## 1. Introduction

Deep convection plays an important role in redistributing moisture and heat in the atmosphere. Deep convective systems, from single-cell thunderstorms to mesoscale convective systems, significantly affect our climate system through the associated heavy precipitation, diabatic heating, and high cloud anvils, which contribute to radiative impacts. Current global climate models with a typical horizontal grid spacing of 100 km, or even at the finer horizontal grid spacing of 25 km in state-of-the-art models, are unable to resolve these features that are typically at the scale of tens of kilometers. Within such models, these convective systems are subgrid-scale processes that require some form of parameterization.

Within these deep convective systems even smaller-scale complex interactions occur between the cloud dynamics and precipitating hydrometeors, such as within strong convective updrafts and downdrafts. Until recently, most global climate models have relied on diagnostic precipitation schemes, in which precipitation is formed and depleted completely over one time step by rainout or evaporation. As horizontal resolution increases and time step decreases, the diagnostic precipitation approach becomes less appropriate. Recently, Gettelman et al. (2015) implemented a two-moment microphysics scheme in the Community Atmosphere Model (CAM), which includes a prognostic treatment of rain and snow mass and number mixing ratios. Even at a 200-km horizontal grid spacing, the prognostic microphysics was found to improve the ratio of accretion to autoconversion process rates. They also found differences in the vertical structure of precipitation that led to these improvements in process rates. The prognostic microphysics scheme determines the gridcell-averaged time tendencies of microphysical processes that depend on cloud liquid water mass mixing ratio (such as autoconversion and accretion) using subgrid variance of cloud liquid water. For other microphysical species, such as rain and snow, precipitation and cloud fractions are used to account for some subgrid variability.

Many studies have discussed the biases that can occur when computing microphysical process rates based on gridcell averages alone and neglecting subgrid-scale variability of microphysical quantities (e.g., Pincus and Klein 2000; Larson et al. 2001; Kogan and Mechem 2014, 2016). Probability density functions (PDFs) have been employed to describe this unresolved variability and to account for its effect on microphysical process rates (e.g., Tompkins 2002; Larson and Griffin 2013; Griffin and Larson 2013; Larson and Schanen 2013; Chowdhary et al. 2015). These microphysical process rates may also depend on the spatial covariability among the variables, particularly for processes with highly nonlinear rates. For example, Kogan and Mechem (2014) have shown that autoconversion rate is more sensitive to this covariability than accretion rate because the former has a stronger nonlinear dependency on droplet number concentration and liquid water content. Predicting covariances and, more generally, joint distributions of microphysical variables is challenging (Larson et al. 2011); even PDF schemes may account for it only crudely, for example, by prescribing corresponding correlation coefficients (Storer et al. 2015).

In the context of convective vertical transport of hydrometeors, the same challenge exists—namely, the vertical fluxes of hydrometeors are represented by their covariances with vertical velocity—which need to be parameterized because, to our knowledge, they are not explicitly predicted by any existing subgrid schemes. Consequently, even assumed PDF schemes that explicitly solve for vertical turbulent fluxes of liquid water potential temperature and total (vapor plus cloud) water mixing ratio, such as for example the Cloud Layers Unified by Binormals (CLUBB) (Golaz et al. 2002; Larson and Golaz 2005), do not predict covariances between vertical velocity and microphysical variables, such as rain or snow mixing ratios. Available information about marginal, or one-dimensional distribution of these variables is typically not used directly in computing the subgrid vertical mixing of microphysical quantities because they lack information on their correlations with vertical velocity. Instead, their turbulent fluxes are assumed to be proportional to the vertical gradient of their grid-mean values and computed using the eddy-diffusion approximation. While the approximation may be reasonable for non- or weakly precipitating shallow clouds, it cannot treat a countergradient transport, which has been shown to occur frequently for precipitating hydrometeors in deep convection (Wong et al. 2015).

