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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/ 
the dewpoint, P vap , to the temperature, T: 
pvap ¿ / Tt \ 
In (1 ) . 
Pt RwT t \ t) 
(A48) 
The vapour pressure is approximately equal to the partial 
pressure of vapour in humid air, P vap =x AV P, computed 
from the total pressure P and the mole fraction x av (A), 
Eq. (2.11). As an analytical estimate to be used below, we 
modify Eq. (A48) by means of the relation In x«l — 1/x: 
P L T AV 
In — « In In x ) v . (A49) 
P t R w T t T t v 
Assuming constant heat capacities, the ideal-gas entropy 
rj(A,T, P) of humid air is determined relative to the triple 
point (7), Pt) of water, 
rj = t] t + A (cp In y - R a In (A50) 
+ (1 — A) ^Cp In y -R W In ^0. 
this state may be computed from A, T and P of a subsat 
urated humid-air parcel having the same entropy and air 
fraction as the final saturated one by calling the functions 
liq_air_icl_si or liq_air_ict_si to determine its 
isentropic condensation level or temperature. 
All Equilibrium conditions for ice and water vapour in 
air (Sect. 5.9) 
To determine conditions for which water vapour in air will 
be in equilibrium with ice, we first expand the two Eq. (5.70) 
(with Eq. 5.71) used to eliminate the Gibbs potential) and 
Eq. (5.72) with respect to small changes of the four indepen 
dent variables: 
- Af™ A A + (/ AV - Af™ -gf)AT (A53) 
= - /” -~i + -vi v 
P 
We insert P from Eq. (A56) into Eq. (A57) and get the isen 
tropic condensation temperature estimate T=Tict(A, p): 
ricT(A,>;)~r t exp 
r,- % (A)-[A« a +(1—A)R w ] In Xy V (A) 
A ( c p-^) + d-A)(4-^) 
(A51) 
Here, at the given air fraction A, the triple-point 
entropy rjt(A)—ri(A,Tt,Pt)——gj W (A,Tt,Pt) is computed 
from Eq. (S 12.2), the mole fraction x AV (A) from Eq. (2.11). 
The constants take the rounded numerical triple-point values 
7t=273.16K, P t =611.654771 Pa, c A =1003.69Jkg -1 K -1 , 
c^=1884.352 Jkg -1 K -1 , R A =R/M A , R w = R/M w , and 
¿=2500915 Jkg -1 is the evaporation enthalpy. The molar 
mass of air is MA=0.02896546kgmol -1 , that of water is 
Mw=0.018015268kg mol -1 , and P=8.314472 Jmol -1 K -1 
is the molar gas constant. 
With A and an estimated T available, we can now pro 
ceed as in case 1 to compute the remaining starting values for 
the iterative solution of the linear system (Eqs. A45-A47) of 
three equations for the four unknowns T, P, p w and p AV us 
ing A 4=0. A fourth equation must be added to the system, 
adjusting the humid-air entropy to the given value, p: 
- f™ AA - f^AT - / A JAp AV = r, + ff (A52) 
This equation is valid for humid air at the dewpoint, i.e. wet 
air with a vanishing liquid fraction. If the sample contains 
a finite amount of liquid water, its entropy must additionally 
be considered in Eq. (A52). 
In particular, the solution of case 4 provides the isentropic 
condensation level P(A, p) of lifted humid air as a function 
of the air fraction and the entropy. 
The equilibrium is computed using this approach with the 
library call set_liq_air_eq_at_a_eta. Alternatively, 
p AV / A J A A + p AV / A J A T - — (A54) 
P 
+ (2/f+P''/* V )iP V = -p, - P V P V 
For the numerical solution, two additional conditions must 
be specified. For example, if we specify temperature and 
pressure then AT=0 and AP=0. Starting values are then 
required for the iterative determination of the remaining 
unknowns. Four important such 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 consid 
ered as an ideal mixture of air and vapour. The partial pres 
sure P vap of vapour is computed from the sublimation pres 
sure of ice at given T by solving Eq. (5.8). The vapour den 
sity follows from Eq. (4.3) as p v =l/g^ (T, P vap ). The dry- 
air density is then estimated as p A — p v x A/(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 ). Using this approach, we obtain 
an estimate for the total pressure, P=P vap +P alr . With A, T 
and P available, the required density estimate of humid air, 
p AV =l/g AV (A, T, P), is easily calculated from the related 
Gibbs function, Eq. (4.37). Using AA=0 and AT =0, the lin 
ear system (Eqs. A53, A54) can now be solved iteratively for 
P 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.