The atmospheric portion of the hydrological cycle depends on the formation and dissipation of precipitation. In a numerical model, precipitation processes are represented by the microphysics process rates. These process rates are highly dependent on the values of hydrometeor fields at any place and time. Hydrometeors (such as rain water mixing ratio) can vary significantly on spatial scales smaller than the size of a numerical model grid box . This means that a good representation of subgrid-scale variability is important for the parameterization of microphysical process rates.

Subgrid-scale variability (but not spatial organization) can be accounted for through use of a probability density function (PDF). PDFs have been used in atmospheric modeling to account for subgrid variability in moisture and temperature e.g., in order to calculate such fields as cloud fraction and mean (liquid) cloud mixing ratio, and have been extended to vertical velocity in order to calculate fields such as liquid water flux . PDFs have been used in microphysics to account for subgrid variability in cloud water and in warm hydrometeor fields in order to calculate warm microphysics process rates. They also have been used to represent cloud ice .

Regarding the PDF's functional form, generality is highly desired. For instance, we would like the PDF to be capable of representing interactions among species, such as accretion (collection) of cloud droplets by rain drops. In addition, the PDF should be able to represent a variety of cloud types, such as cumulus and stratocumulus. Generality in the PDF's functional form is important because it facilitates the formulation of unified cloud parameterizations e.g.,.

Cloud Layers Unified By Binormals (CLUBB) is a single-column model that uses a multivariate PDF to account for the subgrid-scale variability of model fields . The original PDF used by CLUBB consisted of only vertical velocity, w, total water mixing ratio (vapor + liquid cloud), rt, and liquid water potential temperature, θl. The PDF is a weighted mixture, or sum, of two multivariate normal functions. Each one of these multivariate normal functions is known as a PDF component. Although a normal distribution is unskewed, the two-component shape makes it possible to include skewness in model fields.

extended CLUBB's PDF to account for subgrid variability in rain water mixing ratio, rr, and rain drop concentration (per unit mass), Nr. Each of these hydrometeor species was assumed to follow a single lognormal (SL) distribution on the subgrid domain. This treatment worked well for calculating microphysics process rates in a drizzling stratocumulus case . Subsequently, CLUBB's PDF was extended to other hydrometeor species involving ice, snow, and graupel.

However, the single lognormal treatment of hydrometeors is less successful when it is applied to a partly cloudy, precipitating case. The problem is that the single lognormal assumes that a hydrometeor is found (that is, has a value greater than 0) at every point on the subgrid domain. This is not realistic in a partly cloudy regime, such as precipitating shallow cumulus, which has non-zero precipitation over only a small fraction of the domain.

Consider an example in which rain covers 10 % of the grid level. Then the in-precipitation mean of rr is 10 times greater than the grid-mean value. This can cause problems when microphysics process rates are calculated using the SL. The accretion rate of rr is proportional to the value of rr inside cloud. In this example, the SL, which distributes the lognormal around the grid mean, would underpredict accretion rate because it causes rr to be too small in cloud. Likewise, evaporation rate is proportional to the value of rr outside cloud. The SL would overpredict evaporation rate because it spreads rr throughout the domain, including the clear portion.

The solution to this problem is to account for the non-precipitating region of the subgrid domain. This is done by representing the non-precipitating region of the domain with a delta function at a value of the hydrometeor of zero. The in-precipitation portion of the subgrid domain can still be handled by using a single lognormal distribution to represent subgrid variability in the hydrometeor species. The resulting distribution is called a delta-lognormal (DL). In the above example with 10 % rain fraction, the (in-precipitation) lognormal from the DL PDF would be distributed around the in-precipitation mean, as desired, rather than around the grid mean, which is a factor of 10 smaller.

Further improvements in accuracy can be achieved with relatively minor modifications to the PDF. As previously mentioned, CLUBB's PDF contains two components. Each of these components can be easily subdivided into an in-precipitation sub-component and an outside-precipitation sub-component. The result is a delta-lognormal representation of the hydrometeor field in each PDF component. Both delta functions are at zero and represent the region outside of precipitation, but the in-precipitation hydrometeor values are distributed as two lognormals that may have different means and/or variances. When the two lognormals differ in some way, the resulting distribution is called a delta-double-lognormal (DDL). Figure  illustrates the SL, DL, and DDL hydrometeor PDF shapes.

