## 1. Introduction

Shallow marine trade cumulus are one of the most prevalent cloud types on Earth and play an important role in establishing the thermodynamic structure of the lower atmosphere in the trade latitudes. To first order, the trade inversion structure is established through a balance between the cooling and moistening effects accompanying the evaporation of trade cumulus updrafts and warming and drying from subsidence (Riehl et al. 1951; Stevens 2007). By reorganizing moisture through the boundary layer, shallow cumulus (both marine and continental) may play a role in preconditioning the atmosphere for episodes of deeper convection (Rickenbach and Rutledge 1998; Wu et al. 2009). Although precipitation from deep convection has received the most attention, studies have shown that shallow cumulus clouds are the most common type of precipitating clouds (by sheer number) over the western tropical Pacific (e.g., Rickenbach and Rutledge 1998; Johnson et al. 1999). Satellite observations from the National Aeronautics and Space Administration (NASA) Tropical Rainfall Measuring Mission (TRMM) project support this conclusion about the ubiquity of shallow marine cumulus. Short and Nakamura (2008) found modes of cloud-top height at 2–3 km (corresponding to trade cumulus) and near 5 km (the cumulus congestus mode), with the shallow mode nearly always present. The shallow and congestus modes together contribute about 20% of the total tropical oceanic precipitation, although the bulk of the rainfall tends to come from the deeper congestus clouds.

Because shallow cumulus cloud tops typically lie below 4 km in altitude (and below the 0°C isotherm), their microphysical processes are governed by the well-established theory of warm-rain microphysics, which forms the basis of cloud physics parameterizations used in numerical models. The accuracy of the representation of microphysical and dynamical processes in numerical models depends on the grid scale. Short-range (limited area) forecast models give better results when horizontal grid spacing approaches about 1 km. For example, the Met Office model has shown that finer grid spacings lead to substantial improvements in forecasting rainfall events, especially when the synoptic forcing is strong (Lean et al. 2008). Grid spacings of ~1 km may not be sufficient to fully resolve individual convective clouds; nevertheless, higher resolution aids in representing structures such as cold pools (Savic-Jovcic and Stevens 2008; Wang and Feingold 2009) that accompany mesoscale organization. More realistic kinetic energy spectra are also captured with horizontal grid spacing less than 10 km (Terasaki et al. 2009).

The use of high-resolution simulations may be currently practical in the research mode; however, present computer capacity precludes cloud-resolving representations of moist convection in long-term, real-time forecasts over large areas. In addition, higher resolution alone does not guarantee improved forecasts, as low forecasting skill is commonly attributed to inadequacies in parameterizations of moist physical processes such as microphysical and convective parameterizations. It remains vitally important to improve the accuracy of traditional microphysical and convective parameterizations.

Previous microphysical parameterizations have been developed for application at a *point in the cloud*, generally taken to be represented by the model grid point. The larger grid spacings used by mesoscale models [~(2–20) km] may contain substantial unresolved subgrid-scale (SGS) variability. For nonlinear process rates like autoconversion, using grid-mean values of the microphysical variables can lead to substantial bias in process rates (Pincus and Klein 2000; Larson et al. 2001a; Wood et al. 2002; Larson et al. 2012) and radiative (Cahalan et al. 1994; Jeffery and Austin 2003) quantities. This bias is a consequence of Jensen’s inequality [see the comprehensive discussion in Larson et al. (2001a)], which, for either convex or concave functions, is expressed as

Unbiased process rates can be evaluated by integrating over the SGS probability distribution functions (PDFs) for the parameterization variables. Previous studies have established PDFs of atmospheric variables from observations (Grossman 1984; Eck and Mahrt 1991; Price and Wood 2002; Lee et al. 2010; Larson et al. 2001a,b) and numerical models (Lewellen and Yoh 1993; Xu and Randall 1996; Wang and Stevens 2000). For process rates that are a function of multiple variables, the knowledge of the joint probability distribution function (JPDF) of the different variables becomes an essential component of the parameterization.

The magnitude of the SGS variability bias is strongly dependent upon the nonlinearity of the process rates and the character of the SGS cloud variability reflected in the shape of the PDFs. Thus, an accurate cloud parameterization designed for use in a mesoscale model requires that cloud variability be taken into account. SGS cloud variability is most commonly represented using a PDF-based approach, where SGS distributions of different quantities are represented as probability density functions (Tompkins 2002; Golaz et al. 2002; Cheng and Xu 2009). At this point in time, the most sophisticated of these PDF-based approaches is the Cloud Layers Unified by Binormals (CLUBB) parameterization (Golaz et al. 2002; Larson and Golaz 2005). CLUBB assumes a joint PDF of vertical velocity, liquid water potential temperature, and total water.

