The sensitivity of stratocumulus-capped mixed layers to cloud droplet concentration: Do LES and mixed-layer models agree?

The sensitivity of a stratocumulus-capped mixed layer to a c h nge in cloud droplet concentration is evaluated with a large-eddy simulation (LES) and mixed layer model (MLM), to see if the two model types agree on the strength of the second a erosol indirect effect. Good agreement can be obtained if the MLM entrainment closure exp licitly reduces entrainment efficiency proportional to the rate of cloud droplet sedimenta tion at cloud top for cases in which the LES-simulated boundary layer remains well mixed, with a single peak in the vertical profile of vertical velocity variance.. To achieve this agreement, the MLM entrainment closure and t he drizzle parameterization must be modified from their observationally-based defaults . This is because the LES advection scheme and microphysical parameterization significantly b ias the entrainment rate and precipitation profile compared to observational best guesses. Befo re this modification, the MLM simulates more liquid water path and much more drizzle at a given droplet concentration than the LES and is more sensitive to droplet concentration, even und ergoing a drizzle-induced boundary layer collapse at low droplet concentrations. After thi s modification, both models predict a similar decrease of cloud liquid water path as droplet conc entration increases, cancelling 3050% of the Twomey effect for our case. The agreement breaks do wn at the lowest simulated droplet concentrations, for which the boundary layer in the LES is not well mixed. Our results highlight issues with both types of model. Poten tial LES biases due to inadequate resolution, subgrid mixing and microphysics must be c ar fully considered when trying to make a quantitative inference of the second indirect effe ct rom an LES of a stratocumulustopped boundary layer. On the other hand, even slight intern al decoupling of the boundary layer invalidates MLM-predicted sensitivity to droplet concent rations.


Introduction
The indirect effect of anthropogenic aerosol on clouds and thereby on the global radiation balance remains a key uncertainty in climate modeling and prediction. In particular, the cloud droplet number, cloud fraction and liquid water path (LWP) of subtropical marine stratocumulus cloud decks respond to changes in aerosol in ways global models struggle to represent. Low aerosol concentrations lead to reduced cloud fraction, more precipitation and a more decoupled and cumuliform character, as seen both from observations of pockets of open cells (Comstock et al., 2005;Stevens et al., 2005a;;Wood et al., 2008) and large-eddy simulations or LES (Stevens et al., 1998;Ackerman et al., 2003;Xue et al., 2008;Savic-Jovcic and Stevens, 2008). High aerosol concentrations produce small cloud droplets that do not sediment out of the entrainment zone, promoting mixing-induced evaporative cooling that enhances cloud top entrainment and can thin the stratocumulus layer (Ackerman et al., 2004;Bretherton et al., 2007). Thus, ship or volcano tracks of enhanced aerosol concentration and reduced droplet effective radius in nonprecipitating stratocumulus often appear to have lower LWP than the surrounding cloud (Coakley and Walsh, 2002;Gasso, 2008); though there are issues of interpretation (Ackerman et al., 2003).
Models are required to interpret and generalize these limited observations. Two types of models that have been widely used are LES and mixed-layer models (MLMs). LES models realistically represent the interaction of turbulence, cloud processes and radiation within the constraints of grid resolution. However, they are computationally complex. Model intercomparisons (Stevens et al., 2005b;Ackerman et al., 2009) show that LES as a group cannot reliably simulate entrainment into Sc through the sharp capping inversion, typically leading to simulated Sc layers with too small LWP. Furthermore, LES microphysical parameterizations for predicting precipitation development have numerous uncertainties (Ackerman et al., 2009).
MLMs have been used for several studies of Sc sensitivity in which long-term behavior and conceptual simplicity have been emphasized (Baker and Charlson, 1990;Pincus and Baker, 1994;Wood, 2007;Caldwell and Bretherton, 2009). MLMs break down when the PBL becomes decoupled, but many Sc-capped boundary layers are fairly well-mixed even in the presence of drizzle (e.g. the SE Pacific EPIC observations of Bretherton et al. (2004). MLMs usually use empirical, observationally-informed, parameterizations of entrainment and cloud microphysical processes in terms of the bulk variables that MLMs predict (e. g. mean cloud base and inversion height). An MLM provides a simpler and complementary view to LES of Sc-capped mixed layers, albeit one with restricted applicability.
The goal of this paper is to test whether LES and mixed layer models respond similarly to changes in assumed cloud droplet concentration in stratocumulus-capped mixed layers, in which both model types might be expected to apply. We view this as a proxy for their response to aerosol perturbations. To this end, we compare sets of identically forced and initialized nocturnal LES and MLM simulations run out 5 days, focusing on the evolution of LWP.
Our study is quite similar to that of Sandu et al. (2009), hereafter S2009. It was conceived independently and in parallel with S2009. They compared 72-hour LES and MLM simulations of polluted and pristine cloud-topped boundary layers, using forcings idealized from a model intercomparison of the diurnal cycle of stratocumulus in the California coastal zone. Our setup is somewhat simpler (no diurnal cycle) and more idealized. S2009 found that an increase in droplet concentration decreased LWP in their LES, but increased LWP in their MLM. They concluded that an MLM is not a useful tool for looking at cloud-aerosol interactions even for relatively well-mixed boundary layers. We reach the more optimistic conclusion that for nearly non-drizzling stratocumulus-capped mixed layers, the sensitivity of LWP to cloud droplet concentration is quite comparable for our LES and an appropriately configured MLM. In the discussion section, we will trace this difference between the studies mainly to the choice of MLM entrainment parameterization. In particular, we will find that it is vital to consistently build into the MLM the response of entrainment to droplet sedimentation and drizzle.

