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4.2 The Mixed Layer Equations     continued...

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].