Factors influencing cloud area at the capping inversion for shallow cumulus clouds

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Published online 28 April 2009 in Wiley InterScience ( Factors influencing cloud area at the capping inversion for shallow cumulus clouds A. P. Lock* Met Office, UK ABSTRACT: Large-eddy simulations of marine cumulus clouds are performed. Aspects of the simulations are varied in order to identify those parameters that play a role in determining when the cumulus clouds spread out under the capping inversion into stratocumulus. These include the strength of the surface fluxes, which also affects the strength of the cumulus mass transport, the strength of the capping inversion, the moisture content of the free atmosphere, the strength of cloud-top radiative cooling and whether there is significant precipitation or sedimentation of cloud drops. Tests are also performed to evaluate sensitivity to the model resolution. As in previous studies, having sufficient resolution at the capping inversion is found to be important for robust simulation of any stratiform cloud that forms there. In addition, the cloud cover is found to be sensitive to many of the physical processes investigated. In the end, though, a clear and robust dependence emerges on the cloud-top entrainment instability parameter, κ. Forlargeκ, a typical (small) shallow cumulus cloud cover is seen. For κ below the threshold for buoyancy reversal, the cumulus spreads into a solid sheet of stratocumulus beneath the inversion. Consistent with previous studies, there is no abrupt transition and the cloud cover decreases smoothly as κ increases. The conclusion is reached that varying other parameters in the simulations affects the cloud cover through their impact on the size of the jumps across the inversion and thence κ. The implications for turbulent mixing parametrizations in general circulation models are discussed. c Crown Copyright Reproduced with the permission KEY WORDS stratocumulus; entrainment; transition Received 20 August 2008; Revised 28 January 2009; Accepted 26 March Introduction Shallow cumulus clouds are prevalent, especially over tropical and subtropical oceans. Norris (1998) shows not only large areas with significant occurrence of observations of small cumulus clouds but also significant amounts of stratocumulus from spreading cumulus and cumulus under stratocumulus. These latter cloud regimes are often associated with a transition from stratocumulus, typically seen in the eastern Pacific and Atlantic oceans. The difference in albedo between a field of shallow cumulus clouds and shallow cumulus that has spread into a sheet of stratocumulus is huge. It is therefore very important for weather forecast and climate models to be able to distinguish between these regimes accurately. Furthermore, it is increasingly being recognized that much of the variance in estimates of climate sensitivity can be attributed to differences in the modelled responses in regions of low cloud and in transitions between shallow cumulus and stratocumulus over the subtropical oceans in particular (Bony and Dufresne, 2005; Williams and Webb, 2008). In order to represent these cloud regimes realistically, these models should be able to represent accurately not just the radiative properties of each cloud type but also the transports arising from the cumulus and Correspondence to: A. P. Lock, Met Office, FitzRoy Road, Exeter, EX1 3PB, UK. adrian.lock@metoffice.gov.uk stratocumulus elements as well as the structure of the capping inversion. Understanding the source of model differences is hampered, however, by a lack of knowledge of which processes are most important. There have been many studies on the circumstances under which well-mixed stratocumulus sheets might dissipate (Deardorff, 1980; Randall, 1980; MacVean and Mason, 1990; Siems et al., 1990; Duynkerke, 1993). Many of these looked at the potential for enhancement of entrainment mixing across the inversion due to evaporative cooling of entrained air, as will be discussed further below. Other studies have investigated the processes that might be responsible for the decoupling of stratocumulus (Wyant et al., 1997; Stevens, 2000; Lewellen and Lewellen, 2002). This decoupling allows cumulus clouds to form with cloud-base below that of the stratocumulus and so is potentially a first step in the transition across the subtropical oceans, from stratocumulus in the east to shallow cumulus in the trades. The question of which processes might control the point at which cumulus cloud spreads out under its capping inversion into stratocumulus rather than simply dissipating, has, however, received little attention. A notable exception is the study by Stevens et al. (2001). This large-eddy simulation (LES) intercomparison, based on observations made during the Atlantic trade-wind experiment (ATEX), found that clouds in different models ranged from typical of shallow cumulus (around 20% of HMSO. Published by John Wiley & Sons, Ltd.

