R. Feistel et al.: Oceanographic application and numerical implementation of TEOS-IO: Part 1
669
www.ocean-sci.net/6/633/2010/
Ocean Sci., 6, 633-677, 2010
The equilibrium is computed using this approach with the
library call set_ice_air_eq_at_a_t or using the function
ice_air_condensation_pressure_si.
Case 2: Equilibrium at given air fraction, A, and
pressure, P
At given A and P, humid air can approximately be con
sidered as an ideal mixture of air and vapour. The partial
pressure P vap =x AV P of vapour is computed from the total
pressure P and the mole fraction x AV (A), Eq. (2.11). In turn,
the sublimation temperature T=p subl (p va P) 0 f water is com
puted from Eq. (5.8). With A, T and P available, the required
density estimate of humid air, p AV =l/g AV (A, T, P), is eas
ily calculated from the related Gibbs function, Eq. (4.37).
Using A A =0 and AP=0, the linear system (Eqs. A53, A54)
can now be solved iteratively for T and p AV .
In particular, this solution provides the frost point temper
ature P(A, P) of humid air as a function of the air fraction
and the pressure.
The equilibrium is computed using this approach with the
library call set_ice_air_eq_at_a_p or using the function
ice_air_f rostpoint_si.
Case 3: Equilibrium at given temperature, T, and
pressure, P
At given T and P, humid air can approximately be con
sidered as an ideal mixture of air and vapour. The partial
pressure P vap of vapour is computed from the sublimation
pressure of ice at given T by solving Eq. (5.8). The vapour
density follows from Eq. (4.3) as p v =l/g^ (P, p va P) and the
air density from p A =l/g AV (1, T, P—P vap ). The air frac
tion is then available from A—p A / (p A +p v ). With A, T
and P available, the required density estimate of humid air,
p AV =l/g AV (A, T, P), is easily calculated from the related
Gibbs function, Eq. (4.37). Using AT=0 and AP=0, the lin
ear system (Eqs. A53, A54) can now be solved iteratively for
A and p AV .
In particular, this solution provides the specific humid
ity q=l— A(T, P) of saturated humid air below the freezing
point as a function of the temperature and the pressure.
The equilibrium is computed using this approach with the
library call set_ice_air_eq_at_t_p or using the function
ice_air_massf raction_air_si.
Case 4: Equilibrium at given air fraction, A, and
entropy,)]
At given A and rj, we use the approximate Clausius-
Clapeyron equation to relate the partial vapour pressure at
the frost point, p vap , to the temperature, T:
P vap L / T t
~P~ ~~r^tX ~ T
(A55)
The vapour pressure is approximately equal to the partial
pressure of vapour in humid air, P vap =x AV P, computed
from the total pressure P and the mole fraction x av (A),
Eq. (2.11). As an analytical estimate to be used below, we
modify Eq. (A55) by means of the relation In x«l — 1/x:
P L T
In — « In In x£ v (A56)
P t R w T t T t v
Assuming constant heat capacities, the ideal-gas entropy
rj(A,T,P) of humid air is, relative to the triple point (P t ,
P t ) of water,
n = % + A (c A In y - R a In y ^ (A57)
+ (1 - A) (cj, Inj-Rw In y ^.
We insert P from Eq. (A56) into Eq. (A57) and get the isen-
tropic condensation temperature estimate T=P ICT (A, rj):
Tier (A, r)) « 7) exp (A58)
rj - rj t (A) - [AP a + (1 ~ A)R w ] In x AV (A)
“ Tw\) + (1 “ A )( c f “ Tt)
Here, at the given air fraction A, the triple-point
entropy rj t (A) —rj(A,T t , P t ) =—g AV (A, P t , P t ) is computed
from Eq. (S12.2), the mole fraction x av (A) from Eq. (2.11),
the constants take the rounded numerical triple-point values
P t =273.16K, P t =611.654771 Pa, c A =1003.69Jkg“ 1 K“ 1 ,
c^=1884.352Jkg -1 K -1 , R A =R/M A , R w = R/M w , and
¿=2834359 J kg -1 is the sublimation enthalpy. The molar
mass of air is MA=0.02896546kgmol _1 , that of water is
Mw=0.018015268kgmol _1 , and P=8.314472 Jmol -1 K -1
is the molar gas constant.
With A and an estimated T available, we can now proceed
as in case 1 to compute the remaining starting values for the
iterative solution of the linear system (Eqs. A53, A54) of two
equations for the three unknowns T, P and p AV using A A=0.
A third equation must be added to the system, adjusting the
humid-air entropy to the given value, i):
- A A - / A ^ A T - / A J Ap AV = r, + / AV . (A59)
This equation is valid for humid air at the frost point, i.e. ice
air with a vanishing ice fraction. If the sample contains a
finite amount of ice, its entropy must additionally be consid
ered in Eq. (A59).
In particular, the solution of case 4 provides the isentropic
ice condensation level P (A, rj) of lifted humid air as a func
tion of the air fraction and the entropy.
The equilibrium is computed using this approach with
the library call set_ice_air_eq_at_a_eta. Alterna
tively, this state is computed from A, T and P of a sub
saturated humid-air parcel having the same entropy and
air fraction as the final saturated one by calling the func
tions ice_air_icl_si or ice_air_ict_si to deter
mine the isentropic condensation level or temperature.