Any PDF-based approach makes assumptions about the shape of the distribution. Most PDF parameterizations have assumed a prescribed analytical form for the PDF: delta functions (Randall et al. 1992; Lappen and Randall 2001), Gaussian (Sommeria and Deardorff 1977), and double Gaussian (Lewellen and Yoh 1993; Golaz et al. 2002). CLUBB employs joint double-Gaussian distributions in *θ*_{t}*–q*_{t}*–w* space (Golaz et al. 2002), which are constrained from in situ aircraft observations and large-eddy simulation (LES) output (Larson et al. 2002). Cloud observations gathered over a wide variety of conditions show substantial variations in cloud water distributions, which may be unimodal or multimodal, and either symmetrical or skewed (Tompkins 2002).

Although a wide variety of PDFs have been tested, the data sources used to derive the PDFs have significant limitations because of the difficulties in obtaining comprehensive and integrated datasets over the wide range of scales. Most previous observational studies have included either in situ measurements by aircraft (Davis et al. 1996; Wood and Field 2000; Larson et al. 2001a), or retrievals from satellite instrumentation (Wielicki and Parker 1994; Barker 1996; Barker et al. 1996). Each of these data sources has its limitations. For example, aircraft provides a one-dimensional path through a cloud field, possibly leading to problems of undersampling and unrepresentative sampling. The currently available satellite data also have limitations with respect to retrieving vertical profiles of water vapor and cloud microphysical variables. Previous work has concentrated on constraining PDFs of thermodynamic quantities, for example, *θ*_{t}*–q*_{t}*–w* space, with condensate associated with only a portion of that joint parameter space. Until recently (e.g., Larson and Griffin 2013; Lebsock et al. 2013; Boutle et al. 2014), relatively little research has been performed on characterizing distributions of the cloud microphysical variables themselves [e.g., mixing ratios of cloud and rainwater (*q*_{c}, *q*_{r}) and cloud and rain drop concentration (*N*_{c}, *N*_{r}) in order to constrain the joint SGS variability of the microphysical quantities].

Cloud-resolving models (CRMs) can provide more complete datasets that can be used to evaluate the assumptions underlying PDF approaches. Compared to observations, 3D CRM datasets are dynamically balanced^{1} and respond to the specified thermodynamic conditions (Lewellen and Yoh 1993; Xu and Krueger 1991; Xu and Randall 1996). However, CRMs still resolve cloud-scale motions on a rather coarse grid (~1 km), and they neglect cloud variability at scales smaller than this because of the simple “binary” assumption of either zero or overcast fractional cloud cover in each grid cell. Another limitation of typical CRMs is the use of generic bulk microphysics parameterizations, which may not be specifically tuned to peculiarities of particular cloud types of interest (e.g., trade cumulus).

We avoid these drawbacks by employing an LES model, configured with grid spacings an order of magnitude finer than those typically used by CRMs. Furthermore, the LES employed in this study uses a bulk microphysical parameterization^{2} developed from size-resolving (bin) microphysics simulations especially formulated for shallow cumulus clouds (Kogan 2013). Output datasets obtained from the bulk microphysics LES model are used to develop PDFs of microphysical variables and scaling (closure) relationships with grid-mean variables. The closure relationship specifies how grid-mean variables in the mesoscale model determine the particular PDFs used to represent SGS variability. We demonstrate that using JPDFs of cloud microphysical variables obtained from LES results in greatly more accurate process rates for typical mesoscale model grid sizes. We begin by first quantifying 1D PDFs. Although substantial previous work exists along these lines, we nevertheless first explore 1D PDFs because of our finding that some JPDFs can be well expressed by the product of two 1D PDFs.

## 2. Model and dataset

All simulations used to develop and evaluate the PDFs were based on a version of the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS) LES (Kogan et al. 1995; Khairoutdinov and Kogan 1999) called the System for Atmospheric Modeling—Bulk Microphysics (SAMBM; Kogan 2013). The dynamical core of SAMBM consists of the System for Atmospheric Modeling (SAM), developed by M. Khairoutdinov (Khairoutdinov and Randall 2003). The bulk formulation of microphysics in SAMBM uses the microphysical parameterization for shallow cumulus described in Kogan (2013). Similar to the bulk parameterization of Khairoutdinov and Kogan (2000), this new parameterization is based on multiple nonlinear regression of output from LES run with size-resolving (bin) microphysics. The bulk autoconversion, accretion, and self-collection rates, along with expressions for terminal velocity, are all formulated from the bin microphysics results. The new parameterization has been tested against simulations using the CIMMS LES with explicit microphysics (SAMEX; Kogan et al. 2012) in case studies of northeast Atlantic marine stratocumulus [the Atlantic Stratocumulus Experiment (ASTEX); Albrecht et al. 1995] and marine trade cumulus based on the Rain in Cumulus over the Ocean (RICO) field campaign (vanZanten et al. 2011).

