298
Buoyancy Forced Circulation and Cross-Gyre Flow
We can divide the latitude band into (at least) two regions. We may expect that
to the west, where Us > c,, that heating (w* > 0) drives southward flow across
the gyre boundary in the lower layer (with equal northward return flux above
in the upper layer) while cooling does the opposite. In the eastern regions where
we may expect Us < c,, heating leads to deep northward flow across the gyre
boundary. If there is no heating or cooling on the intergyre boundary there is
no flow across the boundary (i.e., 8hj8¢ = 0) unless a third intermediate zone
develops in which Us = c, which is exactly the "window" described in Section 5.2.
Example 1: Subpolar Gyre
Consider now the example discussed by Luyten and Stommel (1986b) in which
a model of the subpolar gyre is driven by both Ekman pumping, WE > 0, and
cooling, w* < 0. For simplicity Luyten and Stommel considered the upper layer
to be thick enough on the eastern boundary that in spite of loss of fluid to the
upper Ekman layer and to the layer beneath it, the upper layer covers the entire
area of the subpolar gyre, i.e. no outcropping of layer 2 occurs. The Ekman
pumping and the cross-isopycnal velocities are both specified as functions only
of latitude. The former is positive, and the latter is negative in the subpolar
gyre, and they are both assumed to increase linearly with distance from the
intergyre boundary where they are both taken to be zero. There is no geostrophic flow across the boundary between the subpolar and subtropical gyre in
this model, and we concentrate on the subpolar gyre for now. Figure 5.3.2
schematically presents the region and the forcing. Note that for the chosen
forcing the ratio of w* to WE is a constant. The linear increase in Ekman
pumping is realistic only for the southern portion of the subpolar gyre, and we
therefore consider the domain of integration to represent the southern portion
of the gyre and allow its northern boundary to be porous. As we see below, the
characteristics at this boundary arrive from the south, and it is thus not necessary to prescribe boundary conditions on the northern edge of the domain
of the calculation.
On the eastern boundary the layer thicknesses H 1 and H 2 are given constants, each different from zero. Therefore, near the eastern boundary c, is
nonzero and since Us is 0 there, characteristics emanating from the eastern
boundary enter the interior of the basin and trend westward. The further
development of the characteristic field depends on the solution of the problem,
and they cannot be described precisely until the calculation is completed in the
manner described above.
On the western boundary the variation of the layer thicknesses with latitude is not constrained except by the Sverdrup relation (5.3.3), which gives
only one constraint between the two variables hand h1• For their calculation
Stommel and Luyten chose to set the layer thicknesses on the western
boundary by assuming that all the flow out of the western boundary current
Buoyancy Forced Circulation and Cross-Gyre Flow
We can divide the latitude band into (at least) two regions. We may expect that
to the west, where Us > c,, that heating (w* > 0) drives southward flow across
the gyre boundary in the lower layer (with equal northward return flux above
in the upper layer) while cooling does the opposite. In the eastern regions where
we may expect Us < c,, heating leads to deep northward flow across the gyre
boundary. If there is no heating or cooling on the intergyre boundary there is
no flow across the boundary (i.e., 8hj8¢ = 0) unless a third intermediate zone
develops in which Us = c, which is exactly the "window" described in Section 5.2.
Example 1: Subpolar Gyre
Consider now the example discussed by Luyten and Stommel (1986b) in which
a model of the subpolar gyre is driven by both Ekman pumping, WE > 0, and
cooling, w* < 0. For simplicity Luyten and Stommel considered the upper layer
to be thick enough on the eastern boundary that in spite of loss of fluid to the
upper Ekman layer and to the layer beneath it, the upper layer covers the entire
area of the subpolar gyre, i.e. no outcropping of layer 2 occurs. The Ekman
pumping and the cross-isopycnal velocities are both specified as functions only
of latitude. The former is positive, and the latter is negative in the subpolar
gyre, and they are both assumed to increase linearly with distance from the
intergyre boundary where they are both taken to be zero. There is no geostrophic flow across the boundary between the subpolar and subtropical gyre in
this model, and we concentrate on the subpolar gyre for now. Figure 5.3.2
schematically presents the region and the forcing. Note that for the chosen
forcing the ratio of w* to WE is a constant. The linear increase in Ekman
pumping is realistic only for the southern portion of the subpolar gyre, and we
therefore consider the domain of integration to represent the southern portion
of the gyre and allow its northern boundary to be porous. As we see below, the
characteristics at this boundary arrive from the south, and it is thus not necessary to prescribe boundary conditions on the northern edge of the domain
of the calculation.
On the eastern boundary the layer thicknesses H 1 and H 2 are given constants, each different from zero. Therefore, near the eastern boundary c, is
nonzero and since Us is 0 there, characteristics emanating from the eastern
boundary enter the interior of the basin and trend westward. The further
development of the characteristic field depends on the solution of the problem,
and they cannot be described precisely until the calculation is completed in the
manner described above.
On the western boundary the variation of the layer thicknesses with latitude is not constrained except by the Sverdrup relation (5.3.3), which gives
only one constraint between the two variables hand h1• For their calculation
Stommel and Luyten chose to set the layer thicknesses on the western
boundary by assuming that all the flow out of the western boundary current