PUBLICATIONS. Journal of Advances in Modeling Earth Systems

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1 PUBLICATIONS Journal of Advances in Modeling Earth Systems RESEARCH ARTICLE /2014MS Key Points: Stratocumulus LWP increases for increase in SST and fixed entrainment Opposite is found if entrainment response is allowed and is strong enough Sign of change in surface sensible heat flux depends on entrainment response Correspondence to: Stephan R. de Roode, Citation: De Roode, S. R., A. P. Siebesma, S. Dal Gesso, H. J. J. Jonker, J. Schalkwijk, and J. Sival (2014), A mixed-layer model study of the stratocumulus response to changes in large-scale conditions, J. Adv. Model. Earth Syst., 6, , doi: /2014ms Received 21 MAY 2014 Accepted 17 NOV 2014 Accepted article online 20 NOV 2014 Published online 17 DEC 2014 Corrected 30 DEC 2014 A mixed-layer model study of the stratocumulus response to changes in large-scale conditions Stephan R. De Roode 1, A. Pier Siebesma 1,2, Sara Dal Gesso 1,2, Harm J. J. Jonker 1, Jer^ome Schalkwijk 1, and Jasper Sival 1 1 Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, Netherlands, 2 KNMI, De Bilt, Netherlands Abstract A mixed-layer model is used to study the response of stratocumulus equilibrium state solutions to perturbations of cloud controlling factors which include the sea surface temperature, the specific humidity and temperature in the free troposphere, as well as the large-scale divergence and horizontal wind speed. In the first set of experiments, we assess the effect of a change in a single forcing condition while keeping the entrainment rate fixed, while in the second set, the entrainment rate is allowed to respond. The role of the entrainment rate is exemplified from an experiment in which the sea surface temperature is increased. An analysis of the budget equation for heat and moisture demonstrates that for a fixed entrainment rate, the stratocumulus liquid water path (LWP) will increase since the moistening from the surface evaporation dominates the warming effect. By contrast, if the response of the entrainment rate to the change in the surface forcing is sufficiently strong, enhanced mixing of dry and warm inversion air will cause a thinning of the cloud layer. If the entrainment warming effect is sufficiently strong, the surface sensible heat flux will decrease, as opposed to an increase which will occur for a fixed entrainment rate. It is argued that the surface evaporation will always increase for an increase in the sea surface temperature, and this change will be enlarged if the entrainment rate increases. These experiments aid the interpretation of results of similar simulations with single-column model versions of climate models carried out in the framework of the CFMIP-GCSS Intercomparison of Large-Eddy and Single-Column Models (CGILS) project. Because in a large-scale models, the entrainment response to changes in the large-scale forcing conditions depends on the details of the parameterization of turbulent and convective transport, intermodel differences in the sign of the LWP response may be well attributable to differences in the entrainment response. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 1. Introduction Stratocumulus cloud fields are frequently present in the areas over the subtropical oceans that are dominated by a synoptically driven large-scale subsidence and relatively high-potential temperatures in the lower free troposphere as compared to the boundary layer underneath [Klein and Hartmann, 1993]. Despite their rather shallow geometric depth, typically of the order of a couple of hundreds of meters, their high albedo (0:520:7) and their large areal coverage cause a significant negative radiative effect on the global energy balance [Hartmann et al., 1992]. Randall et al. [1984] argued that only small changes in the coverage and thickness of stratocumulus clouds are required to offset the warming due to enhanced greenhouse gas concentrations. It is therefore interesting to question how the global stratocumulus cloud amount will change under climate change conditions. Actually, the representation of stratocumulus clouds in climate models is challenging due to the delicate thermodynamic structure of stratocumulus-topped boundary layers, which are typically capped by a thermal inversion layer having a depth of only a few tens of meters across which the temperature may increase by about K [Wood, 2012]. It has been shown by Bony and Dufresne [2005] that subtropical marine boundary clouds, including stratocumulus, as simulated by climate models show the largest spread in the cloud radiative effect as a result of a doubling CO 2 scenario. These results strongly suggest that low clouds are the largest source for uncertainty to climate model sensitivity. Idealized experiments with single-column model (SCM) versions of global climate models and large-eddy simulation (LES) models were performed as part of the CGILS (CFMIP-GCSS Intercomparison of Large-Eddy DE ROODE ET AL. VC The Authors. 1256

