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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/
the dewpoint, P vap , to the temperature, T:
pvap ¿ / Tt \
In (1 ) .
Pt RwT t \ t)
(A48)
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. (A48) by means of the relation In x«l — 1/x:
P L T AV
In — « In In x ) v . (A49)
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 determined relative to the triple
point (7), Pt) of water,
rj = t] t + A (cp In y - R a In (A50)
+ (1 — A) ^Cp In y -R W In ^0.
this state may be computed from A, T and P of a subsat
urated humid-air parcel having the same entropy and air
fraction as the final saturated one by calling the functions
liq_air_icl_si or liq_air_ict_si to determine its
isentropic condensation level or temperature.
All Equilibrium conditions for ice and water vapour in
air (Sect. 5.9)
To determine conditions for which water vapour in air will
be in equilibrium with ice, we first expand the two Eq. (5.70)
(with Eq. 5.71) used to eliminate the Gibbs potential) and
Eq. (5.72) with respect to small changes of the four indepen
dent variables:
- Af™ A A + (/ AV - Af™ -gf)AT (A53)
= - /” -~i + -vi v
P
We insert P from Eq. (A56) into Eq. (A57) and get the isen
tropic condensation temperature estimate T=Tict(A, p):
ricT(A,>;)~r t exp
r,- % (A)-[A« a +(1—A)R w ] In Xy V (A)
A ( c p-^) + d-A)(4-^)
(A51)
Here, at the given air fraction A, the triple-point
entropy rjt(A)—ri(A,Tt,Pt)——gj W (A,Tt,Pt) is computed
from Eq. (S 12.2), the mole fraction x AV (A) from Eq. (2.11).
The constants take the rounded numerical triple-point values
7t=273.16K, P t =611.654771 Pa, c A =1003.69Jkg -1 K -1 ,
c^=1884.352 Jkg -1 K -1 , R A =R/M A , R w = R/M w , and
¿=2500915 Jkg -1 is the evaporation enthalpy. The molar
mass of air is MA=0.02896546kgmol -1 , that of water is
Mw=0.018015268kg mol -1 , and P=8.314472 Jmol -1 K -1
is the molar gas constant.
With A and an estimated T available, we can now pro
ceed as in case 1 to compute the remaining starting values for
the iterative solution of the linear system (Eqs. A45-A47) of
three equations for the four unknowns T, P, p w and p AV us
ing A 4=0. A fourth equation must be added to the system,
adjusting the humid-air entropy to the given value, p:
- f™ AA - f^AT - / A JAp AV = r, + ff (A52)
This equation is valid for humid air at the dewpoint, i.e. wet
air with a vanishing liquid fraction. If the sample contains
a finite amount of liquid water, its entropy must additionally
be considered in Eq. (A52).
In particular, the solution of case 4 provides the isentropic
condensation level P(A, p) of lifted humid air as a function
of the air fraction and the entropy.
The equilibrium is computed using this approach with the
library call set_liq_air_eq_at_a_eta. Alternatively,
p AV / A J A A + p AV / A J A T - — (A54)
P
+ (2/f+P''/* V )iP V = -p, - P V P V
For the numerical solution, two additional conditions must
be specified. For example, if we specify temperature and
pressure then AT=0 and AP=0. Starting values are then
required for the iterative determination of the remaining
unknowns. Four important such cases are considered in the
following.
Case 1: Equilibrium at given air fraction, A, and
temperature, T
At given A and T, humid air can approximately be consid
ered as an ideal mixture of air and vapour. The partial pres
sure P vap of vapour is computed from the sublimation pres
sure of ice at given T by solving Eq. (5.8). The vapour den
sity follows from Eq. (4.3) as p v =l/g^ (T, P vap ). The dry-
air density is then estimated as p A — p v x A/(l—A). The
partial pressure of dry air is computed from Eq. (S5.ll) as
P air =(p A ) 2 / AV (l, T, p A ). Using this approach, we obtain
an estimate for the total pressure, P=P vap +P alr . 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 AA=0 and AT =0, the lin
ear system (Eqs. A53, A54) can now be solved iteratively for
P and p AV .
In particular, this solution provides the pressure P(A, T)
of saturated humid air as a function of the air fraction and the
temperature.