Thermodynamics and Water Vapour - Transport in the Atmosphere-Vegetation-Soil Continuum - page 350

The speciic heats of air at constant volume (denoted by a subscript v, not to be con-
fused with the subscript v used for water vapour) and constant pressure are relat-
ed as:

ccR
p −=

(B.7)

For dry air the values for the speciic heats are: c vd = 717 J kg -1 K -1 and c pd = 1004 J
kg -1 K -1 at a temperature of 285 K (Forsythe, 1954 ). The temperature dependence is
small (1-2 % over the range 250-310 K). For moist air the speciic heat at constant
pressure is a combination of c pd and the speciic heat at constant pressure for water
vapour c pv (1849 J kg -1 K -1 ):

= − ( ) += +

( )

cc 1 qcqc 1 084 . q

(B.8) p pd pv pd

Then the value for c v follows from ( B.7 ) and ( B.8 ):

( )

( ) − + ( ) =+

cc

v = +

1084 . qR 1061 . qc 06114 . qcR . −

(B.9) pd d vd pd d

Deviations of R, c p and c v from their dry air values is of the order of 1-3% depending
on the actual speciic humidity. In many applications these deviation are small enough
to be neglected and hence the dry air values of the thermodynamic properties of air can
be used. However, the effect of water vapour on density, and more speciically density
luctuations, is in many cases not negligible as it affects buoyancy (see Chapter 3 ).
Finally, the irst law of thermodynamics reads:

d qcTp

cT p

= d + d α

v

d (B.10)

= d −

p

ρ

where d q is a differential amount of heat added to the system, which is used both to
increase the temperature by a differential increment d T and to change the speciic
volume by an amount d α (the speciic volume is the inverse of the density: α= −1 ).
To arrive at the second equality, use has been made of Eq. ( B.7 ).

B.2 Hydrostatic Equilibrium
The pressure at a certain height is determined by the weight of the air above that level.
Hence the pressure decreases with height. In the case of hydrostatic equilibrium (that
is, no vertical acceleration) the vertical pressure gradient exactly balances the weight
of the air. For an ininitesimal height increment d z this is:

d pgzz =− ρ ()d (B.11)