644 
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/ 
It is expressed in terms of derivatives of enthalpy by means 
of the Jacobi method and Eqs. (S9.1), (S9.2), as 
a T = 1 d(v,S A ,P) = 1 d(v,S A ,P) fd(T,S A ,P) 
vd(T,s A ,P) vd(r,,s A ,P)/ d(r 1 ,s A ,P) ( ' j 
_ 1 (dv/dr])s p _ h n p 
v(dT/di)) SP h P h vrt ' 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
h v p _ grp 
hph m g P 
(4.13) 
(ii) The thermal expansion coefficient with respect to poten 
tial temperature, a e , is defined as: 
v\d&) s p 
(4.14) 
Similar to Eq. (4.12), with the help of Eq. (4.8) we compute 
e = i d(v,s A ,p) = l d(v,s A ,p) /d(e,s A ,p) 
vd(d,s A ,p) vd(r),s A ,P)/ d(ri,s A ,P) ( ' 
_ 1 (dv/di]) s P _ h n p 
v (96/dr]) s P h P h 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
h n p _ grpgee 
h P h nn gPgTT 
(4.16) 
as a & —c° p a h (IOC et ah, 2010). Conservative temperature, 
0, is potential specific enthalpy, h", Eq. (4.7), expressed in 
terms of an arbitrarily defined temperature unit, &—h e /c° p 
(McDougall, 2003; IOC et ah, 2010); as such, it belongs 
to level 5 of the library where non-basic-SI units and user- 
defined functions are implemented. In contrast, potential en 
thalpy itself is defined at the core level 3 of the SIA library. 
(iv) The isothermal haline contraction coefficient, fi, is de 
fined as: 
1 / dv \ 
T,P 
(4.20) 
Similar to Eq. (4.12) we write Eq. (4.20) in terms of Jaco- 
bians 
1 8(v,T,P) _ 1 8(v,T,P) jd(S A ,T,P) 
vd(S A ,T, P)~~vd(S A ,n,P)/ d(S A ,T],P) ' 
(4.21) 
Expanding the functional determinant in the numerator 
yields, with the help of Eqs. (S9.1) and (S9.2) 
(4.22) 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
Here, g" is the potential Gibbs energy defined as g" = 
g(S A ,6,P r ). 
(iii) The thermal expansion coefficient with respect to po 
tential enthalpy, a h , is defined as: 
\dh 6 )$ p 
(4.17) 
Similar to Eq. (4.12), with the help of Eq. (4.7) we compute 
a_i d(v,s A ,P) _id(v,s A ,P) /d(h 6 ,s A ,p) 
vd(h°,s A ,p) vd(r),s A ,P)/ d(r 1 ,s A ,p) ( ' j 
1 (dv/drj) s P h vP 
v(dh 6 /dri) sp hph 6 n 
Using Table S8, the partial derivatives of h can be substituted 
by those of g, with the result 
a h = Kp = 8tp__ ( 4 . 19 ) 
hph“ gpgTT6 
The thermal expansion coefficient with respect to conser 
vative temperature, a®, is related to Eq. (4.19) by a con 
stant conversion factor, c° p =3991.86795711963 Jkg -1 K -1 , 
hstjh v p — hsph 
hph 
P^rjrj 
gSP 
gp 
(4.23) 
(v) The haline contraction coefficient with respect to poten 
tial temperature, ft 6 , is defined as: 
9 = --(jPj 
v \dS A jQ P 
(4.24) 
Similar to Eq. (4.12) we write Eq. (4.24) in terms of Jaco- 
bians 
i d(v,e,P) _ i d(v,e,P) /d(s A ,e,P) 
vd(S A ,6,P) vd{S A , n ,P)/ d{S A ,r),py ( ' j 
Expanding the functional determinant in the numerator 
yields, with the help of Eqs. (S9.1) and (S9.2) 
1 h sph? m - hr,ph° St] 
hp h d m 
(4.26)