Roughness length

Roughness length (formula_1) is a parameter of some vertical wind profile equations that model the horizontal mean wind speed near the ground. In the log wind profile, it is equivalent to the height at which the wind speed theoretically becomes zero in the absence of wind-slowing obstacles and under neutral conditions. In reality, the wind at this height no longer follows a mathematical logarithm. It is so named because it is typically related to the height of terrain roughness elements (i.e. protrusions from and/or depressions into the surface). For instance, forests tend to have much larger roughness lengths than tundra. The roughness length does not exactly correspond to any physical length. However, it can be considered as a length-scale representation of the roughness of the surface.
Mathematical foundation.
The roughness length formula_1 appears in the expression for the mean wind speed formula_3 near the ground derived using the Monin–Obukhov similarity theory:
formula_4
where
In the simplest possible case (statically neutral conditions and no wind-slowing obstacles), the mean wind speed simplifies to:
formula_12
This provides a method to calculate the roughness length by measuring the friction velocity and the mean wind velocity (at known elevation) in a given, relatively flat location (under neutral conditions) using an anemometer. Of note is that, in this simplified form, the log wind profile is identical in form to the dimensional law of the wall.
If we don't know the friction velocity, one can calculate the surface roughness as follow
formula_13
Due to the limitation of observation instruments and the theory of mean values, the levels (z) should be chosen where there is enough difference between the measurement readings. if one has more than two readings, she can fit the measurements to the above equation and find the roughness length.
When calculating the surface roughness, the displacement height can be neglected.
Application.
As an approximation, the roughness length is approximately one-tenth of the height of the surface roughness elements. For example, short grass of height 0.01 meters has a roughness length of approximately 0.001 meters. Surfaces are rougher if they have more protrusions. Forests have much larger roughness lengths than tundra, for example. Roughness length is an important concept in urban meteorology as the building of tall structures, such as skyscrapers, has an effect on roughness length and wind patterns.
Relation to other roughness measures.
The roughness length is one of many possible measures of the roughness of a surface. For example, in classical mechanics the coefficient of friction is commonly used to measure the roughness of a surface as it relates to the force exerted on another contacted object. And, in fluid dynamics, hydraulic roughness is a measure of the resistance water experiences when flowing over land or through a channel. All of these measures ultimately derive from frictional forces, which result from irregularities on the surfaces of relevance.