2 942 A. P. LOCK cloud cover) to cumulus spreading into a complete stratocumulus layer. This they linked to sensitivities to model resolution and numerics, but they also attributed the extent of the differences to a positive feedback between cloud cover and long-wave (LW) cloud-top cooling. Process that are likely to influence cloud area at the capping inversion include the following: (i) the rate of transport of cloud water into the inversion region by the cumulus clouds; (ii) the rate of dilution by free atmospheric air, entrained as the cumulus clouds hit the inversion and also by turbulent mixing in any stratocumulus clouds; (iii) rainfall production, not least through removal of water from rising cumulus clouds, thereby reducing the transport of water to the inversion region; (iv) radiative processes leading to cloud-top cooling in any stratiform cloud layer, in turn leading to turbulent mixing and additional dilution by the entrainment of free atmospheric air. Given these potentially important processes, it is instructive to consider which aspects of the boundarylayer environment (the mean profiles or large-scale forcing, for example) might play a role in determining the cloud cover. Possible candidates are as follows: (1) the strength of the buoyancy jump across the inversion, through its impact on the ability of clouds to penetrate the inversion and therefore entrain; (2) the moisture content of the free atmosphere entraining drier air would evaporate more cloud water; (3) cloud droplet concentration, through its role in determining droplet sizes and thence microphysical processes such as rainfall production; (4) factors influencing the turbulence intensity in the stratocumulus-cloud layer, and thence entrainment, for example, the strength of the cloud-top radiative cooling or the strength of evaporative cooling during the mixing process. As alluded to above, the role of evaporative cooling in generating entrainment in cloudy boundary layers (contained within point (4)) has been the source of much debate over many decades. Following the notation of Kuo and Schubert (1988), the parameter κ = θ e θ l = 1 + (1) (L/c p ) q t (L/c p ) q t gives some measure of the buoyancy of parcels of air formed from mixtures of cloudy air, with thermodynamic properties at the inversion base, with air from just above the inversion. In Equation (1), θ e is the equivalent potential temperature, θ l = θ e (L/c p )q t, q t the total water mixing ratio, L the latent heat of vaporization of water, c p the specific heat capacity of dry air at constant pressure and denotes the change in a quantity across the capping inversion. For κ greater than a threshold value (which is a weak function of state, but is typically around 0.23), Randall (1980) and Deardorff (1980) showed that mixtures could be negatively buoyant. They therefore postulated a feedback whereby turbulently driven mixing across the cloud top (entrainment) could generate negatively buoyant mixtures that would sink away from the cloud top, generating more mixing and hence more entrainment, a feedback that has become known as cloud-top entrainment instability (CTEI). Because the generation of negative buoyancy requires evaporation of cloud water, they postulated that CTEI could lead to the break-up of stratocumulus clouds. This work sparked significant controversy, not least because many observations of stratocumulus appeared to exist on the unstable side (i.e. κ>0.23; see Kuo and Schubert (1988) or more recently Stevens et al. (2003)). Subsequent authors then published alternative measures that were more restrictive (Siems et al., 1990; Duynkerke, 1993), including MacVean and Mason (1990) who argued that κ 0.7 was required before CTEI became energetically favourable for cloud break-up. This threshold had the advantage of nicely delineating observations of inversions capping stratocumulus. Subsequent work with LES (MacVean, 1993) suggested that in fact there was no threshold but an increase in rate of destruction of cloud water for κ>0.23 that became sufficiently rapid as κ = 0.7 was approached so, as to dominate any other cloud-generating process (such as cloud-top radiative cooling). Grant and Lock (2004), hereafter GL, noted that the formation of negatively buoyant downdraughts was leading to positive buoyancy fluxes in the inversion region in their cumulus simulations. Consistent with this, κ is greater than 0.6 across the whole range of their simulations, suggesting that negatively buoyant mixtures could somehow be important for shallow cumulus. More recently, Heus and Jonker (2008) have diagnosed shells of negatively buoyant downdraughts surrounding cumulus clouds, both in their LES and in observations. In this study, LES will be used to investigate the role of many of the above processes in controlling the cloud area at the capping inversion for shallow cumulus clouds. Although the sensitivities reported in Stevens et al. (2001) suggest that perhaps LES can give little useful guidance in this regime, it will be shown that the conditions in that simulation appear to lie in a sensitive region of parameter space and that in other areas the LES results are robust. This is shown through additional sensitivity tests to increase the resolution, an aspect found to be particularly sensitive by Stevens et al. (2001). 