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    Illustration of the sedimentation procedure. Circles denote the parcel centers. Arrows denote the droplet fluxes through the interface. See text for more detail.

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    Vertical profiles of 〈W ′2〉 in RF01 measured in situ (circles with bars) and by radar (circle–dot) (after Stevens et al. 2005a). The dashed line that effectively approximates the observed data was obtained from the LES simulation UCLA-0 configuration with a vertical spatial resolution of 1 m (after Stevens et al. 2003a, 2005a). This curve has been used in the present model. The range of the results obtained using 10 LES models is marked by different levels of shading (Stevens et al. 2005a). The solid line represents the mean values predicted by the LES models. Horizontal dashed lines delimit the cloud area.

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    The lateral structure function measured by Lothon et al. (2005) during flight RF07 at z = 240 m and used in the model for the calculation of coefficients Dm. Crosses indicate the values of the structure function reproduced by the model. One can see that the model exactly adjusts the measured profiles.

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    The horizontally averaged profiles of the mixing ratios (a), (c) measured during (c), (d) RF01 (no drizzle case) and (a), (b) RF07 (drizzle case) and (b), (d) calculated at different time instances in the corresponding simulations. The profiles are plotted with time intervals of 5 min. (b), (d) Initial profiles (dashed) marked as t = 0. Panel (a) is adopted from Faloona et al. (2005), while (c) is adopted from Stevens et al. (2003b).

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    Vertical profiles of liquid water static energy temperature from (a) RF01 (after Stevens et al. 2003b) and model simulations of (b) RF01 and (c) RF07. The good agreement between the profile obtained in the RF01 run and the observations in RF01 is seen. As follows from the comparison with the value of the potential temperature reported for RF07 (Stevens et al. 2003a), the model overestimates the temperature by about 2°C in this run. Profiles are plotted with increments of 5 min. Dashed lines in (b), (c) indicate the initial profiles at t = 0.

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    Size distributions of dry aerosols in RF01 and RF07 as used in the corresponding simulations. The distributions are a composite of the RDMA and the PCAPS probe subcloud in situ measurements.

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    Vertical profiles of the horizontally averaged values of the LWC (a)–(d) measured and (e), (f) simulated in the (a), (c), (e) RF01 and (b), (d), (f) RF07 cases. Simulated profiles are plotted with increments of 5 min. The dashed line in the (e) depicts the mean LWC profile obtained by averaging the results of 10 state-of-the-art LES models used for the simulation of RF01 (Stevens et al. 2005a).

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    The vertical profiles of the concentration of droplets with radius exceeding 1 μm calculated in (left) RF01, (middle) RF07, and(right) using in situ–measured results during RF07. In RF07, drizzle reaches the surface in both the observations and the simulation.

  • View in gallery

    Fields of LWC in the (a) RF01 and (b) RF07 runs; parcels having different LWCs are marked by gray shades. (c) The field of radar reflectivity in RF07 at t = 125 min, when drizzle takes place in RF07. Vertical profiles of horizontally averaged and rms values are presented as well. The cloud depth in RF07 is larger than that in RF01. The high inhomogeneity of LWC at small distances is seen. The radar reflectivity is nonuniform and has a “columnar” structure.

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    The height–reff scattering diagrams calculated in the (left) RF01 and (middle) RF07 runs for a 60–120-min period, and (right) a diagram calculated using the measured DSDs. See text for more details.

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    (top) The Z–LWC diagrams for clouds (left) simulated in the RF01, (middle) simulated in RF07 cases at the nondrizzling stage of cloud evolution (0–20 min), and (right) calculated using measured DSD in nondrizzling cloud DSDs (Krasnov and Russchenberg 2002; Khain et al. 2008). (bottom) The Z–LWC diagrams for clouds simulated in RF01 and RF07 averaged over 4 h of cloud evolution, and calculated using DSDs measured in drizzling and heavily drizzling clouds (see references above). In diagrams calculated from the model, each parcel is denoted by a point marked in the diagram each 5 min. The location of the parcels in the Z–LWC diagram is shown as well: parcels located in the top half of the cloud layer are marked yellow, parcels located in the bottom half of the cloud layer are marked blue, and those located below the LCL are marked green. Lines 1, 2, and 3 denote regimes of diffusion growth, the transition regime, and the “drizzle” regimes, respectively. These regimes correspond to those marked in Fig. 10. Lines A, S, and F for the calculated values show different approximations of Z(LWC) (see Krasnov and Russchenberg 2002; Khain et al. 2008 for more details).

  • View in gallery

    Radar reflectivity fields calculated in (top) RF07, and (middle) the RF07 and (bottom) RF01 nonsedimentation runs in which droplet sedimentation was turned off. Vertical profiles of horizontally averaged and rms values are presented as well. The dramatic effects of sedimentation on the cloud microphysical structure are seen: in the nonsedimentation runs, the maximum values of the radar reflectivity reach 40 dBZ, indicating the formation of raindrops.

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    (a)–(d) The height–reff scattering diagrams and (e), (f) the Z–LWC diagrams calculated in the simulations with no sedimentation: (left) RF01 and (right) RF07. The height–reff scattering diagrams are plotted for periods of 5–60 and 65–120 min. The Z–LWC diagram is calculated for the period 0–60 min. In the no-sedimentation case raindrops form in both simulations, indicating the crucial role of sedimentation in the microphysics of stratocumulus clouds.

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    Time dependences of the area-averaged LWC in the RF07 runs with different realizations of the velocity field of the same statistical properties.

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Investigation of Droplet Size Distributions and Drizzle Formation Using a New Trajectory Ensemble Model. Part I: Model Description and First Results in a Nonmixing Limit

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  • 1 Department of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel
  • 2 International Research Centre for Telecommunications-transmission and Radar, Faculty of Information Technology and Systems, Delft University of Technology, Delft, Netherlands
  • 3 Weizmann Institute of Science, Rehovot, Israel
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Abstract

A novel trajectory ensemble model of a stratocumulus cloud is described. In this model, the boundary layer (BL) is fully covered by a great number of Lagrangian air parcels that during their motion can contain either wet aerosols or aerosols and droplets. The diffusion growth of aerosols and droplets, as well as drop collisions, is accurately described in each parcel. Droplet sedimentation is taken into account, which allows simulation of precipitation formation. The Lagrangian parcels are advected by the velocity field generated by the turbulent-like flow model obeying turbulent correlation laws. The output of the numerical model includes droplet and aerosol size distributions and their moments, such as droplet concentration, droplet spectrum width, cloud water content, drizzle content, radar reflectivity, etc., calculated in each parcel. Horizontally averaged values are calculated as well.

Stratocumulus clouds observed during two research flights (RF01 and RF07) in the Second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS II) field campaign are simulated. A good agreement between the dynamical and microphysical structures simulated by the model with observations is obtained even in the nonmixing limit. A crucial role of sedimentation for the creation of a realistic cloud microphysical structure is demonstrated. In sensitivity studies, the statistical stability of the model is analyzed.

Applications of the model for the investigation of droplet size distribution and drizzle formation are discussed, as is the possible utilization of the model for remote sensing applications.

Corresponding author address: A. Khain, Dept. of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel. Email: khain@vms.huji.ac.il

Abstract

A novel trajectory ensemble model of a stratocumulus cloud is described. In this model, the boundary layer (BL) is fully covered by a great number of Lagrangian air parcels that during their motion can contain either wet aerosols or aerosols and droplets. The diffusion growth of aerosols and droplets, as well as drop collisions, is accurately described in each parcel. Droplet sedimentation is taken into account, which allows simulation of precipitation formation. The Lagrangian parcels are advected by the velocity field generated by the turbulent-like flow model obeying turbulent correlation laws. The output of the numerical model includes droplet and aerosol size distributions and their moments, such as droplet concentration, droplet spectrum width, cloud water content, drizzle content, radar reflectivity, etc., calculated in each parcel. Horizontally averaged values are calculated as well.

Stratocumulus clouds observed during two research flights (RF01 and RF07) in the Second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS II) field campaign are simulated. A good agreement between the dynamical and microphysical structures simulated by the model with observations is obtained even in the nonmixing limit. A crucial role of sedimentation for the creation of a realistic cloud microphysical structure is demonstrated. In sensitivity studies, the statistical stability of the model is analyzed.

Applications of the model for the investigation of droplet size distribution and drizzle formation are discussed, as is the possible utilization of the model for remote sensing applications.

