An LES model study of the influence of the free tropospheric thermodynamic conditions on the stratocumulus response to a climate perturbation

Abstract

Twenty-five large-eddy simulations are performed to study how free tropospheric thermodynamic conditions control equilibrium state solutions of stratocumulus-topped marine boundary layers. In particular, we systematically vary the lower tropospheric stability (LTS) and a similar measure for the bulk humidity difference between the 700 hPa level and the surface, . For all simulations, a completely overcast boundary layer is obtained in which the turbulence is mainly driven by cloud top radiative cooling. The steady state liquid water path (LWP) is rather insensitive to the LTS, but increases significantly and almost linearly with the free tropospheric humidity. In a second suite of runs, the response of the stratocumulus layer to an idealized global warming scenario is assessed by applying a uniform warming of 2 K to the initial temperature profile including the sea surface while the initial relative humidity profile is kept identical to the control case. The warming of the sea surface acts to increase the latent heat flux, which invigorates turbulence in the boundary layer. The steady state inversion height therefore increases, despite the competing effect of a more humid free troposphere that increases the downwelling radiative flux and hence tends to decrease the entrainment rate. The stratocumulus layer nevertheless thins for all free tropospheric conditions as cloud base rises more than cloud top. This implies a positive stratocumulus cloud-climate feedback for this scenario as thinner clouds reflect less shortwave radiation back to space. The cloud thinning response to the climate perturbation is found to be mostly controlled by the change of .

Key Points:

• Positive stratocumulus cloud-climate feedback found from steady state LESs
• The LWP of top-driven stratocumulus depends mainly on free tropospheric humidity
• Climate response attributable to change of bulk lower tropospheric humidity jump

1 Introduction

Marine low clouds have a net cooling effect on the planet as they reflect more of the incoming solar radiation than the underlying sea surface, while their warming effect due to thermal radiation is small [Randall et al., 1984]. This net cooling effect is strongest for stratocumulus clouds, because of their large cloud cover. Changes in the reflectivity and occurrence of stratocumulus as a result of climate change can therefore amplify or weaken global warming, which makes their accurate representation in climate models essential. However, the presence of stratocumulus clouds depends crucially on turbulence processes. Climate models use coarse resolutions and rely heavily on parameterization schemes to represent such turbulence processes. Therefore, they often fail to properly represent stratocumulus clouds [Williams and Webb, 2009; Nam et al., 2012; Teixeira et al., 2011]. As a consequence, changes of their reflectivity and occurrence remain a major source of uncertainty in future climate projections [Bony and Dufresne, 2005; Soden and Held, 2006; Vial et al., 2013].

The response of low clouds to idealized climate perturbations has recently been investigated using large-eddy simulation (LES) models [e.g., Xu et al., 2010; Rieck et al., 2012; Bretherton and Blossey, 2014]. Most of the turbulent transport in the atmospheric boundary layer is explicitly resolved in such models, making them suitable for the accurate representation of low clouds. A particularly relevant LES study was performed as part of the CGILS (Cloud Feedback Model Intercomparison Project/Global Atmospheric System Study Intercomparison of Large-Eddy and Single-Column Models) project. Idealized cases were designed for the simulation of three low cloud types that are climatologically prevailing over subtropical oceans, namely stratocumulus, cumulus-under-stratocumulus, and trade cumulus [Blossey et al., 2013]. To determine the cloud response to an idealized climate perturbation, the initial temperature was uniformly increased by 2 K. The total specific humidity was furthermore increased, assuming that large-scale processes act to keep the initial relative humidity profile the same as for the control case. This perturbation caused a decrease of the domain-averaged LWP for the stratocumulus case. In a second experiment, the large-scale subsidence velocity was decreased in addition to the warming perturbation to mimic a weakening of the Hadley circulation. The cloud thickening effect due to the decreased subsidence velocity overcompensated the decrease of the LWP in most LES models, resulting in a small net increase of the LWP [Blossey et al., 2013; Bretherton et al., 2013]. The simulations were also run with several single-column model (SCM) versions of climate models. The cloud responses found from the SCMs strongly varied and both positive and negative cloud feedbacks were found [Zhang et al., 2013].

Dal Gesso et al. [2014a, hereinafter DG14a] extended the CGILS cumulus-under-stratocumulus case by considering a phase space spanned by a range of lower tropospheric stabilities (LTSs) and free tropospheric humidities. This setup was motivated by, among others, Klein and Hartmann [1993], who showed that there is a strong correlation between the observed seasonally averaged LTS and low cloud amount, as well as by the strong dependence of the entrainment rate and the LWP of stratocumulus clouds on the free tropospheric humidity [e.g., Chlond and Wolkau, 2000; Ackerman et al., 2004; Lock, 2009]. Using a mixed-layer model (MLM), DG14a showed that the LWP in a steady state increases as the humidity in the free troposphere decreases or as the LTS decreases. Analogous to the CGILS experiment, a second set of simulations was performed for which the temperature was increased while the initial relative humidity profile was kept constant. The large-scale subsidence velocity was not perturbed to allow for a more straightforward interpretation of the results. The cloud layer thinned in response to this perturbation for virtually all cases and the thinning was found to be strongest for dry and relatively warm free tropospheric conditions. The thinning response was caused by the combination of, in the first place, an increased cloud base height caused by the warming of the boundary layer and in the second place, a decreased cloud top height, which could be explained by a smaller entrainment rate as a result of a reduced forcing in terms of the longwave radiative cooling near the cloud top. The latter is due to a larger free tropospheric specific humidity which will tend to increase the atmospheric emissivity, thereby diminishing the longwave radiative jump across the cloud top.

The cases described by DG14a were run with a SCM version of EC-Earth [Hazeleger et al., 2010] in a follow-up study by Dal Gesso et al. [2014b, hereinafter DG14b] to evaluate how accurately the stratocumulus steady states are represented by the parameterization schemes used in this model. Besides stratocumulus, shallow cumulus solutions were found for low LTS and for a relatively dry free troposphere. When averaged over all cases, a positive cloud-climate feedback was found from the SCM results, which is in accord with the MLM study by DG14a. However, the magnitude and even the sign of the cloud feedback changed irregularly from case to case, thereby emphasizing the importance of the phase space setup. This lack of coherence among the SCM simulations prohibited a careful analysis of the mechanisms responsible for the positive cloud response.

