uniform at all. In fact, upwelling of the abyssal ocean is not uniform, and the consequence of the non-uniform upwelling will be discussed later.)
Substituting Eqns. (5.60a) and (5.60b) into (5.61) leads to the vorticity equation dh vr— = — (5.62)
This equation corresponds to the Sverdrup relation in the wind-driven circulation theory. Here the slope of the parabolic free surface, dh/dr, plays a role equivalent to the j-effect in the ocean. Thus, the meridional velocity satisfies vr = -gT (5.63)
rn2 r
Two cases with no net source/sink of water in the interior Because there is no net source/sink, the free surface does not vary with time; thus, Z = 0, so that a(™rr^ = 0, and dX = -d(rd"rr^ = 0. There are two cases. First, if a source/sink is placed near the eastern boundary, there are zonal jets crossing the model basin, as shown in Figure 5.48a. Second, if there is no source/sink at a given "latitude" r, there should be no zonal flow. This is because the eastern wall is a rigid boundary, vx = 0; thus, vx = 0 everywhere. There is an exception to this rule. Near the southern wall, friction is not negligible, so geostrophy breaks down. Water particles leave the southern boundary and enter the interior, as shown in Figure 5.48b, c.
The case with a net source of water For this case, Z > 0, so that vr < 0, which means that water in the interior moves northward! For the case with point sources/sinks, the free surface elevation increases with time Z = X^a1 S, where X0 is the angle of the pie-shaped equipment, and S is the sum of the source(defined positive) and sink (defined negative). Thus, the meridional velocity is
Xow2a2r
Note that this is independent of the location of the source; even if the source is put at the pole, water in the interior still moves toward the pole - toward the source. Of course, there must be a place where this dynamical constraint breaks down. As in many other cases of oceanic circulation, the western boundary layer plays a major role in fulfilling the requirement of mass balance and potential vorticity balance for a closed circulation.
Mass must be balanced in a steady state, and this balance gives rise to the surprising fact that the western boundary current required for mass balance in the sector may flow toward the source. The mass flux across the southern boundary of the controlled volume r
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