Log wind profile

The equation to estimate the mean wind speed (${\displaystyle u_{z}}$) at height ${\displaystyle z}$ (meters) above the ground is:

where ${\displaystyle u_{*}}$ is the friction velocity (m s⁻¹), ${\displaystyle \kappa }$ is the Von Kármán constant (~0.41), ${\displaystyle d}$ is the zero plane displacement (in metres), ${\displaystyle z_{0}}$ is the surface roughness (in meters), and ${\displaystyle \psi }$ is a stability term where ${\displaystyle L}$ is the Obukhov length from Monin-Obukhov similarity theory. Under neutral stability conditions, ${\displaystyle z/L=0}$ and ${\displaystyle \psi }$ drops out and the equation is simplified to,

Zero-plane displacement (${\displaystyle d}$) is the height in meters above the ground at which zero mean wind speed is achieved as a result of flow obstacles such as trees or buildings. This displacement can be approximated as ²/₃ to ³/₄ of the average height of the obstacles.[2] For example, if estimating winds over a forest canopy of height 30 m, the zero-plane displacement could be estimated as d = 20 m.

Roughness length (${\displaystyle z_{0}}$) is a corrective measure to account for the effect of the roughness of a surface on wind flow. That is, the value of the roughness length depends on the terrain. The exact value is subjective and references indicate a range of values, making it difficult to give definitive values. In most cases, references present a tabular format with the value of ${\displaystyle z_{0}}$ given for certain terrain descriptions. For example, for very flat terrain (snow, desert) the roughness length may be in the range 0.001 to 0.005 m.[2] Similarly, for open terrain (grassland) the typical range is 0.01-0.05 m.[2] For cropland, and brush/forest the ranges are 0.1-0.25 m and 0.5-1.0 m respectively. When estimating wind loads on structures the terrains may be described as suburban or dense urban, for which the ranges are typically 0.1-0.5 m and 1-5 m respectively.[2]

In order to estimate the mean wind speed at one height (${\displaystyle {{z}_{2}}}$) based on that at another (${\displaystyle {{z}_{1}}}$), the formula would be rearranged,[2]

where ${\displaystyle u({{z}_{1}})}$ is the mean wind speed at height ${\displaystyle {{z}_{1}}}$.

The log wind profile is generally considered to be a more reliable estimator of mean wind speed than the wind profile power law in the lowest 10–20 m of the planetary boundary layer. Between 20 m and 100 m both methods can produce reasonable predictions of mean wind speed in neutral atmospheric conditions. From 100 m to near the top of the atmospheric boundary layer the power law produces more accurate predictions of mean wind speed (assuming neutral atmospheric conditions).[3]

The neutral atmospheric stability assumption discussed above is reasonable when the hourly mean wind speed at a height of 10 m exceeds 10 m/s where turbulent mixing overpowers atmospheric instability.[3]

Log wind profiles are generated and used in many atmospheric pollution dispersion models.[4]

• Wind profile power law
• List of atmospheric dispersion models