Wind Speed At Height Log Power Law Calculator

Near the ground, wind speed rises with height because surface drag removes momentum and turbulence carries the deficit upward. In the surface layer, roughly the lowest 10 percent of the boundary layer, turbulent stress is nearly constant with height, and dimensional analysis then forces the mean profile to be logarithmic: U(z) = (u_star / kappa) * ln(z / z0), with kappa the von Karman constant near 0.40, u_star the friction velocity, and z0 the aerodynamic roughness length. Expressed as a ratio between two heights the friction velocity cancels, giving the working form used here. Roughness length is not obstacle height but roughly a tenth of it: about 0.0002 m over calm sea, 0.03 m over cropland, 0.3 to 1.0 m over suburbs, and above 1 m in dense city cores. Friction velocity typically runs 0.1 to 0.5 m/s. The power law U(z) = Uref * (z / zref)^alpha is the empirical alternative, requiring no roughness estimate. The one-seventh value alpha = 0.143 fits flat open terrain under neutral stratification; 0.10 to 0.12 is more appropriate offshore and 0.20 to 0.30 over suburbs. The two methods agree closely for extrapolation ratios under about 3 to 1 and diverge beyond that. Both share the same limitation: they assume neutral stability. On clear nights, suppressed mixing produces effective exponents of 0.4 to 0.6 and can support nocturnal low-level jets that break the profile entirely, while unstable afternoons flatten it. Ignoring stability commonly introduces 20 to 40 percent speed error over large height ranges, which matters disproportionately for wind energy because power density scales as 0.5 * rho * U^3 - a 10 percent speed error becomes roughly a 33 percent energy error.

The logarithmic profile descends from the law of the wall in turbulent pipe and boundary layer flow. Ludwig Prandtl introduced the boundary layer concept at the 1904 International Congress of Mathematicians in Heidelberg and later developed mixing length theory, and Theodore von Karman published his similarity analysis in 1930 producing the constant that carries his name. Prandtl and his students extended the result to flow over rough walls, defining the roughness length as the extrapolated height of zero mean velocity. Meteorologists adapted this to the atmosphere through the 1930s and 1940s, with Oscar Sutton and Lewis Fry Richardson contributing to the treatment of atmospheric turbulence and its dependence on thermal stratification. The stability generalization came in 1954, when Alexander Obukhov and Andrei Monin at the Institute of Physics of the Atmosphere in Moscow published the similarity theory that bears their names, showing that surface layer profiles collapse onto universal functions of the ratio z / L, where L is the Obukhov length. Field campaigns validated the correction functions, most influentially the 1968 Kansas experiment analyzed by Businger, Wyngaard, Izumi, and Bradley. The power law has a separate and more pragmatic lineage in engineering, where it was used for wind loading on structures well before wind energy existed. Its prominence grew with the wind industry from the 1980s onward, since hub heights climbed from about 30 m in the early 1990s to 80 to 120 m and beyond while most meteorological records remained at the 10 m standard, making reliable extrapolation a financial necessity. Ground-based lidar and sodar profiling, commercialized during the 2000s, now measure the profile directly and are used to check both laws on site.