Initial profiles, large-scale vertical velocity, and tendencies of temperature and moisture corresponding to average conditions over the 3-week period from 16 December 2004 to 8 January 2005 of the RICO field campaign were used for initialization and forcing of the LES [described in detail in the LES model intercomparison study by vanZanten et al. (2011)]. Initial cloud condensation nuclei (CCN) concentration was assumed to be 104.4 cm^{−3}, a number similar to the concentrations measured by the Passive Cavity Aerosol Spectrometer Probe (PCASP) during the RICO flight RF11 on 7 January 2005 (J. Hudson 2010, personal communication). CCN concentration is a prognostic variable in the SAMBM and thus is allowed to evolve via advection, turbulence, activation, and drop evaporation; it also evolves because of coalescence processing over the course of the simulation [see Eq. (1) in Kogan (2013)]. The model horizontal and vertical grid spacings for the simulation were 100 and 40 m, respectively, with a total of 512 × 512 × 100 grid points (51.2 × 51.2 × 4 km^{3} domain). The dynamical time step was 2 s. The simulation was run for 24 h, and model output from the last 12 h was employed in constructing the PDFs. Over the 24-h simulation, this single case produced thousands of clouds having a wide range of depth, updraft intensity, and precipitation production.

## 3. One-dimensional PDFs

We first obtain the 1D (single variable) PDFs. Analysis of the 1D PDFs helps in understanding the variation of PDFs in the vertical, as well as the dependence of the PDFs on cloud thickness, which is one of the major factors controlling precipitation and the distribution of cloud microphysical parameters. As Fig. 1a shows, precipitation rapidly increases in the second half (12–24 h) of the simulation, a period which contributes about 86% of the total accumulated surface precipitation. Since our focus is on precipitation parameterization, the analysis dataset is selected from clouds evolving during the second half of the simulation. The analysis included a total of 610 clouds, each with a horizontal cross section^{3} larger than 100 grid points.

Figure 1b shows the evolution of the mean and standard deviation of cloud top heights. The bulk effect of the cumulus clouds is to deepen the boundary layer, which is reflected in the increase of cloud top with time. Trade cumulus act to warm and moisten the subcloud layer and to cool and moisten the cloud layer (Siebesma et al. 2003; Stevens 2007). As the environment destabilizes with time, clouds grow higher, produce greater amounts of liquid water content, and lead to larger precipitation rates and total accumulated precipitation.

To evaluate contribution of clouds of different thickness to precipitation, we divide the dataset into four groups, each containing approximately equal numbers of clouds (an exact division is not possible because of discretization of cloud-top heights using the 40-m vertical grid). Distribution of clouds by cloud-top heights is shown in Fig. 2. The first quartile (group 1) includes clouds with tops lower than 2.1 km, the second quartile (group 2) is between 2.14 and 2.26 km, group 3 is between 2.30 and 2.42 km, and group 4 has tops higher than 2.46 km (the four groups are also referred to as G1–G4).

The cloud-thickness discretization introduces up to 20% variation in the number of clouds in each group (Fig. 3a). Figures 3b and 3c show that the mean horizontal cross section and volume of a cloud in the G4 category are more than double that of any of the other cloud categories. The mean precipitation flux at cloud base of a cloud in G4 is about twice that of a cloud in G3, and three times that of a G2 cloud (Fig. 3d). The effects of both larger precipitation rate and cross section of clouds in the G4 category combine to result in a precipitation contribution from the G4 clouds (Figs. 3e,f) substantially larger than the contribution from the other cloud categories. The total precipitation flux produced by all clouds in G1 is only 3.2% of the total amount from G2, G3, and G4 (Fig. 3f). For this reason, the development of our PDF parameterization will be based on the analysis of clouds from G2 to G4. Even though the precipitation contribution from the smaller G1 clouds is insignificant, nevertheless, their contribution to cloud cover is substantial (about 16%; Fig. 3b), and these smaller clouds cannot be neglected in studies addressing cloud albedo. The latter topic, however, is beyond our current investigation.

