48
Homogeneous Models of the Ocean Circulation
(2.8.13)
Clearly this form is taken for its analytical simplicity, and in principle any more
complicated nonlinear form could serve equally well. However, as we see
below, this choice turns out to be particularly fortuitous. The constants y0 and
U in (2.8.13) are completely arbitrary, but we clearly identify U / fJ with the
square of the inertial boundary-layer thickness.
The vorticity integral (2.8.12) with the specification of (2.8.13) can be
solved in general, but it is more illuminating to consider the case where:
so that boundary-layer techniques can be used. In this limit the asymptotic
solution of (2.8.12) which satisfies the conditions that tjJ = 0 on the boundaries
of the basin at x = 0 and xe (the western and eastern boundaries) and at y = y n
and 0 (the northern and southern boundaries) is:
t/1 = U(y _Yo) ( 1 _ e-x/lh _ e-(x,-x)/bi)
- U(yn- Yo)e-(yn-Y)/bi + Uyoe-yfbi
where ()1 = JfJTP.
(2.8.14)
The solution consists of an interior, constant, westward flow of magnitude
U. It impinges on the western boundary where inertial boundary layers of
constant width ()1 tum the flow along the boundary. The interior flow must be
westward to have an inertial boundary layer according to Greenspan's theorem. We can see from (2.8.12) and (2.8.13) that if the interior flow were eastward, a solution could still exist, but the solutions required to satisfy the
boundary conditions on the walls of the basin would be spatially wavelike with
a wavelength ()1. These waves would fill the interior. When U < 0, on the other
hand, the effect of the boundaries is limited to boundary layers on all four
boundaries.
The flow impinging on the western boundary turns northward north of the
line y = Yo and turns southward for streamlines impinging on the wall south of
y0. The flow in the western boundary layer then flows to the northern and
southern boundaries and subsequently flows eastward on both those boundaries. The amount of transport in each of these eastward jets depends on the
position of the bifurcation point y0 which, we remember, is arbitrary. The
eastward transport in the northern boundary layer is U(yn- Yo) while the
eastward transport in the southern boundary layer is Uy0• These boundarylayer flows impinging on the eastern boundary are turned along the eastern
boundary to flow meridionally. The flow decelerates along its path in the
eastern boundary layer and gradually emerges from the eastern boundary layer
and joins the interior. Figure 2.8.2 shows the Fofonoff mode for the two
choices y0 = Yn and 0. In the first case there is no northern boundary layer, and
the circulation is counterclockwise (cyclonic) in the gyre while in the second
Homogeneous Models of the Ocean Circulation
(2.8.13)
Clearly this form is taken for its analytical simplicity, and in principle any more
complicated nonlinear form could serve equally well. However, as we see
below, this choice turns out to be particularly fortuitous. The constants y0 and
U in (2.8.13) are completely arbitrary, but we clearly identify U / fJ with the
square of the inertial boundary-layer thickness.
The vorticity integral (2.8.12) with the specification of (2.8.13) can be
solved in general, but it is more illuminating to consider the case where:
so that boundary-layer techniques can be used. In this limit the asymptotic
solution of (2.8.12) which satisfies the conditions that tjJ = 0 on the boundaries
of the basin at x = 0 and xe (the western and eastern boundaries) and at y = y n
and 0 (the northern and southern boundaries) is:
t/1 = U(y _Yo) ( 1 _ e-x/lh _ e-(x,-x)/bi)
- U(yn- Yo)e-(yn-Y)/bi + Uyoe-yfbi
where ()1 = JfJTP.
(2.8.14)
The solution consists of an interior, constant, westward flow of magnitude
U. It impinges on the western boundary where inertial boundary layers of
constant width ()1 tum the flow along the boundary. The interior flow must be
westward to have an inertial boundary layer according to Greenspan's theorem. We can see from (2.8.12) and (2.8.13) that if the interior flow were eastward, a solution could still exist, but the solutions required to satisfy the
boundary conditions on the walls of the basin would be spatially wavelike with
a wavelength ()1. These waves would fill the interior. When U < 0, on the other
hand, the effect of the boundaries is limited to boundary layers on all four
boundaries.
The flow impinging on the western boundary turns northward north of the
line y = Yo and turns southward for streamlines impinging on the wall south of
y0. The flow in the western boundary layer then flows to the northern and
southern boundaries and subsequently flows eastward on both those boundaries. The amount of transport in each of these eastward jets depends on the
position of the bifurcation point y0 which, we remember, is arbitrary. The
eastward transport in the northern boundary layer is U(yn- Yo) while the
eastward transport in the southern boundary layer is Uy0• These boundarylayer flows impinging on the eastern boundary are turned along the eastern
boundary to flow meridionally. The flow decelerates along its path in the
eastern boundary layer and gradually emerges from the eastern boundary layer
and joins the interior. Figure 2.8.2 shows the Fofonoff mode for the two
choices y0 = Yn and 0. In the first case there is no northern boundary layer, and
the circulation is counterclockwise (cyclonic) in the gyre while in the second