminar flows. TheUULESM is based on a set of fully prognostic 3D nonhydrostatic primitive equations that use the quasicompressible approximation (Zulauf 2001). The model domain was placed at the equator, f 5 08, to eliminate any Coriolis effects. Even in the largest domain simulations, the maximum departure from the equator (50 km) is sufficiently small as to justify not including the Coriolis effect in the model calculations. The horizontal extent ...

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...ertainty in climate forecasts (Dufresne and Bony 2008), mainly because clouds both drive and respond to the general circulation, the hydrological cycle, and the atmospheric radiation budget. Unlike fields of water vapor, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and divergence within cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in dete...

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...rate aL. From Eq. (12), radiative heating increases the mixed-layer gravitational potential energy density at rate adz; 2 ›ln(dz) ›t L 5 2Hgh uyN 2dz2 . (15) From Eq. (14), the rate of loss of potential energy density due to expansion of themixed layer laterally into the clear-sky surroundings is aL; 2 ›ln(L) ›t dz 5 2 Ndz L . (16) The loss rate of potential energy density due to lateral expansion of the mixed-layer aL can alternatively be obtained through the dispersion relation for hydrostatic gravity waves v 5 Nk/m with horizontal wavenumber k; 1/L and vertical wavenumberm; 1/dz (Holton 2004). A dimensionless ‘‘spreading number’’ S can be defined as the ratio ofadz andaL in Eq. (10). For a cloud that is initially at rest, in which case a mixed layer has not yet developed, radiation deposition remains concentrated within the layer dz; h. FromEqs. (15) and (16), we obtain S5adz aL 5 [›ln(dz)/›t]jL [›ln(L)/›t]jdz 5 HgL uyN 3h2 , (17) which is the ratio of two rates of cloud spreading: spreading due to laminar lofting and spreading of a deepening mixed layer. If S . 1, then the potential energy density within the layer L2dz increases owing to radiative flux deposition at a rate adz th...

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... fields of water vapor, clouds evolve quickly, so their radiative forcing and dynamic evolution are highly coupled on time and spatial scales that cannot be easily resolved within global climate models (GCMs). For faithful reproduction of large-scale climate features, resolving radiatively driven motions on subgrid scalesmay be at least as important as accurately representing mean grid-scale fluxes (Cole et al. 2005). Radiative flux convergence and divergence within cloudy air is normally thought to produce vertical lifting and mixing motions (Danielsen 1982; Ackerman et al. 1988; Lilly 1988; Jensen et al. 1996; Dobbie and Jonas 2001). What is often overlooked is that clouds with a finite width also adjust to radiative heating by spreading horizontally, especially if the heating is concentrated in a thin layer at the cloud top or bottom (Garrett et al. 2005, 2006). Such radiatively driven mesoscale circulations have been identified within thin tropopause cirrus, and they are thought to play a role in determining the heating rate of the upper troposphere (Durran et al. 2009) and in stratospheric dehydration mechanisms (Dinh et al. 2010). Jensen et al. (2011) suggest that radiative cooling can help to...

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...72 10 km 1.5 0.68 1418 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70 5. Discussion We have separated the evolutionary response of clouds to local diabatic heating into distinct modes of cross-isentropic lifting, along-isentropic spreading, and evaporation of cloud condensate. A straightforward method has been described for determining how a cloud will evolve based on ratios of the associated rates. The dominant modes of evolution are outlined in Table 5. For example, cirrus anvils begin their life cycle as dense cloud from convective towers that have reached their level of neutral buoyancy (Scorer 1963; Jones et al. 1986; Toon et al. 2010). Such broad optically thick clouds are associated with high values of S owing to their large horizontal extent and high concentrations of cloud ice. Radiative flux convergence is confined to a thin layer at cloud base. Heating is so intense, and the cloud is so broad, that the cloudy heated air cannot easily escape by spreading into surrounding clear air. Instead, large values of S favor the development of a deepening mixed layer. The mixed layer still spreads, but in the form of turbulent density currents rather than laminar motions. However, as the clou...