For PDF calculations, cloud variables in the 4-km model vertical domain were subdivided into ten 400-m-thick layers. Using the 400-m-thick layers substantially increases the statistical confidence of the calculations compared to calculating PDFs for every model vertical level. The clouds in category G2 span four of these analysis layers, while G3 and G4 cloud categories span five and six layers, respectively. Layers 1 and 2 are mostly below cloud base, layers 3–6 cover cloud base and the middle of the cloud layer, and layers 7 and 8 represent regions near cloud top (for G3 and G4 categories).

We calculated PDFs of three cloud variables: cloud drop number concentration *N*_{c}, cloud water mixing ratio *q*_{c}, and rainwater mixing ratio *q*_{r}. These quantities enter the expressions for microphysical autoconversion and accretion rates, and accounting for their variability (as described by PDFs) is an important aspect of calculating unbiased microphysical process rates over a mesoscale NWP grid. PDFs of the different microphysical variables were found by first normalizing by the mean value over each 400-m layer. The PDFs of normalized parameters were binned into 15 regularly spaced categories over the normalized range from 0 to 3.

### a. PDF of cloud-drop concentration

The vertical structure of the *N*_{c} distributions over the 10 vertical layers is shown by contouring values of equal probability (Fig. 4). This approach is conceptually similar to the contoured frequency by altitude diagram (CFAD) plots of Yuter and Houze (1995), though here we apply it to microphysical variables rather than reflectivity. The PDF at each level was calculated if the number of cloudy points in the dataset^{4} exceeded a threshold value of 100. The bottom layer and the upper two layers did not contain sufficient cloud points above this threshold and therefore are not included in the PDF calculations. Qualitatively, the PDFs for G2–G4 exhibit similar structure with a pronounced maximum near cloud base and increasing PDF width toward cloud top, especially pronounced for the deepest clouds (G4). This similarity may explain why the use of a PDF derived for one particular group (G4) provides a good approximation of grid-averaged conversion rates for all clouds in G2–G4 (see section 5).

*R*

^{2}≥ 0.94 for all layers, except for the top-most layer, where

*R*

^{2}= 0.81; see Table 1) of the PDFs for any given layer:

Best-fit parameters and errors for approximations of drop concentration distribution for G2–G4. The variables *A*, Xc, and *W* denote the amplitude, the mode, and the width of the Gaussian function [Eq. (1)], while the numerical subscript refers to the group number. The square of the correlation coefficient (*R*^{2}) is also referred to as the coefficient of determination.

The parameter *y*_{0} here is used to scale the function *f*(*x*) to 1; the other parameters of the distribution shown in Table 1 exhibit consistent height dependence. In particular, the width (*w*) of the distribution increases with height, and the mode (*x*_{c}) and amplitude (*A*) slightly decrease with height.

### b. PDF of cloud water mixing ratio

*N*

_{c}and shows little systematic variation, except perhaps a slightly narrower distribution in the middle of the cloud. Individual PDFs are shown in Fig. 7. We approximate the cloud water PDFs for all layers using a lognormal function:The parameters here have the same definitions as for the Gaussian function [Eq. (1)]. As an example, Table 2 shows the best-fit parameters for the lognormal approximation to the cloud water distribution in layer 5 for G2–G4.

### c. PDF of rainwater mixing ratio

The PDFs of rainwater mixing ratio shown in Figs. 8 and 9 are similar to those for the cloud water mixing ratio. Specifically, the rainwater distributions are asymmetrical with positive skewness. However, the range of rainwater distributions is narrower than for cloud water, and the distributions display a somewhat more pronounced broadening near cloud top, especially for G3 and G4. As for cloud water, the lognormal function provides a good fit (*R*^{2} ≥ 0.95) for the simulation-derived rainwater PDFs. Row 2 of Table 2 shows parameters of the lognormal distribution in layer 5 for G2–G4.

## 4. Two-dimensional PDFs

*q*

_{c}and

*N*

_{c}(input variables for autoconversion) and for

*q*

_{c}and

*q*

_{r}(input variables for accretion). We also assess the extent to which the full two-dimensional PDF can be approximated by the product of two one-dimensional PDFs associated with the individual variables.

