1. Introduction
Stratocumulus cloud layers are frequently found over relatively cold parts of the subtropical oceans and in the presence of large-scale subsidence. These conditions favor the formation of a thermal inversion, which acts to trap moisture, giving rise to extended fields of stratocumulus (Wood 2012). Although the depth of stratocumulus layers is relatively shallow, typically on the order of a few hundreds of meters, they strongly reflect downwelling solar radiation. During the equatorward transport by the prevailing trade winds over increasing sea surface temperatures, the subtropical stratocumulus cloud fields gradually break up and are replaced by shallow cumulus clouds. If a model is not able to capture this stratocumulus-to-cumulus cloud transition (SCT), this will lead to significant errors in the radiative fluxes received at the ground surface. This is a critical problem, as climate models disagree on the change of the subtropical low-cloud amount under a global warming scenario, which gives rise to a considerable amount of uncertainty in projections of the future global-mean temperature (Bony and Dufresne 2005; Webb et al. 2013; Tsushima et al. 2016).
To investigate the change of the low-cloud amount under an idealized warming scenario, Zhang et al. (2013) performed experiments with single-column model (SCM) versions of climate models and large-eddy simulation (LES) models. The LES results point to a reduction of the amount of subtropical marine low clouds in a warmer climate (Blossey et al. 2013; Van der Dussen et al. 2015; Bretherton 2015). The study by Zhang et al. (2013), and follow-up studies by Dal Gesso et al. (2014) and Dal Gesso et al. (2015) report a wide scatter in the change of the steady-state subtropical low-cloud amount in the SCM results. These results actually give rise to the question of how large-scale forcing conditions like the sea surface temperature, free-tropospheric temperature and humidity, and the large-scale subsidence determine control the SCT.
The SCT has been the subject of several observational (e.g., Albrecht et al. 1995; Bretherton et al. 1995; De Roode and Duynkerke 1997; Sandu et al. 2010) and modeling studies (e.g., Krueger et al. 1995; Sandu and Stevens 2011; Van der Dussen et al. 2013). Chung et al. (2012) studied a series of steady-state LESs in the SCT regime, which can be interpreted as an Eulerian view of the transition. These studies helped to develop a conceptual view of this transition. According to this model, the cloud breakup is fundamentally driven by the increasing SST. Convective activity driven by surface evaporation increases as the air advects over warmer waters. The strengthening of convectively driven turbulence enhances the entrainment of warm and dry free-tropospheric air at cloud top, which leads to a higher virtual potential temperature of the stratocumulus cloud layer as compared to the subcloud layer. This stratification prevents surface-driven thermals from reaching the stratocumulus cloud, except if they become saturated. In that case, latent heat release due to condensation of water allows the plumes to rise as positively buoyant cumulus clouds, which may penetrate the stratocumulus cloud layer to inject it with moisture from below (Wang and Lenschow 1995; Miller and Albrecht 1995; De Roode and Duynkerke 1996; Van der Dussen et al. 2014). Meanwhile, the stratocumulus gradually thins if entrainment of relatively warm and dry free-tropospheric air dominates the longwave radiative cooling at cloud top and the moisture supply from below. The stratocumulus finally dissipates into thin and broken patches, penetrated from below by cumulus clouds.
To assess whether LES models are capable of faithfully capturing the dynamics of low clouds, several modeling intercomparison studies have been performed, some of which focused on stratocumulus (Moeng et al. 1996; Duynkerke et al. 1999, 2004; Stevens et al. 2005a; Ackerman et al. 2009), while other studies were dedicated to shallow cumulus (Siebesma et al. 2003; VanZanten et al. 2011) or cumulus-penetrating stratocumulus (Stevens et al. 2001). More recently, four Lagrangian stratocumulus-to-cumulus transition cases were proposed to evaluate how well models do in terms of the transition between the two regimes. This intercomparison study was performed in the framework of the Global Energy and Water Cycle Exchanges Project (GEWEX) Global Atmospheric System Studies (GASS) and the European Union Cloud Intercomparison, Process Study and Evaluation Project (EUCLIPSE). Three of the transition cases were based on the “composite” view of this transition build using state-of-the-art reanalysis and satellite data (Sandu et al. 2010), while a fourth one revisited the SCM intercomparison case based on the ASTEX campaign (Bretherton et al. 1999). While ASTEX offers the opportunity to evaluate models against in situ data, the set of composite transitions represents a more idealized framework for model evaluation, which offers the possibility of comparing the models for a variety of SCT cases, which differ, for example, in terms of amplitude or time scale of the transition.
