Wind Speed At Height Log Power Law Calculator
Our meteorology & atmospheric science calculator computes wind speed at height log power law accurately. Get results you can export or share.
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Where V(z) is speed at target height z, Vref is reference speed at Zref, z0 is roughness length, and alpha is the power law exponent.
Last reviewed: December 2025
Worked Examples
Example 1: Wind Turbine Hub Height Extrapolation
Example 2: Urban Wind Assessment
Background & Theory
The Wind Speed At Height (log Power Law) Calculator applies the following established principles and formulas. Earth science calculators draw on a wide range of measurement scales and physical principles that quantify natural phenomena across geological, atmospheric, and hydrological systems. Earthquake magnitude is most precisely described by the Moment Magnitude Scale (Mw), which replaced the original Richter scale for larger events. Mw is calculated as Mw = (2/3) log10(M0) − 10.7, where M0 is the seismic moment in dyne-centimeters. The Richter scale, while still referenced colloquially, is a local magnitude (ML) measurement derived from peak seismograph amplitude at a standard 100 km distance. Wind intensity is classified using the Beaufort Scale, a 13-point empirical scale (0–12) relating wind speed in knots to observable sea and land effects, with Beaufort 12 corresponding to hurricane-force winds above 64 knots. Tropical cyclone intensity is further categorized by the Saffir-Simpson Hurricane Wind Scale, which assigns Categories 1 through 5 based on sustained wind speed, correlating with expected structural damage. Mineral hardness is quantified on the Mohs scale (1–10), comparing scratch resistance relative to reference minerals from talc (1) to diamond (10). Soil composition analysis measures the proportions of sand, silt, and clay by particle size, alongside organic matter content, bulk density, and porosity, which together determine engineering and agricultural suitability. Seismic wave velocity in rock varies by material: P-waves travel at approximately 5–7 km/s in granite and 1.5 km/s in water, while S-waves travel at roughly 60% of P-wave speeds. Atmospheric pressure decreases with altitude according to the barometric formula: P = P0 × exp(−Mgh / RT), where M is molar mass of air, g is gravitational acceleration, h is altitude, R is the universal gas constant, and T is temperature in Kelvin. Standard sea-level pressure is 101,325 Pa. Tidal calculations use harmonic analysis of gravitational forcing by the Moon and Sun, with the principal lunar semidiurnal tidal constituent (M2) having a period of approximately 12.42 hours.
History
The history behind the Wind Speed At Height (log Power Law) Calculator traces back through the following developments. The systematic study of Earth's structure and processes spans millennia, but the scientific foundations were laid in the seventeenth century. In 1669, Danish naturalist Nicolas Steno published his principles of stratigraphy, establishing the laws of superposition, original horizontality, and lateral continuity — foundational rules for reading rock layers that remain in use today. Scottish geologist James Hutton introduced the concept of uniformitarianism in 1788, proposing that geological processes observable in the present have operated throughout Earth's history at broadly consistent rates. This idea of deep time challenged prevailing biblical chronologies and set the stage for modern geology. Charles Lyell systematized these ideas in his landmark three-volume work Principles of Geology, published beginning in 1830, which directly influenced Charles Darwin's thinking on biological evolution during the voyage of the Beagle. The nineteenth century saw growing curiosity about continental shapes, but a coherent theory awaited Alfred Wegener, a German meteorologist who proposed continental drift in 1912, arguing that the continents had once formed a supercontinent he called Pangaea. His evidence included matching fossil records and geological formations across the Atlantic, but his mechanism was disputed for decades. The theory gained acceptance in the 1960s when seafloor spreading was confirmed through paleomagnetic studies, and plate tectonics emerged as the unifying framework of modern geoscience. The United States Geological Survey was established by Congress in 1879 to classify public lands and examine the geological structure, mineral resources, and products of the national domain. The twentieth century brought instrumental advances, including the global seismograph network deployed after World War II, initially to monitor nuclear tests, which dramatically improved earthquake detection and characterization. Satellite Earth observation began in earnest with the Landsat program launched in 1972, enabling continuous global monitoring of land use, glacier retreat, and vegetation patterns. Today, GPS networks, LIDAR scanning, and ocean-floor mapping provide centimeter-scale precision for tracking tectonic motion, sea level rise, and volcanic deformation in near real time.
Frequently Asked Questions
Formula
Log: V(z) = Vref*ln(z/z0)/ln(Zref/z0); Power: V(z) = Vref*(z/Zref)^alpha
Where V(z) is speed at target height z, Vref is reference speed at Zref, z0 is roughness length, and alpha is the power law exponent.
