Stommel-Arons Theory: Abyssal Flow on the Sphere
397
(7.3.6)
The meridional velocity is always poleward as long as w0 is positive. Thus in the
southern hemisphere the flow is southward while it must be northward in the
northern hemisphere. As long as w0 is finite at the equator, it follows that v
must vanish on the equator. There can therefore be no interior flow across the
equator. Any flow crossing the equator must occur in regions of different
dynamics where the geostrophic Sverdrup balance is upset, i.e., in the western
boundary current.
From (7.3.1b) and the condition that P be constant on the eastern
boundary of the ocean basin at¢= ¢e:
2fJ.R21,
P= - - -
wo(¢,8)sin 2 8d¢'.
H 1>
(7.3.7)
The zonal velocity follows from (7.3.1a):
1 R 8 [ {, (
) . 2
']
u = sin t} H [)t} j 1> Wo l/J, 8 sm 8d l/J
(7.3.8)
in which form it is valid even for eastern boundaries which are not meridians,
i.e., for which ¢e = ¢e(8).
If, for simplicity, the velocity wo is taken to be independent of horizontal
position, the algebra is considerably simplified. For a barotropic model such as
this there is no substantial qualitative effect in considering w0 constant,
although when baroclinic effects are considered, as in Section 7.7, the
geographical distribution of w0 becomes important. For now, however, we
take w0 as a constant. Thus, for example:
(7.3.9)
so that the zonal flow is always eastward.
Consider as a first example, the ocean basin composed of the spherical
sector shown in Fig. 7.3.1. It stretches from the equator to the pole, and its
meridional boundaries are at ¢ = ¢e on the eastern and ¢ = ¢w on the western
boundary. A source of fluid meant to represent the sinking of cold water at the
pole in the North Atlantic is shown at the apex of the sector and has a
magnitude S0 . The streamlines of the interior flow are already known since, as
in the experiment described in the previous section, the interior flow is
completely independent of the position of the sources.
The mass balance for the subsector north of an arbitrary latitude e, shown
by the dotted line in the figure, is found by balancing the sum of the interior
mass flux, the western boundary current's transport, and the source's
contribution, S0, all into the sector, against the flux out of the abyss due to
397
(7.3.6)
The meridional velocity is always poleward as long as w0 is positive. Thus in the
southern hemisphere the flow is southward while it must be northward in the
northern hemisphere. As long as w0 is finite at the equator, it follows that v
must vanish on the equator. There can therefore be no interior flow across the
equator. Any flow crossing the equator must occur in regions of different
dynamics where the geostrophic Sverdrup balance is upset, i.e., in the western
boundary current.
From (7.3.1b) and the condition that P be constant on the eastern
boundary of the ocean basin at¢= ¢e:
2fJ.R21,
P= - - -
wo(¢,8)sin 2 8d¢'.
H 1>
(7.3.7)
The zonal velocity follows from (7.3.1a):
1 R 8 [ {, (
) . 2
']
u = sin t} H [)t} j 1> Wo l/J, 8 sm 8d l/J
(7.3.8)
in which form it is valid even for eastern boundaries which are not meridians,
i.e., for which ¢e = ¢e(8).
If, for simplicity, the velocity wo is taken to be independent of horizontal
position, the algebra is considerably simplified. For a barotropic model such as
this there is no substantial qualitative effect in considering w0 constant,
although when baroclinic effects are considered, as in Section 7.7, the
geographical distribution of w0 becomes important. For now, however, we
take w0 as a constant. Thus, for example:
(7.3.9)
so that the zonal flow is always eastward.
Consider as a first example, the ocean basin composed of the spherical
sector shown in Fig. 7.3.1. It stretches from the equator to the pole, and its
meridional boundaries are at ¢ = ¢e on the eastern and ¢ = ¢w on the western
boundary. A source of fluid meant to represent the sinking of cold water at the
pole in the North Atlantic is shown at the apex of the sector and has a
magnitude S0 . The streamlines of the interior flow are already known since, as
in the experiment described in the previous section, the interior flow is
completely independent of the position of the sources.
The mass balance for the subsector north of an arbitrary latitude e, shown
by the dotted line in the figure, is found by balancing the sum of the interior
mass flux, the western boundary current's transport, and the source's
contribution, S0, all into the sector, against the flux out of the abyss due to