Richardson Criterion with Horizontal Turbulence

In calculating the density difference between a turbulent eddy and the surroundings, we have assumed (12.30) that there is no change of composition of the fluid elements. If one now assumes that the internal composition of an eddy varies during its motion due to the horizontal turbulence, the derivative of the density difference becomes

The difference of the internal and external n gradients can be estimated in the same way as the difference of the T gradients (12.34). We make the substitution


Fig. 12.6 Illustration of the role of the horizontal turbulence, which reduces the composition difference of the vertically moving eddies in a medium with a shear. Adapted from Talon & Zahn [562]

of V by Vp, of K by Dh and of Vadby0.0ne gets [562]

r vi

V + 1 6 Dh where v and i are the typical velocity and mean free path of the shear motions. Dh being larger than K, is smaller than r. With the density gradient (12.61) and Richardson's criterion (12.29), one gets

This expression of the criterion accounts for the reduction of the stabilizing effect both by the thermal diffusion through r and by the horizontal turbulence through r which diminishes the effective p gradient.

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