Models and simulation setup
Our simulations are based on the GEWEX Cloud System Study (GCSS) nocturnal nonprecipitating stratocumulus case specifications for single column models (Zhu et al., 2005). This case was idealized from Research Flight 1 (RF01) of the Second Dynamics and Chemistry of Marine Stratocumulus Experiment, DYCOMS-II . This case featured a well mixed stratocumulus-capped mixed layer in which MLM and LES can reasonably be compared with each other and with observations. Salient details and changes to the case specifications are discussed below. 4

LES
The LES used in this study is version 6.7 of the System for Atmospheric Modeling (SAM), kindly supplied by Marat Khairoutdinov and documented by Khairoutdinov and Randall (2003). The Khairoutdinov (M. F. and Y. L. Kogan) bulk microphysics parameterization is used to for conversion between cloud and rain water. Cloud droplet sedimentation is included following Eqn. (7) of Ackerman et al. (2009), based on a log-normal droplet size distribution with a geometric standard deviation σ g = 1.2. A Deardorff sub-grid turbulent diffusivity with prognostic subgrid TKE is used. All simulations use a uniform 25 m horizontal and 5 m vertical grid spacing over a 2.4×2.4×1.5 km domain with doubly-periodic boundary conditions, and an overlying sponge layer in which the vertical grid spacing rapidly coarsens.

MLM
The MLM, described by Bretherton (C. S.) and Caldwell and Bretherton (2009), predicts the mixed layer moist static energy h, total (vapor plus cloud liquid) water mixing ratio q t and inversion height z i . It allows for continuously varying profiles of radiative heating and precipitation flux within the mixed layer. The precipitation flux includes a sedimentation and drizzle component (Caldwell and Bretherton, 2009). The sedimentation flux is related to the liquid water content exactly as in the LES. The MLM entrainment parameterization is based on Nicholls and Turton (1986): where w e is the entrainment rate, w * is a convective velocity computed from the vertical integral of the buoyancy flux, ∆b i is the inversion jump of virtual potential temperature express in buoyancy units, and A is a nondimensional entrainment efficiency, assumed to have the form Here, a 1 = 0.2 is the entrainment efficiency of a dry convective boundary layer, χ * is the mixing fraction of overlying air needed to evaporate all the water out of the mixed layer air 5 at the cloud top, and χ * ∆b 2 is the resulting buoyancy change to the cloudy mixed layer air; it may be positive or negative ('buoyancy reversal'). The evaporative enhancement a 2 = 25 is a nondimensional coefficient empirically chosen to fit observations, following Caldwell and Bretherton (2009). Following Bretherton et al. (2007), the evaporative enhancement is reduced by an exponential factor dependent on an LES-tuned sedimentation feedback parameter a sed = 9 multiplied by the ratio of the mean droplet sedimentation speed w sed near cloud top to the convective velocity w * . The sedimentation speed is computed from the quadratic dependence of the terminal velocity of small droplets to their radius, integrated across the assumed log-normal droplet size distribution. If the cloud-top liquid water content is q li , we can define a volume-mean droplet radius r i = (3ρ a q li /4πρ w N ) 1/3 and where c = 1.19 × 10 8 m −1 s −1 ; we choose σ g = 1.2 as in the LES.
In an MLM, the liquid water profile is adiabatic and linearly increasing with height, and q li is proportional to LWP 1/2 . With typical values for all coefficients in Eq. (3), The self-consistency of the MLM simulations is tested by computing a decoupling indicator, the buoyancy integral ratio or BIR (Bretherton, C. S.). The BIR is defined as the vertical integral of the negative buoyancy flux in sub-cloud layer to the vertical integral of the positive buoyancy flux over the rest of the mixed layer. A BIR exceeding 0.1-0.2 suggests that the boundary layer will not remain well-mixed and the MLM is no longer appropriate Stevens, 2000).

