More recently, these 1-D models have been extended to 3-D to take into account divergences in the various fluxes resulting from the horizontal inhomogeneity of the winds, and the effect of the dynamic ocean currents that advect momentum, heat and salt on even smaller scales.
In the above transport
equations, we have used an approximation in the turbulent
diffusion terms for the generalized vertical turbulent fluxes of
momentum and heat,
, 
, and 
.
These approximations are obtained via
so-called closure procedures, as is usually done in the simulation
of turbulent flows. A review of these methods for geophysical
flows is given in a survey by Mellor and Yamada
[44]
and Kantha
and Clayson
[34].
An application of first and second order closure
procedures to the oceanic mixed layer was performed in
[66].
For
brevity, we shall outline below only the so-called first-order
closure or eddy diffusion models, where the vertical turbulent
fluxes 
, etc. are approximated as follows
where the K's are computed from relations of the type
with 
being the
turbulence kinetic energy
, l is the so-called ``integral length
scale'' of turbulence, defined as
and 
, 
are stability
factors that have a functional dependence on the Richardson number
(the ratio of the vertical density variation to the vertical
velocity shear squared). In its simplest form, this factor is
approximated as
,
with 
commonly having a value of
.5 and 
, the so-called critical Richardson number (above which
turbulence cuts off), having a value of .25. The turbulence
quantity q is obtained from an equation which balances diffusion,
shear production and buoyancy production due to stratification
against self-decay of turbulence, but it will not be reproduced
here. For a more detailed description of these models, we refer
the reader to the papers of Mellor and Yamada
[44],
Kantha and
Clayson
[34],
and Martin
[40].
