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a No ridge, uniform upwelling b With ridge, uniform upwelling a No ridge, uniform upwelling b With ridge, uniform upwelling

C c) No ridge, non-uniform upwelling d With ridge, non-uniform upwelling

Fig. 5.62 Interfacial displacement (in units of m), the thickness of the abyssal layer is H = 3,000 m, the uniform upwelling rate is W00 = 10-7 m/s, the amplitude of the enhanced upwelling rate over the mid-ocean ridge is W0 = 2 x 10-5 m/s, and the maximum height of the mid-ocean ridge is B = 1,500 m (supercritical).

Fig. 5.62 Interfacial displacement (in units of m), the thickness of the abyssal layer is H = 3,000 m, the uniform upwelling rate is W00 = 10-7 m/s, the amplitude of the enhanced upwelling rate over the mid-ocean ridge is W0 = 2 x 10-5 m/s, and the maximum height of the mid-ocean ridge is B = 1,500 m (supercritical).

This case is different from the previous case of a one-dimensional mid-ocean ridge. In the present case, the topography is two-dimensional, and typical potential vorticity contours are shown in Figure 5.63. When closed potential vorticity contours exist, we cannot obtain the solution by simple integration along characteristics because some of the characteristics are closed, and solutions within such closed characteristics have to be found by invoking higher-order dynamics, such as bottom friction and inertial terms, which are omitted in the simple model discussed here. More elaborate numerical approaches are required for such cases; examples are given by Katsman (2006). Our discussion here is limited to the case of no closed geostrophic contours, and we will call such topography "sub-critical topography."

Flow in the abyss with subcritical topography (maximal height of 400 m) is shown in Figures 5.64 and 5.65. This case again shows that the enhanced upwelling near the topography alone does not change the fundamental structure of the abyssal circulation. Although the localized enhancement of upwelling can alter the zonal velocity near the latitude of the seamount, it does not change the direction of the meridional flow, as shown in Figures 5.64c and 5.65c.

0 10E 20E 30E 40E 50E 60E 0 10E 20E 30E 40E 50E 60E

C Topography (in m; B = 400m) d Topograph (in m; B = 1200m)

C Topography (in m; B = 400m) d Topograph (in m; B = 1200m)

Fig. 5.63 a-d Seamount topography (c, d) and the associated normalized potential vorticity (a, b), q = fH/h (in units of 10-4/s). The maximal height of the seamount is 400 m (a, c) and 1,000 m (b , d).

It is interesting to see that perturbations induced by locally enhanced upwelling associated with an axis-symmetric seamount are nearly symmetric in the meridional direction, and the minor asymmetry is due to the increase of the Coriolis parameter in the meridional direction.

On the other hand, the topographic stretching can induce noticeable change in the meridional velocity pattern near the seamount. In fact, flow near the western slope of the seamount is equatorward (Fig. 5.65b), similar to the case of a mid-ocean ridge discussed above. Perturbations induced by topography are highly asymmetric in the meridional direction, as shown in Figure 5.64b. The negative anomaly in the interfacial displacement has a shape like a golf club.

Therefore, topographic stretching associated with both the mid-ocean ridge and an isolated seamount can induce equatorward flow on the western flank of the topography, opposite to the poleward flow predicted by the Stommel-Arons classical theory.

### Flow over isolated steep bottom topography

In the world's oceans, there are many instances of isolated steep bottom topography, including seamounts and trenches. In terms of an inverse gravity model, most potential vorticity

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