1. Introduction
Turbulent vertical velocity fluctuations play a key role in planetary boundary layer (PBL) and cloud processes, and their representation forms a critical component of weather, climate, and dispersion models. One of the primary mechanisms for vertical transport of heat, moisture, momentum, and trace gases in the atmosphere is through turbulent eddies. For the scales at which most regional and global models operate, turbulent eddies are not resolved and must therefore be parameterized. At typical model gridbox sizes greater than ~1–2 km, the mean vertical velocity of all the updrafts and downdrafts contained within a grid box will be close to zero. However, in order to represent vertical transport and cloud processes correctly in models, it is critical to know how the vertical velocity is partitioned between updrafts and downdrafts within a grid box.
A key control in the formation of cloud droplets is the updraft velocity. The rate at which an air parcel rises controls its cooling and in turn supersaturation and activation of aerosol particles into droplets. A cloud droplet will continue to grow while the supersaturation in a rising, cooling air parcel exceeds the critical supersaturation. The number and size of cloud droplets are fundamental to the optical properties of clouds, as well as for precipitation—the so-called aerosol indirect effects (AIE). These effects constitute one of the greatest sources of uncertainty in modeling climate. Thus, a key to improving the simulation of clouds—and climate more generally—is an improved representation of the subgrid vertical velocity (Lohmann et al. 1999; Donner et al. 2016).
The distribution of subgrid vertical velocity may be incorporated indirectly, such as in mass-flux cumulus models, where the mass flux is the product of vertical velocity, density, and updraft area. Alternatively, the distribution of subgrid vertical velocity may be represented directly, using the approach of assumed probability distribution functions (PDFs). Initially, PDF methods involved only thermodynamic variables, such as temperature and moisture (Sommeria and Deardorff 1977; Mellor 1977), and later incorporated vertical velocity (Randall 1987; Lewellen and Yoh 1993, hereafter LY; Lappen and Randall 2001; Larson et al. 2002; Cheng and Xu 2008; Bogenschutz et al. 2012) to account for cloud dynamics. Some approaches represent the joint PDF, including vertical velocity, as a double-delta function (Randall 1987; Randall et al. 1992; Lappen and Randall 2001). However, owing to difficulties representing the skewed nature of the PDF in cumulus cloud layers with a double-delta function, subsequent approaches have used a double-Gaussian PDF (LY; Larson et al. 2002; Cheng and Xu 2008; Bogenschutz et al. 2012). The double-Gaussian PDF is one of the simplest distributions that will permit both symmetrical and skewed shapes, corresponding to Gaussian and non-Gaussian turbulence statistics, respectively. The ability to represent both regimes in a model is important, as they both occur frequently in the atmosphere.
Stratocumulus clouds typically exhibit Gaussian-like PDFs, whereas cumulus cloud layers with low-cloud fraction (less than ~20%) may be strongly skewed and non-Gaussian. Observations of the vertical velocity PDF in convective boundary layers (CBLs) have shown it is generally positively skewed, with a negative mode (Berg et al. 2017). The shape of the PDF is a result of how the turbulence is typically organized in the CBL: strong updrafts occupying a small horizontal area embedded in subsidence (downdraft) regions with smaller velocities that occupy a larger horizontal area.
Direct representation of the vertical velocity PDF has a longer history in modeling of pollutant dispersion using Lagrangian stochastic models, where Bærentsen and Berkowicz (1984) proposed using a double-Gaussian function to model the PDF of vertical velocity in the CBL. This model was later supported by observations in the PBL (Quintarelli 1990; Du et al. 1994). The means and relative amplitudes of the two Gaussians can vary, which allows a skewed distribution. If the two Gaussians overlap, then the PDF reduces to a single Gaussian. For cumulus, the main mode corresponds to mostly noncloudy downdrafts, while the second mode corresponds to mostly narrow cloudy updrafts. To specify the double-Gaussian PDF, five parameters are required, corresponding to the relative amplitude, and the means and standard deviations of each Gaussian. If a model can predict the first five statistical moments of vertical velocity, then the PDF may be immediately specified. However, computing the higher-order moments is a difficult problem and expensive computationally. Thus, one or more closure assumptions are usually made to reduce the number of required moments.
To account for the skewness of the vertical velocity distribution, a model must be able to provide moments of up to at least third order. Observations of higher-order vertical velocity turbulence statistics (moments greater than 2) in the atmosphere are difficult to obtain, as a long averaging period is required to reduce systematic measurement bias and to reach an acceptable level of accuracy. For this reason there are few observations of higher-order moments reported in the literature. Most observations have also been limited to the subcloud layer or the lowermost layers of cloud (e.g., Hogan et al. 2009; Ansmann et al. 2010; Lenschow et al. 2012; Tonttila et al. 2015; Maurer et al. 2016; Berg et al. 2017). One of the few studies to observe the vertical velocity structure throughout the full layer of cloud is from Ghate et al. (2014). However, they did not analyze the PDF.
