Monin–Obukhov length

The Monin-Obukhov Length is the height over the 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:

$L=-{\frac {u_{*}^{3}{\bar {\theta }}_{v}}{kg({\bar {w}}^{'}{\bar {\theta }}_{v}^{'})}}$

where $u_{*}$ is the frictional velocity, ${\bar {\theta }}_{v}$ is the mean potential virtual temperature, ${\bar {w}}^{'}$ is the perturbation scalar velocity' and $\theta _{*}$ is a potential temperature scale (k). This can be further reduced using the similarity theory approximation:

$({\overline {w^{'}\theta _{v}^{'}}})_{s}\approx -u_{*}\theta _{*}$

to give:

$L={\frac {u_{*}^{2}{\bar {\theta }}_{v}}{kg\theta _{*}}}$

The parameter $\theta _{*}$ is proportional to ${\bar {\theta }}_{v}(z_{r})-{\bar {\theta }}_{v}(z_{0,h})$ the vertical difference in potential virtual temperature. The greater ${\bar {\theta }}_{v}$ at $Z_{0,h}$ in comparison with its value at $Z_{r}$ , the more negative the change in ${\bar {\theta }}_{v}$ 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 $u_{*}$. 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 ${\bar {\theta }}_{v}$ with altitude and stable stratification.