# Fi fi T hi

df1 h1V1

2h2 + hrh2h2 + -2-h2 = 2 + hr + y - (f 1» + f2» - f 1 - fi) (4.226)

v2/2 + h1 + hrh2 = G2 (f2) (4.228) subject to the following boundary conditions:

where hr = h2»/h1» is the ratio of the dimensional layer thickness of the interior solution. This system is a well-behaved system and can be solved easily, and continuous solutions can also be found. For example, two sections of a continuous solution are shown in Figure 4.36. In the physical coordinate x, the solution exponentially matches the interior solution at the outer edge of the western boundary. On the other hand, this matching condition appears in the form of a linear function at a finite distance from the western boundary in the streamfunction coordinate. Since the solution of the inertial western boundary has to be sought through a a b a b

Fig. 4.36 Structure of the inertial western boundary current with two moving layers at two sections, including layer thickness, meridional velocity, and streamfunction; a solutions in the physical coordinate x at y = 0.25; b solutions in the streamfunction coordinate at y = 0.55 (Huang, 1990b).

shooting method, using the streamfunction coordinate has a great advantage in numerical calculations.

Matching the inertial western boundary current with the mid-ocean thermocline

### Motivation

We have discussed thermocline structure in the ocean interior and an inertial western boundary current with two moving layers. It has been argued that mixing/dissipation within the southern part of the western boundary regime is negligible, so we would like to match the thermocline solution for the ocean interior with some kind of western boundary current. This is one more step toward constructing a unified picture for the circulation in a closed basin.

### Model formulation

The model ocean (Fig. 4.37) consists of three layers of constant density, and the lowest layer is very thick and assumed motionless. The subtropical gyre interior is divided into three domains that have slightly different dynamics.

The dynamics of the model is basically the same as the classical ventilated thermocline by Luyten et al. (1983). However, we will assume that potential vorticity in the second layer is uniform for all the subducted water. This is not inconsistent with the observation that potential vorticity in the deep part of the Gulf Stream is practically homogenized. Potential vorticity in the uppermost layer is not uniform because it is directly exposed to surface forcing. Our assumption gives rise to a rather elegant solution for the outcropping line, the western boundary of the shadow zone, and layer thickness.

Fig. 4.37 Sketch of a model ocean for a subtropical gyre: a the inertial western boundary current; b the interior ocean with two moving layers, where xo(y) is the outcrop line for the upper layer, and xs(y) is the boundary of the shadow zone for the second layer; c a meridional section of the model ocean with three layers (Huang, 1990c).

Fig. 4.37 Sketch of a model ocean for a subtropical gyre: a the inertial western boundary current; b the interior ocean with two moving layers, where xo(y) is the outcrop line for the upper layer, and xs(y) is the boundary of the shadow zone for the second layer; c a meridional section of the model ocean with three layers (Huang, 1990c).

### Domain I

North of the outcrop line x0 = x0(y), the first layer vanishes, so only the second layer is in motion. The wind stress is treated as a body force for the upper layer, and the basic equations are

(h2U2)x + (h2V2)y = 0 (4.234) where y2 = g(p3 - p2)/p0. Cross-differentiating gives the Sverdrup relation jh2V2 = - / p0 (4.235)

and the layer thickness satisfies h2 = h2 + (V) (xe - x) (4.236) 2 e p0jY2\ f Jy where he is the constant layer thickness along the eastern boundary of the basin. Since potential vorticity is constant along the outcrop line, f /h2 = f0/he, along x0 (y) (4.237)

Accordingly, the outcrop line satisfies xe - x0 (y) = ^fff h2 (4.238)

Domain II

South of x0 (y) and north of xs (y) both layers are in motion, so the basic equations are

-fhivi = -hi [(yi + Y2) hix + Y2h2x] + tx/Po fhiui = -hi [(yi + Y2) hiy + Y2h2y] (hiui)x + (hivi)y = 0 -fh2V2 = -h2 [Y2hlx + Y2h2x] fh2U2 = -h2 [Y2hly + Y2h2y]

where y1 = g(p2 - p1)/p0. After some manipulations, the layer thicknesses satisfy p2

We further assume that the outcrop line xo (y) has a special shape, so that potential vorticity in the second layer is constant after subduction. This assumption is supported by observations that potential vorticity in the Gulf Stream is nearly constant (Iselin, 1940; Huang and Stommel, 1990). Thus, the layer thickness is hi = ^^ (-Y2 f + Al/2 Yi + Y2 V fo where f

Domain III

South of line xs (y), the second layer is stagnant, so only the first layer is in motion. The upper layer thickness satisfies h2i =

PoPYi V f

Since the second layer is stagnant, its thickness satisfies h2 = he - h1 along the western edge of this domain; thus, this boundary is determined by the following equation xe - x5 (y) = poPYih2e[i - f

Domain IV

This is the southern part of the western boundary regime where two moving inertial boundary currents exist, which can be calculated by the streamfunction coordinate transformation discussed in the previous section.