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Comparatively speaking, in the Warming Case (thin line) GPE increases very slowly. This slowness of increase in GPE reflects the fact that the stratification in this case is relatively stable; thus, the downward penetration of heat and the associated increase in volume and GPE of the model ocean are very slow processes. At the beginning of the experiment, the convective adjustment was almost interrupted (thin line in Fig. 3.46b). Surface heating is now working as a weak source of GPE during the spin-up phase. Overall, the conversion of potential energy to KE is relatively small because the GPE loss to convection is so much smaller than before.

Although our discussion here is limited to a simple model with only surface thermal forcing included, leaving other important forcing conditions untouched, the results obtained from this diagnosis have a much broader implication. Further study along the lines of GPE balance in the ocean should be very promising. In particular, a model including wind stress forcing could provide vitally important information about the balance of GPE in the ocean, and such knowledge will broaden our view of the physics of oceanic general circulation.

3.8 Entropy balance in the oceans

Entropy balance is one of the most fundamental thermodynamic laws governing the universe. Examining the thermohaline circulation from the viewpoint of entropy balance may bring about some new insights, though such approach has seldom been pursued, with a few exceptions. In order to simplify the problem, our discussion here excludes the potential contribution from sea ice.

3.8.1 Entropy production due to freshwater mixing

Dilution heat associated with seawater

When two water parcels of the same temperature but different salinity are adiabatically mixed under constant pressure, the final temperature of the mixture may be different from the original, and the heat loss/gain is called the dilution heat. Depending on the parameters, such as temperature, salinity, and pressure, dilution heat may be positive or negative. Assuming that each parcel has 1 kg of seawater, and using the Taylor expansion, enthalpy of the two water masses with slightly different salinity is hx + h2 = h (T, S + AS,p) + h(T, S — AS, P) = 2h(T, S, P) + AS2dSSh

For seawater over a rather wide range of parameter values, the second derivative is negative, dSSh < 0, thus we have h1 + h2 < 2h0. Seawater is almost incompressible, so that if temperature is maintained constant, changes in enthalpy are approximately equal to the heat received from the environment, and this is called the dilution heat. However, if mixing is adiabatic, mixing water parcels with different salinity generally leads to cooling.

Enthalpy and entropy changes associated with mixing of seawater To explore the process of mixing and transport of freshwater in the ocean, we set up the following virtual experiment under a constant temperature of 15°C and sea-level pressure. A series of boxes numbered by N (where N = 1,..., 37) are filled up with seawater, with the salinity (S) in the N-th box being S = N — 1. At the left-hand end, box N = 1 has no salt; it receives 1 kg of pure water and exports it to the right, and this water mixes with 1 kg of salty water from box 3 with a salinity of 2. The end product of mixing is 2 kg of water with a salinity of 1 (Fig. 3.47a). Similarly, the mass balance in each box N (2 < N < 35) is illustrated in Figure 3.47b. For each box the total fluxes of mass, water, and salt are balanced. Across the boundaries between two boxes, there is a net water flux moving to the right, but there is no net salt flux between boxes. This can be shown as follows. Across the boundary between boxes with salinity S = N and S = N +1 (Fig. 3.47b), the mass balance

S = 0

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