Fig. 4.33 Isolines of potential vorticity in the bottom layer, with the heavy dashed line indicating the outcropping line of the upper layer. Although east of the thin dashed line potential vorticity contours are blocked by the eastern boundary, west of the thin dashed line potential vorticity contours disconnected from the eastern boundary are closed through the western boundary.

boundary, along which geostrophic motion is forbidden. However, there are contours stemming from the western boundary as well. For simplicity, we assume that these contours are closed through the western boundary without being affected by friction. According to the ideal-fluid theory, water is free to travel along these contours.

The essential feature of this vorticity map is that the meridional vorticity gradient has different signs on the two sides of intersecting point P between these two dashed lines, whose location is determined by d n3

Note that near the inter-gyre boundary, yD + h\e & h2e .After some algebraic manipulations, one obtains

4oja2 sin3 60 (6 =6q

Point P is sometimes called the Rossby repellor, where characteristics from the eastern boundary and the western boundary meet; it is a very important singular point in the ther-mocline theory. The characteristic starting from P separates the basin into two regions: the eastern region where vorticity contours start from the eastern boundary and geostrophic motion is forbidden, and the western region where vorticity contours start from the western boundary and fluid is free to travel along these contours, assuming no friction in the western boundary current - an idealization widely used to construct a solution in the ocean interior in ideal-fluid thermocline theory.

Note that the existence of the western region depends on the smallness of h2e and h3e and strong forcing we. If the layers are too thick and forcing is not so strong, P would be located to the west of the western wall, so there would be no closed geostrophic contours. This is the case which has been explored many times before. It is also interesting to observe that for a given forcing we and h2e, we can always choose a h3e so small that the Rossby repellor is located within the basin interior. Therefore, closed potential vorticity contours always exist if the layer thicknesses in the model are chosen properly.

Potential vorticity homogenizationfor the pool regime and unventilated layers The pool regime

The pool region for a ventilated layer is defined as the regime where streamlines start from the outer edge of the western boundary, instead of the outcrop line. Thus, potential vorticity in the pool regime cannot be determined by tracing backward along the streamline to the outcrop line. A more accurate approach would be to include the complicated dynamical processes in the western boundary regime. The dynamics within the pool regime was left untouched in the original ventilated thermocline model by Luyten et al. (1983). However, a 9! = 35o b 9! = 39.2o a 9! = 35o b 9! = 39.2o

Fig. 4.34 The expansion of the shadow zone (thin dashed line is its boundary) and the pool region (west of the solid line) obtained from the ventilated thermocline model with two moving layers, as the outcrop line (thick dashed line) moves northward. The northern boundary of the model is set at 50° N, but the first outcrop line &i is gradually moved toward the northern boundary.

Fig. 4.34 The expansion of the shadow zone (thin dashed line is its boundary) and the pool region (west of the solid line) obtained from the ventilated thermocline model with two moving layers, as the outcrop line (thick dashed line) moves northward. The northern boundary of the model is set at 50° N, but the first outcrop line &i is gradually moved toward the northern boundary.

it can be shown easily that as the outcrop line moves toward the inter-gyre boundary, the area occupied by the pool region also increases quickly, as shown in Figure 4.34. For the case with 01 = 48.2°, the pool regime occupies most of the basin, and we cannot ignore this vast regime. To obtain a dynamically consistent solution for the whole basin, it is thus desirable to include the dynamical theory of the pool regime.

The common feature for the pool regime in the ventilated layer and the regime within the closed geostrophic contours in the unventilated layer is that streamlines in these regimes do not start from the outcrop lines. In the second case, streamlines are closed on their own, and we can extend the potential vorticity homogenization theory postulated by Rhines and Young (1982a, b). In the first case, streamlines are actually started from the outer edge of the western boundary. A comprehensive treatment of such a problem is difficult. As a compromise, one can assume that potential vorticity in the pool regime obeys certain laws. Two of the most frequently used assumptions are the following: first, one can leave the solution in the pool regime undetermined; second, one can assume that potential vorticity in the pool regime is also homogenized very much like it is in the case of the unventilated thermocline.

In addition, the pool region in the western part of the basin can also be treated as a ventilated pool, i.e., all the water in this pool is ventilated water from the uppermost layer which is directly in contact with surface forcing (e.g., Dewar, 1986; Dewar et al., 2005). None of these approaches is perfect, but there are no simple alternatives for such a difficult problem.

The potential vorticity homogenization theory presented below follows the approach by Pedlosky (1996). For the case of a model with three moving layers, as shown in Figure 4.28, the potential vorticity balance in the third layer can be written as

where D3 is the vorticity dissipation term. Integrating Eqn. (4.197) over a closed area gives

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