Radiative heating parameterization
The RF01 case specification included an idealized radiative heating parameterization used by both the LES and MLM. The net upwelling radiative flux at each height in a grid column is 6 specified based on the column liquid water path above and below that height. This specification does not include the influence of droplet number on the radiative heating profile through its effect on the droplet effective radius. Larson and Kotenberg (2007) have described how to account for this influence, which may affect entrainment rate by modulating how much radiative cooling occurs within the entrainment zone. The original specification produces no radiative cooling in clear air columns, which unrealistically ignores the emissivity of water vapor. Since some of our LES simulations produced only partial cloud cover, we enforced a minimum average cooling rate of 2 K day −1 cooling rate between the sea-surface and the inversion in each grid column. If the simulated cloud in the column does not produce at least this average cooling (this requires a column LWP of about 10 g m −2 ), the average cooling is brought up to this threshold by adding a height-independent increment of liquid water within the boundary layer for the radiative cooling profile calculation.
In the RF01 case specification, a strongly stratified layer with strong radiative cooling is specified just above the inversion. In a MLM, free tropospheric temperature and humidity profiles are specified all the way down to the inversion, and it is inconvenient to include a freetropospheric layer with extra radiative cooling. Thus, we did not include the RF01-specified layer of enhanced radiative cooling and stratification above the inversion. Instead, we crudely compensated by increasing the inversion temperature jump 2 K from the RF01 specifications and using a linear temperature profile above that.

Initial state, free-tropospheric profiles and surface fluxes
Following the RF01 specifications, the simulations are initialized with a mixed layer with an inversion height z i = 840 m, a total water mixing ratio q t =9 g kg −1 and a moist static energy h = 317.03 kJ kg −1 , producing a stratocumulus cloud of initial thickness 250 m. The largescale subsidence varies linearly with height z with a horizontal divergence D = 3.75×10 −6 s −1 .
As in RF01, we assume a height-independent free-tropospheric humidity q + (z) = 1.5 g kg −1 . As motivated above, we use a linear above-inversion profile By choosing h + (0) = 303.92 kJ kg −1 , we obtain a linear and roughly moist-adiabatic temperature profile that is 2 K warmer than the RF01 specifications at the initial inversion height z i = 840 m. The above-inversion radiative cooling is specified to balance subsidence warming, and the humidity is height-independent, so the temperature and moisture profiles do not drift above the inversion. Following the RF01 SCM case specification, we calculate surface heat and moisture fluxes using a sea-surface temperature (SST) of 292.5 K. The LES wind profile is forced by the specified geostrophic winds, and the surface fluxes are computed using Monin-Obukhov theory and the simulated lowest-level wind speed. The MLM uses a bulk aerodynamic formula with the surface wind speed of V = 7.35 m s −1 specified in the SCM case description and a transfer coefficient C T = 0.001V for heat and moisture.

