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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).

Thanks in advance

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The way you state this question there won't be one correct answer, so you should flag this to be turned CW or change the question to something more specific, maybe "Where does the logarithmic wind speed profile originate from?" –  Tobias Kienzler Nov 12 '10 at 12:58
    
I 'd prefer to have it CW and have serveral answer. There is not one "origin" except if you are looking for an historical origin (which is not what I am interested in). However, it seems that I need a mod to put it CW. –  robin girard Nov 12 '10 at 15:12
    
I have math processing error with my browser... why ? Also, as I say I can't put it CW myself... no moderators here to do that ? –  robin girard Nov 13 '10 at 7:27
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@marek "obviously only one answer can really be correct" why is this obvious ? –  robin girard Dec 2 '10 at 7:05
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@marek Do you have a satisfactory definition of what are "equivalent explanations" ? In math you can have different arguments to show the same theorem.The difference can be in nature and sometime you can say that one argument is stronger than another, but sometime they are jus radically different. To give you a mathematical image (only to feed your intuition): on a surface, there is not necessarily a unique shortest path between two points. –  robin girard Dec 2 '10 at 11:05
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2 Answers

up vote 4 down vote accepted

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:

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

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Very clear answer, thanks! –  robin girard Dec 1 '10 at 7:10
    
It must be said that a logarithmic velocity profile can be observed not only in the bottom part of the atmospheric boundary layer, but more in general, in turbulent flow close to a bounding surface in no-slip condition. –  menta Dec 3 '10 at 10:46
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My guess is this is a property of turbulent boundary layers over rough surfaces. You may be able to associate the roughness and windspeed to a Strouhal number, and together with the Raynolds number give you at what situation you have this $\log()$ law be dominant.

As for a derivation of this behavior, I am hoping someone with fluids experience can actually answer your question.

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