The horizontal diffusion terms , ,
in equations (73) to (75) are given by
expressions similar to
equations (83) to (85) except that
is replaced by **S**,
and
respectively.

The above expressions for horizontal diffusion quantities are general expressions. It is possible to simplify these expressions by ignoring and terms, since as a rule and except in very shallow coastal waters, and further we can put , the bottom depth.

It is readily apparent that even then, the most
general form of the expressions for horizontal diffusion terms are
complicated and rather expensive to compute. There is therefore a
need for simplification. Besides, there is still a great deal of
uncertainty associated with the choice of horizontal mixings
coefficients and .
More often than not, horizontal diffusion
is regarded as merely a numerical necessity to control subgrid
scale energy pileup in a numerical calculation and the values of
and are prescribed in an * ad-hoc* manner,
governed more by
numerical than physical consideration. Our understanding of
horizontal mixing, especially as they relate to subgrid scale
processes is rather primitive (especially as compared to the much
better understanding of vertical mixing processes due to small
scale turbulence) and we are therefore justified in simplifying
the expressions for , , , , and .