The main purpose of this paper is to present the formulation of an updated multivariate PDF that extends CLUBB's traditional PDF to include the DL and DDL hydrometeor PDF shapes. Additionally, a new method is derived to divide the grid-box mean and variance of a hydrometeor species into PDF component means and standard deviations. A secondary purpose of this paper is to present a preliminary comparison of the new PDF shapes with PDF output by large-eddy simulations (LESs). The SL, DL, and DDL hydrometeor PDF shapes are compared to histograms of hydrometeor data taken from precipitating LESs. Additionally, microphysics process rates are calculated using each of the idealized PDF shapes and compared to microphysics process rates taken from the LES.

The remainder of the paper is organized as follows. Section  gives a detailed description of the new PDF. Section  discusses the PDF parameters and includes the derivation of a new method to divide the grid-box mean and variance into PDF component means and standard deviations for a hydrometeor species. Section  describes the LES setup and the test cases, as well as the driving of CLUBB's PDF for the tests. Section  presents a comparison of hydrometeors between the LES and the SL, DL, and DDL PDF shapes. The comparison includes plots of PDFs, Kolmogorov–Smirnov and Cramer–von Mises scores, and microphysics process rates. Section  contains all conclusions.

We now describe how the multivariate PDF used by CLUBB is modified to improve the representation of hydrometeors. Perhaps the most important modification is the introduction of precipitation fraction, fp, to the PDF. Precipitation fraction is defined as the fraction of the subgrid domain that contains any kind of precipitation (where any hydrometeor species has a positive value). In order to account for any precipitation-less region in the subgrid domain, the PDF is modified to add a delta function at a value of zero for all hydrometeor species. Each PDF component contains its own precipitation fraction. Expressed generally for a PDF of n components, the overall precipitation fraction is related to the component precipitation fractions by fp=∑i=1nξifpi, where fpi denotes precipitation fraction in the ith PDF component, and where 0≤fpi≤1 for all fpi. Additionally ∑i=1nξi=1, where ξi is the relative weight, or mixture fraction, of the ith PDF component, and where 0<ξi<1 for all ξi. A PDF with more than one component requires that each PDF component have a mixture fraction.

Before writing the form of the multi-component PDF, we digress to discuss a special case, the cloud droplet concentration (per unit mass), Nc. In , Nc was introduced to the PDF and was assumed to follow a single lognormal distribution. This assumption for Nc means that when any cloud is found at a grid level, Nc>0 at every point on the subgrid domain. This is unphysical in a partly cloudy situation, because cloud droplets would be found at points where cloud water is not found. Additionally, the single lognormal treatment of Nc can cause problems with the microphysics. The grid-level mean of Nc, denoted Nc‾ (for the remainder of this paper, an overbar denotes a grid-level mean and a prime denotes a turbulent value), is handed to the PDF by the model, and this mean value includes clear air in a partly cloudy situation. This results in a value of Nc‾ that is much smaller than the in-cloud values of Nc. Since the single lognormal in Nc is distributed around Nc‾, Nc is much too small in cloud for cases with small cloud fraction, leading to an excessive autoconversion (raindrop formation) rate. In order to distribute Nc where (and only where) cloud water mixing ratio, rc, is found on the subgrid domain, it cannot use the same method as the other hydrometeors. Hydrometeors such as rr can be found outside clouds where rc is not found, or alternatively hydrometeors might be absent inside cloud where rc is found. Instead the PDF is modified so that a new variable, Ncn, replaces Nc in the PDF. The variable Ncn is a mathematical construct that can be viewed as an extended cloud droplet concentration or even as a simplified, conservative cloud condensation nuclei concentration. It is distributed as a single lognormal over the subgrid domain. At points where cloud water is found, Nc is set equal to Ncn. Otherwise, Nc is set to 0 at points where no cloud water is found (see Eq.  below). The value of Ncn‾ is approximately the in-cloud mean of Nc, and in special cases, is exactly the in-cloud mean of Nc. Please see Appendix  for a more detailed explanation.

The PDF includes all the hydrometeor species found in the chosen microphysics scheme with the exception of rc, which is calculated from other variables in the PDF through a saturation adjustment, and Nc, which is described above. In addition to rr and Nr, a microphysics scheme may include hydrometeor species such as ice mixing ratio, ri, ice crystal concentration (per unit mass), Ni, snow mixing ratio, rs, snowflake concentration (per unit mass), Ns, graupel mixing ratio, rg, and graupel concentration (per unit mass), Ng. The vector containing all the hydrometeor species included in the PDF will be denoted h. The full PDF can be written as Pw,rt,θl,Ncn,h.