In search for an improved parameterization of subgrid-scale transport of hydrometeors, Wong et al. (2015) demonstrated that the shape of the vertical flux profile can be well represented by decomposing the net flux into contributions from four quadrants of the joint hydrometeor mixing ratio–vertical velocity distribution. This step is referred to as conditional sampling. They also showed, using offline testing, that in order for the parameterized fluxes to match those diagnosed from cloud-resolving model (CRM) simulations, the magnitudes of the quadrant means need to be increased to account for intraquadrant correlation between the variables. Although these results demonstrated the potential usefulness of the quadrimodal decomposition of the joint PDF for parameterizing the vertical flux, the proposed approach is not directly applicable to a PDF-based scheme, such as CLUBB, because conditional sampling into quadrants implies the knowledge of the joint PDF, which CLUBB for example does not provide.

The goals of this study are threefold. First, we aim to modify the hydrometeor transport scheme of Wong et al. (2015), so that the conditional sampling is applied to marginal (i.e., one dimensional) PDFs instead of joint (i.e., two dimensional) distributions. Second, we clarify the role of the PDF scaling, which has been empirically introduced to adjust the magnitude of the parameterized vertical fluxes. Finally, we expand the offline testing of the newly proposed transport scheme to additional deep convection cases representing continental midlatitude and tropical oceanic environments.

In section 2, we describe the cloud-resolving model and the four selected deep convective cases used in this study. In section 3, we discuss the parameterization of subgrid-scale hydrometeor transport in the context of an existing assumed PDF higher-order turbulence closure. The proposed parameterization scheme based on marginal distributions and the role of the scaling parameter are described in section 4, followed by the comparison between parameterized and benchmark vertical fluxes (section 5). Finally, a summary is given in section 6.

## 2. Model and case descriptions

The three-dimensional CRM used in this study is the System for Atmospheric Modeling (SAM) (Khairoutdinov and Randall 2003), which is a nonhydrostatic anelastic model with periodic lateral boundary conditions. The dynamical core uses an Adams–Bashforth time-integration scheme and a second-order Eulerian multidimensional positive definite advection transport algorithm (MPDATA; Smolarkiewicz and Grabowski 1990). A two-moment prognostic microphysics scheme by Morrison et al. (2005, 2009) is used to compute mass and number mixing ratios of cloud liquid water, cloud ice, rain, graupel, and snow. In addition to microphysics, a 1.5-order turbulent kinetic energy subgrid-scale closure is used, and the Rapid Radiative Transfer Models (RRTMs) are used for short- and longwave radiation coupling (Iacono et al. 2008; Clough et al. 2005).

High-resolution simulations driven by observations from field campaigns are conducted and serve as benchmark simulations. Three- to four-day periods are selected from three field campaigns: the Atmospheric Radiation and Measurement 1997 (ARM97) Intensive Observation Period (IOP) (Xu et al. 2002; Khairoutdinov and Randall 2003), the Midlatitude Continental Convective Clouds Experiment (MC3E) (Petersen and Jensen 2012; Jensen et al. 2015, 2016), and the Tropical Warm Pool International Cloud Experiment (TWP-ICE) (May et al. 2008; Xie et al. 2010). The first two experiments took place over the southern Great Plains in Oklahoma, United States, and the third experiment was carried out near Darwin, Australia. For the ARM97 case, we simulate a 4-day period from 27 June to 1 July 1997. This period consisted of a series of weak precipitation events followed by a major precipitation event on 29 June with a maximum precipitation rate of 84 mm h^{−1} (Xu et al. 2002; Khairoutdinov and Randall 2003). This stronger convective event will be the focus of our analysis (from 1520 UTC 29 June to 1010 UTC 30 June 1997). For the MC3E case, we simulate a 3-day period from 23 to 26 May 2011. The first selected subperiod (MC3E-1) is from 1500 UTC 23 May to 0300 UTC 24 May 2011 and consists of scattered storms in the region. The second subperiod (MC3E-2) from 1500 UTC 24 May to 0300 UTC 25 May 2011 is selected for the more organized convection that occurred in the form of a squall line. For the TWP-ICE case, we simulate a 3.5-day period from 1200 UTC 21 January to 1200 UTC 24 January 2006, during which Darwin, Australia, experienced a period of active monsoon weather. The subperiod we analyzed for this case includes the development of a major mesoscale convective system that produced intense precipitation from 0300 UTC 23 January to 1200 UTC 24 January 2006 [event C in Fridlind et al. (2012)]. For each of the four subperiods, we define a peak convection instantaneous snapshot (listed in Table 1), defined as the time of maximum rain and snow water paths.

