R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
671 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
The air fraction A—w A /(w A +w y )—w A /(l — uj w — uj 111 ) 
is immediately and exactly computed from the input values. 
Case 3 above provides the algorithm to derive T and P from 
A. 
The equilibrium of wet ice air is computed using this 
approach with the library call 
set_liq_ice_air_eq_at_wa_wl_wi. 
Case 5: Equilibrium at given dry-air fraction, u> A , 
entropy, >/, and the liquid fraction of the 
condensed part, u>=u> w /(u> w +u> Ih ) 
This case is relevant to adiabatically lifted air parcels with 
conservative values of the first two parameters, the dry- 
air fraction, w A , and the entropy, )). The third parame 
ter, w=w w f(w w +w m ), varies between w= 0, i.e., the melt 
ing level (completely frozen condensate), and w= 1, i.e., the 
freezing level (completely molten condensate). Only points 
(w A , rj) selected from the regions shown in Fig. 12 permit 
valid solutions in this case. 
The temperature of wet ice air is only slightly dif 
ferent from the triple-point temperature, 7"~7i=273.16 K, 
which is used here as an initial estimate for the itera 
tive solution. Lacking a better simple estimate, we set 
the initial vapour fraction, w y , to 50% (or another frac 
tion) of the total water fraction, w y +w w +w m =l — w A , 
i.e., w y ?»(l —uj a )/2. The related air fraction of the gas 
phase is then A—w a / (w A +w y )^2w A / (l+w A y The par 
tial pressure of vapour is close to the triple-point pressure, 
Pt=611.654771 Pa. The total pressure is estimated from 
the mole fraction x av (A), Eq. (2.11), as P=P t /x Ay (A). 
With A, T and P available, the required density esti 
mates of liquid water, p w —l/g^(T,P), and of humid air, 
p AV =l/g AV (A, T, P), are calculated from the related Gibbs 
functions, Eqs. (4.2) and (4.37). Finally, the given en 
tropy, rj, and the liquid fraction of the condensed part, 
w—w w f(w w +w m ), are used to adjust the phase fractions 
to the entropy balance p=u) W p w +u) Ih p Ih +(u) A -|-u) V )p AV , 
which can be written in the form 
rj — — w ^1 — 
,A s 
,Ih 
fr ~ (1 - «0 
AV 
(A64) 
Expanded with respect to small changes of the independent 
variables, this equation is added to the system (Eqs. A60- 
A63) in the form 
[w (^A—w a ^J ff T +(l—w) {&—w A ^g l j T +w A (A65) 
AT + (1 — w)^A—w a ^Jg^pAP+w ^A—w a ^J 
The resulting system of five Eqs. (A60)-(A63), Eq. (A65) 
can be solved iteratively for the five unknowns A, T, P, p w , 
p AV , from which in turn all other properties can be evaluated. 
The equilibrium of wet ice air is computed using this 
approach with the library call 
set_liq_ice_air_eq_at_wa_eta_wt or using the func 
tions liq_ice_air_if l_si or liq_ice_air_iml_si. 
A13 Equilibrium conditions for seawater and water 
vapour in air (Sect. 5.11) 
To determine conditions under which water vapour in air ex 
ists in equilibrium with seawater, we first linearize the three 
Eqs. (5.88), (5.90) and (5.92) with respect to small changes 
of the six variables: 
S A g s ss AS A -Af Ay AA+ (A66) 
(/f-<-/r W -^+^r)Ar 
+ /< 
f AV 
(p AV ) 2 / ApAV V pW (p w ) 2 , 
\+f W -f AV +Aff+g s -S A g s s 
A p 
w 
p AV / A J A A + p AV / A J A T - — (A67) 
P 
+ (iff + P A X>P V = ^ - A V /p 4V 
P w /r>T - ££ + (2/7 + (A68) 
_ w f w 
- pW P Jp 
For brevity, / F (T, p w ) is abbreviated here by / w and 
similarly for its partial derivatives. For the numerical 
solution, three additional conditions must be specified. For 
example, salinity, temperature and pressure may be specified 
so AS a =0, AT=0 and AP=0. Appropriate starting values 
for the iterative determination of the remaining unknowns