Surface Layer - Mathematical Formulation

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:

.