Monin-Obukhov similarity

Monin-Obukhov similarity theory states that in the atmospheric surface layer, non-dimensionalised gradients of wind, temperature, and scalars are universal functions of a single parameter: height divided by the Obukhov length.

The Obukhov length $L$ is the height at which buoyant production of turbulence becomes comparable to shear production. Above it buoyancy dominates; below it, shear does. Negative $L$ means unstable, positive means stable, and $|L|\to \infty$ is neutral. Compressing the entire stability problem into one length scale is what makes the theory usable.

The practical consequence is large. Instead of solving the turbulence equations, a model evaluates two empirical stability functions and a pair of roughness lengths, and gets its surface fluxes. Effectively every surface-layer scheme in operational use rests on this, and so does eddy-covariance flux interpretation and most micrometeorological measurement.

The assumptions, and where they fail

The theory requires horizontal homogeneity, stationarity, and a constant-flux layer where the fluxes vary little with height. Real surfaces violate all three, unevenly:

Stable conditions are the serious problem. At night, especially over snow or in light winds, turbulence becomes intermittent and can decouple from the surface entirely. There is no continuous turbulent transport for the theory to describe. Applied literally, the stability functions drive fluxes toward zero and surface temperature into free fall, so models impose artificial limits to prevent it. Those limits are tuning, not physics, and they are why nocturnal near-surface temperature errors are among the more stubborn biases in regional models. The theory is least reliable in precisely the conditions where the strongest temperature gradients occur.

Heterogeneous surfaces have no single roughness length or surface temperature. A grid cell spanning forest, field, and water can be handled with effective parameters or by tiling, and because the flux relationships are nonlinear the two approaches disagree.

Roughness lengths for heat and momentum differ, often by an order of magnitude, and their ratio is poorly constrained. It functions as a tuning parameter that is rarely acknowledged as one.

Very unstable, low-wind conditions are also outside the framework, since free convection has no shear scale to normalise against.

That the theory remains standard despite all this is a fair measure of how well it does in the conditions it does cover. It is worth knowing its failure modes rather than treating a surface-layer scheme as settled physics.

See also: surface-layer schemes that implement it, PBL schemes above it, and the MM5 similarity scheme for one implementation.