The Buoyancy- and Wind-Driven Subtropical Gyre: Analytical Solutions
313
The solution (5.4.4) is valid only west of the boundary given by (5.4.14),
the region which in the adiabatic solution is the ventilated zone. To complete
the solution we must find the solution in the eastern unventilated region.
The Solution in the Unventilated Region
There is no analytical solution currently available for the region east of the
critical streamline given by (5.4.14), but analytical progress of a revealing sort
can be made by considering perturbation solutions for small b, i.e., for cases in
which the buoyancy forcing is small compared to the wind driving. Normally
such perturbation solutions which give rise to small changes in the solution,
contain little of qualitative interest. However, in the region of the old shadow
zone the small velocities that are driven in layer 2 by the heating comprise the
only circulation that exists there since the region is at rest for adiabatic motion.
Therefore a perturbation solution around the state of rest in layer 2 is of
considerable qualitative interest.
In the ventilated zone the solution for is of the form:
f
= 1--+0(b)
/2
( 5.4.23)
and as we noted this leads to a westward shift of O(b) in the eastern boundary
of the ventilated zone. For b = 0 the velocity in layer 2 is zero in the unventilated region and the solution in the eastern region is given by the adiabatic
shadow zone solution:
[
]
1/2
h(O) _ Y2 D2
I
-
0
'
Y1
(5.4.24)
where the superscript (0) refers to the first term in an implicit expansion of the
solution in a parameter which measures the magnitude of b. The potential
vorticity in layer 2 associated with this 0(1) solution is:
( 5.4.25)
The contours of the lowest order potential vorticity are shown schematically in Fig. 5.4.2.
The contours are shown by the dashed lines in the figure. Also shown is the
boundary of the shadow zone when b = 0. This is the curve for which
h(O) = H 2 • The isolines of q!fjl cover the region between this curve and the
eastern boundary. The effect of heating shifts the eastern boundary of the
ventilated zone westward to the curve on which h = H 2 . The narrow space
between them has a width of O(b) and is not reached by q~ 0 ) isolines of the
family emanating from the eastern wall. Each of these curves is identified by its
313
The solution (5.4.4) is valid only west of the boundary given by (5.4.14),
the region which in the adiabatic solution is the ventilated zone. To complete
the solution we must find the solution in the eastern unventilated region.
The Solution in the Unventilated Region
There is no analytical solution currently available for the region east of the
critical streamline given by (5.4.14), but analytical progress of a revealing sort
can be made by considering perturbation solutions for small b, i.e., for cases in
which the buoyancy forcing is small compared to the wind driving. Normally
such perturbation solutions which give rise to small changes in the solution,
contain little of qualitative interest. However, in the region of the old shadow
zone the small velocities that are driven in layer 2 by the heating comprise the
only circulation that exists there since the region is at rest for adiabatic motion.
Therefore a perturbation solution around the state of rest in layer 2 is of
considerable qualitative interest.
In the ventilated zone the solution for
f
/2
( 5.4.23)
and as we noted this leads to a westward shift of O(b) in the eastern boundary
of the ventilated zone. For b = 0 the velocity in layer 2 is zero in the unventilated region and the solution in the eastern region is given by the adiabatic
shadow zone solution:
[
]
1/2
h(O) _ Y2 D2
I
-
0
'
Y1
(5.4.24)
where the superscript (0) refers to the first term in an implicit expansion of the
solution in a parameter which measures the magnitude of b. The potential
vorticity in layer 2 associated with this 0(1) solution is:
( 5.4.25)
The contours of the lowest order potential vorticity are shown schematically in Fig. 5.4.2.
The contours are shown by the dashed lines in the figure. Also shown is the
boundary of the shadow zone when b = 0. This is the curve for which
h(O) = H 2 • The isolines of q!fjl cover the region between this curve and the
eastern boundary. The effect of heating shifts the eastern boundary of the
ventilated zone westward to the curve on which h = H 2 . The narrow space
between them has a width of O(b) and is not reached by q~ 0 ) isolines of the
family emanating from the eastern wall. Each of these curves is identified by its