Abstract
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 proba- bilistic 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 profile, we develop expres- sions 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 para- meter 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, Riso 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 mod- els' 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.
Highlights
To better predict multi-year wind distributions at heights well beyond the atmospheric surface layer (‘ASL’ hereafter), i.e. above 50–100 m, based on measurements at lower heights, the extrapolation of measured statistics demands a model for the wind profile and model(s) for the profile(s) of the long-term wind statistics
Available lidar devices are able to measure wind statistics at these heights, but due to the current cost of lidar technology this is generally not feasible; this is true for observations lasting multiple years, i.e. over time scales needed to reliably characterize the local wind climate
Models for the scalar wind profile have existed for decades, most notably the similarity theory of Monin and Obukhov (1954)
Summary
To better predict multi-year wind distributions at heights well beyond the atmospheric surface layer (‘ASL’ hereafter), i.e. above 50–100 m, based on measurements at lower heights, the extrapolation of measured statistics demands a model for the wind profile and model(s) for the profile(s) of the long-term wind statistics. In the European Wind Atlas (Troen and Petersen 1989, denoted ‘EWA’ hereafter), perturbation theory was applied to the geostrophic drag law, along with M–O similarity in order to extrapolate observed wind statistics This gave a coupled formulation for the profiles of Weibull parameters via extrapolation of both first and second moments of wind speed, as influenced by geostrophicscale stability perturbations in a climatological mean sense. This includes correction of the EWA (Troen and Petersen 1989) estimate of the height of narrowest wind distribution (peak of k), and its incorporation in the adapted Wieringa-like profile for the Weibull shape parameter. The following two sub-sections present the profiles of long-term mean wind speed and longterm wind variance, which are used to give new models for observation-based k(z)
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