R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
655 
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Ocean Sci., 6, 633-677, 2010 
Entropy Bounds of Wet Air 
-10000 
-9000 
-8000 
-7000 
-6000 
-5000 
-4000 
-3000 
-2000 
-1000 
-0 
-1000 
0 10 20 30 40 50 60 70 80 90 100 
Air Fraction in % 
Fig. 9. Valid entropy values of wet air computed from Eq. (5.64) as 
arguments of enthalpy h AW ('w A , 77, P^j are restricted to triangular 
regions, depending on the air fraction w A between 0 and 100% for 
selected pressures P as shown. At the upper entropy bound, the liq 
uid phase is completely evaporated, given by the solution T(A, P) 
of case 2 in Appendix A10, indicated as the “Dewpoint” lines in the 
figure. At the lower bound, the condensate of the wet-air sample 
starts freezing, Eq. (5.64), indicated here as the lines radiating from 
(77, A), and referred to as “Ice Formation” lines. The envelop on 
which the triangles’ air-fraction maxima are located is the triple 
line, shown dashed, where ice, liquid water and vapour coexist in 
the presence of air, as described in Sect. 5.10, Eq. (S28.8). Note 
that the vapour-pressure lowering of water due to dissolved air is 
neglected in the equations. Freezing curves were computed with 
the library functions ice_liqjneltingtemperature_si 
and liq_air_g_entropy_si, dewpoint curves using 
liq_air_dewpoint_si and air_g_entropy_si. 
For running uj a , the triple line is computed by 
calling the sequence set_liq_ice_air_eq_at_a, 
liq_ice_air_temperature_si, 
liq_ice_air_pressure_si and air_g_entropy_si. 
The related library functions are 
liq_air_potenthalpy_si, Eq. (5.67), 
liq_air_pottemp_si, Eq. (5.68), and 
liq_air_potdensity_si, Eq. (5.69). 
Potential enthalpy is a measure of the “heat content” of wet 
air in the sense of von Helmholtz’ (1888) suggestion and was 
introduced into oceanography by McDougall (2003). The 
formula for the computation of the meteorological wet-bulb 
temperature from the enthalpy of humid air is given on page 
1.4-28 of WMO (2008). 
5.9 Equilibrium humid air - ice 
When humid air is in equilibrium with ice, its state is referred 
to as “saturated ice air” or the “frost point”. The thermody 
namic relations for this state are quite similar to those of the 
previous Sect. 5.8 except that the Gibbs function g w of liquid 
water is replaced by the Gibbs function g ,h of ice. The condi 
tion for this equilibrium is equality of the chemical potentials 
of ice, Sect. 2.2, and of water in humid air, Eq. (S12.15): 
8AV - A (1v) r ,,= 8 "’ (570) 
In terms of the Primary Standard functions and their indepen 
dent variables (Sect. 2), Eq. (5.70) is expressed by the system 
g AV = / AV ^A, T, p AV J + P/p AV (5.71) 
(5.72) 
The independent variables in this scheme are the total pres 
sure, P, the humid-air density, p AV , the temperature, T, 
and the air fraction, A. Expressing the chemical potential 
in Eq. (5.70) by means of Eq. (5.71) gives two equations in 
these four unknowns. Once two of the unknowns are spec 
ified, then the system is closed and may be solved numer 
ically. Four important cases are discussed in detail in Ap 
pendix All. 
No matter which of the cases 1-4 considered in Ap 
pendix All is applied to compute the equilibrium between 
ice and humid air, the numerical solution of Eqs. (5.70)- 
(5.72) results in a consistent set of equilibrium values for A, 
T, P and p AV which is then available for the computation of 
any other property of either saturated humid air or ice in this 
state. 
The definitions (Eqs. 5.54, 5.55) and their inverse func 
tions (Eqs. 5.56, 5.57) remain unchanged below the freezing 
temperature except that A sat (T, P) must now be computed 
from Eqs. (5.71), (5.72). The related library functions 
are ice_air_rh_wmo_f rom_a_si, 
ice_air_rh_cct_f rom_a_si, 
ice_air_a_from_rh_wmo_si and 
ice_air_a_f rom_rh_cct_si. 
With “ice air” we refer to a composite system of ice 
and humid air (e.g. a cirrus cloud) with the mass frac 
tions w A of dry air, uj v of vapour and uj 111 of ice satis 
fying u) A +u) V -|-u) Ih =l. The mutual equilibrium requires 
A sat (T, P)=w A /w Ay , with u) AV =ui A -Hn v =l—ui 111 being the 
gaseous mass fraction. The Gibbs function of ice air reads 
(Feistel et al., 2010a), 
g AI (uAr,p) = 
/ W A 
+ 1 - 
A sat (P, P) 
g AW (A sat , T, P) 
(5.73) 
A sat (T,P) 
g lh (T,P), 
and is a linear function of the air fraction, w A . Various ice-air 
properties are available from combinations of partial deriva 
tives of the potential (Eq. 5.73) with A sal (T. P) computed