2 and Single-Column Models) project to study the low-cloud response to an idealized global warming scenario [Zhang et al., 2013]. By applying a constant diurnal mean shortwave radiative flux at the top of the atmosphere and forcing terms like the large-scale advective tendencies of heat and moisture that were constant in time, the simulations were run toward an approximate steady state. In the perturbed climate scenario, the sea surface temperature was increased and changes in the large-scale forcing conditions were taken into account. A wide range of variability with both positive and negative cloud feedback signs was found from the SCM experiments. The LES results showed an overall positive feedback to a global warming scenario for shallow cumulus and cumulus under stratocumulus clouds [Blossey et al., 2013]. A negative feedback was found for the well-mixed stratocumulus regime in case the perturbed climate scenario included a weakening of the subsidence. By contrast, an additional set of LES experiments showed a positive feedback if the subsidence was not changed. Caldwell and Bretherton [2009] used a mixed-layer model (MLM) to show that a weaker subsidence tends to yield a higher stratocumulus cloud top and an overall thicker cloud layer. Because in the CGILS cases the perturbed climate scenario comprises a multitude of changes in forcing conditions, Bretherton et al. [2013] argue that compensating mechanisms may leave only a small net change in the cloud response which in turn may possibly explain the large scatter in low cloud feedbacks predicted by climate models. The CGILS setup was extended by including perturbations in the forcing conditions [Bretherton et al., 2013] and the free tropospheric humidity and temperature conditions [Dal Gesso et al., 2014a, 2014b]. In the study presented here, a MLM is applied to investigate the change in the cloud thickness if a single forcing term is changed. Because the SCM results presented by Zhang et al. [2013] suggest that the change in the vertical mixing at the top of the cloud plays a key role in the sign of the cloud radiative feedback, a set of experiments with, and one without an entrainment response, have been performed. In this way, it will be possible to systematically study how the entrainment response to a change in any of the forcing conditions may amplify, weaken, or even change the sign of the change in the cloud thickness. It will also be explained from analytical solutions how a strong entrainment response to an increase in the sea surface temperature will tend to increase the latent heat flux yet leaving a thinning of the cloud layer. The results bear some resemblance to the MLM study by Betts [1989]. However, the setup of the experiments presented here is very similar to the SCM intercomparison study by S. Dal Gesso et al. (A single-column model intercomparison on the stratocumulus representation in present-day and future climate, manuscript in preparation, 2014), which allows for a straightforward interpretation of the SCM results. 2. Mixed-Layer Model The MLM assumes a vertically well-mixed boundary layer. As the MLM includes the relevant physical processes acting in stratocumulus clouds, it is a very suitable tool to study well-mixed stratocumulus-topped boundary layers [Schubert et al., 1979; Betts, 1983; Nicholls, 1984; Betts, 1989; Bretherton and Wyant, 1997; Stevens, 2002, 2006; Bretherton et al., 2007; Caldwell and Bretherton, 2009; Caldwell et al., 2013; Jones et al., 2014], the indirect aerosol effect in stratocumulus [Wood, 2007], but also clear convective boundary layers [Van Driel and Jonker, 2011] topped by shallow cumulus clouds [Schalkwijk et al., 2013]. Bretherton et al. [2013] performed an elaborate suite of LES sensitivity experiments of the CGILS cases, which they showed could be well interpreted with a MLM model. The main motivation to use a conceptual model like the MLM in this study rather than a much more sophisticated tool like a large-eddy simulation model is that the analytical solutions of MLM equations provide some insight into how the external forcing conditions control the stratocumulus cloud amount Description of the Model Figure 1 displays a schematic representation of the assumed boundary layer structure in a MLM. Conserved variables w 2fq t ; h l g are constant with height, with the total specific humidity, q t 5q v 1q l ; q v and q l are the water vapor and liquid water specific humidities, respectively, and the liquid water potential temperature h l 5h2 Lv q l c p P, with h the potential temperature, L v the latent heat for condensation of water, c p the specific heat of dry air at constant pressure, and P the Exner function. Because the concept of a MLM for stratocumulus is explained in detail by Stevens [2006] and Caldwell and Bretherton [2009], only a short summary will be presented. The tendency of the mean mixed-layer value w ml is governed by its turbulent flux at the surface F w;sfc and the top of the boundary layer F w;t, and the vertical variation of a source term S w which can, DE ROODE ET AL. VC The Authors. 1257

3 Figure 1. Schematic of the vertical profiles of the liquid water potential temperature, total specific humidity, and liquid water specific humidity in the mixed layer model. for example, represent the effect of radiation, a loss of water by precipitation or a contribution due to large-scale horizontal advection. The turbulent flux at the surface is obtained with use of the following bulk formula F w;sfc 5C d Uðw sfc 2w ml Þ; (1) where the subscript sfc denotes the surface value and U the total horizontal wind speed in the boundary layer, and C d 50:001 is a dimensionless aerodynamic bulk exchange coefficient. In accord with the flux-jump relation, the entrainment flux at the top of the mixed layer F w;t is taken proportionally to the entrainment rate (w e ) and the jump of w across the inversion [Lilly, 1968], F w;t 52w e ðwj z 1 i 2w ml Þ: (2) Here zi 1 denotes the height just above the inversion layer, which is assumed to have a vanishingly small thickness. Using the above formulas, the budget equation for w ml can be expressed as [Stevens, z ml i 5C d sfc 2w ml Þ1w e ðwj z 12w i ml Þ2DS w ; (3) where DS w represents the total change of the source terms between the surface and the top of the boundary layer. The tendency equation for the inversion height completes the 5w e1wj zi ; (4) where w indicates the large-scale subsidence velocity. The level at which the saturation specific humidity q sat equals q t;ml defines the cloud-base height z b. The height-dependent liquid water content q l is computed with use of the solution for h l;ml and q t;ml, the hydrostatic equation, the gas law, and the Clausius- Clapeyron relation Entrainment Closure A critical part of the MLM is its closure with a parameterization for the cloud top entrainment. The entrainment rate is controlled by the inversion strength, the net radiative forcing in the cloud layer, and the production of turbulent kinetic energy. Although most entrainment parameterizations take these processes into account, they basically differ in the weight given to each process [Stevens, 2002]. In this study, we calculate the entrainment rate following Nicholls and Turton [1986], with w e 5A w3 z i Db ; (5) Db g h 0 Dh v ; (6) the buoyancy jump across the inversion, g the acceleration due to gravity, h sfc a reference temperature, and h v the virtual potential temperature. The convective velocity scale w is a function of the vertical integral of the buoyancy flux, The entrainment efficiency factor is given by ð z 1 1=3 i w 5 2:5 w 0 b dz! 0 : (7) z50 DE ROODE ET AL. VC The Authors. 1258

