Calculation of roughness length and displacement height

Calculation of Aerodynamic Roughness Length and Zero-Plane Displacement

The aerodynamic surface roughness length, z_{o}, and zero-plane displacement, D, are defined for a neutral wind profile by

U(z) = (u_{*}/k)ln((z-D)/z_{o})

where U is the mean wind speed measured at height z (above ground level), u_{*} is friction velocity, and k=0.4 is the von Karman constant. D and z_{o} are best calculated with a wind profile measured at several levels, but (theoretically) they can also be calculated from two levels of wind data as commonly measured by the ISFF Flux-PAM stations, i.e. with a prop-vane at 10m height and a sonic anemometer at a second height. A good test for neutral stability is zero heat flux <w'tc'> or zero temperature variance <tc'tc'>..

Solving for the two wind profile parameters,

D = z_{1} - (z_{2}-z_{1})/[exp(k(U_{2}-U_{1})/u_{*}) - 1]

Note that if the zero-plane displacement height is known, then the roughness length can be calculated from measurements of wind speed and friction velocity by a singe sonic anemometer.

The sensitivity of these formulas to errors in height, wind speed, and friction velocity measurements are,

where dz, dU, and du_{*} are measurement errors in height, wind speed, and friction velocity. It can be noted that the errors in roughness length dz_{o} are proportional to z_{o}, whereas the errors in zero-plane displacement dD are proportional to (z_{1}-D) and (z_{2}-D). Thus as z_{o} approaches 0, its uncertainty decreases proportionally, whereas the uncertainty in zero-plane displacement does not decrease as D approaches 0. The error terms containing the factor kU/u_{*}=ln((z-D)/z_{0})~ 4-8 can be particularly large. For example, if z_{2} / z_{1 }= 2 and D << z_{1}, then