Geoscience Reference
In-Depth Information
Closure by relating storage to evaporation
In this class of methods the additional equation is obtained by relating
E
to
S
. For instance,
Budyko (1955; 1974, p. 97) took as the additional equation (4.33) with (4.34), inwhich
w
was assumed to represent the water storage
S
in the basin and the value of
w
0
was taken
as a layer of 10-20 cm water, to be obtained by calibration with seasonal and regional
variations. The method can be applied with average monthly values of
E
,
P
,
R
,
S
=
(
S
1
+
S
2
)
/
2 and (
dS
/
dt
)
=
(
S
2
−
S
1
), inwhich the subscripts 1 and 2 refer to the beginning and
the end of the month. The calculations can be carried out by successive approximations
as follows. An initial value of
S
1
is chosen at random for the first month. Substitution
of (4.33) into (4.55) yields an equation for
S
2
and without
E
;with an initial value of
S
1
chosen at random for the first month this produces a first estimate of
S
2
,which when
substituted in(4.33) produces
E
for the first month. The same procedure is carried out for
the second month, with
S
2
of the first serving as
S
1
of the second, and so on. The sum of
all these monthly
E
values can be compared with the total annual value of (
P
R
). The
ratio of the two should allow a proportional adjustment of the assumed value of
S
1
of the
first month, and the process can be started over again and continued until the calculated
annual
E
equals the recorded (
P
−
R
). The main weakness of any method based on a
relationship such as (4.33) is, beside the question of the validity of this proportionality,
first the unknown value of the maximal water content parameter
w
0
, and second the rather
ambiguous meaning of the potential evaporation concept. Budyko's method has been applied
extensively over various regions of the former USSR. A very similar method has also
been proposed by Thornthwaite and Mather (1955; see also Steenhuis and VanderMolen,
1986).
−
Closure by relating storage to stream flow
In a second class of methods the additional equation is obtained by relating
S
with the
runoff
R
from the basin, usually during recessions of the river flow, i.e. during drought
flows in the absence of precipitation, so that
P
does not have to be considered in Equation
(4.55). In past studies (see Tschinkel, 1963; Daniel, 1976; Brutsaert, 1982) this has mostly
been done by means of kinematic functions, which can be written in the form
K
n
R
m
S
=
(4.56)
where
K
n
and
m
are constants (cf. Equation (12.48)). This combination of (4.55) and (4.56)
is another example of the lumped kinematic approach; after elimination of
S
, inprinciple
it should thus be possible to determine
E
from streamflow data. The parameters
K
n
and
m
can be determined by calibration under conditions of negligible
E
. The main drawback
of the application of Equation (4.56) inthis context, is that the storage
S
in(4.56) refers
mainly to groundwater storage and not to near-surface soilmoisture which feeds most of
the evaporative processes in the basin. This means that recession flows are sensitive only to
evaporation from areas, where the roots of the vegetation are indirect contact with the water
table. Hence the evaporation determined this way originates mostly from the riparian zone,
and not from areas further away from the stream channels, where the vegetation and the
groundwater are essentially uncoupled. The
S
variable in Equation (4.56), which drives the
streamflow
R
, can be assumed to represent total basin storage, only after the soilmoisture
has been totally depleted, that is after a long recession.
Thisdifficulty can be avoided, as shown by Dias and Kan (1999), by integrating Equa-
tion (4.55) over sufficiently long budget periods
t
, at the end of which most of the water