4 A5a 1 11a 2 ð12 D m Db Þ ; (8) where the quantity D m 52 Ð 1 0 bðvþdv and v5m 1=ðm 1 1m 2 Þ, the ratio of the mass of air originating from just above the inversion layer (m 1 ) to the total air mass of a mixed air parcel (m 1 1m 2 ) with m 2 the mass of boundary layer air. The factor 2 is included to let D m 5Dh v for cloud-free conditions. It is assumed that all possible mixing fractions have an equal probability to occur. In the absence of cloud liquid water, D m 5Db and consequently A 5 a 1. With a 1 50:2, this parameterization yields the generally accepted entrainment rate for a clear convective boundary layer. By contrast, in case cloudy air is mixed with relatively warm and dry inversion air, evaporative cooling will enhance the entrainment efficiency A as D m < Db in accord with observations [De Roode and Duynkerke, 1997] and direct numerical simulation results [Mellado, 2010]. We use a 2 515, as this value has been found to give the best agreement with observations [Stevens et al., 2005; Bretherton et al., 2007]. Dal Gesso et al. [2014a] performed MLM simulations similar to the ones discussed in this paper except that multiple changes in the forcing conditions were studied. They tested different entrainment parameterizations and found that they all gave qualitatively similar cloud responses Setup of the Experiments We basically follow the same procedure as proposed by Dal Gesso et al. [2014a] and Dal Gesso et al. (manuscript in preparation, 2014). To solve the MLM equations, it is necessary to specify the boundary conditions at height z 1 i, which is just above the inversion layer. In the free troposphere, the variation of the potential temperature with height is set by a fixed vertical gradient c h 56 K/km, hðzþ5h ft;0 1 c h z for z z 1 i ; (9) where the quantity h ft;0 is the extrapolated value of the free tropospheric potential temperature profile to the surface. The lower-tropospheric stability (LTS) is defined as the difference between the potential temperature at 700 hpa and the sea surface. Because in the simulations, the pressure level defining the LTS is very close to a geometric height of z LTS 53 km, we calculate the LTS according to The large-scale subsidence is assumed to vary linearly with height, LTS hj z53km 2h sfc 5h ft;0 1c h z LTS 2h sfc : (10) wðzþ52dz; (11) with D the large-scale divergence (of the horizontal wind). Note that this linear profile differs from Dal Gesso et al. [2014a] and Dal Gesso et al. (manuscript in preparation, 2014c), who used a profile that follows an exponential power law. The fact that a linear subsidence profile allows simple analytic solutions of the MLM equations has strongly motivated our choice. We will also assume horizontally homogeneous conditions such that we can neglect the horizontal advection of heat and moisture. The free tropospheric specific humidity is set to a constant value, qj z 15q ft, such that subsidence does not impose a drying trend above i the cloud layer. It is assumed that the radiative cooling and the subsidence warming are maintaining a constant potential temperature in the free troposphere. Therefore, in summary, both the specific humidity and the potential temperature in the free troposphere are assumed to be constant in time. The drizzle flux is set to zero in this study, S qt 50. The results are therefore particularly representative for a polluted environment in which low clouds have a relatively high cloud droplet concentration and small drizzle fluxes. For a detailed analysis on the response of the stratocumulus cloud thickness to changes in the cloud droplet concentration, we refer to a MLM study by Wood [2007]. The effect of the radiative forcing F rad is included in the source term of the liquid water potential temperature budget equation according to, DS hl 5DF rad =qc p 5ðDF 2kq ft Þ=ðqc p ÞDF; (12) where q is the density of air and DF 562 W m 22. The quantity k 5 8Wm 22 /(g kg 21 ) takes into account the dependency of the net longwave radiative absorption on the free tropospheric specific humidity that is assumed to be constant up to a height of 3 km [Dal Gesso et al., 2014a]. For notational conveniency, we use DF to indicate the net radiative forcing, which has units m K s 21. The cooling due to longwave radiation DE ROODE ET AL. VC The Authors. 1259