In MLMs and SCMs, turbulent transport is calculated from parameterization schemes. By contrast, LES models do not suffer from this limitation as they explicitly resolve the eddies that perform the bulk of the turbulent transport. Furthermore, the effects of processes like latent heat release and longwave radiative cooling on the entrainment rate are typically much better represented. Since the entrainment rate is key to the equilibrium state solutions of the stratocumulus-topped boundary layer [De Roode et al., 2014], this has motivated us to repeat the experiments performed by DG14a and DG14b with an LES model. The present study discusses how equilibrium states of the stratocumulus-topped boundary layer are affected by the free tropospheric conditions, and how the stratocumulus cloud amount changes under an idealized global warming scenario. In a companion paper by S. Dal Gesso et al. (A single-column model intercomparison on the stratocumulus representation in present-day and future climate, submitted to Journal of Advances in Modeling Earth Systems, 2014), the results of an SCM intercomparison are discussed and compared to the LES model results in detail. Although the use of LES constitutes an important step forward from MLMs and SCMs, it should be noted that entrainment is governed by mixing processes at scales much smaller than the LES grid spacing [e.g., Mellado et al., 2013]. Hence, higher-resolution LES or even direct numerical simulation is required to explicitly resolve this mixing, which is currently beyond the reach of the available computational resources.

The following section describes the most relevant aspects of the case setup and the climate perturbation that was applied. The results of the control climate simulations are described in section 3, while the response to the idealized climate perturbation is discussed in section 4. Section 5 contains some discussion and a summary of the main conclusions is given in section 6.

2 Setup

2.1 Case Specifications

DG14a developed a framework based on stratocumulus conditions in the North-East Pacific, within which the liquid water potential temperature and the total water specific humidity in the free troposphere were varied to investigate their effect on the boundary layer structure in a steady state. Each case is identified by the LTS in combination with a similar variable, , that measures the difference between the free tropospheric specific humidity and the saturation specific humidity at the surface:

(1)

(2)

Here the subscripted “ft” denotes the value of a variable at the 700 hPa level, which corresponds to a height of approximately 3 km. The subscripted “0” indicates the value at the surface. The sea surface temperature T₀ and pressure p₀ are constant, while and are nudged toward their initial values. The LTS and are therefore constant in time for every simulation.

In the current work, a set of 25 LESs is performed that includes all combinations of

All simulations are run for 10 days to an approximate steady state, which is only achieved when for a conserved variable the following budget equation is satisfied:

(3)

Here denotes the large-scale horizontal wind components, z is the height, is the vertical turbulent flux of , and accounts for the diabatic sources and sinks due to precipitation and radiation. Furthermore, denotes the large-scale horizontal gradients of , which are assumed to be zero in this study. The large-scale horizontal advection term therefore does not contribute to the heat and moisture budgets, despite the nonzero mean horizontal wind velocity. More specifically, the y coordinate of the domain is aligned with the mean wind, which is constant with height at an initial velocity of −6.74 ms⁻¹ and is equal to the geostrophic wind. For notational convenience the overbars indicating horizontal averaging are omitted for all variables except for turbulent fluxes and the subsidence velocity in the remainder of the article.

In the quiescent free troposphere, turbulent fluxes are negligibly small hence equation 3 can be simplified as:

(4)

DG14b assumed the temperature lapse rate in the free troposphere to follow the moist adiabat, which determines the vertical gradient of . The subsidence profile

(5)

is chosen such that the diabatic cooling due to radiation approximately balances subsidence warming [Bellon and Stevens, 2012]. Here −3.5 mm s⁻¹ is the value to which saturates at heights that are large with respect to the scale height m. For on the other hand, in the absence of horizontal advection, there are no diabatic terms in the free troposphere. Equation 4 can therefore only be satisfied if the subsidence term is zero, which is achieved by setting constant with height up to 3 km. For CGILS, decreased with height above 2 km. The values of the CGILS cases are therefore quite large as compared to the range considered here, at ( ) = (22.4 K, −11 g kg⁻¹) and (25.4 K, −9.5 g kg⁻¹) for the cumulus-under-stratocumulus and the stratocumulus case, respectively [Blossey et al., 2013].

All other boundary conditions and forcings such as the surface temperature and pressure of the control simulations as well as the diurnally averaged solar zenith angle (52°) and the downwelling solar radiative flux at the top of the atmosphere (765.84 W m⁻²) are equal to those prescribed for the S11 case [Blossey et al., 2013]. The information and initial profiles required to perform the simulations are available online (http://www.euclipse.nl/wp3/SteadyStates/main.shtml).

2.2 Idealized Climate Perturbation

In addition to the control climate simulations, a second set of simulations is performed in which the temperature of the atmospheric column and the sea surface temperature are uniformly increased by 2 K, thereby keeping the LTS the same as in the control climate.

The total specific humidity is furthermore increased to keep the initial relative humidity profile identical to that of the control case. As the saturation specific humidity is a convex function of the temperature, the increase of at the surface is typically larger than in the cooler free troposphere. The assumption of a constant relative humidity in a perturbed climate therefore imposes a change in the bulk jump of the total specific humidity , whose magnitude can be derived by first writing the increase of with temperature at any height as follows:

(6)

Here is the specific gas constant for water vapor and is the latent heat of vaporization. Using this equation with equation 2 gives

(7)

from which it can be shown that changes throughout the phase space by between −0.4 g kg⁻¹ K⁻¹ for humid and −0.7 g kg⁻¹ K⁻¹ for relatively dry free tropospheric conditions.

2.3 Model Configuration

The Dutch Atmospheric Large-Eddy Simulation (DALES 4.0) [Heus et al., 2010; Böing et al., 2012] model is used in a Boussinesq mode by specifying a base state density that is constant with height in the momentum equations. A hybrid of a fifth-order upwind scheme [e.g., Wicker and Skamarock, 2002] and a fifth-order weighted essentially nonoscillatory advection scheme [Jiang and Shu, 1996; Blossey and Durran, 2008] is used for the advection of scalars in order to avoid spurious overshoots at the inversion. These overshoots were hypothesized to influence the magnitude of the response of the LWP to the warming perturbation in the LES model intercomparison of the CGILS S12 stratocumulus case [Blossey et al., 2013].

The warm-rain microphysics model of Kogan [2013] is used for the parameterization of autoconversion, accretion, self-collection, and evaporation processes as well as to determine the sedimentation velocities of rainwater specific humidity and rain droplet number concentration . A piecewise-linear semi-Lagrangian advection scheme is used for the sedimentation of and [Juang and Hong, 2009]. The cloud droplet number concentration is set to a constant value of 100 cm⁻³ wherever the liquid water specific humidity and the effect of cloud droplet sedimentation is accounted for using the parameterization of Ackerman et al. [2009].

