# Normal law of distribution of wind velocity vector in polar coordinates

### Abstract

It is impossible to organize wind energy systems without studying of wind speed regime at the surface layer of the atmosphere within a specific area and at climatic scales. Such studies are often accompanied by approximations of probabilities of wind speed performed in the form of normal law of a system of random values presented by a zonal *u *and a meridional *u* which are components of a wind speed vector. It is suggested that, for the purposes of wind energy, display of a wind speed vector in polar coordinates *(r, α)* where *r* is a module of wind speed and *α* is a polar angle appears to be more preferable. The article shows a transform from a normal law of distribution of probabilities with density *f**(u,**u**)* to a normal law distribution with density *f**(r,α)* completed by means of functional transformation with elliptic dispersion in place. Based on a normal law of distribution *f**(r,α)* and through integration with respect to corresponding variables individual distributions of probabilities *f**(r)* and *f**(α)* as well as conditional distributions of probabilities *f**(r/α)* and *f**(α/r)* were obtained in the areas of their existence. The article shows individual distributions in case of circular and elliptic dispersion of a wind speed vector. It shows that an individual distribution of a wind speed probability in case of circular dispersion and in the absence of correlated dependence turns into the Rayleigh's distribution and a conditional distribution of a polar angle degenerates in an even distribution. The cases of distributions with dispersions of a wind speed module having elliptic properties subject to availability of correlated connection between wind speed components were also studied. Calculation of probabilities of a polar angle being within different sections of the area 0≤α≤2π with set values of a wind speed module also took place. Numerical experiments proved the advantage of such modeling of distributions of wind speed vector.

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*Ukrainian Hydrometeorological Journal*, (20), 5-17. https://doi.org/10.31481/uhmj.20.2017.01

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