5 Table 1. Summary of Experimental Settings for the Reference Case and the Magnitude of the Applied Perturbations in the Sensitivity Experiments a Variable u Units Reference Value Perturbation du Definition h sfc K Surface potential temperature D s : Large-scale divergence U ms Horizontal wind velocity h ft;0 K [285,301] 1.0 Free troposphere reference potential temperature, see equation (9) q ft gkg 21 [0,9] 0.1 Free troposphere specific humidity c h Kkm in free troposphere a The SST K, the surface pressure p sfc hpa, which yields a surface saturation specific humidity q sat;sfc 511:5 gkg 21.We used a reference pressure of 1000 hpa for the calculation of the potential temperature. depends critically on the thermal emissivity of the atmosphere above the boundary layer [Bretherton et al., 2013], whose net effect is implicitly taken into account through the k parameter in (12). A cloud layer is present in the initial state, and the model is run until the boundary layer is in a steady state. Table 1 presents the settings of the reference case. The sea surface temperature, horizontal wind speed, and large-scale divergence are representative values for STBLs [De Roode and Duynkerke, 1997; Stevens et al., 2005]. The figures presented show the results as a function of the LTS and the free tropospheric specific humidity. Because the reference experiments were all performed for a constant SST, and because the vertical gradient of the potential temperature in the free troposphere is also fixed, the variation of the LTS is solely determined by changes in the quantity h ft;0. 3. Steady State Solutions Figure 2 displays the steady state solutions of boundary layer properties as a function of the LTS and the specific humidity in the free troposphere, q ft. The solutions in the upper right corner of the phase-space plots represent fog layers. This regime where the cloud-base height reaches the ground surface is stippled. The left part of the plots is also stippled to indicate deep boundary layers with negative surface buoyancy fluxes, which will have a damping effect on turbulence and subsequently on the vertical mixing. This regime is loosely referred to as decoupled since these boundary layers might violate the well-mixed assumption. Figure 2a shows that both quantities have a clear impact on the LWP. Higher LTS values are accompanied by smaller LWP values. For low LTS values, we find boundary layer heights that are deeper than about 1 km (see Figure 2b). The cloud top height is controlled by a balance between the entrainment and subsidence velocities. In particular, it follows directly from equation (4) that in a steady state, the entrainment rate and boundary layer depth must be linearly related, w e 5Dz i : (13) Deep boundary layers are caused by high entrainment rates, which according to flux-jump relation (2) imply rather large entrainment fluxes of heat and moisture. To allow for a steady state, the surface heat flux must become negative to compensate for the strong entrainment warming effect, the magnitude of the latter being proportional to the entrainment rate. Turbulence in the subcloud layer will be damped if the surface buoyancy flux becomes negative, and subsequently the assumption that the boundary layer is vertically well mixed may be violated. For deep boundary layers, the cloud layer has a higher value of h l and a smaller value of q t than the subcloud layer, with shallow cumulus developing at the top of the subcloud layer and stratocumulus present below the capping inversion layer [Bretherton et al., 2013; Van der Dussen et al., 2013]. Figures 2b and 2c show that toward very dry free tropospheric conditions, deep boundary layers and high entrainment rate are found. This is on one hand due to a strong effect of evaporative cooling near cloud top to the production of turbulence and a maximum longwave radiative cooling rate. For the exact opposite reasons, the boundary layer becomes relatively shallow toward high values of q ft. Figures 2d 2f demonstrate that this has a distinct influence on the thermodynamic structure of the boundary layer and on the cloud-base height. Let us consider the mixed-layer budget equation equation (3), for which steady state solution for q t;ml can be written as, DE ROODE ET AL. VC The Authors. 1260

6 Figure 2. Steady state solutions as a function of the lower tropospheric stability (LTS) and the free tropospheric specific humidity (q ft ) for (a) the liquid water path, (b) the inversion height z i, (c) the entrainment rate w e, (d) the liquid water potential temperature h l;ml, and (e) total water specific humidity in the mixed layer q t;ml, respectively, and (f) the cloud-base height z b. The shaded area depicts fog situations in which the cloud-base height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes. DE ROODE ET AL. VC The Authors. 1261

7 q t;ml 5q sat;sfc 1 w eðq ft 2q sat;sfc Þ : (14) w e 1C d U The total specific humidity in the mixed layer thus depends delicately on the ratio of the horizontal wind and entrainment velocities. In particular, if the entrainment velocity is small compared to C d U, we obtain and for C d U=w e! 0, we find lim q t;ml5q sat;sfc ; (15) C d U=w e!1 lim q t;ml5q ft : (16) C du=w e!0 For large LTS, the entrainment rate will be relatively small and the mixed layer will be relatively moist. Last, we note that the results are qualitatively similar to the ones presented by Dal Gesso et al. [2014a], who used a different subsidence profile. In comparison to the behavior of q t;ml ; h l;ml behaves differently in the sense that a warmer free troposphere yields lower h l;ml values. The steady state solution for h l;ml includes a sink term that represents the net radiative cooling in the mixed layer, h l;ml 5h sfc 1 w eðh l j z 12h sfc Þ2DF i : (17) w e 1C d U The liquid water potential temperature just above the inversion height can be eliminated by means of equations (9) and (13) to give h l;ml 5h sfc 1 w eðh ft;0 1c h w e =D2h sfc Þ2DF : (18) w e 1C d U The solutions for the limiting cases of the ratio C d U=w e become, respectively, and lim h l;ml5h sfc 2DF=ðC d UÞ; (19) C d U=w e!1 lim h l;ml5h ft;0 1w e c h =D2DF=w e : (20) C d U=w e!0 Thus, for small entrainment rates, the mixed layer will be relatively cold. Because for small w e, the mixed layer will also be relatively moist, very low cloud-base heights are found in the upper right part of the phase space, where the LTS and q ft are both large. The solutions for which z b 50 can be interpreted as fog. We conclude that for a given LTS, lower cloud-base heights and consequently thicker clouds are found for higher specific humidity values in the free troposphere. 4. The Boundary Layer Response to Perturbations in Large-Scale Forcing Conditions We notice from the analytical solutions (14) and (18) that the equilibrium state of the mixed layer is governed by a set of cloud controlling factors including the dynamic quantities U and D, the thermodynamic quantities h sfc, h ft;0 ; q ft, and c h. The surface saturation specific humidity q sat;sfc depends on the potential sea surface temperature h sfc. Although we can derive analytical steady state solutions for the mixed layer, it is highly nontrivial to derive expressions for LWP changes as a result of changes in the cloud controlling parameters. This is primarily due to the fact that the LWP solutions are expressed in terms of the cloud controlling factors but also in terms of the entrainment velocity, which in itself depends also on the cloud controlling factors in a complex way. More precisely, assume that we know the LWP in terms of n cloud controlling factors u 1 ; ::::; u n and an entrainment velocity w e ðu 1 ; ::::; u n Þ that, in turn, also depends on the cloud controlling factors. The change of the LWP as a result of a variation in a single cloud controlling variable u i can then be written as DE ROODE ET AL. VC The Authors. 1262

