# Roughness Length and Displacement Height

“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”. “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.”

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

$u_z = \frac{u_*}{\kappa} \left[\ln \left(\frac{z-d}{z_0} \right) + \psi(z,z_0,L)\right]$

where $u_*$ is the friction (or shear) velocity (m s−1), $\kappa$ is the Von Kármán constant (~0.41), $d$ is the zero plane displacement, $z_0$ is the surface roughness (in meters), and $\psi$ is a stability term where $L$is the Monin-Obukhov stability parameter. Under neutral stability conditions, $z/L = 0$ and $\psi$ drops out.

In the logarithmic wind profile, the roughness length is the height at which wind speed is zero (indicated by z0). It provides an estimate of the average roughness elements of the surface. With vegetated surfaces, because the vegetation itself provides a certain roughness, the logarithmic wind profile goes to zero at a height equal to the displacement height plus the roughness length.

The displacement height (or zero plane displacement height) of a vegetated surface – usually indicated with d – is the height at which the wind speed would go to zero if the logarithmic wind profile was maintained from the outer flow all the way down to the surface (that is, in the absence of the vegetation). In other words, it is the distance above the ground at which a non-vegetated surface should be placed to provide a logarithmic wind field equal to observed one. By another point of view, it should be regarded as the level at which the mean drag on the surface appears to act (Jackson, 1981). For forest canopies, it is estimated to vary between 0.6 and 0.8 times the height of the canopy (Arya, 1998; Stull, 1988).

z0: Roughness length is defined as the height at which the mean velocity is zero due to substrate roughness. Real walls/ground are not smooth and often have varying degrees of roughness, this parameter (which is determined empirically) accounts for that effect. d: Zero Plane displacement is defined as the height at which the mean velocity is zero due to large obstacles such as buildings/canopy. The two parameters are not the same because they describe the effects of two fundamentally different processes. d can be anywhere from 6 to 20 times larger than z0.