6.5: Density, Potential Temperature, and Neutral Density

During winter, cold water formed at the surface sinks to a depth determined by its density relative to the density of the deeper water. Currents then carry the water to other parts of the ocean. At all times, the water parcel moves to stay below less dense water and above more dense water. The distribution of currents within the ocean depends on the distribution of pressure, which depends on the variations of density inside the ocean as outlined in Section 10.4. So, if we want to follow water movement within the ocean, we need to know the distribution of density within the ocean.

Density and sigma-t

The calculation of water movement requires measurements of density with an accuracy of a few parts per million. This is not easy.

Absolute Density of water can only be measured in special laboratories, and only with difficulty. The best accuracy is \(1 : 2.5 \times 10^{5}\) = 4 parts per million.

To avoid the difficulty of working with absolute density, oceanographers use density relative to density of pure water. Density \(\rho (S, t, p)\) is now defined using Standard Mean Ocean Water of known isotopic composition, assuming saturation of dissolved atmospheric gasses. Here \(S, t, p\) refers to salinity, temperature, and pressure.

In practice, density is not measured, it is calculated from in situ measurements of pressure, temperature, and conductivity using the equation of state for sea water. This can be done with an accuracy of two parts per million.

Density of water at the sea surface is typically \(1027 \ \text{kg/m}^{3}\). For simplification, physical oceanographers often quote only the last 2 digits of the density, a quantity they call density anomaly or \(Sigma \ (S,t,p)\): \[\sigma (S,t,p) = \rho (S,t,p) - 1000 \ \text{kg/m}^{3} \nonumber \]

The Working Group on Symbols, Units and Nomenclature in Physical Oceanography (SUN, 1985) recommends that \(\sigma\) be replaced by \(\gamma\) because \(\sigma\) was originally defined relative to pure water and it was dimensionless. Here, however, I will follow common practice and use \(\sigma\).

If we are studying surface layers of the ocean, we can ignore compressibility, and we use a new quantity called sigma-t (written \(\sigma_{t}\)): \[\sigma_{t} = \sigma(S,t,0) \nonumber \]

This is the density anomaly of a water sample when the total pressure on it has been reduced to atmospheric pressure (i.e., zero water pressure), but the temperature and salinity are in situ values.

Potential Temperature

As a water parcel moves within the ocean below the mixed layer, its salt and heat content can change only by mixing with other water. Thus we can use measurements of temperature and salinity to trace the path of the water. This is best done if we remove the effect of compressibility.

As water sinks, pressure increases, the water is compressed, and the compression does work on the water. This increases the internal energy of the water. To understand how compression increases energy, consider a cube containing a fixed mass of water. As the cube sinks, its sides move inward as the cube is compressed. Recalling that work is force times distance, the work is the distance the side moves times the force exerted on the side by pressure. The change in internal energy may or may not result in a change in temperature (McDougall and Feistel, 2003). The internal energy of a fluid is the sum of molecular kinetic energy (temperature) and molecular potential energy. In sea water, the latter term dominates, and the change of internal energy produces the temperature change shown in figure \(\PageIndex{1}\). At a depth of 8 km, the increase in temperature is almost 0.9\(^{\circ}\)C.

Figure \(\PageIndex{1}\): Left: profiles of in situ \((t)\) and potential \((\theta)\) temperature, and Right: profiles of sigma-t and sigma-theta, in the Kermadec Trench in the Pacific measured by the R/V Eltanin during the Scorpio Expedition on 13 July 1967 at 175.825\(^{\circ}\)E and 28.258\(^{\circ}\)S. Data from Warren (1973).

To remove the influence of compressibility from measurements of temperature, oceanographers (and meteorologists who have the same problem in the atmosphere) use the concept of potential temperature. Potential temperature \(\Theta\) is defined as the temperature of a parcel of water at the sea surface after it has been raised adiabatically from some depth in the ocean. Raising the parcel adiabatically means that it is raised in an insulated container so it does not exchange heat with its surroundings. Of course, the parcel is not actually brought to the surface. Potential temperature is calculated from the temperature in the water at depth, the in situ temperature.

Potential Density

If we are studying intermediate layers of the ocean, say at depths near a kilometer, we cannot ignore compressibility. Because changes in pressure primarily influence the temperature of the water, the influence of pressure can be removed, to a first approximation, by using the potential density.

Potential density \(\rho_{\Theta}\) is the density a parcel of water would have if it were raised adiabatically to the surface without change in salinity. Written as sigma, \[\sigma_{\Theta} = \sigma(S, \Theta, 0) \nonumber \]

\(\sigma_{\Theta}\) is especially useful because it is a conserved thermodynamic property.

Potential density is not useful for comparing density of water at great depths. If we bring water parcels to the surface and compare their densities, the calculation of potential density ignores the effect of pressure on the coefficients for thermal and salt expansion. As a result, two water samples having the same density but different temperature and salinity at a depth of four kilometers can have noticeably different potential density. In some regions the use of \(\rho(\Theta)\) can lead to an apparent decrease of density with depth (figure \(\PageIndex{2}\)), although we know that this is not possible because such a column of water would be unstable.