This paper discusses the representation of the four Lagrangian SCT cases in six different LES models. The Lagrangian approach means that an air mass is followed as it is being advected by the mean wind from the subtropics toward the equator over an increasingly warmer SST. Superposed to this change in the surface forcing, the air mass is being heated by absorption of solar radiation during daytime. The paper is organized as follows. In section 2, the cases and the LES models are introduced. Section 3 discusses the LES results with an emphasis on the development of the two-layer structure of the boundary layer. This decoupled structure motivates us to analyze the thermodynamic budgets of the two layers separately. The contribution of various processes, such as entrainment, turbulent fluxes at the cloud base, and radiation to the stratocumulus cloud-layer evolution is presented in section 4. Section 5 analyzes the heat and moisture budgets of the subcloud layer and explains the time evolution of the surface fluxes of heat and moisture. Section 6 discusses and summarizes the main findings.
2. Setup of the experiments
In this intercomparison case, a so-called Lagrangian approach is applied, which means that an air mass is followed as it is being advected by the mean wind, allowing us to study the SCT in a single simulation (Schubert et al. 1979). The horizontal advection term in the conservation equations for heat and moisture may be assumed to be zero in the simulations as the air parcel is followed along its trajectory. This assumption is acceptable as long as the vertical wind shear is negligibly small.
a. Summary of the Lagrangian stratocumulus transition cases
Three composite cases representing SCTs of varying speed were built based on the observational study of Sandu et al. (2010). In that study, a large number of Lagrangian trajectories of air parcels in four subtropical oceans were computed using the wind fields provided by reanalysis of past observations, and the evolution of the cloud and of its environment along each of these individual trajectories was documented from satellite datasets and meteorological reanalysis [Moderate Resolution Imaging Spectroradiometer (MODIS) level-3 data for cloud properties and European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim; Simmons et al. 2007) for environmental properties]. This study suggested that averaged forcings can be considered as representative of individual trajectories and can therefore be used to initialize numerical simulations of the transition between the two cloud regimes. Building on these findings, a composite of the large-scale conditions encountered along the trajectories for the northeast Pacific (NEP) during June–August 2006 and 2007 were used to set up a case study of the SCT that will be referred to here as the reference case study and is further described in Sandu and Stevens (2011). Two variations of this reference case corresponding to a faster and to a slower transition, respectively, in cloud fraction were also derived for the intercomparison study [and are also described in Sandu and Stevens (2011)]. For that, the transitions analyzed for the NEP during June–August 2006 and 2007 were divided into three categories (fast, intermediate, and slow) on the basis of the mean cloud fraction over the first 48 h. The initial profiles and the large-scale conditions for each of the three cases represent the medians of the distributions of the various properties obtained for the respective subset of trajectories.
The setup of the fourth SCT case is described in detail by Van der Dussen et al. (2013). This case is based on observations collected during the first ASTEX Lagrangian experiment (Albrecht et al. 1995; Bretherton et al. 1995; De Roode and Duynkerke 1997) and large-scale forcing conditions as obtained from ERA-Interim. Since the setup of the composite cases is somewhat idealized, and because the ASTEX case particularly differs from the composite cases in terms of precipitation and its relatively cold and moist free troposphere, we think it is useful to discuss its results along with the results from the composite cases.
