Log wind profile

☆ Save On Wikipedia ↗

The log wind profile is a semi-empirical relationship commonly used to describe the vertical distribution of horizontal mean wind speed within the lowest portion of the planetary boundary layer (PBL).[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 PBL (up to around 1 km) and the troposphere or free atmosphere. In the free atmosphere, geostrophic wind relationships should be used, instead.

Formulation

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

u z = u ∗ κ [ ln ⁡ ( z − d z 0 ) + ψ ( z , z 0 , L ) ] {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)+\psi (z,z_{0},L)\right]} {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)+\psi (z,z_{0},L)\right]}

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

u z = u ∗ κ [ ln ⁡ ( z − d z 0 ) ] {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)\right]} {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)\right]}.

Zero-plane displacement ( d {\displaystyle d} {\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 ( z 0 {\displaystyle z_{0}} {\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 z 0 {\displaystyle z_{0}} {\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 ( z 2 {\displaystyle {{z}_{2}}} {\displaystyle {{z}_{2}}}) based on that at another ( z 1 {\displaystyle {{z}_{1}}} {\displaystyle {{z}_{1}}}), the formula would be rearranged,[2]

u ( z 2 ) = u ( z 1 ) ln ⁡ ( ( z 2 − d ) / z 0 ) ln ⁡ ( ( z 1 − d ) / z 0 ) {\displaystyle u({{z}_{2}})=u({{z}_{1}}){\frac {\ln \left(({{z}_{2}}-d)/{{z}_{0}}\right)}{\ln \left(({{z}_{1}}-d)/{{z}_{0}}\right)}}} {\displaystyle u({{z}_{2}})=u({{z}_{1}}){\frac {\ln \left(({{z}_{2}}-d)/{{z}_{0}}\right)}{\ln \left(({{z}_{1}}-d)/{{z}_{0}}\right)}}},

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

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).[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]

Applications

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

See also

References

  1. Oke, T.R. (1987). Boundary Layer Climates. Methuen.
  2. Holmes JD. Wind Loading of Structures. 3rd ed. Boca Raton, Florida: CRC Press; 2015.
  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.