2. Simulations The simulations performed here are based on the GL equilibrium shallow-cumulus simulations, with surface fluxes and cloud depth typical of marine subtropical cumulus. Most are based on the second simulation in their Table 1, denoted GL2 here, which has conditions

3 FACTORS INFLUENCING CUMULUS CLOUD AREA 943 very similar to those in the trade-wind cumulus model intercomparison of Siebesma et al. (2003). Sensitivity to the strength of the cumulus convection is investigated by additional simulations based on the fourth simulation in GL s Table 1, where the surface fluxes, generated by the specified sea-surface temperature lower boundary condition, are significantly enhanced, giving a cloudbase mass flux approximately 50% larger, denoted GL4 here. The question of forcing is always problematic in LES. GL, for example, took great care to generate forcing profiles that were vertically smooth, constant in time and of a magnitude sufficient to balance the heating and moistening of the shallow cumulus clouds, in order to allow the model to generate steady-state cumulus-cloud layers. Here, the important process of cloud-top radiative cooling is added, using the cloud-water-dependent LW radiation scheme from Stevens et al. (2001). The net radiative flux is calculated from the model s local density, ρ, and liquid-water mixing ratio, q l,as { } F rad = F 0 exp k ρq l dz. (2) z As standard, the same input parameters are used as in Stevens et al. (F 0 = 74 W m 2,k= 130 m 2 kg 1 ). Sensitivity tests are also performed where F 0 is halved to 37 W m 2 and set to zero. Once this additional, realistic, degree of freedom is introduced (so that simulations that form significant cloud cover will have significant additional cooling), it becomes difficult to prescribe forcing without potentially imposing the final result if simulations with greater cloud cover are given different forcing, to what extent is that cloud cover a result of the forcing? Therefore, the large-scale forcing profiles (which represent the effects of large-scale horizontal advection across the subtropics, and so are cool and dry) are left unchanged from GL. In order to maintain the inversion base roughly between its initial level of 1500 m and a height well away from the damping layer (which starts at 2 km), the large-scale divergence, D, is set empirically for each simulation (the values used are given in Table I). Tests showed little sensitivity to small changes in D, but if D is too strong the inversion is pushed downwards and only cumulus clouds form. If D is too weak, on the other hand, the cloud Table I. Summary of parameters for the initial profiles and the large-scale divergence forcing, D, applied in the standard sets of simulations. Simulation Initial Applied LS divergence ( 10 6 s 1 ) θ v (K) κ θ l (K) q t (g kg 1 ) GL Set Set Set Set GL GL4 set

4 944 A. P. LOCK layer can rise into the damping layer. These choices, then, are a necessary compromise between wishing to keep the simulations close to the initially imposed state, thereby allowing the parameter space to be explored, and wanting to give the simulations the freedom to develop a solution that is not significantly dependent on the forcing. Finally, in order to help maintain the θ profile above the inversion close to the initial conditions, an additional cooling is applied above the maximum cloud-top height that balances the applied subsidence and large-scale forcing. The gradients in the moisture profiles above the inversion are sufficiently small that any drift, due to imbalances between the subsidence and large-scale forcing there, is found to be negligible. There is also a Newtonian damping layer above 2 km that relaxes the θ and q v profiles back to their initial conditions. The initial profiles are also taken from GL, but with several changes to allow a wide range of inversion properties to be explored without the simulations drifting too far from their initial state. In particular, preliminary simulations that simply adjusted θ v (the virtual potential temperature) and q t in the free atmospheric from GL were found to give values of κ that evolved significantly during the simulations. As significant layer cloud formed at the inversion, the associated radiative cooling was found to cause the θ (and indirectly the q