Corresponding author address: A. Khain, Dept. of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel. Email: khain@vms.huji.ac.il

1. Introduction

The precipitating, radiative, and reflectivity properties of warm stratiform clouds strongly depend on the shape of droplet size distributions (DSDs), which can vary substantially at the scales of several tens of meters (Korolev and Mazin 1993; Korolev 1994, 1995). Especially strong changes of DSDs are related to drizzle formation (e.g., Stevens et al. 1998; vanZanten et al. 2005; Petters et al. 2006). The investigations of DSD formation mechanisms, as well as those of drizzle formation, is the main objective of a great number of observational and numerical studies, respectively.

Microphysical properties of stratocumulus clouds were measured by research aircraft in the course of several field experiments: for example, the Joint Air–Sea Interaction project (JASIN; Slingo et al. 1982); the Atlantic Stratocumulus Transition Experiment (ASTEX; e.g., Martin et al. 1994; Albrecht et al. 1995; Duynkerke et al. 1995), the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE; Austin et al. 1995), and the second Aerosol Characterization Experiment (ACE-2; Brenguier et al. 2000a; Pawlowska et al. 2000). Using these data, many useful statistical results have been obtained (e.g., Khairoutdinov and Kogan 2000; Wood 2005). The relationships between aerosol concentration and cloud droplet concentration were found using both the observed data (e.g., Martin et al. 1994) and numerical simulations (e.g., Segal and Khain 2006), drizzle parameterizations were formulated for general circulation models (e.g., Pawlowska and Brenguier 2003), and dependences of drizzle fluxes on the mean cloud depth and droplet concentration (e.g., Gerber 1996; Brenguier et al. 2000b) have been proposed.

At the same time, the fundamental mechanisms of DSD and drizzle formation in stratocumulus clouds are still not well understood. As was mentioned by Stevens et al. (2003a), “Although there exists a modest and growing literature on drizzle in the stratocumulus-topped boundary layer (STBL) some very elementary questions remain, including the actual precipitation rates in marine stratocumulus and their relation to ambient aerosol, cloud thickness, and intensity of turbulence.” This situation is related to the lack of both theoretical understanding and appropriate observed data. It is well known that DSDs are formed under a strong influence of the atmospheric boundary layer (ABL) dynamics. However, aircraft measurements of cloud microphysics were very infrequently accompanied by corresponding measurements of the dynamical (including turbulent) properties of the ABL. Even such an important parameter as the vertical velocity at the cloud base is often not measured. Additionally, the state-of-the-art numerical models produce the ABL dynamical structure by “themselves” using “large scale” soundings as the only source of thermodynamic information.

The unique dataset collected during the DYCOMS-II field study, which took place in July 2001 in the Pacific Ocean near California (Stevens et al. 2003a, b, 2005a, b), contains both microphysical and Doppler radar measurements, which allow one to fill many of the gaps mentioned above. Analysis of the observed data indicated that drizzle may be even more prevalent than previously thought: in only two of the seven flights in comparatively thin maritime stratocumulus clouds was there no evidence of drizzle at the sea surface. For the first time, turbulent characteristics such as magnitudes and vertical profiles of vertical velocity variance, σ2w = 〈W ′2〉, and the skewness, 〈W ′3〉/〈W ′23/2, were evaluated (Stevens et al. 2005a; Lothon et al. 2005). The measured drizzle fluxes turned out to be highly nonuniform in the horizontal (Wood 2005; vanZanten et al. 2005) even under horizontally uniform cloud geometry. It was found that drizzle was much more prevalent in clouds having droplets with an effective radius exceeding about 8.5 μm (vanZanten et al. 2005; Twohy et al. 2005), which agrees with results of Yum and Hudson (2002) and is somehow smaller than 14–15 μm found by Gerber (1996) for heavy drizzle formation and by Rosenfeld and Gutman (1994) and Pinsky and Khain (2002) for triggering the raindrop formation in cumulus clouds.

Large eddy simulation (LES) models have emerged as a powerful tool for simulating the microphysical properties of stratocumulus clouds (e.g., Kogan et al. 1994, 1995; Feingold et al. 1994, 1998a, b; Stevens et al. 1996, 1999; Moeng et al. 1996; Khairoutdinov and Kogan 1999; Khairoutdinov and Randall 2003). A detailed analysis of the microphysical and dynamical properties of LES models is beyond the scope of this paper. Limitations of the state-of-the art Eulerian LES models as concerns their ability to accurately represent microphysical processes and the STBL dynamics were described in detail by Feingold et al. (1998a) and Stevens et al. (2005a), respectively. Analyzing the results of recent intercomparisons of 10 state-of-the art very high resolution LES models in simulations of nondrizzling stratocumulus clouds observed in one of the research flights (RF01) during DYCOMS-II, Stevens et al. (2005a) concluded that “in the absence of significant leaps in the understanding of subgrid-scale physics, the appropriate representation of processes at cloud top can only be achieved by a significant refinement in resolution (up to 1 m)—a refinement that, while conceivable given existing resources, is probably still beyond the reach of most centers.”

These conclusions indicate the necessity of looking for other approaches that allow for accurate simulation and analysis of cloud microphysical processes, even at the expense of the utilization of less general and more idealized conditions as compared to those used by the LES models. In this study, we propose a new model of the STBL. To avoid the problem of the mutual effects of a great number of the factors on the microphysics, the microphysical cloud structure is simulated using the turbulent-like ABL dynamics with observed energetic and statistical (correlation) properties. The concept and detailed description of the method are presented in the next section.

The purpose of the study is to investigate the processes of formation and spatial variability of DSDs at small time and spatial scales under a given dynamical structure of the STBL that is typical of nondrizzling and drizzling clouds. We are going to reproduce the classical concept of the formation of stratocumulus clouds, as a result of the vertical mixing of the stratified ABL caused by eddies of different scales. The questions we address in this study are of a fundamental nature and, in particular, are aimed at the reproduction and explanation of some microphysical results obtained in DYCOMSII. Among these questions are the following: (a) What are the processes determining the shape and spatial variability of DSD? (b) How do droplet sedimentation and turbulent mixing affect the DSD? (c) How does drizzle form in stratocumulus? (d) What mechanisms determine the strong variability of drizzle fluxes? In this paper (Part I), we describe the model and illustrate its ability to reproduce the major properties of stratocumulus clouds under nondrizzle and drizzle conditions in neglecting the turbulent mixing between parcels (no mixing limit).

2. Model description

a. The model concept

To address the questions mentioned above, it is desirable to use the Lagrangian approach known as rich in microphysics (mainly, with regard to the representation of diffusion droplet growth/evaporation). This approach is usually used either in single-parcel models (Bower and Choularton 1993; Pinsky and Khain 2002) or in trajectory ensemble models (TEMs; e.g., Stevens et al. 1996; Feingold et al. 1998a; Harrington et al. 2000; Erlick et al. 2005). The advantage of these models from a microphysical point of view is that the equation for aerosol particles (APs) and droplet growth is solved on a variable mass grid, whose bins follow the size of growing drops. This eliminates the arbitrary distinction between APs and droplets and avoids the utilization of simplified parameterization procedures for droplet nucleation. The utilization of the variable mass grid eliminates artificial DSD broadening caused by the necessity of DSD remapping on the regular mass grid after each time step of the diffusion growth (e.g., Khain et al. 2000).

Usually TEMs consist of two submodels: the LES model and the Lagrangian parcel model. The DSDs are calculated in several hundred of Lagrangian cloud parcels moving within the velocity field calculated by an LES model. The microphysical properties of stratocumulus clouds are determined by the statistical analysis of the DSDs formed in these individual parcels. This approach allows one to simulate the DSD formation in parcels having different histories and to investigate the reasons of DSD variability.

Note, however, that in the state-of-the art TEMs cloud parcels are isolated and separated by significant distances. Correspondingly, droplet collisions and precipitation formation are not treated in the TEMs. As a result, the TEMs were applied only for the investigation of droplet diffusion growth when droplet collisions are ineffective (no drizzle formation). The parcel trajectories in the TEMs are usually calculated only above the cloud base determined by the LES model. Thus, the TEMs did not take into account the changes of the aerosol size distributions and DSDs in the collision–sedimentation–evaporation–new drop activation chain. Parcels in the TEMs are treated as “passive scalar” and do not affect the thermodynamical structure of the STBL. At last, the averaging over separate cloud parcels located at the same height in the TEMs is not equal to the averaging over the horizontal levels within the STBL, which makes the comparison of the TEM results with the observations difficult.