Results
First, we compare LES of the modified RF01 case described above with four different specified values of droplet concentration N = 150, 50, 30 and 10 cm −3 . The simulations evolve for 5 days. Compared with the 3-12 hour duration of most previous LES cloud-aerosol interaction studies, these longer simulations allow us to more closely approach an equilibrium response of the boundary layer clouds to the perturbation.
Then, identically initialized and forced MLM simulations are compared with the LES results. We derive a modified MLM entrainment closure that better matches the LES results, which are affected by numerical over-entrainment. We compare the response of the LWP to the droplet concentration predicted by the MLM to that of the LES, screening out cases in which the LES produces a decoupled boundary layer. Fig. 1 shows some domain-averaged statistics from the four LES cases. The initial evolution of the N = 150 case can be compared with the RF01 observations, since the observed cloud droplet concentration was 140 cm −3 . The simulated entrainment rate of around 4.5 mm s −1 only slightly exceeds the observed 4 mm s −1 , but the simulated entrainment is maintained by a cloud whose LWP is less than 25 g m −2 , much less than the observed LWP of 50-60 g m −2 . This is an important systematic error of the LES for cloud-aerosol-precipitation interaction, since too thin a cloud will not precipitate as easily and will be more radiatively susceptible to cloud droplet changes.

LES results
In general, the N = 30 and 50 cases behave similarly to the N = 150 case, with little or no cloud base drizzle and a cloud fraction exceeding 0.8. All three cases maintain a high entrainment rate that deepens the boundary layer and raises the minimum cloud base during the simulation.
There are clear trends between these three cases. The N = 30 case has larger LWP and higher convective velocity than the N = 150 case throughout the simulations. The LWP difference develops within the first few hours, when N = 150 has a larger w e and more entrainment drying. In all three case, as the LWP drops below 25 g m −2 , the cloud fraction decreases. This reduces radiative cooling, acting as a negative feedback on the entrainment. Accordingly, the N = 150 simulation has a total radiative flux divergence across the boundary layer which remains about 5% lower than for N = 50 and 30 throughout the simulation (not shown). Therefore, after a day the N = 150 inversion height becomes lower than N = 30; the same crossover is not seen in LWP.
We attribute these trends to droplet sedimentation effects on entrainment (Bretherton et al., 2007). For a given liquid water content, the mean cloud droplet size, and hence the sedimentation velocity, increases as N decreases. More sedimentation decreases the efficiency of entrainment by removing the water droplets from the cloud top, so there is less water to evaporatively cool mixtures of entrained and boundary-layer air.
Specifically, we can define an LES entrainment efficiency by analogy with the MLM entrainment closure, There are only slight differences in the three cases between the evolution of w e , z i , and the buoyancy jump ∆b i (measured between 25 m above and below z i ) . However, the convective 9 velocity w * (determined from the vertical integral of the LES-derived buoyancy flux) quickly drops to a value about 7% lower for N = 150 than for N = 30, with N = 50 lying in between. This is consistent with a roughly 20% larger entrainment efficiency for N = 150 (A LES ≈ 1.2) than for N = 30 (A LES ≈ 1.0). The reduction in convective velocity is accomplished by decreased LWP, which causes cloud thinning and lower fractional cloud cover of the cloud, both of which diminish the vertically-integrated buoyancy production of turbulence. In particular, we stress that a larger entrainment efficiency need not lead to a much larger sustained entrainment rate, but often instead leads to a cloud layer with lower LWP and slightly less turbulence. In this light, we interpret the low LWP bias of the N = 150 simulation compared to observations as evidence that the LES is entraining too efficiently. The N = 10 case is markedly different from the other three. It maintains a lower entrainment rate, a correspondingly lower inversion and cloud base height, and a lower cloud fraction. Initially, this case produces more than 0.2 mm d −1 cloud base drizzle; this reduces to 0.1 mm d −1 as its LWP drops below that of the other cases. The lower entrainment rate is mainly due to reduced cloud cover, which cuts the radiative cooling of the boundary layer. The N = 10 cloud fraction is lower for a given LWP than for the other cases, perhaps because the subcloud drizzleinduced evaporative cooling is promoting horizontal inhomogeneity of the boundary layer. Fig. 2 shows selected profiles averaged over the period 2-2.25 days, using a normalized vertical coordinate z/z i . For N = 30, 50 and 150, the profiles of the adiabatically conserved variables, liquid static energy s l = c p T + gz − Lq l and total mixing ratio q t , are nearly identical and vertically well mixed within the boundary layer. Thus, these cases are reasonable comparisons for a MLM. All three N 's produce a minimal drizzle flux with a maximum within the cloud layer of less than 0.1 mm d −1 . The N = 30 case shows a much stronger sedimentation flux out of the cloud top than the N = 150 case. Due to the lower entrainment efficiency, the N = 30 cloud is slightly thicker and the buoyancy flux and vertical velocity variance are slightly higher than for N = 150. The N = 50 profiles closely resemble those for N = 30. The N = 10 case has much more drizzle, a kink in the cloud fraction profile indicating more variability in cloud base, vertical gradients in the conserved variable profiles in the upper part of the boundary layer, a level near cloud base at which buoyancy flux drops below zero, and a double-peaked vertical velocity variance. These are all indicators of a less well mixed boundary layer that a MLM cannot be expected to represent well.  Fig. 1. For N = 150, the inversion deepens in the MLM simulation, but less rapidly than for the LES. The MLM maintains roughly double the LWP of the LES throughout the simulation. Its initial LWP is more consistent with the RF01 observational estimate of 60 g m −2 than is the LES, because the MLM entrainment closure has been empirically tuned to match observations. The MLM buoyancy flux profile (magenta line in Fig. 2e) is qualitatively similar to the LES (red line), but with more buoyancy flux in the cloud layer and a larger convective velocity w * . Because the MLM has a lower entrainment efficiency than the LES, its entrainment rate is smaller despite its larger w * .