The initial vertical profiles of the liquid water potential temperature ql, total water specific humidity 
Initial vertical profiles of (a) the liquid water potential temperature 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Initial vertical profiles of (a) the liquid water potential temperature
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Initial vertical profiles of (a) the liquid water potential temperature
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Prescribed SST for the ASTEX, fast, reference, and slow cases. The line styles are according to the legend.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Prescribed SST for the ASTEX, fast, reference, and slow cases. The line styles are according to the legend.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Prescribed SST for the ASTEX, fast, reference, and slow cases. The line styles are according to the legend.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
For the ASTEX case, the large-scale divergence gradually decreases with time, and the observed weakening of the wind velocities is taken into account by a time-varying geostrophic forcing (Van der Dussen et al. 2013). For the composite cases, the large-scale divergence and the geostrophic forcing are constant in time, where the geostrophic winds are the same as the initial profiles of the horizontal wind velocity components shown in Fig. 1. Although the trajectories for the composite cases are simulated during the same period of time, they have slightly different lengths, as their horizontal wind speeds are not the same. The four Lagrangians also assume a constant surface pressure (Table 1). The ASTEX and the three composite cases last 40 h and 3 days, respectively, as these are the time scales during which the bulk of the transition in cloud cover takes place.
Details of the simulations. “Div” represents the large-scale divergence of the horizontal mean wind velocities, which is constant in time and constant up to a height of zDiv, except for the ASTEX case, in which the divergence varies with time.
b. Participating large-eddy simulation models and data output
Table 2 lists the models and their acronyms, along with contributors from each participating group, as well as the main references for the models. The vertical grid resolution in the lower 540 m is Δz = 15 m. To represent the sharp inversion layer capping the cloud layer, the vertical resolution is gradually refined only above this height; between 645 and 2400 m, Δz = 5 m. The horizontal domain size is 4.48 × 4.48 km2, and the number of grid points in the horizontal directions is 
Participating models and contributors.
For each case, six large-eddy simulations, each performed with a different code, are presented. Every code includes a detailed parameterization scheme for radiation and ice-free cloud microphysical processes, where the latter uses a fixed value for the cloud droplet concentration number Nd = 100 cm−3.
Because the lower-tropospheric stability, defined as the difference between the potential temperature at the 700-hPa pressure level and the ground surface (Klein and Hartmann 1993), is key for the evolution of the SCT, a realistic tendency of the free-tropospheric temperature is needed, in particular as the simulations were performed for a period of 2 or 3 days. Therefore, in contrast to many past studies, all models applied a full radiation code.
To compare the modeling results, time series of scalars and hourly mean vertical profiles according to the data protocol proposed by VanZanten et al. (2011) were provided by the modelers. Here it is important to note that liquid water 
3. Evolution of the mean state and turbulence structure
a. Time series
We start our analysis by inspection of the time evolution of the boundary layer, cloud amount, and the surface fluxes of sensible and latent heat (Fig. 3). The time variable in the figure is set such that, at the first occasion of local noon, t = 0. Nighttime periods (denoted by N1, N2, and N3 at the top of Fig. 3h) are indicated by the gray vertical bands in the plots according to the simulation periods summarized in Table 3. For each LES model, and for each daytime and nighttime period, we calculated time-mean results. To get an appreciation of the spread in the modeling results, Table 4 presents the overall LES means and standard deviations. Note that, because during the first 2 h of the simulations the turbulence has not fully developed yet, the results during this spinup period were not used.
Time series of the (a)–(d) lowest cumulus cloud-base height (lower solid lines without symbols) and the mean inversion height (upper solid lines with symbols); (e)–(h) the domain-averaged LWP; (i)–(l) the cloud cover; (m)–(p) the entrainment velocity 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time series of the (a)–(d) lowest cumulus cloud-base height (lower solid lines without symbols) and the mean inversion height (upper solid lines with symbols); (e)–(h) the domain-averaged LWP; (i)–(l) the cloud cover; (m)–(p) the entrainment velocity
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time series of the (a)–(d) lowest cumulus cloud-base height (lower solid lines without symbols) and the mean inversion height (upper solid lines with symbols); (e)–(h) the domain-averaged LWP; (i)–(l) the cloud cover; (m)–(p) the entrainment velocity
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Summary of periods of daytime (D1, D2, and D3) and nighttime (N1, N2, and N3), and the corresponding start and end times in hours from the start of the simulations.
