Monin-Obukhov similarity theory ($U_z=\frac{u_{\ast }}{\kappa z}{\varphi }_m(z/{\ell }_O)$)

A local similarity description for approximately stationary homogeneous surface-layer turbulence with nearly constant momentum and buoyancy fluxes. Besides height and friction velocity, the Obukhov length enters as a stability parameter. The hypothesis fixes the dimensional shear prefactor and function argument, while empirical or dynamical closures determine the functions themselves.

If the similarity functions of a turbulent boundary layer are differentiable at zero and have finite positive neutral values, expanding $f=a+dz/\ell +O((z/\ell {)}^2)$ gives a shear proportional to $1/z-d/(a\ell )$. Integration yields a logarithmic term plus a linear correction. Dimensional reasoning alone does not fix the coefficients or guarantee this regular neutral limit. The gradient Richardson number is of order $z/\ell$, so $|z/\ell |\ll 1$ is a shear-dominated, nearly neutral regime on either side of stability.

In local continuously turbulent scaling at $z/{\ell }_O\gg 1$, dependence on the distance to the wall disappears at leading order. The velocity profile is linear and the buoyancy frequency is constant. Local energy balance gives $ϵ=(c_m-1)|B_0|$ and ${\mathrm{Ri}}_f=1/c_m$, with positive dissipation requiring $c_m>1$. The coefficients depend on a closure, and this regime should not be imposed on fully collapsed turbulence.

The signed wall-turbulence length formed from stress velocity and the vertical turbulent buoyancy flux. With upward buoyancy transport positive, stable cooling has $B_0<0$ and ${\ell }_O>0$; heating from below gives ${\ell }_O<0$. The magnitude marks the transition between shear-dominated and buoyancy-dominated scaling.