In the present study, we describe a new microphysical model of the ABL that, as we believe, is as accurate as is necessary to attribute all model results to physical mechanisms, and not to numerically induced effects. The specific feature of the model is that Lagrangian air parcels cover the entire ABL area. The parcels can be both droplet free and cloudy. At t = 0, the volumes of the air parcels are assumed equal, and parcels are distributed uniformly over the whole area of the BL. At t = 0, the BL is assumed to be cloud free, so that parcels contain nonactivated APs only. The parcels are advected by the time-dependent turbulent-like flow that is generated by the statistical model described in the next section. The velocity field has preset statistical properties, which can be derived from the radar data observed. The turbulent-like flow also includes large eddies with characteristic spatial scales up to the scale of the STBL depth. Such eddies are typical of the STBL. Additionally, the model is able to describe the dynamics of the boundary layer typical of both pure stratiform as well as stratocumulus clouds.

In ascending parcels crossing the lifting condensation level, some fraction of aerosols activates and gives rise to droplet formation. Thus, there exist nonactivated aerosols and droplets in each cloud parcel. In the course of parcel motion, supersaturation in parcels can increase, which may lead to the nucleation of new droplets and to the formation of bimodal and multimodal DSDs. In the event that supersaturation in a parcel is replaced by undersaturation, for instance in downdrafts, droplets evaporate partially or totally. In the latter case, the cloud parcel turns out to be a droplet-free one containing only wet aerosol particles, including those remaining after drop evaporation. The temperature and mixing ratio in the parcels change as a result of condensation–evaporation. The parcels transport the potential temperature and mixing ratio. As a result of microphysical processes within the parcels, as well as parcel motion, the vertical profiles of the temperature and mixing ratio, as well as DSD and AP size distributions, change in the entire STBL. In this way, the model reproduces the interaction between the subcloud and cloud layers. A great improvement of the approach as compared to that used in the state-of-the art TEMs is that the new model takes into account collision between droplets in each parcel and droplet sedimentation, which allows for the simulation of drizzle formation and its fall to the surface.

Stratocumulus clouds are simulated under a given turbulent-like ABL dynamics with energetic, temporal, and spatial correlation properties (see below), which are assumed unchangeable during a few hours of simulations. We see the justification of this simplification in the following. The structure of stratocumulus clouds having the characteristic time scale from several hours to several days is affected by a great number of factors of quite different time and spatial scales. Such factors as surface fluxes, radiation cooling, cloud-top entrainment, mean vertical velocities, etc., change with characteristic time scales determined by the synoptic situation or daily variations. These time scales are much longer than the time scale of the ABL mixing by large eddies (Stevens et al. 2003a). For instance, the characteristic rate of the cloud depth growth is a few meters per hour (Stevens et al. 2003b). The mean entrainment rate found in the first research flight in DYCOMS-II was about 0.38 cm s−1, and the horizontally averaged vertical velocity was of the same order of the magnitude (Stevens et al. 2003b, 2005a). At the same time the standard deviation of the vertical velocity related to large eddies and turbulent fluctuations is of the order of ∼0.5–0.8 m s−1 (Wood 2005; Stevens et al. 2005a). The eddy turnover times (measured by the ratio between the ABL height and the characteristic convective velocity) are of the order of 10 min (Stevens et al. 1996, 2005a, b), which means that the values of supersaturation, the droplet concentration, the shapes of the droplet spectra, the spatial variability of the DSDs, and other microphysical properties are determined mainly by processes with characteristic scales that are much smaller than those determining the mean statistical properties of the ABL dynamics.

Our idea was to simulate cloud formation under the influence of the ABL dynamics observed, for instance, during particular research flights within DYCOMS-II. We believe that the mean statistical properties of the turbulent-like structure change with the synoptic or daily time scales. Further, this structure can be considered stationary (or quasi stationary) at time scales smaller than the “environmental” time scale.

In this part of our study, no turbulent mixing between adjacent parcels is assumed. It will be shown that the model produces reasonable results even in the nonmixing limit. The effects of turbulent mixing will be analyzed in a future study.

We do not take into account the effects of the microphysics on the dynamical (turbulent) structure explicitly. Instead, we generate a turbulent-like dynamical structure that corresponds to that observed in the cloud-topped ABL. In nature, this dynamical structure is formed under the combined effects of latent heat release, radiation, thermal instability, wind shear, surface fluxes, etc. Assimilating the real dynamics, we implicitly take into account all factors affecting the dynamics. We believe that the microphysical structure of clouds corresponds to the ABL dynamics. Simulation of turbulent-like flows corresponding to different thermodynamic situations in the STBL makes it possible to investigate the effects of the ABL dynamics and aerosol properties on the microphysical structure of stratocumulus clouds.

In spite of the model simplifications, we will show that the model reproduces realistically the process of the cloud’s microphysical structure formation at time scales of several turnover times and the spatial scales of a few kilometers.

b. Model dynamics

1) Motion equations

The velocity field in the ABL caused by eddies of different nature and small-scale turbulence is represented as the sum of a great number of harmonics including those representing large eddies with the scales of the ABL depth. The mean horizontal velocity V0(z) is also taken into account. As shown below, the statistics of the velocity field can be tuned to the observations. The expressions for the vertical W(x, z, t) and the horizontal V(x, z, t) velocity components are as follows:
i1520-0469-65-7-2064-e1a
i1520-0469-65-7-2064-e1b
where Cn, Dm are the coefficients described below; M and N are the numbers of harmonics in the horizontal and the vertical directions, respectively; and L and H are the sizes of the computational area in the horizontal and vertical directions, respectively. The function f (z) is used for tuning the velocity field to the experimental data (see below). Random fluctuations of the velocity field with time are described by independent autoregression series of the first-order amn(t) and bmn(t), which represents a discrete version of the Langevin equation widely used to describe turbulent diffusion (Pope 1994):
i1520-0469-65-7-2064-e2a
i1520-0469-65-7-2064-e2b
where ϑn < 1 are the parameters determining the characteristic correlation time of the velocity harmonics having corresponding wavelengths; ϑn increases with the wavelength (the spatial scale) according to turbulent laws. The first terms on the right-hand side of (2) reflect the persistence of the corresponding harmonic, while the second terms indicate their changes with time related to random fluctuations. In (2), εmn and ηmn are the noncorrelated normal white noise with zero mean values and unit variation, so that
i1520-0469-65-7-2064-e3a
i1520-0469-65-7-2064-e3b
where δij is the Kronecker symbol. The correlation functions Rn(τ) of the autoregression sequences (2a) and (2b) are exponents Rn(τ) = exp(−τ/γn) with the characteristic correlation time
i1520-0469-65-7-2064-e4
The sequences amn(t) and bmn(t) obey the following conditions:
i1520-0469-65-7-2064-e5
where the symbol 〈 〉 denotes averaging over a great number of realizations.

The velocity field (1a) and (1b) has the following properties: (a) It obeys the continuity equation: (∂W/∂z) + (∂V/∂x) = 0. (b) The vertical velocity at the upper (z = H) and lower (z = 0) boundaries is equal to zero W(x, 0, t) = W(x, H, t) = 0. These conditions are widely used in numerical and analytical studies of boundary layer stratocumulus clouds, which in our case do not allow parcels to leave the computational area. (c) The following cyclic lateral boundary conditions are employed: W(x, z, t) = W(x + L, z, t); V(x, z, t) = V(x + L, z, t). (d) The quantities averaged over a great number of realizations of the field depend only on z, which indicates the statistical horizontal homogeneity of the field.

The energetic and correlation characteristics of the velocity field given in (1) are as follows:

  • (a) The mean values obey the conditions 〈W〉 = 0, 〈V〉 = V0(z); the velocity variations can be expressed as (see the appendix)
    i1520-0469-65-7-2064-e6a
    i1520-0469-65-7-2064-e6b
    where the coefficients Dm are normalized to obey the condition ΣMm=1D2m = 1. The variations depend only on the vertical coordinate z.
  • (b) The correlation functions along the horizontal direction are given as (see the appendix)
    i1520-0469-65-7-2064-e7a
    i1520-0469-65-7-2064-e7b
    The correlation functions depend only on Δx = x2x1, indicating that the velocity fields are statistically homogeneous in the horizontal direction.
  • (c) The cross-correlation functions of the velocity components along the horizontal direction depend only on Δx = x2x1:
    i1520-0469-65-7-2064-e8
    As a result, correlations between two velocity components at one and the same point are equal to zero. Thus, the field (1a) and (1b) is stationary and statistically uniform in the horizontal direction. The field obeys ergodic properties; namely, averaging with respect to the realizations is equivalent to averaging over a long enough time period.

