**4. Results – details of entropy production in the model**

We describe here the details of entropy production in the model from the final state of the spin-up experiment (Fig. 4(f)). Because entropy production due to the salt transport is negligible (Shimokawa and Ozawa, 2001), local entropy production can be estimated from the first term in equation (6) as

$$A = \frac{\rho \mathbb{C}}{T^2} (A\_x + A\_y + A\_z),\\ A\_x = D\_h (\frac{\mathbb{d}T}{\mathbb{d}x})^2,\\ A\_y = D\_h (\frac{\mathbb{d}T}{\mathbb{d}y})^2,\\ A\_z = D\_v (\frac{\mathbb{d}T}{\mathbb{d}z})^2,\tag{7}$$

where *Dh* denotes horizontal diffusivity of 103 m2 s–1, *Dv* stands for vertical diffusivity of 10–4 m2 s–1 (see section 2), and other notation is the same as that used earlier in the text. It is assumed here that *Fh* = –*k* grad(*T*) = –*ρcDE* grad(T), where *k* = *ρcDE* signifies thermal conductivity and where *DE* represents the eddy diffusivity (*Dh* or *Dv*). Figure 5 shows zonal, depth and zonal-depth averages of each term in equation (7). The quantities not multiplied by d*V* represent the values at the site, and the quantities multiplied by d*V* represent the values including the effect of layer thickness.

It is apparent from the zonal average of *A* (Fig. 5(a)) that entropy production is large in shallow–intermediate layers at low latitudes. This is apparent also in the zonal-depth average of *A*×d*V* (Fig. 5(c)). However, it is apparent from the depth average of *A*×d*V* (Fig. 5(b)) that entropy production is large at the western boundaries at mid-latitudes and at low latitudes. Consequently, entropy production is greatest at the western boundaries at mid-latitudes as the depth average, but it is highest at low latitudes as the depth-zonal average. It is apparent as the figures show of *Ax*, *Ay* and *Az* (Figs. 5(d), (g) and (j)) that *Ax* is large in shallow layers at mid-latitudes, *Ay* is large in shallow-intermediate layers at high latitudes, and that *Az* is large in shallow-intermediate layers at low latitudes. It is also apparent that as the figures show of *Ax*×d*V*, *Ay*×d*V* and *Az*×d*V* (Figs. 5(e), 5(f), 5(h), 5(i), 5(k) and 5(l)) that *Ax*×d*V* is large at the western boundaries at mid-latitudes, *Ay*×d*V* is large at high latitudes, and *Az*×d*V* is large at low latitudes. Additionally, it is apparent that the values of *Az* (*Az*×d*V*) is the largest, and those of *Ax* (*Ax*×d*V*) are smaller than those of *Ay* (*Ay*×d*V*) and *Az* (*Az*×d*V*).

Consequently, there are three regions with large entropy production: shallow-intermediate layers at low latitudes, shallow layers at the western boundaries at mid-latitudes, and shallow-intermediate layers at high latitudes. It can be assumed that the contribution of shallow-intermediate layers at low latitudes results from the equatorial current system. That of western boundaries at mid-latitudes results from the western boundary currents such as Kuroshio, and that of intermediate layers at high latitudes results from the meridional circulation of the global ocean. It is apparent that high dissipation regions at low latitudes expand into the intermediate layer in the zonal averages of *A*×d*V* and *Az*×d*V*. These features appear to indicate that equatorial undercurrents and intermediate currents in the equatorial current system are very deep and strong currents which can not be seen at other latitudes (Colling, 2001). It is also apparent that high dissipation regions at high latitudes in the northern hemisphere intrude into the intermediate layer in the zonal averages of *A*×d*V* and *Ay*×d*V*, and the peak of northern hemisphere is larger than that of southern hemisphere in the zonal-depth averages of *A* and *Ay*. These features appear to represent the characteristics of the circulation with northern sinking (Fig. 4(f)).

Thermodynamics of the Oceanic General Circulation –

Fig. 5. (continued)

right side of each figure.

not a heat engine.

Is the Abyssal Circulation a Heat Engine or a Mechanical Pump? 157

(a) zonal average of *A*, (b) depth average of *A*×d*V*, (c) zonal-depth average of *A*×d*V*, (d) zonal average of *Ax*, (e) depth average of *Ax*×d*V*, (f) zonal-depth average of *Ax*×d*V*, (g) zonal average of *Ay*, (h) depth average of *Ay*×d*V*, (i) zonal-depth average of *Ay*×d*V*, (j) zonal average of *Az*, (k) depth average of *Az*×d*V*, (l) zonal-depth average of *Az*×d*V*

**5. Discussion – Sandström theorem and abyssal circulation** 

The unit for *A* is W K-1 m-3. The unit for *A*×d*V* is W K-1. The unit for *Ax*, *Ay*, and *Az* is K2 s-1. The unit for *Ax*×d*V*, *Ay*×d*V*, and *Az*×d*V* is K2 s-1m3. The contour interval is indicated at the