In order to calculate quantities that depend on saturation, such as rc‾ and cloud fraction, a PDF transformation is required. The PDF transformation is a change of coordinates. The multivariate PDF undergoes translation, stretching, and rotation of the axes . Within each PDF component, a separate PDF transformation takes place. The ith component PDF, Piw,rt,θl,Ncn,h, is transformed to Piw,χ,η,Ncn,h, where χ is an “extended” liquid water mixing ratio that, when the air is supersaturated, has a positive value and furthermore is equal to rc. When the air is subsaturated, χ has a negative value. The variable η is orthogonal to χ. The variables rc and Nc can now be written as rc=χHχandNc=NcnHχ, where Hx is the Heaviside step function.

The general form of a PDF with n components and D variables (whether D includes all the variables in the PDF or any subset of those variables in a multivariate marginal PDF) can be written as Px1,x2,…,xD=∑i=1nξiPix1,x2,…,xD. Of the D variables listed, the first J variables are normally distributed in each PDF component (i.e., w, rt, and θl, or w, χ and η), the next K variables are lognormally distributed (i.e., Ncn), and the last Ω variables are the hydrometeor species, such that D=J+K+Ω. The ith component of the PDF, Pix1,x2,…,xD, accounts for both the precipitating and precipitation-less regions, and is given by Pix1,x2,…,xD=fpiPJ,K+Ωix1,x2,…,xD+1-fpiPJ,Kix1,x2,…,xJ+K∏ϵ=J+K+1Dδxϵ. The subscripts in the ith component, PJ,Ki or PJ,K+Ωi, denote the number of normal variates, J, and the number of lognormal variates, K or K+Ω, used in Eq. ().

Each original PDF component is split into precipitating and precipitation-less sub-components. The component means, variances, and correlations for variables x1…xJ+K do not differ between the precipitating and precipitation-less parts of Eq. (). This greatly simplifies the procedure for parameterizing the component means and variances, given the grid-level means and variances. Additionally, keeping the component means and variances the same between the in-precipitation and outside-precipitation parts of Eq. () allows the PDF to be reduced back to prior versions. For instance, the multivariate PDF in Eqs. () and () reduces to the version given in when all fpi=1 and various PDF parameters are chosen appropriately. Furthermore, when microphysics is not used in a simulation, hydrometeors are not found in the PDF. In this scenario, the PDF reduces to the original version found in .

The PDF does not contain a fraction for each hydrometeor species or type, but rather one precipitation fraction. Each PDF component is split into two sub-components (in-precipitation and outside-precipitation). Including a fraction for each hydrometeor type (rain, snow, etc.) would cause the number of sub-components to grow exponentially with the number of fractions. Using nf hydrometeor fractions increases the number of sub-components to 2nf in each PDF component. This would make setting the PDF parameters associated with each sub-component increasingly difficult.

The multivariate PDF can be adjusted to account for a situation when a variable has a constant value in a PDF (sub-)component. In that situation, the variable can be reduced to a delta function at the (sub-)component mean value. A good example of this would be setting Ncn to a constant value in order to use a constant in-cloud value of cloud droplet concentration. This is also especially useful when dealing with more than one hydrometeor. If one hydrometeor species is found at a grid level, but another hydrometeor species is not found at that level, the hydrometeor that is not found can reduce to a delta function at zero in the precipitating sub-component of Eq. ().

The general form of the m-variate hybrid normal/lognormal distribution in the ith PDF component, Pj,kix1,x2,…,xm, which is found in each sub-component of Eq. (), consists of j normal variates and k lognormal variates, where m=j+k. The first j variables are normally distributed and the remaining k variables are lognormally distributed. The multivariate normal/lognormal PDF is given by Pj,kix1,x2,…,xm=12πm2Σi12∏τ=j+1m1xτ×exp⁡-12x-μiTΣi-1x-μi. Both x and μi are m×1 vectors, where x is a vector of the variables (in normal-space) in the PDF and μi is a vector of the (normal-space) PDF sub-component means. The notation T denotes the transpose of the vector. The m×m (normal-space) covariance matrix is denoted Σi and its determinant is denoted Σi . The advantage of a single multivariate PDF, as opposed to a collection of individual marginal PDFs, is that the multivariate PDF accounts for correlations among the variables in the PDF. This is advantageous when calculating quantities such as rain water accretion rate and rain water evaporation rate.