The tuning of the model dynamics to the observed data is performed as follows: the velocity field (1) contains three groups of parameters to be calculated from the observations: coefficients Cn(n = 1, . . . , N), Dm(m = 1, . . . M), and the characteristic correlation time γn. The function f (z) is used to tune the profile of the vertical velocity variation 〈W ′2o〉 to the observed data.

To calculate Cn, the observed profile 〈W ′2o1/2 has to be reflected antisymmetrically for z < 0, and expanded into the Fourier series within the interval (−H, H). Then, the coefficients Cn can be expressed as
i1520-0469-65-7-2064-e9a
Then, the function f (z) is calculated as
i1520-0469-65-7-2064-e9b
The correlation properties of the vertical velocity in the horizontal direction are determined by the lateral correlation function BWx) (or the lateral structure function DWx) = 〈[W(x1) − W(x2)]2〉) of the vertical velocity in the horizontal direction. Using the correlation or the structure functions obtained from the observations, the coefficients Dm can be calculated as
i1520-0469-65-7-2064-e10

c. Model microphysics

The microphysics of a single Lagrangian parcel is described in detail by Pinsky and Khain (2002). The microphysics includes the diffusion growth equation used for nuclei and water droplets:
i1520-0469-65-7-2064-e11
where r and rN are the droplet and dry CCN radii, respectively; S is the supersaturation with respect to water; and A and B are the coefficients describing the effects of the surface tension and chemical composition of aerosols, respectively. The equation for supersaturation S is
i1520-0469-65-7-2064-e12
where dql/dt is the change of liquid water content (LWC) due to condensation/evaporation. In Eqs. (11) and (12), F, A1, and A2 are the coefficients that are slightly dependent on temperature (Pruppacher and Klett 1997). The vertical velocity W in the point of the parcel location is calculated by Eq. (1a). The size distribution of the cloud particles (both nonactivated nuclei and droplets) is defined on the mass grid containing 500 bins within the 0.01–1000-μm radius range. The mass of each bin in the mass grid changes with time (height) in each parcel according to the equation for diffusion growth. A small 0.01-s time step is used to simulate adequately the growth of the smallest APs, so that the separation between nonactivated nuclei attaining equilibrium (haze particles) and the growing droplets is simulated explicitly (without parameterization) by solving Eq. (11). The processes of drop condensation and evaporation lead to the change of the mixing ratio and the temperature in each parcel. When calculating the density fluctuations, the Boussinesq approximation has been used.

The precise method proposed by Bott (1998) is used to solve the stochastic collision equation. The collision droplet growth was calculated using a collision efficiency table with a high 1-μm resolution in droplet radii (Pinsky et al. 2001). Drop collisions are calculated with 1-s time intervals. Note that the microphysical equations are solved within both the subcloud and cloudy layers of the BL.

The AP budget is calculated in the model. The APs can exist in two “states”: (a) nonactivated wet APs (haze particles) and (b) dissolved APs within droplets. The mass of APs in droplets does not change during the condensation–evaporation process. Collisions of droplets are accompanied by the corresponding increase of the dissolved AP mass in the drop collectors. We use the Bott (1998) approach for remapping the drop masses formed after collisions on the variable mass (bin) grid formed in the course of the diffusion growth. As a result of such remapping, the drop mass turned out to be separated between two neighboring mass bins of the variable mass grid. The masses of dissolved aerosols have been separated in the same proportion as the pure water. In principle, such remapping may lead to the formation of a mass distribution of the dissolver aerosols into droplets of the same mass. As a result, a two-dimensional drop-dissolved aerosols mass grid should be introduced. This approach, used by Bott (2000), is very accurate, but extremely time consuming. Instead, we calculated the mean mass of the dissolved aerosols per droplet in each bin. As a result, each droplet is characterized by the mass (size) and the aerosol mass dissolved in it. In the simulations, the relative humidity exceeds the humidity of deliquescence (about 70% for NaCl), so that droplet evaporation led to the formation of wet APs. Thus, drop collisions lead to changes in the AP size distribution during their recirculation from the subcloud layer to the cloud layer and backward.

The specific feature of the model is the accounting for droplet sedimentation. If no sedimentation takes place, the parcels are adiabatic; that is, the sum of the water vapor and LWC is conserved within a parcel. The algorithm of the sedimentation is as follows:

  • (a) Since DSDs in different parcels are determined at different variable mass grids, at the first step a remapping of all DSDs in all parcels on one and the same supplemental logarithmical mass grid was carried out. This grid contains 500 mass bins. The remapping scheme conserves the mass and concentration of the droplets. The remapping does not lead to any significant artificial droplet spectrum broadening because of the high resolution of the supplemental mass grid and rare (as compared to the diffusion growth) utilization of sedimentation procedure.
  • (b) The location of parcels is determined by the coordinates of their “centers.” The direct calculation of real interfaces between parcels is impossible because of the formation of filaments of a very complicated fractal shape. Hence, the interfaces are determined using a simplified approach; namely, the entire computational area is covered by a supplemental grid with resolution of 5 m. The 5 m × 5 m cells are assigned to the parcel whose center is located closer. As a result, the interfaces between parcels represent a broken line consisting of 5-m-length straight-line elements (Fig. 1). The interface length changes randomly with time in the turbulent flow. The numbers of droplets penetrating from the ith upper parcel to the jth lower parcel during time step Δt are assumed to be proportional to li.jVt(rt, where lij is the length of the horizontal elements of the interface between the parcels (Fig. 1) and Vt(r) is the fall velocities of droplets with radius r.
  • (c) The influxes of droplets were calculated for each parcel. To calculate the changes of the DSDs in each parcel, the drop influxes were normalized by the volume (in our case, the square) assigned initially to each parcel.

This method actually represents an extension of the widely used flux method (or a box method) (e.g., Bryan 1966; Bott 1989) to describe the advection and sedimentation in the Eulerian models with irregular finite-difference grids to the irregular grid formed by the centers of the parcels. The procedure conserves the total numbers and masses of the drops in interacting parcels for each bin. Similarly to the finite-different grids, where the precision depends on the grid resolution, the accuracy of the sedimentation calculations depends on the number of parcels. In other words, while the process of diffusion growth and collisions is performed within a Lagrangian framework, the process of sedimentation is performed within a Eulerian coordinate framework. Such combinations of the Lagrangian and Eulerian approaches can be referred to as the hybrid method.

3. Design of numerical experiments and data assimilation

To justify the model’s ability to simulate adequately both nondrizzle and drizzle stratocumulus clouds, we simulated stratocumulus clouds observed during DYCOMS-II during research flights RF01 (negligible drizzle at the surface) and RF07 (a weakly drizzling cloud). Corresponding numerical simulations will be referenced to as the RF01 run and the RF07 run, respectively. In both cases the cloud-top height was about 850 m (Stevens et al. 2003a, b, 2005a). The ridged upper boundary is identified with the temperature inversion at this level. The model parameters used in the simulations are presented in Table 1. Harmonics of the largest scales represent large eddies, which usually exist in the cloudy and cloud-free ABL (e.g., Ivanov and Khain 1975, 1976; Stevens et al. 1996). To perform the simulations, both dynamical and thermodynamical parameters were adopted.

a. Assimilation of the model dynamics

There are three main groups of parameters of the velocity field in (1) and (2) that should be calculated using the observed data to determine the properties of the velocity field in the model: coefficients Cn and Dm and the characteristic correlation time γn. These values were calculated as follows:

  • (a) The coefficients Cn are determined using the observed profile of the rms vertical velocity deviation 〈W ′2o1/2 [see Eq. (9)], whose magnitude corresponds to either the “dynamical” or “convective” regimes of the BL (e.g., Lenschow et al. 1980; Babb and Verlinde 1999; Kollias and Albrecht 2000) and affects the DSD structure and drizzle-forming processes (Feingold et al. 1996; Stevens et al. 2003b, 2005b). We have used the 〈W ′2o〉 profile measured during RF01 of DYCOMS-II (Stevens et al. 2003b, their Fig. 9) (Fig. 2). The 〈W ′2o1/2 maximum of 0.7 m s−1 is located at z = 500 m. Note that the 〈W ′2o〉 profile measured in RF01 is quite similar to that measured in RF07 (Lothon et al. 2005), so that the profile shown in Fig. 2 was used for both the RF01 and RF07 runs. In Fig. 2, the range of 〈W ′2〉 profiles calculated using 10 LES models (Stevens et al. 2005a) is presented for comparison. One can see that the LES models have underestimated the vertical velocity fluctuations below cloud top and seem to have overestimated them above the cloud.
  • (b) The correlation properties of the velocity field determining the characteristic horizontal scale of vertical velocity variation are described by the lateral structure function DW of the vertical velocity in the horizontal direction. Knowing DW, one can calculate the coefficients Dm using (10). We used the lateral structure function presented by Lothon et al. (2005) in research flight RF07 at z = 240 m (Fig. 3). According to Lothon et al. (2005), the structure function shown in Fig. 3 is quite similar for all DYCOMS-II flights. Respectively, we used the structure function for simulations of RF01 as well.
  • (c) The characteristic time scales of the velocity fluctuations of different spatial scales are also subject to adaptation. In this study, the characteristic time scales were assumed to meet the Kolmogorov relationships:
    i1520-0469-65-7-2064-e13

