The subgrid-scale representation of hydrometeor fields is important for calculating microphysical process rates. In order to represent subgrid-scale variability, the Cloud Layers Unified By Binormals (CLUBB) parameterization uses a multivariate probability density function (PDF). In addition to vertical velocity, temperature, and moisture fields, the PDF includes hydrometeor fields. Previously, hydrometeor fields were assumed to follow a multivariate single lognormal distribution. Now, in order to better represent the distribution of hydrometeors, two new multivariate PDFs are formulated and introduced.

The new PDFs represent hydrometeors using either a delta-lognormal or a delta-double-lognormal shape. The two new PDF distributions, plus the previous single lognormal shape, are compared to histograms of data taken from large-eddy simulations (LESs) of a precipitating cumulus case, a drizzling stratocumulus case, and a deep convective case. Finally, the warm microphysical process rates produced by the different hydrometeor PDFs are compared to the same process rates produced by the LES.

The atmospheric portion of the hydrological cycle depends on the formation and
dissipation of precipitation. In a numerical model, precipitation processes are
represented by the microphysics process rates. These process rates are highly
dependent on the values of hydrometeor fields at any place and time.
Hydrometeors (such as rain water mixing ratio) can vary significantly on spatial
scales smaller than the size of a numerical model grid box

Subgrid-scale variability (but not spatial organization) can be accounted for
through use of a probability density function (PDF). PDFs have been used in
atmospheric modeling to account for subgrid variability in moisture and
temperature

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

Cloud Layers Unified By Binormals (CLUBB) is a single-column model that uses a
multivariate PDF to account for the subgrid-scale variability of model
fields

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

Consider an example in which rain covers 10 % of the grid level. Then the
in-precipitation mean of

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

Further improvements in accuracy can be achieved with relatively minor
modifications to the PDF. As previously mentioned, CLUBB's PDF contains two
components. Each of these components can be easily subdivided into an
in-precipitation sub-component and an outside-precipitation sub-component. The
result is a delta-lognormal representation of the hydrometeor field

A schematic of the single lognormal (SL), delta-lognormal (DL), and
delta-double-lognormal (DDL) hydrometeor PDF shapes. The SL PDF shape
is precipitating over the entire subgrid domain, whereas the DL and DDL
shapes are not. In all three plots of the PDFs (where each PDF is a
function of a hydrometeor species, such as

The main purpose of this paper is to present the formulation of an updated
multivariate PDF that extends CLUBB's traditional PDF to include the DL and DDL
hydrometeor PDF shapes. Additionally, a new method is derived to divide the

The remainder of the paper is organized as follows.
Section

We now describe how the multivariate PDF used by CLUBB is modified to improve the
representation of hydrometeors. Perhaps the most important modification is the
introduction of precipitation fraction,

Before writing the form of the multi-component PDF, we digress to discuss a
special case, the cloud droplet concentration (per unit mass),

The PDF includes all the hydrometeor species found in the chosen microphysics
scheme with the exception of

In order to calculate quantities that depend on saturation, such as

The general form of a PDF with

Each original PDF component is split into precipitating and precipitation-less
sub-components. The component means, variances, and correlations for variables

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

The multivariate PDF can be adjusted to account for a situation when a variable
has a constant value in a PDF (sub-)component. In that situation, the variable
can be reduced to a delta function at the (sub-)component mean value. A good
example of this would be setting

The general form of the

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

When variables are integrated out of the full multivariate PDF, the result is
a multivariate marginal PDF consisting of fewer variables. When all variables
but one are integrated out of the PDF, the result is a univariate marginal or
individual marginal PDF. For any hydrometeor species,

The in-precipitation mean of

The variables that are distributed marginally as binormals use similar notation.
For example,

This paper will use the phrase “PDF parameters” to refer to the PDF

The individual marginal distribution for

When no hydrometeor species are found at a grid level
(

A mean-and-variance-preserving method is used to calculate the
in-precipitation means of the hydrometeor field in the two PDF components,

The grid-level mean value of any function that is written in terms of variables
involved in the PDF can be found by integrating over the product of that
function and the PDF. For example,

When the hydrometeor is not found at a grid level,

When there is precipitation found in both PDF components, further information is
required to solve for the two component means and the two component standard
deviations. The variable

Both the variance of each PDF component and the spread between the means of each
PDF component contribute to the in-precipitation variance of the hydrometeor
(

In order to calculate the value of

In order to calculate the value of

The two remaining unknowns,

When

As the value of

When the value of

This method of closing the hydrometeor PDF parameter equation set produces a
DDL hydrometeor PDF shape when