Model configuration and selected periods from three 250-m benchmark simulations.

The three simulations are conducted using the same set of physics parameterizations and model configuration (Table 1) with a horizontal grid spacing of 250 m and a vertical grid spacing of 25 m near the surface varying up to 250 m over 128 model levels. The model domain is 128 km × 128 km × 28 km. All simulations are driven by horizontally homogeneous large-scale forcing derived from the objective variational analysis of Zhang et al. (2001), available from the ARM Data Archive. Surface latent and sensible heat fluxes are prescribed.

The single-column input to the proposed scheme is diagnosed using instantaneous 3D model fields at 10-min intervals. The output from the scheme is then compared with results computed explicitly using the three-dimensional output.

Figure 1 shows the time evolutions of the domain-mean profiles of cloud condensate (liquid and ice) mixing ratios with superimposed isolines of the vertical velocity variances for the three simulated cases. The four selected subperiods (ARM97, MC3E-1, MC3E-2, and TWP-ICE) are shaded in white. The TWP-ICE tropical maritime convection case shows more consistent cloud cover than ARM97 and MC3E throughout the simulation due to the more humid tropical environment. Our simulations indicate that the cloud condensates reach higher altitudes (up to approximately 18 km) in the tropical environment case than for the continental cases (approximately 15 km). These higher cloud tops are also evident in radar echo-top observations that were found to extend up to 17 km for the most intense deep convective cells during the TWP-ICE field campaign (May et al. 2008).

## 3. Subgrid-scale hydrometeor transport

We now discuss the parameterization of subgrid-scale hydrometeor transport in the context of an existing assumed PDF scheme called CLUBB (Golaz et al. 2002; Larson and Golaz 2005). CLUBB was originally developed to handle subgrid-scale interactions between turbulence and shallow boundary layer clouds (Golaz et al. 2002; Larson and Golaz 2005; Bogenschutz and Krueger 2013; Guo et al. 2014; Z. Guo et al. 2015) but has recently been extended and tested for deep convection in single-column (Storer et al. 2015) and global climate models (Thayer-Calder et al. 2015; H. Guo et al. 2015).

CLUBB explicitly solves the prognostic equations for vertical turbulent fluxes of liquid water potential temperature and total water (vapor + cloud liquid) mixing ratio, among other higher-order moments. We therefore do not apply our scheme to cloud liquid water mixing ratio. Subgrid variability of microphysical processes is handled by coupling CLUBB with a subcolumn approach (Larson and Schanen 2013; Storer et al. 2015; Thayer-Calder et al. 2015). Sedimentation processes are not considered here as they are handled by the microphysics scheme during each call by the subsampler.

*c*= 0.75 is a tunable (dimensionless) constant. The eddy diffusivity is defined as

*e*is the subgrid turbulent kinetic energy,

*L*is CLUBB’s mixing length (Golaz et al. 2002; Larson et al. 2012),

_{w}is the skewness of vertical velocity. This formulation of eddy diffusivity increases

*K*by a factor of 100 for cumulus layers while remaining small for stratocumulus layers, as vindicated by high-resolution numerical studies (Storer et al. 2015). This modified formulation for

*K*was designed to use prognostic variables in CLUBB, namely the variance of the hydrometeor mixing ratios (

_{w}), which are linked to physical attributes of deep convection. The existing parameterization is based on a downgradient formulation and is aimed at parameterizing the covariance.

Let us now describe the covariance from a PDF perspective. CLUBB assumes that the vertical velocity (*w*) marginal distribution has a double-Gaussian shape, and mass mixing ratios for all hydrometeors (*q*_{x}) have marginal distributions of a delta-double-lognormal shape. The delta function is placed at zero to account for hydrometeor-free fraction in the grid cell (Griffin and Larson 2016; Larson and Griffin 2013; Griffin and Larson 2013). The width of each of the double-lognormal hydrometeor distributions depends on its variance, which is approximated as being proportional to the squared mean of the distribution (Storer et al. 2015; Griffin and Larson 2016). The width of each vertical velocity distribution is defined to be proportional to the square root of the vertical velocity variance (a prognostic variable in CLUBB).