Longwave and shortwave radiative fluxes are parameterized using the Rapid Radiative Transfer Model for General Circulation Models (RRTMG) [Iacono et al., 2008], for which a convenient interface was provided by Peter Blossey [Blossey et al., 2013]. The radiation calculations are performed once every 120 s. Following Ackerman et al. [2009], the effects of having a nonmonodisperse cloud droplet size distribution are accounted for in the calculation of the effective radius, which is the ratio of the third to the second moment of the droplet size distribution.

Surface fluxes of and are calculated interactively, using a constant surface roughness length mm. With the exception of the surface scheme, the model configuration is identical to that used for the second phase of the CGILS experiment. Preliminary inspection of the results shows that all participating models respond similarly to the climate perturbations (P. Blossey, personal communication, 2014). It is therefore likely that the results from the simulations presented below are representative for the general behavior of LES models, although the quantitative results may differ.

2.4 Domain Specifications

The vertical spacing of the numerical grid is 10 m up to a height of 1.8 km, above which it is increased by 5% per level. The 3 km high domain is therefore made up of a total of 219 levels. At the top of the domain, and are relaxed toward their initial values to mimic the nudging toward the initial conditions that was used above 3 km height for the SCM simulations performed by DG14.

Note that the vertical resolution is coarser than the 5 m resolutions that are recommended to properly resolve the small-scale mixing in the inversion layer of stratocumulus-topped boundary layers [e.g., Bretherton et al., 1999; Stevens et al., 1999; Yamaguchi and Randall, 2012]. However, the long integration time of 10 days and the large number of simulations that are performed make the study presented here computationally demanding. Using a coarser vertical resolution of 10 m decreases the computational cost by approximately a factor of 4 as compared to a 5 m resolution. The sensitivity tests described in Appendix Appendix A indicate that the use of finer resolutions will yield very similar results in terms of, for instance, the inversion height. The LWP, on the other hand, can be expected to increase by about 25% when the resolution is increased from 10 to 5 m.

In both horizontal directions, the domain consists of 120 grid points that are spaced 50 m apart. This results in a horizontal domain size of 6 × 6 km², which is comparable to or somewhat larger than the domains used in other recent stratocumulus studies [Ackerman et al., 2009; Chung et al., 2012; van der Dussen et al., 2013; Blossey et al., 2013]. Sandu and Stevens [2011] performed several simulations of stratocumulus transitions on a domain of approximately 9 × 9 km² and found that among others cloud cover and albedo differed by less than 5% from smaller 4.5 × 4.5 km² simulations for weakly precipitating cases. Based on these considerations, the domain used is assumed to be sufficiently large for the purposes of the present study.

3 Control Climate

3.1 Inversion Height

Figure 1a shows the inversion height as a function of and LTS. The results in this plot and in the remainder of this study are averages over the tenth day of the simulations, unless stated otherwise. Note that the data presented in this and the following section is included in NetCDF format as supporting information S1 and S2 for the control and the perturbed climate simulations, respectively. For all free tropospheric conditions, a cloud cover of unity is maintained for the entire duration of the simulations. Figure 1b shows time series of for the three simulations indicated by the colored circles in Figure 1a, from which it is clear that the boundary layer height is close to a steady state at the end of the 10 day integration time. Note that is a proxy for the entrainment rate , as in a steady state

(8)

Figure 1

(a) The inversion height in km averaged over the tenth day for 25 simulations in a phase space spanned by the and by . (b) Time series of the inversion height for the three selected simulations that are indicated by the colored circles in Figure 1a.

The entrainment rate can be expected to increase as the stability of the inversion as measured by , the jump of the virtual potential temperature over the inversion, decreases. The virtual potential temperature can be written as

(9)

in which is the water vapor specific humidity and θ is the potential temperature. Furthermore, the constant , where is the specific gas constant for dry air. In the free troposphere, the air is subsaturated, such that

(10)

According to this equation, a decrease of (or similarly of the LTS) of 1 K results in an decrease of of approximately 1 K. Such a decrease weakens the inversion stability and hence causes an increase of the inversion height, which is also obvious in Figure 1a.

From equation 10 it can furthermore be seen that the presence of humidity in the free troposphere decreases and hence weakens the inversion. A drying of the free troposphere by 1 g kg⁻¹ causes a decrease of of approximately 0.2 K. Figure 1a indeed shows that increases for drier free tropospheric conditions as measured by larger values of the bulk humidity jump factor , whose variations are solely due to the variation of as the sea surface temperature is identical for all simulations. However, the increase due to a change of 1 g kg⁻¹ is approximately as large as that due to a LTS decrease of 2 K, which is much larger than is expected on the virtual effect of water vapor alone. The sensitivity to found from the simulations is therefore about 10 times stronger than expected on the basis of the virtual effect of water vapor alone.

This strong dependency of on the free tropospheric humidity can be explained as follows. In the first place, for drier free tropospheric conditions the effect of cloud droplet evaporative cooling due to entrainment and the subsequent mixing of free tropospheric air into the cloud layer will be stronger. A drier free troposphere therefore effectively weakens the inversion stability, which supports larger entrainment rates for larger values of [e.g., Nicholls and Turton, 1986; Chlond and Wolkau, 2000; Yamaguchi and Randall, 2008].

Second, a moister free troposphere generally emits more longwave radiation. More specifically, it is found that the downwelling longwave radiation increases logarithmically with the water vapor path W [e.g., Zhang et al., 2001]:

(11)

where ρ is the density of air. The stratocumulus cloud absorbs this downwelling radiation, which partly offsets the cooling tendency that is due to the longwave radiation it emits.

Figure 2a shows the difference between the net radiative flux directly above the inversion, indicated by a “+,” and at the surface. This total flux is divided into its longwave and shortwave contributions as shown in Figures 2b and 2c, respectively. The sign convention is such that downwelling fluxes are negative. The positive values in Figure 2a indicate a net cooling of the boundary layer by radiative processes, because

(12)

Figure 2

As Figure 1a, but for (a) the total radiative flux divergence over the boundary layer, , that is split into contributions from (b) longwave and (c) shortwave radiation . Blue and red colors indicate cooling and warming tendencies of the boundary layer, respectively.

Here c_p is the specific heat of air at constant pressure and denotes the boundary layer averaged value of . It is clear from Figure 2a that the net cooling in the boundary layer indeed increases as the free troposphere becomes drier. The longwave radiative cooling is predominantly confined to the top of the stratocumulus layer, which destabilizes this layer and promotes the production of turbulence. As a consequence a larger difference between and tends to increase the entrainment rate [Moeng, 2000; Christensen et al., 2013].