8 3 du fflfflfflfflffl{zfflfflfflfflffl} i fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} w e u fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} i total direct i fflfflffl{zfflfflffl} entrainment response ; (21) where the first term on the RHS is referred to as the direct contribution due to the change of LWP through u i for a fixed entrainment rate, while the second term refers to the indirect contribution by the response of w e due to the variation of u i and a multiplier term. The latter is associated with the change in the LWP resulting from a variation in the entrainment rate. This procedure bears some resemblance to the study of Gordon and Norris [2010] who made an observational analysis of how midlatitude oceanic clouds change with temperature when dynamical processes are held constant (i.e., partial derivative with respect to temperature). The direct response and the multiplier term can be computed directly from the analytical MLM solutions. This does not hold for the entrainment response which in general will depend on the physical parameterization scheme applied in the model Fixed Entrainment Rate: The Direct Boundary Layer Response We calculated the direct LWP response to a perturbation of a single cloud controlling factor u i while keeping both the other forcing factors and the entrainment rate fixed to the reference case values. This was achieved using the steady state solutions (14) and (18), from which we computed perturbed values for q t;ml and h l;ml after adding a perturbation du i to the reference value of u i. The values of the perturbations are presented in Table 1. These results can also be directly obtained with aid of partial derivatives and the analytical mixed-layer model results. Note that the partial derivatives only weakly depend on the entrainment parameterization that is used, through the effect of the latter on the reference state itself. The plots shown on the left part of Figure 3 display the direct LWP response to perturbations in the cloud controlling factors h sfc, h ft;0 ; q ft, U, and D. The values of the LTS, and q ft shown on the x and y axes are according to the reference case. Figure 3a shows the direct LWP response after an increase in the sea surface temperature, the latter of which affects both the heat and moisture fluxes from the surface. The overall increase in the LWP suggests that the moistening of the mixed layer dominates the warming effect. Because a fixed entrainment rate implies that the cloud top height is fixed to the reference case value, any increase in the cloud thickness must result from the partial b =@h sfc < 0. The cloud thickening shown in Figure 3a can be understood from the partial derivatives of q t;ml and h l;ml with respect to h sfc, which can be readily obtained from (14) and (18), respectively, Likewise, it follows from equation t;ml 5 C sat;sfc ; sfc w e 1C d l;ml 5 C du < 1: sfc w e 1C d t;ml 5 C du < 1: sat;sfc w e 1C d U The partial derivatives (23) and (24) show that for a fixed entrainment rate, the warming and moistening of the boundary layer will always be less than their respective changes at the sea surface. This is actually shown in Figures 4a and 4b, where we note that an increase dh sfc 51 K corresponds to a change dq sat;sfc 50:76 g/kg in the surface saturation specific humidity. In fact, the factor C d U=ðw e 1C d UÞ is the fractional importance of the bottom versus top boundary conditions. The impact on the mixed-layer value following a perturbation of the surface boundary condition is thus proportional to the relative importance of that boundary. Using the partial derivatives, we can also t;ml =@h sfc sat;sfc sat ; l;ml =@h z b where the last inequality is satisfied if the temperature at the cloud base is lower than the SST. It actually states that for fixed w e and C d U the relative humidity in the subcloud layer will increase following an DE ROODE ET AL. VC The Authors. 1263

9 Figure 3. The response of the LWP to changes in cloud controlling factors which are indicated in the boxes on the left side of the plots. Following equation (21), the direct LWP response is computed with an entrainment rate fixed to the reference case value, the total LWP response includes an entrainment response, and the indirect effect is the difference between the total and direct effects. (a c) The direct, indirect, and total LWP responses to changes in the potential sea surface temperature h sfc, (d f) the potential temperature in the free troposphere h ft;0, (g i) the specific humidity q ft in the free troposphere, (j l) the horizontal wind speed U, and (m o) the large-scale divergence D. The shaded area depicts fog situations in which the cloud-base height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes. DE ROODE ET AL. VC The Authors. 1264

