648 
R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
Ocean Sci., 6, 633-677, 2010 
www.ocean-sci.net/6/633/2010/ 
5.3 Sublimation equilibrium ice-vapour 
The sublimation pressure of ice is usually computed at a 
given temperature P, giving P subl (r). Similarly, the sub 
limation temperature of ice is usually computed at a given 
pressure P, giving r subl (P), which also gives the ice-point 
temperature of vapour at which frost is formed. The defin 
ing condition is equality of the chemical potentials of water 
vapour and ice, 
g v (P,P) = g Ih (P,P). (5.8) 
In terms of the Primary Standard functions and their inde 
pendent variables (Sect. 2), Eq. (5.8) is represented by the 
system 
f (p, P v ) + P v /J (p, P v ) = Z 1 (P, P) (5.9) 
(p v ) 2 /j(p,p v ) =P, (5.10) 
which exploits the relations (Eqs. S2.6 and S2.11) to avoid 
stacked numerical iterations. Specifying any one of P, P 
and p v , completes the system and allows numerical solution 
as discussed in Appendix A5. 
Once the values of P, P and p v are computed from the 
iteration of Eqs. (59) and (5.10) at the given sublimation 
condition, various equilibrium properties can be determined 
from the formulae given in Table S17. 
5.4 Equilibrium seawater-ice: sea ice 
The freezing temperature of seawater with absolute salin 
ity Sa is usually computed at a specified pressure P giving 
P frz (SA, P). Similarly, the brine salinity of sea ice is cal 
culated at a specified temperature P and pressure P giving 
.S' b "" c (T. P), and the melting pressure at which the solid frac 
tion of sea ice disappears is calculated at specified Sa and P 
giving p melt (5 A ,P). The defining condition for each of these 
is equality of the chemical potentials of ice and of water in 
seawater, 
/ 3e sw \ 
g m (T,P) = g sw (S A ,T,P) - (5.11) 
In terms of the Primary Standard functions and their inde 
pendent variables (Sect. 2), Eq. (5.11) is represented as the 
system 
g* = f w + p w + g s - S A g s s (5.12) 
(p W ) 2 / p W = P, (5-13) 
which exploits the relations Eqs. (S2.6), (S2.ll), (4.4) 
and (S7.12) to avoid stacked numerical iterations. The func 
tion / F (P,p w ) is abbreviated here by / w , and similarly for 
its partial derivatives. Equations (5.12) and (5.13) provide 
two conditions for the four unknowns Sa, P, P and p w . To 
complete the system, two of these variables must be speci 
fied, usually out of the triple Sa, P or P. Once two of these 
variables are specified, the system may be solved iteratively 
as discussed in Appendix A6. 
Once the values of Sa, P, P and p w are computed from 
the iteration of Eqs. (5.12) and (5.13) for the specified choice 
of parameters, various single-phase equilibrium properties 
can be determined from the formulae given in Table S18. 
The composite system “sea ice” consisting of seawater and 
ice can be described by a suitable Gibbs function g SI (Ssi, 
P, P) which is available from the equilibrium solution of 
Eq. (5.11) and can be used to compute all thermodynamic 
properties of this two-phase system, in particular its latent 
heat (for details see Feistel and Hagen, 1998): 
g sl (S sl , P, P) = (l-b)g Ih (P, P)+bg sw (S A , T, P). (5.14) 
Here, ¿’=5'si/5'a<1 is the mass fraction of brine and .S’si is 
the given “bulk” or sea-ice salinity, i.e. the mass fraction of 
salt in sea ice, in contrast to the brine salinity Sa(T, P), the 
mass fraction of salt in the liquid part, which is a function 
of temperature and pressure controlled by the equilibrium 
Eq. (5.11). For a compact writing of the partial derivatives 
of Eq. (5.14) it is useful to define a formal latency operator 
of sea ice, 
Asitz] = zSW -^^^ -z Ih - (5.15) 
Here, z is a certain thermodynamic function. For example, 
the equilibrium condition (Eq. 5.11) can be written in the 
form 
Asitg] = 0. 
(5.16) 
The total differential of Eq. (5.16) is commonly known as the 
Clausius-Clapeyron differential equation of this phase tran 
sition: 
dii>A + 
/Msilili 
V 3J> ) SJ 
The first term yields the chemical coefficient, Eq. (S4.6), 
which has a positive sign as can be concluded from the Sec 
ond Law of Thermodynamics (Landau and Lifschitz, 1964; 
aAsitgn dr 
3T J, 
(5.17) 
S,P 
I 3A S i [g]\ 
V dS A )