Note that in the current setup is constant with height in the lower part of the free troposphere. Given the fact that the downwelling longwave radiation received at the top of the cloud layer increases for larger water vapor paths aloft it can therefore be expected that the dependency of on the free tropospheric humidity may be somewhat weaker if the specific humidity in the free troposphere decreases with height.

The steady state inversion height in Figure 1a varies between 0.4 and 1.8 km. The MLM results of DG14a show a similar range and dependency on the free tropospheric thermodynamic conditions. This indicates that the Nicholls and Turton [1986] entrainment parameterization that was used in the MLM of DG14a realistically represents the dependency of the entrainment velocity on the inversion strength, evaporative cooling, and the downwelling longwave radiative flux at the inversion. The SCM used by DG14b, on the other hand, is too insensitive to variations in the free tropospheric conditions and as a consequence it underestimates the inversion height by up to 1000 m for the warm and dry free troposphere regime.

From MLM simulations, De Roode et al. [2014] found stratocumulus cloud deepening in combination with an increased entrainment rate in the relatively moist and cold free troposphere regime. This so-called cloud deepening through entrainment [Randall, 1984] is not found to take place in the LES results, which is likely due to the decoupling of the boundary layer that is discussed below.

3.2 Liquid Water Path

Figure 3a shows the steady state LWP, which ranges between approximately 40 and 80 g m⁻². The time series of the LWP in Figure 3b show that a steady state is achieved after only a few days, which is somewhat faster than for the inversion height (Figure 1b). Similar results are found from LESs by Bretherton et al. [2010]. They argued that this is a manifestation of the separation of the short thermodynamic time scale, which is of the order of a day, and the much longer dynamical time scale that is related to the divergence and can be up to 4 days [see, e.g., Schubert et al., 1979; Jones et al., 2014]. Once a thermodynamic quasi steady state is achieved, changes to the inversion height are accompanied by almost equal changes of stratocumulus base height. The slow evolution of in the second half of the simulations therefore hardly influences the LWP.

Figure 3

As Figure 1, but for the LWP in g m⁻². The boundary layer is initialized with well-mixed and profiles, resulting in a relatively large initial LWP of about 300 g m⁻².

The LWP variations around the steady state are typically less than 5 g m⁻², which is small as compared to the LWP differences among the cases, indicating that the spread in the LWP that is visible in Figure 3a is significant.

The LWP is too low to support significant rain formation. The precipitation rates are therefore low at <0.2 W m⁻² for all cases, so that the effect of precipitation on the budgets of and is negligible.

From Figure 3a, it can be seen that the LWP is predominantly controlled by and to a much lesser extent by the LTS. This is more clearly shown in Figures 4a and 4b, which show the steady state LWP as a function of and LTS, respectively. This sensitivity of the stratocumulus LWP to variations of the free tropospheric humidity has been recognized before [Chlond and Wolkau, 2000; Ackerman et al., 2004; Lock, 2009; van der Dussen et al., 2014].

Figure 4

The LWP in g m⁻² as a function of (a) the total specific humidity difference between the free troposphere and the surface and (b) the lower tropospheric stability LTS. Blue and red colors indicate the control and the perturbed climate simulations, respectively, and the symbols indicate the different values of (a) the LTS and (b) as shown in the legends.

Figures 5a and 5b show that in roughly the top half of the boundary layer is almost constant with height as a result of the mixing induced by the net radiative cooling at the top of the stratocumulus layer. This mixing causes the profiles of to be close to adiabatic, as is shown in Figure 5c. Interestingly, the actual stratocumulus layer is thin as compared to the depth of the well-mixed upper part of the boundary layer. For instance, the stratocumulus layer is about 250 m thick for the deepest case depicted by the yellow lines, while is well mixed over a depth of over 1000 m. Figure 5d shows that the cloud fraction below the stratocumulus layer is zero and hence there is no sign of cumulus updrafts that often occur underneath stratocumulus clouds in relatively deep marine boundary layers [Bretherton and Pincus, 1995; Wood, 2012].

Figure 5

Vertical profiles of (a) liquid water potential temperature , (b) total specific humidity , (c) liquid water specific humidity , and (d) cloud fraction σ averaged over the tenth day of the simulations. The legend indicates the values of and the LTS for each of the cases, and the locations in the phase space are indicated by the correspondingly colored dots in Figures 1a and 3a. The solid and dashed lines show the control and perturbed climate simulations, respectively. The markers in Figures 5a and 5b denote and at the sea surface for the control (circles) and the perturbed climate (triangles). The black lines in Figure 5c denote the adiabatic profiles.

For the deepest cases, the stratocumulus layer is significantly drier than the surface layer, a feature which is commonly found from observations [Nicholls and Leighton, 1986; Albrecht et al., 1995; Park et al., 2004; Wood and Bretherton, 2004] and is referred to as decoupling. This two-layer structure can obviously not be represented by an MLM. If the stratocumulus-topped boundary layer deviates from a well-mixed situation, the stratocumulus layer will typically have a higher and a lower than the subcloud layer, which both act to reduce the cloud liquid water content. DG14a showed that in the MLM a decrease of the free tropospheric specific humidity causes the stratocumulus layer to thicken, which is due to an increase of the steady state inversion height. The enhanced drying accompanying an increased entrainment rate is uniformly spread over the boundary layer and is therefore relatively small in the cloud layer. From the LES results we find a thinning of the stratocumulus layer when the free tropospheric specific humidity decreases, which is opposite to the response of the MLM. This is likely the result of the decoupling of the boundary layer, which causes the enhanced drying accompanying the increased entrainment rate to be mostly confined to the stratocumulus layer. Therefore, the cloud thinning due to enhanced entrainment drying dominates the response of the decoupled stratocumulus-topped boundary layer to a reduction of the free tropospheric humidity in the LES results.

The SCM results of DG14b also indicate an increase of the LWP for larger values for those cases that are completely overcast. In the previous section, it was noted that the boundary layer in the SCM results was too shallow in general and that the inversion height was less sensitive to changes in the free tropospheric conditions as compared to the LES results. The boundary layers in the SCM simulations are therefore rather well mixed. Similar to the MLM results, this likely explains the discrepancy between the LES and SCM in terms of dependence of the LWP on the free tropospheric humidity.

3.3 Surface Fluxes

Figure 6a shows the buoyancy flux at the surface, which is found to be small at, on average, 1 W m⁻². Negative surface buoyancy fluxes are found for relatively humid and warm free tropospheric conditions.

Figure 6

As Figure 1a, but for (a) the turbulent flux of the virtual potential temperature at the surface , (b) the surface sensible heat flux , and (c) surface latent heat flux .