Figure \(\PageIndex{2}\): Vertical sections of density in the western Atlantic. Note that the depth scale changes at 1000 m depth. Upper: \(\sigma_{\Theta}\), showing an apparent density inversion below 3,000 m. Lower: \(\sigma_{4}\) showing continuous increase in density with depth. After Lynn and Reid (1968).

To compare samples from great depths, it is better to bring both samples to a nearby depth instead of to the surface \((p = 0)\). For example, we can bring both parcels to a pressure of 4,000 decibars, which is near a depth of 4 km: \[\sigma_{4} = \sigma(S, \Theta, 4000) \nonumber \]

where \(\sigma_{4}\) is the density of a parcel of water brought adiabatically to a pressure of 4,000 decibars. More generally, oceanographers sometimes use \(\sigma_{r}\) \[\sigma_{r} = \sigma(S, \Theta, p, p_{r}) \nonumber \]

where \(p\) is pressure, and \(p_{r}\) is pressure at some reference level. In \((\PageIndex{2})\) the level is \(p_{r} = 0\) decibars, and in \((\PageIndex{3})\) \(p_{r} = 4000\) decibars.

The use of \(\sigma_{r}\) leads to problems. If we wish to follow parcels of water deep in the ocean, we might use \(\sigma_{3}\) in some areas, and \(\sigma_{4}\) in others. But what happens when a parcel moves from a depth of 3 km in one area to a depth of 4 km in another? There is a small discontinuity between the density of the parcel expressed as \(\sigma_{3}\) compared with density expressed as \(\sigma_{4}\). To avoid this difficulty, Jackett and McDougall (1997) proposed a new variable they called neutral density.

Neutral Surfaces and Density

A parcel of water moves locally along a path of constant density so that it is always below less dense water and above more dense water. More precisely, it moves along a path of constant potential density \(\sigma_{r}\) referenced to the local depth \(r\). Such a path is called a neutral path (Eden and Willebrand, 1999). A neutral surface element is the surface tangent to the neutral paths through a point in the water. No work is required to move a parcel on this surface because there is no buoyancy force acting on the parcel as it moves (if we ignore friction).

Now let’s follow the parcel as it moves away from a local region. At first we might think that because we know the tangents to the surface everywhere, we can define a surface that is the envelope of the tangents. But an exact surface is not mathematically possible in the real ocean, although we can come very close.

Jackett and McDougall (1997) developed a practical neutral density variable \(\gamma^{n}\) and surface that stays within a few tens meters of an ideal surface anywhere in the world. They constructed their variables using data in the Levitus (1982) atlas. The neutral density values were then used to label the data in the Levitus atlas. This prelabeled data set is used to calculate \(\gamma^{n}\) at new locations where \(t, S\) are measured as a function of depth by interpolation to the four closest points in the Levitus atlas. Through this practice, neutral density \(\gamma^{n}\) is a function of salinity \(S\), in situ temperature \(t\), pressure \(p\), longitude, and latitude.

The neutral surface defined above differs only slightly from an ideal neutral surface. If a parcel moves around a gyre on the neutral surface and returns to its starting location, its depth at the end will differ by around 10 meters from the depth at the start. If potential density surfaces are used, the difference can be hundreds of meters, a far larger error.

Equation of state of sea water

Density of sea water is rarely measured. Density is calculated from measurements of temperature, conductivity, or salinity, and pressure using the equation of state of sea water. The equation of state is an equation relating density to temperature, salinity, and pressure.

The equation is derived by fitting curves through laboratory measurements of density as a function of temperature, pressure, and salinity, chlorinity, or conductivity. The International Equation of State (1980) published by the Joint Panel on Oceanographic Tables and Standards (1981) is now used. See also Millero and Poisson (1981) and Millero et al (1980). The equation has an accuracy of 10 parts per million, which is 0.01 units of \(\sigma(\Theta)\).

I have not actually written out the equation of state because it consists of three polynomials with 41 constants (JPOTS, 1991).

Accuracy of Temperature, Salinity, and Density

If we want to distinguish between different water masses in the ocean, and if the total range of temperature and salinity is as small as the range in figure \(6.1.1\), then we must measure temperature, salinity, and pressure very carefully. We will need an accuracy of a few parts per million.

Such accuracy can be achieved only if all quantities are carefully defined, if all measurements are made with great care, if all instruments are carefully calibrated, and if all work is done according to internationally accepted standards. The standards are laid out in Processing of Oceanographic Station Data (JPOTS, 1991) published by UNESCO. The book contains internationally accepted definitions of primary variables such as temperature and salinity and methods for the measuring the primary variables. It also describes accepted methods for calculating quantities derived from primary variables, such as potential temperature, density, and stability.