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  • View in gallery

    The Z–LWC relationship according to in situ measurements in water clouds. The notation of the fitting lines is given in Table 1.

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    The 2D diagrams of Z–LWC relation [with mean (solid line), std devs (dotted curves), and linear fittings (dashed curves)] for (a) nondrizzling clouds, (b) “the cloud with light drizzle” class (“transition” drizzle regime), and (c) “the cloud with heavy drizzle” class (drizzle regime). The categorization has been done using the radar reflectivity to the optical extinction ratio: (a) log10(Z/α) < −1; (b) −1 < log10(Z/α) < 1.8; and (c) log10(Z/α) ≥ 1.8 (after Krasnov and Russchenberg 2002, 2005). The notations of the fitting lines are given in Fig. 1 and Table 1.

  • View in gallery

    Physical processes described by the TEM (Pinsky et al. 2006, 2008). Adjacent Lagrangian parcels containing the full warm microphysics and cloud–aerosol interaction are advected in the boundary layer by the time-dependent turbulent-like wind field generated by a model of turbulent flow. The largest droplets forming in some Lagrangian parcels fall through the interfaces into and collect droplets located within other parcels. As a result, drizzle forms and falls down. When drops fall within undersaturated air they can evaporate partially or totally. In the latter case, new aerosols are formed. In the model, dynamical properties of the boundary layer are assumed to be given. The cloud microphysics is adjusted to the model dynamics and aerosols properties.

  • View in gallery

    Initial size distribution of dry aerosols.

  • View in gallery

    Fields of (top) radar reflectivity and (bottom) vertical velocity at t = 170 min.

  • View in gallery

    Some Z–LWC diagrams calculated by the model at different time instances. Each point in these diagrams represents one cloud parcel. Parcels located in the upper half of the cloud layer are marked yellow, parcels located in the lower half of the cloud layer are marked blue, and parcels located below the lifting condensation level are marked green. Changes in the Z–LWC relationships related to different processes are marked by arrows (see text for detail).

  • View in gallery

    (top right) The field of rain flux formed by different cloud parcels at the developed drizzle stage (160 min). (left six panels) (left) Number dN/d log10r and (right) mass dLWC/d log10r drop distributions are presented for three chosen cloud parcels located at different heights within the cloud layer. (bottom right) The location of the parcels in the Z–LWC diagram is shown. In the Z–LWC diagram, parcels located in the upper half of the cloud layer are marked yellow, parcels located in the lower half of the cloud layer are marked blue, and parcels located below the lifting condensation level are marked green.

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    The time-accumulated (over 4 h) Z–LWC diagram calculated in the model.

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    Possible coupling of the model and observations for improving retrieval algorithms and investigation of cloud microphysics.

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Combined Observational and Model Investigations of the Z–LWC Relationship in Stratocumulus Clouds

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  • 1 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
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Abstract

In situ measurements indicate the complexity and nonunique character of radar reflectivity–liquid water content (Z–LWC) relationships in stratocumulus and cumulus clouds. Parameters of empirical (statistical) Z–LWC dependences vary within a wide range. Respectively, the accuracy of retrieval algorithms remains low. This situation is partially related to the fact that empirical algorithms and parameters are often derived without a corresponding understanding of physical mechanisms responsible for the formation of the Z–LWC diagrams. In this study, the authors investigate the processes of formation of the Z–LWC relationships using a new trajectory ensemble model of the cloud-topped boundary layer (BL). In the model, the entire volume of the BL is covered by Lagrangian parcels advected by a turbulent-like velocity field. The time-dependent velocity field is generated by a turbulent model and obeys the correlation turbulent laws. Each Lagrangian parcel represents the “cloud parcel model” with an accurate description of processes of diffusion growth–evaporation of aerosols and droplets and droplet collisions. The fact that parcels are adjacent to each other allows one to calculate sedimentation of droplets and precipitation (drizzle) formation. The characteristic parcel size is 50 m; the number of parcels is 1840. The model calculates droplet size distributions (DSDs), as well as their moments (e.g., aerosol and drop concentration, mass content, radar reflectivity) in each parcel. In the course of the model integration, Z–LWC relationships are calculated for each parcel, as well as the scattering diagram including all parcels. The model reproduces in situ observed types of the Z–LWC relationships. It is shown that different regimes represent different stages of cloud evolution: diffusion growth, beginning of drizzle formation, and stage of heavy drizzle, respectively. The large scattering of the Z–LWC relationships is found to be an inherent property of any drizzling cloud. Different zones on the Z–LWC diagram are related to cloud volumes located at different levels within a cloud and having different DSD. This finding allows for improvement of retrieval algorithms.

Corresponding author address: Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel. Email: khain@vms.huji.ac.il

Abstract

In situ measurements indicate the complexity and nonunique character of radar reflectivity–liquid water content (Z–LWC) relationships in stratocumulus and cumulus clouds. Parameters of empirical (statistical) Z–LWC dependences vary within a wide range. Respectively, the accuracy of retrieval algorithms remains low. This situation is partially related to the fact that empirical algorithms and parameters are often derived without a corresponding understanding of physical mechanisms responsible for the formation of the Z–LWC diagrams. In this study, the authors investigate the processes of formation of the Z–LWC relationships using a new trajectory ensemble model of the cloud-topped boundary layer (BL). In the model, the entire volume of the BL is covered by Lagrangian parcels advected by a turbulent-like velocity field. The time-dependent velocity field is generated by a turbulent model and obeys the correlation turbulent laws. Each Lagrangian parcel represents the “cloud parcel model” with an accurate description of processes of diffusion growth–evaporation of aerosols and droplets and droplet collisions. The fact that parcels are adjacent to each other allows one to calculate sedimentation of droplets and precipitation (drizzle) formation. The characteristic parcel size is 50 m; the number of parcels is 1840. The model calculates droplet size distributions (DSDs), as well as their moments (e.g., aerosol and drop concentration, mass content, radar reflectivity) in each parcel. In the course of the model integration, Z–LWC relationships are calculated for each parcel, as well as the scattering diagram including all parcels. The model reproduces in situ observed types of the Z–LWC relationships. It is shown that different regimes represent different stages of cloud evolution: diffusion growth, beginning of drizzle formation, and stage of heavy drizzle, respectively. The large scattering of the Z–LWC relationships is found to be an inherent property of any drizzling cloud. Different zones on the Z–LWC diagram are related to cloud volumes located at different levels within a cloud and having different DSD. This finding allows for improvement of retrieval algorithms.

