Due to their coarse horizontal resolution, present-day climate models must parameterize deep convection. This paper presents single-column simulations of deep convection using a probability density function (PDF) parameterization. The PDF parameterization predicts the PDF of subgrid variability of turbulence, clouds, and hydrometeors. That variability is interfaced to a prognostic microphysics scheme using a Monte Carlo sampling method.

The PDF parameterization is used to simulate tropical deep convection, the transition from shallow to deep convection over land, and midlatitude deep convection. These parameterized single-column simulations are compared with 3-D reference simulations. The agreement is satisfactory except when the convective forcing is weak.

The same PDF parameterization is also used to simulate shallow cumulus and stratocumulus layers. The PDF method is sufficiently general to adequately simulate these five deep, shallow, and stratiform cloud cases with a single equation set. This raises hopes that it may be possible in the future, with further refinements at coarse time step and grid spacing, to parameterize all cloud types in a large-scale model in a unified way.

Deep convective clouds are an integral part of the climate system, yet
representing these clouds accurately in climate models is a challenge.
Explicitly resolving these clouds in decades-long climate simulations is not
yet computationally feasible; thus, it is beneficial to parameterize deep
convection

Deep convection has often been parameterized by the use of mass-flux schemes

PDF parameterizations possess several advantages for the purpose of
parameterizing deep convection. First, the use of a PDF that includes
hydrometeor species and vertical velocity allows for a detailed
representation of microphysics and dynamics, and of the coupling between
them. For instance, a PDF parameterization that

Despite these advantages for parameterizing deep clouds, to date PDF
parameterizations have been applied only to shallow clouds, except in higher-resolution cloud-resolving models

The modified version of CLUBB is then used to perform single-column simulations of three deep convective cases: a tropical case, a case of transition from shallow to deep convection, and a midlatitude case. To demonstrate that the modified version of CLUBB can still simulate shallow cloud cases, we also simulate a shallow cumulus layer and stratocumulus layer. To assess the quality of the single-column CLUBB simulations, we compare the deep convective cases to cloud-resolving simulations and the shallow cases to large-eddy simulations.

The remainder of this paper is organized as follows. In Sect.

CLUBB is a single-column model of clouds
and turbulence in the atmosphere

CLUBB's equation set contains various higher-order moments that are unclosed.
CLUBB closes many of these moments using the assumed PDF method. That is, CLUBB assumes a functional form of the
subgrid PDF and integrates over it in order to estimate the unclosed moments.
CLUBB's subgrid PDF is a single, multivariate PDF with a double Gaussian (sum
of two normals) distribution for vertical velocity

In order to drive microphysical processes, CLUBB has the option to call a
microphysics scheme using a variant of traditional Monte Carlo sampling

In summary, CLUBB-SILHS parameterizes subgrid variability as follows

The computational cost of CLUBB-SILHS is acceptable. CLUBB adds 20 % to the
computational cost of version 5 of the Community Atmosphere Model (CAM5,

Both CLUBB and SILHS are contained in a single SVN (Subversion) code repository that is
freely available for noncommercial use at

In order to better simulate deep convection using CLUBB, we have generalized the treatment of subgrid variability. In particular, we have generalized CLUBB's multivariate PDF to include ice, incorporated the effects of ice and rain on subgrid buoyancy production, improved the vertical transport of hydrometeors, and allowed changes in the microphysics to affect scalar fluxes and variances.

CLUBB represents vertical velocity, liquid water potential temperature, and total water mixing ratio using a double Gaussian PDF. However, a double Gaussian is unbounded and hence is not well suited to representing hydrometeors. In this context, hydrometeors include the mixing ratios and number concentrations of cloud ice, snow, graupel, and rain. These quantities are nonnegative but often have small values.

In order to represent hydrometeors realistically and fully, a subgrid PDF should have two features. First, it should be multivariate, so that it can represent the collection of one hydrometeor species by another. Second, the subgrid PDF should include a hydrometeor-free region, so that hydrometeors do not suffer excessive evaporation or sublimation in otherwise clear regions.

