## Ui u i ui p p p p p p3128

In addition, we assume that the Boussinesq approximations hold and the tidal force is omitted. Multiplying Eqn. (3.44) by ui and averaging leads to the equation for the turbulent kinetic energy:

9,-9, —;—- dp _ , .dp' d / , ,\ / , ,\ dut -K' + u-K> = ^a>- - (a + a') ^ + v- (u'ta^ - s - (ui j ^^

where du' dU

Scaling analysis indicates that for steady-state flows the basic balance is between the turbulence production, the dissipation, and the buoyancy work (Turner, 1973; Osborn, 1980):

Defining the flux Richardson number as the ratio of the buoyancy flux to the turbulent production:

R = upi f p , the eddy density diffusivity is defined as

. ijdXj

Then, the balance of turbulent dissipation in Eqn. (3.131) is reduced to

Rf s

Thus, among the turbulent kinetic energy dissipated in the ocean, a small proportion of it can be converted back to the GPE of the large-scale mean flow. This is a very peculiar and important upscale energy cascade in the stratified turbulence. This means that not all the kinetic energy of turbulence is dissipated into heat, and a small percentage of such dissipation is actually fed back to the large-scale flow.

However, this feedback involves many complicated processes, and the mixing efficiency should not simply be taken as a constant. In fact, the efficiency of mixing varies significantly. Peltier and Caulfield (2003) presented the most up-to-date review of the mixing efficiency in stratified shear flows.

In situ observations indicate that in the upper ocean, below the mixed layer, diapycnal diffusivity is on the order of 10-5 m2/s, which is much larger than the molecular diffusivity. In other places in the oceans, diffusivity can be much higher than this background rate (Fig. 3.17). The strong diapycnal (or vertical) mixing in the oceans is driven by strong internal wave breaking and turbulence.

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