In the subcloud layer, the virtual potential temperature flux can be expressed as a linear combination of the fluxes of and as follows:

(13)

Figure 6b shows that the sensible heat flux is negative for all free tropospheric conditions, which is the main cause of the low surface buoyancy flux. From equation 13 it can be seen that the contribution of the surface latent heat flux to the surface buoyancy flux is small. Furthermore, the surface latent heat flux if found to range between 25 and 60 W m⁻² as can be seen from Figure 6c, which is low as compared to typical maritime subtropical boundary layers [e.g., Bretherton and Pincus, 1995; Stevens et al., 2005]. For these reasons, the surface latent heat flux contributes only a few W m⁻² to the surface buoyancy flux.

Low or negative values for the buoyancy flux in the subcloud layer will hardly produce or even dampen turbulence. The transport of moisture from the subcloud layer to the cloud layer is therefore inhibited, which in turn limits the surface latent heat flux as can be seen from its bulk formulation

(14)

Here is the magnitude of the horizontal wind velocity, is a drag coefficient, and “sl” in the subscript denotes the surface layer. The range and variation of the surface latent heat flux within the phase space are remarkably similar to those reported by DG14b for the SCM results.

4 Perturbed Climate

To investigate the response of the cloud layer to a warming of the climate, a uniform temperature increase of 2 K is applied to the initial profiles, while the initial relative humidity profile is kept the same as for the control case. Such a perturbation does not affect the LTS but causes the magnitude of to increase as is described in section 2.2. We define the response of quantities to this perturbation by the difference between the perturbed and the control climate results divided by a temperature change of 2 K. This difference is denoted by a “d.” It is important to keep in mind that, following DG14a and DG14b, the response is plotted against the values of the control climate cases.

4.1 Response of the Surface Fluxes

A robust feature of climate warming scenarios is that the surface latent heat flux increases [Xu et al., 2010; Webb et al., 2013; Bretherton et al., 2013]. Figure 7a shows that this is also the case for the LESs considered here. The increase is similar to that found from the MLM results by DG14a.

Figure 7

The response to the idealized climate perturbation that is described in section 2.2 for (a) the surface latent heat flux , (b) the surface sensible heat flux , and the radiative flux divergence over the boundary layer of (c) longwave and (d) shortwave radiation. The results are shown as a function of the values of the control cases.

Rieck et al. [2012] argued that in response to a warming, the surface latent heat flux over the ocean increases such as to maintain a constant relative humidity . Using equation 14, the change of the surface humidity flux due to a temperature change can be expressed as

(15)

assuming that the wind speed does not change. Using the Clausius-Clapeyron relation to evaluate the change of the saturation specific humidity with temperature in equation 15, it can be readily shown that for typical subtropical conditions the surface humidity flux increases by about 7% K⁻¹ given a constant [Held and Soden, 2000]. For the LES results, the relative increase of the surface evaporation varies from about 6% K⁻¹ at high to 8% K⁻¹ at low LTS, which indicates that changes little with respect to the control case.

The response of the surface sensible heat flux to the climate perturbation is shown in Figure 7b. On average, this flux decreases by approximately 0.1 W m⁻² K⁻¹.

Figure 7c shows that the divergence of longwave radiation and hence the cooling of the boundary layer by the net emission of longwave radiation decreases. The warming due to absorption of shortwave radiation also decreases (Figure 7d), but less strongly. Hence, the net cooling of the boundary layer due to radiation decreases. The net decrease is on average −0.4 W m⁻² K⁻¹. Assuming that the entrainment flux of does not change significantly, the weakened radiative cooling is likely the cause for the small decrease of the surface sensible heat flux.

4.2 Response of the Stratocumulus Layer

Figure 8a shows the change of the shortwave cloud radiative effect ( ) at the top of the atmosphere. It is defined as the difference between the net shortwave radiative flux at the top of the atmosphere for the actual atmospheric profile and for the clear-sky, where the radiative flux is defined positive downward [Cess et al., 1989]. The change of is used here as an indicator for the sign and magnitude of the cloud feedback. For low clouds, is negative, which implies a net cooling of the atmosphere by clouds as a result of their strong ability to reflect shortwave radiation back to space. It can be seen from Figure 8a that the overall increase of varies between 4 and 12 W m⁻² K⁻¹, i.e., the cooling effect due to the stratocumulus clouds decreases and hence the cloud-climate feedback is positive for this idealized climate change scenario. Quantitatively similar changes were found from the SCM results (DG14b) as well as from the CGILS stratocumulus cases for the warming perturbation at constant relative humidity [Bretherton et al., 2013]. The change of the longwave cloud radiative effect is negligibly small at less than 0.1 W m⁻² K⁻¹ and is therefore not shown.

Figure 8

The response to the idealized climate perturbation described in section 2.2 of (a) the shortwave cloud radiative effect at the top of the atmosphere , (b) the liquid water path LWP, (c) the inversion height , and (d) the stratocumulus cloud base height .

DG14a determined from ERA-Interim data the frequency of nighttime occurrence of the LTS- combinations in the months June, July, and August in the area just off the Californian coast. Their results suggested that the frequency of occurrence increases diagonally toward the low LTS and dry free troposphere regime. From the LES results, the change of the cloud radiative effect is smallest in this regime. Hence, on average, 8 W m⁻² K⁻¹ when weighted by frequency of occurrence. Furthermore, 4 W m⁻² K⁻¹ for LTS > 18 K and for the wide range of considered.

Medeiros et al. [2014] diagnosed the change of the CRE in response to a 4 K increase of the sea surface temperature from results of several climate models. For the high-sensitivity models, they showed that dCRE ranges between 2.5 and 5.5 W m⁻² K⁻¹ in the stratocumulus regime, which was identified by the presence of subsidence and by LTS > 18 K. The LES results therefore suggest that the thinning of stratocumulus clouds in response to a warming of the climate is of comparable magnitude as the response diagnosed from the current generation of climate models.

Since the boundary layer remains completely overcast for all perturbed climate simulations, the response of can be mainly attributed to a decrease of the LWP. Figure 8b shows that the LWP exhibits a maximum decrease of about −12 g m⁻² K⁻¹ in the high LTS and moist free troposphere regime. The LWP decrease is, at about −3 g m⁻² K⁻¹, considerably weaker for the driest and warmest free tropospheric conditions considered in this study.

4.2.1 Adiabatic Lapse Rate

It can be shown from thermodynamic arguments that should increase with temperature [Paltridge, 1980]. For the LES results, the effect of this increase on the LWP response can be quantified by first approximating the LWP of a stratocumulus cloud layer as

(16)

in which

(17)

is the lapse rate of the liquid water specific humidity, and is the geometrical cloud thickness of the stratocumulus layer. In response to a climate perturbation, the LWP may change due to a change in or in h as follows:

(18)

where it is assumed that the cloud cover does not change.

