The Equation of State for Seawater

In polar oceans, a layer of cold, less saline water nearly always overlies water that is both warmer and saltier. This negative temperature gradient by itself is destabilizing,4 so stratification is maintained by the negative salinity gradient (i.e., increasing with depth). The equation of state for seawater is by convention expressed as a function of temperature, salinity as units of the practical salinity scale (abbreviated psu,

4 In fresh water, thermal expansion changes sign at about 4 °C, so fresh (or brackish water) may remain stable despite a negative temperature gradient near the surface. This does not hold for seawater with salinities in excess of about 24 units on the practical salinity scale.

a Ice Velocity 98:77.75 (a 78.73, a Ice Velocity 98:77.75 (a 78.73,

0 2D 40 ED BD 100 120 140 160 180 200 x coordinate, km

Fig. 2.8 a Average ice velocity field inferred from RGPS feature tracking of scenes separated by about a day. Circles indicated starting (solid) and ending (open) positions of the SHEBA drift station for the same period. Vectors are drawn every fourth grid point, one axis division is equivalent to 0.2m s_1. b Lower limit of kinematic surface stress curl, obtained by finite differences across the 5-km grid scale of the RGPS analysis (Adapted from McPhee et al. 2005. With permission American Geophysical Union)

0 2D 40 ED BD 100 120 140 160 180 200 x coordinate, km

Fig. 2.8 a Average ice velocity field inferred from RGPS feature tracking of scenes separated by about a day. Circles indicated starting (solid) and ending (open) positions of the SHEBA drift station for the same period. Vectors are drawn every fourth grid point, one axis division is equivalent to 0.2m s_1. b Lower limit of kinematic surface stress curl, obtained by finite differences across the 5-km grid scale of the RGPS analysis (Adapted from McPhee et al. 2005. With permission American Geophysical Union)

corresponding closely but not exactly to the older expression ppt, parts per thousand), and pressure. Common oceanographic usage is to express pressure in terms of the departure from atmospheric pressure at the surface, with units of bars (105 Pa) or dbar (which corresponds reasonably closely with depth in m). The practical salinity scale relates the measured conductivity of seawater to an international standard, and thus provides a unique salinity for given conductivity, temperature, and pressure, all of which can be measured to high accuracy with modern oceanographic instrumentation. The UNESCO formulas for density as a function of the three state variables are given, e.g., by Gill (1982, Appendix 3) and are used in this work.

At low temperatures, the impact of changes in salinity on density is amplified relative to temperature changes because the thermal expansion factor, ¡3t = is small. The haline contraction factor ([3s = ) is relatively insensitive to temperature, as illustrated in Fig. 2.9a, where variation of fir and fis with respect to their values at the freezing point are shown as functions of temperature. Over the range shown, fir increases by over 400% while fis remains within 2% of its freezing value. At constant pressure, the change in density may be expressed as

p Ratios Versus Temperature (S=34)

p Ratios Versus Temperature (S=34)

! ! ! ! !

a

p/ß/v

\

ß Ratios Versus Pressure

VT(p)/ßT(p=0) for T=-1.86

b

^PT(p)/PT(p =

■0)forT=4

---- ! ^

Vs(p)%(p =0) for T= -1.86/

Y

Pressure, bars

40 60

Pressure, bars

Fig. 2.9 a Ratio of expansion and contraction factors to their values for water at freezing temperature (—1.86 °C) as a function of water temperature. At freezing the ratio fis/fir is about 33. At T = 4 °C, it is about 8. b As in a, except ratios relative to the value at surface pressure (p = 0) as a function of pressure. At 400 m, the thermal expansion factor for water at freezing is about 1.5 times as large as at the surface. For water at T = 4 °C, it is only about 1.1 times as large

To offset a change in salinity so that density remains unchanged would require that ST/8s = -Ps/Pt. For the conditions of Fig. 2.9a (S = 34psu), this ratio is about 33 for T = -1.86 °C and about 8 for T = 4 °C. Thus for water close to freezing, density variation is almost exclusively a function of salinity, and temperature may often be treated as a passive scalar contaminant.

In situ density depends on pressure, but in terms of the impact of vertical density gradient on dynamics, generally the pressure dependence is neglected by considering potential density, i.e., p(T, S,p = 0) or o0 = p (T, S,p = 0) - 1000. The reason for this is clear: a well mixed layer with uniform T and S, will have a pressure induced vertical density gradient, but there is negligible work (besides friction) involved in moving a parcel from one level (pressure) to another. Yet there are idiosyncrasies associated with nonlinearities in the equation of state that make this less straightforward than it might at first appear. Consider, for example, the dependence of the expansion and contraction factors on pressure (Fig. 2.9b). Here the ratios of Pt and Ps to their values at surface pressure are plotted as functions of pressure. Ps has very little pressure dependence, but the magnitude of Pt increases with pressure. Plots are shown for two different temperatures to emphasize that the pressure dependence of Pt is much greater for cold water, resulting from the fact that cold water is more compressible than warm.

An example drawn from near Maud Rise in the Weddell Sea (Fig. 2.10) nicely illustrates certain consequences of nonlinearities inherent in the equation of state.

a

—-=-

\

-20 -40 -60 -80 -100 -120 -140 -160 -180 -200

b

1

\

34.5 34.6 Practical Salinity Scale

34.5 34.6 Practical Salinity Scale

Fig. 2.10 YOYO Station 75 from ANZLUX 1994, on the eastern edge of Maud Rise in the Weddell Sea. a Temperature; b salinity; c. 00 (potential density—1,000). Dashed lines are an idealized two-layer system based on the measurements

Contours of o.

Contours of o.

34.55 Salinity, psu

34.65

34.55 Salinity, psu

34.65

Contours of o

0.093

Contours of o

0.093

34.55 Salinity, psu

34.65

Fig. 2.11 Temperature/salinity diagrams with isopycnal contours for density calculated at a surface pressure and b at pressure corresponding to the mixed layer depth. T/S characteristics of the idealized two-layer system from Figs. 2.2 to 2.10 are indicated by symbols (circle for upper, square for lower). See text for further details (see also colorplate on p. 204)

34.55 Salinity, psu

34.65

Fig. 2.11 Temperature/salinity diagrams with isopycnal contours for density calculated at a surface pressure and b at pressure corresponding to the mixed layer depth. T/S characteristics of the idealized two-layer system from Figs. 2.2 to 2.10 are indicated by symbols (circle for upper, square for lower). See text for further details (see also colorplate on p. 204)

Measured T, S, and 00 profiles can be reasonably well represented in the upper 200 m of the water column by a two-layer system with an upper layer thickness of about 93 m. The potential density difference between the two layers is quite small, less than 0.03 kg m-3. In oceanography it is customary to compare water masses via a temperature-salinity diagram, as drawn in Fig. 2.11a. The T/S pairs representing characteristics of the two (idealized) layers from Fig. 2.10 are shown as symbols embedded in contours of 00. The 00 isopyncal passing through the T/S point for the lower layer (with T = 0.13 °C, Si = 34.63psu) is shown in white. The double arrow indicates the increase in salinity needed to raise the potential density of the upper layer to that of the lower. All else being equal, the salt rejected from about 13 cm of additional ice growth (at the time the ice was about 35 cm thick) would accomplish this. The dashed line connecting the modified surface water and the deeper water in Fig. 2.11a is the so-called mixing line, which describes the T/S characteristics of any product from conservative mixing of the two different water masses. Because of the isopycnal curvature, the mixing line lies to the right of the isopycnal passing through both the deep water and modified surface water, so any mixture of the two water types is denser than either of the end members. Since no consideration of pressure was involved in these arguments, the resulting instability arises from the dependence of ¡3t on temperature (Fig. 2.9a) and by convention is called cabbeling.

Figure 2.11b is like Fig. 2.11a, except that here the isopyncals are drawn for density evaluated at the pressure (depth) of the interface between the two layers, about 9.3 bar. At the higher pressure the slope of the isopycnals in T/S space is less than for surface pressure, which means that for a fixed salinity, the change in density associated with a given change in temperature is greater at depth. In this case, the upper layer only needs a salinity increase corresponding to about 10 cm of ice growth to reach the same in situ density as the lower layer. Thus instability will be triggered before the potential density of the upper layer reaches that of the lower layer. The term coined by McDougall (1987) for this pressure effect is thermobaricity.

A method for illustrating thermobaricity presented by Akitomo (1999) provides additional insight into the nonlinear equation of state issues, and is easily applied to the idealized two-layer system. Suppose that enough ice grows to increase the salinity of the upper layer by SS = 0.027psu so that the in situ density of the two layers is the same at the interface, i.e., p(Tu, Su, p93)=p(Tl, Sl, p93)

The difference between the density of the two-layer upper ocean and an ocean with uniform T and S equal to the upper layer values:

is plotted in Fig. 2.12. If a parcel of water from the upper layer (square marker) is displaced downward across the interface, it will be heavier than its ambient surroundings and will continue downward. A parcel displaced across the interface from below (circle) will be lighter than its surroundings and will continue to rise. Consequently thermobaricity is mechanism for enhanced mixing that draws from the potential energy of the destabilizing temperature gradient. Once started, the thermo-baric process is self sustaining, and is probably an important component of mixing in marginally stable polar oceans like much of the Weddell in late winter. As indicated by Fig. 2.11, it is the curvature of the isopycnals in T/S space that leads to mixing driven by nonlinearities in the equation of state. To separate cabbeling and thermobaricity conceptually may be a question more of semantics than physics, but the important point is that whenever the temperature profile is in itself destabilizing, it is important to consider pressure effects.

If there are widespread regions in the Weddell where upper ocean structure is such that only a few decimeters of ice growth could trigger deep-reaching thermo-baric instability, why does an ice cover exists there at all? Or put another way, why is the Weddell Polynya not a quasi-permanent feature? The answer apparently lies with what Martinson (1990) termed the "thermal barrier." Whenever heat is mixed up from below, it rapidly warms the mixed layer to the point where ocean heat flux to the ice undersurface exceeds conduction through the ice cover or loss from open water, and the ice begins melting. This introduces positive buoyancy that effectively

Fig. 2.12 Diagram of in situ density of the two-layer system minus the density of an upper ocean with uniform upper layer characteristics. Displacement of a water parcel with upper layer characteristics (square) downward makes it denser than its surrounding, while upward displacement from below the interface has the opposite tendency. The system is thermobarically unstable (see Akitomo 1999)

Fig. 2.12 Diagram of in situ density of the two-layer system minus the density of an upper ocean with uniform upper layer characteristics. Displacement of a water parcel with upper layer characteristics (square) downward makes it denser than its surrounding, while upward displacement from below the interface has the opposite tendency. The system is thermobarically unstable (see Akitomo 1999)

limits mixing driven from the surface, forming a new, shallower near surface layer. Thermobaric mixing below this layer may continue, driven by the nonlinearity, but is no longer affecting surface exchanges.5

The fact remains that if melting at the ice/ocean interface is too weak or too slow to counteract the combined effects of surface buoyancy loss from cooling and the cabbeling/thermobaricity mechanism at the base of the mixed layer, then convection will continue (McPhee 2003). Once the ice cover is gone and the air remains cold, there is nothing except horizontal advection of ice or fresh water to quell deep mixing, and essentially a direct connection between the abyssal ocean and the atmosphere is established. The Weddell Polynya demonstrated that such an event can have large, even global, impact.

A factor often ignored in the "mixing line" argument for instabilities arising from mixing of adjacent water masses with similar density but different T/S characteristics is that even in a fairly turbulent regime, diffusivities of heat and salt may differ. In Section 2.7 we described an upwelling event observed in March 1998, at the

5 The conjecture that subsurface well mixed layers like that in the ANZFLUX profile in Fig. 2.10 between 100 and 180 m are remnants of mixing events where thermobaricity contributed is discussed further in Chapter 8.

Temperature Salinity

Fig. 2.13 SHEBA profiler temperature a and salinity b profiles bracketing the upwelling event on day 78. The dashed lines are the displacement of the mean isotherm a and isohaline b corresponding to measurements at 17.7 m on the turbulence mast at time 78.5

SHEBA station where isopycnals were observed to rise about 13 m above their ambient level in the undisturbed ocean, apparently in response to concentrated surface stress curl. Here we examine that event is more detail, as a possible example of double diffusion in a fully turbulent flow. Figure 2.13 shows the temperature and salinity profiles used to construct the Oq profile in Fig. 2.7. The upward displacement of the isotherm in the bracketing profiles to match the temperature observed on the bottommost TIC at time 78.5 is indicated by the dashed line in Fig. 2.13a, and similarly for the matching isohaline in Fig. 2.13b. They differ by 2.4 m and both are less than the isopycnal displacement shown in Fig. 2.7. The logical explanation is that heat was mixed more efficiently than salt so that as the pycnocline fluid moved upward in response to Ekman pumping, its temperature lowered faster than its salinity. The loss in buoyancy from this additional cooling is why the isopyncnal displacement is about 0.4 m more than the isohaline displacement. In Fig. 2.14, the T/S properties of the water observed at 17.7 m at the maximum upwelling are compared with the ambient profiler T/S properties from all stations on day 79, averaged in 0.1 salinity bins. This strongly suggests that the upwelling event was capable of extracting heat from the upper pycnocline faster than salt.

Fig. 2.14 Temperature/salinity diagram for all the SHEBA profiler casts on day 79, averaged in 0.1 psu bins. Sizes of the crosses correspond to twice the standard deviations of both T and S in each bin. The circle marks the T/S value for the 3-h average centered at time 78.5 at 17.7 m

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