In limited testing, the value of the tunable parameter

There is insufficient data from observations to calculate all the fields that
need to be input into CLUBB's PDF. However, this data can be supplied easily
and plentifully by a LES. In this paper, LES output of precipitating cases is
simulated by the System for Atmospheric Modeling (SAM)

In order to assess the generality of the different hydrometeor PDF shapes for
different cloud regimes, SAM was used to run three idealized test cases – a
precipitating shallow cumulus case, a drizzling stratocumulus case, and a deep
convective case. The use of cases from differing cloud regimes help avoid
overfitting the parameterizations of PDF shape. The setup for the
precipitating shallow cumulus test case was based on the Rain in Cumulus over the Ocean (RICO) LES intercomparison

The RICO simulation was run with SAM's implementation of the

The setup for the drizzling stratocumulus test case was taken from the LES
intercomparison based on research flight 2 (RF02) of the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field experiment

The setup for the deep convective test case was taken from the LES
intercomparison based on the Large-Scale Biosphere–Atmosphere (LBA)
experiment

The LBA case requires a microphysics scheme that can account for ice-phase
hydrometeor species. The LBA simulation was run with

CLUBB's hydrometeor PDF shapes will be compared to histograms of hydrometeors
produced by SAM LES data. Our goal is to isolate errors in the PDF shape
itself. In order to eliminate sources of error outside of the PDF shape and
provide an “apples-to-apples” comparison of CLUBB's PDF shapes to SAM data,
we drive CLUBB's PDF using SAM LES fields, rather than perform interactive CLUBB
simulations. The following fields are taken from SAM's statistical profiles and
are used as inputs to CLUBB's PDF:

Additionally, covariances that involve at least one hydrometeor are added to the
above list and are used to calculate the PDF component correlations of the same
two variables. These covariances are

Owing to differences between the KK and Morrison microphysics schemes in SAM,

Although

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

PDFs of rain in the RICO precipitating shallow cumulus case at an
altitude of 380 m and a time of 4200 min. The SAM LES results are in
red, the DDL results are blue solid lines, the DL results are green
dashed lines, and the SL results are magenta dashed-dotted lines.

Figure

Each of the CLUBB hydrometeor PDF shapes has a lognormal distribution within
precipitation in each PDF component. Taking the natural logarithm of every
point of a lognormal distribution produces a normal distribution, and so the
plot of the PDF of

The plot of the PDF of

Figure

The plot of the PDF of RICO

Joint PDF of

Figure

PDFs of rain in the DYCOMS-II RF02 drizzling stratocumulus case at an
altitude of 400 m and a time of 330 min. The SAM LES results are in
red, the DDL results are blue solid lines, the DL results are green
dashed lines, and the SL results are magenta dashed-dotted lines.

Figure

PDFs of rain in the LBA deep convective case at an altitude of 2424 m
and a time of 330 min. The SAM LES results are in red, the DDL results
are blue solid lines, the DL results are green dashed lines, and the SL
results are magenta dashed-dotted lines.

In order to assess how well the PDF shapes are able to capture ice PDFs as well
as liquid PDFs, we turn to the LBA case. In LBA, liquid and ice appear at
different altitudes and times. Figure

PDFs of ice in the LBA deep convective case at an altitude of 10 500 m
and a time of 360 min. The SAM LES results are in red, the DDL results
are blue solid lines, the DL results are green dashed lines, and the SL
results are magenta dashed-dotted lines.

To indicate whether the three PDF shapes work for ice-phase hydrometeors, we
compare marginal PDFs involving

Why does the DDL PDF shape match LES output better than the DL shape in the
aforementioned figures? The PDFs (in Gaussian space) for the LES of RICO and
LBA show a broad, flat distribution of hydrometeor values from the LES. The DL
shape is too peaked in comparison to the LES data. The DDL PDF is able to
spread out the component means and thereby represent the platykurtic shape more
accurately. However, even the DDL PDF fails to capture the far left-hand tail
of the LES PDF. In the RICO, DYCOMS-II RF02, and LBA cases, between about 5
and 20 % of the LES PDF is found to the left of the DDL PDF (see
Figures

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

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

While a lot can be learned by looking at plots of the hydrometeor PDFs, they are
anecdotal and cannot tell us how well the idealized PDF shapes work generally.
To obtain an

Both the K–S and C–vM tests compare the cumulative distribution function (CDF)
of the idealized distribution to the CDF of the empirical data (in this case,
SAM LES data). Both tests require that the CDFs be continuous.
Therefore, the scores are calculated using only the in-precipitation portion
of the hydrometeor PDF in Eq. (

The K–S score is the greatest difference between the empirical in-precipitation
CDF,