Owing to the lack of data, direct numerical simulation (DNS) and LES studies have been used to provide further insight, for example, Moeng and Rotunno (1990), Lenschow et al. (2012), and Waggy et al. (2016). Again, these studies were limited to the subcloud layer, although Guo et al. (2008) compared LES and observations of higher-order vertical velocity statistics within a marine stratocumulus cloud layer. Moyer and Young (1991) and Ching et al. (2010) observed vertical velocity skewness within a marine stratocumulus cloud but did not look at the PDF. Larson et al. (2002) compared their parameterized vertical velocity PDF with airborne observations within stratocumulus and cumulus clouds, although only at a single level. Chu et al. (1996) and Liu et al. (2011) examined the vertical velocity PDF from observations and compared them with several parameterizations; however, their analysis was limited to the surface layer.
The aim of the present work is to examine the vertical velocity PDF throughout the depth of the lower atmosphere, including the cloud layer and above. To our knowledge this is the first study to do so. Several existing schemes to model the vertical velocity PDF are compared with LESs of neutral and unstably stratified boundary layers, topped by stratocumulus and shallow cumulus cloud layers, respectively. These schemes are now in use in some major climate models, for example, CESM (http://www.cesm.ucar.edu) and the Energy Exascale Earth System Model (E3SM) (http://e3sm.org). A new closure is suggested that improves performance within the cloud layer, where higher-order vertical velocity turbulence statistics are found to be markedly different from the subcloud layer.
The paper is organized as follows. Section 2 introduces the LES cases and model configuration. Section 3 examines vertical velocity statistics from the LES and compares them with observations and intercomparison model results, where available. Sections 4 and 5 discuss double-Gaussian PDF closures for vertical velocity, and examine the parabolic relationship between skewness and kurtosis. A new closure is also introduced. Section 6 shows the resulting vertical velocity PDF within the cloud layer for the different model closures. Section 7 examines the updraft probability estimated by the different model closures, and section 8 presents results using the double-Gaussian PDFs to close higher-order moments.
2. Large-eddy simulations
We analyze the vertical velocity (w) PDF in four boundary layer cloud cases using LES. The Advanced Research version of WRF (WRF-ARW), versions 3.6 and 3.8.1, is run in LES mode using the WRF LES package released by Takanobu Yamaguchi (Yamaguchi and Feingold 2012; available at http://esrl.noaa.gov/csd/staff/tak.yamaguchi/code/) and updated to WRF, version 3.6 and 3.8.1, by the author (version 3.6 available from the same location). WRF is a nonhydrostatic fully compressible model and uses the covariant velocities (u, υ, w), dry air mass, geopotential, potential temperature θ, and subgrid-scale (SGS) turbulent kinetic energy (TKE) as the basic prognostic variables (Yamaguchi and Feingold 2012). Sensitivity to the acoustic time step is found using potential temperature as a prognostic variable, which is remedied by instead using the moist potential temperature in version 3.8.1 simulations (Xiao et al. 2015). SGS turbulence is parameterized with a 1.5-order TKE closure (Klemp and Wilhelmson 1978; Deardorff 1980) and surface fluxes are computed using Monin–Obukhov similarity. For microphysics, this study uses the Morrison (Morrison et al. 2009) scheme, which prognoses mass mixing ratios of nonprecipitating water (vapor, water, and ice) and precipitating water (rain, snow, and graupel). Here, diagnosed number concentrations are used. The lateral boundary conditions are periodic and the upper boundary is rigid. A Rayleigh relaxation layer controls gravity wave reflection from the upper boundary.
A wide range of boundary layer cloud cases are examined: two marine shallow cumulus cases, a continental shallow cumulus case, and a marine stratocumulus case. The first is based on the Barbados Oceanographic and Meteorological Experiment (BOMEX) (Holland and Rasmusson 1973) and is a case of nonprecipitating trade wind cumulus. It is run using the same configuration as the Global Energy and Water Cycle Exchanges project (GEWEX) Global Atmospheric System Studies (GASS) (formerly GEWEX Cloud System Studies) intercomparison case (Siebesma et al. 2003). However, here the size of the domain is increased to 16.9 km × 16.9 km with the model top at 7480 m to increase the turbulence statistics collected. The simulation is run using WRF, version 3.8.1, with horizontal and vertical resolutions of 100 and 40 m, respectively, and a 1-s time step. The case is run for 6 h, with results shown averaged over the last 3 h.
The second case is from the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007) and is a precipitating trade wind cumulus case. It is run using the same configuration as the GASS intercomparison case (VanZanten et al. 2011). The simulation is run using WRF, version 3.6, with a horizontal and vertical resolution of 100 and 40 m, respectively, and a 1-s time step. The domain size is 12.9 km × 12.9 km with the model top at 4000 m. The acoustic time step is set to 0.125 s. The cloud droplet number concentration is prescribed at 70 cm−3. The case is run for 24 h, with results shown averaged over the last 4 h.
