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Buoyancy Forced Circulation and Cross-Gyre Flow
5.4 The Buoyancy- and Wind-Driven Subtropical Gyre:
Analytical Solutions
Solution in the Ventilated Region
Stimulated by the numerical calculations of Luyten and Stommel (1986b),
Pedlosky (1986) described an analytical approach to the combined wind- and
buoyancy-driven circulation. The approach has the added advantage of being
generalizable to a system of more than two layers, but for simplicity the present
discussion is limited to the same two-layer model described in the last section.
The governing system of equations is the Sverdrup balance (5.3.3) and the
potential vorticity equation written in terms of the layer thicknesses, for example, (5.3.8).
When the cross-isopycnal velocity is zero, the solution in the region south
of the outcrop line in the ventilated region is given by (4.4.14a,b) in which each
individual layer thickness is given in terms of the total depth, h, in the form:
hi = Cl>(f)h
hz = (1- Cl>(f))h.
(5.4.1a,b)
In the case of adiabatic motion for which the cross-isopycnal flux is zero the
function Cl>(f) is determined at the outcrop line and is given by (4.4.14a,b):
Cl>(f) = ( 1 - f;) for w. = 0.
(5.4.2)
We can generalize this approach to the case when w. -1- 0 in the following
way. Suppose we adopt (5.4.1) as a trial solution and insert it into the Sverdrup
balance (5.3.3) and the governing partial differential equation for hz, (5.3.8).
Note that HI = 0 and H = Hz. Thus (5.3.3) yields:
hz [ 1 + ~: Cl>(f)z] = D~ + H~
(5.4.3)
or:
h z_ D~+H~
-
z
1+r12Cl>
( 5.4.4)
which should be compared with (4.4.18). In (5.4.4) the notation r12 = yifyz is
again used.
To determine Cl> the proposed solution (5.4.1) should be inserted into
(5.3.8). If we use the relation:
8h 8hz 8h 8hz - h 8h dCl>
8e 8cfJ - 8cfJ 7iii - 8cfJ 8e
(5.4.5)