The marginal subgrid PDF shape that we choose for the hydrometeor variates is
a sum of a delta function and a multivariate lognormal. The delta function
represents the portion of a grid box that is devoid of all hydrometeors
(except liquid cloud droplets). The multivariate lognormal represents the
hydrometeor-filled portion, i.e., the precipitation fraction. Details of the
formulation of the lognormal are provided by the source code and in

At altitudes above the freezing level, the precipitation fraction is set
equal to the fraction of a grid box that is supersaturated with respect to
ice. Below the freezing level, the precipitation fraction is determined by a
calculation similar to that of

The assumed PDF method requires that variances of each variate be provided.
The within-hydrometeor standard deviation (

The prescribed ratio (

For multivariate PDFs, correlations among hydrometeor species must also be
provided. In these simulations, the correlations are assumed to be constant
within cloud throughout the course of the simulations. In order to assign
values for these correlations, we examined the range of values found within
cloud-resolving model simulations (which vary in space and time), and
prescribed correlations that were representative. The correlations used for
cloudy levels are shown in Table

Hydrometeor correlations used for all five cases. The variables
included in the correlation matrix are, from left to right or top to bottom:
extended liquid water mixing ratio, orthogonal extended liquid water mixing
ratio, vertical velocity, extended cloud drop number concentration, and the
mixing ratios and number concentrations of rain, cloud ice, snow, and
graupel. Shown are the within-cloud values. For below cloud another
correlation table is used, and for our purposes it is identical except for the
correlation between

Turbulence kinetic energy and turbulent fluxes of heat and moisture are
influenced by the buoyancy of air parcels in local updrafts and downdrafts.
The buoyancy variable in CLUBB is the virtual potential temperature

In prior versions of CLUBB, the only source of latent heating

For the present simulations, we have incorporated a crude method to account
for the latent heating and water loading associated with rain and ice
hydrometeors in the four aforementioned buoyancy generation terms. Namely, we
assume that for the purpose of calculating the four buoyancy generation
terms, rain and all ice hydrometeors are assumed to be perfectly collocated
in space with cloud liquid water mixing ratio. With this simple assumption,
CLUBB's calculation of the four buoyancy generation terms remains unchanged,
except that total condensate – including cloud liquid water, rain and ice
mixing ratios – is input into the calculation in place of cloud liquid
water. Conservation of heat and moisture for

Turbulent updrafts and downdrafts transport hydrometeors in the vertical, thereby acting to broaden the vertical extent of the hydrometeor profiles. In CLUBB, the tendency of turbulence to broaden the hydrometeor profiles is modeled by eddy diffusion. Although it is not obvious a priori that an eddy diffusion model is appropriate for cumulus layers, our large-eddy simulations indicate that eddy diffusion parameterizes the sign of hydrometeor transport satisfactorily in cumulus layers, but that the magnitude of the eddy diffusion is 1 or 2 orders of magnitude larger in cumulus than in stratocumulus layers (not shown).

CLUBB models the transport of a generic hydrometeor,

Microphysical processes influence not only grid box means, but also the

These effects are estimated using SILHS subcolumns. Each subcolumn is fed
into the microphysics separately, and each produces a separate microphysical
update to

For a generic variable

In our simulations, the following prognostic moments are updated by
microphysical source terms:

Options for SAM simulations of the five cases.

The following analysis compares two models, a single-column model
(CLUBB-SILHS), and a 3-D cloud-resolving/large-eddy model (SAM;

In order to assess the feasibility of unified parameterization, CLUBB-SILHS's
configuration of all shallow and deep cloud layers is identical, other than
case-specific options for droplet number concentration and radiative transfer
that are described below. For all cloud cases, CLUBB-SILHS uses a 1 min
computational time step and a 128-level stretched vertical grid, with a
vertical grid spacing of approximately 100 m at an altitude of 1000 m.
CLUBB-SILHS's microphysics is identical to the Morrison microphysics option
in SAM, which is a two-moment scheme that predicts cloud water, rain, cloud
ice, snow, and graupel

We compare each CLUBB-SILHS single-column simulation with a three-dimensional
simulation performed by SAM. In order to help isolate model errors in
CLUBB-SILHS, we configure SAM and CLUBB-SILHS identically in a number of
aspects. Namely, SAM uses the same two-moment microphysics scheme, the same
value of prescribed within-cloud cloud droplet number concentration, and the
same treatment of radiative transfer. SAM is set up similarly in shallow and
deep cases, except that SAM uses smaller grid spacing and time step for the
shallow cases. More SAM options for all cases are described below in
Table

