Binormal Model of Ensemble Partial Cloudiness

W. S. Lewellen Department of Physics and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania

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S. Yoh Department of Physics and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania

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Abstract

A new formulation of partial cloudiness parameterization has been introduced that agrees with that for a random model in one limit and approaches the simple updraft/downdraft model of larger-scale models in the limit of very highly skewed flow. Each of the conserved variables, liquid potential temperature and total humidity, along with the vertical velocity, are assumed to have probability distributions that may be parameterized as combinations of two multivariate normal distributions. This allows the skewness of the variables to be controlled by the bias between the means of the two normals and their relative fractions. It also provides a smooth transition between the normal distribution and the two limiting delta function distribution of the updraft/downdraft model. Comparisons with large-eddy-simulation data show this new model to be valid over a much wider range of conditions than the single normal distribution.

When a simple cloud-top entrainment instability (CTEI) analysis is made using the new binormal model, variations in the dynamic characteristics, here represented by the skewness in the extended liquid water function, s, are found to mask the variation with respect to the ratio in the thermodynamic jump conditions. This helps to explain the observed poor correlation of empirical cloud fraction with this jump condition. On the other hand, the analysis suggests that the ratio of the mean value of the extended liquid water variable, s, to the square root of its variance, may be expected to show a much better correlation with the empirical cloud fraction.

Abstract

A new formulation of partial cloudiness parameterization has been introduced that agrees with that for a random model in one limit and approaches the simple updraft/downdraft model of larger-scale models in the limit of very highly skewed flow. Each of the conserved variables, liquid potential temperature and total humidity, along with the vertical velocity, are assumed to have probability distributions that may be parameterized as combinations of two multivariate normal distributions. This allows the skewness of the variables to be controlled by the bias between the means of the two normals and their relative fractions. It also provides a smooth transition between the normal distribution and the two limiting delta function distribution of the updraft/downdraft model. Comparisons with large-eddy-simulation data show this new model to be valid over a much wider range of conditions than the single normal distribution.

When a simple cloud-top entrainment instability (CTEI) analysis is made using the new binormal model, variations in the dynamic characteristics, here represented by the skewness in the extended liquid water function, s, are found to mask the variation with respect to the ratio in the thermodynamic jump conditions. This helps to explain the observed poor correlation of empirical cloud fraction with this jump condition. On the other hand, the analysis suggests that the ratio of the mean value of the extended liquid water variable, s, to the square root of its variance, may be expected to show a much better correlation with the empirical cloud fraction.

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