If Za > Zb, geostrophic constraint requires that the meridional flow integrated over the western flank of the mid-ocean ridge should move poleward, as predicted by the classical Stommel-Arons theory; this pattern of circulation is called subcritical.
On the other hand, if Za < Zb, the meridional flow integrated over the western flank of the mid-ocean ridge should move equatorward, opposite to the classical Stommel-Arons theory; the circulation and the topography are called supercritical. For the present case, the corresponding constraint is fo2(wooD + 21 ) < f2(wooD +1) (5.107)
Since woo ^ w0, omitting the contribution due to the weak background upwelling reduces this relation to V2/o
Using Eqn. (5.101), we obtain the critical condition
For the general case, Eqn. (5.1o7) is reduced to a cubic equation for the critical height ratio x = B/H.
According to the classical theory of abyssal circulation of Stommel and Arons (196oa), under the assumptions of uniform upwelling and with no bottom topography, flow in the ocean interior is poleward, as shown in Figure 5.62a. A western boundary current is required in order to close the circulation; however, this is omitted from our discussion here.
For the case of enhanced upwelling over a meridional strip in the middle of the ocean and with no bottom topography, the circulation becomes stronger, but its pattern remains similar (Fig. 5.62c).
For the case of a ridge taller than the critical height under uniform upwelling, water over the western flank of the mid-ocean ridge moves equatorward, although water over the eastern flank or west of the mid-ocean ridge still moves poleward (Fig. 5.62b).
For the case with supercritical topography and enhanced upwelling over the mid-ocean ridge there is a strong circulation over the ridge (Fig. 5.62d).
This example demonstrates that topographic stretching is the primary force driving the model toward an equatorward circulation in the western flank of the ridge, opposite to the poleward flow predicted by the classical Stommel-Arons theory.
As another example, we study a sinusoidal-shaped seamount with the mean depth of the abyssal layer H = 3,000 m. The seamount topography and upwelling are in the following forms
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