We choose to simulate three deep convective cases that have been studied in
prior model intercomparisons, in part because doing so allows us to compare
our 3-D simulations with those intercomparison results, building confidence
in our 3-D simulations. We configure the deep convective simulations as per
previous model intercomparisons of those cases. SAM simulations examined here
are comparable to previous simulations and observations in general
characteristics such as timing, precipitation, and liquid water path. We
conclude, therefore, that they provide good reference simulations for
evaluation of CLUBB-SILHS. For deep convective cases, both SAM and
CLUBB-SILHS prescribe droplet number concentration to be 100 cm

Time series of liquid water path (upper left), rain water path (upper right), ice water path (lower left), and snow water path (lower right) from the TWP-ICE deep convective simulation. Liquid water path is too high in the CLUBB-SILHS simulation, suggesting a low precipitation efficiency; however, the other hydrometeors follow SAM fairly closely.

The first deep convective case we present is from the Tropical Warm Pool
International Cloud Experiment (TWP-ICE), which took place near Darwin,
Australia, in early 2006

The second case was taken from the Tropical Rainfall Measuring Mission
Large-Scale Biosphere-Atmosphere (TRMM-LBA) experiment which took place in
Brazil in early 1999

Third, we examine a midlatitude summertime deep convective case that
occurred during the 1997 intensive observation period of the Atmospheric
Radiation Measurement (ARM) program (ARM97). ARM97 is a 4-day simulation
starting on 26 June that exhibits two weak precipitating convective events
followed by a strong convective event. We configure the case as in a previous
intercomparison

The shallow cases are configured as in previous intercomparison studies,
including the appropriate options for radiation and cloud drop number
concentration. The vertical grid spacing and time step for CLUBB-SILHS are
identical to that used for the deep convective cases. SAM's configuration is
summarized in Table

Profiles of liquid water potential temperature (

The first shallow case we
present is a 3-day simulation of drizzling trade-wind cumulus clouds observed
during the Rain in shallow Cumulus over the Ocean (RICO) field campaign

Profiles of rain water mixing ratio (upper left), cloud ice mixing
ratio (upper right), snow mixing ratio (lower left), and graupel mixing ratio
(lower right) from the TWP-ICE deep convective simulation, averaged over the
same time period as Fig.

The second shallow case was taken from the Dynamics and Chemistry of Marine
Stratocumulus (DYCOMS-II) field campaign

This section presents time series and profiles from our tropical deep
convective case (TWP-ICE), our shallow-to-deep transitional case (LBA), and
our midlatitude deep convective case (ARM97). For each of the three cases,
we plot time series of the following vertically averaged quantities: liquid
water path (LWP), rain water path (RWP), cloud ice water path (IWP), and snow
water path (SWP). In addition, we plot profiles from selected time periods of
liquid water potential temperature (

Time series of the four vertically integrated quantities from the TWP-ICE
simulation are shown in Fig.

Time series of liquid water path (upper left), rain water path (upper right), ice water path (lower left), and snow water path (lower right) from the LBA deep convective simulation. Rain forms too late in the CLUBB-SILHS simulation, as there is an excess of ice that is not growing fast enough to form precipitation sized particles; however, the timing and magnitude of liquid water path is simulated adequately.

Figure

Profiles of liquid water potential temperature (

LBA is a difficult case to simulate because it evolves substantially over a
short (6 h) time period as it transitions from shallow to deep
convection. Hence it is a challenge to simulate the timing of ice and
precipitation formation. Time series from LBA are shown in
Fig.

Profiles of rain water mixing ratio (upper left), cloud ice mixing
ratio (upper right), snow mixing ratio (lower left), and graupel mixing ratio
(lower right) from the LBA deep convective simulation, averaged over the same
time period as Fig.

Figure

Time series of liquid water path (upper left), rain water path (upper right), ice water path (lower left), and snow water path (lower right) from the ARM97 deep convective simulation. CLUBB-SILHS's simulation of the first event produces cloud liquid water but no precipitation. However, CLUBB-SILHS's simulation of the other two events match SAM well in timing and magnitude.

Time series plots for ARM97 are shown in Fig.

Mean profiles for ARM97 are shown in Fig.

Profiles of liquid water potential temperature (upper left), total
water mixing ratio (vapor

Deep convection can have important effects on large-scale dynamics through
various feedbacks with microphysics

Two of the changes to CLUBB described in
Sect.

First, improving the PDF for hydrometeors by including a nonunity
precipitation fraction (Sect.

