Log-Wind Profile and Surface Layer Scaling
In the neutrally stratified surface layer, the mean wind profile follows:
\[\frac{\partial \bar{u}}{\partial z} = \frac{u_*}{\kappa z}\]Integrating gives the logarithmic wind profile $\bar{u}(z) = (u_/\kappa)\ln(z/z_0)$ where $u_$ is friction velocity, $\kappa \approx 0.4$ is the von Kármán constant, and $z_0$ is the aerodynamic roughness length.
Monin-Obukhov Similarity Theory
Stability corrections modify the log profile through dimensionless stability functions $\phi_m(\zeta)$ where $\zeta = z/L$ and $L$ is the Obukhov length:
\[L = -\frac{u_*^3 \overline{\theta}_v}{\kappa g \overline{w'\theta_v'}}\]The Richardson number $Ri = N^2 / (\partial u/\partial z)^2$ measures the ratio of buoyant suppression to shear production of turbulence. Turbulence is sustained when $Ri < Ri_c \approx 0.25$.
Turbulent Kinetic Energy Budget
The TKE equation balances shear production $P$, buoyant production/consumption $B$, transport $T$, and dissipation $\varepsilon$:
\[\frac{\partial e}{\partial t} = P + B + T - \varepsilon\]where $e = \tfrac{1}{2}(\overline{u’^2}+\overline{v’^2}+\overline{w’^2})$. The Brunt-Väisälä frequency $N^2 = (g/\theta)(\partial\theta/\partial z)$ governs the oscillation frequency of buoyancy-displaced parcels and sets the scale of internal gravity waves generated at the ABL top.