Figure 10 shows the JPDF of cloud mixing ratio *q*_{c} and cloud drop number concentration *N*_{c}, the microphysical variables employed in some autoconversion parameterizations. For comparison we also show the 2D PDF, which is a product of the two one-dimensional PDFs of *q*_{c} and *N*_{c}. The importance of using the fully two-dimensional JPDFs is obvious. The PDFs in the right panels exhibit distinct ridges aligned parallel to the *x* and *y* axes, which are artifacts of approximating the 2D PDF as the products of the two 1D PDFs [i.e., the functional form *f*(*x*)*g*(*y*)]. The 2D JPDFs on the left panels, on the other hand, are fully two-dimensional PDFs of the form *h*(*x*, *y*) and do not contain the artificial ridge structures. The shape of the PDFs shows an elongated distribution on the *x* axis (*q*_{c}), compared to more narrow distribution of the *N*_{c} parameter on the *y* axis. The difference between *f*(*x*)*g*(*y*) and *h*(*x*, *y*) indicates that the two quantities covary and thus are not independent. For example, the modal value of *q*_{c} increases with increasing *N*_{c}, whereas the modal value of *q*_{c} is fixed in the products of the two 1D PDFs. We note that the 2D JPDFs are rather consistent across all four layers, suggesting the possibility that cloud variability in *q*_{c} and *N*_{c}, might be approximated by a single, height-independent JPDF.

The fully two-dimensional JPDFs of *q*_{c} and *q*_{r} (the quantities used to calculate accretion in bulk models) in Fig. 11 show a somewhat different behavior than for *q*_{c} and *N*_{c}. The JPDFs exhibit greater variation with height, a result of the differences in the *q*_{r} distribution at different layers. In spite of the more complicated vertical variability, representing the 2D PDF by a product of the individual 1D PDFs is a better approximation for *q*_{c} and *q*_{r} than for *q*_{c} and *N*_{c}. (Fig. 11, right panels). This result reflects the weaker correlation between *q*_{c} and *q*_{r} compared to that between *q*_{c} and *N*_{c}. From the physical point of view, the increase in *q*_{c} due to condensation frequently involves droplet activation and a corresponding increase in *N*_{c} at the same point. On the other hand, growth of *q*_{r} proceeds through the coalescence of small droplets (autoconversion), sedimentation of the embryonic precipitation drops, and collection of cloud water while falling through the cloud. As a result of these processes, the maximum in the *q*_{c} profile occurs in the upper third of the cloud, whereas the *q*_{r} profile maximum occurs lower in the cloud. The correlation between cloud and rainwater may further weaken in subsaturated downdraft regions where cloud drops evaporate more easily, but rain drops continue to grow by coalescence (Kogan et al. 2012). This spatial decorrelation between *q*_{c} and *q*_{r} over small grid volumes (the current LES uses 100-m horizontal and 40-m vertical grid) does not contradict the fact that deeper clouds with more cloud water produce, as a whole, more rain (see Fig. 3). *CloudSat* data, on the other hand, exhibit a robust correlation between cloud and rainwater (Lebsock et al. 2013). This apparent dilemma as to the correlation or noncorrelation of *q*_{c} and *q*_{r} is ultimately an issue of scale, specifically, the sampling volume over which the correlation is evaluated. Small sampling volumes such as our LES grid volume or the sampling volumes associated with aircraft observations (Boutle et al. 2014) tend to exhibit weak correlation between *q*_{c} and *q*_{r}. The larger analysis volume used by Lebsock et al. is based on *CloudSat* observations; the latter are characterized by a vertical sampling interval of 480 m and a horizontal pixel size at the surface of 1.4 × 1.7 km^{2}. This *CloudSat* sampling volume is three orders of magnitude larger than our LES grid volume and has an order of magnitude greater grid spacing in the vertical. The finer grids resolve the vertical decorrelation between *q*_{c} and *q*_{r}; the coarser sampling volumes do not.

In the next section, we apply these PDFs to evaluate grid-mean microphysical process rates. The qualitative difference between the two PDF formulations (the “exact” JPDF and the 2D PDF approximated by the product of two 1D PDFs) is especially pronounced for the variables *q*_{c} and *N*_{c}, resulting in a substantial impact on the grid-averaged autoconversion rates. Compared to autoconversion, the accretion rate, which employs *q*_{c} and *q*_{r}, is less sensitive to details about how the PDF is formulated.