Corresponding author address: Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel. Email: khain@vms.huji.ac.il

1. Introduction

Warm stratiform and stratocumulus clouds cover enormous areas of the earth and play a crucial role in the radiation balance of the earth affecting the climate and climatic changes, and contribute significantly to precipitation. The radiative and precipitating properties of clouds depend on their microphysical structure determined by droplet size distributions (DSDs). Microphysical properties of stratiform clouds were measured by research aircraft in the course of several national and international field experiments: the Atlantic Stratocumulus Transition Experiment (ASTEX; Martin et al. 1994; Albrecht et al. 1995; Duynkerke et al. 1995; Frisch et al. 1995; Ramanathan et al. 2001), the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE; Austin et al. 1995), the Cloud Lidar and Radar Experiment (CLARE’98; ESA 1999), the second Aerosol Characterization Experiment (ACE-2; Brenguier et al. 2000), the Baltex Bridge Cloud Campaign (Crewell et al. 2004), and the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS-II; e.g., Stevens et al. 2003a, 2005; Lothon et al. 2004; VanZanten et al. 2005; VanZanten and Stevens 2005).

Despite many aircraft measurements performed during these scientific campaigns, which provide important but local information, remote sensing remains the key tool to evaluate and monitor cloud properties, their spatial and temporal evolution, and general contribution of the clouds to the radiation balance and precipitation. Over the last decade, cloud radars have been recognized as important instruments for the study of microphysical properties of these clouds. Detailed observations are done routinely now at several observatories in the world (e.g., LeMone 1990; Fox and Illingworth 1997; Kollias and Albrecht 2000; Wolf et al. 2000; Krasnov and Russchenberg 2002, 2005; Crewell et al. 2004; O’Connor et al. 2004).

Similar to traditional weather radar systems, where the radar reflectivity factor is used to estimate the rainfall rate, attempts to find the relationships between the radar reflectivity Z and the cloud liquid water content (LWC) have been made (Atlas 1954; Sauvageot and Omar 1987; Fox and Illingworth 1997). In spite of significant improvements in LWC retrievals, the uncertainties in deriving LWC remain significant. It is seen both from large scattering on Z–LWC diagrams as well as in the different parameterizations reported by different authors. We suppose that the major limitations of the statistical Z–LWC relationships are related to the fact that these relationships have been searched without detailed consideration of the corresponding cloud processes. The lack in such analysis hinders the improvement of retrieval algorithms. We are going to show new opportunities arising by utilization of an appropriate numerical model. Numerical modeling is a potentially powerful tool for the investigation of physical mechanisms determining the DSDs and drizzle formation, and, consequently, the Z–LWC relationships. Numerical models properly reproducing the DSDs (and drizzle) formation can serve as a connecting link between microphysical and radiative (e.g., radar reflectivity) properties of the clouds. The purpose of this study is to reproduce Z–LWC diagrams derived from in situ measurements in warm stratocumulus clouds and to reveal physical mechanisms leading to different regimes in the Z–LWC diagrams using a numerical model with spectral bin microphysics.

The structure of the paper is the following: In section 2 the Z–LWC diagram based on in situ measurements is discussed. Section 3 briefly presents the numerical model with the spectral bin microphysics. Conditions of numerical simulation are designed in section 4. The results are presented in section 5. A summary and conclusions can be found in section 6.

2. Aircraft observations

In this section, we present the Z–LWC diagram, which has been designed using microphysical data measured in situ during four field campaigns in different geographical regions, and under different meteorological conditions. The brief description of the measurements is presented below.

a. The CLARE’98 campaign

The Cloud Lidar and Radar Experiment took place near Chilbolton (United Kingdom) in October 1998. This extensive cloud campaign included airborne and ground-based radar and lidar observations as well as in situ aircraft measurements of the drop size distributions (see ESA 1999 for details). The DSDs in clouds were in situ measured from a C-130 aircraft of the Met Office Meteorological Research Flight with a forward scattering spectrometer (FSS) and a two-dimensional cloud (2DC) probe measuring sizes in the range between 1- and 23.5-μm radii and 6.25- and 406.25-μm radii, respectively. The available data have a 5-s interval of averaging.

b. The DYCOMS-II campaign

The DYCOMS-II field campaign took place in July 2001 in the Pacific Ocean near California (Stevens et al. 2003a). It was directed to collect data to study nocturnal marine stratocumulus. The main measuring part of the campaign was made during 10 research flights of the National Center for Atmospheric Research’s (NCAR) Research Aviation Facility EC-130Q. On this aircraft cloud droplet spectrums were measured using a set of probes: the PMS-PCASP 100, the PMS-FSSP-100, the PMS-FSSP-300, the PMS-260X, the PMS-2DC, and the PMS-2DP in the different size ranges between 0.045- and 786-μm radii. For in situ measurements of LWC on aircraft, two King hot-wire probes that were installed on different wings and the Gerber’s Particulate Volume Monitor (PVM-100A) were used. The available data have been obtained using a 1-s interval averaging.

