Geoscience Reference
In-Depth Information
water vapour, change at the same rate when the pressure or temperature changes. As
a result, their ratio is constant under those changes.
The relationship between the mixing ratio and the vapour pressure is derived from
Dalton's law. Using the equation of state for dry air and water vapour yields:
R
R
e
pe
=
d
v
r
−
(B.18)
The ratio of the speciic gas constant for dry air and water vapour is an important
number in meteorology. It equals 0.621 and is close to 5/8. The relationship between
q
and
e
is:
R
R
R
R
e
p
e
p
e
p
5
8
q
=
≈
d
≈
(B.19)
v
v
For the deinition of relative humidity the saturated vapour pressure
e
sat
is needed. The
saturated vapour pressure is the water vapour pressure in a gas that is in equilibrium
(at a given temperature) with the liquid phase: the number of molecules that leave the
liquid phase equals the number of molecules that leave the gas phase and thus rejoin
the liquid phase. The empirical approximations for
e
sat
(
T
) proposed by WMO (
2008
)
is (see
Figure B.1
):
aT
bT
(
−
273 15
.
)
(B.20)
eT e
()
=
exp
sat
sat,0
+
where
e
sat,0
is the
e
sat
at 0 °C (
e
sat,0
= 611.2 Pa.). The value of the constants
a
and
b
depends on the surface over which the saturated vapour pressure needs to be deter-
mined. For water surfaces
a
= 17.62 K
-1
and
b
= -30.03 K, whereas for ice surfaces
the values are
a
= 22.46 K
-1
and
b =
-0.53 K Generally, a subscript 'w' or 'i' is used
to indicate whether saturated values over water or ice are referred to. Here, we always
refer to the saturated vapour pressure over water and omit the subscript 'w' (i.e.,
where
e
sat
is written,
e
sat,w
is intended). Note that Eq. (
B.20
) was originally stated with
the temperature in °C.
In evaporation theory the derivative of
e
sat
(
T
) to temperature is used. From Eq.
(
B.20
) this can be determined to be:
d
d
e
T
eT
ab
bT
()
(
+
273 15
2
.
)
(B.21)
sT
()
≡ =
sat
sat
(
+
)
In
Figure B.1
both
e
sat
(
T
) and
s
(
T
) are depicted.
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