Geoscience Reference
In-Depth Information
constant, and either if the atmosphere is neutral, or if the effect of stability as reflected
in
ζ
isnegligible or constant.
Example 4.1. Mass-transfer coefficient in neutral atmosphere
Under neutral conditions by virtue of (2.41) and (2.44), the water vapor transfer coeffi-
cient, as it appears in Equation (4.3), issimply
k
2
Ce
=
(4.4)
ln [(
z
2
−
d
0
)
/
z
0v
]ln[(
z
1
−
d
0
)
/
z
0
]
inwhich
z
1
and
z
2
are the heights of the measurement of the wind speed and of the specific
humidity, respectively, and inwhich
d
0
can be taken to be zero over a water surface.
Within a certain range of normal wind speeds, neutral conditions are apparently often
satisfied over ocean and sea surfaces. Indeed, numerous experimental determinations
have shown that on average the ocean transfer coefficients are of the order of Ce
10
(
=
Ch
10
)
=
10
−
3
, inwhich the subscript indicates that the measurements are
1
.
2(
±
0
.
30)
×
taken at
z
1
=
z
2
=
10 m above the surface. Generally, the corresponding drag coefficient
isalittle larger, and of the order of Cd
10
=
1
.
4(
±
0
.
3)
×
10
−
3
, on average; it also tends
to be more sensitive to the sea state.
The scatter among many of the experimental estimates of the transfer coeffi-
cients Ce
10
,
Ch
10
and Cd
10
over water is considerable. This means that when accurate
results are required the use of some average coefficient may not be adequate and it
may be necessary to include the effects of atmospheric stability and of the roughness
lengths, and therefore in the case of water surfaces, also of sea state. Numerous expres-
sions have been proposed relating Cd
10
to wind speed or surface shear stress for large
water surfaces (see Brutsaert, 1982). Over water surfaces of limited size, such as small
lakes, Ce can be expected to depend on fetch, that is the distance from the upwind
shore. However, in the case of medium size lakes, with fetches of the order of 1-10 km,
Ce isquite insensitive to fetch, provided the specific humidity of the air and the wind
speed are determined over the center of the lake surface. Thus Equation (4.3) with
Ce
10
=
10
−
3
can also be used for such conditions as a first approximation. For
more accurate results, however, it may be advisable to calibrate Ce in(4.3) for each
individual lake.
1.2
×
The form of (4.3) is, in a sense, also suggestive of many other types of mass transfer
equations, mostly empirical, which have been proposed in the past. One such evaporation
equation, originally proposed by Stelling in 1822 (see Brutsaert, 1982) and still in use
today, can be written as
E
=
(
a
+
b u
1
)(
e
s
−
e
2
)
(4.5)
where
e
is the mean vapor pressure and the subscripts refer to the heights of the mea-
surements. From the definition of the specific humidity
q
,with the equation of
state for water vapor (2.5) and for bulk air(2.6), it follows that to a good approximation
=
ρ
v
/ρ
q
=
0.622
e
/
p
(4.6)
