### Significance of σ_{t} Surfaces

The density of sea water at atmospheric pressure, expressed as σ_{t} = (ρ_{s,ϕ,0} − 1) × 10^{3}, 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

[Equation]

*p, p*

_{0}is the standard pressure, and κ = 1.4053 is the ratio of the two specific heats of an ideal gas (

*c*). 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.

_{p}/c_{v}Consider two air masses, one of temperature ϕ_{1} at pressure *p*_{1}, and one of temperature ϕ_{2} at pressure *p*_{2}. If both have the same potential temperature, it follows that

[Equation]

[Equation]

_{2},

*p*

_{2}is brought adiabatically to pressure

*p*

_{1}, its temperature has been changed to ϕ

_{1}, and, similarly, that the air mass which originally was characterized by ϕ

_{1},

*p*

_{1}attains the temperature ϕ

_{2}if brought to pressure

*p*

_{2}. 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 *D*_{1} be characterized by salinity S_{1} and temperature ϕ_{1}, and another water mass at geopotential depth *D*_{2} be characterized by salinity S_{2} and temperature ϕ_{2}. 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 *D*_{1} is moved adiabatically to the geopotential depth *D*_{2}. During this process the temperature of the water mass will change adiabatically from ϕ_{1} to θ_{1} and the density *in situ* will be σ_{s1,θ1,D2}. Moving the other water mass adiabatically from *D*_{2} to *D*_{1} will change its temperature from ϕ_{2} to θ_{2}. If the two water masses are interchanged, the conditions

[Equation]

*S*

_{1}=

*S*

_{2}, ϕ

_{1}= ϕ

_{2}, and

*D*

_{1}=

*D*

_{2}. This is best illustrated by a numerical example. Assume the values

[Equation]

The adiabatic change in temperature between the geopotential depths of 200 and 700 dyn meters is 0.09°, and thus θ_{1} = 13.82, θ_{2} = 8.01. By means of the Hydrographic Tables of Bjerknes and collaborators, one finds

[Equation]

*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 S_{1} = 36.01 ‰, ϕ_{1} = 13.82°,

_{2}= 34.60 ‰, ϕ

_{2}= 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 S

_{1}= 36.01 ‰, ϕ

_{1}= 13.73°,

*D*

_{1}= 200 dyn meters, and S

_{2}= 34.60 ‰, ϕ

_{2}= 8.01°, and

*D*

_{2}= 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.