Comparison among empirical probability density functions of the vertical velocity in the surface layer based on higher order correlations

Vertical velocity fluctuations were measured in the atmospheric surface layer by means of an ultrasonic anemometer and higher order correlations were calculated on two time series, recorded in unstable and neutral conditions, and selected for the whole measurement period on the basis of the inversion test (stationarity test). Comparisons have been made between observed and predicted correlations by considering Gaussian joint-PDF and Gram-Charlier series expansions truncated to the fourth and sixth order as done earlier by Frenkiel and Klebanoff. A bi-Gaussian PDF, given by a mixture of two Gaussian PDFs, has also been considered. This last has been constructed assuming that either the first three or the first four moments are given, and the relationships between correlation functions of different order are derived. The departure from Gaussian behaviour in both stability conditions is derived. Though Gram-Charlier series expansions show a good correspondence to experimental reality, their use as non-Gaussian probability distributions cannot be suggested in theoretical approaches and should be considered with care in practical applications, due to possible occurrences of small negative probabilities. The results shown in this paper support the applicability of the bi-Gaussian PDF created using up to the fourth moment.