On the Atmospheric Turbulence, 3rd Paper

On the generalized logarithmic formula of wind velocity distribution

Abstract

A formula of vertical wind velocity distribution recently presented by Deacon where, u denotes the wind velocity at the height z, z₀ is the roughness parameter, and α and β are constants, seems to settle the old problem whether the velocity is represented by an exponential formula of the type u_??_az^1/n (2) or by a logarithmic one u_??_alnz+b (3) That is, the velocity can not be represented by simple formulae such as (2) and (3), but by (1) which may be called a generalized exponential formula. But, in this paper, the author shows that the velocity can be represented also by a formula which contains logarithmic terms only and may be called, so to speak, a generalized logarithmic formula.
As can be seen from recent results of wind velocity measurements, such as Deacon's or Pasquill's, the velocity curve shown on the u/U-lgz chart is concave or convex upwards This seems to be an extreme defect of the formula, but what is important in this case is the actual height of (9). If we adopt as the extremities of β which was given by Deacon as β=1.13 (in the unstable case) and β=0.79 (in the stable case), and estimate the heights from (9) substituting E by 1-β, we have z=2210m for unstable leger and z=0.0086m for stable lager. So this becomes trivial in the actual case.
There still remain some uncertainties as to the form of the logarithmic formula in the adiabatic case. For Rossby and Montgomery and also Sverdrup adopted the type u=alnz+z₀/z₀, while Paeschke and Thornthwaite used u=alnz-d/z₀, where, d is a length characteristic to the roughness. But in view of the pure empirical nature of the formula the author considers it sufficient to adopt, as in the 1st report of this paper, u=alnz/z₀, and only when experiments (made near the roughness height) show some deviation from this formula we should use u=alnz±d/z₀ in agreement with the conclusion obtained by Deacon that in conditions of neutral stability the logarithmic law can represent the profile between heights of 1m and 13m over the grass of various lengths with great accuracy, provided that both z₀ and d are chosen independently to give the best fit. (Sutton: Atmospheric Turbulence, p. 57).