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Parametrization of
non-convective condensation processes
May 1987
European Centre for Medium-Range
Weather Forecasts
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1 . Thermodynamics of moist air
A brief review of the general thermodynamics of moist air is given here.
Air can be considered as a mixture of ideal gases. The equation of state is
|
(1) |
where
is pressure, 
is density, 
is absolute temperature and 
is the gas constant
|
(2) |
and 
are the heat capacities
at constant pressure and at constant volume, respectively. Equation (1)
is also valid for moist air, provided that the actual temperature is replaced
by the virtual temperature
|
(3) |
The virtual temperature
is given by
|
(4) |
where
and 
is the ratio of the gas constants for dry air, and for water vapour
. 
is the specific humidity 
(
is the density of water vapour and 
is the density
of moist air). The equation of state for moist air is valid since it is
assumed that water vapour also behaves as an ideal gas
|
(5) |
where the subscript `vap' refers to water vapour.
The first law of thermodynamics is
|
(6) |
is the specific entropy, 
is the specific internal energy (
), 
is the specific enthalpy (
), 
is the heating rate by external sources and 
is the specific
volume (
)From (6) we get
|
(7) |
For dry adiabatic processes, i.e. at
, the potential temperature ? is conserved
|
(8) |
The specific entropy can be expressed in terms of potential temperature as
|
(9) |
The dry adiabatic lapse rate is
|
(10) |
where
is the gravity
of the earthIf the exchange of heat through condensation processes is considered, the thermodynamic equation (6) becomes
|
(11) |
where
is the latent
heat and 
is the saturation mixing ratio.The Clausius-Clapeyron equation gives the slopes of the curves for the saturation water vapour pressure
(Fig. 1 ).
|
(12) |
where
and 
are the specific volumes for water vapour and for water, respectively.
Integration of equation (12) gives (assuming that 
, 
and 
) the Tetens formula (Murray (1967)
|
(13) |
where
|
plane. The inset shows the vapour-ice equilibrium
(lower curve) and the vapour-liquid equilibrium (upper curve) at subfreezing
temperatures.
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12.06.2002 |
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