MLM results with standard entrainment closure
The N = 50 MLM simulation generates enough cloud base drizzle to stabilize the boundary layer, significant reducing buoyancy production of turbulence and entrainment. Thus, initially the boundary layer moistens, the cloud thickens and LWP increases faster than for N = 150, though ultimately a steady state is approached. For N = 30, the drizzle feedback is even stronger and leads to a runaway decrease of entrainment and increase of drizzle and LWP. Entrainment drops to zero after 8 hours, after which the MLM is clearly inapplicable and no results are plotted. Well before this, at about 5 hours, the BIR has climbed above 0.2, indicating significant negative subcloud buoyancy fluxes that imply that the mixed layer assumption is no longer self-consistent.
For N = 10, the MLM run on the initial state produces 6 mm d −1 of cloud base drizzle. The resulting subcloud cooling completely stabilizes the mixed layer, preventing convection and entrainment and rendering the MLM inconsistent from the start. Hence no N = 10 results are plotted on Fig. 3.
In summary, the sign of the response (an increase in LWP for a decrease in N ) is the same for the MLM as for the well-mixed regime of the LES. However, the MLM simulations predict higher LWP and stronger feedbacks between cloud thickness, drizzle and droplet concentration 11 than do the LES simulations.

Retuning the entrainment closure to the LES
We have suggested that the discrepancies between the MLM and LES simulations may result from the LES entraining too efficiently compared to the observations underlying the MLM entrainment closure. For dry convective boundary layers, LES, observations, and our entrainment closure all agree that the entrainment efficiency A ≈ a 1 = 0.2 (Stull, 1976). The choice a sed = 9 was based on prior simulations with our LES (Bretherton et al., 2007). In this section, we will retune the remaining nondimensional parameter in our entrainment closure, the evaporative enhancement a 2 , to fit our LES results. We then test if this produces better agreement between the MLM and LES. We start by computing the entrainment efficiency A LES for each hour of the LES simulations using Eq. (7). Our MLM entrainment closure predicts that The left hand side, the sedimentation-modified evaporative enhancement factor, isolates a 2 . The sedimentation ratio w sed /w * on the left hand side is computed from the LES output based on LWP and N using Eq. (5), exactly as in the MLM. Representative values are w sed /w * = 0.012, 0.024, 0.035 and 0.067 for N = 150, 50, 30 and 10. All terms on the right-hand side can also be computed from LES output. The saturation mixing fraction χ * is computed as in the MLM by using the hourly-mean LWP to infer an adiabatic cloud-top liquid water content and combining this with the inversion jumps in q t and s l . Fig. 4 shows a scatterplot of computed hourly averages of the right hand side vs. the relative evaporative cooling potential c 1 = χ * (1−∆b 2 /∆b i ). As anticipated in the entrainment closure, the results segregate by droplet concentration, and do not show residual dependence on c 1 . The horizontal lines show the left hand side with a 2 = 110, a sed = 9, and the values of the sedimentation ratio w sed /w * given above; this is a reasonable fit to the four simulations. It is reassuring that the choice a sed = 9 made by Bretherton et al. (2007) based on the same LES (but a different set of cases) also fits the sensitivity of our simulations to sedimentation efficiency. The vertical scatter about these lines is due to uncertainty in estimating parameters and hourly entrainment rate variability unexplained by our entrainment closure. Note that the LES-tuned a 2 , although it works across our simulations, cannot be expected to be a universal constant. It depends on the numerical formulation of the LES, its grid spacing near the cloud top and possibly the thermodynamic jumps at the inversion (insofar as they affect the subgrid and numerical diffusion). In the next section, we rerun the MLM with a 2 increased to 110 to see if it better matches the LES results.

