The Nondissipative Model
345
8pi = o(UI) « 1.
8y
Uz
( 6.4.17)
If this were the case, then to lowest the order (6.4.12) would imply that:
(6.4.18a)
or when integrated, and matched to the midlatitude solution at y = Yn:
h + r12hi = h(x, Yn) + r12hi (x, Yn)·
Thus:
hi (x, y) =hi (x, Yn) + [h(x, Yn) - h(x, y)]/r12.
(6.4.18b)
(6.4.19)
The two attempts at specifying hi in terms of h are very crude. The first
(6.4.14) allows no deformation of the interface between the two layers and
therefore no shear in the geostrophic zonal velocity. The second (6.4.19) establishes the other extreme. In (6.4.19) the interface between the two moving
layers moves so much that there is complete compensation of the pressure in
the upper layer so that its geostrophic velocity is zero, and the zonal flow is due
entirely to the west wind drift at the equator forced by the wind stress. If the
stratification between the first two layers is large, i.e., in the limit of very large
r I2 the two specifications of hi become equivalent. This equivalence does not
imply that either of the specifications is correct, and the incomplete character
of the treatment of the upper layer remains a serious weakness of the simple
inertial model of the EUC. However, calculations described below show very
little qualitative difference between the solution for the undercurrent in layer 2
between the two choices, and the structure of the current seems insensitive to
the details of the specification of hi. Attempts to deal with the dynamics of
layer 1 in a more deductive are taken up in Section 6.6.
In either case, once hi is related to h, h2 is also determined in terms of h,
and (6.4.13a,b) becomes a set of two equations in two unknowns ifthe function
Qz(Bz) is known. The determination of the functional relation between potential vorticity and Bernoulli function becomes the central dynamical issue.
The function Bz is constant on streamlines, and therefore the specification
of Qz(Bz) merely identifies the relation of the potential vorticity with its
streamline. For flow in the ventilated portion of the subtropical gyre the potential vorticity is set at the time of subduction. If the flow is conservative that
value of potential vorticity is maintained as the flow enters the domain of the
equatorial flow. The potential vorticity becomes partitioned differently between
its parts near the equator as the contribution of the planetary vorticity becomes
weaker, and the relative vorticity becomes important as the streamline approaches the equator. Similarly, since the pressure field is scaled with the
Coriolis parameter, its contribution to the Bernoulli function becomes weaker
in the equatorial region while the kinetic energy, comparatively negligible at
higher latitudes, becomes significant at the equator. Each streamline remains a
345
8pi = o(UI) « 1.
8y
Uz
( 6.4.17)
If this were the case, then to lowest the order (6.4.12) would imply that:
(6.4.18a)
or when integrated, and matched to the midlatitude solution at y = Yn:
h + r12hi = h(x, Yn) + r12hi (x, Yn)·
Thus:
hi (x, y) =hi (x, Yn) + [h(x, Yn) - h(x, y)]/r12.
(6.4.18b)
(6.4.19)
The two attempts at specifying hi in terms of h are very crude. The first
(6.4.14) allows no deformation of the interface between the two layers and
therefore no shear in the geostrophic zonal velocity. The second (6.4.19) establishes the other extreme. In (6.4.19) the interface between the two moving
layers moves so much that there is complete compensation of the pressure in
the upper layer so that its geostrophic velocity is zero, and the zonal flow is due
entirely to the west wind drift at the equator forced by the wind stress. If the
stratification between the first two layers is large, i.e., in the limit of very large
r I2 the two specifications of hi become equivalent. This equivalence does not
imply that either of the specifications is correct, and the incomplete character
of the treatment of the upper layer remains a serious weakness of the simple
inertial model of the EUC. However, calculations described below show very
little qualitative difference between the solution for the undercurrent in layer 2
between the two choices, and the structure of the current seems insensitive to
the details of the specification of hi. Attempts to deal with the dynamics of
layer 1 in a more deductive are taken up in Section 6.6.
In either case, once hi is related to h, h2 is also determined in terms of h,
and (6.4.13a,b) becomes a set of two equations in two unknowns ifthe function
Qz(Bz) is known. The determination of the functional relation between potential vorticity and Bernoulli function becomes the central dynamical issue.
The function Bz is constant on streamlines, and therefore the specification
of Qz(Bz) merely identifies the relation of the potential vorticity with its
streamline. For flow in the ventilated portion of the subtropical gyre the potential vorticity is set at the time of subduction. If the flow is conservative that
value of potential vorticity is maintained as the flow enters the domain of the
equatorial flow. The potential vorticity becomes partitioned differently between
its parts near the equator as the contribution of the planetary vorticity becomes
weaker, and the relative vorticity becomes important as the streamline approaches the equator. Similarly, since the pressure field is scaled with the
Coriolis parameter, its contribution to the Bernoulli function becomes weaker
in the equatorial region while the kinetic energy, comparatively negligible at
higher latitudes, becomes significant at the equator. Each streamline remains a