Fig. 5.139 Overturning circulation driven by horizontal differential heating applied to the upper boundary, Pr = 8, Ra = 5 x 108, and aspect ratio a = 1 (courtesy of Liang Sun).

Fig. 5.139 Overturning circulation driven by horizontal differential heating applied to the upper boundary, Pr = 8, Ra = 5 x 108, and aspect ratio a = 1 (courtesy of Liang Sun).

107 108 109

Fig. 5.140 The cell thickness as a function of the Rayleigh number (courtesy of Liang Sun).

107 108 109

Fig. 5.140 The cell thickness as a function of the Rayleigh number (courtesy of Liang Sun).

The numerical study of flow forced by horizontal differential heating in such a two-dimensional tank is much easier than laboratory experiments, and can provide very useful insights into the physics of the thermal circulation. However, some essential differences exist between such a model and the circulation in the oceans.

First and foremost is the huge difference in Rayleigh number between the two. As the Rayleigh number increases, the flow may go through transitions from one dynamical regime to the next. For example, W. Wang and Huang (2005) found a regime transition around Ra — 5 x 108. Other transition regimes may exist at much higher Rayleigh numbers. For example, Roche et al. (2001) reported a transition in the classical Rayleigh-Benard convection through experiments with low-temperature liquid helium in the vicinity of Ra ~ 1012. Whether the flow under the horizontal differential heating has similar transition remains unknown. Assuming that the ocean basin has a horizontal dimension on the order of 107 m, the corresponding Rayleigh number based on the horizontal dimension is on the order of 1030. Numerical simulation of turbulence with such a Rayleigh number remains a major challenge. Furthermore, even if we could carry out physical experiments with such a huge dimension, the character of the resulting flow might be completely different from what we have seen under the low range of Rayleigh number.

Second, many of the physical processes that exist in the oceans have been excluded from such a two-dimensional model, such as rotation and external sources of mechanical energy that support a diffusivity over 100 times stronger than the molecular diffusivity. Therefore, the results obtained from such two-dimensional model must be interpreted with caution.

Zonally averaged model for the thermohaline circulation

The description of complicated three-dimensional circulation in the ocean can be greatly simplified and studied if a zonally averaged two-dimensional numerical model is utilized. Consider an ocean of width L and depth H; the steady circulation is represented by the following zonally averaged momentum equations:

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