MLM simulations with a 2 = 110
Fig. 5 shows time series from the MLM run with a 2 = 110 for the three cases N = 30, 50 and 150 which the LES predicts are well mixed. In all cases, the evolution of the LWP over the first two days is much closer to the LES than in the case a 2 = 25. This shows the control of entrainment efficiency on cloud thickness. Nevertheless, the N = 30 simulation has much more cloud base drizzle and less entrainment than their LES analogues. This initiates a drizzleinduced boundary layer collapse by the same entrainment-mediated mechanism we saw in the a 2 = 25 MLM simulation.
The modified entrainment closure greatly improves the agreement of MLM and LES as long as the MLM predicts less than 0.15 mm day −1 of cloud base drizzle. Once the MLM-simulated cloud starts to drizzle even a little, the MLM starts to diverge from the LES. This difference can be traced to the difference between LES and MLM in simulated drizzle for a given amount of LWP. The MLM turns cloud base drizzle on gradually as the 1.75 power of LWP/N .
This is qualitatively similar to a similar fit P cb ∝ LWP 3.7 N −2.3 obtained by Geoffroy et al. (2008) from an LES with a related bulk microphysical scheme. Compared with the MLM drizzle parameterization (the dashed red line on the figure), our LES has a more threshold-13 like behavior, with very little cloud base drizzle at low LWP/N . This makes the MLM more susceptible than the LES to reduction of entrainment by drizzle at moderate LWP.

MLM simulations with a 2 = 110 and LES-tuned cloud base drizzle
If we use Eq. (9) in place of the default MLM cloud-base drizzle parameterization, in addition to using the LES-tuned entrainment parameter a 2 = 110, the resulting MLM simulations of LWP and cloud-base drizzle for N = 30, 50 and 150 (Fig. 7) match the LES results more closely. Now the MLM, like the LES, does not exhibit drizzle-induced boundary layer collapse even for N = 30, remaining self-consistently coupled with BIR< 0.05 over the 5-day simulation period. The sensitivities of LWP and w * to N are comparable in the two models. Both have a 20-30% increase in LWP and a 5-10% increase in w * from N = 150 to N = 30. The LES-tuned MLM also reproduces key LES vertical profiles fairly well, as shown in Fig. 2  for N = 150. This is particularly noteworthy for the liquid water profile (Fig. 2c) and buoyancy flux profile (Fig. 2e), which are quite sensitive to both vertical and horizontal inhomogeneity in boundary-layer air properties. The LES buoyancy flux profile is rounded off within the cloud layer compared to the MLM because the MLM assumes 100% cloud cover above the mean cloud base and no cloud below this level, while the LES cloud fraction transitions more gradually due to the range of saturation levels between the updrafts and downdraft. Nevertheless, the MLM captures the overall structure of the LES buoyancy flux profile quite well.
The MLM does not show as much sensitivity of inversion height to N as does the LES. We interpret this as due to cloud fraction-radiation interaction. The separation of LES inversion heights for different N occurs between 1.5 and 2 days, when the cloud in the N = 150 simulation becomes thin and the cloud fraction decreases to 80%, reducing the boundary layer radiative forcing for turbulence. A similar sensitivity cannot occur in the MLM, where cloud fraction must be 100% if any cloud is present.
For N = 10, the MLM still drizzles enough initially to undergo a quick evolution to high BIR and then zero entrainment. Since the MLM immediately diagnoses decoupling (initial BIR = 0.25), this is consistent with the decoupled boundary layer simulated by the LES.
The MLM drizzle parameterization is an empirical fit encapsulating observed relationships, and the LES microphysics can be significantly biased compared to observations, as other intercomparisons have shown (Wyant et al., 2007). Thus, as with entrainment, the LES predictions are arguably less plausible than the MLM as long as the boundary layer remains well mixed.

