Top-Hat Representation of Turbulence Statistics in Cloud-Topped Boundary Layers: A Large Eddy Simulation Study

Large eddy simulation is used to study top-hat parameterizations of second- and third-order scalar statistics in cumulus and stratocumulus cloud-topped boundary layers (CTBLs). Although the top-hat parameterizations based on commonly used conditional sampling methods are a useful approach to modeling the vertical fluxes in the simulated CTBLs, they fail to realistically represent the scalar variances. The reason is that the common sampling methods are based at least in part on the sign of vertical velocity, but not on the sign of the scalars whose variances are represented and that scalars and velocity are not perfectly correlated. Furthermore, the self-correlation nature for a variance means that all the fluctuations contribute to its value, while the top-hat models completely ignore the deviations from the top-hat means and thus considerably degrade the representation of the variance. For the fluxes, however, only the coherent convective elements make the most contribution. Analysis of analytic models and “toy” time series indicates in a more generic setting that the effect of poor correlations between the signal upon which the sampling is based and the signal whose variance is to be represented tends to degrade the ability of top-hat parameterizations to capture the variance. The analysis of toy time series also indicates that variability among individual events within a composite degrades the top-hat representation of the variance more than variability within events. For the vertical velocity–scalar-related third-order moments, the top-hat model gives reasonable estimates for the cumulus CTBL but not for the stratocumulus CTBL. These differences are explained by structural differences (tied to circulation differences in the two CTBLs) in their respective joint probability density functions of vertical velocity and various scalars.