R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
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Ocean Sci., 6, 633-677, 2010 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
o6 h % hl i p ~ h SP h m gTp{gST-g 6 S6 )-gsPgTT 
P = ;—jo = • ( 4 - 27 ) 
hph a m gpgTT 
(vi) The haline contraction coefficient with respect to poten 
tial enthalpy, /1®, is defined as: 
,0= -"Gr) 
V \ 0 ¿A / h° ,P 
(4.28) 
Similar to Eq. (4.12) we write Eq. (4.28) in terms of Jaco- 
bians 
1 d(v,h 9 ,P) 
v d (S A ,h e ,P) 
1 d(v,h 9 ,P) d(S A ,h 6 ,P) 
vd(S A ,r ] ,P) / d(S A ,ri,P) 
(4.29) 
can be inferred. Hence, when enthalpy is used as an inde 
pendent thermal variable in combination with salinity and 
pressure, the responsible thermodynamic potential function 
is entropy, r) SW (S A ,h,P). Note that the superscript “SW” 
on r) is included here to indicate its use as a thermodynamic 
potential function for seawater, consistent with the inclusion 
of “SW” on both the Gibbs function g sw and the enthalpy 
/i sw when used as a potential function for seawater. To ob 
tain this function value numerically from its arguments, the 
Eq. (S8.2) 
h — h sw (S A , rj, P) (4.33) 
must be solved for rj. Because of Eq. (4.7), if the potential 
enthalpy value h 9 is given, the same algorithm can be used 
to get the related entropy from 
h e =h sw (S A ,rj,P r ). (4.34) 
The inversions of Eqs. (4.33) and (4.34) give respectively 
Expanding the functional determinant in the numerator 
yields, with the help of Eq. (4.7) 
1 hsph e v — h v ph e s 
h P h 9 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
p & _ h< s h >iP— h Sph®_g S TgTp-gspgTT-g 6 s gTp/9 
hph 9 gpgTT 
(4.31) 
The latter equalities in Eqs. (4.13), (4.16), (4.19), (4.23), 
(4.27) and (4.31) are the results given earlier in Table S7. 
The potential quantities written in terms of the enthalpy of 
seawater are listed in Table S10. 
Entropy as a function of salinity, temperature and pressure 
is available from Eq. (S7.2). Potential temperature is defined 
by the relation rj (S A , T, P) —rj (S A ,0, P r ), therefore the same 
function (Eq. S7.2) can be used to compute entropy as a func 
tion of salinity, potential temperature and reference pressure. 
Since the cases (i) to (vi) above, Eqs. (4.13), (4.16), (4.19), 
(4.23), (4.27) and (4.31), specify the different expansion and 
contraction coefficients as functions of entropy, these coeffi 
cients are available as functions of potential temperature, too, 
by means of Eq. (S7.2). 
From the enthalpy definition Eq. (4.5) and the differential 
Eq. (3.7) of the Gibbs function, the relation 
rj = rj SW (S A ,h,P) (4.35) 
and 
rj = rj SW (S A ,h e ,P r ), (4.36) 
which are really the same functions with different ar 
guments. The iterative inversion algorithm is straight 
forward and is implemented as the library function 
sea_eta_entropy_si. It provides entropy i) in the form 
of either rj SW (S A ,h, P) or rj SW (S A ,h 9 , P r ), from which in 
turn all properties listed in Tables S9, S10 can be determined. 
Note, however, that we have not implemented an explicit 
routine for entropy, Eq. (4.35), as a potential function in the 
library. That is, the function sea_eta_entropy_si pro 
vides entropy as a function of salinity, enthalpy and pres 
sure, but it does not provide the partial derivatives of entropy 
with respect to those variables, nor does it take any orders of 
derivatives as input parameters. As such, the thermodynamic 
potential “entropy” is not available in the present SIA library 
version in the same form as the other potential functions that 
are summarised in Table 1. Nevertheless, various properties 
(Tables S9, S10) derived from it are implemented at level 3 
and evaluated from indirect algorithms, just as if the potential 
“entropy” were available. The corresponding routines can be 
identified in the implementation of the library discussed in 
Part 2 (Wright et ah, 2010a) by an _eta_ instead of an _h_ 
in the function names given in Table S10, which indicates 
the implicit use of entropy as the potential function. Conse 
quently, these routines take enthalpy or potential enthalpy as 
the thermal input parameter rather than entropy. 
4.4 Gibbs function of humid air 
dr] sw = jAh - jAP - j;AS a 
In many practical situations the pressure rather than the 
(4.32) density of humid air is available from observations. For