**1.5 Sandström theorem**

Related to a closed steady circulation such as abyssal circulation, there is an important thermodynamic postulate: Sandström's theorem (Sandström, 1908, 1916)1.

Sandström considered the system moving as a cycle of the heat engine with the following four stages (see Fig. 2).


When the system moves anti-clockwise (expansion in stage 2 and contraction in stage 4), i.e., the heating source (d>0; α is a specific volume that is equal to the volume divided by the mass) is located at the high-pressure side and the cooling source (d<0) is located at the low-pressure side (Fig. 2a; *P*heating > *P*cooling), the work done by the system is positive:

d 0. *P α* (2)

In contrast, when the system moves clockwise (contraction in stage 2 and expansion in stage 4), i.e., the cooling source is located at the high-pressure side and the heating source is located at the low-pressure side (Fig. 2b; *P*heating < *P*cooling). Therefore, the work done by the system is negative:

$$
\oint Pdx \ll 0.\tag{3}
$$

Consequently, 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 the cooling source.

Regarding the atmosphere, the heating source is located at the ground surface and the cooling source is located at the upper levels because the atmosphere is almost transparent to shortwave radiation of the sun, which heats the ground surface directly. Then heat is transferred from the heated surface by vertical convection. Therefore, the atmosphere can be regarded as a heat engine.

<sup>1</sup> An English translation of Sandström (1906) is available as an appendix in Kuhlbrodt (2008), but the Sandström papers are written in German, and are not easy to obtain. Other explanations of Sandström's theorem can be found in some textbooks of oceanic and atmospheric sciences: Defunt (1961), Hougthon (2002), and Huang (2010).

up water from the deep layer to the surface is needed, provided that sinking can occur in the cold saline (i.e. dense) region of the North Atlantic. Drake Passage is located in the region of westerly wind band where water upwells from below to feed the diverging surface flow. Because net poleward flow above the ridges is prohibited (there is no east–west side wall to sustain an east–west pressure gradient in the Antarctic circumpolar current region), the upwelled water must come from below the ridges, i.e., from depths below 1500–2000 m. In addition, very little mixing energy is necessary to upwell water because of weak

Related to a closed steady circulation such as abyssal circulation, there is an important

Sandström considered the system moving as a cycle of the heat engine with the following

2. Adiabatic change (expansion or contraction) from the heating source to the cooling

4. Adiabatic change (contraction or expansion) from the cooling source to the heating

When the system moves anti-clockwise (expansion in stage 2 and contraction in stage 4), i.e.,

 d 0. *P α* (2) In contrast, when the system moves clockwise (contraction in stage 2 and expansion in stage 4), i.e., the cooling source is located at the high-pressure side and the heating source is located at the low-pressure side (Fig. 2b; *P*heating < *P*cooling). Therefore, the work done by the

 d 0. *P α* (3) Consequently, 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 the

Regarding the atmosphere, the heating source is located at the ground surface and the cooling source is located at the upper levels because the atmosphere is almost transparent to shortwave radiation of the sun, which heats the ground surface directly. Then heat is transferred from the heated surface by vertical convection. Therefore, the atmosphere can be

1 An English translation of Sandström (1906) is available as an appendix in Kuhlbrodt (2008), but the Sandström papers are written in German, and are not easy to obtain. Other explanations of Sandström's theorem can be found in some textbooks of oceanic and atmospheric sciences: Defunt (1961), Hougthon

low-pressure side (Fig. 2a; *P*heating > *P*cooling), the work done by the system is positive:

>0; α is a specific volume that is equal to the volume divided by the

<0) is located at the

thermodynamic postulate: Sandström's theorem (Sandström, 1908, 1916)1.

1. Expansion by diabatic heating under constant pressure

3. Contraction by diabatic cooling under constant pressure

mass) is located at the high-pressure side and the cooling source (d

stratification near Antarctica.

**1.5 Sandström theorem** 

four stages (see Fig. 2).

source

source

the heating source (d

system is negative:

cooling source.

regarded as a heat engine.

(2002), and Huang (2010).

Fig. 2. Heat engines of two types discussed by Sandström (1916): (a) anti-clockwise and (b) clockwise.
