664 
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/ 
An estimate of the freezing temperature T in K as a func 
tion of the absolute pressure P in Pa is obtained from a cor 
relation fit between 252 and 273 K: 
T 
1 
T t 
?» a\ 
- - 1 
(A 17) 
and the specific gas constant of water, 
Pw=461.518 05 Jkg -1 K _1 . 
The density p v in kg m -3 of the condensing vapour is es 
timated as a function of absolute temperature T in K from 
the Clausius-Clapeyron, Eq. (A21), in combination with the 
ideal-gas equation: 
with an rms error equal to 1.4E-5 in T/T t . The constants 
are P t =611.654 771 007 894 Pa, P t =273.16K, 
ai=-1.673 297 591 763 51E-07, 
a 2 =—2.022629 299 996 58E-13. 
An estimate of the density p w in kg m -3 of the freezing 
liquid as a function of absolute temperature T in K is ob 
tained from a correlation fit between 252 and 273 K: 
ft 
RwT 
exp 
Ah / 1 
Rw \?t 
The constants are the same as for Eq. (A21). 
(A22) 
A6 Equilibrium conditions for ice in seawater 
(Sect. 5.4) 
p w /t \ /t \ 2 /t \ 3 
(A 18) 
with rms error equal to 1.2E^1 in p w /p t w . The 
constants are p t w =999.792520031 621 kgm -3 , 
ai=-1.785 829 814 921 13, a 2 =-12.232 508 430 673 4, 
a 3 =—52.823 693 643 3529. 
A5 Equilibrium conditions for ice and water vapour - 
sublimation (Sect. 5.3) 
To determine the conditions under which ice exists in equi 
librium with seawater, we first linearize the two Eqs. (5.12), 
(5.13) with respect to small changes of the four unknowns 
Sa, T, P and p w to obtain: 
S A g s ss AS A - (/f + g S T - S A g s ST - g : r h ) AT (A23) 
— (pW + Sp ~ Sagsp ~ AP 
— fW _1_ R _1_ „S n S Ih 
J + -w + 8 ~ Sa 8s ~ 8 
P 
To determine conditions for sublimation, we first linearize 
the two Eqs. (5.9, 5.10) with respect to small changes of the 
three unknowns T, P and p v to obtain: 
(/ r v - i ?)Ar+(i-«^Ai>=« Ih -/ v ~ (A 19) 
- p V /7 p AT + ^ - (2fj + P v f pp ) Ap V (A20) 
The function / F (T, p v ) is abbreviated here by / v , and simi 
larly for its partial derivatives. To iteratively solve the system 
(Eqs. 5.9, 5.10) for T, P and p v using Eqs. (A19), (A20), a 
third equation must be added which specifies an addition 
ally imposed condition, usually A T=0 (if the temperature is 
specified) or AP=0 (if the pressure is specified). 
Auxiliary empirical equations are used to determine initial 
estimates for T, P and p v . 
The sublimation temperature T in K as a function of the 
absolute pressure P in Pa is estimated using the Clausius- 
Clapeyron equation (for details see Feistel and Wagner, 
2007): 
I ~ I _ in P_ 
T~ T t Ah P t ' 
(A21) 
The constants are P t =611.654771007894Pa, P t =273.16K, 
the sublimation heat, Ah=2 834 359.445 433 54 Jkg -1 
- p*7 r >r + ^ - (2/7 + „»/») V- (A24) 
_ w f w 
-P J P p W- 
To iteratively solve the system (Eqs. 5.12, 5.13) for S A , T, 
P and p w using Eqs. (A23), (A24), two further equations 
must be added which specify an additionally imposed pair 
of conditions, commonly taken to be AT =0 and AS,\ =0 (if 
the temperature and the salinity are specified) or AP=0 and 
AS A =0 (if the pressure and the salinity are specified). 
Auxiliary empirical equations are used to determine initial 
estimates forS A , T, P and p w . 
In the oceanographic range, the pure-water part Eq. (A25) 
of the International Equation of State of Seawater EOS-80 
provides a very good estimate of the density p w — l/v w as a 
function of temperature and pressure (Millard, 1987): 
u w = 
1 
,w 
En* 
2=0 
1 
V 
TT 
4 3 2 
J2 k i x ' a i x ' +x 2 J2 b i x ' 
/=0 /=0 /=0 / 
(A25) 
The reduced variables are x—{T^—273.15 K)/(l K) and 
tt= (P—101325 Pa)/ (lO 5 Pa), and the coefficients are given 
in Table A2.