Monin-Obukhov Length

The Monin-Obukhov Length is the height above ground, where mechanically produced (by vertical shear) turbulence is in balance with the dissipative effect of negative buoyancy, thus where Richardson number equals to 1:

where $u\_*$ is the frictional velocity, is the mean potential virtual temperature, is the perturbation scalar velocity' and $heta\_*$ is a potential temperature scale (k). This can be further reduced using the similarity theory approximation:

$(overline\{w^\text{'}\; heta^\text{'}\_v\})\_sapprox-u\_*\; heta\_*$

to give:

The parameter $heta\_*$ is proportional to the vertical difference in potential virtual temperature. The greater at $Z\_\{0,h\}$ in comparison with its value at $Z\_r$ , the more negative the change in with increasing height, and the greater the instability in the of the surface layer. In such cases, L is negative with a small magnitude, since it is inversely proportional to $heta\_*$. When L is negative with a small magnitude, $frac\{z\}\{L\}$ is negative with a large magnitude. Such values of $frac\{z\}\{L\}$ correspond to large instability due to buoyancy. Positive values of $frac\{z\}\{L\}$ correspond to increasing with altitude and stable stratification.