Modeling the Distribution of Sub-Grid Moisture Variability for Cloud Parameterizations in Large Scale Models

Peter M. Norris1,2 (peter.norris@nasa.gov), Lazaros Oreopoulos3, and Arlindo M. da Silva2

1. UMBC Goddard Earth Sciences & Technology Center 2. NASA/GSFC Global Modeling and Assimilation Office 3. NASA/GSFC Climate & Radiation Branch

Large Scale Models (LSM) with statistical sub-grid cloudparameterizations require realistic parametric forms forthe distribution of sub-grid moisture. A vertical column ofsuch distributions, together with vertical overlap assump-tions, can be used to generate an ensemble of sub-columncloud fields for each LSM grid-column. These can then beoperated on by Independent Column Approximation (ICA)radiative transfer to yield more accurate radiative aver-ages for each grid-column.

Statistical Cloud Schemes

Key Questions• Which simple probability density function (PDF) is mostrealistic for describing the subgrid-scale variability oftotal moisture on scales smaller than an LSM grid-box?• What effect do large-scale moisture trends have on theform of these subgrid-scale PDFs?

MethodologyWe examine various candidate layer probability densityfunctions (PDFs) for modeling total moisture content froma 20 day Cloud Resolving Model (CRM) simulation over theARM SGP site (Zeng et al., JAS v64, 2007). We includeboth symmetric (Gaussian) and skewed (Generalized Ex-treme Value) distributions (see Norris et al., QJRMS v134,2008 ― the GEV distribution, with its exponential-typetails, tends to be more realistic than say the Beta distrib-ution, with its polynomial-type tails). A maximum likelihood(ML) method chooses which PDF and set of parametersmost likely produces the observations. The results shownare for 1-km resolution and 32 x 32 subsets of the fulldomain, mimicking a ¼º LSM.

We also consider the effect of grid-scale trends on thesePDFs. Such trends can sometimes dominate subgrid small-scale variability, particularly outside the boundary layer.We consider a 2D linear trend, whose PDF must be convol-ved with the subgrid-scale PDF on the assumption that thesubgrid- and grid-scale variability are not correlated.

Symmetrical vs. Skewed PDFs Trend vs. No TrendExamples of 3 horizontallayers of excess waterpath (XWP = total minussaturation) and the cor-responding distributiondensities (PDFs) in grey.The red lines are maxim-um likelihood (ML) fitsof the normal distribut-ion and the standard(GEV) & reverse (rGEV)Generalized ExtremeValue distributions (3params). Marked skew-ness (of both signs) isevident.

Three more horizontallayers of water path andtheir empirical PDFs.Notice the grid-scaletrends in the layers. Thered lines are maximumlikelihood fits for thePDF of a 2D linear trendconvolved with the PDFsof the reverse & regularGEV distributions. Thetrends tend to flattenthe PDF top (a puretrend yields a truncatedtriangular distribution).

Adding Skewness Improves the FitEvery 3 hours during the20 day simulation, an MLfit was performed for 3pressure bands (edges@ 400, 700 hPa). Thebetter the fit, the lowerthe maximum absoluteCDF difference betweenthe empirical and fittedPDFs. Allowing maximumlikelihood selection be-tween normal, GEV andrGEV distributionstends to produce abetter fit (sometimesmuch bet-ter) than usingthe nor-mal distributionalone.

Bias in the mean XWP isscaled by the standarderror in the mean. Theabsolute value of thatbias is not significantlyimpacted by includingGEVs in the fit. Conver-sely, absolute bias in thestandard deviation (scal-ed by its standarderror) becomes larger.This is a potential prob-lem with the GEV dist-ributions ― the extend-ed tails may producesome very large valueswhich will need to belimited in an LSM imp-lementation.

Distribution of Arnold &Groeneveld (AG) skew-ness measure for theML fits. Mid- and upper-level regions have abroader distribution ofskewness. Mid-levels areespecially skewed in thepositive direction.

Adding Trends is Near Neutral

Conclusions(1) The GEV should be further tested as a suitable PDF forstatistical cloud parameterizations. It is able to capturequite well the frequently observed positively and negativelyskewed PDFs of moisture, although it may lead to positivebiases in variance unless tail-limited. (2) So far, addinglinear trends seem not to be particularly important.

First we add a 2D lineartrend, but fixed by thelarge scale trends be-tween LSM gridboxes.As expected, it has verylittle effect on theoverall goodness of fit,since large-scale trendstend to smear out feat-ures at the gridboxscale. Next we let trendslopes enter the ML fit(as in the 3 examples atthe top). This seems toworsen the fit on someoccasions.

More precisely, althoughclose to half the casesimprove and half deter-iorate, the deteriorat-ions can be 2x the mag-nitude of the improve-ments (i.e., the change inmax(|ΔCDF|) is in therange -0.05 to 0.1. TheML fitting proceduresfor the combined trend+ small-scale need morework, and maybe quad-ratic trends should betried. But so far, theinclusion of trends is notparticularly beneficial.