Profiles of rain water mixing ratio (upper left), cloud ice mixing
ratio (upper right), snow mixing ratio (lower left), and graupel mixing ratio
(lower right) from the ARM97 deep convective simulation, average over the
same time as Fig.

CLUBB-SILHS simulations of TWP-ICE (above), LBA (middle), and ARM97 (below) demonstrating the ability of CLUBB-SILHS to simulate dynamics for the deep convective cases. For each case, the variance of vertical velocity is shown on the left, the turbulent flux of liquid water potential temperature is in the middle, and the turbulent flux of total water mixing ratio is on the right (averaged over the same time periods as in previous figures). CLUBB-SILHS matches the CRM well in variance of vertical velocity, but underestimates the magnitude of some of the turbulent fluxes, particularly in the more convectively active cases.

Second, it turns out to be important for the simulations of deep convective
cases to have strong turbulent transport of hydrometeors. The changes
introduced to the eddy diffusivity calculation in
Sect.

The other two methodological changes – adding latent heating to subgrid
moments (Sect.

Aside from the errors explicitly mentioned above, the CLUBB-SILHS results presented here agree well with SAM. However, the configuration of CLUBB-SILHS used – with its fine vertical resolution, short time step, and numerous sample points – is computationally expensive. In this section, we present results that use configurations that are more computationally affordable.

Simulations presented in prior sections use a 1 min time step.
Figure

LBA simulations showing the sensitivity to the inclusion of the new precipitation fraction. The black line is the SAM CRM simulation, the red dashed line is the CLUBB-SILHS (default) simulation presented in Sect. 3.2, which includes the precipitation fraction, and the blue dotted line is CLUBB-SILHS with the same configuration as in the red line, but with the option for the precipitation fraction set to false. Above are profiles of cloud liquid water mixing ratio (left) and cloud ice mixing ratio (right), averaged over the last hour of the simulation (minutes 301–360). Below are time series of liquid water path (left) and rain water path (right). Nonzero precipitation fraction is important for LBA, because it increases precipitation efficiency, allowing more rain to leave the atmosphere, thereby removing excess cloud liquid water aloft and reducing excessive ice formation and growth.

TWP-ICE (above), LBA (middle), and ARM97 (below) simulations showing the sensitivity to the boosted eddy diffusivity for convective cases. For each case, graupel mixing ratio is shown on the left and snow mixing ratio on the right. The black line is the SAM CRM simulation, the red dashed line is the CLUBB-SILHS simulation configured exactly as that presented earlier in Sect. 3, and the blue dotted line is CLUBB-SILHS with the same configuration as in the red line, but without the new diffusivity calculation. The new formulation for eddy diffusivity smooths the hydrometeor profiles and transports hydrometeors farther aloft.

It is possible to use CLUBB-SILHS with any (even) number of subcolumns. Using
more subcolumns leads to better sampling and hence more accurate estimates of
averages of SGS variability

Computational cost can also be reduced by coarsening the vertical resolution.
To test the effects of this, Fig.

That the results of the CLUBB-SILHS simulations degrade some when
(vertical/temporal/sampling) resolution is decreased is not surprising. Such
sensitivities are typical in all scales of models. For example, CAM has been
shown to be sensitive to both horizontal and vertical resolution and the
physics time step used

Using the same configuration as used for the deep convective cases, we also simulate two shallow cloud cases. We find that these two cases are not degraded by the modifications introduced into CLUBB-SILHS in order to improve deep convective simulations.

CLUBB-SILHS simulations of TWP-ICE (above), LBA (middle), and ARM97
(below) showing sensitivity of simulations to the time step. For each case,
liquid cloud fraction is shown on the left (averaged over the same time
periods as in previous figures) and a time series of snow water path is shown
on the right. The black line is the SAM CRM simulation, the red dashed line
is the (default) CLUBB-SILHS simulation presented earlier in
Sect.

CLUBB-SILHS simulations of TWP-ICE (above), LBA (middle), and ARM97
(below) showing sensitivity of simulations to the number of subcolumns
utilized in SILHS. For each case, liquid cloud fraction is shown on the left
(averaged over the same time periods as in previous figures) and a time
series of snow water path is shown on the right. The black line is the SAM
CRM simulation, the red dashed line is the CLUBB-SILHS simulation presented
earlier in Sect.

Results from the RICO shallow cumulus case are shown in
Fig.

