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.
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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