c. The CAMEX-3 campaign

The third field campaign in the Convection and Moisture Experiment Series (CAMEX-3) took place in the Florida coastal zone in August–September 1998 (Rizvi et al. 2002). The objective of the field program was data collection for research in tropical cyclones using National Aeronautics and Space Administration (NASA)-funded aircraft ER-2 and DC-8, and ground-based remote sensing. For this study it was important that all research flights took place in strong cumulus clouds. For measurement of the DSDs FSS (the size range between 0.42- and 23.67-μm radii) and 2DC (the sizes range between 17.75- and 762.50-μm radii), probes have been used that were mounted on the DC-8. The available data have a 60-s interval of averaging.

d. The BBC-1 campaign

The first Baltex Bridge Cloud (BBC-1) Campaign (August–September 2001) was organized to obtain a comprehensive and synergistic dataset of cloud properties for the study of cloud processes and to provide the background information to improve cloud parameterizations in numerical models (Crewell et al. 2004). Ground-based observations such as cloud lidars, radars, thermodynamic soundings, radiation, and boundary layer properties were obtained at the experimental site of Cabauw, the Netherlands. At appropriate times the set of continuous observations was augmented by coordinated aircraft flights involving the Météo France Merlin aircraft. Cloud microphysics were measured with three droplet-spectrometer probes, FSSP-100, OAP-2DC, and OAP-2DP, at the size range between 1.75- and 1370-μm radii with 10-s averaging interval.

To obtain complete information about the cloud droplet size distributions, the distributions measured using different particle probes have to be properly matched (Baedi et al. 2000). Briefly, the procedure of the calculation of DSD was as follows: the concentrations of droplets measured in each cell of FSSP-100 and 260X probes were used to calculate spectral densities in each of these bins. Then, the values of the spectral density were put on the same axis in the order of the increase of mean cell radius. The boundaries of new cells in the matched DSD were placed in the middle between adjacent cells.

Since cloud drops at typical working frequency of the cloud radars act as Rayleigh scatterers, the radar reflectivity factor Z and LWC were calculated, respectively, as
i1558-8432-47-2-591-e1
i1558-8432-47-2-591-e2
where N0 is the droplet concentration, (m−3), ρw is the water density, Ni is the number of particles measured in the ith bin normalized by the bin width (m−3 × mm−1), ri and Δri are the midradius and width of the ith bin (mm), and the angular brackets stand for the averaging over the whole size spectrum.
The Z–LWC diagram calculated using the in situ data for the four campaigns is shown in Fig. 1. The diagram shows a nonunique relationship between Z and LWC and, therefore, cannot be used to parameterize the whole set of the water clouds data without additional information about droplet size distributions, or without revealing specific regimes of cloud formation. The Z–LWC relationship is commonly expressed in the form of a power law:
i1558-8432-47-2-591-e3
where parameters a and b are constants that can be estimated theoretically using commonly used assumptions about droplet size distributions in water clouds or empirically using a statistical technique to fit the in situ measured data. The Table 1 presents the set of such parameterizations. Although nondrizzling and drizzling clouds have quite close values of the power b in the Z–LWC relationship, it is necessary to remember that their droplet size distributions are very different, which is reflected by the difference in the factor a by four orders of magnitude. This statistical approach provides little information about physical mechanisms responsible for the existence of so many parameterizations.
To reveal different physical regimes on the Z–LWC diagram, Krasnov and Russchenberg (2002, 2005) introduced an additional classification parameter, namely, the extinction coefficient
i1558-8432-47-2-591-eq1
which is proportional to the second DSD moment. To determine parameter α, the same DSD dataset was used, which was applied for the calculation of Z. This parameter α characterizes the part of the DSD related to smaller droplets. Small ratios Z/α correspond to the lack of large drops and to the presence of a large amount of small droplets (i.e., can be attributed to pure condensation stage of the cloud evolution). In contrast, the large values Z/α indicate the existence of drizzle. Thus, utilization of this additional classification parameter allows one to separate the Z–LWC diagram into three zones, corresponding to different microphysical processes in clouds. Figure 2 shows Z–LWC diagrams corresponding to different ranges of the ratio log10(Z/α). One can see that utilization of the additional classification parameter α allows one to separate the entire zone of the Z–LWC diagram into three zones, which were assumed to be related to different physical processes or stages of stratocumulus cloud evolution—diffusion growth, the beginning of drizzle formation, and heavy drizzle. One of the purposes of the numerical simulations was the reproduction of these regimes, as well as the Z–LWC diagram as a whole.

To improve the physical understanding of such properties of the Z–LWC diagrams and retrieval algorithms, several questions should be answered: 1) What are physical processes in clouds that lead to the Z–LWC diagrams shown in Fig. 1? 2) What are the reasons for the wide scattering, especially pronounced in the transition regime? Is this scattering related to the difference of environmental conditions and differences in cloud properties in different field campaigns or is the scattering an inherent property of a “single” stratocumulus cloud that develops under homogeneous environmental conditions? 3) Do Z–LWC relationships depend on the height within a cloud? 4) Can different regimes coexist at the same time or do they fully replace one another during cloud evolution? Under what conditions is one regime replaced by another one?

These questions will be addressed in numerical simulations using the model described in the next section. We will present a physical basis allowing one to understand the nature of each regime, as well as its characteristic properties.