v ) profiles in the cloud layer to drift away from the mixing-line structure imposed in GL (from Betts (1986), where the pressure derivative of the lifting-condensation-level pressure, β, isfixedat 1.3). Instead, these simulations tended towards a profile of β that decreased approximately linearly with height. In order to reduce this drift, then, the initial θ and q v profiles in the cloud layer are generated by imposing an approximation to this β profile (β decreasing linearly from 1.3 at cloud base to 0.4 at the inversion base, at 1500 m), whilst keeping the GL θ v profile fixed. The result of this is to give a saturated stratocumulus layer at the top of the cumulus-cloud layer, between approximately 1300 and 1500 m. A side-effect is for all simulations to start with a solid sheet of stratocumulus at the inversion base and so experience the cloud-top radiative-cooling feedback discussed by Stevens et al. (2001). The final change, compared with GL, is to sharpen the capping inversion to occur between adjacent grid-levels. The jumps in θ l and q t across the inversion are adjusted to give a range of specified values of the virtual potential temperature jump, θ v, and specified values of κ ranging from 0.1 (i.e. stable to CTEI by all measures) to 0.7 (approximately equal to MacVean and Mason s stability threshold), as listed in Table I. Most of the sensitivity tests are performed for set 2 of the simulations, with θ v = 4.5 K. Throughout this article the simulations will be referred to by the initial, specified, values of θ v and κ. In the final part of section 4, values of the jumps are required to be calculated from time-mean profiles during the simulations. For these, the inversion base is identified as the level of maximum buoyancy of a moist adiabatic parcel lifted from cloud base and the inversion top as the level where the total moisture flux drops to 2.5% of its maximum value. Because the profiles are idealized (i.e. with close to linear profiles above and below the inversion), there is very little sensitivity to these choices. The standard simulations are performed without any precipitation processes, as in Stevens et al. (2001). To investigate the role such processes might play, a further set of simulations is performed using the dual-moment microphysics scheme described in Abel and Shipway (2007), using the functions found in that study to give the best agreement with the aircraft observations from the Rain In Cumulus over Oceans (RICO) field campaign. A cloud droplet number concentration of 70 cm 3 was used to be representative of a clean marine environment and so give non-trivial precipitation production. Finally, recent simulations of stratocumulus (Bretherton et al., 2007; Ackerman et al., 2009) have highlighted a sensitivity in this regime to the inclusion of the sedimentation of cloud water. A set of simulations is included to test the impact of this process here, using the parametrization in Ackerman et al. with their recommended setting of the parameter σ g = 1.2. Three-dimensional simulations are run for 12 hours with initial random perturbations of up to 0.2 K to θ and 0.05 g kg 1 to q v below 1500 m. There is a uniform geostrophic wind of 10 m s 1 and the Coriolis parameter is set to be appropriate for 15 N. The standard simulations use a uniform horizontal grid size, x, in both dimensions of 50 m and a domain size of 6.4 km (as used by Stevens et al., 2001). This resolution is already higher than that used in GL or as standard in Stevens et al., both of which used a 100 m horizontal grid. In the vertical, a uniform grid size, z, of20mup to approximately 2 km is used, above which the grid is stretched to the model top at 3 km. Stevens et al. also used a 20 m grid while GL used 40 m. Stevens et al. showed significant sensitivity of the cloud cover to resolution. Therefore, higher resolution simulations are also performed here, with x = 25 m and z = 12.5 m below 2 km, with the same domain size. The simulations with stronger inversions also tend to have sharper inversions, which therefore require a finer grid in order to be resolved. For this reason the set 4 simulations (with θ v = 9 K) are only performed at the enhanced resolution. 3. Results Figure 1 shows profiles from the start (dotted lines) and final three hours of the GL2 cumulus simulation, from which these simulations are derived, together with three of the standard simulations with κ = 0.7 (with θ v = 3, 4.5 and 6 K (requiring κ = 0.7 and θ v = 9 K results in negative humidity above the inversion, and so this simulation is omitted). Despite the changes to the initial profiles in the cloud layer and the differences above the inversion, the cloud and sub-cloud layer θ profiles in these simulations all tend towards

