Several authors have reported diel oscillations in streamflow records and have hypothesized that these oscillations are linked to evapotranspiration cycles in the watershed. The timing of oscillations in rivers, however, lags behind those of temperature and evapotranspiration in hillslopes. Two hypotheses have been put forth to explain the magnitude and timing of diel streamflow oscillations during low-flow conditions. The first suggests that delays between the peaks and troughs of streamflow and daily evapotranspiration are due to processes occurring in the soil as water moves toward the channels in the river network. The second posits that they are due to the propagation of the signal through the channels as water makes its way to the outlet of the basin. In this paper, we design and implement a theoretical model to test these hypotheses. We impose a baseflow signal entering the river network and use a linear transport equation to represent flow along the network. We develop analytic streamflow solutions for the case of uniform velocities in space over all river links. We then use our analytic solution to simulate streamflows along a self-similar river network for different flow velocities. Our results show that the amplitude and time delay of the streamflow solution are heavily influenced by transport in the river network. Moreover, our equations show that the geomorphology and topology of the river network play important roles in determining how amplitude and signal delay are reflected in streamflow signals. Finally, we have tested our theoretical formulation in the Dry Creek Experimental Watershed, where oscillations are clearly observed in streamflow records. We find that our solution produces streamflow values and fluctuations that are similar to those observed in the summer of 2011.

Several authors have observed daily fluctuations in streamflow
during periods of little or no rain

While observing streamflow at the outlet of the Dry Creek Experimental
Watershed in Idaho during July of 2011 (described in Sect.

In this paper, we aim to design and implement a theoretical model to test the
hypotheses that attribute time delays to flows in the river network. We start
by assuming a particular baseflow pattern in each river link of a given river
network. Then, we work with simplified routing equations that assume constant
velocity and give rise to a linear transport equation that allows us to
develop an analytic solution for the flow at any given point along the river
network. By fixing the baseflow pattern, we remove the dependence of
streamflow properties (e.g., amplitude and time delay) on soil processes. If
the resulting streamflows along the river network exhibit oscillations with
different time delays and amplitudes, then we conclude that the effects
described in

The paper is structured in the following way: in Sect.

The Dry Creek Experimental Watershed in Idaho is a 28 km

The left panel shows the Dry Creek watershed in Idaho. The right panel shows temperature (top), streamflow at gauge C2E near the center of the watershed (second), streamflow at gauge C2M (third), and streamflow at the outlet of the watershed (bottom). To demonstrate the delay in phases, a vertical line at the beginning of each day is included in each graph.

The left panel of Fig.

Let us now assume that the total subsurface runoff from each hillslope into a
river link in a given river basin is oscillatory and its amplitude undergoes
exponential decay (as seen for baseflow under dry conditions). Then, we
define the runoff by the formula

The left panel shows how runoff enters the river
network as lateral flow from each hillslope to its adjacent link. The right
panel shows a sample baseflow pattern given by Eq. (

A sample baseflow pattern with parameter values

In this paper, the streamflow at the outlet of a river link is defined as the
solution to the system of ordinary differential equations, which has been
derived from the mass conservation equation in the river links of the
network, given by

To determine the streamflow at the river network outlet, we first consider
the influence of runoff from a single hillslope and how that runoff signal
propagates downstream; see Sect.

To determine the solution at any point, we consider runoff on only one hillslope (adjacent to link 1 in this case), and we trace the effects of that runoff downstream with no additional runoff from any subsequent hillslopes.

As mentioned above, we first apply runoff

When the runoff entering link

At step 2, when the runoff has traversed

The goal of this section is to determine the equation for the streamflow at a
given point of calculation along the river network, in particular at the
network outlet. We take the parameters representing properties of each river
link to be uniform over all links in the network (i.e., same parameter

A small sample network to describe how total streamflow is computed.

First, we will focus on the outlet of link

For a general network whose width function is given by

To thoroughly interpret the components of Eq. (

In the limiting case of

It is apparent that the second sum of Eq. (

In order to test the competing hypotheses by

We compute the streamflow solution for the Mandelbrot–Vicsek tree of
magnitude 14, as shown in Fig.

In the case of uniform velocities, the streamflow at the outlet is given by
the solution to Eq. (

The Mandelbrot–Vicsek tree of magnitude 14. The magnitude of each link is written next to the link. One link of each magnitude is distinguished by the dots along the network.

Sample runoff pattern (top) and resulting
streamflow solution at the outlet in the uniform case (bottom) for

Flows at the outlet of each magnitude link using
different

Recall that

While the effect of varying

The results of simulating streamflow in the Mandelbrot–Viscek tree using
different values of

From Fig.

Average flows at different locations along the Mandelbrot–Vicsek tree.

