# Log wind profile

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The Log wind profile is a semi-empirical relationship commonly used to describe the vertical distribution of horizontal mean wind speeds within the lowest portion of the planetary boundary layer. The relationship is well described in the literature.[1]

The logarithmic profile of wind speeds is generally limited to the lowest 100 m of the atmosphere (i.e., the surface layer of the atmospheric boundary layer). The rest of the atmosphere is composed of the remaining part of the planetary boundary layer (up to around 1000 m) and the troposphere or free atmosphere. In the free atmosphere, geostrophic wind relationships should be used.

## Definition

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

${\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)+\psi (z,z_{0},L)\right]}$

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

Zero-plane displacement (${\displaystyle d}$) is the height in meters above the ground at which zero wind speed is achieved as a result of flow obstacles such as trees or buildings. It is generally approximated as 2/3 of the average height of the obstacles. For example, if estimating winds over a forest canopy of height h = 30 m, the zero-plane displacement would be 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, and is between 1/10 and 1/30 of the average height of the roughness elements on the ground. Over smooth, open water, expect a value around 0.0002 m, while over flat, open grassland ${\displaystyle z_{0}}$ ≈ 0.03 m, cropland ≈ 0.1-0.25 m, and brush or forest ≈ 0.5-1.0 m (values above 1 m are rare and indicate excessively rough terrain).

## Limits

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).[2]

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]

## Applications

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

## References

1. ^ Oke, T.R. (1987). Boundary Layer Climates. Methuen.
2. ^ Cook, N.J. (1985). The designer's guide to wind loading of building structures: Part 1. Butterworths.
3. ^ Cook, N.J. (1985). The designer's guide to wind loading of building structures: Part 1. Butterworths.
4. ^ Beychok, Milton R. (2005). Fundamentals Of Stack Gas Dispersion (4th ed.). author-published. ISBN 0-9644588-0-2.