5 FACTORS INFLUENCING CUMULUS CLOUD AREA 945 those of GL2, consistent with a mean state in balance with the applied forcing the surface fluxes are also negligibly different. Even after 12 hours, though, the q t profiles and thence relative humidity (RH) are still somewhat moister than GL2 in the cloud layer. There is, however, a gradual drying trend (not shown), so this would appear to be indicative of a slower adjustment time-scale for moisture compared with temperature. It can also be seen from Figure 1 that varying the strength of the capping inversion (while keeping κ constant) has almost no impact on the evolution within the boundary layer. The only difference is that the stronger inversions are also sharper, presumably because cumulus clouds can penetrate the free atmosphere less deeply. The stronger inversion also restricts the mass entrained into the boundary layer (note from Table I the decrease in subsidence forcing required to maintain the inversion close to 1500 m), but note that keeping κ fixed results in a drier free atmosphere. Thus, for this subset of simulations, as the cloud-top entrainment rate reduces with a strengthening inversion, the entrained air becomes drier. The net result is apparently almost exactly the same ventilation of the boundary layer and so the same (small) cloud fraction. Figure 2 shows the impact on the mean profiles of varying κ for the simulation set 2, with θ v = 4.5 K.As κ decreases (the humidity above the inversion increases and θ l also decreases slightly) so the RH, the cloud water content and cloud cover increase just under the inversion. In other words, as dilution of the boundary-layer humidity through entrainment reduces, as measured by κ, so the stratocumulus layer becomes more persistent and extensive. As a result there is more cloud-top radiative cooling (not shown) that drives more turbulent mixing in the stratocumulus layer, as seen in the turbulent kinetic energy (TKE) profiles in Figure 2. This increasing TKE is then the likely source of the additional entrainment across the inversion during the simulation as a whole. This can be deduced from the significant increase in subsidence forcing required to maintain the inversion close to 1500 m as the cloud cover increases (see Table I). Although not investigated in any detail in this study, it is likely that the simulations with higher κ initially entrain more rapidly, depleting water from the initial stratiform cloud layer until an equilibrium state is achieved. The same is also found to be true for the two other values of θ v tested here, and the complete set of results is encapsulated in Figure 3. This shows the time evolution of the shaded cloud-area coverage (the percentage of grid columns containing any cloud). Independent of the inversion strength, then, for small values of κ ( 0.3) the cloud cover remains close to 100%, while for large κ ( 0.6) the cloud rapidly decreases to a typical shallow cumulus cloud cover of around 20%. For intermediate κ ( 0.5), cloud cover decreases more steadily down to the typical shallow cumulus amount. Figure 1. Initial profiles (dotted lines) and averages over the last three hours from GL2 (white) and, in black, the three standard simulations with κ = 0.7 and θ v = 3 (solid), 4.5 (dashed) and 6 K (dash dotted).

6 946 A. P. LOCK Figure 2. Profiles averaged over the last three hours from set 2 of the simulations (with θ v = 4.5) andκ = 0.1 (solid), 0.2 (dotted), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted). Figure 3. Time series of shaded cloud cover for the standard simulations. Colours denote the initial value of θ v (6 K black, 4.5 K grey, 3 K white) and the initial values of κ are 0.1 (solid), 0.2 (dotted), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted) Impact of resolution Stevens et al. (2001) reported a significant dependence of cloud cover on resolution in their simulation of tradewind cumulus under a strong inversion, so simulations from set 2 (with θ v = 4.5 K) have been performed with a finer grid. In Figure 4, significant differences can be seen to occur for values of κ = 0.2 and 0.3, with the enhanced resolution leading to a substantially greater average cloud cover in the stratocumulus layer. Where this happens, it leads to greater cloud-top radiative cooling (not shown), stronger turbulence within the stratocumulus layer and a sharper and higher inversion (note the subsidence and other forcings are unchanged). Note that the time-averaging means that the maximum of the cloud fraction profiles somewhat underestimates the shaded cloud area, which is always greater than 90% for κ<0.3 at high resolution. This sensitivity to resolution is the same asthat reportedinstevenset al. Their simulation had κ 0.4 (with θ = 9Kand q v = 5.75 g kg 1 ), and such an intermediate value of κ is found here to be in the most sensitive part of the parameter space.