Worked Examples
Example 1: Wind Turbine Hub Height Extrapolation
Problem: Weather station measures 10 m/s at 10 m. Extrapolate to 80 m hub height over agricultural terrain (z0=0.03 m, alpha=0.143).
Solution: Log: V(80) = 10*ln(80/0.03)/ln(10/0.03) = 10*7.888/5.809 = 13.58 m/s\nPower: V(80) = 10*(80/10)^0.143 = 10*1.349 = 13.49 m/s\nu* = 10*0.4/ln(10/0.03) = 0.689 m/s\nPower density = 0.5*1.225*13.58^3 = 1534 W/m2
Result: Log: 13.58 m/s | Power: 13.49 m/s | Density: 1534 W/m2
Example 2: Urban Wind Assessment
Problem: Estimate wind at 50 m over urban area (z0=1.0 m, alpha=0.30) given 5 m/s at 20 m.
Solution: Log: V(50) = 5*ln(50/1.0)/ln(20/1.0) = 5*3.912/2.996 = 6.53 m/s\nPower: V(50) = 5*(50/20)^0.30 = 5*1.316 = 6.58 m/s\nu* = 5*0.4/ln(20/1.0) = 0.668 m/s\nPower density = 0.5*1.225*6.53^3 = 171 W/m2
Result: Log: 6.53 m/s | Power: 6.58 m/s | Density: 171 W/m2
Frequently Asked Questions
What is the logarithmic wind profile?
The logarithmic wind profile describes how wind speed increases with height above ground in the atmospheric surface layer the lowest 50 to 200 meters. Derived from Monin-Obukhov similarity theory under neutral stability conditions it states that speed at height z equals friction velocity divided by the von Karman constant times the natural log of z over roughness length. This profile is most accurate in the constant-flux layer where wind direction is approximately constant and mechanical turbulence dominates. It is preferred for wind energy resource assessment air pollution dispersion modeling and meteorological analysis when roughness information is available.
What is the power law wind profile?
The power law is an empirical approximation relating wind speed at a target height to a reference using V(z) = Vref times (z/Zref) to the power alpha. It is simpler than the log profile requiring only reference speed two heights and an exponent without roughness length. The standard exponent of one-seventh or 0.143 was derived for flat terrain with moderate roughness. The two methods agree well for height extrapolation ratios up to about 3:1 but can diverge significantly for larger ratios or non-neutral stability. The power law is used when detailed surface characterization is unavailable while the log law is preferred for rigorous engineering.
How does atmospheric stability affect wind profiles?
Atmospheric stability significantly modifies the vertical wind profile from its neutral logarithmic form by changing turbulent mixing intensity. In unstable conditions like sunny afternoons enhanced mixing brings momentum downward reducing shear and creating more uniform profiles. In stable conditions like clear nights suppressed mixing leads to stronger shear and potentially very light surface winds even with strong winds aloft. Monin-Obukhov theory accounts for stability through correction functions based on height to Obukhov length ratio. Ignoring stability can introduce 20 to 40 percent errors in wind speed extrapolation especially for large height differences.
Why is wind speed extrapolation important for wind energy?
Modern wind turbine hub heights have increased from about 30 m in the 1990s to 80 to 120 m or more but most weather stations measure wind at 10 m height. Extrapolating speed to hub height is essential for wind resource assessment and project feasibility analysis. Since wind power is proportional to the cube of speed even small errors in extrapolation produce large energy estimate errors. A 10 percent speed overestimate translates to roughly 33 percent power overestimate affecting project financial viability. Modern campaigns deploy met towers or lidar at hub height to reduce uncertainty in speed extrapolation.
What is wind power density?
Wind power density is the kinetic energy rate through a unit area perpendicular to the wind calculated as 0.5 times air density times speed cubed in watts per square meter. It is the fundamental metric for comparing wind energy potential across locations because it incorporates the cubic speed relationship. NREL classifies resources into seven classes with Class 1 below 100 W/m2 at 10 m being poor and Class 7 above 800 W/m2 being outstanding. Because of the cubic relationship power density at 80 to 120 m hub heights can be 2 to 4 times greater than at 10 m measurement height. This justifies the economics of tall turbine towers despite higher construction costs.
How do coastal wind profiles differ from inland?
Coastal and offshore profiles differ substantially due to dramatically different surface roughness of water versus land. Ocean roughness lengths of 0.0001 to 0.001 m produce much weaker wind shear and more uniform profiles than rougher land surfaces. The power law exponent over open water is typically 0.10 to 0.12 compared to 0.14 to 0.20 over land terrain. At coastal locations profiles change abruptly at the shoreline as the internal boundary layer adjusts from sea to land roughness over several kilometers. Thermal land-sea contrasts create sea breezes producing complex non-logarithmic profiles near the coastline.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy