Fig. 2.12 Meridional distribution of potential density along 30.5° W (kg/m3), using two different reference levels.

unstable stratification is artificial. Were the equation of state linear, there would be no such artificial problems.

Apparently, the vertical gradient of a0 changes signs at about 4 km depth. Thus, the water column at this depth range seems gravitationally unstable. A column of water taken from depth range 4-4.5 km would be unstable if it were adiabatically brought to the sea surface. Under the adiabatic assumption, at the sea surface, water from 4 km depth would be heavier than water from 4.5 km depth. However, the water column at 4-4.5 km is actually quite stable to small perturbations. The only problem for such a seemingly unstable case is that such extremely large perturbations do not happen in the ocean. As a matter of fact, water below 4.5 km belongs to the Atlantic Bottom Water formed near Antarctica, while water at depth 4 km belongs to the Atlantic Deep Water formed near the Greenland/Norwegian Sea.

In fact, the water column in the depth range of 4-4.5 km is stable for small perturbations in the vertical direction, and the seemingly unstable situation inferred from the vertical gradient of a0 is an artifact due to the nonlinearity of the equation of state. To overcome this difficulty, we often use potential density defined at different levels. For example, a0 (or a&) is used for the upper ocean; a2 (using 2,000 db as the reference pressure, which corresponds to the in situ pressure at the depth of slightly less than 2 km) is used for the mid-depth circulation; and a4 (using 4,000 db as the reference pressure) is used for analyzing the deep circulation. If we want to study a process that takes place over a large depth range, the neutral surface can be used, as discussed in the next subsection.

Note that using a 2 improves the situation, but there may arise some minor problems near the sea floor; thus, for abyssal circulation, it is better to use a4. A simple approach relying on choosing a simple reference pressure does not seem to work for the whole depth range of the ocean, and this is a technical difficulty which oceanic general circulation models based on density coordinates must overcome. To avoid the potential problems associated with such artificial instabilities, the commonly used MICOM (Miami Isopycnal Coordinate Ocean Model) model is based on a2 as a compromise.

Using potential temperature we can compare the density variation due to contributions from both the pressure and potential temperature changes. As we have seen above, the range of adiabatic temperature is about 0.6°C within the top 4 km; the corresponding density change due to this effect is about 0.10 kg/m3. For the same range of depth change, the density increases from 27 kg/m3 to 45 kg/m3, i.e., an increase of 18 kg/m3. A large part of the vertical density variance in the oceans is due to change in pressure. This part of density variance is dynamically inactive, except for keeping the water column stable. This is an important reason why potential density is so widely used in the study of oceanic circulation.

As an example, we examine the case of a water parcel at the sea surface with a temperature of 2°C. If this water parcel is moved downward, without exchanging heat and salt with the environment, its potential temperature is kept constant; however, its in situ temperature increases with pressure, indicated by the dashed line in Figure 2.11a. The in situ density also increases (Fig. 2.11b); such an increase in density is due to the compressibility of seawater, as explained above. The corresponding potential density does not change, as indicated by the solid line in Figure 2.11c; however, if the in situ temperature were vertically constant, T = 2°C, the corresponding potential density would increase with depth, as indicated by the dashed line in Figure 2.11c.

In order to separate the effects of density change due to temperature and pressure, in this subsection water density is defined as a function of potential temperature, salinity, andpres-sure. Since density can change solely due to pressure change, using potential temperature, instead of temperature, is more convenient in the discussion of thermobaric effect. Accordingly, the commonly used thermal expansion coefficient is defined as a = (dp/d©)PS. To the first-order approximation, seawater density is linearly proportional to changes in ©, S, P; in a Taylor expansion, we have p d 2p

S- - 1 = -aS© + P8S + ySP +-— 8©8P + ••• (2.86)

where y = Kn. The second-order term on the right-hand side is the so-called thermobaric effect, i.e., density changes due to the combination of temperature and pressure. Owing to the contribution from the second-order term in Eqn. (2.86), a increases when pressure is enhanced. This will be shown in the following discussion.

As an example, changes in specific volume and specific internal energy of two water parcels under adiabatic compression are shown in Figure 2.13. It is readily seen that cold

Changes in specific volume Changes in internal energy

Changes in specific volume Changes in internal energy

Fig. 2.13 Changes in a specific volume and b internal energy as a function of potential temperature and pressure.

Fig. 2.13 Changes in a specific volume and b internal energy as a function of potential temperature and pressure.

water is more compressible, and thus the internal energy of a cold-water parcel also increases more quickly with the increase of pressure than a warm-water parcel.

The different behaviors of compressibility y and thermal expansion coefficient a can best be illustrated by following the two examples shown in Figure 2.14. By definition d (ln p) d (ln p)

thus

By da

It is well known that a is very small at low temperature. For example, a ~ 0.5 x 10-4/°C at © = 0°C and S = 35. As T increases, a increases almost linearly (Fig. 2.14). The fact that the thermal expansion coefficient is very small at nearly freezing temperature has crucial dynamical implications for high-latitude oceanography. Since the haline contraction coefficient j is nearly constant, a very small a means that salinity is the dominating factor in regulating the stratification and flow there.

Another critical fact is that a increases with pressure, i.e., da/dP > 0, as shown in Figures 2.14a and 2.15a. According to Eqn. (2.87), this is consistent with the fact that dY/d© < 0, as shown in Figure 2.14b. Furthermore, the derivative da/d© declines with pressure and temperature (Fig. 2.15b). Thus, cold water is more compressible than warm water: the so-called thermobaric effect. Although sea surface temperature in most parts of the world's oceans is relatively warm, water in the subsurface ocean and deep ocean is

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