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5.4 Transformation to Sigma Coordinates     continued...

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 .