3. A model of stratocumulus cloud

a. General characteristics

The model is described by Pinsky et al. (2006, 2008) in detail. Here only a short model description will be presented. The trajectory ensemble model (TEM) of the cloud-topped BL developed by Pinsky et al. (2006, 2008) combines the advantages of the Lagrangian and the Eulerian models. The main physical processes described by the model are schematically shown in Fig. 3. The specific feature of the model is that Lagrangian air parcels cover the entire boundary layer area and can be both droplet-free and cloudy. It contrasts with the state-of-the-art TEM (e.g., Feingold et al. 1998; Stevens et al. 1998; Harrington et al. 2000; Erlick et al. 2005), where Lagrangian parcels are only cloudy ones and are separated by significant distances. At t = 0 air parcels are distributed randomly over the whole area of the boundary layer and contain nonactivated aerosols only.

In ascending parcels crossing the condensation level, some fraction of aerosols activates and gives rise to droplet formation. In the course of parcel motion supersaturation can be replaced by undersaturation, for instance in downdrafts, and droplets evaporate partially or totally (denucleation). In addition to droplets, the parcels also transport potential temperature and humidity. In this way the model reproduces the interaction between the subcloud and cloud layers. Representation of the BL by a set of adjacent parcels allows us to take into account droplet sedimentation and effects of drizzle falling from above on the DSD in the parcels located below.

Both the dynamical (statistical and energy properties of turbulent-like velocity field) and thermodynamical characteristics of the boundary layer (temperature and air humidity) are assimilated from the observed data. Condensation (evaporation) changes temperature and humidity within parcels and leads to changes of the DSD. Motion of parcels changes the thermodynamic structure of the BL. This way, dynamical (turbulent like), thermodynamic, and microphysical processes turned out to be closely related.

As was mentioned above, the stratocumulus cloud is simulated in the model under given turbulent-like atmospheric boundary layer (ABL) dynamics with characteristic energy, time, and spatial correlation properties (see below for details). In the present study these properties 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 mechanisms of quite different time and spatial scales. Such factors as surface fluxes, radiation cooling, cloud-top entrainment, mean vertical velocities, and so on change with characteristic time scales (determined by synoptic situation) that are much longer than the time scale of the ABL mixing by large eddies (Stevens et al. 2003a). The characteristic rate of the cloud depth growth is a few meters per hour (Stevens et al. 2003b). The mean entrainment rate in the typical maritime stratocumulus (say, during the first research flight in DYCOMS-II) is about 0.4 cm s−1; the horizontally averaged vertical velocity was of the same order of the magnitude (Stevens et al. 2003a, b). At the same time the standard deviation of the vertical velocity related to large eddies and turbulent fluctuations is of approximately 0.5–0.8 m s−1 (Stevens et al. 2003a). The eddy turnover times (measured by the ratio between the ABL height and the characteristic convective velocity) are on the order of 10 min (Stevens et al. 1996, 2003a, b). It means that the values of supersaturation, droplet concentration, shape of droplet spectra, spatial variability of DSD, and other microphysical properties are determined mainly by the processes with characteristic scales much smaller than those determining the mean statistical properties of the BL dynamics. Besides, the cloud microphysical structure is often affected by the “macroscale” factors not directly, but rather indirectly via the influence on the dynamical (turbulent) structure of the BL.

We believe that the turbulent-like structure of the ABL rapidly adjusts to the environmental situation. So, this structure can be considered stationary (or quasi stationary) at time scales smaller than the “environmental” time scale.

In the current model version no interaction between parcels is assumed except the exchange related to the droplet sedimentation. Respectively, the question arises as to the time during which parcels conserve their identification. This problem was addressed in all TEM models (e.g., Feingold et al. 1998; Erlick et al. 2005). In these studies it was shown that the characteristic life time of cloud parcels is 10–20 min (reasonable results have been obtained even when the parcels were simulated during 40 min). In the present study we simulate stratiform clouds within quite a narrow cloud layer and the turnover time of about 10 min. It means that the mean residential time of parcels in the cloud layer is about 10 min. Below the cloud layer droplets evaporate and give rise to the formation of aerosol particles (AP) of another size distribution. It means that the cloudy parcel becomes a droplet-free one. When this new droplet-free air parcel ascends and new droplets form, the cloud parcel can be regarded as a new one. These considerations allow us to neglect turbulent mixing between parcels at the first stage of the model development.

We do not take into account the effects of latent heat release on the dynamic (turbulent) structure explicitly. Instead, we generate a turbulent-like dynamical structure that corresponds to that observed in the cloud-topped boundary layer (CTBL). In nature this dynamical structure is formed under the combined effect of latent heat release, entrainment, surface fluxes, and so on. Thus, assimilating the real dynamics, we take into account all factors affecting dynamics. We believe that the microphysical structure of clouds corresponds to the BL dynamics. Simulation of turbulent-like flows corresponding to different thermodynamic situations in the CTBL makes it possible to investigate the effects of the BL dynamics and aerosol properties on the microphysical structure of stratocumulus clouds.

Despite these limitations, Pinsky et al. (2006, 2008) showed that the model reproduces realistically the process of the cloud microphysical structure formation at time scales of several turnover times and the spatial scales of a few kilometers. We formulate the model dynamics, thermodynamics, and microphysics as follows.

b. Model dynamics

The velocity field in the BL caused by eddies of different nature is represented as the sum of a great number of harmonics. The mean horizontal velocity V0(z) is also taken into account. The expressions for the vertical W(x, z, t) and the horizontal V(x, z, t) velocity components are as follows:
i1558-8432-47-2-591-e4a
i1558-8432-47-2-591-e4b
where Cn and Dm are the coefficients described below, and M and N are the numbers of harmonics in the horizontal and the vertical directions, respectively. Time random fluctuations of the velocity field are given by independent autoregression series of the first-order am(t) and bm(t) as described by Pinsky et al. (2006, 2008).

