**3. Entropy production rate calculation**

According to Shimokawa & Ozawa (2001) and Shimokawa (2002), the entropy increase rate for the ocean system is calculable as

$$\begin{split} \frac{\mathbf{dS}}{\mathbf{d}t} &= \mathbb{I} \frac{\mathbf{1}}{T} \frac{\hat{\mathbf{c}}(\rho cT)}{\hat{\mathbf{c}}t} + \text{div}(\rho cT \, v) + p \text{div}(v) \mathbf{J} \, \mathbf{V} + \mathbb{I} \frac{F\_h}{T} \mathbf{d}A \\ &- ak \mathbb{I} \left[ \frac{\hat{\mathbf{c}} \mathbf{C}}{\hat{\mathbf{c}}t} + \text{div}(\mathbf{C} \, v) \right] \text{lnC } \mathbf{d}V - ak \mathbb{I} \, F\_\mathbf{S} \ln \mathbf{C} \, \mathbf{d}A \end{split} \tag{4}$$

where *ρ* stands for the density, *c* denotes the specific heat at constant volume, *T* signifies the temperature, *α* = 2 is van't Hoff's factor representing the dissociation effect of salt into separate ions (Na+ and Cl–), *k* is the Boltzmann's constant, *C* is the number concentration of salt per unit volume of seawater, *F*h and *F*s are the heat and salt fluxes per unit surface area respectively, defined as positive outward, and d*V* and d*A* are the small volume and surface elements, respectively.

If we can assume that the seawater is incompressible (div *v* = 0) and that the volumetric heat capacity is constant (ρc = const.), then the divergence terms in (4) disappear. In this case, we obtain

$$\frac{\text{d}S}{\text{d}t} = \text{[}\frac{\rho c}{T}\frac{\partial T}{\partial t}\text{d}V + \text{[}\frac{F\_h}{T}\text{dA} - ak\text{]}\frac{\partial C}{\partial t}\text{lnC}\text{d}V - ak\text{[}F\_\text{S}\text{lnC}\text{d}A\text{]}\,. \tag{5}$$

The first two terms in the right-hand side represent the entropy production rate attributable to heat transport in the ocean. The next two terms represent that attributable to the salt transport. The first and third terms vanish when the system is in a steady state because the temperature and the salinity are virtually constant (*T*/*t* = *C*/*t* = 0). In the steady state, entropy produced by the irreversible transports of heat and salt is discharged completely into the surrounding system through the boundary fluxes of heat and salt, as expressed by the second and fourth terms in equation (5).

The general expression (4) can be rewritten in a different form. A mathematical transformation (Shimokawa and Ozawa, 2001) can show that

$$\frac{\text{dS}}{\text{d}t} = \text{[}^{\text{F}\_{\text{h}}} \cdot \text{grad}(\frac{1}{T})\text{d}V + \text{[}\frac{\text{\{\}}}{T}\text{d}V - ak\text{]} \frac{\text{F}\_{\text{s}} \cdot \text{grad}(\text{C})}{\text{C}}\text{d}V\text{}\_{\text{s}}\tag{6}$$

where *Fh* and *Fs* respectively represent the flux densities of heat and salt (vector in threedimensional space) and *Ф* is the dissipation function, representing the rate of dissipation of kinetic energy into heat by viscosity per unit volume of the fluid. The first term on the righthand side is the entropy production rate by thermal dissipation (heat conduction). The second term is that by viscous dissipation; the third term is that by molecular diffusion of salt ions. Empirically, heat is known to flow from hot to cold via thermal conduction, and the dissipation function is always non-negative (*Ф* ≥ 0) because the kinetic energy is always dissipated into heat by viscosity. Molecular diffusion is also known to take place from high to low concentration (salinity). Therefore, the sum should also be positive. This is a consequence of the Second Law of Thermodynamics.

According to Shimokawa & Ozawa (2001) and Shimokawa (2002), the entropy increase rate

d 1 ( ) [ div( ) div( )]d d <sup>d</sup>

*S ρcT F <sup>ρ</sup>cT v p v V A tT t <sup>T</sup>*

*C*

*t*

[ div( )]ln d ln d

*αk Cv C V αkF C A*

where *ρ* stands for the density, *c* denotes the specific heat at constant volume, *T* signifies the temperature, *α* = 2 is van't Hoff's factor representing the dissociation effect of salt into separate ions (Na+ and Cl–), *k* is the Boltzmann's constant, *C* is the number concentration of salt per unit volume of seawater, *F*h and *F*s are the heat and salt fluxes per unit surface area respectively, defined as positive outward, and d*V* and d*A* are the small volume and surface

If we can assume that the seawater is incompressible (div *v* = 0) and that the volumetric heat capacity is constant (ρc = const.), then the divergence terms in (4) disappear. In this case, we

<sup>d</sup> d d ln d ln d

*ST C <sup>ρ</sup><sup>c</sup> <sup>F</sup> V A <sup>α</sup>k CV <sup>α</sup>kF C A*

The first two terms in the right-hand side represent the entropy production rate attributable to heat transport in the ocean. The next two terms represent that attributable to the salt transport. The first and third terms vanish when the system is in a steady state because the temperature and the salinity are virtually constant (*T*/*t* = *C*/*t* = 0). In the steady state, entropy produced by the irreversible transports of heat and salt is discharged completely into the surrounding system through the boundary fluxes of heat and salt, as expressed by

The general expression (4) can be rewritten in a different form. A mathematical

d 1 grad( ) grad( )d d <sup>d</sup>

*t TT C*

*<sup>S</sup> <sup>Φ</sup> F C F VV <sup>α</sup>k V*

where *Fh* and *Fs* respectively represent the flux densities of heat and salt (vector in threedimensional space) and *Ф* is the dissipation function, representing the rate of dissipation of kinetic energy into heat by viscosity per unit volume of the fluid. The first term on the righthand side is the entropy production rate by thermal dissipation (heat conduction). The second term is that by viscous dissipation; the third term is that by molecular diffusion of salt ions. Empirically, heat is known to flow from hot to cold via thermal conduction, and the dissipation function is always non-negative (*Ф* ≥ 0) because the kinetic energy is always dissipated into heat by viscosity. Molecular diffusion is also known to take place from high to low concentration (salinity). Therefore, the sum should also be positive. This is a

*h*

*t Tt T t* 

*h*

*S*

. (5)

*s*

, (6)

, (4)

*S*

**3. Entropy production rate calculation** 

for the ocean system is calculable as

elements, respectively.

d

the second and fourth terms in equation (5).

d

transformation (Shimokawa and Ozawa, 2001) can show that

*h*

consequence of the Second Law of Thermodynamics.

obtain