10 Figure 4. The direct response of mixed-layer properties to changes in the potential sea surface potential temperature h sfc for the mixed-layer (a) liquid water potential temperature h l;ml, (b) total water specific humidity q t;ml, and (c) the cloud-base height z b. The shaded area depicts fog situations in which the cloud-base height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes. increase in the SST. This occurs because the increase in the boundary layer specific humidity is dictated by the Clausius-Clapeyron relation at sea surface temperature, whereas the saturation specific humidity at the cloud-base height will increase by a smaller amount if the cloud-base temperature is colder than the sea surface. An increase in the subcloud layer relative humidity will result in a descending cloud base. For the case in which the SST increases with a fixed entrainment rate, the partial derivatives state that the effect of boundary layer moistening dominates the warming effect causing a thickening of the cloud layer. A warming of the free troposphere through an increase of h ft;0 has a warming effect on the boundary l;ml w e 5 > 0: ft;0 w e 1C d U If the entrainment rate is fixed, the boundary layer humidity does not change, and the boundary layer warming causes the cloud layer to thin (see Figure 3d). Figure 3g shows the change in the LWP for an increase in the free tropospheric humidity. A moistening of the free tropospheric humidity causes a more humid boundary t;ml 5 w e > 0: ft w e 1C d U This effect alone will obviously lead to a cloud thickening. However, because the radiation depends on the free tropospheric humidity according to (12), the change in the net radiative cooling also affects h ft @q ft 1 k > 0: (28) w e 1C d U qc p Figure 3g shows that these two competing effects on the LWP cause a net cloud thinning for high values of the LTS but a thickening for low LTS and low q ft. The effect of the horizontal wind speed can also be evaluated from the partial derivatives " # " 52C w e ðh l j z 12h sfc Þ2DF i d ðw e 1C d UÞ 2 5Cd 2 U h sfc2h l;ml ðw e 1C d UÞ 2 ; (29) " 52C w e ðq ft 2q sat;sfc Þ d ðw e 1C d UÞ 2 > 0: (30) In the last step of (29), we have substituted the steady state solution of equation (3). An increase in the horizontal wind speed U will moisten the boundary layer and will warm the boundary layer in the regime with DE ROODE ET AL. VC The Authors. 1265

11 positive sensible heat fluxes, each of which have opposing effects on the cloud-base height. Figure 3j illustrates that except for the high LTS and high free tropospheric humidity values an increase in the horizontal wind speed will cause a thickening of the cloud. Finally, Figure 3m shows that the LWP tends to decrease if the large-scale divergence is increased while the entrainment rate is kept constant. This is predominantly due to a descent of the cloud top height, which is in line with large-eddy simulations of stratocumulus studied by Blossey et al. [2013] The Entrainment Multiplier Figure 5 shows the multiplier e. Interestingly, in the upper left part of the phase space, i.e., for conditions in which the free troposphere is relatively cold and e > 0. In every part of the phase space, the cloud-base height must increase due to a warming and drying of the mixed layer according to, 5 C c duðh ft;0 2h sfc Þ1DF1w h e D ðw e 12C d e ðw e 1C d UÞ 2 > 0; t;ml 5 C duðq ft 2q sat;sfc e ðw e 1C d UÞ 2 < 0; (32) both of which follow straightforwardly from the analytical steady state solutions (14) and (18). In the regime e > 0, the rise of the cloud-base height is smaller than that of the cloud top, leaving a thicker cloud layer and a higher LWP. Randall [1984] discussed the change in the cloud layer thickness as a result of a change in the entrainment velocity only, and derived the thermodynamic jump conditions that will result in a cloud deepening through entrainment. The findings in Figure 5 are a variation of Randall s study in the sense that in our experiments other forcing terms like surface fluxes of heat and moisture, and radiative cooling are included The Total LWP Response According to equation (21), the response of the entrainment rate to changes in the cloud controlling factors may either amplify, dampen, or even change the sign of the total LWP response as compared to the direct response. To assess the total response of the boundary layer, a second suite of experiments was performed in which the entrainment rate was allowed to respond to the imposed changes in the large-scale forcing conditions. It is recognized that the entrainment response in a large-scale model will in general depend on the details of the turbulent transport scheme used. The entrainment response is therefore a property that is key to interpret the cloud feedback found in a model [Dal Gesso et al., 2014a]. The total LWP responses are shown in Figures 3c, 3f, 3i, 3l, and 3o. We first notice that differences between the direct and total LWP response are very small for changes in q ft, U, and D. The change in the LWP as a result of an increase in the free tropospheric moisture shows quite some variations which is due to the fact that the entrainment rate depends on the surface evaporation, cloud thickness, radiative cooling as well as the value of q ft itself, where all these quantities change for a change in q ft. Second, the LWP response to a change in h sfc is approximately similar but with opposite sign to a similar change in h ft;0. This is due to the fact that a change in either quantity has an equal but opposite effect on the LTS, where its weakening will support higher entrainment rates. The key result of these experiments is that the entrainment response can change the sign of the LWP change. A close inspection of the lower right corner of the phase space, which represents a relatively warm and dry free troposphere, reveals that for both quantities, as well as for q ft, the direct and total LWP responses have opposite signs (e.g., a and c, d and f, or g and i). Specifically, in this part of the phase space, the direct sfc > 0, whereas the total response dlwp=dh sfc < 0. The different signs can be understood from the multiplier being e < 0, while the results in Figure 6 show that the entrainment response to an increase in h sfc is e =@h sfc > 0. The Nicholls and Turton [1986] entrainment parameterization triggers larger entrainment rates for larger h sfc, such that the indirect entrainment effect counteracts the direct effect. A similar reasoning applies to the results for a change in the free tropospheric temperature or humidity. DE ROODE ET AL. VC The Authors. 1266