*P*(

*w*,

*q*

_{x}) is the joint distribution of

*w*and

*q*

_{x}. Instead of defining

*P*(

*w*,

*q*

_{x}), it is possible to compute the covariance directly from the variances and the correlation coefficient (

*w*and

*q*

_{x}. The variances may be provided by a PDF-based scheme (e.g., as described earlier for CLUBB); however, the correlation coefficient is typically not predicted or known.

*P*(

*w*,

*q*

_{x}) (i.e., much of the impact of correlation on the fluxes) can be captured by decomposing the joint PDF into a small number of sections, such as quadrants as was done in Wong et al. (2015). We refer to this step as conditional sampling. The remaining correlation that exists within these quadrants can then be further represented by scaling the means of the quadrant PDFs. Equation (3) can then be approximated in the form of

*s*(⋅) is the scaling function applied to PDFs in each section

*i*with a weight of the flux contribution,

*σ*

_{i}. Wong et al. (2015) defined each quadrant using the conditions (i)

*P*(

*w*,

*q*

_{x}) is known a priori.

Here, we reformulate the Wong et al. approach into using only the marginal PDFs of vertical velocity and hydrometeor mass mixing ratios; that is, *P*(*w*, *q*_{x}) is replaced by *P*(*w*) and *P*(*q*_{x}) in the first and second integrands of Eq. (4), respectively. Using the product of only the marginals inherently sets the correlation within each section

## 4. Scheme description

### a. Input fields

The transport scheme is intended to compute vertical fluxes assuming that only marginal PDFs of subgrid-scale vertical velocity and hydrometeor mass mixing ratios are known. In CLUBB, a double-Gaussian marginal PDF of vertical velocity is selected based on prognostic variance and the third-order moment, and a delta-double-lognormal marginal PDF for each mixing ratio is constructed using the predicted mean and diagnosed variance. These marginal PDFs are different at each time step and grid cell.

To isolate the scheme’s ability to parameterize the vertical fluxes from possible errors due to the assumed shapes of the marginal PDFs in CLUBB, we obtain our input PDFs directly from the CRM benchmark simulations. The 3D fields of vertical velocity and hydrometeor mass mixing ratios are explicitly binned to generate discrete marginal PDFs. The vertical velocity distributions use all 512 × 512 grid cells at each model level. For hydrometeor mass mixing ratios, only grid cells with nonzero mixing ratios are used (grid cells with no hydrometeor present are assumed to be represented by the delta function in the assumed PDF). The vertical velocity and hydrometeor mass mixing ratio distributions are binned into 160 bins ranging from −40 to 40 m s^{−1} in increments of 0.5 m s^{−1} for *w* and 180 bins from 10^{−16} to 10^{2} g kg^{−1} with 10 bins per decade for *q*_{x}. An example of a joint two-dimensional PDF of vertical velocity and rain mass mixing ratio together with their respective marginal PDFs is shown in Fig. 2. Testing CLUBB’s ability to model the correct marginal PDFs is outside of the scope of this study but will be performed in the future with interactive tests of the transport algorithm.

### b. Parameterization using scaled marginal PDFs

#### 1) Conditional sampling

*i*represents the flux type defined using conditional sampling,

*σ*

_{i}is the fractional area and sum of which over all

*i*equals 1 and

*u*), downdraft (

*d*), and quiescent or stratiform (

*s*) regions. Compared to the Wong et al. (2015) method based on sampling fluxes in quadrants of the joint distributions, here the fractional area for each flux type is determined using threshold-based sampling of a marginal PDF of vertical velocity

*P*(

*w*):

