Vertical Velocity P.d.f. In Different AtmosphericTurbulence Conditions Assessed By A NewFormulation Of The Gram-Charlier Method

Abstract

In the 1966 Kampe de Fernet suggested to study the dynamical of turbulence through Gram-Charlier expansion. The Gram-Charlier expansion allows to approximate a probability densities function throught a series expansions involving the normal density and its derivatives. In this work a modified formulation of Gram-Charlier expansion is applied to estimate the probability density function of vertical wind velocity. The theoretical progress consists of a new choice of the parameters of the Gaussian density (standard deviation and the mean values) that appears in the expansion. In fact in the classic Gram-Charlier expansion the random variable is standardized and the Gaussian parameters are equal to zero and one. This assumption is sometimes too restrictive and abetter performance can be obtained changing the parameters values. The probability density function of vertical wind velocity is calculated from an experimental parametrization of moments, using as input data the first three moments. These moments are function of the dimensionless ratio (Z/Zi) and atmospheric stability, represented by the ratio U*/W* (where U* is the friction velocity and W* is the convective velocity scale). The new methodology is used to determine the values of the Gaussian parameters in Gram-Charlier expansion, that better reproduce the probability density function, for different Z/Zi levels (from 0 to 1 with a step of 0.01) in three different atmospheric turbulence conditions (U*/W*=0.2, 0.4 and 0.6). Further the analytical trends of both mean and standard deviation versus Z/Zi are calculated.