The third case is of nonprecipitating continental shallow cumulus and is based on observations made at the Southern Great Plains (SGP) site of the Atmospheric Radiation Measurement (ARM) program. The same configuration as for the GASS intercomparison is used (Brown et al. 2002). The simulation is run using WRF, version 3.8.1, with horizontal and vertical resolutions of 66.7 and 40 m, respectively, and a 0.5-s time step. The domain size is 6470 m × 6470 m with the model top at 4400 m. The case is run for 14.5 h, with results shown averaged over 8.5–10.5 h, corresponding to 1400–1600 local time (LT).
The final and fourth case is from the first research flight (RF01) of the Second Dynamics and Chemistry of Marine Stratocumulus field study (DYCOMS-II) (Stevens et al. 2003) and is a marine stratocumulus case. Again, the same configuration as for the GASS intercomparison is used (Stevens et al. 2005). The simulation is run using WRF, version 3.6, with horizontal and vertical resolutions of 35 and 5 m, respectively, and a 0.1-s time step. The domain size is 3395 m × 3395 m with the model top at 1500 m. The acoustic time step is set to 0.01 s. To maintain the cloud deck over time in line with the intercomparison models, the cloud droplet number concentration is prescribed at 120 cm−3. The case is run for 4 h and the results shown are averaged over the last hour.
The sensitivity of higher-order moments of vertical velocity to the model resolution was examined in BOMEX and ARM, representing two very different shallow cumulus cases displaying strongly non-Gaussian turbulence statistics. In simulations doubling both the vertical and horizontal resolutions, differences in peak kurtosis in the cloud layer were found to be at most 2% for BOMEX and 18% for ARM. The sensitivity to resolution was not examined for DYCOMS RF01, as the high resolution of the standard intercomparison case was found to compare well with observations, and higher-order moments are of less interest for this case relative to the shallow cumulus cases.
For all simulations, the instantaneous three-dimensional vertical velocity field is saved every 5 min of simulation. The results shown are averaged from this output.
3. Vertical velocity statistics
The turbulent structure of the PBL is reflected in the vertical velocity statistical moments. The variance, 
The mean profiles of vertical velocity variance, skewness, and kurtosis are shown in Figs. 1 and 2 for each case. For DYCOMS RF01, a maximum in the vertical velocity variance is seen around cloud base (Fig. 1a). The simulation is more energetic than most of the intercomparison models throughout most of the PBL and is much closer to observations than the intercomparison mean. Observations of the third moment (Fig. 1b) show negative values throughout most of the cloud layer and in the upper subcloud layer, reflecting strong downdrafts driven by cloud-top radiative cooling, which are typical for a stratocumulus cloud. Again, the simulation results are much closer to the observations than the intercomparison mean, which displays the wrong sign throughout most of the PBL. The latter is indicative of decoupling, where the turbulence has become more surface driven, resulting in positive values of the third moment. The skewness in the simulation (Fig. 1c) reflects similar properties as the third moment. Comparatively small values are seen relative to the shallow cumulus cases, indicative of largely symmetric, Gaussian turbulence. A maximum in skewness is seen at cloud top. Model intercomparison and observational data are not available for the skewness for this case. The kurtosis (Fig. 1d) displays largely Gaussian values (around 3), apart from a sharp peak at cloud top. Similar to the skewness, this indicates mostly symmetric, Gaussian turbulence. Observational or model intercomparison data are not available for the kurtosis for any of the cases.
Vertical mean profiles from LES for the DYCOMS RF01 case for (a) variance of w, (b) third moment of w, (c) skewness of w, and (d) kurtosis of w. Horizontal dashed lines indicate the cloud layer, defined as any point with 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles from LES for the DYCOMS RF01 case for (a) variance of w, (b) third moment of w, (c) skewness of w, and (d) kurtosis of w. Horizontal dashed lines indicate the cloud layer, defined as any point with
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles from LES for the DYCOMS RF01 case for (a) variance of w, (b) third moment of w, (c) skewness of w, and (d) kurtosis of w. Horizontal dashed lines indicate the cloud layer, defined as any point with
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 1, but with the LES results for the (a)–(c) BOMEX, (d)–(f) RICO, and (g)–(i) ARM cases. (left) The variance of w, (center) the skewness of w, and (right) the kurtosis of w.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 1, but with the LES results for the (a)–(c) BOMEX, (d)–(f) RICO, and (g)–(i) ARM cases. (left) The variance of w, (center) the skewness of w, and (right) the kurtosis of w.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 1, but with the LES results for the (a)–(c) BOMEX, (d)–(f) RICO, and (g)–(i) ARM cases. (left) The variance of w, (center) the skewness of w, and (right) the kurtosis of w.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
For the BOMEX shallow cumulus case, the vertical velocity variance (Fig. 2a) shows a peak around the middle of the subcloud layer, a minimum around cloud base, and a second peak in the cloud layer. The peak in the subcloud layer is due to surface fluxes, whereas the peak in the cloud layer is related to enhanced turbulence from latent heat release. The simulation here is somewhat more energetic than the intercomparison cases in the upper subcloud layer and lower cloud layer. The peak in the cloud layer is about the same as the intercomparison mean. There are no observations for comparison in this case. The skewness (Fig. 2b) is positive throughout the PBL (apart from some small negative values in the upper cloud layer), indicative of stronger updrafts. The skewness increases through the cloud layer to reach a maximum value of around 3 near the center of the cloud. The increase in skewness reflects updrafts becoming stronger and narrower with height. There are no intercomparison or observational data available for the skewness. The kurtosis for this case (Fig. 2c) exhibits Gaussian values near the surface and above the cloud layer. It gradually increases away from the surface in the subcloud layer, in line with previous observations in the CBL, for example, Lenschow et al. (2012). Values of around 6 are seen just below cloud base. Above this, the kurtosis increases sharply within the cloud layer to reach a maximum of around 48 near the center. Few reports exist on the vertical velocity kurtosis within shallow cumulus, although large values in the center of the cloud layer are also found in the LES performed by Zhu and Zuidema (2009, their Fig. 3). A similar kurtosis profile within a shallow cumulus cloud is seen in Fig. 13 of Bougeault (1981) in his numerical simulation, although with a lower peak value of around 16. The high kurtosis is a result of low-cloud fraction typical of shallow cumulus, with predominantly small fluctuations about the mean in clear air, coupled with strong, narrow updrafts in cloudy air that form the long tail in the vertical velocity distribution. There are no observational data for comparison for this case.