## 5. Testing of the PDF parameterization

### a. General formulation

*ϕ*and

*ψ*, defined aswhere

*q*

_{c},

*q*

_{r}, and

*N*

_{c}over the cloudy areas at each 400-m layer. The grid-mean conversion rates averaged over the cloudy areas are then given by double integrals; for example, the autoconversion rate is expressed asHere

*q*

_{c}and

*N*

_{c}, and

*φ*and

*ψ*space defined from relationshipIntroducing the PDF enhancement factor

*D*

_{au}, which is a direct measure of subgrid variability,and we can rewrite Eq. (5) as

*q*

_{c}is given in kilograms per kilogram and

*N*

_{c}is given per cubic centimeters. With

*α*≈ 4.2 and

*β*≈ −3, we can expect substantial difference between the exact expression given by Eq. (8) and the approximation based on grid-averaged parameters, which neglects contribution from the PDF enhancement factor

*D*

_{au}:

*q*

_{c}and

*q*

_{r}(Kogan 2013):The derivation for accretion rate is similar to the autoconversion rate:where

*q*

_{c}and

*q*

_{r}.

*q*

_{c}and

*q*

_{r}is weaker than correlation between

*q*

_{c}and

*N*

_{c}. This is evidenced in Fig. 11 by the greater similarity (especially over the lower layers) between the JPDF

*q*

_{c}and

*q*

_{r}. As a result, the general function

*γ*and

*δ*in the formulation of the accretion rate are close to 1, which makes the dependence Eq. (11) nearly linear. In this case, approximation (14) can be further simplified:We evaluate the formulation (14) of using the 1D individual PDFs to approximate JPDF of

*q*

_{c}and

*q*

_{r}and the formulation (15) where SGS variability is completely neglected in the section below using data from our simulation output.

### b. Testing of different PDF approximations

The impact of using different PDF formulations in calculations of the grid-mean microphysical process rates was evaluated by first calculating the conversion rates at each grid point and then averaging them for each layer. These calculations (one for autoconversion and one for accretion) establish the benchmark process rates. Each analysis layer consists of 512 × 512 × 10 points; however, for the RICO cloud system, the cloud fraction is about 0.1, so the number of cloudy points in each layer is only about 2 × 10^{5}. Based on the values of *q*_{c}, *N*_{c}, and *q*_{r} at each grid point, we calculate the exact grid-point autoconversion and accretion rates, which are then averaged for each 400-m layer to provide exact conversion rates in each layer. Applying this procedure for each hourly simulation output resulted in about 200 data points. These exact values of grid-mean autoconversion and accretion are compared with values calculated based on PDFs and layer-averaged parameters according to Eqs. (4)–(15).

The parameterized conversion rates are calculated for each of the three following approaches: 1) based on the JPDF; 2) based on the PDF approximated by a product of two 1D PDFs; and 3) based on the layer-mean variables *q*_{c}, *N*_{c}, and *q*_{r}. While calculations 1 and 2 use one or another formulation of PDFs for obtaining grid-averaged conversion rates, calculation 3 completely neglects SGS variability and, therefore, does not use PDFs at all.

Figure 12 shows the comparison of autoconversion and accretion process rates calculated using these three approaches. Accounting for the full variability via using the JPDF results in the most accurate values for both autoconversion and accretion rates (au2D and ac2D). Using a 2D PDF formed as a product of two 1D PDFs (au1D and ac1D) overestimates the autoconversion rates by an order of magnitude. Neglecting variability completely (auMean and acMean), on the contrary, underestimates the autoconversion rate (Fig. 12a).

The differences between accretion rates using the three different approaches are smaller (Fig. 12b), even in approach 3, where SGS variability is neglected. As mentioned before, this weaker sensitivity to SGS variability relative to the sensitivity found in the autoconversion calculation stems from the fact that 1) the correlation between variables *q*_{c} and *q*_{r} is weaker than between *q*_{c} and *N*_{c} and 2) the accretion process rate in Eq. (12) is given by a nearly linear function of *q*_{c} and *q*_{r}.

The errors of various approximations can be more clearly assessed in a plot showing ratios of approximated averages to the exact values (Fig. 13). The full JPDF provides the most accurate conversion rates, with only 7.5% and 10.5% mean error for autoconversion and accretion rates, respectively.^{5} Approximating the JPDF as a product of the individual 1D PDFs produces a substantial deterioration in autoconversion rates but shows a smaller effect on accretion (Fig. 13b). The range of bias increases for larger autoconversion rates, where it may lead to substantial errors in precipitation rate. Neglecting variability altogether (Fig. 13c) results in about 60%–80% underestimation of autoconversion, while large values of accretion are underestimated by up to 40% and small values are overestimated by up to 100%.