Mean values and their standard deviations during the daytime and nighttime periods according to Table 3. Standard deviation is rounded to one significant digit. However, for compact notation, we express, for example, 
In brief, the results show that for all cases the cloud-topped boundary layer is gradually deepening with time, while the cumulus cloud-base height reaches an approximate steady state. The effect of the diurnal variation of the solar radiation is clearly found from the time series of the LWP. Because of the absorption of solar radiation in the cloud layer, the LWP has reduced values during daytime. The cloud layer breaks up during the second daytime period (D2) for the fast case, although it tends to recover to a closed cloud deck during the second night (N2), except for MOLEM. The slow case appears to maintain an almost closed cloud deck during the entire simulation period. For all SCTs, the entrainment velocity is much larger during nighttime than during the day. Finally, for the composite cases, the surface evaporation gradually increases, whereas the sensible heat flux remains rather small.
A closer inspection reveals that during local noon the growth of the inversion height becomes very small for the composite cases, which is because of a reduced cloud-top entrainment rate, whereas the subsidence keeps pushing down the boundary layer top (Figs. 3a–d). The variation in the boundary layer depth as represented by the standard deviation 
By contrast, the intermodel spread in the cloud liquid water path (LWP) is relatively large, particularly during the night (Figs. 3e–h), similar to what was found in the stratocumulus model intercomparison study by Stevens et al. (2005a). The LESs agree fairly well in terms of the representation of the diurnal variation of the LWP, although the amplitude is larger in the MPI/UCLA, DHARMA, and EULAG models. The latter model explains a significant part of 
MOLEM and EULAG have a consistently different longwave radiative forcing for the three composite cases, as compared to the other LES models, for which results are very similar. For example, during the first night of the composite cases, the longwave radiative flux divergence in the cloud layer is about 5 W m−2 smaller in MOLEM and about 10 W m−2 larger in EULAG. The effect of the differences in the longwave radiative cooling on the cloud-layer evolution is discussed in detail in section 5. Figures 3i–l show the time evolution of the cloud cover. Only in the EULAG model is a solid cloud maintained for all SCTs, which possibly results from the imposed stronger cloud longwave radiative cooling. In the other models, the stratocumulus starts to break up some hours after sunrise because of the absorption of solar radiation in the cloud layer (Nicholls 1984). Most of the time, the stratocumulus is able to recover to a closed-cell cloud deck after sunset. The difference between the three composite cases becomes clear as the cloud cover tends to reduce more rapidly for the fast case compared to the reference or slow cases, which is in a rough agreement with estimations of cloud cover from MODIS. However, the intermodel differences in the daytime cloud cover are rather large. For example, for the fast and reference cases, the standard deviation of the cloud cover has maximum values during the third daytime period (D3).
The absorption of the solar radiation leads to the warming and the thinning of the cloud layer. The absorption of solar radiation in the cloud layer counteracts the longwave radiative cooling at the cloud top. The stabilization of the cloud layer during daytime tends to weaken the buoyancy production of turbulence, which in turn causes a reduction in the entrainment velocity. If we compare the entrainment velocity for the four cases, we find smaller values for a stronger thermal stratification as measured by the inversion jump values of 
The LES models give SHF values that are less than 10 W m−2 (Figs. 3q–t). The LHF tends to increase with time (Figs. 3u–x), except for the ASTEX case, for which a flattening of the temporal SST increase and a weakening geostrophic forcing yields lower wind velocities and consequently lower LHF values. The composite cases exhibit a gradual increasing trend in the LHF, with an imposed diurnal cycle in which the flux increases faster during the night than during the day. The standard deviation of the LHF is within 10 W m−2.
