662 
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/ 
Critical Region: Auxiliary Cubic Equation of State 
Fig. A3. Selected isotherms of the auxiliary cubic equation 
of state, Eq. (A4), in the critical region 623.15<7’<650K, 
16.529< P <35 MPa. Shown in bold is the saturation-pressure curve 
of IAPWS-95, separating the single-phase region above from the 
two-phase region below. The critical point is at 7c =647.096 K, 
PC-322 kg m -3 , 7 > c-22.064MPa. Given a line of constant sub- 
critical pressure P and an isotherm T, their intersection points with 
positive slopes provide either the density estimate for p V (T, P) of 
vapour (on the left branch) or for p v (T, P) of the liquid (on the 
right), separated from each other by the unstable region of negative 
slopes. 
IAPWS-95 critical point (Fig. A3) through the specifications 
of 7c, Pc, and pc- 
The initial densities for the iteration, Eq. (A3), in the 
critical region are computed from the intersection points 
of the horizontal isobars with the isotherms as shown in 
Fig. A3. In the subcritical range, T<Tq and P<Pq, there 
exist three solutions, the vapour density to the left of the 
isotherm maximum, the liquid density to the right of the min 
imum, and an extraneous unstable solution in between the 
extrema. The curve (not shown) connecting the minima and 
the maxima of adjacent isotherms, which passes smoothly 
through the critical point, is the spinodal of the auxiliary 
equation. Beneath the spinodal, the compressibility is neg 
ative, (dp/dP) T <0, thermodynamic stability is violated and 
no stable single-phase states can exist. By means of this sta 
bility gap, the spinodal separates low-density vapour from 
high-density liquid on the particular isotherm. At the criti 
cal point, maximum, minimum and inflection point coincide, 
and at supercritical temperatures only one fluid solution ex 
ists for any given pressure. Below the critical temperature, a 
single solution from the liquid branch is computed for P>Pc 
which is considered a supercritical fluid state according to 
our numerical definition of the liquid and vapour functions 
(Eqs. A1 and A2). Very close to the critical point, initial 
densities computed from the auxiliary cubic equation of state 
may falsely be located inside the spinodal of IAPWS-95 and 
Table Al. Coefficients of the auxiliary critical equation of state, 
Eq. (A4). 
i 
j 
a ij 
i 
j 
a ij 
0 
3 
-0.602044738250314 
2 
0 
118.661872386874 
l 
0 
-7.60041479494879 
2 
1 
186.040087842884 
l 
1 
-17.463827264079 
2 
2 
25.5059905941023 
l 
2 
0.69701967809328 
2 
3 
14.4873846518829 
l 
3 
30.8633119943879 
thus prevent convergent iteration. In this highly specialized 
case, applications may need better starting values than those 
from the cubic polynomial, e.g. find exact densities at the gas 
and liquid spinodal points from the condition (d P/dp) T =0 
and use one of them to confine p (T, P) for a bisection itera 
tion method such as the secant or Brent algorithms. Details 
of the universal critical properties are available from Stan 
ley (1971), Anisimov (1991), Kurzeja et al. (1999), Skripov 
and Faizullin (2006), or Ivanov (2008). 
A2 Seawater temperature from salinity, entropy and 
pressure (Sect. 4.3) 
To compute the specific enthalpy potential and its partial 
derivatives from the Gibbs function g sw (SA, T, P) of sea 
water, the value of T appearing in the expression for the en 
thalpy, Eq. (4.5), must be obtained from knowledge of the 
entropy, rj, along with the salinity and pressure values. The 
required temperature is obtained by numerically solving the 
Eq. (4.6) to give T—T (Sa, P, P)- 
To solve Eq. (4.6) we first linearize i)——with respect 
to small changes of temperature to obtain the equation 
- g™AT = r, + g| w , (A5) 
which can be used to iteratively update the value of T at given 
values of Sa, >], P- Since the heat capacity of water is rather 
constant under different oceanic conditions and Eq. (A5) has 
an unambiguous solution in the region of oceanographic in 
terest, the simple linear estimate 
T = 273.15 K H 
4000 J kg“ 1 K“ 2 
(A6) 
provides a sufficiently accurate initial temperature to ensure 
convergent iteration of Eq. (A5). 
A3 Saturated water vapour conditions (Sect. 5.1) 
To numerically determine the conditions corresponding to 
the saturation point (frequently referred to as the boiling 
point or dewpoint) of pure water, we first linearize the three 
Eqs. (5.2)-(5.4) with respect to small changes of the four