12 Figure 5. The partial derivatives of the liquid water path with respect to the entrainment rate. The shaded area depicts fog situations in which the cloud-base height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes Surface Flux Response According to Webb and Lock [2013], the change in the surface evaporation plays a critical, yet poorly understood role in the cloudclimate feedback. It is therefore interesting to ask how the surface evaporation will change for a change in the sea surface temperature in our MLM framework, and also how this change is affected by the entrainment response. The sensitivity of the surface latent heat flux to a change in the sea surface temperature can be readily assessed from equation (1), df qt;sfc 5C d U 12 dq t;ml : (33) dq sat;sfc dq sat;sfc The question how q t;ml changes with respect to q sat;sfc is thus relevant for understanding the change in the latent heat flux. The total change in q t;ml as a result of a change in q sat;sfc as dictated by the SST can also be split into different components, With aid of equation (22) it follows that dq t;ml t;ml dh sfc w sfc h sfc e : sfc sfc : (35) Since the increase in q t;ml is smaller than in q sat;sfc, the direct effect of a sea surface temperature increase is an enhanced surface humidity flux. The effect of the indirect term can be predicted with the results presented before. The entrainment multiplier term was already evaluated in equation (32) and was found to be negative. Figure 6 shows a positive entrainment response to an increase in the SST such that the indirect effect will cause a further increase in the moisture difference between the surface and the mixed layer. In other words, an enhanced entrainment mixing of dry free tropospheric air will act to dry the boundary layer. The total derivative of q t;ml with respect to h sfc is displayed in Figure 7b. As dq sat;sfc 50:76 g/kg, the moistening of the boundary layer is significantly smaller than the increase in the saturation specific humidity at the surface. An increased humidity difference between the surface and the boundary layer will cause an increase in the surface humidity flux. Along the same line, the change in the surface sensible heat flux can be assessed by evaluating the total change in h l;ml as a result of a change in the SST, dh l;ml e 3 : (36) dh fflfflfflfflffl{zfflfflfflfflffl} sfc fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} w e h fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} fflfflfflffl{zfflfflfflffl} sfc total direct multiplier entrainment response If the entrainment rate does not respond to a change in h sfc, then the change in the sensible heat flux will be controlled by the direct term in (36). According to equation l;ml =@h sfc < 1, which predicts that the direct effect of an increase in the SST is an increase in the surface sensible heat flux. It is therefore interesting to note that if the entrainment rate is allowed to respond to the change in the SST, dh l;ml =dh sfc > 1, as shown in Figure 7. This means that the warming of the mixed layer is larger than that of the sea surface, which is primarily due to an enhanced entrainment of warm inversion air. As an explanation, equation (31) states that the multiplier term in (36) is positive, and Figure 6 shows a positive entrainment response. This DE ROODE ET AL. VC The Authors. 1267

13 example shows that the sign of the change in the sensible heat flux strongly depends on the entrainment response. Figure 6. The partial derivative of the entrainment rate with respect to the potential sea surface temperature h sfc. The shaded area depicts fog situations in which the cloud-base height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes. Zhang et al. [2013] studied the response of a well-mixed stratocumulus regime to an idealized global warming scenario with SCMs, and found that for most SCMs, the sensible heat flux decreased while the latent heat flux increased in the perturbed climate. Although these simulations have a somewhat different setup as in our MLM study, the MLM results suggest that an increase in the entrainment rate could be a likely cause of the opposite signs in the changes of the sensible and latent fluxes. Rieck et al. [2012] used a LES model to study the effect of a warmer atmosphere on a shallow cumulus cloud field, and reported that despite an increase in the surface evaporation, a stronger cloud top mixing causes fewer clouds. Following the same reasoning, the MLM predicts an increase in the latent heat flux if the SST is increased, which however may not prevent the stratocumulus cloud layer to thin in case the entrainment response to the change in the surface forcing causes a strong drying and warming tendency. Finally, it is worthwhile to mention that opposite changes in the sensible and latent heat fluxes were also found from a shallow cumulus case modeling study in case the horizontal wind speed is increased [Nuijens and Stevens, 2012]. 5. Conclusions This study is motivated by a suite of experiments carried out with single-column model versions of climate models by Zhang et al. [2013] to assess the effect of an idealized global warming scenario on the change in the low cloud amount. For the well-mixed stratocumulus regime, models simulated opposite signs in the change of the cloud radiative effect. This was explained by differences in the way models represent mixing at top of the cloud layer. To study the effect of entrainment on the response of stratocumulus clouds to changes in the large-scale forcing conditions, we perturbed equilibrium state solutions of stratocumulustopped boundary layers using a mixed-layer modeling approach. We first performed a control experiment, after which we perturbed the large-scale forcing conditions one at a time. To disentangle the effect of the perturbed large-scale forcing from the entrainment response, the perturbed experiments were performed with an entrainment rate fixed to the control value and another set in which the entrainment rate was allowed to respond to the change in the forcing. The MLM results show that for a fixed entrainment rate, the LWP will increase if the SST is increased, irrespective of the free tropospheric conditions. However, if the entrainment rate response is sufficiently large, the enhanced mixing of dry and warm free tropospheric air can cause a thinning of the cloud and a subsequent reduction of the LWP. These results bear some analogy with the findings of the CGILS project [Zhang et al., 2013], which found that a thinning of the stratocumulus deck under perturbed climate conditions occurred in SCMs with an active shallow convection scheme, and vice versa. Furthermore, we demonstrate that an increase in the SST while keeping the entrainment rate fixed will unambiguously cause an increase in the surface sensible heat flux. By contrast, if the entrainment response is sufficiently large, an enhanced entrainment warming will diminish the temperature gradient near the surface causing a decrease in the sensible heat flux. Furthermore, an increase in the SST will increase the surface latent heat flux, which will be further enhanced if the entrainment rate increases. DE ROODE ET AL. VC The Authors. 1268