The velocity field (4a)(4b) obeys the continuity equation; the vertical velocities obey boundary conditions W(x, 0, t) = W(x, H, t) = 0. The field obeys the cyclic conditions at the lateral boundaries. The velocity field is homogeneous in the horizontal direction (the correlation functions depend only on Δx = x2x1); it is statistically stationary in time and obeys ergodic properties—namely, the values averaged with respect to a great number of samples are equal to the mean values along the horizontal direction. The correlation functions of the velocity field in the horizontal direction can be expressed using the coefficients Cn and Dn.

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 (see Pinsky et al. 2008)
i1558-8432-47-2-591-e5
There are several empirical formulas for vertical profiles of 〈W ′2〉 (e.g., Lenschov et al. 1980). For current simulations, we have used profile 〈W ′2〉 = 3.2(z/H)2.2 × (1 − 0.8z/H)1.4. The 〈W ′21/2 maximum of 0.6 m s−1 is located at 500 m. Note that this profile is quite similar to that observed in the BL during field experiment DYCOMS II research flight 2 when both weak and heavy drizzle were observed (VanZanten and Stevens 2005).
The correlation properties of the velocity field determining the characteristic horizontal scale of velocity variation are determined by the correlation function of the vertical velocity in the horizontal direction. The shape of the correlation function is determined by coefficients Dm. In the present study, Dm has been chosen in such a way that the correlation function BWx, z) should meet the Kolmogorov theory,
i1558-8432-47-2-591-e6
where ε is the dissipation rate. The calculation of coefficients Dm was also performed using the expansion into the Fourier series.

The parameters of the flow can be efficiently tuned to obey the observations in the boundary layer. As a result, the dynamics of the model BL will correspond to the observed dynamics. The main parameters used in simulations of cloud formation are presented in Table 2.

c. Model microphysics and thermodynamics

The main microphysical components of a single Lagrangian parcel are described by Pinsky and Khain (2002). The microphysics of the parcel model includes the diffusion growth equation used for aerosol nuclei and water droplets, the equation for supersaturation, and the stochastic collision equation that is solved using the precise method proposed by Bott (1998). The size distributions of cloud particles (both nonactivated aerosols and droplets) are defined on a mass grid containing several hundred bins within the radius range of 0.01–1000 μm. The mass corresponding to each bin changes with time (height) according to the diffusion growth equation. The separation between droplets and nonactivated aerosols is reproduced automatically, when the largest aerosol particles start rapidly growing, while smaller particles remain to be in equilibrium with the environment. The Lagrangian approach used eliminates the artificial spectrum broadening typical state-of-the-art Eulerian models (see Khain et al. 2000), which use immovable mass grids. Collisions of droplets are calculated using a collision efficiency table with the resolution of 1 μm (Pinsky et al. 2001). The microphysical equations are solved both in the subcloud and cloudy layers of the BL. Droplet sedimentation and corresponding changes in droplet spectra in the parcels are calculated using the flux method. The differences between influxes and outfluxes calculated for each bin determine the changes of DSD in each parcel. The droplet fluxes are calculated taking into account the change of the interface lengths between the adjacent parcels during their motion.

As a result of diffusion growth–evaporation, temperature and humidity in parcels change. The motion of parcels in the BL together with heating and cooling of air in parcels due to condensation and evaporation leads to a well-mixed BL, in which the horizontally averaged lapse rate is close to the dry adiabatic one in the subcloud layer, and to the moist adiabatic within the cloud layer. At the same time, the temperatures and humidity are different in different parcels.

Initial AP distribution is represented either by a three-modal lognormal distribution (Hobbs et al. 1985; Respondek et al. 1995) or by tables taken from observations. As was shown in many studies (e.g., Segal et al. 2004; Segal and Khain 2006), aerosol size distributions observed under different conditions (both maritime and continental) can be approximated well by the proper choice of the mode parameters.

Supplemental simulations indicate that the model is able to reproduce realistically both nondrizzling and drizzling clouds.

4. Design of numerical simulation

We discuss below results of a simulation in which stratocumulus cloud develops and transfers from a nondrizzling to drizzling regime. This simulation allows us to investigate the mechanisms leading to the formation of regimes (areas) seen at the Z–LWC diagram of Figs. 1 and 2. The dynamical and microphysical parameters of the model used in these simulations are presented in Table 2. The ridged upper boundary is identified with the temperature inversion assumed to be at z = 1250 m. The relative humidity near the surface was set at 90% (q = 12.5 g kg−1).

The concentrations of cloud condensational nuclei (CCN) with the radii < 0.03, > 0.1, and > 1 μm were 4070, 130, and 1 cm−3, respectively. No giant CCN were assumed in the simulations. The size distributions of dry aerosols were assumed the same in all parcels. The initial size distribution of dry CCN is shown in Fig. 4. These AP parameters lead to the formation of cloud parcels with droplet concentration of a few hundred per cubic centimeter. This concentration is typical for the continental stratocumulus clouds.

5. The Z–LWC relationship formation

In the simulation, the mean depth of the cloud layer was about 550 m, which is large enough for drizzle formation. Analysis of DSD shape, as well as the field of the radar reflectivity, showed that drizzle started forming in ∼1.5 h and rain flux reaches its maximum at about 150 min.