The comparisons shown in Figs. 12 and 13 were made using 1D and 2D joint formulations of PDFs, which demonstrated the superior performance of using a JPDF. However, both the JPDF and the less accurate formulation based on 1D PDFs were, nevertheless, calculated using the *exact finescale* distributions of cloud parameters at *each time step* and at *each layer*. In short, the PDFs employed in the above calculations were both time and height dependent, that is, they were ideally tuned to the clouds they were attempting to represent. In large-scale models, accurate, *time-dependent* information about PDFs is not available, necessitating the use of a simplified, a priori prescribed form of a PDF. We made sensitivity tests to evaluate the usefulness and accuracy of formulating such simplified JPDFs. In one case, we assumed the JPDFs were constant in time but varied in height. To obtain these fixed-in-time JPDFs, we used all 13 hourly data outputs from the 12–24-h simulation and calculated 10 PDFs for each 400-m layer. In the second case, the JPDF was constant in both time and height; specifically, JPDFs

Both simplified formulations of JPDFs were used in calculations of conversion rates that were compared with calculations based on the most accurate JPDFs, which depend both on time and height (referred to as exact JPDF). The results shown in Fig. 14a demonstrate that using a JPDF that is fixed in time but variable in height results in accuracy comparable to using exact JPDFs. The errors in estimating autoconversion rates (mean of 10.1% with standard deviation of 8.5%) are about the same as using the exact time- and height-dependent JPDF (7.5% ±7.8%). The errors in estimation of accretion rates are also comparable (13.9% ±10.7% versus 10.5% ±8.2% in the exact JPDF case). The use of an even simpler JPDF—fixed in both time and altitude (Fig. 14b)—produces rather accurate results for autoconversion rates (mean of 10.0% with standard deviation of 7.5%). For accretion rates, the “fixed height” JPDF formulation results in greater error (mean of 34.8% with standard deviation of 35.8%), which is unevenly distributed. For the smallest 50% accretion rates, the errors can be as large as 100%, while for larger accretion rates, the errors are in the 20%–30% range.

## 6. Conclusions

Unbiased calculations of microphysical process rates such as autoconversion and accretion in mesoscale NWP models require that SGS variability over the model grid volume be taken into account. This variability can be expressed as PDFs of microphysical variables, and the process rates can be integrated over these PDFs in order to obtain the unbiased grid-averaged process rates. Using dynamically balanced LES results from a case of shallow marine trade cumulus, we develop PDFs of the individual variables *q*_{c}, *N*_{c}, and *q*_{r}, as well as joint PDFs (JPDFs) of pairs of variables (*q*_{c}, *N*_{c}) and (*q*_{c}, *q*_{r}). The latter are used to calculate autoconversion and accretion, respectively, in bulk models. Both 1D PDFs and 2D JPDFs are best formulated as functions of their nondimensional parameters, that is, normalized using the layer-mean parameter values. Nondimensionalizing the variables reduces the height dependence and, hence, allows for more robust parameterization of PDFs.

One-dimensional PDFs are useful for microphysical parameterizations based on single moments of the drop size distribution. We demonstrate that these 1D PDFs can be approximated using analytical functions and analyze the variation of their parameters in the vertical. The 1D PDF results anchor our study to the previous observational and numerical modeling studies (discussed in section 1) characterizing PDFs. The 1D PDFs of droplet concentration *N*_{c} can be well approximated by a Gaussian function, which broadens somewhat with height. Cloud water (*q*_{c}) and rain (*q*_{r}) water mixing ratio distributions are highly skewed and can be represented by lognormal distributions, with relatively weak height dependence.

We find that JPDFs of *q*_{c} and *N*_{c} (inputs to the autoconversion) cannot be represented by a simple product of their individual 1D PDF; only the “exact” JPDF captures correctly the covariation between *q*_{c} and *N*_{c}. The product of the individual 1D PDFs of *q*_{c} and *q*_{r} (inputs to the accretion rate formula), on the other hand, is a better representation of the full joint JPDF of *q*_{c} and *q*_{r}. This lack of strong correlation between *q*_{c} and *q*_{r} arises from the vertical decorrelation of the two fields, since cloud water tends to be at a maximum over the upper part of the cloud and rainwater is greatest in the lower part of the cloud. For shallow boundary layer clouds, these details in the vertical distribution of *q*_{c} and *q*_{r} are not resolved by large sampling volumes such as those from *CloudSat*.

The process rates calculated using different formulations of the PDFs are compared to the benchmark process rate calculated using the point-by-point variability over the LES grid. Specifically, we calculate the bias in autoconversion and accretion rate. Using the JPDFs that are both height and time dependent produces the most accurate values for both autoconversion and accretion. Approximating the 2D JPDF by using a product of individual 1D PDFs overestimates the autoconversion rates by an order of magnitude, whereas neglecting the SGS variability altogether results in a drastic underestimate of the autoconversion rate. Differences in accretion rate among the different PDF assumptions are small compared to autoconversion rate differences, a result largely arising from the weaker correlation between *q*_{c} and *q*_{r}, and the near-linearity of the accretion rate formula.