14 Figure 7. The total response of the mixed-layer (a) liquid water potential temperature h l;ml and (b) the total water specific humidity q t;ml, respectively, to a perturbation of the potential sea surface temperature h sfc. The shaded area depicts fog situations in which the cloudbase height is at the sea surface, and the points in the phase space that are left of the dashed line have negative surface buoyancy fluxes. Though the results show that a change in a single forcing term can have a significant impact on the LWP, the results also suggest that the net effect of simultaneous changes on the LWP may be small [Bretherton et al., 2013]. For example, the increase in the SST in the perturbed CGILS experiments is accompanied by an equal increase in the free tropospheric temperature. The MLM experiments show that their individual effects on the LWP are approximately opposite. A similar set of quantities that have opposing effects on the LWP change are the horizontal wind speed and the large-scale divergence. Both quantities can be expected to change simultaneously, for example, if the strength of the Hadley circulationwoulddecrease.inthatcase,areducedwindspeed and a smaller large-scale divergence will each affect the LWP in an opposite way. The results presented in this paper serve as a diagnostic tool to analyze and explain results from a similar series of numerical experiments carried out with SCM versions of climate models and a large-eddy simulation model (Dal Gesso et al., 2014b, also manuscript in preparation, 2014) as part of the European Union Cloud Intercomparison, Process Study and Evaluation Project (EUCLIPSE) project. Acknowledgment This work has received funding from the European Union, Seventh Framework Programme (FP7/ ) under grant agreement The data obtained with the MLM will be provided by the authors on request. References Betts, A. K. (1983), Thermodynamics of mixed stratocumulus layers: Saturation point budgets, J. Atmos. Sci., 40, Betts, A. K. (1989), Idealized stratocumulus models and cloud layer thickness, Tellus, Ser. A, 41, , doi: /j tb00379.x. Blossey, P. N., C. S. Bretherton, M. Zhang, A. Cheng, S. Endo, T. Heus, Y. Liu, A. P. Lock, S. R. de Roode, and K.-M. Xu (2013), Marine low cloud sensitivity to an idealized climate change: The CGILS LES intercomparison, J. Adv. Model. Earth Syst., 5, 1 25, doi: / jame Bony, S., and J.-L. Dufresne (2005), Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models, Geophys. Res. Lett., 32, L20806, doi: /2005gl Bretherton, C. S., and M. C. Wyant (1997), Moisture transport, lower-tropospheric stability, and decoupling of cloud-topped boundary layers, J. Atmos. Sci., 54, Bretherton, C. S., P. N. Blossey, and J. Uchida (2007), Cloud droplet sedimentation, entrainment efficiency, and subtropical stratocumulus albedo, Geophys. Res. Lett., 34, L03813, doi: /2006gl Bretherton, C. S., P. N. Blossey, and C. R. Jones (2013), Mechanisms of marine low cloud sensitivity to idealized climate perturbations: A single LES exploration extending the CGILS cases, J. Adv. Model. Earth Syst., 5, , doi: /jame Caldwell, P., and C. S. Bretherton (2009), Response of a subtropical stratocumulus-capped mixed layer to climate and aerosol changes, J. Clim., 22, Caldwell, P., Y. Zhang, and S. Klein (2013), CMIP3 subtropical stratocumulus cloud feedback interpreted through a mixed-layer model, J. Clim., 26, Dal Gesso, S., A. P. Siebesma, S. R. de Roode, and J. M. van Wessem (2014a), A mixed-layer model perspective on stratocumulus steady states in a perturbed climate, Q. J. R. Meteorol. Soc., 140, , doi: /qj Dal Gesso, S., A. P. Siebesma, and S. R. de Roode (2014b), Evaluation of low-cloud climate feedback through single-column model equilibrium states, Q. J. R. Meteorol. Soc., in press. De Roode, S. R., and P. G. Duynkerke (1997), Observed Lagrangian transition of stratocumulus into cumulus during ASTEX: Mean state and turbulence structure, J. Atmos. Sci., 54, Gordon, N. D., and J. R. Norris (2010), Cluster analysis of midlatitude oceanic cloud regimes: Mean properties and temperature sensitivity, Atmos. Chem. Phys., 10, DE ROODE ET AL. VC The Authors. 1269

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