Although the multivariate PDF allows for the calculation or specification of the (horizontal) correlation between any two variables at the same grid level, the PDF does not contain information about vertical correlations. Vertical correlations can arise in calculations of radiative transfer, diagnosed hydrometeor sedimentation, or other processes that involve the correlation of a variable with itself at different vertical levels. Such processes are excluded from this study, and hence information about vertical correlations is not needed here. For one possible method to parameterize vertical correlations, see .

When variables are integrated out of the full multivariate PDF, the result is a multivariate marginal PDF consisting of fewer variables. When all variables but one are integrated out of the PDF, the result is a univariate marginal or individual marginal PDF. For any hydrometeor species, h, found in the full multivariate PDF in Eq. (), the univariate marginal distribution is Ph=∑i=1nξifpiPLih+1-fpiδh, where PLih is a lognormal distribution in the ith PDF component, which is given by PLih=12π12σ̃hihexp⁡-ln⁡h-μ̃hi22σ̃hi2.

The in-precipitation mean of h in the ith PDF component is μhi. This is the mean of the ith lognormal of h. However, μ̃hi, as in Eq. (), is the normal-space component mean of h. It is the in-precipitation mean of ln⁡h in the ith PDF component and is given by μ̃hi=ln⁡μhi1+σhi2μhi2-12, where σhi is the in-precipitation standard deviation of h in the ith PDF component. The quantity σhi is the standard deviation of the ith lognormal of h. The normal-space component standard deviation of h is σ̃hi, as found in Eq. (). It is the in-precipitation standard deviation of ln⁡h in the ith PDF component and is given by σ̃hi=ln⁡1+σhi2μhi2.

The variables that are distributed marginally as binormals use similar notation. For example, μwi is the mean of w in the ith PDF component, or the mean of the ith normal. Likewise, σwi is the standard deviation of w in the ith PDF component, or the standard deviation of the ith normal.

This paper will use the phrase “PDF parameters” to refer to the PDF component means, standard deviations, and correlations involving variables in the PDF, as well as the mixture fractions and the PDF component precipitation fractions. The PDF parameters are calculated from various grid-mean input variables. In this paper, the component means, standard deviations, and correlations involving w, rt, and θl, and the mixture fractions, ξ1 and ξ2, are calculated according to the Analytic Double Gaussian 2 (ADG2) PDF, as described in Sect. (e) of the Appendix of . ADG2 requires the following quantities as input: the overall (grid-box) mean, variance, and third-order central moment of w (w‾, w′ 2‾, and w′ 3‾, respectively), the overall mean and variance of rt (rt‾ and rt′ 2‾, respectively), and the overall mean and variance of θl (θl‾ and θl′ 2‾, respectively). ADG2 preserves the values of these input variables, meaning that the PDF parameters can be used to successfully reconstruct the values of the input variables. Additionally, ADG2 requires and preserves the overall covariance of w and rt (w′rt′‾), the overall covariance of w and θl (w′θl′‾), and the overall covariance of rt and θl (rt′θl′‾). All of the aforementioned quantities are prognosed or diagnosed in CLUBB and are not the subject of this paper.

The individual marginal distribution for Ncn is specified to be a single lognormal over the entire subgrid domain. This requires that both PDF component means equal the overall (grid-box) mean (μNcn1=μNcn2=Ncn‾). Likewise, this requires that both PDF component standard deviations equal the overall standard deviation (σNcn1=σNcn2=Ncn′ 2‾ 1/2).

When no hydrometeor species are found at a grid level (h=0), fp=fp1=fp2=0. Otherwise, if any hydrometeor species in h is found at a grid level (has a value greater than 0), fptol≤fp≤1, where fptol is the minimum value allowed for precipitation fraction when hydrometeors are present. We now describe how CLUBB parameterizes fp1 and fp2, given fp. First, we note that fp=ξ1fp1+ξ2fp2. A tunable parameter, υ* (where the * subscript denotes a tunable or adjustable parameter), is introduced and is defined as the ratio of ξ1fp1 to fp, where 0≤υ*≤1. The precipitation fraction of PDF component 1 is solved by fp1=min⁡υ*fpξ1, 1. The precipitation fraction of PDF component 2 can now be solved by fp2=min⁡fp-ξ1fp1ξ2, 1. When fp1 calculated by Eq. () is small enough to force fp2 calculated by Eq. () to be limited at 1, the value of fp1 is recalculated (with fp2=1) and is increased enough to satisfy Eq. ().

