Geophysical Research Letters

Geomorphological origin of recession curves



[1] We identify a previously undetected link between the river network morphology and key recession curves properties through a conceptual-physical model of the drainage process of the riparian unconfined aquifer. We show that the power-law exponent, α, of −dQ/dt vs. Q curves is related to the power-law exponent of N(l) vs. G(l) curves (which we show to be connected to Hack's law), where l is the downstream distance from the channel heads, N(l) is the number of channel reaches exactly located at a distance l from their channel head, and G(l) is the total length of the network located at a distance greater or equal to l from channel heads. Using Digital Terrain Models and daily discharge observations from 67 US basins we find that geomorphologic α estimates match well the values obtained from recession curves analyses. Finally, we argue that the link between recession flows and network morphology points to an important role of low-flow discharges in shaping the channel network.

1. Introduction

[2] River networks exhibit remarkable morphological and structural similarities, which can be expressed quantitatively through traditional and fractal geometry [Horton, 1945; Hack, 1957; Marani et al., 1994; Rodríguez-Iturbe and Rinaldo, 1997]. One intriguing and important question is how the spatial organization of river networks is reflected in the hydrologic response to rainfall inputs. The geomorphological theory of the hydrologic response [Kirkby, 1976; Rodríguez-Iturbe and Valdés, 1979; Rinaldo and Rodríguez-Iturbe, 1996] links the geomorphological properties of the hillslope-network system and its responses [e.g., Rinaldo et al., 2006a, 2006b], but requires the complete specification of travel times for the local groundwater flows in order to deal with recession hydrographs. In fact, to our knowledge, few contributions have attempted to connect recession hydrographs and the morphological features of the drainage network producing it [Marani et al., 2001; Vivoni et al., 2008], even though recession curves have long been studied [e.g., Brutsaert and Nieber, 1977].

[3] Through a simple model of the drainage process and discharge observations from 67 US basins we provide here evidence that key properties of low-flow regimes are determined by the time-varying geometry of saturated channelled sites.

2. Active Drainage Network and the Source Function

[4] We assume that the recession hydrograph at an outlet is dominated by drainage of the unconfined aquifer by the channel network instantaneously intersecting it, the Active Drainage Network (ADN). ADN geometry varies over time because drainage of the aquifer produces the progressive ‘desaturation’ of the ephemeral low-order streams (see Figure 1).

Figure 1.

Idealized recession process. Source streams are indicated with closed circles. Solid lines represent active streams, dotted lines represent dry streams.

[5] Within this conceptual framework we make the following assumptions:

[6] 1. The recession flow varies slowly in time and can be described through a succession of steady states. The total discharge at the outlet, Q(t), is thus equal to the total drainage discharge delivered by the aquifer to the network (i.e., propagation effects along the network are negligible):

equation image

where G(t) is the total length of the ADN and q is the discharge per unit length drained by the ADN;

[7] 2. q is spatially constant.

[8] 3. The rate at which source links recede, c = dl/dt, is constant in space and time (a source link is defined as a saturated link with no upstream saturated links).

[9] Under these assumptions one finds:

equation image

where N(t) is the number of channel sources in the ADN configuration at time t (see Figure 1). The last equality in equation (2) recognizes that the rate of change of the total ADN length, dG/dt, is equal to the rate at which individual saturated source links recede, multiplied by the number of source links in the current ADN configuration. dQ/dt in the recession phase thus depends both on N(t), controlled by the desaturation of channel reaches, and on dq/dt, controlled by aquifer depletion. One can now envision two end-member cases. In the first case dQ/dt is dominated by the rate of change of the ADN length (likely to occur in sloping basins). In the second case dQ/dt is dominated by the rate of change of q (more likely to happen in relatively flat basins). Brutsaert and Nieber [1977] focus their attention on this latter case, and study the time dependence of dq/dt in a non-sloping channel. Here we focus on the time dynamics of the ADN and make the following assumption:

[10] There exists a regime in the recession phase in which the geometry of the ADN varies quickly, such that the term in dq/dt can be neglected and equation (2) becomes:

equation image

If the recession phase starts at t0, an arbitrary channel segment in the initial ADN contributes to the recession flow for a time interval τ = tt0 which is directly proportional to the distance, l, separating the segment considered from the farthest upstream source: tt0 = l/c. Because of this correspondence between time and length along the network we can trade time for length, such that G(l) is the total length of the ADN at a distance greater or equal to l downstream of the sources of the initial ADN configuration. N(l), here termed the Source Function, is defined as the number of sites located at a distance l from the sources of the initial ADN configuration (see Figure 1, top). At time t one thus finds:

equation image

Equations (4) link network morphology to recession curve properties and can be tested by studying −dQ/dt as a function of Q. One generally finds a power-law relation of Brutsaert and Nieber [1977]:

equation image

This is typically justified by assuming a power-law relation between the volume of water stored within the aquifer, V, and discharge: VQβ. In fact, upon consideration of the continuity equation dV/dt = −Q, one readily finds equation (5) with α = 2 − β [Kirchner, 2009]. Equations (4) suggest, on the other hand, that the form of the observed −dQ/dt vs. Q curves can be traced to the underlying geomorphologic relation N(l) vs. G(l). In fact, because of (4), equation (5) holds if:

equation image

with αg = α. Relations (5) and (6) reveal the geomorphic origin of recession curves.

3. Observational and Geomorphic Recession Curves

[11] We use daily average discharge data obtained from USGS for a set of 67 basins in the U.S. sampling a wide variety of different areas, relief, soil type and use. We note that the daily resolution is appropriate for the present scopes as we are here interested in recession curves, i.e in timescales of the order of several days. Following Brutsaert and Nieber [1977], we plot −dQ/dt ≈ −(QiQi−1)/Δt vs. Q ≈ (Qi + Qi−1)/2, where Δt = 1 day. We discard winter observations (December to February), to avoid mixing snow-melt and aquifer contributions, and consider as recession curves only periods of monotonically decreasing streamflow lasting at least 5 days (but experiments with different thresholds above 5 days yielded very similar results). We also consider only recessions occurring after discharge peak values larger than the average discharge of the stream, to avoid considering minor events which may not have significantly increased the average soil saturation, thus not triggering a significant response of the groundwater. −dQ/dt vs. Q plots (see Figure 2a for a representative example: Paluxy basin − TX − drainge area = 1062 Km2) exhibit different regimes. Relatively high values of Q (A–B in Figure 2a) correspond to parts of the hydrograph with considerable contributions from surface processes. The relation between dQ/dt and Q is not one-to-one, as may be expected in the case of fast, non-stationary, surface flows. Only for slightly lower discharge values (B–C) we observe a scaling behaviour of −dQ/dt vs. Q. We interpret this regime (which is present in all hydrographs analyzed from all the basins considered) as the result of the dominance of drainage processes through the ADN. For increasingly smaller values of Q (C–D) the hydrographs tend to be quite ‘flat’, which we interpret as being connected with a decrease in drained discharge per unit length (q) due to the depletion of the aquifer (rather than to changes in the geometry of the ADN). Interestingly, the observed exponent in −dQ/dt vs. Q curves (α in equation (5)) remains fairly constant in the intermediate scaling regime across a large number of different events for the same basin (Figure 2b). However, the curves for different events tend to be shifted, such that, for a given value of Q, more intense events (characterized by large peak discharges, on the right in Figure 2b) exhibit a smaller value of −dQ/dt with respect to less intense events. This finding strongly suggests that the one-to-one relation VQβ, assumed in previous studies to justify equation (5) [e.g., Tague and Grant, 2004; Kirchner, 2009], does not hold. Furthermore, the simultaneous fit of the observations from all hydrographs produces a severe underestimation of the actual value of α characterizing individual recession hydrographs (thin solid line in Figure 2b). The estimation of the scaling exponent must, on the contrary, be performed by separately analyzing each available hydrograph to produce a frequency distribution of observed α values. Figure 2c illustrates the outcome of such a procedure applied to 336 events for the Paluxy basin. Mean, median and standard deviation of the distributions obtained for a significant number of hydrographs (between 22 and 468) for the entire dataset are shown in Table S1 of the auxiliary material.

Figure 2.

(a) A sample recession curve for the Paluxy basin. Different regimes are shown as AB, BC and CD. (b) BC portions of recession curves for selected representative hydrographs and their trend lines (α varies between 2.05 and 2.28). The solid thin line is the, very different, trend line (slope = 1.19) obtained by fitting all data points simultaneously. (c) Frequency distribution of α values computed from all hydrographs for the Paluxy basin.