As temperature increases, , so the first term of equation 18 will cause an increase of the LWP in a warmer climate. The magnitude of the LWP response as a result of this lapse rate effect depends on temperature and on the depth of the cloud layer and is between 1 and 1.5 g m⁻² K⁻¹ for the current setup as can be seen from Figure 9a. From LESs of cumulus-topped boundary layers, Rieck et al. [2012] also found this increase of the in-cloud liquid water content. Nevertheless, the domain-averaged LWP decreased in their case mainly due to a decrease of the cloud cover, which was attributed to a decrease of . Therefore, the sign of the cloud feedback was positive.

Figure 9

The normalized LWP response (a) due to the change of the lapse rate of liquid water specific humidity and (b) due to the change of the geometrical cloud thickness h according to equation 18.

For the stratocumulus cases considered here, the LWP decreases as well, which is due to a decrease of the geometrical thickness h of the stratocumulus layer. This effect is described by the second term on the right-hand side of equation 18. Figure 9b shows that the decrease of the LWP due to the decrease of h is much stronger than the LWP increase due to the change in . The decrease of h is the result of an increase of stratocumulus cloud base height relative to the inversion height . Below, the responses of and are discussed individually.

4.2.2 Inversion Height

An increase of the inversion height in the perturbed climate can be expected as a result of the increased surface latent heat flux that invigorates the turbulence in the boundary layer [Bretherton and Wyant, 1997; Bretherton and Blossey, 2014]. This effect has been found in several cumulus studies [Xu et al., 2010; Nuijens and Stevens, 2011; Rieck et al., 2012]. However, as a result of the assumption that the initial relative humidity in the free troposphere is not affected by the climate perturbation, the total specific humidity in the free troposphere is increased. Since this causes the downwelling longwave radiative flux at the top of the stratocumulus layer to increase and consequently the radiatively induced cloud top cooling to decrease, the latter effect will act to reduce the entrainment rate.

In their MLM study, DG14a separated these mechanisms by first applying a warming perturbation of 2 K to the sea surface and the atmosphere at constant initial relative humidity, without changing the prescribed net radiative cooling rate of the boundary layer. Under that assumption, the inversion height increased significantly for all free tropospheric conditions. The radiative flux divergence over the boundary layer was decreased by 1.5 W m⁻² K⁻¹ in a second set of simulations, in addition to the climate warming perturbation. In this experiment, decreased for all cases. In the LES results, the response of the net radiative flux divergence over the boundary layer is weaker than that imposed in the MLM experiments of DG14a, at approximately 0.5 W m⁻² K⁻¹ as was deduced from Figures 7c and 7d. This modest radiative response causes the increase of the inversion height due to the increased latent heat fluxes to be the dominating mechanism and hence increases by between 10 and 25 m K⁻¹ as can be seen from Figure 8c. In contrast, a decrease was found from the LES results of the CGILS S12 stratocumulus case for those simulations in which the subsidence velocity was not perturbed [Bretherton et al., 2013]. Almost all LES models that participated in the CGILS intercomparison study agree on this response [Blossey et al., 2013]. The response of the net radiative flux divergence was evaluated at −1.0 to −1.5 W m⁻² K⁻¹, which is at least 2 times as large as for the present simulations. Therefore, it is possible that for the CGILS S12 case, the response due to the change of the net radiative flux divergence dominates the total response of the inversion height, causing it to decrease.

Admittedly, the increases that were found are not large compared to the vertical grid spacing of 10 m. The response is significant, however, as the differences are persistent during the entire 10 day simulation period.

4.2.3 Stratocumulus Cloud Base Height

Figure 8d shows that the increase of cloud base height is approximately twice as large that of . Hence, the stratocumulus layer thins for all free tropospheric conditions. The increase of is related to a decrease of the relative humidity , vertical profiles of which are shown in Figure 10a. The vertical coordinate has been nondimensionalized by dividing by , to simplify the comparison of the boundary layer structure between the perturbed and the control climate results. Near the surface changes little, as was already deduced from the analysis of the surface latent heat flux response. Throughout the upper part of the boundary layer, decreases slightly, with the largest decrease located in the middle of the boundary layer. For the cumulus simulations of Rieck et al. [2012], decreased as well, affecting the LWP mainly through a decrease of the cloud fraction. For the stratocumulus layers considered here, the response causes an LWP decrease through an increase of the stratocumulus cloud base height.

Figure 10

(a) The relative humidity , (b) the liquid water potential temperature , and (c) the total specific humidity as a function of dimensionless height . Solid and dashed lines show the control and the perturbed climate results, respectively. The initial perturbations have been subtracted from the perturbed climate results in Figures 10b and 10c. The black line in Figure 10a indicates the saturation level, .

To assess the effect of the response of and on the decrease of the relative humidity in the cloud layer, Figures 10b and 10c show their vertical profiles as a function of the normalized height . To simplify the comparison with the control case (indicated by solid lines), the initial perturbations of and have been subtracted from the steady state results of the perturbed climate simulations (dashed lines). Clearly, the shape of the profile is hardly affected by the climate perturbation. The profiles of on the other hand are more decoupled in the perturbed simulations, suggesting that the decrease of the relative humidity is mainly due to a drying of the upper part of the boundary layer.

4.3 Inversion Properties

Figure 11 shows a scatterplot of the initial inversion jumps of and in light numbers for the control (black) and the perturbed climate simulations (blue). The solid lines connect the initial and steady state inversion jumps. The base and the top of the inversion layer are determined by finding the layer in which the variance of exceeds 5% of its peak value [Yamaguchi et al., 2011], and the inversion jump is defined as the difference of a variable across this layer. To provide a reference within the and phase space, the buoyancy reversal criterion line as derived by Randall [1980] and Deardorff [1980] is also shown in Figure 11. It was suggested in these studies that solid stratocumulus cloud decks could not persist at the left side of this line as a result of a runaway entrainment mechanism, which would dry and warm the stratocumulus layer thereby causing it to rapidly break up. However, stable stratocumulus layers have often been observed for such conditions [e.g., Kuo and Schubert, 1988; Stevens et al., 2003]. It was furthermore argued by van der Dussen et al. [2014] that stratocumulus clouds can persist far into the buoyancy reversal regime if the cloud building processes, such as the humidity flux from the surface, are sufficiently strong.