There is insufficient data from observations to calculate all the fields that need to be input into CLUBB's PDF. However, this data can be supplied easily and plentifully by a LES. In this paper, LES output of precipitating cases is simulated by the System for Atmospheric Modeling (SAM) . SAM uses an anelastic equation set that predicts liquid water static energy, total water mixing ratio, vertical velocity, and both the south–north and west–east components of horizontal velocity. Additionally, it predicts hydrometeor fields as directed by the chosen microphysics scheme. A predictive 1.5-order subgrid-scale turbulent kinetic energy closure is used to compute the subgrid-scale fluxes . SAM uses a fixed, Cartesian spatial grid and a third-order Adams–Bashforth time-stepping scheme to advance the predictive equations of motion. It uses periodic boundary conditions and a rigid lid at the top of the domain. The second-order MPDATA (multidimensional positive definite advection transport algorithm) scheme is used to advect the predictive variables .

In order to assess the generality of the different hydrometeor PDF shapes for different cloud regimes, SAM was used to run three idealized test cases – a precipitating shallow cumulus case, a drizzling stratocumulus case, and a deep convective case. The use of cases from differing cloud regimes help avoid overfitting the parameterizations of PDF shape. The setup for the precipitating shallow cumulus test case was based on the Rain in Cumulus over the Ocean (RICO) LES intercomparison . The horizontal resolution was 100 m, and 256 grid boxes were used in each horizontal direction. The vertical resolution was a constant 40 m and 100 grid boxes were used in the vertical. The model top was located at 4000 m in altitude. The model time step was 1 s and the duration of the simulation was 72 h. A vertical profile of level-averaged statistics was output every minute and a three-dimensional snapshot of hydrometeor fields was output every hour.

The RICO simulation was run with SAM's implementation of the hereafter KK warm microphysics scheme. KK microphysics predicts both rr and Nr. SAM's implementation of KK microphysics uses a saturation adjustment to diagnose rc, and cloud droplet concentration is set to a constant value (which is 70cm-3 for RICO).

The setup for the drizzling stratocumulus test case was taken from the LES intercomparison based on research flight 2 (RF02) of the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field experiment . The horizontal resolution was 50 m and 128 grid boxes were used in each horizontal direction. An unevenly spaced vertical grid was used containing 96 grid boxes and covering a domain of depth 1459.3 m. The model time step was 0.5 s and the duration of the simulation was 6 hours. A vertical profile of level-averaged statistics was output every minute and a three-dimensional snapshot of hydrometeor fields was output every 30 min. The DYCOMS-II RF02 simulation was also run with SAM's implementation of KK microphysics and used a constant cloud droplet concentration of 55cm-3.

The setup for the deep convective test case was taken from the LES intercomparison based on the Large-Scale Biosphere–Atmosphere (LBA) experiment . The horizontal resolution was 1000 m, and 128 grid boxes were used in each horizontal direction. An unevenly spaced vertical grid was used, containing 128 grid boxes and covering a domain of depth 27 500 m. The model time step was 6 s and the duration of the simulation was 6 hours. A vertical profile of level-averaged statistics was output every minute and a three-dimensional snapshot of hydrometeor fields was output every 15 min for the final 3.5 h of the simulation.

The LBA case requires a microphysics scheme that can account for ice-phase hydrometeor species. The LBA simulation was run with microphysics, which predicts the mixing ratio and number concentration (per unit mass) of rain, cloud ice, snow, and graupel. SAM's implementation of Morrison microphysics diagnoses rc using a saturation adjustment right before the microphysics is called and then allows microphysics to update the value of rc, which in turn is used to update the value rt. Cloud droplet concentration was set to a constant value of 100cm-3.