Updraft and downdraft regions so defined approximate Wong et al.’s (2015) quadrants I and IV, respectively, while weakly precipitating regions associated with upward and downward motions (their quadrants II and III, respectively) are combined into one flux type (denoted as *s*). These simplifications are possible because based on the CRM simulations, vertical velocities *w*_{1} = 1.5 and *w*_{2} = 1.5 m s^{−1} are typically associated with hydrometeor mixing ratios of values smaller than the domain mean (Fig. 3), whereas vertical velocities with magnitudes greater than 1.5 m s^{−1} are associated with mixing ratios greater than the domain mean. Figure 3 shows the logarithmic difference in the frequency of occurrence (*f*) of any particular vertical velocity value between regions with *q*′ (greater than mean, denoted as *f*_{gtm}) and *q*′ < 0 (less than mean, denoted as *f*_{ltm}). Warm positive values indicate that the corresponding vertical velocity at that particular height occurs more frequently in regions with *q*′ > 0, whereas cool negative values indicate that the vertical velocity occurs more frequently in regions with *q*′ < 0. We note that the dark red regions found for strong updrafts and downdrafts are typically bins where *f*_{ltm} = 0 and *f*_{gtm} > 0, and vice versa for the dark blue regions.

As shown, strong updrafts and downdrafts are typically associated with greater-than-mean mixing ratios. The vertical velocity thresholds of −1.5 and 1.5 m s^{−1} roughly encompass the area where the frequency is much greater in *q*′ < 0 than in *q*′ > 0. All four hydrometeor types show similar behavior, perhaps except for rain above approximately 5 km. However, rain mass mixing ratios are small above that altitude at which the freezing level typically resides, whereas the magnitude of the ratio log(*f*_{gtm}/*f*_{ltm}) may still be large. At the lower model levels, the blue regions indicate that the stronger positive vertical velocities are associated with below-mean hydrometeor mixing ratios. As will be shown later, this is consistent with negative correlations found in updrafts at these model levels and will be accounted for in the parameterization using a lower model-level adjustment described later in this section. We also varied the vertical velocity thresholds from 1 to 2.5 m s^{−1} and found only small variations in the resulting fluxes.

#### 2) Scaled means

The products of the scaled means in the heavily precipitating updraft and downdraft regions were found to give a more accurate representation of the vertical hydrometeor fluxes than simply the product of the means as it accounts for subplume (intraquadrant) correlations (Wong et al. 2015). The subplume correlations indicate that greater mass mixing ratios are typically associated with stronger updrafts and downdrafts (e.g., as shown in the top-right and top-left sections, respectively, in Fig. 2), likely owing to growth processes from dynamical–microphysical feedbacks.

*P*(ϕ), where ϕ ∈ [

*w*,

*q*

_{x}], the scaled mean over a flux type

*i*is

*α*is a (tunable) scaling parameter and

*ϕ*for flux type

*i*, both of which are discussed in detail below. The overbars, as before, denote the average over each flux type, except in the case of

*α*, the heavier the weights are placed on the larger absolute values in the marginal PDFs. The heavier weights represent the greater contribution from these larger absolute values to the total flux. A renormalized PDF,

*p*

_{i}(ϕ), is used in Eq. (7) and is calculated from the marginal PDF,

*P*(ϕ), using

The normalization factor [denominator in Eq. (7)] ensures that the resulting scaled mean is within the limits of the original marginal distribution. If the PDFs provided vary in space and time, such as those provided by CLUBB, the vertical fluxes will reflect the prognostic subgrid-scale variability.

The mean of vertical velocity in each section is defined using the same set of *w*_{1} and *w*_{2} thresholds as for the fractional areas (i.e., −1.5 and 1.5 m s^{−1}, respectively). For the stratiform components, we use α = 0 (i.e., there is no scaling of the mean for this flux type), and note that when the magnitude of the flux is small, the scaling has little effect.