The vertical velocity field (1a) with the correlation properties described above has zero skewness, which reflects the normality of the generated turbulent-like field. According to the measurements during DYCOMSII, the magnitudes of the skewness are quite close to zero at all heights in the STBL. At the same time, the LES simulations predicted significant deviations of the skewness from zero (especially within the cloud layer) (Stevens et al. 2003b, 2005a). Thus, the velocity field we use in the simulations has statistical properties that agree with the observations better than those produced by very high resolution LES models. The main dynamical parameters used in the simulations are presented in Table 1.

b. Initial temperature and humidity profiles and aerosol distributions

The specific feature of the flights under consideration was that many parameters such as droplet concentrations, sea surface temperatures, cloud-top heights, etc., were quite similar (see Table 2). The main difference between the conditions was the difference in the mean specific humidity: 9 g kg−1 in RF01 versus 10 g kg−1 in RF07, which determined the difference in the cloud-base height: 585 m in RF01 and 310 m in RF07. As was mentioned above, the purpose of the simulation was to form clouds in the initially noncloudy BL by vertical mixing with vortices of different scales. Correspondingly, at t = 0 the relative humidity (RH) in all parcels was set to less than 100%. Hence, we could not use the vertical temperature and humidity profiles observed in the cloudy BL as the initial ones. In contrast, the model has to generate by itself the profiles close to the observed ones as a result of the BL mixing. Since the amount of total water content (water vapor + liquid water) is conserved in the BL in the process of vertical mixing and condensation, the initial mixing ratio profiles were chosen to have mean values of 9 and 10 g kg−1 to simulate the clouds in RF01 and RF07, respectively. Similar considerations were applied for the choice of initial temperature profiles. The most important consideration in such a choice was to provide RHs within the BL and cloud-base heights that were close to the observations. Supplemental simulations indicate that the sensitivity of the temperature profile after model “spinup” to the choice of the initial temperature gradient is low, because vertical BL mixing leads to a dry-adiabatic temperature gradient below cloud base and nudges the temperature gradient to the moist adiabatic aloft. The initial profiles of the mixing ratio and liquid water static energy temperature used in the model are shown in Figs. 4 and 5, respectively.

Figure 6 shows the aerosol size distributions derived from the observations and used as the initial conditions in the RF01 and RF07 simulations. At t = 0, the aerosol size distributions in all parcels were assumed similar. The initial aerosol distributions in the RF01 and RF07 runs were taken from in situ measurements. To restore the aerosol distribution, a composite spectrum was created using the data from two instruments— the Radial Differential Mobility Analyzer (RDMA) and the Passive Cavity Aerosol Spectrometer Probe (PCASP)—which measure in diameter ranges of 0.008 < D < 0.128 μm and 0.1 < D < 3.0 μm, respectively (as described in Twohy et al. 2005). For each flight area below cloud, a segment was selected and the average distribution was used.

4. Results of simulations

a. Reproduction of basic thermodynamic and microphysical quantities

Figures 4a–d shows the profiles of the specific humidity in the observations and as calculated using the model. One can see that after a comparatively short transition period (during which parcels mix the BL) the horizontally averaged model profiles actually coincide with the measured profiles. Figure 5 shows vertical profiles of liquid water static energy temperature from RF01 (after Stevens et al. 2003b), and the model RF01 and RF07 runs. Good agreement between the profile obtained in the RF01 run and that observed in RF01 is seen. In the comparison with the potential temperature reported for RF07 (Stevens et al. 2003a), the model overestimates the potential temperature for the RF07 by about 2°C. Note that the data about temperature in RF07 somehow differ. For instance, the temperature presented in a low-rate file (LRT) on the National Center for Atmospheric Research/Earth Observing Laboratory (NCAR/EOL) Web site (not shown) is a little higher than those simulated in the RF07 run.

Figure 7 compares the measured and simulated vertical profiles of the horizontally averaged LWC. Simulated profiles are plotted in increments of 5 min. The dashed line in Fig. 7e depicts the LWC profile obtained by averaging the results obtained from 10 state-of-the-art LES models used for the simulation of RF01 (Stevens et al. 2005a). One can see that the mean LWC in the RF07 case exceeds that in the RF01 case in both the observations and the simulations. The model reproduces the observed LWC profiles quite well. The model tends to slightly overestimate the maximal values of LWC measured by the Gerber and King probes (especially in the RF01 run), which can be attributed to the lack of turbulent mixing between the parcels. At the same time, the LWC values calculated by the model reproduce the observations much better than do the most state-of-the-art LES models that significantly underestimate the LWC in RF01 (Fig. 7e). We attribute this result first of all to the better representation of the boundary layer dynamics (including that in the upper part of the BL), which was derived from observations in the present model.

Figure 8 shows the vertical profiles of the concentration of droplets with radius exceeding 1 μm calculated in the RF01 in the RF07 runs, as well as those calculated using in situ–measured DSDs in RF07. The algorithm of the matching of DSDs measured by different devices to get DSDs within the whole range of drop sizes is described by Krasnov and Russchenberg (2002) and Khain et al. (2008). In the RF07 case, drizzle reaches the surface in both the observations and the simulation. The maximum droplet concentration simulated in the RF01 run exceeds that in the observations by about 15% (see Table 2). The droplet concentration simulated in the RF07 case is in good agreement with the observations. Note that the mean droplet concentration simulated in the model does not change significantly with height within the cloud layer. This feature is in agreement with the observations in stratocumulus clouds [see Fig. 9, as well as Brenguier et al. (2000b); Pawlowska et al. (2000)]. Table 2 summarizes the comparison of the basic values measured and calculated for these two research flights.

b. Fine features of the microphysical cloud structure

We present some results that illustrate the microphysical structure of the simulated clouds and we indicate the possible ways that the model can be applied for investigation of drizzle formation. Figures 9a and 9b shows fields of LWC in the RF01 and RF07 runs at t = 125 min, when drizzle takes place in the RF07 run. The vertical profiles of the horizontally averaged values and standard deviations are plotted as well. Cloudy parcels form the structure typical of stratiform clouds with a relatively uniform cloud-base level. The cloud depth, as well as the maximum values of the LWC, is larger in the RF07-run. One can see a significant variability of LWC at scales of several tens of meters. DSDs in neighboring parcels also can differ significantly (not shown). Such variability was found to be a typical feature of stratocumulus clouds (Korolev 1995). The significant variability of the LWC and concentration is often attributed to nonadiabatic processes, such as parcel dilution by dry air entrainment through the cloud top and drizzle loss (Stevens et al. 1998; Wood 2005). Note that in the RF01 run all of the parcels are close to adiabatic (very weak precipitation and no turbulent mixing with the neighboring parcels). The LWC and concentration variability along any height level within a cloud in this case can be attributed to the fact that the parcels come from different height levels having different initial values of the mixing ratio. In addition, the parcels cross the lifting condensation level with different velocities and experience different updrafts during their motion. This determines the difference in the droplet concentrations in parcels reaching the same level. Note that the level of cloud adiabaticity is usually evaluated using horizontally averaged LWC (e.g., Wood 2005). Such an approach leads to an adiabaticity that differs from 1 even in the RF01 run case, where all parcels are close to adiabatic by definition. Figure 9c shows the field of radar reflectivities at t = 125 min in the RF07 run. The radar reflectivities in the particular parcels reach 5–7 dBZ, which are values typical of weak drizzle. The high inhomogeneity of the radar reflectivity reflects the high spatial inhomogeneity of the drizzle within the cloud. In spite of the fact that the velocity field is statistically uniform in the horizontal direction, drizzle takes place in a comparatively small fraction of the cloud. The spatial inhomogeneity of the drizzle flux and the vertical belt structure of the enhanced reflectivity are typical features of stratocumulus clouds seen in many radar images (e.g., Krasnov and Russchenberg 2006; Stevens et al. 2003a, b; Wood 2005). Such a belt structure can be attributed to the fact that maximum drizzle fluxes (especially in the lower parts of clouds and below clouds) are located mainly in a zone of downdrafts caused by large eddies.

