R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
639 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
Density - Temperature Diagram of Dry Air 
-14-13-12-11-10-9 -8 -7 -6 -5-4-3-2-101234 
Density lg( p /(kg nr 3 ) ) 
a) Dry-Air Fraction Range above the Freezing Point 
Fig. 4. Validity range of the Helmholtz function for humid air, 
Eq. (2.7). For oceanographic and meteorological applications it is 
unnecessary to consider liquid or solid air. Thus, we restrict consid 
eration of Eq. (4.37) as follows: (i) for temperatures above the crit 
ical temperature of dry air, r>r c =132.5306K, all density values 
occurring between the pressure bounds are permitted; and (ii) for 
subcritical temperatures T<T C , only densities below the dewpoint 
curve of dry air, indicated by “Dewpoint” are permitted. The re 
sulting validity boundary for dry air is shown in bold. “CP” is the 
critical point of dry air. To consider humid air, virial coefficients 
are required. The validity range in temperature of the third virial 
coefficients is shown by horizontal lines. Additionally, the pressure 
on saturated humid air is restricted to 5 MPa (Hyland and Wexler 
1983), not shown. 
virial coefficients are valid is from —80 to +200 °C, Fig. 4, 
(Hyland and Wexler, 1983). Consequently, the most limit 
ing conditions for the validity of Eq. (2.7) are the tempera 
ture restrictions on the viral coefficients and the requirement 
for validity of the truncated virial expansion, i.e. the omit 
ted terms of / AV proportional to A 3 (l — A)p 3 , A 2 (\—A) 2 p i 
and A(\— A) 3 p 3 must be negligibly small in comparison to 
the retained terms. A rough estimate for a maximum valid 
density is 100 kg m -3 as concluded from a comparison with 
experimental data for saturated air in which substantial frac 
tions of both vapour and air are present (Feistel et al., 2010a; 
Fig. 8). When significant amounts of both air and water 
vapour are present, the valid temperature range is determined 
by the validity range for the virial coefficients. As the den 
sity of either the air or vapour component is decreased, the 
contribution from the virial coefficients decreases and the va 
lidity range in temperature extends to higher values, reaching 
873 K when water vapour is eliminated and 1273 K when air 
is eliminated. 
The air fraction is bound between 0 and 1 but is addition 
ally limited by the vapour saturation condition, Fig. 5. At 
high total pressures, the restriction to vapour pressures below 
the saturation value represents a significant limitation on the 
upper limit of 1—A that can be achieved in thermodynamic 
b) Dry-Air Fraction Range below the Freezing Point 
Temperature t /°C 
Fig. 5. Saturation curves A sat (r,P) of humid air at 
the pressures 101.325, 50, 20 and lOkPa, as indicated. 
Panel (a) shows results for temperatures above the freez 
ing point, computed by solving Eq. (5.48) using the library 
function liq_airjnassfraction_air_si, Eq. (S21.9), and 
panel (b) shows results for temperatures below the freez 
ing point, computed by solving Eq. (5.70) using the func 
tion ice.airjnassfraction_air_si, Eq. (S25.10). Valid 
air fraction values A are located above the particular satura 
tion curve, A>A sat (T,P), in the region indicated by “HU 
MID AIR”. In the presence of ice-free seawater, the validity 
range for A is more restricted, A>A cond (SA, T, P)>A sat (T, P), 
by the condensation value A cond , computed from the function 
sea_air jnassf raction_air_si, Eq. (S29.1). 
equilibrium. For total pressures below the vapour pressure 
of liquid water or the sublimation pressure of ice at the given 
temperature, the value of A may take any value between 0 
and 1. 
The Helmholtz function f A (T, p) for dry air together with 
its partial derivatives is implemented as the library function