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Ocean Sci., 6, 633-677, 2010 
and the potential density from Eq. (S14.1), pe, library func 
tion air_potdensity_si, 
-l 
Pg 
dh AW (A, i],P r )\ 
SPr ) A ' V 
(4.45) 
In terms of the Primary Standard functions and their indepen 
dent variables (Sect. 2), Eq. (5.1) is expressed by the system 
^Jr 1 ( 7' ^ W \ ^ 
/ F (i> w ) + P7„ F (r,„'*') 
= / F (r,p'j+p'7J(r,p'j 
Evidently, for any fixed reference pressure, P r , the value of 
/z AV (A,r), P r ) and its partial derivatives, as well as any other 
arbitrary function depending on this triple of variables, re 
main unchanged during isentropic processes (rj = const) at 
constant specific humidity (A = const). 
Physically reasonable values of the entropy to be used 
as an independent variable of the enthalpy are restricted to 
ranges depending on humidity and pressure, between the par 
ticular limits given by dry and saturated air, see Sect. 5.8. 
5 Level 4: Phase equilibria and composite systems 
Equilibrium properties at phase transition boundaries or of 
coexisting phases are often characterized by drastic spatial 
or temporal changes, and large values of latent heat exchange 
or volume expansion, e.g. if seawater freezes or evaporates. 
Such multi-phase and multi-component properties are avail 
able from combinations of the thermodynamic potentials if 
they are consistently adjusted to reference state conditions 
which fix the absolute energies and entropies of the sub 
stances involved (Feistel et al., 2008). Gibbs functions can 
be constructed for composite systems such as sea ice (Feistel 
and Hagen, 1998, Sect. 5.4) that contain two stable phases 
(e.g. ice and seawater). When the temperature, the volume 
or the pressure of a composite system is changed, mass is 
transferred from one phase to the other; for example if sea 
water freezes or evaporates, brine salinity or vapour pressure 
adjust to the new conditions imposed and the heat capacity 
or the thermal expansion of the whole system exhibits very 
large changes resulting from the changes in latent heat contri 
butions. By utilizing mutually consistent potential functions, 
rigorous mathematical formulae can be determined for the 
numerical calculation of latent properties depending on the 
particular conditions such as isobaric, isochoric or isentropic 
processes. 
5.1 Equilibrium liquid water-vapour: saturation 
The saturation point of pure water is usually computed at 
a given temperature T, providing the vapour pressure P — 
P vap (r), or at a given pressure P providing the boiling tem 
perature T — r boll (P). The defining condition is equality of 
the chemical potentials of liquid and vapour, which equal the 
Gibbs functions in the case of pure phases, 
g w (P,P) =g w (T,P). (5.1) 
(p W f fp( T ’P W ) = P (5-3) 
(p v ) 2 /j(r,p v )=P (5.4) 
which exploits the relations (Eqs. S2.6 and S2.ll) to avoid 
stacked numerical iterations. Eq. (5.2) is equivalent to 
Eq. (5.1) and is also known as the “Maxwell condition” in 
p w 
the form / |Vj—^-Jdp=0. Equations (5.2)-(5.4) provide 
p v 
three equations for the four unknowns T, P, p v and p w . 
Any one of these quantities can be specified independently 
to complete the system and permit the numerical solution as 
discussed in Appendix A3. 
Once the values of T, P, p v and p w are computed from 
the iteration of Eqs. (5.2)-(5.4) at the specified saturation 
condition, various equilibrium properties can be determined 
from the formulae given in Table S15. 
5.2 Equilibrium water-ice: melting and freezing 
The melting pressure of ice is usually computed at a given 
temperature T, giving P melt (r). Similarly, the freezing tem 
perature of water is normally determined at a given pressure 
P, giving P frz (P), which also gives the melting temperature 
of ice. In either case, the defining condition is equality of the 
chemical potentials of liquid water and ice, 
g w (T,P) =g m (T,P). (5.5) 
In terms of the Primary Standard functions and their indepen 
dent variables (Sect. 2), Eq. (5.5) is represented as the system 
f (P, p w ) + p w /J (p, p w ) = g lh (T, P) (5.6) 
(p W f fp( T 'P W ) = P ’ (5 ' 7) 
which exploits the relations (Eqs. S2.6 and ES2.11) to avoid 
stacked numerical iterations. Equations (5.6) and (5.7) sup 
ply two equations for the three unknowns P, P and p w . 
Specifying any one of these quantities completes the deter 
mination of the system which can then be solved as discussed 
in Appendix A4. 
Once the values of P, P and p w are computed from the 
iteration of Eqs. (5.6), (5.7) at the specified melting condi 
tion, various equilibrium properties can be determined from 
the formulae given in Table S16.