646

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/

this  purpose,  the  appropriate  thermodynamic  potential  is  the
Gibbs  function  g AV  (A,  T,  P)  of  humid  air,  computed  from
the  Helmholtz  function  / AV  (A,  T.  p)  of  humid  air,  Eq.  (2.7),
by  the  Legendre  transform  (Alberty,  2001)

g AV  =  / AV

—  V

3/ AV
dv

/*AV  ,  „
=  /  +P
A,T

9/ AV \
)  A,T

(4.37)

and  the  subsequent  substitution  of  the  independent  variable
p  by  P,  obtained  from  solving  numerically  the  equation

P  =  p

2

9/ AV \
3P  /  A,T

(4.38)

For  oceanographic  and  meteorological  applications  it  is  not
necessary  to  consider  liquid  or  solid  air.  Therefore,  we  restrict ­
  consideration  of  Eq.  (4.37)  to  the  following  regions:
(i)  for  temperatures  above  the  critical  temperature  of  dry
air,  T>r c =132.5306K,  all  pressures  in  the  range  shown  in
Fig.  4  are  included;  and  (ii)  at  subcritical  temperatures  T<T C
only  temperatures  higher  than  the  dewpoint  temperature  7b,
i.e.  the  condensation  point  of  liquid  air,  are  included.  The
function  T\)  is  available  from  Lemmon  et  al.  (2000)  and  is
shown  in  Fig.  4.
As  a  starting  value  for  the  density  iteration  of  Eq.  (4.38)
at  given  pressure  P,  air  fraction  A  and  temperature  T,  the
ideal-gas  equation  is  suitable:

p  «

Mav04) p
7IT

(4.39)

The  molar  mass  of  humid  air  Mav  is  given  by  Eq.  (2.8),  and
77=8.314472  Jmol -1  K -1  is  the  molar  gas  constant.  Inserting ­
  the  numerical  result  for  p  into  Eq.  (4.37)  provides  the
required  function  value  of  g AV  at  given  A,  T,  P.  For  the  numerical ­
  computation  of  partial  derivatives  of  the  Gibbs  function, ­
  algebraic  combinations  of  analytical  derivatives  of  the
Helmholtz  function  are  implemented  as  given  in  Table  S11.
Thermodynamic  properties  of  humid  air  at  given  A,  T,
P  are  computed  from  the  Gibbs  function  (Eq.  4.37)  and  its
partial  derivatives,  Table  S11,  as  given  in  Table  S12.  The
deviation  of  the  compressibility  factor  Zav  from  unity  describes ­
  the  non-ideal  behaviour.  The  adiabatic  lapse  rate  is
given  with  respect  to  pressure  rather  than  altitude  and  refers
to  subsaturated  humid  air,  often  referred  to  as  “dry-adiabatic”
in  the  meteorological  literature.  The  air  contraction  coefficient, ­
  ft=—^(§j) TP ,  is  the  relative  density  increase  if  a
small  mass  of  vapour  in  a  sample  is  replaced  by  air.  Additional ­
  equations  for  humid-air  properties  are  discussed  in
Feistel  et  al.  (2010a).

4.5  Enthalpy  of  humid  air

When  humid  air  is  lifted  adiabatically  from  the  surface  to  a
certain  pressure  level,  its  air  fraction  and  its  entropy  can  often ­
  be  considered  as  conservative  during  this  process.  Thus,
the  entropy  i)  rather  than  the  temperature  T  is  known  for  a

parcel  at  some  given  altitude  if  the  initial  entropy  was  computed ­
  at  the  surface.  For  this  application  purpose,  the  appropriate ­
  thermodynamic  potential  is  the  specific  enthalpy
/z AV  (A,  ij,P)  of  humid  air,  computed  from  the  Gibbs  function ­
  g AV  (A,  T,  P)  of  humid  air,  Eq.  (4.37),  by  the  Legendre
transform  (Alberty,  2001)
h AV  =  g AV_ T ^^^  (4.40)

The  subsequent  substitution  of  the  independent  variable  T  by
r)  is  obtained  numerically  from  solving  Eq.  (S12.2)  for  T:

n  =  -

3g AV \
3  T  ) a ,p

(4.41)

As  a  starting  value  for  the  iterative  solution  of  Eq.  (4.41)  for
T  at  given  pressure  P,  air  fraction  A  and  entropy  rj,  we  use
an  ideal-gas  approximation  of  Eq.  (S12.2)  in  the  vicinity  of
the  IAPWS-95  triple  point  (7),  P t )  of  water:

T  ~7"|  exp

n  -  m  +  Raw  ln  (P/P t )
A(c A  +  77a)  +  (1  —  j4)(c)>  +  Rw)

(4.42)

This  expression  does  not  depend  on  the  particular
choice  made  for  the  adjustable  coefficients  of  the  entropy. ­
  The  constants  are  rj t =rj(A,T t ,  P t )  computed
from  Eq.  (S12.2)  at  P t =273.16K,  P t =611.654771  Pa,
c A =1003.69  Jkg -1  K“ 1 ,  c v=1884.352Jkg- 1  K“ 1 ,
77a=77/Ma,  77w=77/Mw  and  Raa/—R/Mav-  The  molar ­
  mass  of  air  is  MA=0.02896546kgmol _1 ,  that  of  water  is
Mw=0.018015268kgmol _1 ,  Mav  is  given  by  Eq.  (2.8),  and
77=8.314472  J  mol -1  K -1  is  the  molar  gas  constant.
Once  the  value  of  T  has  been  determined  from  solving
Eq.  (4.41),  the  partial  derivatives  of  /z av (A,^,P)  are  obtained ­
  from  those  of  g AV (A,P,  P)  as  given  in  Table  S13.
Thermodynamic  properties  as  given  in  Table  S14  can  be  calculated ­
  from  algebraic  combinations  of  these  derivatives.
If  a  humid-air  parcel  is  moved  adiabatically  to  some  reference ­
  pressure  P=P r  below  its  isentropic  condensation  level
(ICL,  Emanuel,  1994;  Feistel  et  al.,  2010a),  all  its  thermodynamic ­
  properties  given  in  Table  S14  can  be  computed  at  that
reference  level  from  the  partial  derivatives  of  /z av (A,  rj,  P r )
and  the  in  situ  entropy  rj  (A,  T,  P),  Eq.  (S  13.1).  As  discussed
for  seawater  in  Sect.  4.3,  properties  derived  from  the  potential ­
  function  /z AV  at  the  reference  pressure  (frequently  specified ­
  as  the  surface  pressure)  are  commonly  referred  to  as  “potential” ­
  properties  in  meteorology.  Examples  are  the  potential ­
  enthalpy,  hg,  library  function  air_potenthalpy_si,

/z 0  =/z AV (A,rç,P r ),

(4.43)

the  potential  temperature  from  Eq.  (S14.2),  6,  in  °C,  library
function  air_pottemp_si,

T o  +  0  =

dh AW (A,  rç,P r )\
3*1  /  A,P,

(4.44)