124
Vertical Structure: Baroclinic Quasi-Geostrophic Models
If we choose fo = 10- 4 s- 1 , f3 = 10- 13 cm- 1 s- 1 , H = 1 km, and Ld = 40 km,
the critical value of Wo is 1.6 x w- 4 cmjs, well within the range of realistic
values of W£. The example thus suggests that the wind forcing present in the
Sverdrup interior may be strong enough to wrap the geostrophic contours
around on themselves and detach them from the eastern boundary, at least in a
part of the region, and allow motion in layers deeper than the one directly
forced by the Ekman pumping. What remains to be seen is what that region is
for more realistic forcing in which the Sverdrup transport requires a western
boundary layer, and how the motion, if it exists within the domain of closed
geostrophic contours, can be determined.
3. 7 Determination of the Recirculation
When the motion is nearly inviscid, the local dynamics is determined by the
conservation of potential vorticity in layer 2. When the geostrophic contours
are attached to the eastern boundary, free geostrophic motion is blocked, and
only weak motion directly forced by the frictional coupling to the upper layer
takes place. In the weak dissipation limit this motion is negligible. However, in
the region of closed geostrophic contours a free geostrophic, recirculating
mode is possible. In the inviscid limit a recirculating motion of arbitrary
amplitude is possible as long as the streamlines coincide with the geostrophic
contours. As long as:
(3.7.1)
the requirements of inviscid dynamics are consistently satisfied. The paths of
the motion have been laid down by the formation of the closed geostrophic
contours, and only a weak forcing, acting persistently over time, is required to
set the fluid in motion and to determine both its amplitude and detailed form
subject, always, to (3.7.1). That weak forcing comes from the weak frictional
coupling of layer 2 to layer 1 through viscous stresses. We can think of the
motion in layer 2 as the resonance of a free, nonlinear, steady mode and its
resonant character therefore requires only a tiny forcing to produce an order 1
motion. We normally think of a resonance producing a large response to an
0(1) forcing. In the oceanic recirculation problem it is instead the production
of an 0( 1) motion by infinitesimally small forcing that is relevant.
The mathematical expression of this comes from the integral of the
vorticity equation over a closed contour of constant ln. We may rewrite
(3.5.10b) as:
(3.7.2)
The dominant term in (3.7.2) is locally the inertial term on the left side. The
condition that it must self-cancel to within the order of the small dissipative
Vertical Structure: Baroclinic Quasi-Geostrophic Models
If we choose fo = 10- 4 s- 1 , f3 = 10- 13 cm- 1 s- 1 , H = 1 km, and Ld = 40 km,
the critical value of Wo is 1.6 x w- 4 cmjs, well within the range of realistic
values of W£. The example thus suggests that the wind forcing present in the
Sverdrup interior may be strong enough to wrap the geostrophic contours
around on themselves and detach them from the eastern boundary, at least in a
part of the region, and allow motion in layers deeper than the one directly
forced by the Ekman pumping. What remains to be seen is what that region is
for more realistic forcing in which the Sverdrup transport requires a western
boundary layer, and how the motion, if it exists within the domain of closed
geostrophic contours, can be determined.
3. 7 Determination of the Recirculation
When the motion is nearly inviscid, the local dynamics is determined by the
conservation of potential vorticity in layer 2. When the geostrophic contours
are attached to the eastern boundary, free geostrophic motion is blocked, and
only weak motion directly forced by the frictional coupling to the upper layer
takes place. In the weak dissipation limit this motion is negligible. However, in
the region of closed geostrophic contours a free geostrophic, recirculating
mode is possible. In the inviscid limit a recirculating motion of arbitrary
amplitude is possible as long as the streamlines coincide with the geostrophic
contours. As long as:
(3.7.1)
the requirements of inviscid dynamics are consistently satisfied. The paths of
the motion have been laid down by the formation of the closed geostrophic
contours, and only a weak forcing, acting persistently over time, is required to
set the fluid in motion and to determine both its amplitude and detailed form
subject, always, to (3.7.1). That weak forcing comes from the weak frictional
coupling of layer 2 to layer 1 through viscous stresses. We can think of the
motion in layer 2 as the resonance of a free, nonlinear, steady mode and its
resonant character therefore requires only a tiny forcing to produce an order 1
motion. We normally think of a resonance producing a large response to an
0(1) forcing. In the oceanic recirculation problem it is instead the production
of an 0( 1) motion by infinitesimally small forcing that is relevant.
The mathematical expression of this comes from the integral of the
vorticity equation over a closed contour of constant ln. We may rewrite
(3.5.10b) as:
(3.7.2)
The dominant term in (3.7.2) is locally the inertial term on the left side. The
condition that it must self-cancel to within the order of the small dissipative