Results from the DYCOMS-II RF02 drizzling stratocumulus simulation are shown
in Fig.

Although further study is necessary, the fact that reasonable results can be obtained for shallow convection with the same model configuration as for deep convection hints that a unified PDF parameterization may indeed be possible, with one equation set representing all turbulence and cloud types.

CLUBB-SILHS simulations of TWP-ICE (above), LBA (middle), and ARM97
(below) showing sensitivity of simulations to the number of vertical grid
levels. For each case, liquid cloud fraction is shown on the left (averaged
over the same time periods as in previous figures) and a time series of snow
water path is shown on the right. The black line is the SAM CRM simulation,
the red dashed line is the (default) CLUBB-SILHS simulation presented in
Sect.

Results from the RICO simulation of shallow cumulus clouds. Above are time series of liquid water path (upper left) and rain water path (upper right). Below are mean profiles of liquid water potential temperature (middle left), total water mixing ratio (middle right), liquid cloud fraction (lower left), and cloud liquid water mixing ratio (lower right). Profiles are averaged over the last 2 h of the simulation (minutes 4301–4320). The CLUBB-SILHS simulation looks comparable to SAM, though there is too little cloud water produced.

This paper presents single-column simulations of deep and shallow cloud layers. To simulate these clouds, our model contains one component (CLUBB) that estimates the subgrid PDF of clouds, turbulence, and hydrometeors, and a second component (SILHS) that draws samples from this PDF and feeds them into a microphysics scheme. This methodology provides a detailed representation of subgrid dynamics and hydrometeors, and also provides a mechanism to couple the two at the subgrid scale. The detailed coupling is particularly important for deep convection, which has strong interactions between dynamics and microphysics.

Results from the DYCOMS-II RF02 simulation of stratocumulus clouds. Above are time series of liquid water path (upper left) and rain water path (upper right). Below are mean profiles of liquid water potential temperature (middle left), total water mixing ratio (middle right), liquid cloud fraction (lower left), and cloud liquid water mixing ratio (lower right). Profiles are averaged over the last hour of the simulation (minutes 301–360). CLUBB-SILHS does a reasonable job of simulating these shallow stratocumulus clouds, though the amount of liquid is too low, which is related to the coarse vertical grid spacing.

Most of the simulated fields are satisfactory, although there exist a few
deficiencies. One is that rain forms 100 min too late in the LBA case of
transition from shallow to deep convection (see
Fig.

Nevertheless, CLUBB-SILHS produces quite acceptable overall results for both
deep and shallow simulations. In particular, most profiles of clouds and
hydrometeors match those of SAM acceptably, given the stochastic nature of
convective precipitation. What ingredients in CLUBB-SILHS allow it to
simulate deep convection? One is CLUBB-SILHS's detailed representation of and
coupling between turbulence and microphysics and, in particular, ice
microphysics. CLUBB-SILHS uses a delta–lognormal subgrid PDF of hydrometeors,
which allows for the possibility of a hydrometeor-free region of a grid box and
also allows the hydrometeors to fall preferentially through liquid cloud
water. This, in turn, allows more precipitation to reach the ground, reducing
cloud water aloft (see Fig.

These simulations suggest that – with improvements at coarse time steps and vertical grid spacings – it is feasible to develop a unified parameterization of deep convection, shallow convection, stratiform clouds, and turbulence. Such a parameterization would parameterize all cloud types in a large-scale model with a single equation set. A unified parameterization would avoid the artificial categorization of clouds that is inherent in parameterization suites that use separate schemes for separate regimes. Use of a unified parameterization would also make it easier to ensure consistency of assumptions throughout a model. Greater consistency, in turn, would instill more confidence in a model's formulation.

CLUBB includes predictive equations for the horizontal winds (

Coauthors from the University of Wisconsin–Milwaukee acknowledge support by the Office of Science, US Department of Energy, under grants DE-SC0008668 (BER) and DE-SC0008323 (Scientific Discoveries through Advanced Computing, SciDAC). P. Rasch was supported by SciDAC, and M. Wang was supported by SciDAC and the DOE Atmospheric System Research (ASR) Program. The Pacific Northwest National Laboratory is operated for DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. The authors would like to thank Mikhail Ovchinnikov and Steven Ghan for helpful discussions. In addition, the authors would like to thank the two anonymous reviewers who provided helpful comments which improved the original manuscript. Edited by: R. Neale