R.  Feistel  et  al.:  Oceanographic  application  and  numerical  implementation  of  TEOS-IO:  Part  1

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www.ocean-sci.net/6/633/2010/

Ocean  Sci.,  6,  633-677,  2010

Here,  the  latency  operators  are  defined  as
A  Ailh]  =  h AW  -A  -  h m ,  (5.95)
AsiM  =  h sw  -  S A  -  h lh .  (5.96)
Here,  A  is  the  saturation  air  fraction  from  Eq.  (5.70),  S A
the  brine  salinity  from  Eq.  (5.11),  D A  and  Ds  are  the
chemical  coefficients  Eqs.  (S4.6),  (S12.16),  uj av =uj a /A  and
w sw —w s /S A  are  the  gaseous  and  the  liquid  fractions,  and
w A  and  w s  are  the  given  constant  mass  fractions  of  air  and
of  salt  in  the  sea-ice-air  sample.
6  Summary  and  short  discussion
The  mutually  consistent  formulations  of  thermodynamic  potentials ­
  for  liquid  water,  water  vapour,  ice,  seawater  and  humid ­
  air  are  now  available  and  permit  the  numerical  computation ­
  of  a  wealth  of  thermodynamic  properties  of  the  geophysical ­
  fluids,  their  mixtures,  composites  and  phase  transitions. ­
  The  new  seawater  standard  TEOS-IO  (IOC  et  al.,
2010)  together  with  its  collection  of  background  papers  developed ­
  by  WG127  in  cooperation  with  IAPWS  is  based  on
this  physically  and  mathematically  rigorous  building-block
concept  (Feistel  et  al.,  2008).  To  support  the  practical  use
and  general  implementation  of  TEOS-10,  WG127  has  developed ­
  a  source  code  library  that  provides  easy  access  to  a  large
selection  of  properties  and  may  serve  as  a  guide  for  writing
customized  application  code  using  the  new  standard.
The  library  is  hierachically  organized;  all  available  properties ­
  are  computed  exclusively  from  the  Primary  Standard,
i.e.,  level  1  of  the  code,  by  merely  mathematical  and  numerical ­
  methods.  The  concept  of  the  Primary  Standard  is
intentionally  similar  to  axiomatic  systems  in  mathematics
which  possess  the  general  properties  of  consistency,  independence ­
  and  completeness.  These  properties  ensure  that
the  Primary  Standard  contains  all  necessary  but  no  redundant
components,  and  prevents  the  computation  of  contradicting
results.  The  higher  levels  obey  the  conditions  of  a  mathematical ­
  semi-order  structure;  code  of  a  given  level  does  not
refer  to  code  of  higher  levels,  thus  avoiding  direct  or  indirect
recursion.
In  the  case  of  seawater,  it  would  be  most  natural  to  provide
access  to  only  the  saline  component  of  the  Gibbs  function
(Eq.  2.2)  at  level  1  and  not  permit  access  to  the  individual  coefficients ­
  (Eqs.  2.3-2.5)  of  the  salinity  expansion.  However,
it  is  necessary  to  have  access  to  the  individual  temperature
and  pressure  dependent  coefficients  in  order  to  rigorously
consider  numerical  limits  as  S A  tends  to  zero.  Thus,  these
fundamental  building  blocks  are  made  individually  available
at  level  1.  To  obey  the  independence  rule  for  level  1  routines,

it  is  then  necessary  to  place  the  Gibbs  function  (Eq.  2.2)  at
level  2,  which  is  not  subject  to  this  condition.  A  similar  situation ­
  appears  in  the  case  of  humid  air.  The  Primary  Standard ­
  provides  the  Helmholtz  function  of  dry  air  (Eq.  2.6)
together  with  the  air-water  virial  coefficients  as  the  fundamental ­
  information  from  which  the  properties  of  humid  air
can  be  computed.  To  ensure  independence  for  level  1  routines, ­
  the  Helmholtz  function  of  humid  air,  Eq.  (2.7),  and
the  cross-over  Helmholtz  function  (Eq.  2.13)  are  then  implemented ­
  in  level  2  of  the  library.  Note  that  while  the  library
is  constructed  to  strictly  adhere  to  the  development  based  on
axiomatic  results  at  level  1,  we  have  discussed  the  potentials
of  seawater  and  humid  air  together  with  the  level-1  functions
in  Sect.  2  of  this  paper  because  of  their  close  logical  relations.
In  addition  to  the  Primary  Standard,  the  library  provides
easy  access  to  other  thermodynamic  potential  functions  derived ­
  from  the  Primary  Standard.  Available  are  Helmholtz
functions  that  are  computed  from  temperature  and  density,
Gibbs  functions  computed  from  temperature  and  pressure,
enthalpy  functions  computed  from  entropy  and  pressure,  and
implicitly  entropy  as  a  potential  computed  from  enthalpy  and
pressure.  A  list  of  explicitly  implemented  potential  functions
is  given  in  Table  1.  From  each  of  these  potential  functions,
all  thermodynamic  properties  of  the  particular  system  can  be
computed;  the  library  provides  an  extensive  but  still  selective
set  of  relevant  properties.  For  additional  composite  systems
such  as  seawater  with  humid  air,  several  properties  are  available ­
  from  the  library  even  though  related  potential  functions
were  not  implemented  explicitly.
Further  details  on  organization,  content  and  access  to  the
library  are  contained  in  the  companion  paper  (Wright  et  al.,
2010a).
Appendix  A
Al  Densities  of  liquid  water  and  water  vapour
(Sect.  4.1)
As  discussed  in  the  text  of  Sect.  4.1,  there  cannot  exist  a
single-valued  Gibbs  function  g(T,P)  that  fully  represents
the  properties  of  the  Helmholtz  function  f F (T,p)  of  fluid
water.  Rather,  there  are  two  different  Gibbs  functions,
g w (T,P)  —  f F (r,p W ^j  +  P/p W  (Al)
for  liquid  water  and
g v (7\P)  =  / F (r,p v )  +P/p w  (A2)
for  vapour.
To  implement  the  above  expressions  for  the  Gibbs  functions ­
  we  must  determine  the  liquid  and  vapour  densities  corresponding ­
  to  the  temperature  and  pressure  inputs.  This  requires ­
  iterative  solution  of  Eq.  (Al),  with  considerable  care
required  to  select  the  appropriate  root  for  each  case.