Monin–Obukhov similarity theory
Monin–Obukhov (M–O) similarity theory describes the non-dimensionalized mean flow and mean temperature in the surface layer under non-neutral conditions as a function of the dimensionless height parameter,[1] named after Russian scientists A. S. Monin and A. M. Obukhov.
Similarity theory is an empirical method that describes universal relationships between non-dimensionalized variables of fluids based on the Buckingham π theorem.
Similarity theory is extensively used in boundary layer meteorology since relations in turbulent processes are not always resolvable from first principles.
[2] An idealized vertical profile of the mean flow for a neutral boundary layer is the logarithmic wind profile derived from Prandtl's mixing length theory,[3] which states that the horizontal component of mean flow is proportional to the logarithm of height.
M–O similarity theory further generalizes the mixing length theory in non-neutral conditions by using so-called "universal functions" of dimensionless height to characterize vertical distributions of mean flow and temperature.
M–O similarity theory marked a significant landmark of modern micrometeorology, providing a theoretical basis for micrometeorological experiments and measurement techniques.
is a length parameter for the surface layer in the boundary layer, which characterizes the relative contributions to turbulent kinetic energy from buoyant production and shear production.
The Obukhov length was formulated using Richardson's criterion for dynamic stability.
perturbations of vertical velocity and virtual potential temperature, respectively.
M–O similarity theory parameterizes fluxes in the surface layer as a function of the dimensionless length parameter
From the Buckingham Pi theorem of dimensional analysis, two dimensionless group can be formed from the basic parameter set
can be determined to empirically describe the relationship between the two dimensionless quantities, called a universal function.
The eddy diffusivity coefficients for momentum and heat fluxes are defined as follows,
, In reality, the universal functions need to be determined using experimental data when applying M–O similarity theory.
Based on the results of the 1968 Kansas experiment, the following universal functions are determined for horizontal mean flow and mean virtual potential temperature,[7] Other methods which determine the universal functions using the relation between
[8][9] For sublayers with significant roughness, e.g. vegetated surfaces or urban areas, the universal functions must be modified to include the effects of surface roughness.
[6] A myriad of experimental efforts was devoted to the validation of the M–O similarity theory.
Field observations and computer simulations have generally demonstrated that the M–O similarity theory is well satisfied.
The 1968 Kansas experiment found great consistency between measurements and predictions from similarity relations for the entire range of stability values.
[7] A flat wheat field in Kansas served as the experiment site, with winds measured by anemometers mounted at different heights on a 32 m tower.
Results from the Kansas field study indicated that the ratio of eddy diffusivities of heat and momentum was approximately 1.35 under neutral conditions.
A similar experiment was conducted in a flat field in northwestern Minnesota in 1973.
This experiment used both ground and balloon-based observations of the surface layer and further validated the theoretical predictions from similarity.
[10] In addition to field experiments, analysis of M–O similarity theory can be conducted using high-resolution large eddy simulations.
However, the velocity field shows significant anomalies from M–O similarity.
[11] M–O similarity theory, albeit successful for surface layers from experimental validations, is essentially a diagnostic empirical theory based upon local first order turbulence closure.
When applied to vegetated areas or complex terrains, it can result in large discrepancies.
The basic parameter set of the M–O similarity theory includes buoyancy production
It is argued that with such a parameter set, the scaling is applied to the integral features of the flow, whereas an eddy specific similarity relationship prefers the usage of energy dissipation rate
[12] This scheme is able to explain anomalies of M–O similarity theory, but involves non-locality to modeling and experiments.
