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If we are interested only in the long-term stability of the model, only the first two modes, sin 0 and cos 0, are required. For example, the corresponding equations for the

VSF loop are

£2 = — a1 a i = Ub1 — k ai b 1 = Ua1 — k b1 — X/2 (5.254)

Using a linear transformation, we can rewrite Eqn. (5.254) as a simple set of ordinary differential equations, the water-wheel equation (Huang and Dewar, 1996):

This system is very similar to the well-known Lorenz equation:

If k = 1, the water-wheel equation and the VSF loop model under the present simple forcing are equivalent to the Lorenz system with b = 1. The model's behavior is very similar to that of the well-known Lorenz model (Sparrow, 1982) (Figs. 5.136 and 5.137).

Both the water-wheel model and the loop model are idealizations of the much more complicated three-dimensional thermohaline circulation in the oceans. Thus, the above described chaotic behavior of such simplified one-dimensional models suggests that thermohaline circulation in the world's oceans is as complicated as the chaotic motions in the atmospheric circulation.

5.4.4 Two-dimensional thermohaline circulation

Thermal circulation driven by horizontal differential heating The next level of complexity in the simulation of the thermohaline circulation is the thermally forced circulation in a two-dimensional rectangular tank. The laboratory study of thermal circulation in a quasi-two-dimensional tank has been discussed in Chapter 3 in connection with the energetics of the oceanic circulation. In this section, we focus on the numerical study of the two-dimensional thermohaline circulation.

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