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5.2 Governing Equations     continued...

Vertical mixing coefficients and in the fully turbulent mixed layers at the surface and the bottom are obtained by second order closure model of turbulence based on the work of Mellor and Yamada [43,44] ;see also the work of Galperin, Kantha, Hassid and Rosati [18]. The turbulence in this so-called level 2.25 model is characterized by two quantities, the turbulence kinetic energy and the turbulence macroscale l. This two-equation model of turbulence is governed by the following equations:

and

Second order closure (see [18]) relates the vertical mixing coefficients , and to the turbulence scales q and l:

where , and are stability functions determined from algebraic relations derived analytically from simplifications made to the full second moment closure model:

and

where

, , , and are constants that determine the ratios of various turbulence length scales to the turbulence macroscale l. The turbulence closure assumes that all turbulence length scales are proportional to one another.

Note that and are functions of , which is in turn a function of the buoyancy gradient. The term multiplying in equation (44) is a wall proximity function inserted empirically to assure log-law behavior near solid boundaries and L is given by

where D is the depth of the fluid column.

Terms , , and are empirical constants determined by appealing to well-known laboratory experiments on turbulence, as are constants , , and :

is the well-known von Karman constant (=0.4) that occurs in the log-law governing the velocity profile adjacent to a boundary in a turbulent boundary layer.

(See exercise 7.)