R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1 
643 
www.ocean-sci.net/6/633/2010/ 
Ocean Sci., 6, 633-677, 2010 
4.2 Gibbs function of seawater 
The Gibbs function of seawater, Eq. (2.1), is reproduced here 
as, 
g sw (S A ,T,P) =g w (T,P) +g s (S A ,T,P), (4.4) 
and is directly available from the sum of the Gibbs function 
of pure water computed at level 3, Table S6, and the saline 
part from the Primary Standard, level 1, Eq. (2.2). Properties 
of seawater can be computed from the partial derivatives of 
g s and g sw as given in Tables 4 and 7. 
The Gibbs function g sw (S A , T, P) of seawater, Eq. (4.4), 
together with its first and second partial derivatives is imple 
mented as the library function sea_g_si. 
4.3 Enthalpy of seawater 
Besides the Gibbs and the Helmholtz functions, the specific 
enthalpy h sw (S A ,rj, P) of seawater, expressed in terms of 
Absolute Salinity S A , specific entropy, rj, and absolute pres 
sure P is a third important thermodynamic potential, useful 
in oceanography in particular for the computation of proper 
ties related to adiabatic processes (Feistel and Hagen, 1995; 
McDougall, 2003; Feistel, 2008; IOC et al., 2010). 
To compute this potential and its partial derivatives from 
the Gibbs function g sw (S A , T, P) of seawater, the indepen 
dent variable T appearing in the expression for the enthalpy, 
some reference pressure P— P r , the thermodynamic proper 
ties given in Table S9 can be computed at that reference level 
from the partial derivatives of h sw (S A , rj, P r ). Such proper 
ties derived from the potential function /; sw at the reference 
pressure are commonly referred to as “potential” properties 
in meteorology and oceanography. Originally introduced by 
von Bezold (1888), potential temperature is defined as the 
temperature that a fluid parcel takes if it is moved adiabat- 
ically from its in situ pressure to a reference pressure level, 
which is often specified as the ocean surface. Analogous def 
initions hold for the potential density and potential enthalpy 
(IOC et al., 2010). 
The potential enthalpy, h e , is obtained from Eq. (S8.2), 
h e =h sw (S A ,r],P r ), (4.7) 
the absolute potential temperature, 6, in K, is obtained from 
Eq. (S9.2), 
e = 
dh^(s A , n ,p r ) 
dr] 
= K 
S,Pr 
(4.8) 
and the potential density, p e , is obtained from Eq. (S9.1), 
(p 6 )- 1 =v° = 
dh™(s A ,r],P) 
dp 
S,V 
=Pr = h 
p* 
(4.9) 
jsw = i sw_ r ^j («) 
must be determined from knowledge of salinity, entropy and 
pressure. Given values of S A , i] and P, the corresponding 
value of T is obtained by numerically solving the equation 
n = - 
9g SW 
dT 
S,P 
(4.6) 
to provide the implicit relation T—T (S,\. >]. P). Details on 
the iterative solution method used in the libraries are given in 
Appendix A2. 
The specific enthalpy h sw (S A ,rj,P) of seawater, 
Eq. (4.5), as a thermodynamic potential is implemented in 
the library as the function sea_h_si. 
Once the value of T has been determined as described in 
the appendix, the partial derivatives of h sw (S A ,)]. P) are ob 
tained from those of g sw (S A . T, P) as given in Table S8. 
From the enthalpy and its derivatives, all thermodynamic 
properties can be computed. A selection is given in Table S9 
and additional quantities are given in Table S10 after so- 
called “potential” properties are introduced. 
Many oceanic processes like pressure excursions of a sea 
water parcel conserve salinity and entropy to very good ap 
proximation. In particular, if a parcel is moved this way to 
Evidently, for any fixed reference pressure, P r , the values 
of h sw (S A ,rj, P r ) and its partial derivatives, as well as any 
other arbitrary function depending on this triple of variables, 
remain unchanged during isentropic (r] = const) and isohaline 
(5a = const) processes. 
Derived from Eqs. (4.7) and (4.8), three kinds of thermal 
expansion and haline contraction coefficients are important 
for numerical models (IOC et al., 2010) and these are con 
sidered below as the cases (i) to (vi). In these cases, we have 
omitted the superscripts SW on the seawater potential func 
tions for simplicity of the expressions. As well, we have al 
ways regarded the reference pressure P r as a constant value 
in each derivative considered here, without explicitly indicat 
ing this in the formulas. This implies that potential enthalpy, 
Eq. (4.7), and potential temperature, Eq. (4.8), are pressure- 
independent functions of salinity and entropy, and in partic 
ular, that any derivatives taken at constant (5a, rj) can equiv 
alently be taken at constant (5a, 9) or constant (5a, h e ). An 
example is the isentropic compressibility, 
1 / \ _ 1 / 3u \ _ 1 / 3u \ 
3p / 
S,17 
v \ dP 
s,e 
dP ) 
(4.10) 
S,h 
(i) The thermal expansion coefficient, « , is defined as: 
T 1 / dv 
a — — 
v \dT/ s P 
(4.11)