78
Homogeneous Models of the Ocean Circulation
8 3 ¢
8¢
Er¢
UI 8x3 + f3 ax = AH 8x4
(2.13.2)
where use has been made of the inequality 8¢j8x » 8¢j8y to simplify the
equation. U1 is the Sverdrup zonal flow. In the limit of interest, ?h > [JM,
approximate wavelike solutions of (2.13.2) can be found in the form:
¢ = (y) cos(kx) exp( -xj £)
where the wavelength, 2, is given by the standard formula (Pedlosky 1987) for
the stationary Rossby wave:
(
)
1/2
). = 2: = 2n ~~
(2.13.3)
and the frictional decay scale is given by:
(2.13.4)
Moore and later Pedlosky (1987) suggested this as an approximation to the
western boundary layer structure in the region of eastward Sverdrup flow,
U1 > 0. However, there are severe deficiencies with this picture as a description
of the western boundary layer. Most clearly, the Moore solution is only a
partial approximation to the full problem. Even were it valid, it would apply
only in the transition zone for the boundary layer as it merges into the interior.
Nothing in the Moore solution recognizes either the dynamics deep within the
layer or the boundary condition that must be satisfied at the western boundary.
Why should such a wavelike solution be manifest in the experiments only in the
case of no-slip conditions if the Moore argument has general validity as a
picture of the boundary layer? More devastating are the results of Ierley and
Ruehr (described in Section 2.9) that showed that no boundary layer can exist
in an outflow region of a flow this nonlinear. No boundary-layer structure
matching the interior with a damped wavelike structure (or any other) was
found to be possible. Thus, the Moore picture should probably be regarded less
as a model of the boundary layer in the northern part of the gyre and more in
the nature of a description of a dynamical regime of its own, existing in the
transition zone between the small recirculation gyre in the northwest corner of
the flow (which does not satisfy the fundamental boundary-layer approximation, "d¢/"dx ~ "d¢/"dy) and the region of the Sverdrup interior. We
can speculate that it shows up only in the no-slip calculation because only in
this case is the recirculation limited enough, for these parameter values, to
allow a large enough region of Sverdrup interior zonal flow to support the
stationary Rossby wave. The strong recirculations evident in the three other
examples of Fig. 2.12. 7 leave little room for a Ross by wave field in the northern
part of the gyre.
Homogeneous Models of the Ocean Circulation
8 3 ¢
8¢
Er¢
UI 8x3 + f3 ax = AH 8x4
(2.13.2)
where use has been made of the inequality 8¢j8x » 8¢j8y to simplify the
equation. U1 is the Sverdrup zonal flow. In the limit of interest, ?h > [JM,
approximate wavelike solutions of (2.13.2) can be found in the form:
¢ = (y) cos(kx) exp( -xj £)
where the wavelength, 2, is given by the standard formula (Pedlosky 1987) for
the stationary Rossby wave:
(
)
1/2
). = 2: = 2n ~~
(2.13.3)
and the frictional decay scale is given by:
(2.13.4)
Moore and later Pedlosky (1987) suggested this as an approximation to the
western boundary layer structure in the region of eastward Sverdrup flow,
U1 > 0. However, there are severe deficiencies with this picture as a description
of the western boundary layer. Most clearly, the Moore solution is only a
partial approximation to the full problem. Even were it valid, it would apply
only in the transition zone for the boundary layer as it merges into the interior.
Nothing in the Moore solution recognizes either the dynamics deep within the
layer or the boundary condition that must be satisfied at the western boundary.
Why should such a wavelike solution be manifest in the experiments only in the
case of no-slip conditions if the Moore argument has general validity as a
picture of the boundary layer? More devastating are the results of Ierley and
Ruehr (described in Section 2.9) that showed that no boundary layer can exist
in an outflow region of a flow this nonlinear. No boundary-layer structure
matching the interior with a damped wavelike structure (or any other) was
found to be possible. Thus, the Moore picture should probably be regarded less
as a model of the boundary layer in the northern part of the gyre and more in
the nature of a description of a dynamical regime of its own, existing in the
transition zone between the small recirculation gyre in the northwest corner of
the flow (which does not satisfy the fundamental boundary-layer approximation, "d¢/"dx ~ "d¢/"dy) and the region of the Sverdrup interior. We
can speculate that it shows up only in the no-slip calculation because only in
this case is the recirculation limited enough, for these parameter values, to
allow a large enough region of Sverdrup interior zonal flow to support the
stationary Rossby wave. The strong recirculations evident in the three other
examples of Fig. 2.12. 7 leave little room for a Ross by wave field in the northern
part of the gyre.