The microphysical structure of simulated clouds is illustrated in Fig. 10, where the droplet effective radii (reff) in parcels in the RF01 and RF07 runs are plotted as a function of height for the period 60–120 min. These diagrams will be references to the height–reff scattering diagrams below. In the diagrams, the reff in the parcels are marked by a point at each 5 min. The height–reff scattering diagram calculated using the measured DSDs is also presented in Fig. 10. Three different stages of cloud evolution correspond to three zones in the diagrams: (a) The stage of the diffusion growth (the zone 1), when reff grows to roughly 12 μm in the RF01 run and to 14–15 μm in the RF07 run. In the simulations, reff ≈ 12 μm serves as a threshold value separating the clouds with negligible drizzle at the surface from the clouds producing drizzle at the surface. (b) The stage of the drizzle formation (regime 2) reflects the efficient drizzle growth by collisions. The increase in the drizzle drop size with the decrease in the height indicates the drop collector growth by collection of small droplets. (c) The stage of the developed drizzle (zone 3) corresponds to the time when the drizzle reaches the cloud base and falls below the cloud base. These drizzle drops reach the largest size, which, however, does not increase during drizzle fall because of evaporation and rare drizzle collisions below the cloud base. One can see that the height–reff scattering diagram calculated in the RF07 run agrees well with that calculated using the measured DSDs. For instance, the size of the drizzle drops falling on the surface is centered at 100 μm in both the simulations and the observations. The vertical profile of drizzle sizes (determining the drizzle flux) calculated in the model is quite similar to that in the observations (see also Wood 2005). However, according to the observed data, the drizzle formation process in the real cloud was somewhat more intense than was simulated in the RF07 run; according to the measured DSDs, the maximum of the reff reaches about 200 μm, while it reaches only 160 μm in the RF07 run. One can also see that the real DSDs with reff exceeding 20 μm are formed in the RF07 case at a higher level than in the simulation. The comparison of the height–reff scattering diagrams for the RF01 and RF07 runs showed a much lower drizzle rate and smaller drizzle drop size in the RF01 run. While in the RF07 run (within the time period 60–120 min) most DSDs below cloud base have effective radii of about 100 μm, the reff in the RF01 run varies within a wide range. This result can be attributed to the fact that in the RF01 run parcels descending from above into zone of undersaturated air contain droplets and a relatively small number of drizzle drops, most of which completely or partially evaporate, producing air parcels with very low LWC and droplet concentrations. In the RF07 run, parcels below cloud base contain a relatively large amount of drizzle drops, which remain relatively large during the fall to the surface.

c. The Z–LWC relationships

Another efficient tool for investigating microphysical processes in clouds is the analysis of radar reflectivity–LWC diagrams (Z–LWC; Krasnov and Russchenberg 2002; Khain et al. 2008). Figure 11 shows the Z–LWC diagrams for clouds simulated in the RF01 and RF07 runs at the nondrizzling stage (0–20 min), as well as the diagrams calculated for the whole 4-h period of cloud evolution. The Z–LWC diagrams are compared with those presented by Krasnov and Russchenberg (2006) and Khain et al. (2008), as calculated using the measured DSDs in different nondrizzling, as well as in drizzling (including heavy drizzling), clouds. The Z–LWC relationships obtained in the DYCOMS-II RF07 are included in the figures as well. In the diagrams derived from the model simulations, each parcel is denoted by a point. Analysis of the Z–LWC diagrams (see Khain et al. 2008 for more details) allows one to distinguish three stages (or three different zones on the diagrams): the diffusion growth, transition regime, and “drizzle” regimes (Krasnov and Russchenberg 2002; Khain et al. 2008). These stages denoted by the numbers 1, 2, and 3 are the same as were found by analyzing the height–reff scattering diagrams in Fig. 10. One can see that the dependence of Z on LWC is monotonic only at the nondrizzling stage. During the initial period of a few tens of minutes, the Z–LWC diagrams in the RF07 and RF01 runs are quite similar. However, in the RF07 run LWC reaches larger values. At t > 30 min, collisions become effective in the RF07 run, which leads to the formation of the first large drops in parcels located in the upper layer of stratocumulus clouds having maximum LWC (these parcels are marked yellow). As they fall, these drops grow to drizzle size by collection of droplets within the parcels located below. This stage corresponds to the transition zone (zone 2) seen in Figs. 10 and 11.

Drizzle drops reach their maximum size in the lower part of cloud layer and below clouds forming the zone of the radar reflectivity maximum denoted as the drizzle zone (zone 3) in the Z–LWC and the height–reff scattering diagrams. Corresponding parcels are marked green in Fig. 11.

According to the “radar lags” data (flown above the cloud in each flight during DYCOMS-II; information online at http://www-das.uwyo.edu/~vali/dycoms/dy_rept/circle_img/circles.html), the reflectivity maxima measured in RF01 and RF07 were −12 and 4–5 dBZ, respectively. The values calculated by the model actually coincide with the observations (see Fig. 11). Thus, the model reproduces quite well the measured Z–LWC diagram at all stages of cloud evolution, including the slopes depicting different microphysical regimes, as well as the diagrams bounding from above and from below and the maximum values of reflectivity. The good agreement of the in situ–measured and simulated height–reff and Z–LWC scattering diagrams indicates that the model is able to reproduce very fine microphysical features of real clouds, which allows us to apply the model for further investigation of drizzle formation processes.

d. Effects of droplet sedimentation

In contrast to the state-of-the-art TEM, the model discussed in our study includes drop collisions and drop sedimentation. To investigate the role of droplet sedimentation, supplemental simulations RF01_no_sed and RF07_no_sed have been carried out, which were similar to the RF01 and RF07 runs, except that sedimentation was excluded. Figure 12 compares the radar reflectivity fields calculated in the RF07 run with those in the runs with no sedimentation. Dramatic effects of sedimentation on cloud microphysical structure are clearly seen: in the no-sedimentation runs, the maximum values of the radar reflectivity in particular parcels reach 40 dBZ, indicating the formation of raindrops. While the horizontally averaged radar reflectivity in the RF07 run is maximum at the cloud base (where the drizzle drop size reaches its maximum), the reflectivity maximum in the no-sedimentation runs takes place near cloud top. Enormous rms values of the reflectivity in these runs indicate that while in some parcels raindrops are formed, other parcels contain only small droplets and/or aerosols.

Figure 13 shows the height–reff and the Z–LWC scattering diagrams calculated during simulations RF01_no_sed and RF07_no_sed. Comparison of the height–reff diagrams in Fig. 13 with those in Fig. 10 indicates a dramatic difference in the microphysics caused by turning off the drop sedimentation. In some parcels raindrops of 1000-μm radius form in both of the no-sedimentation runs. The maximum raindrops size is determined by the maximum size of the mass grid used. At the same time, many parcels in the nonsedimentation runs contain droplets with effective radii of a few microns. The drizzle zone disappears in the RF07_no_ sed run, so that only a few parcels have DSD with an effective radius of 100–200 μm. Turning off sedimentation leads to dramatic differences in the Z–LWC diagrams too. In simulations with the sedimentation turned off, the maximum values of the radar reflectivity exceed 20 dBZ to the end of the first hour. In agreement with the height–reff diagrams, the Z–LWC diagrams do not reveal the drizzle stage. In the no-sedimentation cases large drops cannot leave the parcels and continue to collect other small droplets until raindrops form and no cloud droplets remain. Vertical arrows in Figs. 13e and 13f indicate that collisions lead to the formation of large drops without changes to the LWC in parcels. One can see that raindrops form in the parcels containing LWC values of about 1 g m−3. Raindrops in the RF07 no sedimentation simulation are seen to be forming earlier than in the RF01 no sedimentation run because of the larger LWC in the first case.