As an example of the model output fields we present Fig. 5, showing the fields of radar reflectivity and vertical velocity at t = 170 min. One can see that 1) the radar reflectivity reaches +10 dBZ in zones of intense drizzle and 2) heavy drizzle tends to fall within the cloud columns. The formation of the columnar structure of the radar reflectivity is a widely observed feature of drizzling clouds (see, e.g., observations performed during DYCOMS-II). We attribute this feature to the following: note first that typical updrafts and downdrafts in stratocumulus clouds are of several tens of centimeters per second and are related to large eddies. This means that cloud droplets and even drizzle can hardly fall down within cloud updrafts. According to analysis of DSD, large droplets and small drizzle form first near cloud top and then fall down within the zone of cloud downdrafts (see the lower panel in Fig. 5). As a result, drizzle falls with velocities that may significantly exceed drop terminal fall velocities and reaches the surface during the period shorter that the lifetime of corresponding downdraft (which has the turnover time of about 10 min), and vertical bands of enhanced radar reflectivity arise. The separation distance between such columns of intense drizzling is about 1–3 km (Stevens et al. 2003a). Note that drizzle experiences also the influence of horizontal velocity, so that the location of drizzle near the surface is shifted in the horizontal from the location of maximum downdrafts.

We start analyzing the microphysical process leading to the formation of the radar reflectivity–LWC diagram discussed in section 2 (Figs. 1, 2a–c) with Fig. 6 showing the Z–LWC diagrams calculated by the model at different time instances. Each point in these diagrams represents one cloud parcel. Parcels located in the upper half of the cloud layer are marked yellow, parcels located in the lower half of the cloud layer are marked blue, and parcels located below lifting condensation level are marked green. One can see that during the first ∼80-min period, the Z–LWC diagram is similar to that presented in Fig. 2a. This type of radar reflectivity–LWC diagram is observed in nondrizzling clouds (Krasnov and Russchenberg 2005) and is related to the diffusional growth of droplets by condensation. By ∼80–90 min, the first drizzle forms in some “lucky” parcels located near the cloud top (i.e., these parcels ascend from the cloud base to cloud top experiencing supersaturation). These lucky parcels have maximum LWC and the widest DSD, showing larger radar reflectivity than the other parcels (see Fig. 6).

Further cloud development indicates the beginning of the second regime referred to above as the transition regime. Since drizzle has a significant fall velocity, drizzle sediments from one parcel (donor parcel) to other parcels located below (acceptor parcels).

Three main microphysical processes determine further evolution of the Z–LWC diagram:

  1. The first process is collisions of drizzle with smaller droplets within parcels–donors. This process leads to the formation of larger drizzle (and larger radar reflectivity) under nearly the same LWC. On the Z–LWC diagram this process corresponds to a shift of lucky parcels having the maximum LWC (yellow dots) in the vertical direction toward larger Z, as is shown in Fig. 6 by arrow 1 at t = 145 min.
  2. The second mechanism is as follows: drizzle falls down from parcels–donors and grows by collection of small droplets in parcels–acceptors. In this process, parcels–acceptors located in the middle and lower part of cloud layer get drizzle from above and, having relatively low LWC, increase their radar reflectivity. On the Z–LWC diagram this process is schematically shown by vertical arrows 2 at t = 145 min. When drizzle first falls into the subcloud layer, it forms relatively high radar reflectivity at very low values of LWC. This situation corresponds to parcels marked by green in Fig. 6 at t = 145 min. Such parcels start forming the upper boundary on the Z–LWC diagram. Drizzle size usually reaches its maximum at cloud base. When parcels containing large drizzle and having also large LWC descend to cloud-base level or below, they form the upper boundary of the Z–LWC diagram corresponding to the heavy drizzle regime (cf. at t = 170 min with Fig. 2c).
  3. The third process leading to the formation of the transition regime on the Z–LWC diagram is the drizzle (as well as LWC) loss by parcels–donors. As a result, both LWC and Z decrease. At the Z–LWC diagram this process is expressed schematically by the dashed arrow 3 in Fig. 6 at t = 145 min. It is clear that combined processes are also possible, when cloud parcels serve both as donors and acceptors.

One can see that the reason of the large scattering on the Z–LWC during transition and heavy drizzle regimes is the existence of several mechanisms of the DSD evolution after the first drizzle formation. Random mutual location of parcels–donors and acceptors may lead to very different DSDs with a wide range of ratios of smaller droplets and drizzle. Thus, the large scattering of the Z–LWC diagram is an inherent feature of all drizzling stratocumulus clouds. A one-to-one correspondence between Z and LWC does not exist in this stage in principle.

The relationship between the spatial location of cloud parcels in the BL and their location at the Z–LWC diagram at the developed drizzle stage is illustrated in more detail in Fig. 7. The upper-right panel shows the field of rain flux at 160 min. Number and mass drop size distributions are presented for three chosen cloud parcels located at different heights within the cloud layer. Note that DSDs presented in the figure include also wet aerosols (haze) with radii below 1 μm. The location of these parcels at the Z–LWC diagram is shown as well. One can see that DSD and mass size distributions in the parcel located at the upper level (parcel one) are quite wide and contain many cloud droplets and a tail of drizzle corresponding to the second LWC maximum in the mass distribution. This parcel is a parcel donor and has both large LWC and radar reflectivity. Large cloud droplets of approximately 20–25 μm contribute substantially to Z in this parcel. The second parcel located in the middle of the cloud layer has smaller concentration of cloud droplets but larger amount of drizzle than the first parcel. This parcel is the parcel–acceptor (and also donor for parcels located below). LWC in this parcel is smaller than that of the first one, but radar reflectivity is larger, because of the contribution of drizzle. The third parcel is located just below cloud base. It contains a small number of cloud droplets (i.e., droplets with radii below ∼25 μm) with negligible mass, but significant drizzle with radii of 100–200 μm, which is a dominating contribution to LWC. This parcel is an acceptor. Despite the fact that its LWC is approximately a factor of 20 less than that of the first one, its radar reflectivity is the highest, because of the larger size of drizzle. Such parcels form at the heavy drizzle stage. Most of the parcels forming the upper boundary of the Z–LWC diagram are formed by parcels located at cloud base or below cloud base (marked green). These parcels contain large drizzle, so that they produce large radar reflectivity even if their LWC is small. One can see that at this time all regimes coexist, which causes a significant scattering of points on the Z–LWC diagram (see Fig. 7), with a wide variation of Z from −30 to 10 dBZ, and LWC from 0 to 1.8 g m−3.