CLUBB's hydrometeor PDF shapes will be compared to histograms of hydrometeors produced by SAM LES data. Our goal is to isolate errors in the PDF shape itself. In order to eliminate sources of error outside of the PDF shape and provide an “apples-to-apples” comparison of CLUBB's PDF shapes to SAM data, we drive CLUBB's PDF using SAM LES fields, rather than perform interactive CLUBB simulations. The following fields are taken from SAM's statistical profiles and are used as inputs to CLUBB's PDF: rt‾, θl‾, w′ 2‾, rt′ 2‾, θl′ 2‾, w′rt′‾, w′θl′‾, rt′θl′‾, w′ 3‾, fp, rr‾, rr′ 2‾, Nr‾, and Nr′ 2‾. For the LBA case, we add ri‾, ri′ 2‾, Ni‾, Ni′ 2‾, rs‾, rs′ 2‾, Ns‾, Ns′ 2‾, rg‾, rg′ 2‾, Ng‾, and Ng′ 2‾. Another input to CLUBB's PDF is w‾. The value of w‾ from large-scale forcing is set according to case specifications in both SAM and CLUBB. CLUBB's PDF is generated at every SAM vertical level and at every output time of SAM level-averaged statistical profiles.

Additionally, covariances that involve at least one hydrometeor are added to the above list and are used to calculate the PDF component correlations of the same two variables. These covariances are rt′rr′‾, θl′rr′‾, rt′Nr′‾, θl′Nr′‾, and rr′Nr′‾. Please see Appendix  for more details on the calculation of PDF component correlations. The values of the component correlations do not affect the individual marginal PDFs of the hydrometeors. They are included for the calculation of microphysics process rates (see Sect. ).

Owing to differences between the KK and Morrison microphysics schemes in SAM, fp used by CLUBB's PDF is computed slightly differently depending on which microphysics scheme is used by SAM. The differences are due to the number of hydrometeor species involved in the microphysics, the thresholding found internally in the microphysics codes, and the variables that are output to statistics by SAM. KK microphysics contains only rain, and SAM's implementation of KK microphysics clips any value of rr (and with it Nr) below a threshold value in clear air. Therefore, it is simple to set fp to the fraction of the domain occupied by non-zero values of rr and Nr. Morrison microphysics predicts rain, ice, snow, and graupel. For each of these species, SAM outputs a fraction. To provide an apples-to-apples comparison with CLUBB, fp is approximated as the greatest of these four fractions at any particular grid level.

Although fp is provided by the LES for this study, it can be diagnosed based on the cloud fraction using a method such as that of . If the cloud fraction, in turn, is diagnosed based on the omnipresent prediction of means, variances, and other moments – as in higher-order moment parameterizations such as CLUBB – then the onset of partial cloudiness is well defined and indeterminacy about the time of cloud initiation is avoided. In contrast, parameterizations that diagnose cloud fraction based on, e.g., cloud water mixing ratio, lack crucial information in cloudless grid boxes, as discussed in . The well-defined onset of CLUBB's cloud fraction is inherited by the precipitation fraction.

We first evaluate the shape of the idealized PDFs directly against SAM LES output data. Histograms of SAM LES data are generated from the three-dimensional snapshots of hydrometeor fields. One histogram is generated at every vertical level for each hydrometeor field. A histogram of a SAM hydrometeor field is compared to the CLUBB marginal PDF of that hydrometeor field at the same vertical level and output time. The comparison is done with each of the SL, DL, and DDL PDF shapes.

Figure  compares marginal PDFs involving rr and Nr for the RICO case at an altitude of 380 m and a time of 4200 min. For the plot of the PDF of rr in Fig. a, the delta function at rr=0 has been omitted. The SAM data are divided into 100 bins, equally sized in rr, that range from the largest value of rr to the smallest positive value of rr. (In what follows, all histograms use 100 equal-size bins, arranged from smallest to largest value.) The SL hydrometeor PDF shape significantly overpredicts the PDF at small values of rr and significantly underpredicts it at large values of rr. These errors are an expected consequence of the single lognormal's attempt to fit the precipitation-less area. The DL and DDL PDF shapes provide a much closer match qualitatively to the SAM data. A quantitative assessment of the quality of the fit will follow in Sect. .

Each of the CLUBB hydrometeor PDF shapes has a lognormal distribution within precipitation in each PDF component. Taking the natural logarithm of every point of a lognormal distribution produces a normal distribution, and so the plot of the PDF of ln⁡rr in Fig. b is a normal distribution in each PDF component for each of the DDL, DL, and SL PDF shapes. The plot of the PDF of ln⁡rr (hereafter referred to as the PDF of rr in Gaussian space) complements the aforementioned plot of the PDF of rr (Fig. a). The plot of the PDF of rr is log-scaled on the y axis, accentuating the small values of Prr that are found at large values of rr. The plot of the PDF of ln⁡rr accentuates the PDF at small values of rr.

