Geoscience Reference
In-Depth Information
mixing processes in catchments, suggesting that analyses of residence time distributions based on steady
state assumptions over the longer term (see Section 11.8) should be interpreted with some care.
11.8
Residence Time Distribution Models
The types of mixing model outlined above have concentrated on short time scale contributions to storm
hydrographs. It is not necessary in those analyses to estimate how long the stored pre-event water has been
in the catchment. To do so, it is necessary to have measurements of concentrations of inputs and outputs
over much longer periods of time and to assume some model for the full residence time distribution of
water in the catchment while making some similar assumptions about the spatial homogeneity of input
concentrations and (in the simplest cases) temporal stationarity of the residence time distribution. As
with the simple mixing models and end member mixing analysis methods decribed above, it is necessary
to have some differences between the concentrations in the inputs and outputs to make the analysis
tractable. In general, because of the damping effects of storage in a catchment the variability in the input
concentrations are greater than the variability in the output concentrations. Some signal in the output
concentration is still necessary, however. If all the input variability is damped out, it will not be possible
to infer anything about the residence times in the catchment except that they are very long relative to the
time scale of variability in the inputs and that the storage in the catchment must be very large relative to
the mass flux.
For the isotopes of oxygen and hydrogen/deuterium, a structured signal in the input concentrations
happens naturally because of the fractionation that takes place when condensation takes place in the
atmosphere. Thus, the heavier isotopes are preferentially released in the formation of precipitation.
The fractionation process depends on temperature. The lower the temperature, the heavier the isotopic
concentration in precipitation. In humid temperate climates, this results in a general seasonal pattern in
the isotopic concentrations in precipitation, with generally heavier concentrations in winter. However,
individual events can vary significantly from this general seasonal trend and there can also be marked
concentration variations within events in both time and space. In other climate regimes, the seasonal
patterns can also be more complex, such as the monsoonal effects reported by Xie
et al.
(2011).
It is possible to provide a general framework for the analysis of residence time distributions from
input precipitation to output discharges within a catchment (or representative element watershed, see
Chapter 9) storage.
The mass balance equation for the storage can be written as:
for the water
dS
(
t
)
dt
=
P
(
t
)
−
E
(
t
)
−
Q
(
t
)
(11.2)
and for an arbitrary tracer
dM
(
t
)
dt
=
P
(
t
)
C
P
(
t
)
−
E
(
t
)
C
E
(
t
)
−
Q
(
t
)
C
Q
(
t
)
−
R
(
t
)
(11.3)
where
S
(
t
) is storage at time
t
,
P
(
t
) is precipitation input,
E
(
t
) is evapotranspiration output,
Q
(
t
)is
discharge output,
M
is mass of tracer,
C
is tracer concentration, and
R
(
t
) is a source or sink term for
mass of tracer. The transport of tracer within the storage volume can then be thought of most simply as
a form of time variable convolution so that
t
Q
(
t
)
=
P
(
τ
)
f
(
t
−
τ
|
t
)
dτ
(11.4)
0
and
f
(
t
t
) is the residence time distribution for inputs reaching the stream outlet which is expected,
in this general case, to vary with time as the catchment wets and dries and therefore conditional on
t
.
−
τ
|