It should be noted that the upper boundary of the Z–LWC diagram corresponding to the developed drizzle stage indicates the functional dependence between the radar reflectivity and the LWC as in the diffusional growth stage (nondrizzling stage). This feature can be attributed to the fact that this dependence is caused by parcels located at and below cloud base, where the mass of cloud droplets is negligible (see DSD at Fig. 7), so that both LWC and radar reflectivity are determined by drizzle mode in the DSD.

The time-accumulated Z–LWC diagram is shown in Fig. 8. A comparison of Fig. 1 and Fig. 8 indicates that 1) the model is able to reproduce the observed Z–LWC diagrams with a high accuracy, including the reproduction of slopes of lines at different regimes and 2) different zones in the Z–LWC diagram often correspond to parcels located at different cloud levels. At a given Z, minimum LWC will take place below and near cloud base, intermediate LWC will take place in the lower and middle fraction of the cloud layer, and the maximum LWC will be within the upper part of the cloud layer.

The question may arise concerning the validity of the comparison of Z–LWC diagrams plotted using in situ data collected along horizontal flight legs of typically 10–15-km length with the model data calculated within the computational zone of 4-km width. Note that the characteristic size of large eddies determining to a large extent the variability of cloud microphysics is 500–1000 m. The computational zone includes several such large eddies. An aircraft crosses cloud zones with different DSD corresponding to different cloud stages. The model run continues for 4 h, during which the cloud evolves and passes through all stages of its evolution. Thus, in both cases statistical series contain different DSD corresponding to different cloud stages. Both statistics contain a wide range of all possible DSD. We believe, therefore, that both statistics are similar, which allows the comparison of model and observed Z–LWC diagrams.

6. Discussion and conclusions

A novel trajectory ensemble model is applied to the investigation of microphysical mechanisms leading to the formation of Z–LWC relationships derived from observations. It is shown that the model reproduces well all regimes (or classes) seen at the experimental diagrams. The results allow us to answer questions addressed in the study. The analysis shows that 1) different zones of the Z–LWC diagram are formed at different stages of cloud evolution. At the same time, at the transition period, and especially at the final stage (heavy drizzle), all regimes can coexist. 2) Different levels within the stratocumulus-topped BL can be characterized by different Z–LWC relationships. 3) A significant scattering at the Z–LWC diagrams at “transition” and heavy drizzle regimes reflect an inherent property of stratocumulus cloud related to the variability of DSD. This variability is related to the process of collisions of drizzle within parcels–donors and process of droplet collection by drizzle within parcels–acceptors by drizzle falling from parcels–donors. DSD arising at these stages have different relationships between amounts of relatively small and large drops. 4) Since the observed Z–LWC diagram was obtained under different thermodynamic and aerosol conditions, the excellent reproduction of the main features of the Z–LWC diagram in the numerical simulation (thermodynamic conditions differ somehow from those in different observations) indicates that the Z–LWC diagram is determined by general properties of DSD.

Note that the upper and lower boundaries of the diagram of the model Z–LWC relationship are very sharp with the slopes corresponding to parameter b in (3) close to 2, which agrees with observations for nondrizzle and heavy drizzle cases (see Table 1). We believe that these dependences contain important information about the general properties of DSD expressing the relationships between the third (mass) and sixth (radar reflectivity) moments of DSDs. At present there are three main approaches to derive parameterization dependencies Z(LWC): (i) in situ data, (ii) choice and justification of appropriate DNS shapes (e.g., Krasnov and Russchenberg 2002, 2005), and (iii) utilization of the numerical model providing DSD in each point of the boundary layer. Deriving improved parameterizations requires significant efforts and will be investigated in a separate study. We believe that the utilization of additional information like height dependencies of radar reflectivity obtained from radar measurements can be used for deriving vertical profiles of LWC(Z).

The results of the numerical simulation indicate a significant importance of the numerical analysis of microphysical processes for remote sensing, and in particular, for the retrieval of LWC using the radar reflectivity. For instance, the results showed that some decrease in the scattering of Z–LWC diagrams can be achieved, if the diagrams will be separated with respect to height within the BL (or better, with respect to the distance above cloud base). The results showed, however, that it is impossible to find a functional Z(LWC) dependence without significant scattering at transition and heavy drizzle regimes by any classification algorithms using the data within the entire cloud layer. It can be attributed to the fact that the existence of the significant scattering reflects an inherent property of clouds, namely that the relationships between the amount of small and large drops in DSD (i.e., between Z and LWC) start to be substantially random within a cloud layer at drizzle stage.