7 FACTORS INFLUENCING CUMULUS CLOUD AREA 947 Figure 4. The impact of resolution on set 2 simulations ( θ v = 4.5): profiles averaged over the last three hours and time series of shaded cloud cover at standard resolution (white lines) and enhanced (black lines), with initial values of κ = 0.1 (solid), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted) Impact of the strength of the cumulus transport The simulation GL4 had much stronger surface fluxes, and 60% larger cloud-base mass flux than GL2, and so essentially had stronger cumulus convection. To test sensitivity to the cumulus strength, then, simulations based on GL4 were performed (set up using the same methodology as in the standard simulations) with θ v = 4.5 Kandκ = 0.2, 0.3 and 0.5. The enhanced strength of turbulent mixing with GL4 can be seen from the TKE profiles in Figure 5, which are some 50% greater than they were for GL2 (compare with Figure 4). The time series in Figure 5 shows that, at standard resolution, the cloud cover decreases to no higher than 50%, even for small initial κ (κ = 0.5 still gives essentially shallow cumulus, as for GL2), but at enhanced resolution cloud cover of close to 100% is seen for κ 0.3. In contrast to the results from the more weakly forced cumulus simulations shown in Figure 4, where the cloud cover can remain very close to 100% throughout, there is an initial drop during the spin-up phase of the simulations (during the first hour) that is particularly severe at the standard resolution. At enhanced resolution, the small κ simulations gradually recover virtually complete cloud cover, but it is possible that the initial burst of vigorous cumulus clouds makes this more difficult at the standard resolution. This difference will be returned to in section Impact of precipitation processes Precipitation might be expected to have an impact primarily through changing the vertical distribution of the water transported by the cumulus clouds. It will reduce the transport of liquid water to the inversion, presumably tending to reduce cloudiness there directly. It will also change the distribution of latent heating in the cloud layer, potentially stabilizing the cloud layer and thus affecting the dynamics of the cumulus clouds. Both processes appear to be operating for set 2 simulations in Figure 6, where the precipitation flux is of similar magnitude to the turbulent flux of liquid water (cloud plus rain) but will act in the opposite sense, depleting cloud water. The precipitating simulations can therefore be seen to have higher θ in the upper part of their cloud layer, suggestive of reduced evaporative cooling. They also have higher q v at the cloud top, however, and the inversion is significantly lower (largescale divergence has been kept the same), indicating reduced entrainment of dry air. This result is consistent with the modelling study of precipitating cumulus by Stevens and Seifert (2008). In none of the precipitating simulations does the time-averaged cloud cover exceed 20%, and so the cloud-top radiative cooling is all but removed (not shown). The warming in the θ profiles, then, arises from a combination of processes.

8 948 A. P. LOCK Figure 5. The impact of resolution on the GL4 simulations ( θ v = 4.5): profiles averaged over the last three hours and time series of shaded cloud cover at standard (white) and enhanced (black) resolution, and with κ = 0.2 (dotted), 0.3 (dashed) and 0.5 (dash dotted). Sensitivity to resolution was also tested in these precipitating simulations for set 2: see Figure 7. At enhanced resolution, the cloud cover in simulations with precipitation is much more similar to the equivalent nonprecipitating ones (compare with Figure 4), although the inversion height is still somewhat lower (not shown). For initial κ 0.3, a stratocumulus layer is produced at the cumulus top, with cloud cover generally greater than 80%. At enhanced resolution, then, the effects of precipitation on the cloud cover are small. The substantial sensitivity at coarser resolution, however, is still noteworthy and will be returned to in section 4. Results from non-precipitating simulations from set 2 with cloud water sedimentation included were very similar to the standard simulations, although they did show slightly increased water contents in the stratiform layer when one formed (not shown). This indicates a slight reduction in entrainment across the cloud top when sedimentation of cloud water is included, consistent with the results of Bretherton et al. (2007), but with no quantitative impact on the cloud cover Impact of cloud-top radiative cooling It can be seen from Table I that the simulations with smaller κ required a larger divergence to stop the inversion from rising rapidly. This additional entrainment is generated by turbulence in turn generated by radiative cooling at the top of the stratiform cloud layer. As introduced in section 2, whether this additional mixing plays a role in determining the cloudiness is investigated through additional simulations from set 2 where the potential radiative cooling is halved and switched off completely, via the input parameter F 0. In order to maintain the inversion at a similar height when this additional entrainment is reduced, it was found that D needed to be reduced. For set 2 simulations with F 0 halved, values of D = 4, 4, 3.5, 3 and 2 ( 10 6 s 1 ) were used, for the initial κ values from 0.1 to 0.7, respectively. For F 0 = 0, D was set to s 1 for all simulations. These settings then give inversion tops that are reasonably consistent between all the set 2 simulations, namely within 50 m of 1720 m by the end. The impact of reducing F 0 is found to be to weaken the TKE in the stratocumulus layer significantly, and the stratocumulus also becomes significantly thinner (not shown). The cloud cover also reduces somewhat (see Figure 8), but the sensitivity to the initial κ is the same as in the control simulations. This suggests that, while the radiative-cooling feedback can amplify the amount of cloud, it has little influence on the initial formation of a significant cloud fraction at the inversion.

