Significance of σ_t Surfaces

The density of sea water at atmospheric pressure, expressed as σ_t = (ρ_s,ϕ,0 − 1) × 10³, is often computed and represented in horizontal charts or vertical sections. It is therefore necessary to study the significance of σ_t surfaces, and in order to do so the following problem will be considered: Can water masses be exchanged between different places in the ocean space without altering the distribution of mass?

The same problem will first be considered for the atmosphere, assuming that this is a perfect, dry gas. In such an atmosphere the potential temperature means the temperature which the air would have if it were brought by an adiabatic process to a standard pressure. The potential temperature, θ, is

where ϕ is the temperature at the pressure p, p₀ is the standard pressure, and κ = 1.4053 is the ratio of the two specific heats of an ideal gas (c_p/c_v). In a dry atmosphere in which the temperature varies in space and in which the vertical gradient differs from the gradient at adiabatic equilibrium, it is always possible to define surfaces of equal potential temperature. One characteristic of these surfaces is that along such a surface air masses can be interchanged without altering the distribution of temperature and pressure and, thus, without altering the distribution of mass.

Consider two air masses, one of temperature ϕ₁ at pressure p₁, and one of temperature ϕ₂ at pressure p₂. If both have the same potential temperature, it follows that

or

---

The latter equations tell that if the air mass originally characterized by ϕ₂, p₂ is brought adiabatically to pressure p₁, its temperature has been changed to ϕ₁, and, similarly, that the air mass which originally was characterized by ϕ₁, p₁ attains the temperature ϕ₂ if brought to pressure p₂. Thus, no alteration of the distribution of mass is made by an exchange, and such an exchange has no influence either on the potential energy of the system or on the entropy of the system. In an ideal gas the surfaces of potential temperature are therefore isentropic surfaces.

With regard to the ocean, the question to be considered is whether surfaces of similar characteristics can be found there. Let one water mass at the geopotential depth D₁ be characterized by salinity S₁ and temperature ϕ₁, and another water mass at geopotential depth D₂ be characterized by salinity S₂ and temperature ϕ₂. The densities in situ of these small water masses can then be expressed as σ_s1,ϑ1,D1 and σ_s²,ϑ2,D2.

Now consider that the mass at the geopotential depth D₁ is moved adiabatically to the geopotential depth D₂. During this process the temperature of the water mass will change adiabatically from ϕ₁ to θ₁ and the density in situ will be σ_s1,θ1,D2. Moving the other water mass adiabatically from D₂ to D₁ will change its temperature from ϕ₂ to θ₂. If the two water masses are interchanged, the conditions

must both be fulfilled if the distribution of mass shall remain unaltered. These conditions can be fulfilled, however, only in the trivial case that S₁ = S₂, ϕ₁ = ϕ₂, and D₁ = D₂. This is best illustrated by a numerical example. Assume the values These values represent conditions encountered in the Atlantic Ocean, but at a distance of about 50° of latitude.

The adiabatic change in temperature between the geopotential depths of 200 and 700 dyn meters is 0.09°, and thus θ₁ = 13.82, θ₂ = 8.01. By means of the Hydrographic Tables of Bjerknes and collaborators, one finds

Thus, the conditions (XII, 10) are not both fulfilled and the two water masses cannot be interchanged without altering the distribution of mass.

It should also be observed that the mixing of two water masses that are at the same depth and are of the same density in situ, but of different temperatures and salinities, produces water of a higher density. If, at D = 700 dyn meters, equal parts of water S₁ = 36.01 ‰, ϕ₁ = 13.82°,

---

and S₂ = 34.60 ‰, ϕ₂ = 8.10°, respectively, are mixed, the resulting mixture will have a salinity S = 35.305 ‰, and a temperature ϕ = 10.96°. The density in situ of the two water masses was identical (σ_s,ϕ,D = 30.24), but the resulting mixture has a higher density, 30.29. Similarly, if equal parts of the water masses S₁ = 36.01 ‰, ϕ₁ = 13.73°, D₁ = 200 dyn meters, and S₂ = 34.60 ‰, ϕ₂ = 8.01°, and D₂ = 200 dyn meters are mixed, the density in situ of the mixture will be 27.98, although the densities of the two water masses were 27.97 and 27.92, respectively.

This discussion leads to the conclusion that in the ocean no surfaces exist along which interchange or mixing of water masses can take place without altering the distribution of mass and thus altering the potential energy and the entropy of the system (except in the trivial case that isohaline and isothermal surfaces coincide with level surfaces). There must exist, however, a set of surfaces of such character that the change of potential energy and entropy is at a minimum if interchange and mixing takes place along these surfaces. It is impossible to determine the shape of these surfaces, but the σ_t surfaces approximately satisfy the conditions. In the preceding example, which represents very extreme conditions, the two water masses were lying nearly on the same σ_t surface (σ_t1, = 27.05, σ_t2 = 26.97).

Thus, in the ocean, the σ_t surfaces can be considered as being nearly equivalent to the isentropic surfaces in a dry atmosphere, and the σ_t surfaces may therefore be called quasi-isentropic surfaces. The name implies only that interchange or mixing of water masses along σ_t surfaces brings about small changes of the potential energy and of the entropy of the body of water.