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R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-10: Part 1 
Ocean Sci., 6, 633-677, 2010 
www.ocean-sci.net/6/633/2010/ 
We begin with a first guess for the density and linearize 
Eq. (3.3) with respect to changes of density under the as 
sumption that our first guess is sufficiently near the desired 
root that the linearization is valid. This gives 
P/P + Pfp + (2 fp + Pfpp) A p (A3) 
which, by Newton iteration, permits the computation of a 
density improvement Ap from a given estimate p at fixed 
values of T and P. The iteration will converge to a sin 
gle fluid value for supercritical conditions, and to one or the 
other of the distinct vapour and liquid density values, de 
pending on the initial density estimate, for subcritical con 
ditions. Once the solution p is known, all thermodynamic 
properties of fluid, liquid or vapour, can be computed by ei 
ther of two formally different methods: 
(i) For the direct access to liquid water or vapour proper 
ties, the required function of T and p is called from 
Sect. 3.1, Table S2. 
(ii) For the indirect use of water properties as a part of 
e.g. seawater properties, the Gibbs function, Eqs. (Al) 
or (A2) and its derivatives must be made available to 
the related calling functions, see Eq. (2.1), Sect. 4.2 and 
Table S7. 
To determine liquid or vapour solutions of Eq. (Al) where ei 
ther one or both of these may exist along with possible spuri 
ous numerical solutions, the choice of an initial starting point 
must be made carefully to lie inside the “convergence radius” 
of the desired attractor. It is therefore useful to consider liq 
uid and vapour separately in each of the subcritical range, 
the critical region and the supercritical range, as shown in 
Fig. Al. 
Liquid and vapour can be distinguished from each other 
by their different densities and entropies in the vicinity of 
the saturation line which is the curve connecting the triple 
point (TP) with the critical point (CP) in Fig. Al. On the 
saturation line, both phases can coexist in physical space, 
separated by an interface (the “water surface”) across which 
the properties change abruptly. The saturation line is defined 
by equal chemical potentials, i.e. equal specific Gibbs ener 
gies (Eqs. Al, A2) of the two phases. Except for this mutual 
equality, there is no particular distinguishing property within 
either phase which might separate the saturation state from 
its surrounding T — P states (Landau and Lifschitz, 1964). In 
the vicinity of the saturation line, the phase with the lower 
Gibbs energy (Eqs. Al, A2) is stable, the other state exists, 
but is metastable. Here, metastable means stable with re 
spect to infinitesimal fluctuations but unstable with respect 
to certain macroscopic perturbations, namely the emergence 
of finite volumes of the coexisting phase (nucleation of su 
percritical bubbles or droplets). At a greater distance from 
the saturation line, the state with higher Gibbs energy may 
a) Liquid Water: Initial Density Estimates 
Temperature T /K 
b) Water Vapour: Initial Density Estimates 
Temperature T /K 
4 
3 
2 
1 
0 
-1 
-2 
-3 
-4 
Fig. Al. Initial estimates used in the library for the numerical com 
putation of density from pressure by iteratively solving Eq. (A3), 
either for liquid water, panel (a), or water vapour, panel (b). The 
region surrounding the critical point (CP) is treated separately, as 
shown in Fig. A3. Here we make use of the Gibbs functions 
“g(p, T)” in two of the five regions defined in IF-97, region 1 (liq 
uid/fluid) and region 2 (vapour/fluid) as shown in Fig. A2. Pan 
els (a) and (b) differ only for subcritical conditions T<T C and 
P<P C \ otherwise there is only one solution corresponding to the 
unique Gibbs function for fluid water. TP is the ice-liquid-vapour 
triple point. The saturation curve connects TP with CP and sepa 
rates liquid above from vapour below. To the far left, ice Ih (ICE) is 
separated from the liquid by the melting curve, above TP, and from 
the vapour by the sublimation curve, below TP. Note that the melt 
ing curve above 200 MPa belongs to forms of ice other than Ih, the 
ambient hexagonal phase. For vapour below 273.15 K, the ideal-gas 
equation is used.