The plot of the PDF of ln⁡rr is a plot of only the in-precipitation portion of the distribution, omitting all zero-values. The in-precipitation portion of the PDF is divided by fp, which allows the area under the curve to integrate to 1. The PDF shown in Fig. b is the Gaussianized form of Eq. ().

Figure b shows that the SL hydrometeor PDF shape significantly misses the mark, for its peak is located too far to the left of the bulk of the SAM LES data. This shift of the peak to excessively small values is to be expected of a continuous PDF shape that tries to include a delta function at zero. The DL PDF shape is far too peaked in comparison to the SAM LES data, which is spread out broadly in Gaussian space. The DDL PDF shape is able to achieve a spread-out shape because it has two different means within precipitation. This allows it to better fit the more platykurtic shape of the SAM LES data in Gaussian space.

The plot of the PDF of RICO Nr is found in Fig. c and the Gaussian-space plot of Nr is found in Fig. d. Similar to rr, the SL shape overpredicts the PDF at small values of Nr and underpredicts it at large values of Nr. In Gaussian space, it is easy to see that SL's peak is located too far to the left. The DDL shape provides a better fit than the DL shape to SAM LES data in Fig. c. Again, the DL shape is too peaked in Fig. d, whereas the bimodal DDL is able to spread out, which provides a better match to SAM LES data.

Figure  contains scatter plots that show the bivariate PDF of rr and Nr for both SAM LES and CLUBB's PDF in RICO at the same altitude and time as Fig. . The CLUBB PDF scatter points were generated by sampling the DDL PDF using an unweighted Monte Carlo sampling scheme. This demonstrates the advantages of the multivariate nature of CLUBB's PDF. The hydrometeor fields are correlated the same way in CLUBB's PDF as they are in SAM LES.

Figure  compares marginal PDFs involving rr and Nr for the DYCOMS-II RF02 case at an altitude of 400 m and a time of 330 min. All three hydrometeor PDF shapes provide a decent match to the SAM LES data. In Fig. a and c, the SL and DL PDF shapes dip a little below the SAM LES line in the middle of the data range for rr and Nr, respectively. The DDL PDF shape stays closer to the SAM LES line in this region. Additionally, the SL PDF shape overestimates the SAM LES line close to the y axis. In Fig. b and d, the Gaussian-space plots show that the two components of the DDL shape superimpose more than they did for the RICO case, owing to the reduced in-precipitation variance in the drizzling stratocumulus case.

In order to assess how well the PDF shapes are able to capture ice PDFs as well as liquid PDFs, we turn to the LBA case. In LBA, liquid and ice appear at different altitudes and times. Figure  compares marginal PDFs involving rr and Nr for the LBA case at an altitude of 2424 m and a time of 330 min. Compared to SAM's PDF, the DDL hydrometeor PDF shape is too bimodal, but it still provides the best visual match of the three hydrometeor PDF shapes to SAM data. The fit will be quantified in Sect. .

To indicate whether the three PDF shapes work for ice-phase hydrometeors, we compare marginal PDFs involving ri and Ni for the LBA case at an altitude of 10 500 m and a time of 360 min (Fig. ). Similar to the rr and Nr plots for RICO and LBA, Fig. a and c show that the SL PDF shape overpredicts the PDF at small values of ri and Ni and underpredicts it at large values of ri and Ni. The DL shape provides a better fit than the SL, and the DDL has a slightly better fit than the DL. The Gaussian-space plots in Fig. b and d show that the SAM LES distribution of ln⁡ri and ln⁡Ni is again platykurtic. The SL PDF shape has a peak that is shifted to the left. The DDL hydrometeor PDF shape is able to spread out the most to cover the platykurtic shape of the LES in Gaussian space.

Why does the DDL PDF shape match LES output better than the DL shape in the aforementioned figures? The PDFs (in Gaussian space) for the LES of RICO and LBA show a broad, flat distribution of hydrometeor values from the LES. The DL shape is too peaked in comparison to the LES data. The DDL PDF is able to spread out the component means and thereby represent the platykurtic shape more accurately. However, even the DDL PDF fails to capture the far left-hand tail of the LES PDF. In the RICO, DYCOMS-II RF02, and LBA cases, between about 5 and 20 % of the LES PDF is found to the left of the DDL PDF (see Figures b, b, b, and b). However, these values of hydrometeor mixing ratios are small. They are roughly a factor of 20 or more smaller than the median value. By combining these factors, we see that the percentage contribution of hydrometeor mixing ratios that are omitted on the left-hand tail is only about 1 %.