Although the bulk features of the time variation of the cloud structure and the differences between the four cases are consistently represented, the variation in the cloud cover and the LWP leads to a rather large value for the standard deviation of the net shortwave radiation at the surface, with a maximum value of 80 W m−2 during the third daytime period (D3) for the fast case. During the entire simulation period, the standard deviation of the net longwave radiation at the surface is within 10 W m−2.
b. Boundary layer decoupling
Hourly mean vertical profiles of 
Vertical profiles of (a) the liquid water potential temperature 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Vertical profiles of (a) the liquid water potential temperature
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Vertical profiles of (a) the liquid water potential temperature
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The different evolutions in 
Evolution of (a) the (liquid water) potential temperature and (b) the (total) specific humidity just above the inversion 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Evolution of (a) the (liquid water) potential temperature and (b) the (total) specific humidity just above the inversion
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Evolution of (a) the (liquid water) potential temperature and (b) the (total) specific humidity just above the inversion
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Figure 6 compares the decoupling parameters 
The decoupling parameters (a) 
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The decoupling parameters (a)
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The decoupling parameters (a)
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Large values of the decoupling parameters indicate that the cloud layer is relatively warm and dry with respect to the subcloud layer. Because a high temperature or a low total water amount in the cloud tends to reduce the cloud liquid water content, we will now take a closer look at the time evolution of the decoupling parameters. In particular, we will inspect the results for the slow case, which shows a rather large scatter in the nighttime LWP values among the six LESs. The gradual deepening of the boundary layer is reflected in the gradual increase of 
The time evolution of the decoupling parameters of (a)–(d) 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The time evolution of the decoupling parameters of (a)–(d)
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The time evolution of the decoupling parameters of (a)–(d)
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
c. Turbulence
The 
Instantaneous fields in the vertical plane for (a) the total water specific humidity, (b) the liquid water potential temperature, and (c) the vertical velocity as obtained 36 h from local noon from the DALES ASTEX run. The thick solid black lines indicate the contours of the cloud edges. See main text for an explanation of the areas that are indicated by the encircled numbers.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Instantaneous fields in the vertical plane for (a) the total water specific humidity, (b) the liquid water potential temperature, and (c) the vertical velocity as obtained 36 h from local noon from the DALES ASTEX run. The thick solid black lines indicate the contours of the cloud edges. See main text for an explanation of the areas that are indicated by the encircled numbers.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Instantaneous fields in the vertical plane for (a) the total water specific humidity, (b) the liquid water potential temperature, and (c) the vertical velocity as obtained 36 h from local noon from the DALES ASTEX run. The thick solid black lines indicate the contours of the cloud edges. See main text for an explanation of the areas that are indicated by the encircled numbers.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The findings presented so far suggest that the intermodel spread in the LWP during nighttime can be linked to the various strengths of the decoupling between the cloud and the subcloud layer. Stevens et al. (2005b) reported similar findings for the DYCOMS II nighttime stratocumulus LES intercomparison case. They found a strong link between the buoyancy flux profile, the vertical velocity variance, and the degree of decoupling. It is therefore instructive to repeat their analysis by inspecting the turbulence profiles for the SCTs. Figure 9 shows hourly mean vertical profiles of the vertical velocity variance 
Hourly mean turbulence statistics for the slow case at four selected times. The profiles at 12 and 36 h from local noon are at midnight, and 24 and 48 h represent conditions during local noon. (a)–(d) The vertical velocity variance, (e)–(h) the virtual potential temperature flux, (i)–(l) the total water specific humidity flux, and (m)–(p) the turbulent kinetic energy. The line colors and symbols are shown in the legend in (d). The thin black vertical line in the plots showing the virtual potential temperature flux indicates a zero value for easy reference.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Hourly mean turbulence statistics for the slow case at four selected times. The profiles at 12 and 36 h from local noon are at midnight, and 24 and 48 h represent conditions during local noon. (a)–(d) The vertical velocity variance, (e)–(h) the virtual potential temperature flux, (i)–(l) the total water specific humidity flux, and (m)–(p) the turbulent kinetic energy. The line colors and symbols are shown in the legend in (d). The thin black vertical line in the plots showing the virtual potential temperature flux indicates a zero value for easy reference.