Roughness length ( z 0 {\displaystyle z_{0}} ) is a parameter used when modeling the horizontal mean wind speed near the ground. In wind vertical profile such the log wind profile, the roughness length (with dimension of length and SI unit of metres) 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 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 z 0 {\displaystyle z_{0}} appears in the expression for the mean wind speed u z {\displaystyle u_{z}} near the ground derived using the Monin–Obukhov similarity theory:
u z = u ∗ κ [ ln ( z − d z 0 ) + ψ ( z − d − z 0 L ) ] , {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)+\psi \left({\frac {z-d-z_{0}}{L}}\right)\right],}
where
u ∗ {\displaystyle u_{*}} is the friction velocity
κ {\displaystyle \kappa } is the Von Kármán constant
z {\displaystyle z} is the elevation (as measured from the ground)
d {\displaystyle d} is the elevation of the displacement plane (as measured from the ground), which is an offset that accounts for wind-slowing obstacles such as buildings, trees, or any other structures which impede flow
L {\displaystyle L} is the Monin-Obukhov length (which is defined to be the height at which buoyancy and wind shear are equally effective at creating turbulence)
ψ {\displaystyle \psi } is a correction factor for stability, with ψ = 0 {\displaystyle \psi =0} indicating statically neutral conditions. Conditions are statically neutral when the temperature of the air monotonically increases with elevation. In the simplest possible case (statically neutral conditions and no wind-slowing obstacles), the mean wind speed simplifies to:
u z = u ∗ κ ln ( z z 0 ) . {\displaystyle u_{z}={\frac {u_{*}}{\kappa }}\ln \left({\frac {z}{z_{0}}}\right).}
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. In this simplified form, the log wind profile is identical in form to the dimensional law of the wall. If the friction velocity is unknown, one can calculate the surface roughness as follows
z 0 = exp ( u ( z 2 ) ln ( z 1 ) − u ( z 1 ) ln ( z 2 ) u ( z 2 ) − u ( z 1 ) ) {\displaystyle z_{0}=\exp \left({\frac {u(z_{2})\ln(z_{1})-u(z_{1})\ln(z_{2})}{u(z_{2})-u(z_{1})}}\right)}
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