As stated in section 1.5, Sandström suggested that a closed steady circulation can only be maintained in the ocean if the heating source is located at a higher pressure (i.e. a lower level) than that of the cooling source. Therefore, he suggested that the oceanic circulation is

Huang (1999) showed using an idealized tube model and scaling analysis that when the heating source is at a level that is higher than the cooling source such as the real ocean, the circulation is mixing controlled, and in the contrary case, the circulation is frictioncontrolled. He also suggested that, within realistic parameter regimes, the circulation requires external sources of mechanical energy to support mixing to maintain basic stratification. Consequently, oceanic circulation is only a heat conveyer, not a heat engine. Yamagata (1996) reported that the oceanic circulation can be driven steadily as a heat engine only with great difficulty, considering the fact that the efficiency as a heat engine of the

Strictly speaking, we should consider dissipation in a mixed layer and dissipation by convective adjustment for entropy production in the model. Dissipation in a mixed layer can be estimated from the first term in (6) as

$$B = \frac{\rho \mathbf{C}}{T^2} \frac{\{T\_r \text{ - } T\_s\text{ }\}}{\Delta t\_r} \text{ },\tag{8}$$

where *T*r signifies restoring temperature (Fig. 3(c)), *T*s is the sea surface temperature in the model, and Δ*t*r stands for the relaxation time of 20 days (see section 2). It is assumed here that *Fh* = –*k* grad(*T*) = – *ρcDM* grad(*T*), where *k* = *ρcDM* is thermal conductivity, DM = Δ*z*<sup>r</sup> 2 /Δ*t*<sup>r</sup> represents diffusivity in the mixed layer, and Δ*z*r is the mixed layer thickness of 25 m (see section 2). The estimated value of *B* is lower than that of *A* by three or four orders: it is negligible. Dissipation by convective adjustment can be estimated from the first term in (5) such that

$$\mathbf{C} = \frac{\rho \mathbf{C}}{T\_b} \frac{(T\_b \ - T\_a)}{\Delta t} \,\mathrm{\,\,\,\mathrm{\,\,\,} \tag{9}$$

where *Tb* is the temperature before convective adjustment, *Ta* is the temperature after convective adjustment, and Δ*t* is the time step of 5400 s (see section 2). In fact, *Tb* is identical to *Ta* at the site where convective adjustment has not occurred. The value of *C* is negligible because the effect of convective adjustment is small in the steady state.

Fig. 5. Entropy production in the model.

Fig. 5. (continued)

Strictly speaking, we should consider dissipation in a mixed layer and dissipation by convective adjustment for entropy production in the model. Dissipation in a mixed layer can

> ( *r s <sup>2</sup> <sup>r</sup> <sup>ρ</sup><sup>C</sup> T -T <sup>B</sup> <sup>T</sup> <sup>Δ</sup><sup>t</sup>*

where *T*r signifies restoring temperature (Fig. 3(c)), *T*s is the sea surface temperature in the model, and Δ*t*r stands for the relaxation time of 20 days (see section 2). It is assumed here that *Fh* = –*k* grad(*T*) = – *ρcDM* grad(*T*), where *k* = *ρcDM* is thermal conductivity, DM = Δ*z*<sup>r</sup>

represents diffusivity in the mixed layer, and Δ*z*r is the mixed layer thickness of 25 m (see section 2). The estimated value of *B* is lower than that of *A* by three or four orders: it is negligible. Dissipation by convective adjustment can be estimated from the first term in (5)

> *b <sup>ρ</sup><sup>C</sup> T -T <sup>C</sup>*

because the effect of convective adjustment is small in the steady state.

where *Tb* is the temperature before convective adjustment, *Ta* is the temperature after convective adjustment, and Δ*t* is the time step of 5400 s (see section 2). In fact, *Tb* is identical to *Ta* at the site where convective adjustment has not occurred. The value of *C* is negligible

( ) *b a*

)

, (8)

*<sup>T</sup> <sup>Δ</sup><sup>t</sup>* , (9)

2 /Δ*t*<sup>r</sup>

be estimated from the first term in (6) as

Fig. 5. Entropy production in the model.

such that

(a) zonal average of *A*, (b) depth average of *A*×d*V*, (c) zonal-depth average of *A*×d*V*, (d) zonal average of *Ax*, (e) depth average of *Ax*×d*V*, (f) zonal-depth average of *Ax*×d*V*, (g) zonal average of *Ay*, (h) depth average of *Ay*×d*V*, (i) zonal-depth average of *Ay*×d*V*, (j) zonal average of *Az*, (k) depth average of *Az*×d*V*, (l) zonal-depth average of *Az*×d*V* The unit for *A* is W K-1 m-3. The unit for *A*×d*V* is W K-1. The unit for *Ax*, *Ay*, and *Az* is K2 s-1. The unit for *Ax*×d*V*, *Ay*×d*V*, and *Az*×d*V* is K2 s-1m3. The contour interval is indicated at the right side of each figure.