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Hourly mean turbulence statistics for the slow case at four selected times. The profiles at 12 and 36 h from local noon are at midnight, and 24 and 48 h represent conditions during local noon. (a)–(d) The vertical velocity variance, (e)–(h) the virtual potential temperature flux, (i)–(l) the total water specific humidity flux, and (m)–(p) the turbulent kinetic energy. The line colors and symbols are shown in the legend in (d). The thin black vertical line in the plots showing the virtual potential temperature flux indicates a zero value for easy reference.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The surface buoyancy fluxes are positive. Toward the top of the subcloud layer, the buoyancy flux decreases and can even become negative, indicating that, on average, rising plumes are negatively buoyant. If the plumes become saturated with water vapor, the latent heat release due to condensation enables them to rise farther as positively buoyant clouds. The negative buoyancy fluxes just above the top of the cloud layer are due to entrainment of warm free-tropospheric air. Longwave radiative cooling in the cloud-top regions leads to buoyancy production, and, as the cooled cloud parcels become heavier than the surrounding air, they start sinking, leading to a positive buoyancy flux.
The imposed solar radiative heating of the cloud layer during daytime has a distinct effect on the turbulence structure of the boundary layer. In particular, the signature of a decoupled boundary layer structure is clearly visible from the double-peak structure in 
The time evolution of the flux ratios for the (a)–(d) buoyancy 
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The time evolution of the flux ratios for the (a)–(d) buoyancy
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The time evolution of the flux ratios for the (a)–(d) buoyancy
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Mean values of the flux ratios 
Likewise, the flux ratio 
4. Stratocumulus LWP budget
Time evolution of the inversion jumps of (a)–(d) 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time evolution of the inversion jumps of (a)–(d)
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time evolution of the inversion jumps of (a)–(d)
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Time evolution of the dominant terms in the LWP budget, with the variables displayed on the vertical axes denoting LWP tendencies (g m−2 h−1) due to (a)–(d) longwave radiative cooling 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time evolution of the dominant terms in the LWP budget, with the variables displayed on the vertical axes denoting LWP tendencies (g m−2 h−1) due to (a)–(d) longwave radiative cooling
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Time evolution of the dominant terms in the LWP budget, with the variables displayed on the vertical axes denoting LWP tendencies (g m−2 h−1) due to (a)–(d) longwave radiative cooling
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The entrainment drying and warming effects are represented by 
The budget analysis indicates that the imbalance of a couple of rather large contributions to the LWP tendency determines the actual LWP tendency. It also clarifies the role of entrainment. The fast case has the smallest inversion jumps of 
Van der Dussen et al. (2013) showed from additional sensitivity experiments for the ASTEX case that the difference in the LWP is mainly attributable to differences in the precipitation rate. They also found that stronger precipitating stratocumulus had less entrainment of warm and dry inversion air at its top. During daytime, model differences in LWP are also diminished by solar radiative heating of the cloud layer. This mechanism is particularly clear during the third daytime period (D3) of the fast and reference case simulations by EULAG. The LWP in this model is much higher than in the others (Figs. 3f,g), which causes a much stronger cloud-thinning tendency because of the absorption of solar radiation (Figs. 12f,g).
5. Subcloud-layer heat and moisture budgets
The behavior of the surface SHF and LHF during the transitions is very different in the sense that the SHF becomes approximately constant at about 10 W m−2, whereas the LHF tends to increase with time during the Lagrangian advection of the cloudy air mass (Figs. 3q–x). A classical framework to explain the time evolution of surface fluxes is the mixed-layer model (MLM), which assumes a vertically well-mixed boundary layer (Lilly 1968; Schubert et al. 1979; Nicholls 1984). The values of the decoupling parameters 
a. Evolution of the subcloud-layer height
b. Analysis of the results
Summary of the boundary conditions used for the subcloud mixed-layer model, its time scales, and the definitions of the constants 
Table 8 presents the time scales for the SCT cases, based on the average subcloud-layer values from all the LES models. The tendency of the SST was obtained from a linear regression. For all SCT cases, the mean value of 
Average values as obtained during the entire run and from all the LES models, except for the surface fluxes, which represent the analytical results at the end of the simulations.