*β*coefficient is based on guidance from the CRM benchmark simulations. Using the definitions from Wong et al. (2015), we have found that the ratio

*σ*

_{I}/(

*σ*

_{I}+

*σ*

_{IV}) may be a possible surrogate for the ratio

*β*coefficient, we take the maximum of [

*σ*

_{u}/(

*σ*

_{u}+

*σ*

_{d}),

*σ*

_{d}/(

*σ*

_{u}+

*σ*

_{d}). When either

*σ*

_{u}= 0 or

*σ*

_{d}= 0,

*β*is set to 1 so that the scaled mean mixing ratios are unchanged. We will show in the next section that the assumption that the updraft scaled mean mixing ratios are greater than those in downdrafts requires some adjustment at the lower model levels.

#### 3) Scaling parameter

A scaling parameter (*α*) is used to account for the subplume correlation between the dynamics and microphysics. The larger the scaling parameter, the greater the correlation between the magnitudes of the vertical velocity and hydrometeor mixing ratios is assumed. One can determine the best estimate of the *α* parameter for which a scaled flux formulation matches the benchmark flux. Graphically, pairs of *w*–*q*_{x} plot, as illustrated in Fig. 6 for a convective updraft flux at a particular height. Pairs of scaled means *α* likewise can be presented in a *w*–*q*_{x} plot (dashed line in Fig. 6). The intersection of the two lines gives the best estimate of *α*, for which the products of the scaled means reproduce the benchmark quadrant fluxes exactly. Vertical profiles of the best estimates of *α* at peak of convective activity from the four selected cases are shown in red in Figs. 7 and 8 for the scaled flux formulations presented in Wong et al. (2015) and in the proposed marginal PDF approach, respectively.

Vertical profiles of subplume correlation coefficients in convective updrafts and downdrafts as defined in the two approaches are shown in black in Figs. 7 and 8, respectively. The correlation coefficients are computed using the 250-m benchmark simulations at the same peak convection times as the best estimates of *α*. It is interesting that the *α* profiles in both approaches are remarkably similar to the profiles of the subplume correlation for each flux type. These results indicate that the *α*-scaling parameter is related to the subplume correlations. A caveat in the *α*-scaling method is that it only accounts for positive (negative) correlations in the updraft (downdraft) plumes. However, as shown in both Figs. 7 and 8, the positive correlations in downdraft plumes are rare and typically small, and the negative correlations in updraft plumes are typically confined to lower model levels and are accounted for using an adjustment scheme described in the next section.

The subplume correlations vary in height and time. Although in the future it may be possible to develop a height-dependent *α*, here we assume a constant parameter α = 0.4 for the convective updraft and downdraft fluxes. As shown in Fig. 8, the best-estimate *α* values using the marginal PDF approach are larger than those used for the quadrant fluxes defined in Wong et al. (2015) (Fig. 7). The increase is found to be necessary to match the benchmark fluxes computed using the new decomposition of the flux based on marginal *w* PDF rather than the quadrant decomposition tested in Wong et al. (2015). The increased scaling is likely needed because the correlation between the variables changes with the redefinition of the decomposition.

#### 4) Adjustment in lower model levels

As mentioned above, the scheme assumes that both strong updrafts and downdrafts are associated with the same part of the mass mixing ratio PDFs above the mean. Analysis of the CRM simulations demonstrates that at lower altitudes this assumption becomes invalid. Typically, lower levels of strong ascents correspond to early periods of formation of precipitating hydrometeors. For example, rain may only begin to form in strong updrafts when cloud water content reaches sufficiently large values, and raindrops may begin to convert into graupel above the freezing level. In these cases, the initial hydrometeor mixing ratios are small. At the same time, neighboring downdrafts at similar altitudes are carrying larger loads of older hydrometeor species from aloft that have grown to larger sizes. The reduced correlation between positive vertical velocities and hydrometeor mass mixing ratios at lower levels can be seen in Figs. 7 and 8 for all four cases. Consistent with the reduced correlation near the lower model levels, positive vertical velocities are also found to be more frequently associated with small hydrometeor mass mixing ratios (*q*′ < 0) than large hydrometeor mass mixing ratios (*q*′ > 0), whereas downdrafts are more frequently associated with the latter (Fig. 3).