The RICO shallow cumulus case shows overall qualitatively similar vertical velocity statistics as the BOMEX case (Figs. 2d–f). There is again a double peak in the variance, with the simulation here more energetic than the intercomparison models. This case is more energetic in the cloud layer than the BOMEX case. The skewness is mostly close to the intercomparison mean and peaks in the center of the cloud layer around 3, similar to BOMEX, although it decreases less rapidly around the peak. The kurtosis is mostly close to Gaussian in the subcloud layer and above the cloud, and increases with height in the subcloud layer. It peaks around 38 in the center of the cloud layer. There are no observational data for comparison for this case.
The ARM case differs from the other shallow cumulus cases in that it is driven by stronger surface fluxes over land. More energetic turbulent eddies result, reflected in the vertical velocity variance profile (Fig. 2g). Higher values are seen in both the subcloud and cloud layers relative to the other cases. The skewness (Fig. 2h) for this case does not exhibit the same well-defined peaks as for BOMEX and RICO in the cloud layer and varies more gradually. A maximum skewness of around 1.2 is seen at the top of the subcloud layer and in the center of the cloud layer. The kurtosis (Fig. 2l) peaks at the top of the cloud layer, unlike BOMEX and RICO, which peak closer to the center of the cloud. The kurtosis reaches a maximum of about 40 for ARM.
4. Modeling the vertical velocity PDF
If a model was able to predict an infinite number of statistical moments, then the vertical velocity PDF could be exactly defined (Shohat and Tamarkin 1943). However, owing to computational expense, only a very limited number may be predicted. Thus, any relationships between moments that can be used to diagnose higher-order moments not predicted by the model are very beneficial. Most atmospheric models in current use predict up to the third-order moments at most. However, this leaves the fourth-order moments that still need to be diagnosed to close the third-order moment equations. The earliest approaches used the Millionshchikov hypothesis (Millionshchikov 1941; Monin and Yaglom 1971, 1975), which states that the fourth-order moments can be assumed to be quasi normal (Gaussian). This assumption has been used in many models (e.g., André et al. 1976; Moeng and Randall (1984); Canuto et al. (1994). However, subsequent studies have found this assumption to fail for skewed convective boundary layers (Ilyushin and Kurbatskii 1997; Canuto et al. 2001; Gryanik and Hartmann 2002, hereafter GH; Alberghi et al. 2002). The results shown in Fig. 2 for the shallow cumulus cases corroborate these studies, where the fourth-order vertical velocity moments are very large and far from Gaussian. Prior studies have been limited to the subcloud layer, and one of the aims of the present work is to examine fourth-order closures for the entire depth of the lower atmosphere, including the cloud layer. As we have seen in Fig. 2, extreme departures from Gaussian values are seen for the kurtosis (normalized fourth-order moment) in the shallow cumulus cloud layer.
Tampieri et al. (2000) proposed 
Figure 3a shows the relationship between S and K for all the LES cases, for points within the subcloud layer for the time periods specified in section 2 for each case. The best nonlinear least squares fit for the LES data (with the constraint when 
The relationship between skewness and kurtosis for every LES grid point over the time period specified in section 2, for (a) points within the subcloud layer and (b) points within the cloud layer. Results for BOMEX, RICO, ARM and DYCOMS RF01 are shown in red, blue, black, and magenta, respectively. The green curve shows the best fit to the LES, while the other curves are previously suggested relations discussed in the text.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The relationship between skewness and kurtosis for every LES grid point over the time period specified in section 2, for (a) points within the subcloud layer and (b) points within the cloud layer. Results for BOMEX, RICO, ARM and DYCOMS RF01 are shown in red, blue, black, and magenta, respectively. The green curve shows the best fit to the LES, while the other curves are previously suggested relations discussed in the text.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The relationship between skewness and kurtosis for every LES grid point over the time period specified in section 2, for (a) points within the subcloud layer and (b) points within the cloud layer. Results for BOMEX, RICO, ARM and DYCOMS RF01 are shown in red, blue, black, and magenta, respectively. The green curve shows the best fit to the LES, while the other curves are previously suggested relations discussed in the text.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Within the cloud layer (Fig. 3b), however, the kurtosis increases more rapidly with skewness than in the subcloud layer and the aforementioned relationships break down. The different behavior in the cloud layer is due to enhanced turbulence from latent heat release. The best nonlinear least squares fit to the LES data (with the constraint when 
The new relationships will subsequently be used to diagnose the kurtosis from the skewness, in both the subcloud and cloud layers, and to define a closure for specifying the double-Gaussian PDF parameters for vertical velocity, discussed in the next section.
