Geoscience Reference
In-Depth Information
of the Obukhov length
L
in(2.46). This first estimate of
L
allows next the calculation
of a second estimate of the fluxes by means of (2.50)-(2.52) (or (2.54)-(2.56)), which
in turn allow the calculation of a second estimate of
L
, and so on. The iteration can be
stopped when
s
uccessive estimates cease to change appreciably. When measurements
of
u
are available at more than two elevations, at each iteration
u
∗
,
E
and
H
can be obtained as the slopes from (2.50)-(2.52) (or (2.54)-(2.56)) by least squares
regression through the origin.
,
q
,
and
θ
Example 4.2. Evaporation by profile method in a neutral atmosphere
In a neutrally stratified atmosphere, the turbulent heat flux is relatively small; therefore the
Obukhov length
L
, defined in(2.46), is large, and thus
-functions
become negligible. As a result, Equations (2.50)-(2.52) reduce to the logarithmic pro-
file equations (2.40) and (2.43). Combination of these two equations allows the direct
calculation of the evaporation rate by means of the following expression
ζ
small, so that the
k
2
ρ
(
u
2
−
u
1
)(
q
3
−
q
4
)
=
ln
z
2
−
ln
z
4
−
E
(4.8)
d
0
d
0
z
1
−
d
0
z
3
−
d
0
in terms of measurements of the wind speed at levels
z
1
and
z
2
, and measurements of
specific humidity at levels
z
3
and
z
4
above the ground. An equation similar to this result
was first presented by Thornthwaite and Holzman (1939). While this derivation provides
a good didactic illustration of the profile method, it should be noted that Equation (4.8) is
of limited practical applicability, because over land the atmosphere is only rarely neutral.
Thus, in most cases the profile method requires solution of the full set of equations
(2.50)-(2.52) (or (2.54)-(2.56)).
The second method consists of using the known mean profile and the surface flux of
another but similar scalar, in addition to the mean profile of the scalar under consider-
ation. The requirement of similarity refers inthis context to the equality of the transfer
coefficients Ce and Ch in Equations (2.33) and (2.35) or in(2.36) and (2.38) for the
scalars; inthis sense, it also refers to the equality of the functions
v
in the
profile equations (2.51) and (2.52) (or (2.55) and (2.56)). Probably the oldest application
of thisprinciple is the Bowen ratio (Bowen, 1926)
h
and
Bo
=
H
/
L
e
E
(4.9)
inwhich
L
e
is the latent heat of vaporization of water. Hence, ifsimilarity isvalid, this
ratio, which is used mostly in the energy budget method (see Section 4.3) can also be
written in terms of profile measurements as follows
θ
1
−
θ
2
)
L
e
(
q
1
−
c
p
(
Bo
=
(4.10)
q
2
)
Ov
er
water the surface values
θ
s
and
q
s
are commonly used instead of values in the
air
θ
1
and
q
1
, respectively. The Bowen ratio concept thus leads to a simple expression
