R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
663 
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Ocean Sci., 6, 633-677, 2010 
unknowns T, P, p v and p w , which gives: 
available from the cubic correlation polynomial 
= /W _ /V + ( _ 
A P 
- o w f w AT + — 
P JTp 1X1 + pW 
_ w f w _ 
P J D 
,w 
(A12) 
with an rms error of 1 K. The constants are pc=322kg m 3 , 
r c =647.096 K, Ai=—7.340 173 295 988 58E-02, 
<32=5.705 164 877 110 65E-03, 
(A8) a 3 =—4.313 138 469559 49E-04. 
For T<50K, an initial estimate of vapour density p v on 
the saturation curve is available from the correlation polyno 
mial 
- P V /7 P AT + ^ - (2fj + p v // p ) Ap v (A9) 
For brevity, / F (T, p w ) is abbreviated here by / w , and sim 
ilarly for / v as well as their partial derivatives. To obtain 
Eq. (A7), Eq. (5.2) was first expanded and then simplified 
by using Eqs. (A8) and (A9). When the equilibrium point is 
reached, Eq. (A7) takes the form of the Clausius-Clapeyron 
equation as its right-hand side vanishes. 
To solve the system (Eqs. A7-A9) for T, P, p v and p w , 
a fourth equation must be added which specifies an addition 
ally imposed condition, usually one of AT =0 (for specified 
temperature) or AP=0 (for specified pressure). 
Auxiliary empirical equations are used to determine initial 
estimates for T, P, p v and p w . 
For T <640 K, an initial estimate of the boiling tempera 
ture or the vapour pressure on the saturation curve is esti 
mated from the Clausius-Clapeyron-type correlation polyno 
mial 
m^d-O+^f-i) (Al0) 
with an rms error in ln(P/P t ) equal to 0.01. The constants 
are P t =611.654 771 007 894 Pa, P t =273.16K, 
«i=—19.873 100570911 6, a 2 =-3.089 754 373 529 98. 
For T <350 K, an estimate of the liquid density p w on the 
saturation curve is available from the correlation polynomial 
with an rms error of 0.002 kg m 3 . The constants 
are p t w =999.792520031 621 kgm“ 3 , 
<31 = 1.80066818428501E-02, <3 2 =-0.648 994 409 718 973, 
<33=1.565 947 649 083 47, <3 4 =-3.18 116 999 660964, 
<35=2.985 90977093295. 
Over the higher temperature range 300K<P<Tq, an es 
timate of the liquid density p w on the saturation curve is 
(A13) 
with an rms error of 0.01 in In (p v /p t v ). The constants 
are p t v =4.854575 724778 59 x 10“ 3 kg m“ 3 , 
<3i=-19.223 508 686 606 3, <3 2 =-6.157 701 933 029 55, 
<3 3 =—4.965 736 126 494. 
For 550 K<P < 7(\ an estimate of the vapour density p v 
on the saturation curve is available from the cubic correlation 
polynomial 
with an rms error of 0.4 K. The constants are 
<31 =-0.237 216 00218 091, <3 2 =0.186 593 118 426 901, 
<3 3 =—0.258 472 040 504 799. 
A4 Melting and freezing conditions for pure water 
(Sect. 5.2) 
To iteratively determine freezing and melting conditions for 
pure water, we first linearize the two Eqs. (5.6), (5.7) with 
respect to small changes of the three unknowns T, P and 
p w to obtain: 
+ ,A15) 
- P/r>3 + ^ - (2/7 + f W C) < A16 > 
_ w fW 
P Jp p w 
The function / F (T, p w ) is abbreviated here by / w , and sim 
ilarly for its partial derivatives. To iteratively solve the sys 
tem (Eqs. 5.9, 5.10) for T, P and p w using Eqs. (A15), 
(A16), a third equation must be added which specifies an ad 
ditionally imposed condition, usually A T=0 (if the temper 
ature is specified) or AP=0 (if the pressure is specified). 
Auxiliary empirical equations are used to determine initial 
estimates for T, P and p w .