Editing Surface layer (section)

==Mathematical formulation==
A simple [[Mathematical model|model]] of the surface layer can be derived by first examining the turbulent [[momentum flux]] through a surface.<ref name="Holton">{{cite book |last=Holton |first=James R. |url=https://books.google.com/books?id=fhW5oDv3EPsC |title=Dynamic Meteorology |edition=4th |series=International Geophysics Series |volume=88 |year=2004 |publisher=Elsevier Academic Press |location=Burlington, MA |pages=129–130 |chapter=Chapter 5 - The Planetary Boundary Layer |isbn=9780123540157 }}</ref>
Using [[Reynolds decomposition]] to express the horizontal flow in the <math>x</math> direction as the sum of a slowly varying component, <math>\overline{u}</math>, and a turbulent component, <math>u'</math>,

:<math> u = \overline{u} + u'</math> <ref name="reynolds">{{cite news|url=http://www.eng.fsu.edu/~dommelen/courses/flm/flm00/topics/turb/node2.html |title=Reynolds Decomposition | publisher=[[Florida State University]] | date=6 December 2008 | access-date=2008-12-06}}</ref>

and the vertical flow, <math>w</math>, in an analogous fashion,

:<math> w = \overline{w} + w' </math>

we can express the flux of turbulent momentum through a surface, <math>u_*</math>, as the time-averaged magnitude of vertical turbulent transport of horizontal turbulent momentum, <math>u'w'</math>:

:<math> u_*^2  = \left|\overline{(u'w')_s} \right|</math>.

If the flow is [[Homogeneity and heterogeneity|homogeneous]] within the region, we can set the product of the vertical gradient of the mean horizontal flow and the eddy viscosity coefficient <math>K_m</math> equal to <math>u_*^2</math>:

:<math>K_m\frac{\partial \overline{u}}{\partial z} = u_*^2 </math>,

where <math>K_m</math> is defined in terms of [[Ludwig Prandtl|Prandtl]]'s mixing length hypothesis:

:<math>K_m = \overline{\xi'^2}\left |\frac{\partial\overline{u}}{\partial z}\right |</math>

where <math>\xi'</math> is the mixing length.

We can then express <math>u_*</math> as:

:<math>\frac{\partial \overline{u}}{\partial z} = \frac{u_*}{\overline{\xi'}}</math>.