646 
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/ 
this purpose, the appropriate thermodynamic potential is the 
Gibbs function g AV (A, T, P) of humid air, computed from 
the Helmholtz function / AV (A, T. p) of humid air, Eq. (2.7), 
by the Legendre transform (Alberty, 2001) 
g AV = / AV 
— V 
3/ AV 
dv 
/*AV , „ 
= / +P 
A,T 
9/ AV \ 
) A,T 
(4.37) 
and the subsequent substitution of the independent variable 
p by P, obtained from solving numerically the equation 
P = p 
2 
9/ AV \ 
3P / A,T 
(4.38) 
For oceanographic and meteorological applications it is not 
necessary to consider liquid or solid air. Therefore, we re 
strict consideration of Eq. (4.37) to the following regions: 
(i) for temperatures above the critical temperature of dry 
air, T>r c =132.5306K, all pressures in the range shown in 
Fig. 4 are included; and (ii) at subcritical temperatures T<T C 
only temperatures higher than the dewpoint temperature 7b, 
i.e. the condensation point of liquid air, are included. The 
function T\) is available from Lemmon et al. (2000) and is 
shown in Fig. 4. 
As a starting value for the density iteration of Eq. (4.38) 
at given pressure P, air fraction A and temperature T, the 
ideal-gas equation is suitable: 
p « 
Mav04) p 
7IT 
(4.39) 
The molar mass of humid air Mav is given by Eq. (2.8), and 
77=8.314472 Jmol -1 K -1 is the molar gas constant. Insert 
ing the numerical result for p into Eq. (4.37) provides the 
required function value of g AV at given A, T, P. For the nu 
merical computation of partial derivatives of the Gibbs func 
tion, algebraic combinations of analytical derivatives of the 
Helmholtz function are implemented as given in Table S11. 
Thermodynamic properties of humid air at given A, T, 
P are computed from the Gibbs function (Eq. 4.37) and its 
partial derivatives, Table S11, as given in Table S12. The 
deviation of the compressibility factor Zav from unity de 
scribes the non-ideal behaviour. The adiabatic lapse rate is 
given with respect to pressure rather than altitude and refers 
to subsaturated humid air, often referred to as “dry-adiabatic” 
in the meteorological literature. The air contraction coef 
ficient, ft=—^(§j) TP , is the relative density increase if a 
small mass of vapour in a sample is replaced by air. Ad 
ditional equations for humid-air properties are discussed in 
Feistel et al. (2010a). 
4.5 Enthalpy of humid air 
When humid air is lifted adiabatically from the surface to a 
certain pressure level, its air fraction and its entropy can of 
ten be considered as conservative during this process. Thus, 
the entropy i) rather than the temperature T is known for a 
parcel at some given altitude if the initial entropy was com 
puted at the surface. For this application purpose, the ap 
propriate thermodynamic potential is the specific enthalpy 
/z AV (A, ij,P) of humid air, computed from the Gibbs func 
tion g AV (A, T, P) of humid air, Eq. (4.37), by the Legendre 
transform (Alberty, 2001) 
h AV = g AV_ T ^^^ (4.40) 
The subsequent substitution of the independent variable T by 
r) is obtained numerically from solving Eq. (S12.2) for T: 
n = - 
3g AV \ 
3 T ) a ,p 
(4.41) 
As a starting value for the iterative solution of Eq. (4.41) for 
T at given pressure P, air fraction A and entropy rj, we use 
an ideal-gas approximation of Eq. (S12.2) in the vicinity of 
the IAPWS-95 triple point (7), P t ) of water: 
T ~7"| exp 
n - m + Raw ln (P/P t ) 
A(c A + 77a) + (1 — j4)(c)> + Rw) 
(4.42) 
This expression does not depend on the particular 
choice made for the adjustable coefficients of the en 
tropy. The constants are rj t =rj(A,T t , P t ) computed 
from Eq. (S12.2) at P t =273.16K, P t =611.654771 Pa, 
c A =1003.69 Jkg -1 K“ 1 , c v=1884.352Jkg- 1 K“ 1 , 
77a=77/Ma, 77w=77/Mw and Raa/—R/Mav- The mo 
lar mass of air is MA=0.02896546kgmol _1 , that of water is 
Mw=0.018015268kgmol _1 , Mav is given by Eq. (2.8), and 
77=8.314472 J mol -1 K -1 is the molar gas constant. 
Once the value of T has been determined from solving 
Eq. (4.41), the partial derivatives of /z av (A,^,P) are ob 
tained from those of g AV (A,P, P) as given in Table S13. 
Thermodynamic properties as given in Table S14 can be cal 
culated from algebraic combinations of these derivatives. 
If a humid-air parcel is moved adiabatically to some refer 
ence pressure P=P r below its isentropic condensation level 
(ICL, Emanuel, 1994; Feistel et al., 2010a), all its thermody 
namic properties given in Table S14 can be computed at that 
reference level from the partial derivatives of /z av (A, rj, P r ) 
and the in situ entropy rj (A, T, P), Eq. (S 13.1). As discussed 
for seawater in Sect. 4.3, properties derived from the poten 
tial function /z AV at the reference pressure (frequently speci 
fied as the surface pressure) are commonly referred to as “po 
tential” properties in meteorology. Examples are the poten 
tial enthalpy, hg, library function air_potenthalpy_si, 
/z 0 =/z AV (A,rç,P r ), 
(4.43) 
the potential temperature from Eq. (S14.2), 6, in °C, library 
function air_pottemp_si, 
T o + 0 = 
dh AW (A, rç,P r )\ 
3*1 / A,P, 
(4.44)