# [Physics] logarithmic wind speed profile

aerodynamicsatmospheric sciencefluid dynamics

Under some atmospheric stability condition, over flat terrain, it has been observed for a while that the ratio between wind speed at height $h_1$ above the earth and the wind speed at height $h_0$ is $\log\frac{h_1}{h^*}/\log\frac{h_0}{h^*}$ where $h^*$ is related to the terrain (called roughness length). (see for example http://en.wikipedia.org/wiki/Log_wind_profile)

What are the theories (with some details or references please) that explain this rule. Please put only your prefered theory (and hence one per post).

Logarithmic profile for wind speed regards the bottom part of atmospheric boundary layer (say, about the bottom 100 m, on a boundary layer about 1000 m high). It can be deducted doing some non obvious but reasonable assumptions.

A) Vertical flux of horizontal momentum due to turbulence must be uniform in the lowest part of the atmosphere. Let's consider a reference frame where the average velocity $\overline{u}$ is directed along x axis. Let's decompose velocity in its average and random (turbulent) parts, according to Reynolds decomposition: x component of velocity is given by

$u = \overline{u} + u'$

Vertical component is:

$w = w'$

where $\overline{u'} = 0$, and $\overline{w'} = 0$, but in general $\overline{u'w'} \neq 0$: u' and w' are covariant. Vertical flux of horizontal momentum is given by $\overline{u'w'}$. Thus the first assumption can be expressed as follows:

1: $\overline{u'w'} = constant$

B) Prandtl hypothesis: random part of horizontal velocity u' is proportional to vertical wind shear:

2: $u' = l' \frac{\partial \overline{u}}{\partial z}$

where l' is the "mixing length": we can suppose that an air particle maintains its original horizontal speed during its random motion for a length l', before mixing with the surrounding air.

C) Vertical length scale of turbulent eddies is comparable to their horizontal length scale, thus the random part of vertical velocity is of the same order as the horizontal one:

3: $w' \approx l' \frac{\partial \overline{u}}{\partial z}$

Using expressions 2 and 3 in 1:

4: $\overline{u'w'} = \overline{l'^2} \left(\frac{\partial \overline{u}}{\partial z}\right)^2$

D) At the bottom part of the atmosphere the absolute value of mixing length l' is proportional to high z: this is reasonable because random motion are limited at the bottom by the earth surface. The hypothesis is: $(|l'| = kz)$ where k is Kàrmàn constant. Substituting l' in expression 4 we obtain:

5: $\overline{u'w'} = (kz)^2 \left(\frac{\partial \overline{u}}{\partial z}\right)^2$

Extracting the square root of 4 and separating the variables we obtain:

6: $d\overline{u} = \frac{\sqrt{\overline{u'w'}}}{k} \frac{dz}{z}$

Integrating we obtain the logarithmic profile:

$\Delta \overline{u} = \frac{\sqrt{\overline{u'w'}}}{k} \log{\frac{z}{z_0}}$