Figure 11

Scatterplot of and for each of the control (black) and the perturbed climate simulations (blue). The experiments are numbered consecutively starting from the simulation with the lowest LTS and the largest . The initial conditions are indicated by the numbers in regular font, while the bold-faced numbers indicate the inversion jumps averaged over the tenth day of the simulations. The initial and final states of the simulations are connected by solid lines and the gray dotted line denotes the buoyancy reversal criterion [Kuo and Schubert, 1988] as a reference.

The initial value of can be related to the bulk tropospheric jump

(19)

in which is the initial boundary layer value of . Because we assume that the initial relative humidity does not change in a perturbed climate, the increase of in the boundary layer is larger than in the colder free troposphere, explaining the larger initial value of for the perturbed climate simulations. However, the initial differences of between the control and the perturbed climate cases have been reduced significantly at the end of the simulations. These reductions can only be caused by a stronger drying of the stratocumulus cloud layer in the perturbed relative to the control climate, since is constant with height and time in the lower part of the free troposphere. Much of the drying is due to entrainment, which can be expressed as the product of the entrainment rate and the inversion jump of humidity [Lilly, 1968]

(20)

Hence, a larger inversion jump of increases the potential to dry the boundary layer by entrainment.

The steady state inversion jumps collapse remarkably well onto a line in the phase space in Figure 11. The imaginary line onto which the perturbed climate results approximately collapse is shifted to the lower left-hand corner in Figure 11 with respect to the control climate. For the perturbed climate simulations, is up to 1 K smaller. This decrease of is mainly due to enhanced radiative cooling of the lower free troposphere as a result of the increase in the specific humidity. van der Dussen et al. [2014] showed that a colder and a drier free troposphere (i.e., a smaller value of and a larger value of , respectively) are typically associated with a stronger cloud thinning tendency due to entrainment. For the current set of simulations, the stronger entrainment drying tendency is to a large extent balanced by an increase of the surface latent heat flux such that the LWP can reach a new equilibrium state.

5 Discussion

5.1 Correlation Between Change of and LWP Response

The response of the stratocumulus layer to the idealized climate perturbation can be summarized as follows. In the first place, the stratocumulus base height increases due to a drying of the upper part of the boundary layer, that is mostly related to the increase of the magnitude of . Second, the inversion height increases, which is related to the competition between the increase of the surface latent heat flux and the increase of the downwelling longwave radiation. This competition is qualitatively accounted for in the change of the value of as it is defined as the difference between and .

The important mechanisms determining the response of the stratocumulus layer can therefore mostly be correlated with changes in . Figure 4a shows the LWP as a function of the actual value of for the control (blue shades) as well as for the perturbed climate simulations (red shades). The increase of the magnitude of as a result of the climate perturbation shifts the location of each simulation in the plot to the left with respect to the control simulations. The perturbed climate cases, with the exception of the high LTS ones, fall approximately on the line that can be fitted through the control climate results. This suggests that much of the LWP decrease can indeed be attributed to the change of .

5.2 Radiation Versus Surface-Driven Boundary Layers

It was shown that for most cases considered in this research, the surface buoyancy flux is rather small (Figure 6c). Furthermore, the vertical profiles of the total specific humidity show that the boundary layer structure is decoupled for the cases with the deepest boundary layers. Figures 12a and 12b show profiles of the buoyancy flux and of the vertical velocity variance , respectively. The buoyancy flux is small or negative in the subcloud layer, but large in the stratocumulus layer as a result of net radiative cooling. This causes , which constitutes an important part of the turbulence kinetic energy, to be much larger in the stratocumulus layer than at the surface. Figure 12c furthermore shows that the vertical velocity skewness

(21)

is mostly negative throughout the boundary layer, apart from the spike close to the top of the boundary layer that is often found for stratocumulus clouds [Moeng and Rotunno, 1990]. A negative value for S_w indicates that the turbulence in the boundary layer is determined mostly by downdrafts and that the boundary layer dynamics are predominantly radiatively driven.

Figure 12

As Figure 5, but for (a) the turbulent flux of the virtual potential temperature in W m⁻² as a proxy for the buoyancy flux, (b) the vertical velocity variance , and (c) the skewness of the vertical velocity S_w as defined by equation 21.

The results of an SCM intercomparison study will be discussed in a companion paper (Dal Gesso, submitted manuscript, 2014), which includes a detailed comparison with the LES results discussed here. In many SCMs, the vertical transport is parameterized in terms of updrafts forced from the surface. As the current setup results in predominantly top-driven boundary layers it can be expected to be particularly challenging for such SCMs. This is illustrated by the results of the EC-Earth SCM used by DG14b, which, among others, show much less dependence of the steady state inversion height on the free tropospheric conditions as compared to the LES results.

The MLM results of DG14a indicated that for well-mixed boundary layers, the sign of the stratocumulus cloud feedback is positive. The present study shows that in the other extreme, namely a decoupled situation with weak surface forcing, the sign is positive as well. Similarly, Bretherton et al. [2013] found that the sign of the feedback for both well-mixed and decoupled stratocumulus cases is positive. The sign of the cloud-climate feedback therefore does not seem to depend on the degree of decoupling of the boundary layer. It is reassuring that despite the different setup of CGILS as compared to the present experiments, the sign of the cloud feedback is found to be consistent and that the thinning of stratocumulus clouds is a robust response to a climate warming perturbation at constant initial relative humidity. The magnitude of the response may however be affected by the details of the setup.

The LES results show an increase of the inversion height in the perturbed climate. In contrast, a decrease is found the CGILS stratocumulus experiments (−30 to 0 m K⁻¹) [Blossey et al., 2013] as well as from the MLM experiments by DG14a (−40 to −10 m K⁻¹). A possible cause for this discrepancy is the relatively weak response of the net radiative divergence over the boundary layer for the present simulations as was discussed in section 4.2.2. Furthermore, the decoupling of the boundary layer may play a role. To investigate the sensitivity of the cloud response to boundary layer decoupling and its effect on the response of the inversion height, the current setup could be adjusted to allow for positive surface sensible and larger surface latent heat fluxes by prescribing cooling and drying tendencies due to horizontal advection in the boundary layer. This was for instance done for the CGILS experiments [Blossey et al., 2013] as well as for the steady state simulations of Chung et al. [2012], both of which have significantly higher surface fluxes and better mixed boundary layers for their stratocumulus simulations. This approach could make the case setup more realistic. The downside of a less idealized setup is that it introduces additional degrees of freedom and potentially complicates the interpretation of the response of the cloud layer.

