Geoscience Reference
In-Depth Information
11.8.3
Fractal Residence Time Distributions
An interesting development in the analysis of transit and residence time distributions originates with the
work of Kirchner
et al.
(2000, 2001), which was based on the analysis of three years of daily chloride data
in rainfalls and discharges collected at the Plynlimon experimental catchments in Wales. Their analysis
showed that the simplest assumption of an exponential residence time distribution in this catchment
was not a good one, and that a much better fit to the data could be obtained by using a two-parameter
gamma distribution (see Box 11.2). The gamma distribution can take on a wide range of forms; with
shape parameter values
1, there is greater weight in the tails of the distribution. In an analysis of
22 catchments using a spectral filtering method, Godsey
et al.
(2010) suggest that the majority of those
catchments have shape factors significantly less than 1 (see also Kirchner
et al.
, 2010). Four catchments
with the highest values of the shape parameter (
<
0.65), and therefore closest to exponential distribution
(with shape factor = 1), all had catchment lakes, which might provide more complete mixing behaviour.
There are a number of interpretations of this type of heavy-tailed behaviour. One is that the residence
time distributions are “fractal” (Kirchner
et al.
, 2000), which means that there is a not a single represen-
tative mean residence time but a distribution of storages with different mean residence times. Another is
that rather than a lumped storage in the system, it is behaving more like a storage with inputs distributed
in space that are then advected and dispersed according to the ADE (Box 11.1; Kirchner
et al.
, 2001). A
catchment does, of course, have distributed inputs over its hillslopes, though the advection and disper-
sion processes are likely to be much more complex than the simple ADE representation and may involve
multiple storages and heavy-tailed velocity distributions (Lindgren
et al.
, 2004; Davies
et al.
, 2011).
>
11.8.4
Issues in the Interpretation of Residence Time Distributions
The Kirchner
et al.
(2000, 2001) studies raise a number of interesting issues:
•
how the form of numerical analysis chosen can affect the apparent residence times in the catchment
(see also Kirchner, 2005);
•
that there may be multiple sets of assumptions and consequent explanations of the type of filtering of
an input that results from the dynamics of the catchment;
•
that how far we can distinguish between those dynamics might depend on the uncertainties in sam-
pling the signal, both in the frequency and representativity of the sampling and in the length of the
sampling period. Very long residence times might be poorly represented by short periods of record
(and might depend on the environmental tracer used: see Lyon
et al.
, 2009; Kirchner
et al.
, 2010;
Stewart
et al.
, 2010).
The original chloride data from Plynlimon used by Kirchner
et al.
(2000) was three years of daily
data in a catchment that responds very rapidly to rainfall. The daily time series might therefore not be an
adequate sample of the mass flux of chloride out of the catchment. Some of evidence of this was suggested
by the study of Page
et al.
(2007) which attempted to simulate the same data using a catchment scale
model based on TOPMODEL (Box 6.1). The catchment model also requires an estimate of chloride
inputs and the data available suggested that there was a gross imbalance of about 30% in chloride, with
the inputs in rainfall being much lower than the outputs (it is possible that more detailed sampling of the
chloride in discharge might have made this imbalance worse). The Plynlimon catchments lie in an area
where there can be dry deposition of chloride in westerly airstreams from the Atlantic and the Irish Sea
and scavenging of chloride from low clouds that often cover the upland forest and grassland. Page
et al.
(2007) produced a model of the inputs to correct for the mass imbalance based on the meteorological
conditions, but were only partially successful in reproducing the chloride concentrations in discharge.
In fact, analysing the simulated concentrations using similar spectral filtering methods to Kirchner
et al.