5. Specifying double-Gaussian PDF parameters
Assuming the host model is able to predict the first three moments of vertical velocity, an additional two relations are required to specify the five parameters of the PDF. A number of closures have been suggested in the literature. Here, we examine some of the most recently proposed closures and compare them with the unparameterized double-Gaussian “best fit” PDF from the LES data. The required input moments for the closures are obtained directly from LES. This approach enables a fair comparison between the model closures, and the results reflect the best-case scenario.
The unparameterized double-Gaussian PDF parameters are determined from the LES data using an expectation maximization (EM) algorithm (Dempster et al. 1977). It is an iterative method that finds the maximum likelihood estimates of the PDF parameters, that is, the set of parameters most likely to have generated the given data. It has previously been used by Perraud et al. (2011) and Firl and Randall (2015) to determine double-Gaussian PDF parameters for clouds. Here, the EM algorithm that is part of the mixtools package in R is used (Benaglia et al. 2009) to determine the best-fit PDF parameters. It is applied at each model level in the LES over the time period specified in section 2 for each case. The results shown are averaged over the same time period.
Figure 4 shows an example of the best-fit double-Gaussian PDF determined using the EM algorithm within a stratocumulus cloud layer (DYCOMS RF01 case) and a shallow cumulus cloud layer (BOMEX case). For the stratocumulus case, Gaussian 1 represents the bulk of the cloudy points. Gaussian 2 supplements Gaussian 1 where the skewness or kurtosis departs from Gaussian. For the shallow cumulus case, Gaussian 1 instead represents the bulk of the noncloudy points. Gaussian 2 represents the tails of the distribution, including the long updraft tail representing the bulk of the cloudy points. Note that this contrasts with previous studies (e.g., Bærentsen and Berkowicz 1984; Luhar et al. 1996), where Gaussian 2 is assumed to represent primarily the updraft tail only. Figure 4 also highlights that while a single-Gaussian PDF provides a good approximation for stratocumulus, it does not for shallow cumulus.
The vertical velocity PDF within the cloud layer for (a) DYCOMS RF01 and (b) BOMEX. Black shows results from LES, with cloudy points shown in dark blue, and clear points shown in light blue. Purple shows the single-Gaussian PDF with a standard deviation equal to the LES. Red shows the double Gaussian best fit, which is the sum of Gaussian 1 (orange), and Gaussian 2 (green).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The vertical velocity PDF within the cloud layer for (a) DYCOMS RF01 and (b) BOMEX. Black shows results from LES, with cloudy points shown in dark blue, and clear points shown in light blue. Purple shows the single-Gaussian PDF with a standard deviation equal to the LES. Red shows the double Gaussian best fit, which is the sum of Gaussian 1 (orange), and Gaussian 2 (green).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The vertical velocity PDF within the cloud layer for (a) DYCOMS RF01 and (b) BOMEX. Black shows results from LES, with cloudy points shown in dark blue, and clear points shown in light blue. Purple shows the single-Gaussian PDF with a standard deviation equal to the LES. Red shows the double Gaussian best fit, which is the sum of Gaussian 1 (orange), and Gaussian 2 (green).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the standard deviation of Gaussian 1 and 2 for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Black shows the double Gaussian best fit, and the closures of LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Note for CLUBB, 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the standard deviation of Gaussian 1 and 2 for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Black shows the double Gaussian best fit, and the closures of LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Note for CLUBB,
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the standard deviation of Gaussian 1 and 2 for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Black shows the double Gaussian best fit, and the closures of LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Note for CLUBB,
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Figure 5 shows vertical mean profiles of 
For the marine shallow cumulus cases, BOMEX and RICO, the new closure for 
For the continental shallow cumulus case, ARM, the models generally overestimate 
Finally, for the stratocumulus case, DYCOMS RF01, the new closure and LY provide a good fit to the LES for 
6. Cloud-layer vertical velocity PDFs
Here we examine the vertical velocity PDF within the cloud layer for the different cases, comparing the LES PDF with the various double-Gaussian closures. Figures 6 and 7 show the vertical velocity PDF for each case at a level in the cloud where skewness is around maximum, to highlight differences in the double-Gaussian closures. The right-hand column has a log scale to highlight differences in the tails.