Unlike the K–S test, which only considers the greatest difference between the
CDFs, the C–vM test is based on an integral that includes the differences
between the CDFs over the entire distribution. The integral
is

The K–S and C–vM test scores are produced at every LES vertical level and
three-dimensional statistical output time for every hydrometeor species. This
results in a large number of scores. We desire that each hydrometeor species
have a single K–S score and a single C–vM score in order to more easily compare
the DDL, DL, and SL hydrometeor shapes. We calculate this score by averaging
the individual level scores over multiple levels and multiple output times. For
K–S this is simple, and the result is

After inspecting profiles of SAM LES results for mean mixing ratios in height
and time, regions were identified in height and time where the mean mixing ratio
of a species was always at least

The LBA case contains both liquid and frozen-phase hydrometeor species that
evolve as the cloud system transitions from shallow to deep convection. The
various hydrometeor species develop and maximize at different altitudes and
times, so different periods and altitude ranges are chosen for averaging test
scores for each species. LBA test scores for

Kolmogorov–Smirnov statistic averaged over multiple grid levels and statistical output timesteps comparing each of DDL, DL, and SL hydrometeor PDF shapes to SAM LES results. The best (lowest) average score for each case and hydrometeor species is listed in bold. The DDL has the lowest average score most often, and the DL has the second-lowest average score most often.

The results of

Profiles of mean microphysics process rates in the RICO precipitating
shallow cumulus case time-averaged over the last 2 hours of the
simulation (minutes 4200 through 4320). The SAM LES results are red
solid lines, the DDL results are blue solid lines, the DL results are
green dashed lines, and the SL results are magenta dashed-dotted lines.

Profiles of mean microphysics process rates in the DYCOMS-II RF02
drizzling stratocumulus case time-averaged over the last hour of the
simulation (minutes 300 through 360). The SAM LES results are red
solid lines, the DDL results are blue solid lines, the DL results are
green dashed lines, and the SL results are magenta dashed-dotted lines.

Profiles of mean warm microphysics process rates in the LBA deep
convective case time-averaged over the last hour of the simulation
(minutes 300 through 360). The SAM LES results are red solid lines,
the DDL results are blue solid lines, the DL results are green dashed
lines, and the SL results are magenta dashed-dotted lines.

We note the important caveat that, as compared to DL, DDL has more adjustable parameters. A parameterization with more free parameters would be expected to provide a better fit to a training data set. Therefore, although DDL matches the LES output more closely than does DL, we cannot be certain, based on the analysis presented here, that DDL will outperform DL on a different validation data set. For a deeper analysis, one could use a model selection method that penalizes parameterizations with more parameters. We leave such an analysis for future work.

A primary reason to improve the accuracy of hydrometeor PDFs is to improve the accuracy of the calculation of microphysical process rates. In this section, we compare the accuracy of calculations of microphysical process rates based on the SL, DL, and DDL PDF shapes.

In the simulations of RICO and DYCOMS-II RF02, both SAM LES and CLUBB use KK
microphysics. The process rates output are the mean evaporation rate of

Figure

Figure

Figure

In the simulation of LBA, Morrison microphysics was used in both SAM LES and
CLUBB. In order to account for subgrid variability in the microphysics,
sample points from the PDF are produced at every grid level using the Subgrid
Importance Latin Hypercube Sampler (SILHS)

Normalized Cramer–von Mises statistic averaged over multiple grid levels and statistical output timesteps comparing each of DDL, DL, and SL hydrometeor PDF shapes to SAM LES results. The best (lowest) average score for each case and hydrometeor species is listed in bold. The DDL has the lowest average score every time, and the DL has the second-lowest average score every time.

Figure

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

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

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

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

The CLUBB code is freely available for non-commercial use after registering for
an account on the website

In Section

The PDF

The variable that needs to be solved for is

The equation for

When

This method of back-solving for the component correlations was used to calculate
the PDF component correlations of

The PDF

This equation is integrated and reduced, resulting in

When

The variable that needs to be solved for is

This method of back-solving for the component correlations was used to calculate
the PDF component correlation of

The relationship between

The evaluated integral for

It is important to be able to back-solve

The value of

The relationship between

In a situation where CLUBB is using SILHS with a microphysics scheme that
predicts

The authors gratefully acknowledge financial support from the National Science Foundation under grant no. AGS-0968640 and the Office of Science (BER), US Department of Energy under grant no. DE-SC0008323 (Scientific Discoveries through Advanced Computing, SciDAC). Some simulations presented here were performed on the Avi high-performance computer cluster at the University of Wisconsin – Milwaukee. We thank two anonymous reviewers for their comments.Edited by: O. Boucher