R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
659 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
Here, the latency operators are defined as 
A Ailh] = h AW -A - h m , (5.95) 
AsiM = h sw - S A - h lh . (5.96) 
Here, A is the saturation air fraction from Eq. (5.70), S A 
the brine salinity from Eq. (5.11), D A and Ds are the 
chemical coefficients Eqs. (S4.6), (S12.16), uj av =uj a /A and 
w sw —w s /S A are the gaseous and the liquid fractions, and 
w A and w s are the given constant mass fractions of air and 
of salt in the sea-ice-air sample. 
6 Summary and short discussion 
The mutually consistent formulations of thermodynamic po 
tentials for liquid water, water vapour, ice, seawater and hu 
mid air are now available and permit the numerical compu 
tation of a wealth of thermodynamic properties of the geo 
physical fluids, their mixtures, composites and phase tran 
sitions. The new seawater standard TEOS-IO (IOC et al., 
2010) together with its collection of background papers de 
veloped by WG127 in cooperation with IAPWS is based on 
this physically and mathematically rigorous building-block 
concept (Feistel et al., 2008). To support the practical use 
and general implementation of TEOS-10, WG127 has devel 
oped a source code library that provides easy access to a large 
selection of properties and may serve as a guide for writing 
customized application code using the new standard. 
The library is hierachically organized; all available prop 
erties are computed exclusively from the Primary Standard, 
i.e., level 1 of the code, by merely mathematical and nu 
merical methods. The concept of the Primary Standard is 
intentionally similar to axiomatic systems in mathematics 
which possess the general properties of consistency, inde 
pendence and completeness. These properties ensure that 
the Primary Standard contains all necessary but no redundant 
components, and prevents the computation of contradicting 
results. The higher levels obey the conditions of a mathe 
matical semi-order structure; code of a given level does not 
refer to code of higher levels, thus avoiding direct or indirect 
recursion. 
In the case of seawater, it would be most natural to provide 
access to only the saline component of the Gibbs function 
(Eq. 2.2) at level 1 and not permit access to the individual co 
efficients (Eqs. 2.3-2.5) of the salinity expansion. However, 
it is necessary to have access to the individual temperature 
and pressure dependent coefficients in order to rigorously 
consider numerical limits as S A tends to zero. Thus, these 
fundamental building blocks are made individually available 
at level 1. To obey the independence rule for level 1 routines, 
it is then necessary to place the Gibbs function (Eq. 2.2) at 
level 2, which is not subject to this condition. A similar situ 
ation appears in the case of humid air. The Primary Stan 
dard provides the Helmholtz function of dry air (Eq. 2.6) 
together with the air-water virial coefficients as the funda 
mental information from which the properties of humid air 
can be computed. To ensure independence for level 1 rou 
tines, the Helmholtz function of humid air, Eq. (2.7), and 
the cross-over Helmholtz function (Eq. 2.13) are then imple 
mented in level 2 of the library. Note that while the library 
is constructed to strictly adhere to the development based on 
axiomatic results at level 1, we have discussed the potentials 
of seawater and humid air together with the level-1 functions 
in Sect. 2 of this paper because of their close logical relations. 
In addition to the Primary Standard, the library provides 
easy access to other thermodynamic potential functions de 
rived from the Primary Standard. Available are Helmholtz 
functions that are computed from temperature and density, 
Gibbs functions computed from temperature and pressure, 
enthalpy functions computed from entropy and pressure, and 
implicitly entropy as a potential computed from enthalpy and 
pressure. A list of explicitly implemented potential functions 
is given in Table 1. From each of these potential functions, 
all thermodynamic properties of the particular system can be 
computed; the library provides an extensive but still selective 
set of relevant properties. For additional composite systems 
such as seawater with humid air, several properties are avail 
able from the library even though related potential functions 
were not implemented explicitly. 
Further details on organization, content and access to the 
library are contained in the companion paper (Wright et al., 
2010a). 
Appendix A 
Al Densities of liquid water and water vapour 
(Sect. 4.1) 
As discussed in the text of Sect. 4.1, there cannot exist a 
single-valued Gibbs function g(T,P) that fully represents 
the properties of the Helmholtz function f F (T,p) of fluid 
water. Rather, there are two different Gibbs functions, 
g w (T,P) — f F (r,p W ^j + P/p W (Al) 
for liquid water and 
g v (7\P) = / F (r,p v ) +P/p w (A2) 
for vapour. 
To implement the above expressions for the Gibbs func 
tions we must determine the liquid and vapour densities cor 
responding to the temperature and pressure inputs. This re 
quires iterative solution of Eq. (Al), with considerable care 
required to select the appropriate root for each case.