**2. Numerical model and method**

The numerical model used for this study is the Geophysical Fluid Dynamics Laboratory's Modular Ocean Model (Pacanowski, 1996). The model equations consist of Navier–Stokes equations subject to the Boussinesq, hydrostatic, and rigid-lid approximations along with a nonlinear equation of state that couples two active variables, temperature and salinity, to the fluid velocity. A convective adjustment scheme is used to represent the vertical mixing process. Horizontal and vertical diffusivity coefficients are, respectively, 103 m2 s-1 and 10-4 m2 s-1. The time-step of the integration is 5400 s.

The model domain is a rectangular basin of 72° longitude by 140 latitude with a cyclic path, representing an idealized Atlantic Ocean (Fig. 3(a)). The southern hemisphere includes an Antarctic Circumpolar Current passage from 48°S to 68°S. The horizontal grid spacing is 4 degrees. The ocean depth is 4500 m with 12 vertical levels (Shimokawa & Ozawa, 2001). All boundary conditions for wind stress, temperature and salinity are arranged as symmetric about the equator (Figs. 3(b), 3(c), and 3(d)). The wind stress is assumed to be zonal (eastward or westward direction, Fig. 3(b)). A restoring boundary condition is applied: The surface temperature and salinity are relaxed to their prescribed values (Figs. 3(c) and 3(d)), with a relaxation time scale of 20 days over a mixed layer depth of 25 m. The corresponding fluxes of heat and salt are used to calculate *Fh* and *Fs* at the surface. The initial temperature distribution is described as a function of depth and latitude. The initial salinity is assumed to be constant (34.9‰). The initial velocity field is set to zero. Numerical simulation is conducted for a spin-up period of 5000 years.

Figure 4 shows a zonally integrated meridional stream function at years 100, 1000, 2000, 3000, 4000, and 5000, after starting the calculations. At year 100, the circulation pattern is almost symmetric about the equator. The sinking cell in the southern hemisphere does not develop further because of the existence of the Antarctic Circumpolar Current. In contrast, the sinking cell in the northern hemisphere develops into deeper layers, and the circulation pattern becomes asymmetric about the equator. The oceanic circulation becomes statistically steady after year 4000. Temperature variations are shown to be less than 0.1 K after year 4000. In the steady state, the northern deep-water sinking cell is accompanied by an Antarctic bottom-water sinking cell and by a northern intrusion cell from the south. The flow pattern is apparently a basic one in the idealised Atlantic Ocean.

Thermodynamics of the Oceanic General Circulation – Is the Abyssal Circulation a Heat Engine or a Mechanical Pump? 153

152 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

universal principle for time evolution of non-equilibrium systems (see reviews of Kleidon and Lorenz, 2005; Lorenz, 2003; Martyushev & Seleznev, 2006; Ozawa et al., 2003; Whitfield, 2005). However, although some attempts have been made to seek a theoretical framework of

As described above, the problem of whether the abyssal circulation is a heat engine or mechanical pump and how it is related to the Sandström theorem are important for better understanding of the oceanic general circulation. In the following sections, we discuss the problem referring to the results of numerical simulations of the oceanic general circulation. In section 2, a numerical model and method are described. In section 3, a calculation method of entropy production rate in the model is explained. In section 4, details of entropy production in the model are described. In section 5, by referring to the results, the problem of whether the abyssal circulation is a heat engine or mechanical pump and how it is related

The numerical model used for this study is the Geophysical Fluid Dynamics Laboratory's Modular Ocean Model (Pacanowski, 1996). The model equations consist of Navier–Stokes equations subject to the Boussinesq, hydrostatic, and rigid-lid approximations along with a nonlinear equation of state that couples two active variables, temperature and salinity, to the fluid velocity. A convective adjustment scheme is used to represent the vertical mixing process. Horizontal and vertical diffusivity coefficients are, respectively, 103 m2 s-1 and 10-4

The model domain is a rectangular basin of 72° longitude by 140 latitude with a cyclic path, representing an idealized Atlantic Ocean (Fig. 3(a)). The southern hemisphere includes an Antarctic Circumpolar Current passage from 48°S to 68°S. The horizontal grid spacing is 4 degrees. The ocean depth is 4500 m with 12 vertical levels (Shimokawa & Ozawa, 2001). All boundary conditions for wind stress, temperature and salinity are arranged as symmetric about the equator (Figs. 3(b), 3(c), and 3(d)). The wind stress is assumed to be zonal (eastward or westward direction, Fig. 3(b)). A restoring boundary condition is applied: The surface temperature and salinity are relaxed to their prescribed values (Figs. 3(c) and 3(d)), with a relaxation time scale of 20 days over a mixed layer depth of 25 m. The corresponding fluxes of heat and salt are used to calculate *Fh* and *Fs* at the surface. The initial temperature distribution is described as a function of depth and latitude. The initial salinity is assumed to be constant (34.9‰). The initial velocity field is set to zero. Numerical simulation is

Figure 4 shows a zonally integrated meridional stream function at years 100, 1000, 2000, 3000, 4000, and 5000, after starting the calculations. At year 100, the circulation pattern is almost symmetric about the equator. The sinking cell in the southern hemisphere does not develop further because of the existence of the Antarctic Circumpolar Current. In contrast, the sinking cell in the northern hemisphere develops into deeper layers, and the circulation pattern becomes asymmetric about the equator. The oceanic circulation becomes statistically steady after year 4000. Temperature variations are shown to be less than 0.1 K after year 4000. In the steady state, the northern deep-water sinking cell is accompanied by an Antarctic bottom-water sinking cell and by a northern intrusion cell from the south. The

flow pattern is apparently a basic one in the idealised Atlantic Ocean.

MEP (e.g., Dewar, 2003, 2005), we remain uncertain about its physical meaning.

**1.7 Main contents of this chapter** 

to the Sandström theorem is discussed.

**2. Numerical model and method** 

m2 s-1. The time-step of the integration is 5400 s.

conducted for a spin-up period of 5000 years.

Fig. 3. (a) Model domain, and forcing fields of the model as functions of latitude, (b) forced zonal wind stress (N m-2) defined as positive eastward, (c) prescribed sea surface temperature (oC), and (d) prescribed sea surface salinity (‰).

Fig. 4. The zonally integrated meridional stream function at years (a) 100, (b) 1000, (d) 2000, (e) 3000, (d) 4000, and (e) 5000 after starting the numerical calculations. The contour line interval is 2 SV (106 m3 s-1). The circulation pattern reached a statistically steady-state after year 4000.

Thermodynamics of the Oceanic General Circulation –

the first term in equation (6) as

of *Ay* (*Ay*×d*V*) and *Az* (*Az*×d*V*).

of the circulation with northern sinking (Fig. 4(f)).

2

values including the effect of layer thickness.

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

Is the Abyssal Circulation a Heat Engine or a Mechanical Pump? 155

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

> ddd ( ), ( ) , ( ) , ( ) ddd *x y zx h y h z v <sup>ρ</sup><sup>C</sup> TTT A A A AA D A D A D T xyz*

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

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

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

, (7)

222
