Effects of Velocity Shear in Advective Mixed-Layer Models

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Abstract
A general multidimensional model of the upper mixed layer of oceans and lakes is presented. The density profile is approximated as uniform over the depth of the layer. Such an assumption is not made for the distribution of the horizontal velocity component, as there is no observational evidence for it. In fact, many observations indicate that important velocity shears exist in regions where advection can be expected to play an important role in the dynamics of the mixed layer (such as lakes, near upwelling and frontal areas, and in equatorial oceanic regions). It is shown that vertical shear in the horizontal velocity field combined with a horizontal density gradient can result in the production or absorption of mechanical energy. Production results if the velocity shear tends to destabilize the density profile, consumption if the tendency is stabilizing so that work against gravity must be done to mix the stabler density profile to uniformity. In the mechanical energy equation, this effect of the velocity shear is represented by a well-defined term. Comparison with other terms shows that for many relevant examples the shear mechanism contributes substantially to the energy balance. As a consequence, the deepening characteristics of a mixed layer with shear dispersion are different from those in a slab layer. These considerations culminate in the formulation of a generalized entrainment law that is nonlinearly coupled to the vertically integrated heat transport equation. The system is solved exactly for several examples with an imposed velocity structure. One of the important results is that vertical shear in the horizontal velocity components can lead to a considerable enhancement or reduction of the speed with which density anomalies propagate horizontally. For example, if an Ekman spiral is embedded in the mixed layer and cooler water is upstream, that speed turns out to be approximately a factor 2 − 1/π larger than the average (“slab”) velocity.

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