Geoscience Reference
In-Depth Information
the magnitudes of the fluxes involved, energy flux units of W m
−
2
can be interchanged
with hydrologicunits of millimeters per month (mm mo
−
1
)ofliquid water evaporation.
When
Q
n
and either
H
or
E
can be determined independently, Equation (4.13)
provides directly the remaining unknown flux. Usually, however, both
H
and
E
are
unknown, and an indirect method must be used. From the methodological point of
view, these indirect energy budget methods are analogous to the mean profile meth-
ods of Section 4.2.2. In both, essentially three equations are used which contain
three unknowns
E
,
u
∗
and
H
implicitly. In the profile methods these are the equa-
tions for
q
,
u
and
θ
. In the energy budg
et
methods, (4.13) is used either with equations
for
q
and
θ
,orwith equations for
u
and
θ
or
q
,aswill be shown next.
With Bowen ratio (EBBR)
When
Q
n
is known, the combination of the energy budget equation (4.13) with the
Bowen ratio defined in Equation (4.9) produces
Q
ne
E
=
(4.16)
1
+
Bo
Similarly, for the sensible heat flux one has
Bo
Q
ne
1
H
e
=
(4.17)
+
Bo
Bo can be determined as shown in(4.10), from profile data of temperature and specific
humidity in the atmospheric surface layer. As discussed in Section 4.2.1, these data
should be taken as averages over 15-30 min, approximately. Equation (4.16) shows that
the energy budget with Bowen ratio (EBBR) method is most accurate when Bo is small.
Both (4.16) and (4.17) produce a singularity when Bo
1; but, as pointed out by
Tanner (1960), over an active vegetation this is not a problem, as thissituation usually
occurs when
H
is low, around sunrise, sunset and occasionally at night. The situation
does occur more often over cold water, and it may be necessary to use an alternative
method when
=−
0.5toavoid the problem of a very small denominator in
Equations (4.16) and (4.17). Tanner (1960) suggested the use of a bulk-transfer method
for these special conditions. Another way consists of using mean values of Bo corrected
by means of wind measurements, as outlined by Webb (1964); this method is especially
useful when some terms in the available energy
Q
n
are only known for daily periods or
longer.
The EBBR method has the advantage that no similarity functions for the atmospheric
turbulence appear explicitly in the formulation. With Equation (4.10) no measurements
of turbulence or of the mean wind speed are required, and the formulation, as written
in(4.16) with (4.10), is independent of atmospheric stability. In addition, when Bo is
small, the EBBR method may be less susceptible, albeit not immune, to imperfect fetch
conditions, than mean profile methods, inwhich such effects are more directly apparent.
The validity of the EBBR method depends critically on the similarity of the temperature
and humidity profile; for the surface layer this requires the equality of the terms in the
square brackets of Equations (2.51) and (2.52) (or (2.55) and (2.56)). The latest evidence
−
1
<
Bo
<
−
