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a cos 0

Cross-differentiating Eqns. (4.176) and (4.177), subtracting and using the continuity equation (4.178) leads to a potential vorticity equation for layer 2 after subduction:

This equation applies after subduction because now the second layer is shielded from the Ekman pumping; thus, along a streamline, a water parcel maintains its potential vorticity. This is different from the case before subduction, when the second layer is directly forced by Ekman pumping, so its potential vorticity is not conserved, as stated by Eqn. (4.173). According to Eqns. (4.176) and (4.177), the flow in the second layer follows the constant h line, i.e., the h contours are streamlines. Using the law of potential vorticity conservation, Eqn. (4.179), we come to the conclusion that the h contours are also potential vorticity contours.

Note that potential vorticity in the second layer, sin 0/h2, should be a function of h only, which we denote as G(h), i.e., sin 0/h2 = G (h). This function G(h) is set along the line 0 = 01 (the outcrop line for layer 1 and also the subduction line for the second layer):

sin 0

sin 01 sin 01

The last equals sign above is due to the fact that h is constant along trajectories in the second layer after subduction. Thus, following each streamline started from the outcropping line, the potential vorticity conservation law gives us the following relation:

sin 0 sin 01

Accordingly, the thicknesses for these two moving layers obey sin 0 f h2 =-h = - h (4.182)

sin 01 f1

Note that layer thickness ratio is entirely determined by the planetary potential vorticity. This simple relation gives rise to a simple analytical solution for the ventilated thermocline. Were the outcrop line not zonal, the solution would be in a much more complicated form, and this is the beauty of the ventilated thermocline model, as proposed by Luyten et al. (1983).

To determine the still unknown total layer depth h in this zone, we can use the Sverdrup relation for the barotropic mass flux. The momentum equations for the first layer are

a cos 0dX

where y = . The mass conservation equation is

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