The vertical velocity PDF within the cloud layer for (a),(b) BOMEX and (c),(d) RICO. The PDF with a log scale is shown in (b) and (d). The PDF from LES is shown in black, and the double- and single-Gaussian best fits are shown in red and green, respectively. The closures from LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Results with the new prognostic and diagnostic closure are shown in solid and dashed orange, respectively.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The vertical velocity PDF within the cloud layer for (a),(b) BOMEX and (c),(d) RICO. The PDF with a log scale is shown in (b) and (d). The PDF from LES is shown in black, and the double- and single-Gaussian best fits are shown in red and green, respectively. The closures from LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Results with the new prognostic and diagnostic closure are shown in solid and dashed orange, respectively.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The vertical velocity PDF within the cloud layer for (a),(b) BOMEX and (c),(d) RICO. The PDF with a log scale is shown in (b) and (d). The PDF from LES is shown in black, and the double- and single-Gaussian best fits are shown in red and green, respectively. The closures from LY, L96/ADG2, and CLUBB are shown in magenta, purple, and blue, respectively. Results with the new prognostic and diagnostic closure are shown in solid and dashed orange, respectively.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 6, but for (a),(b) ARM and (c),(d) DYCOMS RF01.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 6, but for (a),(b) ARM and (c),(d) DYCOMS RF01.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
As in Fig. 6, but for (a),(b) ARM and (c),(d) DYCOMS RF01.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The shallow cumulus cases are characterized by strongly non-Gaussian turbulence statistics, with PDFs displaying high skewness and kurtosis. The amplitude of Gaussian 1 is significantly overestimated by LY and L96/ADG2 for BOMEX (Figs. 6a,b), whereas CLUBB and the new closure provide a good estimate of the amplitude. As discussed in the previous section, LY and L96/ADG2 underestimate 
Similar differences between the models are seen for RICO, although there is less overestimation of the amplitude of Gaussian 1 for LY (Figs. 6c,d). CLUBB and, to a lesser degree, the new closure somewhat overestimate the amplitude. The updraft tail is a little longer and fatter than the BOMEX case, and all the models capture this. The new closure performs well for the updraft tail. It overestimates the downdraft tail for this case, although it should be noted that the double-Gaussian best fit also overestimates the downdraft tail, pointing to some fundamental limitations of the double-Gaussian formulation.
The ARM case is driven by stronger surface fluxes over land, relative to the BOMEX and RICO cases. The PDF has a greater variance and longer tails as a result (Figs. 7a,b). All the models estimate the amplitude of Gaussian 1 reasonably well. However, as with BOMEX and RICO, the length of the tails is underestimated for LY, L96/ADG2, and CLUBB. The new closure performs well, both for Gaussian 1 and for the tails. The minor mode seen in CLUBB is more prominent in this case than in BOMEX and RICO, and results in an overestimation of the probability of w values between 0.8 and 2 m s−1.
Finally, for DYCOMS RF01, LY, L96/ADG2, and the new closure perform well, with minimal differences seen (Figs. 7c,d). The single Gaussian also provides a good estimate of the PDF. However, CLUBB exhibits a bimodal PDF, with the antimode occurring at the modal value of the LES PDF. The formulation of CLUBB does not allow it to collapse to a single Gaussian for low skewness as with the other closures. It also underestimates the probability in the tails.
7. Updraft probability
The integral of the w PDF over a particular interval is the probability of w values occurring within that interval. For cloud and aerosol physics, we are particularly interested in the updraft portion of the PDF, where cloud forms. Here, we examine the PDF integrals for positive w, which provides an overall measure of the quality of the PDF.
The shallow cumulus cases show similar updraft probability characteristics overall from the LES (Fig. 8). In the subcloud layer, the probability of an updraft lies between 0.4 and 0.5. This probability is equivalent to the updraft fractional area over the domain, an important component of mass-flux cumulus models. The probability decreases with height from the surface and reaches a minimum in the mid- to upper half of the subcloud layer. From this point it increases again until inside the cloud layer, where the probability lies around 0.5. For the marine shallow cumulus cases (BOMEX and RICO), the updraft probability is slightly less than 0.5 for much of the cloud layer. It increases slightly above 0.5 in the upper cloud layer, before becoming approximately constant around 0.5 above. The continental shallow cumulus case (ARM) shows a greater variation in updraft probability in the cloud layer as a result of stronger fluxes.
Vertical mean profiles of the updraft probability 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the updraft probability
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the updraft probability
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
The stratocumulus case (DYCOMS RF01) displays opposite updraft probability characteristics to the shallow cumulus cases. For this case, the updraft probability gradually increases with height within the subcloud layer, from about 0.48 near the surface to 0.54 at the base of the cloud. The probability increases above 0.5 around the midpoint of the subcloud layer. The probability stays roughly constant in the lower half of the cloud layer, at around 0.54, and then decreases with height to the top of the cloud layer, where it reaches a minimum of 0.45. This minimum is a reflection of the cloud-top radiative cooling driving negative buoyancy. Above the cloud layer, the updraft probability increases again to around 0.5.