Discussion
Our results can be used to estimate the relative importance of the second and first indirect effects in our simulations, which Wood (2007) quantified in terms of an indirect effect ratio. For a well-mixed stratocumulus-capped boundary layers, Wood's analysis implies that This formula is strictly applicable to a MLM, in which the cloud is horizontally homogeneous, but should also be a reasonable estimate for the LES in a well-mixed, nearly fully cloud-covered regime. Wood focused on MLM simulations with significant drizzle and found that in that case, R IE is typically positive (i. e. second indirect effect reinforcing the first) and can exceed two. He also noted that R IE is particularly sensitive to the simulated cloud base, and becomes negative for his parameter choices when the simulated cloud base exceeds 400 m (as in our simulations). Our LES and MLM simulations of nearly nondrizzling stratocumulus have negative R IE , since LWP decreases as N increases. The MLM R IE depends on whether its parameters are tuned to observations or to LES. Here we compare the LES-tuned MLM to the LES simulations to test whether a suitably tuned MLM can quantitatively reproduce the R IE of the LES. Comparing N = 30 to N = 150 (δ ln N = 1.6), δ ln LWP ≈ −0.2 for the MLM and -0.3 for the LES from 16 hours onward, once the simulations have gone beyond their initialization transients. This implies an R IE of -0.3 (MLM) or -0.5 (LES) for these simulations, i. e. a significant partial cancellation of the first indirect (Twomey) effect by the LWP changes associated with sedimentation feedbacks. Our results support the use of an MLM to estimate R IE in this regime, as long as sedimentation feedback is included in the entrainment closure, the MLM-predicted buoyancy flux remains positive throughout the subcloud layer, and the LWP exceeds 20 g m −2 ( a rough threshold for cloud cover to remain close to 100%).
For the cases simulated here, the observationally-tuned MLM would have a more negative R IE (at least before the BIR criterion predicts decoupling) than the LES-tuned MLM, due to the higher propensity of the observationally-tuned MLM to drizzle and hence a stronger microphysical feedback on entrainment. Further, this more negative R IE may actually be more realistic than the LES results, because of the LES entrainment and microphysical biases we have noted. That is, we should be wary of current LES predictions of R IE in stratocumulus regimes.
Our results contrast with those of S2009, who found that their MLM tended to produce a thicker cloud if droplet concentration is higher. We believe that the difference lies mainly in their choice of entrainment closure. Crucial physical feedbacks built into our entrainment closure were not included in theirs.
Much of S2009's paper is an excellent tutorial of how to evaluate whether a MLM is an adequate representation of an LES simulation of a stratocumulus-capped boundary layer. S2009, like us, tried to carefully select an entrainment closure that matched their LES. They selected from a set of entrainment closures that have been developed for use in real stratocumulus cloudtopped mixed layer. They selected the original Nicholls-Turton closure (with no sedimentation feedback and a 2 = 60) because it gave a prediction of w e that had the smallest bias averaged over all of their cases compared to the LES, while predicting the temporal and case-to-case variation of entrainment rate comparably well to other closures that they considered (see their  Table 4). Other closures, including a variation of the one used in this paper, better predicted the case-to-case variation of entrainment rate but were rejected because they produced a mean entrainment rate bias compared to the LES. This is an appealing approach, but it makes the tacit assumption that their LES gives an unbiased representation of a stratocumulus-capped boundary layer and hence should satisfy the same entrainment closure as derived from observations. As we showed in Section 3.3, our LES entrains too efficiently and clearly does not satisfy this assumption. We must use a 2 = 110 in our version of the Nicholls-Turton closure to match the LES results. Recent observational studies that have evaluated the Nicholls-Turton closure adopted by S2009 using modern observations have concluded that one should take a 2 = 15-30 Caldwell et al., 2005). This is much less than the a 2 = 60 originally suggested by NT86 on the basis of a very small set of aircraft observations of stratocumulus, some of which were not even in mixed layers. Viewed from this perspective, S2009's LES also appears to entrain too efficiently compared to observations, requiring an a 2 = 60 that is more than twice as large as the observational consensus (though not as extreme as the a 2 = 110 we require to fit the SAM LES). Overefficient entrainment is an almost universal feature of LES models at current resolution, as shown by their underestimation of LWP in the GCSS RF01 intercomparison (Stevens et al., 2005b).
Given this bias in LES entrainment, it is no surprise that observationally-based entrainment closures tend to underestimate the LES entrainment rate, as seen in the Bretherton et al. (2007) and Lilly and Stevens (2008) closures of S2009's Table 4. This does not mean those closures should be rejected. Instead, if our goal is to interpret an LES simulation using a mixed layer model, we may want to use different, retuned versions of those closures that match the LES, as we have done in this paper. Unfortunately, some aspects of entrainment closure, such as the sensitivity of entrainment efficiency to droplet sedimentation, are very hard to constrain using current observations (perhaps ship tracks could be used for this purpose), rendering LES a crucial tool for understanding such effects. The sedimentation correction to entrainment is the primary mechanism for a second indirect effect for the nearly nondrizzling stratocumulus layers in our LES and MLM simulations. Fig. 8 explores how removing the sedimentation correction to entrainment (i. e. taking a sed = 0 in our MLM) affects the response of the MLM to a droplet concentration change. Without the sedimentation correction, the MLM LWP has almost no sensitivity to droplet concentration at high N or to the suppression of drizzle.
We concur with S2009 that some other effects of droplet sedimentation (such as its effect on the vertical structure of the buoyancy flux profile) are captured in a MLM without a sedimentation correction to the entrainment closure, but these effects clearly have a negligible impact on the simulated LWP in the simulations shown in Fig. 8.
The MLM of S2009 also did not include the effect on w * of drizzle that evaporates below cloud base, another key effect that decreases turbulence and entrainment in our MLM (and in the LES).
This still does not explain S2009's finding that their MLM produces a higher LWP when N is larger. According to their Fig. 8, this effect builds during the night, then goes away during the day. Interestingly, their LES also shows the same diurnal cycle of the LWP difference between high and low N , albeit superposed on a different daily-mean response. This result deserves further study. It may arise from dependence of radiative heating rates on N , an effect included in their study and not in ours.

