670 
R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-10: Part 1 
Ocean Sci., 6, 633-677, 2010 
www.ocean-sci.net/6/633/2010/ 
A12 Equilibrium conditions for liquid water, ice and 
water vapour in air (Sect. 5.10) 
To determine equilibrium conditions for liquid water, ice 
and water vapour in air, we first expand the resulting four 
Eqs. (5.83; two equations), (Eqs. 5.85 and 587) with respect 
to small changes of the five independent variables: 
AfP A AA + (/» - Affi - /») 
AT 
P 
(A60) 
p AV pw) AP +(/p AV A fZ -^ AV ^ 
Ap 
AV 
-1/7- 
(P w r 
| Ap W —P 
1 
1 
p w p AV 
+ / W - / AV + a/ AV 
+ (/r AV - Af% ~ *?) 
AT 
Ap AV = g Ih - / AV - + Aff 
P 
(P AW Y 
(A61) 
The total pressure is estimated from the mole fraction 
x av (A), Eq. (2.11), as P=P t /x AV (A). With A, T and P 
available, the required density estimates for liquid water, 
p w =l/ g J(T, P), and for humid air, p AV =l/g AV (A, T, P), 
are easily calculated from the related Gibbs functions, 
Eqs. (4.2) and (4.37). 
The equilibrium of wet ice air is computed using this ap 
proach with the library call set_liq-ice_air_eq_at-a. 
Case 2: Equilibrium at given pressure, P 
The temperature of wet ice air is only slightly different 
from the triple-point temperature, T — r t =273.16K, which 
is used as an initial estimate. The partial pressure of vapour 
is close to the triple-point pressure, P t =611.654771 Pa. From 
the related mole fraction estimate, xv=Pt/P, the mass frac 
tion A is computed, Eq. (2.9). With A, T and P available, the 
required density estimate for liquid water, p w —l/g^(T, P), 
and for humid air, p AV =l/g AV (A, T, P), are easily calcu 
lated from the related Gibbs functions, Eqs. (4.2) and (4.37). 
The equilibrium of wet ice air is computed using this ap 
proach with the library call set_liq-ice_air_eq_at-p. 
Case 3: Equilibrium at given temperature, T 
For brevity, / F (P, p w ) is abbreviated here by / w and 
similarly for its partial derivatives. For the numerical 
solution, one additional condition is needed, such as 
specification of temperature or pressure, AT=0 or AP=0. 
Then appropriate starting values are required to initialize 
the iterative determination of the remaining unknowns. 
Three important cases are considered in the following. The 
solution of Eqs. (A60)-(A63) does not provide the relative 
mass fractions of the three phases. Two more conditions are 
required to fix the latter quantities. Cases 4 and 5 address 
this issue. 
Case 1: Equilibrium at given dry-air fraction of the 
humid-air part, A 
The temperature of wet ice air is only slightly different 
from the triple-point temperature, T=P t =273.16 K, which is 
used as an initial estimate. The partial pressure of vapour 
is close to the triple-point pressure, P t =611.654771 Pa. 
At the temperature T, the pressure of wet ice air equals 
the melting pressure of ice, P=P melt (P), as the solution of 
Eq. (5.5). The partial pressure of vapour is close to the triple 
point pressure, P t =611.654771 Pa. From the related mole 
fraction estimate, xy=Pt/P, the mass fraction A is com 
puted, Eq. (2.9). With A, T and P available, the required 
density estimate for liquid water, p w —l/g^(T,P), and for 
humid air, p AV =l/g AV (A, T, P), are easily calculated from 
the related Gibbs functions, Eqs. (4.2) and (4.37). 
The equilibrium of wet ice air is computed using this ap 
proach with the library call set_liq-ice_air_eq_at-t. 
In the cases 1-3 above, the solution of Eqs. (A60)-(A63) 
defines the intensive properties A, T, P of the equilib 
rium but does not provide the relative mass fractions of 
the three phases present. The nonnegative fractions of dry 
air, w A , vapour, w y , liquid water, w w , and ice, w lh , are 
subject to only two equations, u) A +u) V -|-u) W +u) Ih =l, and 
w A l{w A +w y )—A. Thus, two additional conditions beyond 
those used in cases 1-3 are required to specify the state of 
the parcel completely. 
Alternatively, three conditions independent of the cases 1- 
3 may be given. Two important cases, 4 and 5, are considered 
in the following. 
Case 4: Equilibrium at given dry-air fraction, u> A , liquid 
fraction, u) w and ice fraction, u> Ih 
In this case, the fractions of the sample’s phases are given 
and the necessary T — P conditions are calculated.