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is the relative vorticity. The solution is subject to boundary condition f = 0 along the boundaries of the basin.

In order to find solutions in analytical form, the wind stress is assumed to have a special form x w x y y w x y

Thus, the wind-stress curl averaged over the whole layer is

Introduce the following nondimensional variables:

H p then the nondimensional vorticity equation, after dropping primes, is

where two small parameters are introduced w R

Steady, linear solution in the ocean interior

In this case, we drop the time-dependent term and the nonlinear advection terms, and the vorticity equation is reduced to sVff + fx = - sin x sin y (4.364)

The corresponding solution is f = 1+4^ {2e si"x + cosx + ^nu^T^ [(* + e^2) e°1x - (l + e*»1) eD2x] ) siny where

2s 2s

Numerical solutions

The nonlinear solutions can be found by a perturbation method when Ro is very small. By expanding in terms of a Taylor series in Ro, we can find the lowest-order solution with Ro = 0 in analytical forms. However, the general and more accurate solutions have to be calculated through numerical integration.

As the inertial term becomes more important, the circulation develops a recirculation region in the northwest corner. However, for the case of very strong nonlinearity, the circulation becomes virtually symmetric in the east-west direction, and there is no longer the western intensification, described in terms of the inertial run-away problem discussed in Section 4.1.5. As discussed in previous sections, pushing the shallow-water model to the limit of strong forcing and weak dissipation, but without considering other vital dynamical effects such as the outcrop phenomenon or the formation of closed geostrophic contours in the subsurface layers, may not be a good way to simulate the oceanic circulation.

Solving a two-layer quasi-geostrophic model for the wind-driven circulation, we can obtain solutions that are characterized by the existence of strong recirculation in the northwestern corner of the model basin. An example is shown in Figure 4.65.

In order to examine the dynamics of recirculation, we will focus on a box in the northwestern corner of the model basin (—L/a < x < L/a, —L < y < L) where a is the aspect ratio of the model basin. The quasi-geostrophic vorticity equation is a balance between advection and dissipation

4.4.4 Potential vorticity homogenization applied to recirculation

Numerical experiment

A barotropic model in a ß -plane box

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