Thus, while in the cases when drop sedimentation is taken into account, the drizzle production is crudely compensated for by drizzle loss, in the nonsedimentation cases there is no stationary (or quasi-stationary) state until all condensate mass is concentrated in a small number of raindrops. Thus, drop sedimentation is one of the most important processes determining the microstructure of stratiform clouds (at least in the zero-mixing limit).

e. Time evolution of clouds and sensitivity to different parameters

Verification of the model has been performed in several supplemental sensitivity runs. The first set of simulations examined the determination of the characteristic time scale of cloud formation and the sensitivity of cloud parameters at the mature stage to the initial conditions. Figure 14 shows the time dependences of the LWC averaged over the computational area in the simulations performed under the same conditions as in the RF07 run, with the exception that other realizations with the same statistical characteristics of the turbulent velocity field in (1a) and (1b) were used. One can see that the averaged LWC in the runs in different realizations of the velocity field converge to nearly the same quasi-stationary values. The LWC approaches the values typical of the mature stage during the transition period of 20–35 min. The same characteristic time of adaptation of the mean vertical profiles of temperature and humidity can be seen in Figs. 4 and 5, where the calculated profiles are plotted with time increments of 5 min. The result shows that the thermodynamic structure and the cloud microphysics adjust to the dynamics quite rapidly. The results support the assumption that cloud microphysics and cloud structure correspond to (or are controlled to a large extent by) the BL dynamics. It means that if the dynamical and thermodynamical properties of the BL are similar in two cases, the statistical characteristics of the main parameters of the stratiform clouds formed in these cases will be similar (or nearly similar).

In real clouds, the microphysics affects the dynamics as well (e.g., Stevens et al. 2005b; Petters et al. 2006), so that mutual adaptation takes place. At the same time the effect of the microphysics on the dynamics in many cases is not strong (at least at the nondrizzle stage) because of comparatively low rates of the latent heat release in stratocumulus clouds.

The differences between the LWC values in the different velocity field realizations are caused by differences in the initial amplitudes of the largest harmonics (large eddies) having the longest lifetime and maximum intensity. It is interesting to note that in different flow realizations drizzle starts at different time instances and has different durations. This result can explain a dramatic spatial inhomogeneity of drizzle fluxes in stratocumulus clouds, which are visually quite uniform in the horizontal direction (Stevens et al. 2003a; Wood 2005). This allows us to suppose that the formation of weak drizzle has a random nature, when some factors exceed their thresholds. This hypothesis was supported in a supplemental simulation of a stratiform cloud using a computational area of 20-km length, that is, 7 times larger than that used in other simulations. In spite of the fact that the velocity field was statistically homogeneous, the drizzle formed only within the zones of 1–6-km width.

In other sets of sensitivity simulations, it was found that the decrease of the mean linear parcel size from 40 to 20 m (with the corresponding increase in the parcel number from 1344 to 5376) increases somewhat the drizzle rate and the radar reflectivity maximum. However, this decrease in the parcel size did not affect the general cloud microphysical structure significantly. It was also found that turbulent fluctuations (harmonics) with characteristic scale below ∼30 m affect the general microphysical cloud parameters only slightly.

5. Discussion and conclusions

A novel trajectory ensemble model (TEM) of a cloud-topped boundary layer is described. In this model the BL is fully covered by a great number of Lagrangian air parcels with linear sizes of about 40 m that contain either wet aerosol particles or aerosol particles and droplets. In each parcel, the diffusion growth of aerosols and droplets, as well as collisions, is accurately described. For the first time, the droplet sedimentation in the TEM is taken into account, which allowed the simulation of precipitation formation. The Lagrangian parcels are advected by the velocity field generated by the model of a turbulent-like flow obeying the energetic and correlation properties assimilated from the observations. The model calculates the DSDs and the aerosol size distributions, and their parameters and moments such as the effective radius, concentration, cloud and rainwater content, droplet spectrum width, radar reflectivity, etc. in each parcel. The temporal and spatial variabilities of the aforementioned parameters are calculated as well.

Two simulations of stratocumulus clouds observed during the DYCOMS-II field campaign’s research flights RF01 (negligible drizzle at the surface) and RF07 (weak drizzle) were performed using measured size distributions of aerosols. The model reproduced well the vertical profiles of the horizontally averaged quantities such as absolute air humidity and the droplet concentration. With regard to the LWC, the model reproduces its vertical profiles’ shapes and magnitudes better than the state-of-the-art LES models used by Stevens et al. (2005a) in their test simulations. In spite of small differences in the thermodynamic and aerosol conditions of the RF01 and RF07 cases, the model managed to distinguish RF01 cloud with negligible drizzle at the surface from weak-drizzling (RF07) clouds. Moreover, the averaged drizzle flux calculated in the RF07 run agrees well with the observations. The height-effective radius scattering diagrams, as well as the Z–LWC diagrams simulated by the model in the nondrizzle and drizzle cases, agree well with results derived from in situ–measured DSDs in corresponding research flights. This agreement concerning the reproduction of fine microphysical features indicates that the model describes microphysical characteristics of clouds well even in the nonmixing limit, when turbulent mixing between parcels is not taken into account. The analysis of the diagrams allows one to distinguish the main regimes of cloud evolution: diffusional growth, stage of intense droplet collisions, and the stage of drizzle fall below cloud base. The diagrams differ significantly in the RF01 and the RF07 runs, which makes these diagrams an efficient tool for investigating the process of drizzle formation.

It is shown that rapid drizzle formation starts when the effective radius exceeds 14–15 μm, which is in agreement with the results of Gerber (1996). In this case, drizzle reaches the surface. The formation of drizzle near cloud base requires the drop effective radii in some parcels be larger than ∼10 μm.

Significant spatial variabilities among the LWC, drop concentration, and DSD shape were found at scales of several tens to hundred meters. The spatial variation of the cloud microstructure at a certain cloud level is caused by the difference in the histories of the parcels. The parcels reaching the same height level may have different initial values of mixing ratios and temperatures, as well as different vertical velocities and aerosol size distributions, which may lead to the formation of DSDs with different shapes, to the formation of bimodal and even multimodal DSDs in some fraction of the parcels. Such variability seems to obey the observations (Korolev 1995). The ability of the model to reproduce fine microphysical characteristics, such as the size of drizzle drops at the surface, the height dependence of the effective radius, the slopes at the Z–LWC diagrams, the maximum values of the radar reflectivity, the locations of zones of different microphysical regimes, etc. (see Table 2), indicates that the model can be efficiently used for the investigation of drizzle formation processes.

Simulations in which droplet sedimentation was turned off showed a crucial effect of drop sedimentation on cloud microstructure. In cases of no sedimentation, collisions within the parcels containing large LWC led to the formation of raindrops producing reflectivities of up to 40 dBZ both in drizzling and nondrizzling cases. Thus, sedimentation is one of the most important processes involved in the formation of realistic microphysics of stratiform clouds.

Sensitivity studies support the model concept that cloud microphysics adjusts rapidly to the ABL dynamics. Sensitivity studies also indicate that the ABL dynamics and thermodynamics, as well as aerosol properties, determine to a large extent the microphysical and geometrical (e.g., cloud depth) properties of stratocumulus clouds. At the same time, the formation of drizzle (at least weak drizzle) seems to be a very fine phenomena of, possibly, random nature. This hypothesis seems to be supported in the simulation with 20-km computational area length, where drizzle formed only within narrow zones of 1–6-km width, in spite of a statistically homogeneous velocity field. This result suggests that in clouds with horizontally uniform statistical properties, some parameters randomly exceed their threshold values, in particular (sometimes quite narrow), cloud regions, giving rise to the drizzle formation there.

We would like to mention that good agreement with the observations found in many aspects of the microphysical and geometrical cloud structures was obtained by neglecting the turbulent mixing between parcels, that is, in the nonmixing limit. The detailed investigation of the role of the turbulent mixing will be presented in a future study.

As was discussed earlier in this paper, the energetic and correlation properties of the BL wind field can be assimilated from the observations. Both aircraft and radar and lidar data can be applied (e.g., Krasnov and Russchenberg 2002, 2006). As a result, the investigations of stratocumulus clouds can be performed within the closed “observation–simulation–observation” cycle. After prior analysis, the dynamical data observed (first of all, the series of the Doppler vertical velocity and the lidar data) should be assimilated. The assimilation of the dynamics means that the parameters of the model velocity field will have the same energetic and correlation properties as those of the velocity field observed. The observed aerosol size distributions and the sounding data can also be easily assimilated by the model. The results of the simulations should be compared with the simultaneously measured microphysical, radar, and lidar cloud characteristics. Thus, the model can serve as an efficient connecting link between the observed data of different types (radar, lidar, aircraft, etc.). The results of these comparisons, on the one hand, will foster a better understanding of cloud microphysical processes and, on the other hand, will make it possible to improve both the interpretation of remote sensing data and the retrieval algorithms. Some evidence of the model applicability for remote sensing purposes can be seen in the very good representation of the height-effective radius and the Z–LWC diagrams in the numerical simulations.