Since the updraft and downdraft fluxes are both parameterized using scaled mean hydrometeor mixing ratios based on the distribution where *Z*_{1}), such that the net flux is parameterized by the downdraft flux. The parameterized updraft fluxes are then linearly increased from zero at the freezing level to their full values at the level of maximum updraft flux (*Z*_{2}). The linear increase is justified based on the profiles of the benchmark fluxes.

Similarly, for rain, the net flux is dominated by the downdraft fluxes (Fig. 9a) below the liquid cloud base, defined here as the lowest level at which the domain-mean liquid cloud water mixing ratio, ^{−1}. Consequently, we set *Z*_{1} to be the cloud base level and assign the parameterized updraft fluxes to zero below *Z*_{1}. The updraft fluxes are then linearly increased from zero at the cloud base to their full values at the level of maximum updraft flux (*Z*_{2}). Finally, for ice (Fig. 9c), a similar adjustment procedure is applied, except the parameterized updraft fluxes are set to zero below the ice cloud base—that is, the lowest level at which the domain-mean ice cloud water, ^{−1} (*Z*_{1}). The updraft fluxes are again linearly increased to their full values between *Z*_{1} and *Z*_{2}. Herein, “adjustment” will refer to the above mentioned modifications to the updraft fluxes.

## 5. Comparisons of parameterized fluxes with benchmark fluxes

Parameterized vertical fluxes using the proposed PDF-based hydrometeor transport scheme with adjustment are shown in Fig. 10 (red solid lines). In the same figure, the parameterized vertical fluxes without the adjustment are shown in red dotted lines. Benchmark vertical fluxes are shown in black solid lines. All vertical hydrometeor fluxes are time averaged over the selected period in each case (Table 1). Adjustment of the updraft fluxes greatly improves the shapes of the profiles, especially for rain, snow, and graupel below the levels of their maximum fluxes. The limited impact of the adjustment on cloud ice fluxes is likely because of the weaker downdraft fluxes at the lower altitudes (Fig. 9c). We note that it is the flux divergences based on these flux profiles that govern the vertical advection tendencies. This implies that, for example, the positive vertical advection tendencies of rain from the surface to near the freezing level will not be captured in the scheme without adjustment and, similarly, those just below the freezing level for graupel and snow. As mentioned, the hydrometeors at these levels are associated with strong downdrafts, which may lead to important subsequent diabatic feedback.

Without scaling (orange lines in Fig. 10), the shapes of the flux profiles are similarly captured but the magnitudes are typically underestimated [also found in Wong et al. (2015)]. The effect of the updraft flux adjustment on the unscaled fluxes is comparable to its effect on the scaled fluxes discussed above.

Also plotted in Fig. 10 are the parameterized fluxes using the modified downgradient eddy flux formulation for deep convection [Eq. (1)]. To obtain the parameterized downgradient eddy fluxes, we follow the procedure as outlined in Wong et al. (2015), where all the variables on the right-hand side of Eq. (1) are obtained from the 250-m simulation. The computations are repeated for all four periods (ARM97, MC3E-1, MC3E-2, and TWP-ICE) and the parameterized fluxes averaged over each period are shown as dashed cyan lines in Fig. 10. Wong et al. (2015) tested this formulation for the ARM97 case and showed that the downgradient eddy fluxes tend to remove too much mass from the altitude with maximum microphysical content (as expected from the downgradient approximation). In their study, a comparison was made between the eddy-diffusion fluxes and quadrant-based fluxes, although the latter were based on some a priori knowledge of the covariance between vertical velocity and hydrometeor mass mixing ratio (e.g., exact fractional areas from the benchmark solutions were used; pairs of vertical velocity and mixing ratio were subsampled into quadrants using a priori knowledge that would not be accessible to CLUBB). Here, the comparison is more direct since both approaches are based on input assumed to be available within CLUBB, albeit diagnosed for the presented cases from the CRM.

