R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
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Ocean Sci., 6, 633-677, 2010 
The equilibrium is computed using this approach with the 
library call set_ice_air_eq_at_a_t or using the function 
ice_air_condensation_pressure_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 sublimation temperature T=p subl (p va P) 0 f water is com 
puted from Eq. (5.8). With A, T and P available, the required 
density estimate of humid air, p AV =l/g AV (A, T, P), is eas 
ily calculated from the related Gibbs function, Eq. (4.37). 
Using A A =0 and AP=0, the linear system (Eqs. A53, A54) 
can now be solved iteratively for T and p AV . 
In particular, this solution provides the frost point temper 
ature P(A, P) of humid air as a function of the air fraction 
and the pressure. 
The equilibrium is computed using this approach with the 
library call set_ice_air_eq_at_a_p or using the function 
ice_air_f rostpoint_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 sublimation 
pressure of ice at given T by solving Eq. (5.8). The vapour 
density follows from Eq. (4.3) as p v =l/g^ (P, p va P) and the 
air density from p A =l/g AV (1, T, P—P vap ). The air frac 
tion is then available from A—p A / (p A +p v ). 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 AT=0 and AP=0, the lin 
ear system (Eqs. A53, A54) can now be solved iteratively for 
A and p AV . 
In particular, this solution provides the specific humid 
ity q=l— A(T, P) of saturated humid air below the freezing 
point as a function of the temperature and the pressure. 
The equilibrium is computed using this approach with the 
library call set_ice_air_eq_at_t_p or using the function 
ice_air_massf raction_air_si. 
Case 4: Equilibrium at given air fraction, A, and 
entropy,)] 
At given A and rj, we use the approximate Clausius- 
Clapeyron equation to relate the partial vapour pressure at 
the frost point, p vap , to the temperature, T: 
P vap L / T t 
~P~ ~~r^tX ~ T 
(A55) 
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. (A55) by means of the relation In x«l — 1/x: 
P L T 
In — « In In x£ v (A56) 
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, relative to the triple point (P t , 
P t ) of water, 
n = % + A (c A In y - R a In y ^ (A57) 
+ (1 - A) (cj, Inj-Rw In y ^. 
We insert P from Eq. (A56) into Eq. (A57) and get the isen- 
tropic condensation temperature estimate T=P ICT (A, rj): 
Tier (A, r)) « 7) exp (A58) 
rj - rj t (A) - [AP a + (1 ~ A)R w ] In x AV (A) 
“ Tw\) + (1 “ A )( c f “ Tt) 
Here, at the given air fraction A, the triple-point 
entropy rj t (A) —rj(A,T t , P t ) =—g AV (A, P t , P t ) is computed 
from Eq. (S12.2), the mole fraction x av (A) from Eq. (2.11), 
the constants take the rounded numerical triple-point values 
P t =273.16K, P t =611.654771 Pa, c A =1003.69Jkg“ 1 K“ 1 , 
c^=1884.352Jkg -1 K -1 , R A =R/M A , R w = R/M w , and 
¿=2834359 J kg -1 is the sublimation enthalpy. The molar 
mass of air is MA=0.02896546kgmol _1 , that of water is 
Mw=0.018015268kgmol _1 , and P=8.314472 Jmol -1 K -1 
is the molar gas constant. 
With A and an estimated T available, we can now proceed 
as in case 1 to compute the remaining starting values for the 
iterative solution of the linear system (Eqs. A53, A54) of two 
equations for the three unknowns T, P and p AV using A A=0. 
A third equation must be added to the system, adjusting the 
humid-air entropy to the given value, i): 
- A A - / A ^ A T - / A J Ap AV = r, + / AV . (A59) 
This equation is valid for humid air at the frost point, i.e. ice 
air with a vanishing ice fraction. If the sample contains a 
finite amount of ice, its entropy must additionally be consid 
ered in Eq. (A59). 
In particular, the solution of case 4 provides the isentropic 
ice condensation level P (A, rj) of lifted humid air as a func 
tion of the air fraction and the entropy. 
The equilibrium is computed using this approach with 
the library call set_ice_air_eq_at_a_eta. Alterna 
tively, this state is computed from A, T and P of a sub 
saturated humid-air parcel having the same entropy and 
air fraction as the final saturated one by calling the func 
tions ice_air_icl_si or ice_air_ict_si to deter 
mine the isentropic condensation level or temperature.