9 FACTORS INFLUENCING CUMULUS CLOUD AREA 949 Figure 6. The impact of precipitation: profiles from set 2 simulations ( θ v = 4.5) with (black) and without (white) precipitation, with κ = 0.1 (solid), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted). Figure 7. The impact of resolution on set 2 precipitating simulations: time series of shaded cloud cover at standard resolution (white) and enhanced (black), with initial κ = 0.1 (solid), 0.2 (dotted), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted). 4. Discussion In the Introduction, it was asked which factors might exert some influence over the cloud cover at the inversion in marine shallow cumulus. LES has been used to look for sensitivity to a variety of factors, such as the atmosphere s thermodynamic and moisture structure, the strength of transport by the cumulus clouds, precipitation processes, and the strength of cloud-top radiative cooling, as well as sensitivity to the LES resolution. The results presented in the previous section have shown some sensitivity to most of these processes, with the interesting exception of the buoyancy jump across the inversion. Only when the static stability of the inversion is combined with the humidity jump through the CTEI parameter, κ, does some pattern start to emerge, suggesting that the relative dryness of the mass of air that can be entrained is important. Figure 9 summarizes the results, plotting shaded cloud cover against the initial value of the CTEI parameter, κ, for all simulations reported in this article. In the majority of cases, and particularly at enhanced resolution, the initial κ provides some indication of when stratocumulus is likely to form at the cumulus cloud top, typically for κ 0.4. However, several groups of simulations fail

10 950 A. P. LOCK Figure 8. The impact of varying cloud-top radiative cooling on time series of shaded cloud cover from set 2 simulations ( θ v = 4.5) with κ = 0.1 (solid), 0.2 (dotted), 0.3 (dashed), 0.5 (dash dotted) and 0.7 (dash triple-dotted). to fit this pattern, giving smaller cloud cover at low values of κ. In particular, cloud cover is sensitive to precipitation (squares in Figure 9), the strength of the underlying cumulus convection (triangles) and cloudtop radiative cooling (crosses). The previous studies of stratocumulus by MacVean (1993), later developed by MacVean and Bretherton (personal communication) and Yamaguchi and Randall (2008), concluded that CTEI was best not thought of as a threshold that once crossed would imply immediate cloud break-up. Rather, the fate of the cloud layer depended on the balance between competing processes. Once the buoyancy reversal threshold (of κ 0.23) is passed a new process must be added to the mix, but the time-scales for cloud breakup are typically sufficiently slow that other, cloud-building processes, can dominate. Only as κ 0.7 was approached, they found, did the CTEI time-scale become sufficiently short that this process dominated and the cloud rapidly broke up. So, does the sensitivity to precipitation, cumulus strength and cloud-top cooling seen here imply that these processes dominate over CTEI in determining the cloud evolution? As was noted in section 3, these sensitivities were associated with drifts in the cloud-layer profiles, implying drifts in the jumps across the inversion and hence κ. Figure 10 shows shaded cloud cover, now plotted against contemporary, rather than initial, values of κ. Now all the data have collapsed on to a relatively narrow curve, implying that only a very limited range of cloud covers can be supported in these simulations at a particular value of κ. The changes in the cloudlayer profiles must imply sometimes subtle changes in the inversion jumps such that κ drifts. The cloud cover must then respond to these changes on a relatively short Figure 9. Shaded cloud cover plotted against the initial value of κ for standard simulations (diamonds), with rain (squares), based on GL4 (triangles) and with reduced F 0 (crosses). White (black) symbols denote standard (enhanced) resolution simulations and symbol sizes increase with time. The initial conditions and subsequent three-hourly averaged data are shown. time-scale (certainly smaller than the three-hour averaging period). For example, it can be seen that the impact of precipitation at coarser resolution is to move the simulations towards greater κ (there are no squares in the right-hand panel of Figure 10 with κ<0.2) and thence to a state of reduced cloud cover. The same was also seen when larger subsidence rates were applied in the initially small κ simulations the increased warming and drying at the inversion base again increased κ to the point where it appears impossible for stratiform cloud to form.

