Weibull-k Revisited: “Tall” Profiles and Height Variation of Wind Statistics

The Weibull distribution is commonly used to describe climatological wind-speed distributions in the atmospheric boundary layer. While vertical profiles of mean wind speed in the atmospheric boundary layer have received significant attention, the variation of the shape of the wind distribution with height is less understood.

Previously we derived a probabilistic model based on similarity theory for calculating the effects of stability and planetary boundary-layer depth upon long-term mean wind profiles. However, some applications (e.g. wind energy estimation) require the Weibull shape parameter (k), as well as mean wind speed.

Towards the aim of improving predictions of the Weibull-$$k$$ k profile, we develop expressions for the profile of long-term variance of wind speed, including a method extending our probabilistic wind-profile theory; together these two profiles lead to a profile of Weibull-shape parameter. Further, an alternate model for the vertical profile of Weibull shape parameter is made, improving upon a basis set forth by Wieringa (Boundary-Layer Meteorol, 1989, Vol. 47, 85–110), and connecting with a newly-corrected corollary of the perturbed geostrophic-drag theory of Troen and Petersen (European Wind Atlas, 1989, Risø National Laboratory, Roskilde).

Comparing the models for Weibull-k profiles, a new interpretation and explanation is given for the vertical variation of the shape of wind-speed distributions. Results of the modelling are shown for a number of sites, with a discussion of the models’ efficacy and applicability. The latter includes a comparative evaluation of Wieringa-type empirical models and perturbed-geostrophic forms with regard to surface-layer behaviour, as well as for heights where climatological wind-speed variability is not dominated by surface effects.