The Buoyancy- and Wind-Driven Subtropical Gyre: Analytical Solutions
317
we obtain:
(1)
Y2
1 aM 1 l
v - - - - - - -
2 - f R cos 0 8 1
WE [ 1 •
1
= ( 1 -f) PH2 }! bdf .
(5.4.41)
In the subtropical gyre WE< 0, and therefore the sign of (5.4.41) depends
on the sign of the cross-isopycnal velocity. If there is heating so that w* > 0,
then b < 0, and the integral is positive. Thus in the whole region of the adiabatic shadow zone, i.e., east of the line h( 0 ) = H2 , the fluid must move northward. This is the eastern branch of the buoyancy-driven recirculation, east of
the critical characteristic and calculated numerically by Luyten and Stommel
and discussed in Section 5.3.
The flow in this region impinges on the boundary of the shaded region
given by the edge of the old shadow zone. Once it crosses this boundary, it
must return southward in the shaded sliver in which our solution does not
apply.
The flow cannot cross the western boundary of the shaded zone, for that
boundary is a streamline of the ventilated flow when buoyancy is included. The
situation is shown schematically in Fig. 5.4.3. The northward flow in the old
shadow zone is slow and O(b) compared to the wind-driven motion in the
ventilated region. It flows northward, and its right boundary is the eastern wall
on which h = H2, so that h(l) is zero there. When it meets the eastern boundary
of the shaded region, as shown by the streamlines, it must return southward.
The western boundary of the shaded sliver also has h = H2 so that the total
depth across any section, i.e., AA', resembles the cross section shown above the
plan view. h has a minimum at the point P which is the intersection of the line
AA' with the eastern edge of the sliver. It then rises to the value H2 on the
western edge of the sliver. The entire geostrophic transport recirculates across
the section, going northward to the east of the point P in the old shadow zone
and flowing southward to the west of the point P within the sliver.
The northward transport in the eastern branch is O(b) since the velocity is
O(b) and the width of the region is 0(1). The return southward flow takes
place in a region of width O(b) and therefore the southward velocity must be
0(1) to balance the transport. Since the velocities are 0(1), the weak heating is
unable at 0( 1) to affect the southward flow in the sliver. Thus in the sliver the
southward flow conserves, to lowest order, potential vorticity so that
f/h2 = Q2(h) in the southward flow. Thus the dynamics in this southward
branch of the recirculation is dynamically similar to the ventilated flow in the
sense that q2 is preserved to lowest order for weak heating. However, the
functional relation between potential vorticity and total depth, h, for this flow
is determined along the eastern boundary of the sliver as the buoyancy-driven
flow impinges on that boundary with known values of hand h2. In particular,
317
we obtain:
(1)
Y2
1 aM 1 l
v - - - - - - -
2 - f R cos 0 8 1
WE [ 1 •
1
= ( 1 -f) PH2 }! bdf .
(5.4.41)
In the subtropical gyre WE< 0, and therefore the sign of (5.4.41) depends
on the sign of the cross-isopycnal velocity. If there is heating so that w* > 0,
then b < 0, and the integral is positive. Thus in the whole region of the adiabatic shadow zone, i.e., east of the line h( 0 ) = H2 , the fluid must move northward. This is the eastern branch of the buoyancy-driven recirculation, east of
the critical characteristic and calculated numerically by Luyten and Stommel
and discussed in Section 5.3.
The flow in this region impinges on the boundary of the shaded region
given by the edge of the old shadow zone. Once it crosses this boundary, it
must return southward in the shaded sliver in which our solution does not
apply.
The flow cannot cross the western boundary of the shaded zone, for that
boundary is a streamline of the ventilated flow when buoyancy is included. The
situation is shown schematically in Fig. 5.4.3. The northward flow in the old
shadow zone is slow and O(b) compared to the wind-driven motion in the
ventilated region. It flows northward, and its right boundary is the eastern wall
on which h = H2, so that h(l) is zero there. When it meets the eastern boundary
of the shaded region, as shown by the streamlines, it must return southward.
The western boundary of the shaded sliver also has h = H2 so that the total
depth across any section, i.e., AA', resembles the cross section shown above the
plan view. h has a minimum at the point P which is the intersection of the line
AA' with the eastern edge of the sliver. It then rises to the value H2 on the
western edge of the sliver. The entire geostrophic transport recirculates across
the section, going northward to the east of the point P in the old shadow zone
and flowing southward to the west of the point P within the sliver.
The northward transport in the eastern branch is O(b) since the velocity is
O(b) and the width of the region is 0(1). The return southward flow takes
place in a region of width O(b) and therefore the southward velocity must be
0(1) to balance the transport. Since the velocities are 0(1), the weak heating is
unable at 0( 1) to affect the southward flow in the sliver. Thus in the sliver the
southward flow conserves, to lowest order, potential vorticity so that
f/h2 = Q2(h) in the southward flow. Thus the dynamics in this southward
branch of the recirculation is dynamically similar to the ventilated flow in the
sense that q2 is preserved to lowest order for weak heating. However, the
functional relation between potential vorticity and total depth, h, for this flow
is determined along the eastern boundary of the sliver as the buoyancy-driven
flow impinges on that boundary with known values of hand h2. In particular,