For rain, the downgradient formulation is able to reproduce the general shape of the explicit benchmark fluxes, including the negative fluxes below the freezing level. These results are consistent with those found in Wong et al. (2015). For graupel and snow, the negative fluxes near the freezing level are well represented, but the positive flux divergence (negative advection tendency) extends too high up. These levels coincide with the average maximum microphysical content (blue lines in Fig. 10). For cloud ice, the behavior of the downgradient approximation is most evident, where there is a tendency to advect hydrometeor mass from the peak-content altitude to above and below that level. The explicit fluxes indicate that this is not the case. The scaled-PDF fluxes are also unable to capture the magnitude of the ice vertical fluxes, although the profile shapes are in much closer agreement to the benchmark. Although we show only the temporal averages of the flux profiles, we note that improvement in the temporal evolution of the vertical fluxes as parameterized by the scaled marginal PDF scheme, as compared to those parameterized using the downgradient formulation, is also evident.

## 6. Summary

In this study, we have demonstrated a new three-step approach to conditionally sample and scale marginal PDFs of vertical velocity and hydrometeor mass mixing ratios in order to parameterize subgrid-scale vertical hydrometeor fluxes. These marginal PDFs are available as input from assumed PDF schemes, such as in the version of CLUBB that includes the hydrometeor PDFs (e.g., Storer et al. 2015; Thayer-Calder et al. 2015). The conditional sampling step helps to account for the different correlations between the main elements of a deep convective system: that is, updrafts, downdrafts, and quiescent/stratiform regions. A power-law scaling of the mean variables helps to account for correlation within each region, which is needed for a better estimate of the flux magnitudes. Finally, a lower-level adjustment is applied to the updraft flux to correct for the assumption that strong updrafts and downdrafts carry comparable hydrometeor mass mixing ratios, which does not hold at those altitudes.

The mean flux profiles for mass mixing ratios of four hydrometeor types (rain, cloud ice, snow, and graupel) are computed using marginal PDFs of vertical velocity and mass mixing ratios taken from the CRM simulations. The parameterized flux profiles are in good agreement with the fluxes computed by the CRM directly for all four considered episodes of deep convection in continental (ARM97, MC3E-1, and MC3E-2 cases) and tropical maritime (TWP-ICE case) environments. The general shape of the flux profiles and the magnitude of the fluxes in the new parameterization are significantly improved over those computed using the eddy-diffusion approximation used in a current parameterization.

This study presents an important extension of Wong et al. (2015), who used bivariate conditional sampling of joint (two dimensional) PDFs of vertical velocity and hydrometeor mass mixing ratios to parameterize vertical hydrometeor fluxes due to unresolved vertical motions. Here it is demonstrated that comparable improvement in parameterized fluxes can be achieved by conditionally sampling the marginal PDFs of these quantities, making the scheme appropriate for implementation in PDF-based schemes which do not explicitly predict joint distributions. The proposed parameterization relies on the power-law PDF scaling, which has been empirically introduced by Wong et al. (2015) to adjust the magnitude of the parameterized vertical hydrometeor fluxes. Our study illustrates that the exponent (*α*) in the PDF scaling [Eq. (7)] is closely related to the correlation between *w* and *q*_{x} within the quadrants of the joint *w*–*q*_{x} PDF.

Robust features of the 2D PDFs, based on CRM simulations, are used to guide the parameterization development. These features will need to be evaluated against high-resolution spatial and temporal observations (e.g., scanning radars) as they become available. We note that the scheme uses distributions of meteorological variables (namely vertical velocity and hydrometeor mass mixing ratios), which are “native” variables that can be more readily measured and modeled within a domain. This facilitates a more direct comparison among models and observations. On the contrary, updraft mass fluxes and core velocities, as are often used in traditional convection schemes, are more difficult to compare because of differing definitions.

The scaled-PDF method is computationally less expensive than incorporating prognostic equations for the higher-order moments of hydrometeors in CLUBB. Based on this study, the parameterization scheme has shown great potential as a way to couple the convective vertical velocities with microphysics in existing cumulus schemes, assuming that the schemes can provide reasonable marginal distributions of these variables.

## Acknowledgments

This research is based on work supported by the U.S. Department of Energy, Office of Science, Biological and Environmental Research under the Atmospheric System Research (ASR) Program. Computing resources for the simulations are provided by the National Energy Research Scientific Computing Center (NERSC). Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC06-76RLO1830.

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