The greatest differences between the models are seen for the BOMEX and RICO cases. LY, L96/ADG2, and the new closure slightly overestimate the updraft probability from the surface to the upper subcloud layer. Above this height agreement is better, although the results diverge again around the cloud base, where the new closure (with predicted kurtosis) performs the best. CLUBB underestimates the updraft probability throughout the subcloud layer, by up to 10% for RICO. Going into the cloud layer, the models diverge further, with all underestimating the probability through much of the cloud layer. L96/ADG2 underestimates the probability the most, by up to 45% in both cases. LY underestimates the probability to a lesser degree, by up to 35% in BOMEX and 25% in RICO. CLUBB performs similarly for BOMEX and RICO, underestimating the probability by up to 20%. The new prognostic closure performs best, underestimating the updraft probability by up to 6%. Around cloud top and above, the models perform similarly to the LES, although L96/ADG2 and CLUBB slightly overpredict the updraft probability near the top of the cloud for RICO.
The models show somewhat similar characteristics for ARM as BOMEX and RICO within the subcloud layer. However, model performance is generally improved overall for the cloud layer. Here, the updraft probability is underestimated by up to 20%, 17%, 14%, and 4% for L96/ADG2, CLUBB, LY, and the new prognostic closure, respectively. Above the cloud layer, LY and the new closure perform best, with results similar to LES. CLUBB and L96/ADG2 underestimate the probability by up to 10%.
Of all the cases, the models perform best for DYCOMS RF01, with less variation between them. The better agreement results from the low skewness for this case. CLUBB shows the greatest deviation from the LES, with the updraft probability overestimated by up to 6% near cloud base and underestimated by up to 16% near cloud top. LY, L96/ADG2, and the new closure deviate from LES from the upper subcloud layer to the upper half of the cloud layer, with the updraft probability underestimated by up to 6%. No model performs best for cloud base, with LY, L96/ADG2, and the new closure underestimating, and CLUBB overestimating the probability, by approximately the same degree. At cloud top, the new prognostic closure performs best, closely matching LES. Above the cloud layer, the models perform similarly to LES, with greater deviations seen in CLUBB.
For the strongest updrafts, where 
Vertical mean profiles of the updraft probability for 
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the updraft probability for
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the updraft probability for
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
For ARM, the overestimation of the probability is generally reduced compared to BOMEX and RICO. The probability is overestimated up to a factor of 2.4, 2.2, and 2.2, for CLUBB, LY, and L96/ADG2, respectively, within the cloud layer. The overestimation is significantly reduced for the new closure, with the probability overestimated by up to 20% and 47% for the prognostic and diagnostic closures, respectively. Overall, the new prognostic closure provides the best estimate of the probability of strong updrafts, and it most closely follows the shape of the LES profile.
As before, the models show better agreement for DYCOMS RF01. LY, L96/ADG2, and the new closure all provide a reasonable estimate of the probability within most of the cloud layer. However, the models diverge near the cloud top, with the new closure in good agreement with LES, and CLUBB, LY, and L96/ADG2 overestimating the probability by up to 70%, 50%, and 42%, respectively. Apart from around cloud top, CLUBB significantly underestimates the probability relative to LES and the other models by around 60%.
8. Closure of higher-order moments
Figure 10 shows 
Vertical mean profiles of the fourth moment of w for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Curves are as in Fig. 7. Horizontal dashed lines indicate the cloud layer.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the fourth moment of w for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Curves are as in Fig. 7. Horizontal dashed lines indicate the cloud layer.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
Vertical mean profiles of the fourth moment of w for (a) BOMEX, (b) RICO, (c) ARM, and (d) DYCOMS RF01. Curves are as in Fig. 7. Horizontal dashed lines indicate the cloud layer.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0149.1
In the subcloud layer, the model results converge, with the new prognostic closure, LY, and L96/ADG2 performing slightly better than CLUBB for BOMEX and RICO. The new prognostic closure performs best in the upper subcloud layer and around cloud base. For the ARM case, LY, L96/ADG2, and the new closure somewhat overestimate the peak in 
For DYCOMS RF01, the new closure, LY, and L96/ADG2 perform similarly within the cloud layer, giving good agreement with LES. However, CLUBB underestimates the peak by 30%. In the subcloud layer, the new prognostic closure, LY, and L96/ADG2 perform slightly worse than in the cloud layer. The new diagnostic closure underestimates 
9. Discussion and conclusions
We have examined the vertical velocity probability distribution function (PDF) throughout the depth of the lower atmosphere, including the cloud layer and above, in large-eddy simulations (LESs) of four GASS boundary layer cloud intercomparison cases. To our knowledge, this is the first study to do so. Two marine shallow cumulus cases (BOMEX and RICO), a continental shallow cumulus case (ARM), and a marine stratocumulus case (DYCOMS RF01) are studied. DYCOMS RF01 exhibits largely Gaussian vertical velocity statistics, typical of stratocumulus cloud, and, in this case, representing the vertical velocity PDF with a single Gaussian will suffice. However, the shallow cumulus cases are characterized by non-Gaussian turbulence statistics with large departures from Gaussian values for higher-order moments such as skewness and kurtosis in the cloud layer. For these cases, a PDF that is able to represent such non-Gaussian turbulence statistics is required. A double-Gaussian PDF is one of the simplest distributions that will permit non-Gaussian values of higher-order moments, and it has been employed to represent turbulence by LY, Luhar et al. (1996), Larson et al. (2002), and Larson and Golaz (2005) among others. These model closures for the vertical velocity PDF have been examined, with the required input statistical moments obtained directly from LES. This approach enables a fair comparison between the model closures, and the results reflect the best-case scenario. We do not examine the impact of inaccuracies in the various “host” models (i.e., the higher-order turbulence models used to predict the required moments to construct the double-Gaussian PDF in an atmospheric model). Neither do we test the applicability of the model closures across a range of model grid sizes. Bogenschutz et al. (2010) found little sensitivity to grid size for double-Gaussian PDFs applied to shallow cumulus.
