where ( dp/dP)nS = 1/c2 (c is the speed of sound); the hydrostatic equilibrium gives dP/dz = —pg. Thereby, the buoyancy frequency is reduced to

p dz c2

2.4.14 Thermodynamics of seawater based on the Gibbs function

Due to the nonlinear property of the seawater, many thermodynamic quantities of seawater had to be determined through experimental measurements, and some other quantities had to be introduced through different thermodynamic relations. The use of different means of measurement and formulae can give rise to some inconsistency within the thermodynamic quantities calculated from different means. Fofonoff (1962) proposed to overcome such problems by defining one set of unified formulae in terms of the Gibbs function; thereby all other thermodynamic variables can be defined from the combination of the Gibbs function and its derivatives. About 30 years later, his idea was carried out by Feistel (1993,2003) and Feistel and Hagen (1995). By a least-square fitting to all available seawater thermodynamic properties, the Gibbs function has been expressed in terms of a power series of (T, S, P) with 100 double-precision coefficients. Since all thermodynamic functions are derived from the same set of polynomials, the calculated thermodynamic variables are self-consistent within the error bounds. This formulation is now available in both FORTRAN and Matlab (Feistel, 2005); thus, by using the standard functions, one can calculate all the thermodynamic properties, such as density, specific heat, specific enthalpy, and specific entropy. Table 2.5 lists some of the most commonly used thermodynamic functions defined through the Gibbs function. Here s denotes the mass fraction of salt, as it is the notation used for introducing the thermodynamics functions.

Although entropy is one of the fundamental thermodynamic variables of seawater, it has not been widely used in descriptive or theoretical studies of physical oceanography. Since no reliable formulae for calculating the entropy of seawater were available, there were very few studies related to entropy.

This situation changed considerably thanks to the recent work of Feistel and his colleagues, e.g., Feistel (1993,2003). Using standard subroutines based on the Gibbs function, all thermodynamic functions, in particular entropy, can now be calculated.

According to the formula, entropy depends on the temperature almost linearly, and it increases slightly with salinity (Fig. 2.18a). In fact, a linear function, n = c\T + c2, fits the more accurate calculation based on the standard subroutine within 1.5%, depicted as the thin

Table 2.5. Thermodynamics functions/variables in terms of Gibbs function

Thermodynamic functions

Formula based on Gibbs function

Specific enthalpy (heat content)

Specific free energy (Helmholtz free energy, available work)

Specific internal energy

Specific chemical potential

Specific chemical potential of water in seawater

Specific chemical potential of salt in seawater

Specific entropy Specific volume

Specific heat capacity (at constant pressure)

Haline contraction coefficient Thermal expansion coefficient Isothermal compressibility Speed of sound Adiabatic lapse rate

Vertical stability (Brunt-Vaisala frequency)

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