Acknowledgments

The study was performed under the support of the Israel Academy of Sciences (Grants 140/07 and 950/07) and the Israel Ministry of Sciences (the German–Israel collaboration in Water Resources, Grant WT 0403). The research of in situ DSDs has been done through the support of the Netherlands Space Agency (SRON) and the Dutch National Research Program on Climate Changes Spatial Planning. The authors express their gratitude to J. Snider and M. Petters, for help in the determination of the aerosol size distributions. The authors express deep gratitude to Prof. B. Stevens, who kindly provided the access to the airborne measurements data obtained from the NSF–NCAR RAF EC-130Q aircraft during the DYCOMS-II campaign, as well as for the interest in this study and useful recommendations.

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APPENDIX

Statistical Properties of the Velocity Field

Variation of the vertical velocity

Taking the square of Eq. (1a) and averaging over a great number of realizations gives
i1520-0469-65-7-2064-ea1
Taking into account that coefficients amn(t) and bmn(t) obey the correlation relationships (5), (A1) yields
i1520-0469-65-7-2064-eqa1
Taking into account the normalization condition ΣMm=1 D2m = 1 and Eq. (9b), the final expression can be written as
i1520-0469-65-7-2064-ea2
Analogously, one can derive an expression for the horizontal velocity variation:
i1520-0469-65-7-2064-ea3

Correlation functions of the velocity components in the horizontal direction

Multiplying the values of the vertical velocity at point (z, x1) by the value of the vertical velocity at point (z, x2) and averaging the expression over a great number of realizations, we get
i1520-0469-65-7-2064-ea4
Using the correlation relationships (5), one can simplify the expression (A4) as
i1520-0469-65-7-2064-ea5
Analogously, the correlation function for the horizontal velocity can be written as
i1520-0469-65-7-2064-ea6
Note that BW and BV depend only on the distance between two points, Δx = x2x1, and does not depend on the coordinates of the points themselves, which indicates the statistical horizontal homogeneity of the flow. Analogously, one can obtain expressions for cross-correlation functions, BVW. This function also depends on Δx = x2x1 that is, BVW(x1, x2, z) = φx, z), and is an antisymmetrical one The latter means that the components V and W are noncorrelated in one and the same point; that is, BVW(0, z) = 0.
Fig. 1.
Fig. 1.

Illustration of the sedimentation procedure. Circles denote the parcel centers. Arrows denote the droplet fluxes through the interface. See text for more detail.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 2.
Fig. 2.

Vertical profiles of 〈W ′2〉 in RF01 measured in situ (circles with bars) and by radar (circle–dot) (after Stevens et al. 2005a). The dashed line that effectively approximates the observed data was obtained from the LES simulation UCLA-0 configuration with a vertical spatial resolution of 1 m (after Stevens et al. 2003a, 2005a). This curve has been used in the present model. The range of the results obtained using 10 LES models is marked by different levels of shading (Stevens et al. 2005a). The solid line represents the mean values predicted by the LES models. Horizontal dashed lines delimit the cloud area.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 3.
Fig. 3.

The lateral structure function measured by Lothon et al. (2005) during flight RF07 at z = 240 m and used in the model for the calculation of coefficients Dm. Crosses indicate the values of the structure function reproduced by the model. One can see that the model exactly adjusts the measured profiles.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 4.
Fig. 4.

The horizontally averaged profiles of the mixing ratios (a), (c) measured during (c), (d) RF01 (no drizzle case) and (a), (b) RF07 (drizzle case) and (b), (d) calculated at different time instances in the corresponding simulations. The profiles are plotted with time intervals of 5 min. (b), (d) Initial profiles (dashed) marked as t = 0. Panel (a) is adopted from Faloona et al. (2005), while (c) is adopted from Stevens et al. (2003b).

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 5.
Fig. 5.

Vertical profiles of liquid water static energy temperature from (a) RF01 (after Stevens et al. 2003b) and model simulations of (b) RF01 and (c) RF07. The good agreement between the profile obtained in the RF01 run and the observations in RF01 is seen. As follows from the comparison with the value of the potential temperature reported for RF07 (Stevens et al. 2003a), the model overestimates the temperature by about 2°C in this run. Profiles are plotted with increments of 5 min. Dashed lines in (b), (c) indicate the initial profiles at t = 0.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 6.
Fig. 6.

Size distributions of dry aerosols in RF01 and RF07 as used in the corresponding simulations. The distributions are a composite of the RDMA and the PCAPS probe subcloud in situ measurements.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 7.
Fig. 7.

Vertical profiles of the horizontally averaged values of the LWC (a)–(d) measured and (e), (f) simulated in the (a), (c), (e) RF01 and (b), (d), (f) RF07 cases. Simulated profiles are plotted with increments of 5 min. The dashed line in the (e) depicts the mean LWC profile obtained by averaging the results of 10 state-of-the-art LES models used for the simulation of RF01 (Stevens et al. 2005a).

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 8.
Fig. 8.

The vertical profiles of the concentration of droplets with radius exceeding 1 μm calculated in (left) RF01, (middle) RF07, and(right) using in situ–measured results during RF07. In RF07, drizzle reaches the surface in both the observations and the simulation.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 9.
Fig. 9.

Fields of LWC in the (a) RF01 and (b) RF07 runs; parcels having different LWCs are marked by gray shades. (c) The field of radar reflectivity in RF07 at t = 125 min, when drizzle takes place in RF07. Vertical profiles of horizontally averaged and rms values are presented as well. The cloud depth in RF07 is larger than that in RF01. The high inhomogeneity of LWC at small distances is seen. The radar reflectivity is nonuniform and has a “columnar” structure.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 10.
Fig. 10.

The height–reff scattering diagrams calculated in the (left) RF01 and (middle) RF07 runs for a 60–120-min period, and (right) a diagram calculated using the measured DSDs. See text for more details.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 11.
Fig. 11.

(top) The Z–LWC diagrams for clouds (left) simulated in the RF01, (middle) simulated in RF07 cases at the nondrizzling stage of cloud evolution (0–20 min), and (right) calculated using measured DSD in nondrizzling cloud DSDs (Krasnov and Russchenberg 2002; Khain et al. 2008). (bottom) The Z–LWC diagrams for clouds simulated in RF01 and RF07 averaged over 4 h of cloud evolution, and calculated using DSDs measured in drizzling and heavily drizzling clouds (see references above). In diagrams calculated from the model, each parcel is denoted by a point marked in the diagram each 5 min. The location of the parcels in the Z–LWC diagram is shown as well: parcels located in the top half of the cloud layer are marked yellow, parcels located in the bottom half of the cloud layer are marked blue, and those located below the LCL are marked green. Lines 1, 2, and 3 denote regimes of diffusion growth, the transition regime, and the “drizzle” regimes, respectively. These regimes correspond to those marked in Fig. 10. Lines A, S, and F for the calculated values show different approximations of Z(LWC) (see Krasnov and Russchenberg 2002; Khain et al. 2008 for more details).

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 12.
Fig. 12.

Radar reflectivity fields calculated in (top) RF07, and (middle) the RF07 and (bottom) RF01 nonsedimentation runs in which droplet sedimentation was turned off. Vertical profiles of horizontally averaged and rms values are presented as well. The dramatic effects of sedimentation on the cloud microphysical structure are seen: in the nonsedimentation runs, the maximum values of the radar reflectivity reach 40 dBZ, indicating the formation of raindrops.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 13.
Fig. 13.

(a)–(d) The height–reff scattering diagrams and (e), (f) the Z–LWC diagrams calculated in the simulations with no sedimentation: (left) RF01 and (right) RF07. The height–reff scattering diagrams are plotted for periods of 5–60 and 65–120 min. The Z–LWC diagram is calculated for the period 0–60 min. In the no-sedimentation case raindrops form in both simulations, indicating the crucial role of sedimentation in the microphysics of stratocumulus clouds.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Fig. 14.
Fig. 14.

Time dependences of the area-averaged LWC in the RF07 runs with different realizations of the velocity field of the same statistical properties.

Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2486.1

Table 1.

The main parameters used in the preliminary simulations of cloud formation.

Table 1.
Table 2.

Comparison of calculated values with those measured during RF01 and RF07.

Table 2.
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