Neumann (1960) described two solutions of how the change of the bottom topography can influence ocean currents, one according to Ekman (1923) and the other one to Sverdrup (1941). In Ekman's solution, a friction-driven circulation is deflected to the right when the depth decreases. The magnitude of the deflection is independent of the water depth itself, but depends on the slope of the bottom. A net displacement of the streamlines takes place.
Sverdrup deduced that the streamlines are deflected on the upstream side of a ridge, but turn back on the downstream side of the ridge, which he could explain with the Coriolis force.
Sverdrup assumed a geostrophic current and neglected the effects of frictional forces.
Besides friction, the key process of the interaction between currents and bathymetry is the conservation of potential vorticity. Recent publications (e.g. Gill and Schumann, 1979, Song, 2002) emphasise the influence of potential vorticity on the currents in upwelling regions.
From the results of the long-term run (E0) a relative vorticity can be calculated, which is given by:
=v
x−u
y (8.10)
A idealised two layered sea is assumed. For simplification the first layer is defined as the upper half of the water column and the second layer is defined as the lower half of the water column, hence the interface between both layers describes not a horizontal plane. The interesting motion of the upwelling circulation is related to the deeper layer. The lower half of the water column is of interest, because it approximately reflects the water mass, which will be moved upward.
The simulated currents of the model (E0) are used to calculate the relative vorticity.
Fig. 8.5 illustrates the distribution of the relative vorticity in the lower half of the water column for state 1 (fig. 8.5a), state 3 (fig. 8.5b), state 5 (fig. 8.5c) and state 6 (fig. 8.5d), being the states with the highest vertical velocities. In the figure, the locations that show increased positive vertical velocities are marked with blue ellipses. The change of the alongshore bathymetry is visualised with brown dashed lines, which represent the crestlines of the ridge (11° 40' N) and the beginning of the Sunda shelf (chapter 2). Remarkable features are highlighted as in chapter 5.
In state 1 (fig. 8.5a), on the northern shelf, the distribution of relative vorticity is irregular (F1). As it can be seen in the horizontal velocities (fig. 5.1c, 5.1d), the flow of the current is strongly disturbed by mesoscale bathymetric characteristics, like sea mountains, resulting in instabilities in the flow. Off the shelf, the vorticity is cyclonic (F2), which can be seen as well in the distribution of horizontal currents. Noticeable is the anticyclonic relative vorticity of the currents in the upstream direction of the two upwelling areas (F3, F4), which are congruent with the change in the alongshore depth (brown dashed lines).
The connection between the anticyclonic rotation in these regions and the upwelling downstream has to be discussed. Because the velocity of the current in the zonal direction is small in comparison to the velocity in the meridional direction, u can be set to zero. Results from the long-term run of the model provide a ratio v/u of O(20).
With the neglect of external forcing, meaning no change of /t and u = 0, the change of the relative vorticity is:
D
Dt =v⋅
y (8.11)
In the regions upstream from the positive vertical velocities, the shelf becomes shallower to the south. This means d/y is positive. In a southward current, the velocity v in equation (8.11) is negative, and thus, equation (8.6) can only be true if the relative vorticity is negative, as well. This may happen under two conditions: the first is, that v/x is negative, the second is, that u/y is positive.
Assuming the current is not rotating downstream the upwelling. If the water is becoming shallower, v/x can become negative, thus a shear on the the right side of the current appears. The second possibility, hence u/y is positive illustrates that the current tends to rotate anticyclonicly in the direction to the coast, coming into shallower water, thus the flow crosses the isobaths. The change of the relative vorticity could be an explanation for the distribution of the vertical velocities in the alongshore section and the sections perpendicular to the shore of state 1.
The distribution of relative vorticity in state 3 (fig. 8.5b) shows the same scattering in the northernmost part (F1) of the VUA. Offshore from the region of increased upwelling, an
anticyclonic rotation (F2) of the current can be found. The southward current, which is present in the lower half of the water column, rotates in the onshore direction, thus into shallower water. In the upwelling center, south of Cam Ranh (F3), the currents are directed northward south of the ridge and southward north of the ridge. The cyclonic rotation south of the ridge and the anticyclonic circulation north of the ridge comply well with the onshore rotation of the horizontal currents in larger depths. Anticyclonic currents can also be found downstream from the third region of upwelling (F4) of this state. The explanation of this distribution is the same as for state 1, except for the cyclonic rotation south of the ridge.
However, during state 3, no significant wind forcing is present, thus the effect of the Potential Vorticity doesn't have to exceed the influence of the local wind forcing to induce upwelling. Also, the inshore recirculation (fig. 5.3a) may be explained with this process.
The distribution of relative vorticity of state 5 can be seen in fig. 8.5c. As in winter, on the northern shelf, the relative vorticity has a disordered structure. Remarkable is the positive relative vorticity of the flow upstream from the region of enhanced upwelling. A positive relative vorticity causes a cyclonic rotation, hence, considering the northward direction of the main current, the flow rotates towards the shore. As in state 1, d/y is positive, but now the relative vorticity has to be positive, because the velocity v is positive. As in the case of the southward boundary current, an inshore deacceleration and an onshore rotation flow can be seen in the horizontal circulation.
These findings were confirmed by the model of Song et al. (2001), who simulated the formation of upwelling centers downstream from ridges. Recapitulating, it can be said, that in a flow with the coast to the left, the lower part of the water column rotates cyclonicly where the shelf deepens, resulting in increased onshore flow. Hence, the vertical velocities and the offshore flow of the upper layer are intensified in these regions.
The second summer state (state 6, fig. 8.5d) shows a more scattered distribution of the relative vorticity, which is not surprising, considering the complex horizontal circulation of this state. The most obvious feature is the positive vorticity off the shelf (F1, F2) due to the shear induced by the southward boundary current and the anticyclonic circulation in the deep basin. East of the upwelling (F3), negative relative vorticity occurs. This can be seen as well, in the horizontal circulation in depths of 50 m and 80 m in fig. 5.6b and fig. 5.6c, respectively. In this region and these depths, the boundary current rotates anticyclonicly, which results in a strong onshore flow. At F4, the relative vorticity is positive, which, due to the northward flow, results in an onshore flow of the deeper part of the water column.
Fig. 8.5: Horizontal distribution of the relative vorticity in the lower half of the water column, for state 1 (a), state 3 (b), state 5 (c) and state 6 (d). Unit: 1×10-5[s-1].
F3F3
F4F4
F1F1 F2F2
F3F3
F4F4 F1F1
F4F4
F3 F3 F2F2
F5F5 F1F1
F1 F1
F3F3
F4
F4 F2F2
F2F2
a) b)
c)
d)
The appearance of increased upwelling is correlated with the distribution of relative vorticity, hence the conservation of potential vorticity is a significant process in the VUA.
Moreover, if the change of the alongshore depth, is as high as in the VUA, this process can exceed the effects of both two-dimensional upwelling processes.
This section showed, that the strong change in the alongshore depth resulting in a onshore rotation of the current, thus in shallower water and with that upwelling is increased during summer or appears during winter.
Summarized, this and the section before showed, that the theoretical mechanism illustrated in chapter 8.2 is plausible the process to induce upwelling due to the interaction of the southward boundary current and the bathymetry.