At the link of magnitude 1, the phase shift has little influence on the amplitude and only has an influence on the timing of the wave. At the outlet of a magnitude-2 link, the two upstream links are “in phase”, meaning they have the same time delay as each other since they are the same topological distance from the point at which we compute streamflow. Therefore, these two will exhibit constructive interference. When they are combined with the downstream link, however, the different values of phase shift can result in constructive or destructive interference, although they never completely destroy the oscillations. The phase shift that produces the maximum streamflow is zero because this represents the face that all three streamflows that feed into this outlet are completely in phase.

As we examine the streamflows in links with greater magnitude, the shape of the network (described by the width function) becomes important because the flows from all links of a given distance will reach the outlet at the same time. Being out of phase with links of other distances can cause some reduction in the amplitude of the streamflow oscillations, but the oscillations will not be completely destroyed.

In this section, we apply the analytic streamflow solution for uniform
conditions to the river network of the Dry Creek river basin to study the
effects of scale on streamflow amplitude and timing. In the previous section,
we also examined the flow at different scales (see Fig.

In our theoretical examples, we assumed the length of each link to be uniform
over the river network, so that changes in velocity directly correspond to
changes in the transport constant

Observed streamflow and streamflow fitted
using Eq. (

[t]

For a comparison with available data, we revisit information from the Dry
Creek Experimental Watershed in Idaho. Using streamflow data from LG, the
gauge nearest the watershed outlet along with topological data retrieved
using the program CUENCAS proposed in

Using the observed streamflow time series at several upstream gauges in the
Dry Creek watershed, we can test our analytic solution with the parameters
determined above. If we treat these locations as the outlets of smaller
embedded watersheds, we can again apply Eq. (

Although the predicted streamflow given by our solution does not fit the data
as well for C2E as it does for C2M, we can see that the magnitude of our
predicted streamflow is very close to observed streamflow at either location.
Furthermore, the timing of the oscillations is nearly identical for both C2M
and C2E. In the presence of heterogeneity on the hillslope and along the
river network, we must be flexible about the amount of data we can reasonably
expect to fit well. For example, we show in Appendix

Because our solution fits the data reasonably well at several locations along
the river network where runoff is uniformly enforced, we can be assured of
the internal validity of using a solution such as that given in
Eq. (

Observations of oscillatory streamflow during low-flow conditions have highlighted the magnitude and time delay caused by the diel signal that represents evapotranspiration. Several current hypotheses suggest that the properties of the oscillatory streamflow signal can be attributed to different methods of water movement through the subsurface, although another hypothesis suggests that flow along the river determines the timing and amplitude of oscillations. In this paper, we provide evidence to support the latter argument.

First, we select a mathematical function according to streamflow observations at the catchment scale to represent baseflow patterns at the hillslope scale. The selected baseflow pattern is applied as input to a linear transport equation for all links in a river network that are assumed to have uniform properties and parameter values. For this uniform situation, we develop an analytic solution to represent streamflow at any point in a river network. We compute the solution by separately determining the partial streamflow at the outlet from each river link and then taking the sum over all river links in the river network. In order to include the geomorphology of the river network, we use the width function to compute the complete streamflow solution. We have also extended the streamflow solution to include nonuniform links in the river network.

The solution for streamflow contains a collection of sine functions, each of
which exhibits a phase shift determined by the topological distance of the
corresponding hillslope from the outlet. We have shown that these phase
shifts alone can cause constructive or destructive interference along the
river link but that the physical parameters that determine the phase shift
have a greater impact on the streamflow as it propagates downstream. The
streamflows computed using different physical parameters demonstrate that the
decreasing amplitude and increasing time delay in observed streamflows can be
attributed to the decreasing velocity in the river network during dry
conditions, and they are not necessarily due to soil–water processes, as was
previously thought, which supports the hypothesis of

As a next step, we propose to test the analytic solutions herein in networks with different geomorphological structures in order to compare the resulting streamflow amplitudes and emphasize the dependence upon network geometry. We suggest subsequently comparing our analytic solutions with the numerical results obtained using nonlinear transport equations, which will demonstrate the relationship between link propagation at the hillslope scale and streamflow at the catchment scale. Careful field experiments would be necessary to provide a definitive conclusion about the attribution of time delays.

Data available at

In order to simplify our calculations below, we will use the notation

We prove Eq.

Using these newly defined quantities from Eqs. (

Observed streamflow and streamflow fitted
using Eq. (

Because the Dry Creek Experimental Watershed includes stream gauges at seven
different locations, we sought to compare our solution at all of these
locations. One such gauge (called Treeline) did not experience streamflow
during the duration of our observations. Three other gauges – Con1East,
Con1West, and Bogus South (labeled C1E, C1W, and Bogus,
respectively) – recorded streamflows which can be found in
Fig.

As can be seen in Fig.

The observed and predicted streamflow at the location C1E can be found in the
left panel of Fig.

The center panel of Fig.

This material is based on work supported by the National Science Foundation under grant number NSF DMS-1025483 and financial support from the Iowa Flood Center. The authors also want to acknowledge Witold Krajewski from the University of Iowa and IIHR for helpful discussions and feedback during the preparation of the manuscript.Edited by: T. Hengl