The latent heat flux as a function of time and for different values of 
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The latent heat flux as a function of time and for different values of
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The latent heat flux as a function of time and for different values of
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
In summary, the MLM analysis of the subcloud-layer evolution during its Lagrangian advection well explains the LES results. For a decoupled boundary layer with a constant subcloud-layer height and a fixed value for 
6. Conclusions
Four Lagrangian stratocumulus-to-shallow-cumulus transition experiments were performed with six different LES models. The cases differ predominantly in terms of the amplitude and time scale of the transition. The LES models agree remarkably well in the representation of the evolution of the mean states. For all cases, the structure of the boundary layer transforms from a vertically well-mixed layer to one in which the subcloud and cloud layers appear as two separated mixed layers, with the stratocumulus layer being warmer and drier relative to the subcloud layer, a situation that is referred to as decoupling (Nicholls 1984; Bretherton and Wyant 1997). The difference in the thermodynamic state of the subcloud and cloud layers increases for deeper boundary layers, which is found to be in a qualitative agreement with aircraft observations analyzed by Wood and Bretherton (2004). The general good agreement between the models in the representation of the boundary layer evolution can be partly explained by drizzle and solar heating of the cloud layer. Thicker cloud layers, such as those found for the ASTEX case, will produce more precipitation and will absorb more solar radiation during daytime, and vice versa. In this way, both processes act to diminish intermodel differences in the LWP. For the composite cases, the earliest timing of the breakup of the stratocumulus layer is found for the fast case, which is predominantly because of a slightly stronger entrainment warming and drying as compared to the reference and slow cases.
Superposed to this picture where the boundary layer is deepening due to increasing SSTs, there is a diurnal cycle associated with the absorption of solar radiation within the cloud layer. The models agree well in terms of LWP during the day, but less so in terms of LWP at night. The opposite is true for the cloud cover, which varies considerably among the LES models during daytime. The EULAG model tends to maintain a closed cloud deck that can be attributed to its radiation scheme, which gives a somewhat stronger longwave radiative cooling in the cloud layer. SHF is small and on the order of 10 W m−2, whereas the LHF tends to increase with time for all cases.
The time evolution of the surface heat fluxes can be well explained by means of a simple mixed-layer model that is applied to the subcloud layer and that uses generic bulk features found from the LES results as boundary conditions. Specifically, the model makes use of the facts that the subcloud-layer depth becomes almost constant in time and that the buoyancy flux at the top of the subcloud layer tends to approach a fixed negative fraction of the surface value, similar to what is found for the dry convective boundary layer and cumulus-topped boundary layers. The critical quantity that controls the magnitude of the change in the surface evaporation is the moisture flux at the top of the subcloud layer. The fact that the specific humidity in the subcloud layer increases with time indicates that, on average, the surface moisture flux is larger than the value at the top of the subcloud layer. The LWP budget analysis shows that, during periods with stronger turbulence (i.e., during nighttime), a stronger injection of subcloud-layer moisture into the stratocumulus cloud base is accompanied by a stronger entrainment drying.