Because of the impracticality of using PDFs that are evolving in time during the mesoscale model simulation, we evaluated two additional PDF approximations: 1) fixed in time but height dependent and 2) fixed in time and fixed in height. The latter option would be the easier to implement in mesoscale NWP because it would necessitate only a single JPDF form for each of the two pairs of microphysical variables. Results suggest that use of a single (fixed in time and height) JPDF form for each pair of variables gives an acceptable level of accuracy, especially for autoconversion. We thus recommend, and supply in the appendix, two 15 × 15 tables corresponding to JPDFs of the variable inputs for the autoconversion and accretion rate calculations. This recommendation is justified based on Fig. 14, which shows the errors are essentially the same as using the exact JPDFs for the autoconversion rates. Similarly, small error values are associated with large values of accretion, but small accretion rates are accompanied by substantially greater error. The influence of this accretion rate error on cloud evolution cannot be determined simply by comparing instantaneous error values; rather, the influence of the errors needs to be evaluated over the course of the full time-dependent mesoscale numerical simulation.

More accurate results can be obtained by using height-dependent JPDFs. In a numerical model one would need to use six tables, one for each 400-m-thick layer ranging from cloud base to the highest cloud top (layers 3–8). This approach would be straightforward in current NWP models. For brevity, we do not show these tables in the appendix but can supply them on request.

As a final comment, we note that the PDFs developed in this paper have a characteristic scale associated with the LES domain from which they are derived. In this case, the domain size corresponds to a horizontal domain size of 51.2 × 51.2 km^{2}. Our results (not shown here) demonstrated that major cloud system properties (cloud cover, time series of surface accumulated precipitation, and mean turbulent kinetic energy) are very similar in simulations with domains of 51.2 × 51.2 km^{2} and 25.6 × 25.6 km^{2} (256 points versus 512 points in each horizontal direction). Future efforts will characterize the scale-dependent variability of microphysical variables in more detail using several representative mesoscale model grid sizes.

The authors are grateful to three anonymous reviewers for many constructive comments. This investigation was supported by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreements NA17RJ1227 and NA08OAR4320904, the U.S. Department of Commerce, ONR Grants N00014-11-10439 and N00014-11-1-0518, the National Oceanic and Atmospheric Administration (NOAA) Climate Program Office (CPO) Climate Prediction Program for the Americas/Earth System Science Program (CPPA/ESS) Grant NA10OAR4310160, and the Department of Energy Office of Science Grant DE-SC0006736. The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma.

# APPENDIX

## Tabulated Values of the Joint Probability Distribution Functions

As results shown in Fig. 14b demonstrated, using JPDFs that are fixed in time and height provides accuracy comparable to using “exact” JPDFs for autoconversion rates (errors about 10%) and about 30% errors for the whole range of accretion rates. Over the most important range of middle and large values of accretion rates, the accuracy is higher and comparable to that of the exact JPDF formulation. The most simple and practical way to implement PDF approach in mesoscale models, therefore, is to use values of JPDFs

Joint PDF *φ* from 0 to 3 with an increment of 0.2; rows 1–15 correspond to values of *ψ* from 0 to 3 with the same increment of 0.2.

Joint PDF *φ* from 0 to 3 with an increment of 0.2; rows 1–15 correspond to values of *χ* from 0 to 3 with an increment of 0.2.

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^{1}

By “dynamically balanced” we mean a microphysical dataset that is produced across a realistic distribution of thermodynamical parameters: the latter can be generated in a 3D CRM or LES simulation.

^{2}

The “bulk” microphysics parameterization is based on the prediction of moments of the drop size distribution function, which represent bulk cloud characteristics such as cloud/rain liquid water content and cloud/rain drop concentration.

^{3}

Horizontal cross section is defined from liquid water path field using a threshold of 40 g m^{−2}.

^{4}

Note that the dataset included output from 13 time levels from the 12–24-h simulation.

^{5}

These remaining errors are due to the use of rather coarse resolution (only 15 × 15 bins) and limiting the PDF-normalized variables’ range to 3 times the mean value. Discretizing the PDF in greater detail (higher resolution and a wider range) would reduce the errors, but at the expense of a greater number of PDF bins to be integrated over. The selected number of bins (15) is a good compromise between accuracy and complexity of the discretized PDF.