Conclusions
We find that for a well-mixed stratocumulus-topped boundary layer, an LES and a suitably configured mixed layer model predict a similar decrease of cloud liquid water path as droplet concentration increases. In both models, this effect produces cancellation of 30-50% of the Twomey effect for our case.
However, LES advection schemes and microphysical parameterizations can significantly bias the entrainment rate and precipitation profile compared to observational best guesses. To match LES simulations, the entrainment and drizzle parameterizations of the MLM must be carefully tuned to the LES. This causes the MLM-simulated cloud layer to be thinner and drizzle less compared to a simulation with the observationally-based default MLM parameterizations. In some cases, an MLM simulation with the default parameterizations will produce a drizzling boundary layer which collapses due to drizzle-entrainment feedback, while the LES-tuned MLM simulation produces a thin, nondrizzling cloud evolving toward a steady state. This suggests that potential LES biases must be carefully considered when trying to make a quantitative inference of the second indirect effect from an LES of a stratocumulus-topped boundary layer.
On the other hand, several points must also be kept in mind when inferring the second indirect effect from MLM simulations. First, the MLM is only valid when the boundary layer remains well mixed, as diagnosed using a BIR criterion. Drizzle and insolation both promote decoupling, and the MLM should not be trusted when BIR exceeds 0.1. Second, the MLM entrainment closure plays a crucial role in diagnosing when decoupling is likely as well as in the representation of entrainment-sedimentation feedback, and there is no consensus on the choice of closure. Third, if the simulated cloud gets too thin, then the MLM assumption of 100% cloud cover is no longer reasonable. That is, the transition from a non-cloudy to a cloudy boundary layer is smoother in reality than an MLM allows.
These considerations underline the need for LES with advection schemes that accurately represent entrainment and microphysical schemes that can accurately represent observed relationships between cloud thickness, cloud morphology, and precipitation rate. Recent GCSS intercomparisons (Stevens et al., 2005b;Ackerman et al., 2009) suggest a few LES may be close to achieving these goals. Careful observationally-grounded comparisons of such LES with simpler parameterization approaches such as MLMs and single-column versions of climate models will continue to be needed to underpin our understanding of aerosol indirect effects.