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.)