Subgrid-scale (SGS) variability of cloud microphysical variables over the mesoscale numerical weather prediction (NWP) model has been evaluated by means of joint probability distribution functions (JPDFs). The latter were obtained using dynamically balanced Large Eddy Simulation (LES) model dataset from a case of marine trade cumulus initialized with soundings from Rain in Cumulus Over the Ocean (RICO) field project. Bias in autoconversion and accretion rates from different formulations of the JPDFs was analyzed. Approximating the 2-D PDF using a “generic” (fixed-in-time), but variable-in-height JPDFs give an acceptable level of accuracy, whereas neglecting the SGS variability altogether results in a substantial underestimate of the grid-mean total conversion rate and producing negative bias in rain water. Nevertheless the total effect on rain formation may be uncertain in the long run due to the fact that the negative bias in rain water may be counterbalanced by the positive bias in cloud water. Consequently, the overall effect of SGS neglect needs to be investigated in direct simulations with a NWP model.

Formulation of microphysical processes in meso and large scale models requires accounting for subgrid-scale (SGS) variability; its neglect can lead to substantial bias in calculations of microphysical process rates (Pincus and Klein, 2000; Larson et al., 2001, 2012; Kogan and Mechem, 2014, KM14 hereafter, Nelson et al., 2016). SGS microphysical variability can be represented using probability distribution functions (PDFs). The greatest challenge in this approach is linking the PDF parameters to the grid-mean prognostic variables (so called “closure” relationships). KM14 present a detailed review of the various recent PDF approaches, the most sophisticated of which employs joint analytic PDFs of liquid water potential temperature, total water, and vertical velocity (Cloud Layers Unified by Binomials [CLUBB], Golaz et al., 2002; Larson and Golaz, 2005). The CLUBB approach uses theoretical considerations and a number of a-priori assumptions about the shape of the distributions, obtained from LES output and aircraft observations.

KM14 use a series of Large Eddy Simulations (LES) of

The model employed in our simulations is a version of the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS) LES (Kogan et al., 1995; Khairoutdinov and Kogan, 1999) called SAMBM (System for Atmospheric Modeling – Bulk Microphysics) (Khairoutdinov and Kogan, 2000; Kogan, 2013). This new parameterization has been tested against simulations using the CIMMS LES with explicit microphysics (SAMEX, Kogan et al., 2012) in case studies of northeast Atlantic marine stratocumulus (the Atlantic Stratocumulus Experiment [ASTEX], Albrecht et al., 1995) and marine trade cumulus based on the Rain in Cumulus over the Ocean (RICO) field campaign (vanZanten et al., 2011).

The simulations were performed in the integration domain

The KM14 methodology includes two distinct elements for deriving the PDF parameterization. The first is the use of a layered approach in formulating the PDF under which we subdivide the 4 km model vertical domain into ten, 400-m-thick layers. The resulting increase of the data volume in each layer leads to the increase of statistical confidence of the PDF calculations. The second element is the formulation of the PDFs in non-dimensional space by normalizing the microphysical variables by their layer-mean values.

We consider two joint probability distribution functions (JPDFs) which enter
the expressions for microphysical autoconversion and accretion rates. The
first JPDF is used for calculation of autoconversion; it depends on
cloud-drop number concentration,

As was shown in KM14, the JPDFs vary in the vertical and depend on parameters characterizing cloud system environment; the latter change as cloud system evolves with time during the simulation. Figure 1 shows that domain averaged TKE and LWP increased by 50 % over the course of 12 h, while rain water path (RWP) increased four-fold. Both LWP and RWP vary significantly during the course of simulation. We archive hourly datasets and derived time dependent JPDFs from these datasets. The JPDF variation in time, therefore, corresponds to changing environmental conditions.

The evolution of domain averaged liquid water path (LWP), rain water path (RWP), and turbulent kinetic energy (TKE) during the second half of the simulation. All parameters are normalized by their corresponding value at 12 h.

The fine resolution of the LES model allows us to calculate the “exact”
conversion rates which serve as a benchmark for evaluating different
approximations. Based on the values of

In addition to the time dependent JPDFs, we consider the case when the time dependence (and hence varying environmental conditions) is neglected. These JPDFs are calculated using the entire data set from the whole simulation; they may be referred to as a “generic” shallow cumulus JPDFs. For comparison we also consider an approximation when the JPDFs are neglected altogether, and the conversion rates are calculated using only the layer-mean variables. This case serves as the measure of improvement when SGS variability is included.

The errors of different JPDF approximations are defined as:

The cumulative

The error dependence on the rate value.

Remarkably, the errors coming from JPDF which are derived from the entire 12 h dataset and, therefore, neglect time-dependence of JPDF, are not significantly larger than the errors produced by using the time-dependent JPDFs (compare black and red curves in Fig. 2). These simplified fixed in time “generic” JPDFs can be implemented in the NWP models as a proxy for trade wind shallow cumulus clouds.

Neglecting SGS altogether introduces significant negative bias for
autoconversion rates with errors in the

Important information about the effect of the errors on rain production can
be gained when considering their dependence on the value of the rate itself
(Fig. 3). For autoconversion, the errors increase with the increase of the
value of the rate, e.g., the small autoconversion rates have the smallest
errors (in the

The total conversion rate error as a function of the rate. Panel

The negative bias is eliminated when JPDFs are employed. The bottom panel in
Fig. 4 shows the bias for the total conversion rates using “generic” JPDF.
For large values of the rate, the overall bias is decreased and varies in
the range of

While results in Fig. 4a may suggest that the bias for the total conversion
rate is mostly negative for the larger values of the rate, nevertheless it
does not warrant a conclusion that neglecting the SGS variability will
result in underestimation of rain formation. This is because one needs to
consider the fact that when total (underestimated) conversion rate is being
added to the rain water mixing ratio

Previously in KM14 we evaluated the absolute total errors on various PDF
formulations. In this study we evaluate the errors in more detail and analyze
how they may affect precipitation formation at different stages of cloud and
rain formation. It might be expected that neglecting SGS variability may lead
to slow rain initiation because of the significant underestimation of
autoconversion rates, however, this effect may be somewhat compensated by the
overestimation of accretion rates at the beginning. At the later stages of
rain development both autoconversion and accretion rate are underestimated,
and, thus, lead to smaller values of the rain water mixing ratio. However,
the underestimated values of the conversion rates which are subtracted from
the cloud water mixing ratio, will result in its overestimation, therefore,
counterbalancing the negative bias in

Sounding data for simulations of RICO trade wind convective
clouds is available at

The authors declare that they have no conflict of interest.

This investigation was supported by the ONR grant N00014-16-1-2487. The computing for this project was performed at the University of Oklahoma Supercomputing Center for Education and Research (OSCER). The constructive comments of the anonymous reviewer are appreciated. Edited by: D. Reinert Reviewed by: two anonymous referees