Why does SAM LES data have a platykurtic shape in Gaussian space in these cases? One possible cause is the partly cloudy (and partly rainy) nature of these cases. In these partly rainy cases, a relatively high percentage of the precipitation occurs in “edge regions” near the non-precipitating region. These regions usually correspond to the edge of cloud or outside of cloud. Evaporation (or less accretion) occurs in these regions, increasing the area occupied by smaller amounts of rain. Yet, there is also an area of more intense precipitation near the center of the precipitating region, which produces larger amounts of rain. Collectively, the areas of small and large rain amount produce the large spread in the hydrometeor spectrum.

The DYCOMS-II RF02 PDFs from the LES tend not to share the platykurtic shape seen in the other cases. The RF02 case is overcast, so there are not as many “edge” regions of precipitation as found in partly rainy cases. There is much less in-precipitation variance in the RF02 case. The simpler PDF shape is easier to fit by all the PDF shapes (SL, DL, and DDL). To further illuminate the physics underlying the PDF shapes produced by LESs, further study would be needed.

The multivariate PDF used by CLUBB has been updated to improve the subgrid representation of hydrometeor species. The most important update is the introduction of precipitation fraction to the PDF. The precipitating fraction contains any non-zero values of any hydrometeor species included in the microphysics scheme. The remainder of the subgrid domain is precipitation-less and is represented by a delta function where every hydrometeor species has a value of zero. When a hydrometeor is found at a grid level, its representation in the precipitating portion of the subgrid domain is a lognormal or double-lognormal distribution. The introduction of precipitation fraction increases accretion and decreases evaporation in cumulus cases, allowing more precipitation to reach the ground.

Additionally, a new method has been developed to calculate the in-precipitation mean and standard deviation of a hydrometeor species in each component of CLUBB's two-component PDF. This method preserves the grid-box mean and variance of the hydrometeor species. By simply changing the values of tunable parameters, CLUBB's marginal PDF for a hydrometeor can be changed from a delta-double-lognormal (DDL) to a delta-lognormal (DL) or to a single-lognormal (SL) shape.

In order to compare the effectiveness of the three hydrometeor PDF shapes, three simulations – a precipitating shallow cumulus case (RICO), a drizzling stratocumulus case (DYCOMS-II RF02), and a deep convective case (LBA) – were run using SAM LES. Statistical output values from the LES for the grid-level mean and turbulent fields were used to drive the PDF for each hydrometeor PDF shape. The idealized PDF shapes were compared to the SAM LES results. The DDL PDF shape produced the lowest average K–S and average normalized C–vM scores when compared to SAM LES results, followed by the DL PDF shape. Both produced lower scores than the original SL PDF shape. However, for DYCOMS-II RF02, all three PDF shapes were in almost equal agreement with SAM LES results.

The DL and DDL PDFs possess three important properties: (1) they are multivariate, and hence can represent interactions among multiple hydrometeor species; (2) they admit a precipitation-less region, which is necessary to permit realistic process rates in cumulus cloud layers; and (3) they have realistic tails, as evidenced by the comparisons with LES shown here. Because of these three properties, the DL and DDL PDFs may be general enough and accurate enough to adequately represent hydrometeor variability over a range of important cloud types, including shallow cumulus, deep cumulus, and stratocumulus clouds. This generality, in turn, may help enable parameterization of these clouds types in a more unified way. Indeed, an early version of the DDL PDF has already been used to represent hydrometeor subgrid variability in some interactive simulations with a unified cloud parameterization. Namely, the DDL PDF was used in the interactive single-column simulations of these cloud types by and in the global simulations by . Further testing would be required, however, to better understand the limits of the DL and DDL PDFs. Better understanding is particularly desirable in, for instance, mixed-phase and glaciated clouds. This has been left for future work.

The CLUBB code is freely available for non-commercial use after registering for an account on the website http://clubb.larson-group.com. The specific version of CLUBB used in this paper is available in the SVN repository located at http://carson.math.uwm.edu/repos/clubb_repos/tags/Hydromet_PDF_shapes. In the repository is a file README_Hydromet_PDF_shapes which gives instructions for generating the results found in this paper.