R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
667 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
and expand the resulting three Eqs. (5.48), (5.50) and (5.52) 
with respect to small changes of the five variables: 
AT 
pAV p 
A p+ /AV- A/ av 
(p AV r 
(A45) 
Ap AV - /7 - 
Ap 
w 
= ' > I^-^) + / w -/ av + *A av 
P AW fìp AA + P AV /^Ar - — (A46) 
P 
+ (2) f + AP AV = -A - P«/f 
p»/ r >r-^+(2/>p»/»)A i ,»' ( A47) 
— JL- _ n w f w 
- pW p h ■ 
For brevity, / F (r, p w ) is abbreviated here by / w as 
well as its partial derivatives. For the numerical solution, 
two additional conditions are required, such as specifying 
the temperature and pressure, so that AT=0 and AP=0. 
Appropriate starting values of the remaining unknowns must 
be specified for their iterative determination. Four important 
special cases are considered in the following. 
Case 1: Equilibrium at given air fraction, A, and 
temperature, T 
At given A and T, humid air can approximately be con 
sidered as an ideal mixture of air and vapour. The partial 
pressure P vap of vapour is computed from the vapour pres 
sure of liquid water at given T by solving Eq. (5.1). The 
vapour density follows from Eq. (4.3) as p v =l/g^ (T, P vap ). 
For the dry-air density we have p A =p v xA/(l— A). The 
partial pressure of dry air is computed from Eq. (S5.ll) 
as P air =(p A ) 2 / AV (l, t, p A ). This provides an estimate 
for the total pressure, P=P vap +P a ". With A, T and 
P available, the required density estimate of liquid water, 
p w =l/ g y(T,P), and of humid air, p AV =l/g AV (A, T, P), 
are easily calculated from the Gibbs functions, Eqs. (4.2) 
and (4.37). Using AA=0 and AP=0, the linear system 
(Eqs. A45-A47) can now be solved iteratively for P, p w and 
p AV . 
In particular, this solution provides the pressure P(A,T) 
of saturated humid air as a function of the air fraction and the 
temperature. 
The equilibrium is computed this way by the library 
call set_liq_air_eq_at_a_t or by the function 
liq_air_condensationpressure_si. 
Case 2: Equilibrium at given air fraction, A, and 
pressure, P 
At given A and P, humid air can approximately be con 
sidered as an ideal mixture of air and vapour. The partial 
pressure P vap =x AV P of vapour is computed from the total 
pressure P and the mole fraction x AV (A), Eq. (2.11). In turn, 
the boiling temperature T = p boll (p va P) 0 f water is computed 
from Eq. (5.1). With A, T and P available, the required den 
sity estimate of liquid water, p w —l/g^ (P, P), and of humid 
air, p AV = 1 /g AV (A ,T,P), are easily calculated from the re 
lated Gibbs functions, Eqs. (4.2) and (4.37). Using AA=0 
and AP=0, the linear system (Eqs. A45-A47) can now be 
solved iteratively for T, p w and p AV . 
In particular, this solution provides the dewpoint temper 
ature T(A,P) of humid air as a function of the air fraction 
and the pressure. 
This approach is used to compute the equilibrium with the 
library call set_liq_air_eq_at_a_p or using the function 
liq_air_dewpoint_si. 
Case 3: Equilibrium at given temperature, T, and 
pressure, P 
At given T and P, humid air can approximately be con 
sidered as an ideal mixture of air and vapour. The partial 
pressure P vap of vapour is computed from the vapour pres 
sure of liquid water at given T from solving Eq. (5.1). The 
vapour density follows from Eq. (4.3) as p v =l/g^ (P, p vap ) 
and the air density from p A =l/g AV (l, T, P—P vap ). Now the 
air fraction is available from A—p A /(p A +p v ). With A, T 
and P available, the required density estimate of liquid water, 
p w =l/ g y(T,P), and of humid air, p AV =l/g AV (A, T, P), 
are easily calculated from the related Gibbs functions, 
Eqs. (4.2) and (4.37). Using AT=0 and AP=0, the linear 
system (Eqs. A45-A47) can now be solved iteratively for A, 
p w and p AV . 
In particular, this solution provides the specific humidity 
q=1 — A(T. P) of saturated humid air as a function of the 
temperature and the pressure. 
The equilibrium is computed using this approach with the 
library call set_liq_air_eq_at_t_p or using the function 
liq_air_massf raction_air_si. 
Case 4: Equilibrium at given air fraction, A, and 
entropy, )] 
At given A and p, we use the approximate Clausius- 
Clapeyron equation to relate the partial vapour pressure at