638 
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/ 
equation without the cancelling terms increases speed and ac 
curacy. In such cases a function is more naturally (and easily) 
implemented by calling the separate functions (Eqs. 2.3-2.5) 
rather than their combination in g s (S A , T, P). 
The expansion terms gi(T,P), Eqs. (2.3-2.5), together 
with their partial derivatives are available from the library 
function sal_g_term_si. 
2.4 Humid air 
For a correct description of the thermodynamic properties at 
the ocean-atmosphere interface a thermodynamic potential of 
humid air is required and available from the literature (Feis 
tel et al., 2010a). A related document is in preparation by 
IAPWS (2010). The Helmholtz function for dry air of Lem 
mon et al. (2000) has the form of the molar Helmholtz en 
ergy, / A,mo1 (T, p mo1 ), depending on absolute temperature T 
(ITS-90) and molar air density, p mo1 . For its conversion to 
the specific Helmholtz energy, / A , depending on the mass 
density, p, 
f A (T,p) = -L/ A ’ mol (V,— 
M A J V M A 
(2.6) 
the molar mass of air, Ma=28.965 46 g mol -1 , is computed 
from the recent highly accurate air composition model of Pi 
card et al. (2008). The dry-air part (Eq. 2.6) can be combined 
with the vapour part, / V =/ F (IAPWS-95, Sect. 2.1), in 
volving the second virial coefficient B A w(T) of Harvey and 
Huang (2007) and the third virial coefficients C AA w(T) and 
Caww ( 7 ) of air-vapour interaction reported by Hyland and 
Wexler (1983), to obtain the Helmholtz function of humid 
rAV 
air, / , as 
/ AV (A, T, p) = ( 1 - A) / v (T, ( 1 - A) p) + A/ a (T, Ap) (2.7) 
+2A(1-A)p 
RT 
M A M W 
B A w(T) + -p 
A (1—A) 
-rr- C A aw(T)-\———C A ww(T) 
M A «W 
Here, p is the density of humid air, A is the mass frac 
tion of dry air in humid air, q—\—A is the specific hu 
midity, (1—A)p is the absolute humidity, and r—(\—A)/A 
the humidity ratio or mixing ratio (van Wylen and Sonntag, 
1965; Gill, 1982; Emanuel, 1994). R=8.314 51 J mol -1 K -1 
is the molar (or universal) gas constant used by Lem 
mon et al. (2000), rather than the most recent value 
of R=8.314 472 J mol -1 K -1 (Mohr et al., 2008), and 
Mw=0.018015268kgmol -1 is the molar mass of pure wa 
ter (IAPWS, 2008b). The effective molar mass of humid air 
Mav depends on the mass fraction A in the form 
1 
M AV = (2.8) 
(1 - A)/M w + A/M a 
The mass fraction A of air is computed from the mole frac 
tion x A of dry air as 
x A M A 
x A M A + (1 - x A )M w 
(2.9) 
_ XA 
~~ 1 - (1 — x A ) (1 — M w /M A ) ’ 
and it follows that the mass fraction of vapour is given by 
_ ^ _ 1 x A M A 
x A M A + (1 - x A )M w 
1 ~ * A 
1 — Xa( i _m a /M w )‘ 
(2.10) 
The inverse function of Eq. (2.9), i.e., the mole fraction of air 
as a function of the mass fraction of air, is 
= x _ (1 - A)/Mw 
(1 - A)/M w + A/M a 
A(M w /M a ) 
1 - A(1 - M W /M A ) 
(2.11) 
and the related mole fraction of vapour is 
(1 - A)/My 
(1 - A)/M w + A/M a 
1 - A 
(2.12) 
1 - A(1 - M W /M A )' 
The Helmholtz potential (Eq. 2.7) is formally symmetric in 
the fractions of air and of water vapour. We note that the 
Helmholtz functions / v and f A that we have chosen to use 
in Eq. (2.7) are complete expressions rather than truncated 
expansions in terms of powers of density. Consequently, 
they include contributions corresponding to higher powers 
of density than included in the cross-virial terms represented 
by the third term in Eq. (2.13), / mlx =/ AV —A/ a —(1—A)/ v . 
Equation (2.7) is thus an inhomogeneous approximation for 
mula with respect to the powers of density and the related 
correlation clusters. However, its validity is not restricted to 
small specific humidity, q={\—A), such as some 1-3% of 
ten assumed for empirical equations used in meteorology. It 
can even be applied to physical situations in which air is the 
minor fraction, such as condensers of desalination plants or 
headspaces over subglacial lakes. The mass fraction A rather 
than the specific humidity q is chosen as a composition vari 
able of humid air for its analogy to Absolute Salinity; the two 
describe the amount of natural mixtures, gases or salts, con 
tained in ambient water in either gaseous or liquid form. This 
leads to thermodynamic equations that are formally similar 
in A and 5 a (Feistel et al., 2010a). 
The range of validity is bounded by the simultaneous va 
lidity of the vapour formula (IAPWS-95), of the dry-air for 
mula (Lemmon et al., 2000) and of the cross-virial expan 
sion. The dry-air function correctly describes reliable ex 
perimental data for pressures up to 70 MPa and for tempera 
tures from 60 to 873 K; the maximum air density in this re 
gion is 1035.8 kg/m 3 . The temperature range where all three