Existing model PDF closures are found to perform well overall in the subcloud layer for the shallow cumulus cases. However, they break down in the cloud layer, where vertical velocity statistics become strongly non-Gaussian and high skewness and kurtosis are seen. Inaccuracies are seen in both the amplitude and standard deviation of the first Gaussian, representing primarily clear air, and the second Gaussian, representing the tails. The latter includes the updraft tail, consisting mostly of cloudy air. The tails of the PDF decay too rapidly, underestimating extreme values. At intermediate values within the updraft tail, the probability tends to be overestimated. For updrafts greater than 
For the stratocumulus case, the majority of the models perform well overall, in both the subcloud and cloud layer. There is also much better agreement between the models, with the exception of CLUBB. The latter produces a bimodal PDF, owing to its formulation, which does not collapse to a single Gaussian with low skewness. However, it still produces a reasonable overall probability of updrafts, apart from updrafts greater than 
To diagnose the source of bias in existing model closures and to guide development of an improved closure, an unparameterized double-Gaussian best fit to the LES was obtained using an expectation maximization (EM) algorithm. This analysis highlighted shortcomings in specifying the standard deviation of the second Gaussian 
The new closure relies on knowledge of the vertical velocity kurtosis, or the fourth moment of vertical velocity. If the host model is able to predict the fourth moment, then it may be immediately used to specify the double-Gaussian PDF. The best results are obtained with this method. However, most higher-order turbulence models predict only the third moment at most. In this case, the kurtosis must be diagnosed from lower-order moments. A parabolic relationship between skewness and kurtosis has been found for this purpose (Mole and Clarke 1995; GH; Waggy et al. 2016). These relationships are in good agreement with the LES results within the subcloud layer. However, they break down in the cloud layer, where the kurtosis increases more rapidly with skewness than in these relations. The parabolic relationship is modified here for use in the cloud layer, which provides a much better estimate of the kurtosis. To our knowledge, this relationship has not been previously examined beyond the subcloud layer. The different behavior within the cloud layer is due to enhanced turbulence from latent heat release. The new parabolic relationship is split into two regimes, one for the cloud layer and one outside the cloud layer. If no information is available regarding the cloud layer, then the appropriate regime may also be identified based on only the skewness, with lower values of skewness assumed outside the cloud layer. The resulting relationship appears to perform well for the cases studied here; however, it is recommended more cases be studied in future. A preliminary analysis of the LES ARM Symbiotic Simulation and Observation (LASSO) dataset (Gustafson et al. 2017) has also revealed the different nature of the parabolic relationship in cumulus cloud layers. Model studies of vertical velocity kurtosis within shallow cumulus layers are scarce, although the few available are qualitatively in agreement with the current work. In addition, observations of vertical velocity skewness and kurtosis within cloud layers are particularly in short supply to verify the results seen here.
Estimation of the vertical velocity PDF is much improved with the new closure for the shallow cumulus cases, for both the mode and the tails. The overall updraft probability is also in much better agreement with LES, with the probability underestimated by no more than 10%. In addition, the probability of updrafts greater than 
Once the double-Gaussian PDF is specified, it may be used to diagnose higher-order moments, such as 
The main drawback with the new closure is that it requires numerically solving the moment equations to find the amplitudes and means of the Gaussians after the standard deviations have been specified. As a result, it is more computationally expensive than an analytical formulation. As with existing closures, it relies on the ability of the host model to accurately predict the higher-order moments of vertical velocity, in particular 
Acknowledgments
I would like to express my appreciation to Steven Ghan and the Atmospheric Sciences and Global Change Division at PNNL for their hospitality and support during my visit, when much of this work was carried out. I would also like to thank Grant Firl and one anonymous reviewer for their constructive comments. Partial funding is gratefully acknowledged from the EU APPLICATE Project 727862, and the Climate Model Development and Validation program funded by the Office of Biological and Environmental Research in the U.S. Department of Energy (DOE) Office of Science. The Pacific Northwest National Laboratory (PNNL) is operated for the DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. Additional funding is acknowledged from the Bolin Centre for Climate Research. Supercomputing facilities were provided by PNNL and NCAR.
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