11 FACTORS INFLUENCING CUMULUS CLOUD AREA 951 Figure 10. As Figure 9. but shaded cloud cover is plotted against contemporary values of κ and the first three hours are excluded. data, there is some tentative support from the aircraft measurements made during the First Lagrangian of the Atlantic Stratocumulus Transition Experiment (ASTEX). Values of κ can be calculated from the data in Table 1 of de Roode and Duynkerke (1997), and show an increase from values in the stratocumulus regime of around 1 (flights 2 and 3) to 0.6 in the final flight, which was made in a very patchy cloud pattern of cumulus clouds penetrating the thin and broken stratocumulus above. These observations are, then, at least consistent with Figure 10. Similarly, observations made in trade cumulus clouds with small cloud cover, such as those compiled by Kuo and Schubert (1988), show large values of κ 0.7. On the other hand, Kuo and Schubert also show several observations of persistent solid stratocumulus with κ also approaching 0.7. As discussed above, however, it has been suggested that whether the stratiform cloud layer breaks up depends on the balance between competing processes. A possible explanation for a more rapid transition as κ increases, then, is that the additional generation of cloud-top mixing as shallow cumulus clouds penetrate the inversion alters this balance compared with the stratocumulus regime, in favour of more rapid evaporation of the cloud. Figure 11. As Figure 10, but shaded cloud cover is plotted against contemporary values of RH. It was hypothesized above that κ works well as a predictor of cloud cover because it depends both on the static stability of the inversion, and hence on the rate of mixing across it, and on the humidity jump, and hence the dryness of the entrained air. The static stability itself has already been shown to be inadequate as a predictor of cloud cover (see Figures 1 and 2). Figure 11 shows also that it is not simply the dryness of the air in the free atmosphere (measured as RH, the jump in RH across the inversion) that dictates cloud cover, with the whole range of cloud covers being possible for any free atmospheric RH. What observational support is there for this simple dependence of cloud cover at the capping inversion of shallow cumulus on κ? Despite the paucity of relevant 5. Conclusions There are complex interactions occurring at the inversion capping shallow cumulus clouds, between cumulus turbulent transport, cloud-top radiative cooling, stratocumulus overturning and cloud-top entrainment, microphysical processes and large-scale subsidence. Out of all this emerges a cloud cover that is crucially important to the radiative fluxes and so to a host of applications, from local forecasting to the cloud feedback in climate simulations. From a wide range of LES, a simple dependence of the cloud cover on the CTEI parameter, κ, has been identified. For κ 0.2, where buoyancy reversal does not occur, cloud covers of 80% or more are seen. This suggests that without a reduction in the buoyancy of entrained air through evaporative cooling, and the associated additional turbulent mixing, an extensive stratocumulus sheet is likely to form. For κ 0.5, only shallow cumulus clouds with cloud cover around 20% are formed, again indicating how the generation of significant negative buoyancy by mixing at cloud edges and subsequent evaporative cooling is crucial to the evolution of these clouds (as has recently been noted by Heus and Jonker, 2008). Only for intermediate κ (0.2 κ 0.5) is intermediate cloud cover seen, where individual cumulus clouds begin to spread into a stratiform layer but this layer cloud subsequently dissipates. Consistent with previous studies, no abrupt transition is seen at a particular value of κ, and the cloud cover decreases smoothly as κ increases. Processes that might be considered important to the cloud cover were investigated. Particularly sensitive was the impact of allowing the clouds to precipitate, which led to a significant reduction in cloud cover, although the effect was much reduced at higher resolution. Even

12 952 A. P. LOCK at coarser resolution, it is found that the impact of the precipitation is to change the structure of the cloud layer such that the jumps across the inversion, and thence κ, change. The cloud cover is then found to respond to this new κ on a short time-scale (much shorter than the threehour averaging period used). Whilst it would appear crucial to an accurate representation of cloud cover in numerical weather prediction (NWP) and climate models to represent this sensitivity to κ correctly, it is not proposed here that such a dependence be included explicitly. Rather, it would be hoped that the turbulent mixing, microphysics and radiation parametrizations would in themselves be sufficiently general and accurate to be able to reproduce the dependence observed here. However, there are several reasons to be pessimistic. Firstly, as demonstrated with the LES here, the stratiform cloud forms in a very thin layer beneath the capping inversion and so high vertical resolution is needed to resolve it. Typically the stratocumulus layers seen here were m thick. This is much thinner than traditional GCM vertical grids, especially those used for climate studies, but is becoming within reach at least of many NWP forecast models. Secondly, there are serious numerical difficulties in accurately and consistently representing the fluxes across capping inversions from different processes (turbulence, radiation, subsidence) on finite-difference grids that few models have attempted to address (Suarez et al., 1983; Grenier and Bretherton, 2001; Lock, 2001 being exceptions). Finally, the process of buoyancy reversal is only represented explicitly in a few cloud-turbulence parametrizations (e.g. Lock et al., 2000; Grenier and Bretherton 2001). 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