654 
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/ 
g AW (w A ,T,P^j = 
A sat (T,P) 
g AV ( A sat ; T p) 
(5.58) 
+ 1 - 
w 1 
A sat (7\ P) 
g w (T,P) 
and is a linear function of the air fraction, w A . Various 
wet-air properties are available from combinations of par 
tial derivatives of the potential (Eq. 5.58) with A si "(T. P) 
computed from Eqs. (5.48)-(5.52) as in case 3 from Ap 
pendix AIO, see Table S21. For the computation of the 
partial T — P derivatives of g AW , the first derivatives of 
A sat (T, P) are required. Taking the respective derivatives of 
Eq. (5.48) we get the isobaric drying rate, 
_ ^AawW 
V 3t ) P d a ’ 
and the isothermal drying rate, 
(5.59) 
This is not a trivial task because a good analytical estimate 
for T AW (w A , rj, P) is not available, and the Newton iteration 
of Eq. (5.64) tends to be unstable so that the range of starting 
parameters that yields convergent solutions of Eq. (5.64) is 
rather restricted. An interval method like Brent’s algorithm 
appeared to be the best choice in this case, applied between 
upper and lower temperature bounds. These limits follow 
from the physical conditions that wet air can only exist be 
tween freezing and complete evaporation of the liquid water 
part. Thus, the lower temperature bound 7 nlm (i/ ,A . P) for the 
solution of Eq. (5.64) is the freezing temperature 7”" cll ( P) of 
water under the pressure P, 
g W (T rrin ,P) = g lh (T min ,P), (5.65) 
computed from Eq. (5.5). The upper temperature bound 
Tmax(wi A ,P) is computed from Eq. (5.48) in case 2 from 
Appendix A10 for a vanishing liquid fraction, Eq. (S21.9), 
i.e., the air fraction A of humid air equal to that of wet air, 
w a =A: 
3^ sat \ ^sat^AwM 
3P ) T D a 
(5.60) 
of humid air, i.e. the decrease of its saturated air fraction A sat 
due to heating or compression. The chemical coefficient D A 
is defined in Eq. (S12.16). The latency operator Aaw of wet 
air used here is defined for the specific entropy, rj AW ——g AW , 
of the form 
Aaw W = rj AW - A 
3 h AW \ 
3 A ) j p 
(5.61) 
and for the specific volume u AW =g AW of the form 
AawM = u AV — A 
dv AV 
dA 
— v 
w 
T,P 
(5.62) 
The partial derivatives of the Gibbs function g AW (w A , T, P), 
Eq. (5.58), of wet air are given in Table S20. Properties of 
wet air computed from this Gibbs function are given in Ta 
ble S21. 
For the description of isentropic processes such as the up 
lift of wet air in the atmosphere, enthalpy /z AW (w A , rj, P) 
computed from the Gibbs function (Eq. 5.58) is a useful ther 
modynamic potential: 
=g W (7max,.P) (5.66) 
A—vA, T—T max 
The entropy range corresponding to the interval 7 nun -7 nlax is 
shown in Fig. 9. 
The partial derivatives of the enthalpy h AW (w A , rj, P) are 
computed from those of the Gibbs function, Table S20, as 
given in Table S22. 
Selected properties of wet air computed from the en 
thalpy (Eq. 5.63) and its partial derivatives are given in Ta 
ble S23. 
Many meteorological processes such as adiabatic uplift 
of a wet-air parcel conserve specific humidity and entropy 
to very good approximation. In particular, if a parcel is 
moved this way to some reference pressure P=P r , all its 
thermodynamic properties given in Table S23 can be com 
puted at that reference level from the partial derivatives of 
/z AW (w A ,)). P r ). Such properties derived from the potential 
function /z AW at the reference pressure are commonly re 
ferred to as “potential” properties in meteorology (von Be- 
zold 1888, von Helmholtz 1888). Examples are the potential 
enthalpy, he, 
h e =h AW (w A ,t],P r y (5.67) 
h AW = AW_ T i^\ (5.63) 
V 3T ) wA p 
For this purpose, the temperature T corresponding to a given 
entropy rj, must be determined to evaluate the right side of 
Eq. (5.63). The appropriate value of T must be obtained by 
numerically solving the equation 
n = - 
3g AW \ 
dT J w a p 
(5.64) 
the potential temperature, 0, in °C, obtained from Eq. (S23.2) 
7o + 0 = 
' dh m (w A ,rj,P T )' 
3 n 
(5.68) 
vA,Pr 
and the potential density, pe, obtained from Eq. (S23.1) 
3 h m (w A ,rj,P v )\ 
/ VJ A ,Pr 
Pe 
3 P r 
(5.69)