6 Conclusions

The influence of the free tropospheric conditions on the steady states of radiatively driven stratocumulus-topped boundary layers was investigated by running a set of 25 large-eddy simulations in a phase space spanned by a range of free tropospheric temperatures and humidities. The response to an idealized climate perturbation was furthermore assessed by uniformly increasing the initial temperature profile of the control simulations by 2 K, while the total specific humidity was increased to keep the initial relative humidity the same as for the control simulations. The results complement an earlier conceptual study with a mixed-layer model [Dal Gesso et al., 2014a] as well as simulations with a single-column model version of the EC-Earth climate model [Dal Gesso et al., 2014b].

6.1 Control Climate

The control climate simulations show that the steady state inversion height increases as the LTS decreases, i.e., as the free troposphere becomes warmer. Furthermore, for a dry free troposphere, the downwelling longwave radiative flux absorbed by the stratocumulus layer is relatively low, such that the cloud top cooling gets enhanced thereby increasing the entrainment rate and the boundary layer depth. The LWP depends mainly on and decreases as the free troposphere becomes drier.

For the MLM results by DG14a, the opposite was found, which is due to the inability of the MLM to represent a decoupled, two-layer boundary layer structure. The LES results indicate only a weak forcing of turbulence from the surface, causing the dynamics of the boundary layer to be mainly driven by radiative cooling at the cloud top. This results in a building up of moisture in the subcloud layer and a relatively strong drying of the cloud layer by entrainment.

6.2 Perturbed Climate

In the perturbed climate simulations, the surface latent heat flux is approximately 7% K⁻¹ larger than in the control cases, as is expected on the basis of Clausius-Clapeyron scaling. This stronger surface evaporation flux invigorates turbulence in the boundary layer and hence tends to increase the entrainment rate [Rieck et al., 2012]. On the other hand, the increased specific humidity in the free troposphere enhances the downwelling longwave radiative flux, which tends to decrease the entrainment rate. The net effect is a small increase of the inversion height of between 10 and 25 m K⁻¹.

The drying tendency due to entrainment is shown to increase as a result of the climate perturbation, causing an increase of stratocumulus base height that is greater than the increase of stratocumulus top height. Hence, the stratocumulus layer thins, which results in a decrease of the LWP that is largest at −12 g m⁻² K⁻¹ for high LTS and relatively humid free tropospheric conditions. As a result of the thinning of the cloud layer, the shortwave cloud radiative effect weakens for all free tropospheric conditions by on average 8 W m⁻² K⁻¹, indicating that the sign of the cloud feedback is positive, which is consistent with recent similar studies [Blossey et al., 2013; Bretherton et al., 2013]. In comparison, the SCM results of DG14b overall indicated a positive feedback as well, but the sign and the magnitude of the feedback varied irregularly throughout the phase space.

An important finding is that the change of the bulk humidity difference between the free troposphere and the surface in a perturbed climate is key to the change in the stratocumulus cloud amount, in particular since it determines the drying of the cloud layer through entrainment. This process is responsible for the change of stratocumulus base height. The change of furthermore controls the response of the downwelling longwave radiation that is absorbed by the cloud as well as the response of the surface latent heat flux. These processes together determine the change of the inversion height.

Acknowledgments

The investigations were done as part of the European Union CLoud Intercomparison, Process Study and Evaluation (EUCLIPSE) project, funded under Framework Program 7 of the European Union. The work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities. The model output used to generate the figures is included as supporting information, but can also be obtained from the corresponding author upon request (e-mail: [email protected]). We thank the two anonymous reviewers whose comments helped to improve the manuscript.

Appendix A: Sensitivity to Vertical Resolution

In many of the recent intercomparison studies focused on stratocumulus (transitions), grids were employed with fine, often 5 m, vertical resolutions at the inversion in order to make sure that the inversion gradients are well resolved. In order to reduce computational costs, a vertical resolution of 10 m is chosen for the simulations presented in the current research. Two additional sensitivity runs were conducted to test the dependence of the results on resolution. The case in the center of the phase space, with ( , LTS) = (−5.8 g kg⁻¹, 22.1 K) is used for this sensitivity experiment. The simulation details can be found in Table 1. For all simulations, the aspect ratio of the grid boxes in the part of the domain with uniform vertical grid spacing.

Table 1. Numerical and Domain Details for the Reference as Well as the Sensitivity Simulationsa

	Low Res	Reference	High Res
	75 m	50 m	25 m
	15 m	10 m	5 m
Grid aspect ratio	5	5	5
Nx, Ny	80	120	240
Nz	136	219	356

• ^a The aspect ratio of the grid boxes and the domain size are equal for all simulations.

Figure 13a shows vertical profiles of for the three experiments described in Table 1, averaged over hours 18–24 of the simulations. The effect of a change of the resolution on is clearly small. At lower resolutions, the inversion does tend to be more spread out, but the effect on the entrainment rate is limited as the inversion height varies by only 10 m among the simulations. Figure 13b shows that the upper part of the boundary layer is moister at high resolution, leading to a thicker cloud layer as can be seen in Figure 13c. The LWP increases considerably from 54 g m⁻² at 15 m, to 64 g m⁻² at 10 m, to 80 g m⁻² at 5 m vertical resolution. The sensible heat flux profiles in Figure 13d are very similar for all simulations, but the latent heat flux increases as the resolution is increased (Figure 13e). Figure 13f shows the resolved vertical velocity variance profiles. The three simulations are all strongly top-driven, judging from the large peak at the top of the boundary layer. The magnitude of this peak is well captured by the reference simulation, although the high-resolution simulation has a higher in the subcloud layer.

Figure 13

Vertical profiles of (a) the liquid water specific humidity , (b) the total humidity , (c) the cloud fraction σ, the total (resolved + subfilter scale) turbulent fluxes, (d) , (e) , and (f) the resolved vertical velocity variance averaged over hours 18–24, for the case in the center of the phase space where  = −5.8 g kg⁻¹ and the LTS = 22.1 K. Additional information about the simulations can be found in Table 1.

Figure 14 shows the filtered and the subfilter-scale contributions to the virtual potential temperature flux in W m⁻² as a function of height. The relative contribution of subfilter scales to the total turbulent flux is clearly low for each of the cases and hence most of the turbulence in the simulations is explicitly resolved.

Figure 14

As Figure 13, but for the resolved (solid lines) and the subfilter-scale contributions (dashed lines) to , the turbulent flux of the virtual potential temperature in W m⁻².

The results shown in Figures 13 and 14 provide confidence that the steady state behavior of the simulations with the 10 m is qualitatively comparable to the higher-resolution simulations. The main quantitative differences are likely found in the results for the LWP, which was found to increase by as much as 25% when the vertical resolution was increased from 10 to 5 m.