As it was discussed, the model dynamics (energetic and correlation properties of the wind field in the BL) can be tuned to observations. The intensity of velocity fluctuations and correlation properties of the velocity field can be adapted to the observations in the cloud-topped BL. Both in situ measured as well as radar and lidar data can be applied. As a result, the investigations can be performed within a closed “observation–simulation–observation” cycle as illustrated in the scheme presented in Fig. 9. After prior analysis the dynamical data observed (first of all, the series of Doppler vertical velocity and lidar data) should be adopted. The adaptation of the dynamics means that the parameters of the model velocity field will be calculated to have the same energetic and correlation properties as the velocity field observed. The parameters of aerosol size distributions and sounding data (the temperature, the dewpoint, and the wind velocity profiles) will also be assimilated by the model. As a result, the model will be tuned to the observational input. The results of simulations should be compared with corresponding measured microphysical, radar, and lidar cloud characteristics in the course of cloud evolution. Thus, the cloud-topped BL model will be used as an efficient connecting link between observed data of different type (radar, lidar, aircraft, etc.). Despite the fact that the Z–LWC diagram shown in Fig. 8 is of general nature and reflects the inherent properties of all stratocumulus clouds, the particular Z–LWC diagrams will be different for different environmental conditions and different stages of cloud development (as is shown in Fig. 6). For instance, in case of high aerosol concentration only the first regime corresponding to the diffusional growth will exist. At the same time, lower aerosol concentration or higher humidity will determine the heavy drizzle formation with larger values of Z and LWC. Dynamical, thermodynamical, and aerosol properties of the BL have to determine cloud depth, droplet concentration, vertical profiles of liquid water content, and other cloud properties, such as the averaged drizzle rate, spatial variability of DSD, and so on. So, the results of comparison, on the one hand, will foster better understanding of cloud microphysical processes, and, on the other hand, will allow one to improve both the interpretation of remote sensing data and retrieval algorithms.

Acknowledgments

The data used in this paper were collected during the CLARE’98 campaign carried out under the auspices of the European Space Agency (ESA 1999). The FSSP and 2DCP datasets for this campaign were kindly provided by P. Francis from the Met Office. We express our gratitude to Bjorn Stevens, who provided us with access to the site with the airborne measurements obtained from the NSF/NCAR RAF EC-130Q aircraft during the DYCOMS-II project. The CAMEX-3, DC-8 FSSP, and 2DC data are provided by the GHRC at the Global Hydrology and Climate Center, Huntsville, Alabama. The study was performed under the support of the Netherlands Space Agency (SRON) and the Dutch National Research Program Climate Changes Spatial Planning, the Israel Academy of Sciences (Grants 173/03 and 950/07), and the Israel Ministry of Sciences (the German–Israel collaboration in Water Resources; Grant WT 0403).

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Fig. 1.
Fig. 1.

The Z–LWC relationship according to in situ measurements in water clouds. The notation of the fitting lines is given in Table 1.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 2.
Fig. 2.

The 2D diagrams of Z–LWC relation [with mean (solid line), std devs (dotted curves), and linear fittings (dashed curves)] for (a) nondrizzling clouds, (b) “the cloud with light drizzle” class (“transition” drizzle regime), and (c) “the cloud with heavy drizzle” class (drizzle regime). The categorization has been done using the radar reflectivity to the optical extinction ratio: (a) log10(Z/α) < −1; (b) −1 < log10(Z/α) < 1.8; and (c) log10(Z/α) ≥ 1.8 (after Krasnov and Russchenberg 2002, 2005). The notations of the fitting lines are given in Fig. 1 and Table 1.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 3.
Fig. 3.

Physical processes described by the TEM (Pinsky et al. 2006, 2008). Adjacent Lagrangian parcels containing the full warm microphysics and cloud–aerosol interaction are advected in the boundary layer by the time-dependent turbulent-like wind field generated by a model of turbulent flow. The largest droplets forming in some Lagrangian parcels fall through the interfaces into and collect droplets located within other parcels. As a result, drizzle forms and falls down. When drops fall within undersaturated air they can evaporate partially or totally. In the latter case, new aerosols are formed. In the model, dynamical properties of the boundary layer are assumed to be given. The cloud microphysics is adjusted to the model dynamics and aerosols properties.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 4.
Fig. 4.

Initial size distribution of dry aerosols.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 5.
Fig. 5.

Fields of (top) radar reflectivity and (bottom) vertical velocity at t = 170 min.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 6.
Fig. 6.

Some Z–LWC diagrams calculated by the model at different time instances. Each point in these diagrams represents one cloud parcel. Parcels located in the upper half of the cloud layer are marked yellow, parcels located in the lower half of the cloud layer are marked blue, and parcels located below the lifting condensation level are marked green. Changes in the Z–LWC relationships related to different processes are marked by arrows (see text for detail).

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 7.
Fig. 7.

(top right) The field of rain flux formed by different cloud parcels at the developed drizzle stage (160 min). (left six panels) (left) Number dN/d log10r and (right) mass dLWC/d log10r drop distributions are presented for three chosen cloud parcels located at different heights within the cloud layer. (bottom right) The location of the parcels in the Z–LWC diagram is shown. In the Z–LWC diagram, parcels located in the upper half of the cloud layer are marked yellow, parcels located in the lower half of the cloud layer are marked blue, and parcels located below the lifting condensation level are marked green.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 8.
Fig. 8.

The time-accumulated (over 4 h) Z–LWC diagram calculated in the model.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Fig. 9.
Fig. 9.

Possible coupling of the model and observations for improving retrieval algorithms and investigation of cloud microphysics.

Citation: Journal of Applied Meteorology and Climatology 47, 2; 10.1175/2007JAMC1701.1

Table 1.

Some fittings of parameters of the Z–LWC relationship Z = aLWCb using in situ measured data.

Table 1.
Table 2.

The main parameters used in simulations of cloud formation.

Table 2.
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