Fig. 3.3 Efficiency of the tube model: dependency of the circulation rate (U, in 10-7 m/s) on mixing coefficient k (in 10-4 m2/s). Numbers on the left-hand side of each panel indicate the nondimensional relative position d of heating/cooling source; numbers in the right-hand side of each panel indicate the non-dimensional frictional parameter r (Huang, 1999).

Fig. 3.3 Efficiency of the tube model: dependency of the circulation rate (U, in 10-7 m/s) on mixing coefficient k (in 10-4 m2/s). Numbers on the left-hand side of each panel indicate the nondimensional relative position d of heating/cooling source; numbers in the right-hand side of each panel indicate the non-dimensional frictional parameter r (Huang, 1999).

the circulation is almost linearly proportional to the mixing coefficient k but is insensitive to the friction parameter r. Thus, the circulation is mixing-controlled, and resembles in some respects the meridional circulation in the ocean.

To obtain a circulation on the order of 10-7 m/s, we need a mixing coefficient on the order of 10-4 m2/s. When mixing is very weak, on the order of 10-7 m2/s (molecular diffusion), the nondimensional velocity will be on the order of 10-3 (10-10 m/s or 3 mm/yr, dimensionally), which is very difficult to observe. Thus, we can say that although both Sandstrom's theorem and Jeffreys' argument are correct on their own terms, both of their statements are incomplete and inaccurate in some way.

For the cases of d > 0.5 (where the cooling source is higher than the heating source) shown in the right-hand panel of Figure 3.3, the circulation is insensitive to the diffusivity. Here again, our discussion is focused on the realistic range of diffusivity in the oceans, k < 10-4 m2/s. When the value of k varies 1,000 times, the corresponding circulation rate changes only slightly. However, the circulation rate is very sensitive to the friction parameter r. For example, an increase of one order of magnitude of r will give rise to the same order of change in the circulation. Thus, we can say that when the cooling source is at a level higher than the heating source, the circulation is frictionally controlled.

3.2.2 Where does Sandstrom's theorem stand?

Pure thermally driven circulation

A closer examination shows that circulation driven solely by thermal forcing can be classified into the following three types.

• Type 1: The heating source is located at a pressure level higher than the cooling source; it is well known that in this case there is a strong circulation. The Rayleigh-Benard thermal convection discussed in many textbooks belongs to this category.

• Type 2: The heating source is located at a pressure level lower than the cooling source. Jeffreys (1925) argued that wherever there is a horizontal density difference, there should be circulation; but he did not state how fast the circulation is. Although Type 2 thermal circulation can be weak, it is detectable, as will be discussed shortly.

• Type 3: Heating and cooling sources located at the same pressure level. This is also referred to as horizontal differential heating or horizontal convection. This type of heating/cooling resembles the situation in the ocean, where heating/cooling takes place primarily at the upper surface, neglecting the penetration of solar radiation and geothermal heating. This type of thermal circulation is very weak, as will be seen shortly.

Paparella and Young (2002) discussed horizontal convection in a rectangular model ocean based on the Boussinesq approximations. Their approach can be extended to the case of flow governed by two-dimensional, non-Boussinesq equations, including conservation of mass, momentum, and thermal energy, plus the equation of state (Wang and Huang, 2005). The basic equations (the continuity equation, the momentum equations, the thermal energy equation, and the equation of state) are:

DT 2

where p (p0) is density (mean density) of the fluid, g is gravitational acceleration, T is temperature, T0 is a constant reference temperature, p is pressure, cp is the specific heat, a is the thermal expansion coefficient, and $ is the dissipation function

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