Figure 14 presents a schematic of the main findings of the Lagrangian SCTs. The SHF remains rather small during the equatorward advection of the air mass, while the LHF gradually increases. During nighttime, the longwave radiative cooling acts to destabilize the cloud layer, which tends to generate more turbulence and a higher entrainment rate at the cloud top. Because of stronger turbulence in the cloud layer during the night, subcloud-layer moisture is transported toward the stratocumulus at a rate that exceeds the surface evaporation during the first night of the three composite cases and also during the second night of the slow case. This enhanced moisture flux feeds the stratocumulus with liquid water, thereby competing against the cloud-thinning tendency by increased entrainment of warm and dry air from just above the inversion. Overall, we find that the nocturnal stratocumulus cloud deck is able to recover from a broken structure to a closed structure. During daytime, the cloud layer is heated by absorption of solar radiation. This stabilizes the cloud layer with respect to the subcloud layer, which hinders the vertical turbulent transport of layer moisture to the cloud layer. The warming by the sun and the reduced moisture input at the base of the stratocumulus causes it to thin and to break up.
Schematic showing the gradual breakup of a stratocumulus cloud layer during its Lagrangian advection over an increasing SST. The vertical arrows represent the sensible and latent heat fluxes. During the night, turbulence in the cloud layer intensifies, causing larger humidity fluxes at cloud base and cloud top.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Schematic showing the gradual breakup of a stratocumulus cloud layer during its Lagrangian advection over an increasing SST. The vertical arrows represent the sensible and latent heat fluxes. During the night, turbulence in the cloud layer intensifies, causing larger humidity fluxes at cloud base and cloud top.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
Schematic showing the gradual breakup of a stratocumulus cloud layer during its Lagrangian advection over an increasing SST. The vertical arrows represent the sensible and latent heat fluxes. During the night, turbulence in the cloud layer intensifies, causing larger humidity fluxes at cloud base and cloud top.
Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0215.1
The representation of the moisture transport from the top of the subcloud mixed layer to the stratocumulus layer, and the entrainment of free-tropospheric dry air at the top of the stratocumulus, are essential ingredients to capture the SCT. In fact, in a study on the representation of the SCT in large-scale models by Neggers (2015), it is found that SCMs favor a breakup of stratocumulus for inversion conditions that are unique to each individual model. The presence of such modes may be indicative of a local hydrological cycle that is distinctively different among the models. The finding that the degree of decoupling has an important consequence for the LWP suggests that the decoupling parameters can be a helpful quantity in evaluating parameterization schemes for cloud-topped boundary layers (Dal Gesso et al. 2014). The 3D instantaneous LES (thermo-) dynamic fields may be further used to evaluate parameterizations used in global models.
SCT cases such as those discussed here have been simulated to study the effect of changes in the large-scale forcing conditions in the Hadley cell under climate change conditions to assess its possible impact on the pace of the transition. For example, Bretherton and Blossey (2014) investigated and explained the effect of a perturbed radiative forcing, the overall tropical warming, and changes in the inversion stability on the SCT. Likewise, Van der Dussen et al. (2016) used the LWP budget equation to investigate why a decrease in the large-scale subsidence extends the lifetime of stratocumulus despite an increase in the entrainment rate. In addition, both studies investigated the effect of applying a uniform insolation (constant in time) on the SCTs, which showed that the bulk evolution of the SCT in terms of boundary layer deepening is rather similar. Kazil et al. (2015) investigated the effect of the wind speed on the SCT. They found that a higher wind speed leads to a larger entrainment rate and a faster growth of the boundary layer, caused by an enhanced buoyant production of turbulence kinetic energy (TKE) from latent heat release in cloud updrafts.
Acknowledgments
These investigations were done as part of the European Union Cloud Intercomparison, Process Study and Evaluation Project (EUCLIPSE), funded under Framework Program 7 of the European Union. The setup of the composite transition cases and the LES simulations with MPI/UCLA were supported by the Alexander von Humboldt Foundation and the Max Planck Institute for Meteorology. The simulations with the Dutch LES model were sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities. DHARMA simulations used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-05CH11231, and the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The simulations with SAM were supported by NOAA MAPP Grant NA13OAR4310104. EULAG simulations used resources of the National Center for Atmospheric Research, which is sponsored by the National Science Foundation. We thank Erwin de Beus for his kind technical assistance and Chris Bretherton and three reviewers for their suggestions, which helped to improve the manuscript.
APPENDIX
A Mixed-Layer Model for the Subcloud Layer
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