Mathematical Formulation
A simple model of the surface layer can be derived by first examining the turbulent momentum flux through a surface. Using Reynolds Decomposition to express the horizontal flow in the direction as the sum of a slowly varying component,, and a turbulent component,,:
and the vertical flow, in an analogous fashion:
we can express the flux of turbulent momentum through a surface, as the time averaged magnitude of vertical turbulent transport of horizontal turbulent momentum, :
.
If the flow is homogeneous within the region, we can set the product of the vertical gradient of the mean horizontal flow and the eddy viscosity coefficient equal to :
,
where is defined in terms of Prandtl's mixing length hypothesis:
where is the mixing length.
We can then express as:
.
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