[12] We then analyzed the DTM's (30 m resolution) for all the 67 basins considered and extracted the drainage networks using flow accumulation thresholds [O'Callaghan and Mark, 1984]. We also experimented with different delineation criteria (e.g., thresholds on upstream distance to the source, on cumulated area plus concavity condition, and the area-slope criterion [Tarboton et al., 1991]) with no appreciable difference in the results that follow. For each pixel in the extracted channel network the length, l, to the farthest source is computed to construct N(l) and G(l). Plotting N(l) vs. G(l) (Figure 3a) shows a distinct scaling regime, corresponding to ADN configurations which are intermediate between the fully saturated condition (l = 0, A in Figure 3a)) and an ADN including just the main channel branches (C in Figure 3a). The transition toward the scaling regime (B in Figure 3a) may be interpreted as the sign of a transition from unchannelled (hillslope) to channelled sites. It is indeed quite remarkable that an intermediate scaling regime is present in all cases analyzed (see Figure 3b, for a representative sample) as hypothesized in equation (6). The resulting scaling exponents are summarized in Table S1 of the auxiliary material for the entire set of available basins.

Figure 3.

(a) Geomorphic recession curves of the Paluxy basin with flow accumulation threshold (FA = 1 open circles; FA = 100 triangles). N(l) vs. G(l) curves for the two cases are indistinguishable. (b) Geomorphic recession curves of representative basins extracted from the dataset.

[13] If the exponent of the geomorphic recession curve, αg, corresponds to the exponent derived from observed recession curves, αo, the widely observed relation (5) can now be justified purely in terms of the geomorphic properties of the channel network, without the need of a one-to-one relation between discharge and water volume stored in the basin. We thus study a αg vs. αo plot (Figure 4), which allows a detailed comparison of geomorphological and recession-flow exponents. We separately analyze basins which are relatively steep (based on DTM relief and by locating forested areas in satellite images) and do not have significant reservoirs (which largely control recession discharge values) and the remaining basins. Figure 4 supports the conclusion that αgαo for relatively steep basins. Basins with modest relief do not show such a correlation, suggesting that, in this case, dQ/dt may be dominated by the term in dq/dt in equation (2). The mean of the frequency distribution of the residuals αoαg obtained for the ‘steep’ basins (see inset in Figure 4 and Table S1 of the auxiliary material) is quite small (mean error = −0.19, standard deviation of the error = 0.20, the standard deviation of αo being 0.18), further supporting a good agreement between the exponent obtained from recession curves and its geomorphic estimates.

Figure 4.

αg vs. αo. Black circles refer to sloping basins and show a good agreement between geomorphic and recession curve exponents. Gray, smaller, circles refer to flat basins, which exhibit a lack of correlation between αg and αo, likely because dQ/dt is in this case dominated by dq/dt rather than by changes in the ADN length. The inset shows the frequency distribution of the residuals, αoαg, between hydrograph-derived recession curve scaling exponents and their geomorphic estimates for the sloping basins.

4. Universal Properties of Recession Curves: Hack's Exponent

[14] The geomorphic power-law relation (6) can be linked to Hack's law, expressing the distance from the outlet to the divide as a function of the basin area: lAh. h is Hack's exponent [Rigon et al., 1996]). The total length of the ADN, G(l), for a given configuration indexed by a value l, can be expressed as the total length of the stream network at full saturation, L = G(0), multiplied by the probability, P(equation image > l), that a site chosen at random within the network exhibit a distance to the divide, equation image, larger than l:

equation image

The probability distribution P(equation image > l) is known to be power-law over a large range of scales, where P(equation image > l) ∝ lγ [e.g., Rigon et al., 1996]. Therefore, Qlγ and, because l = c(tt0), −equation imagel−(γ+1), such that −equation imageequation image. Comparing this latter relationship with equation (5) yields:

equation image

Rigon et al. [1996] show that γ = β/h, where:

equation image

P[A > a] is the probability that the contributing area A at a point in a drainage network exceed an arbitrary value a. Finally, considering that h + β = 1 [Maritan et al., 1996], we find:

equation image

Interestingly, the value of Hack's exponent obtained from (10), by using the median values of αo and αg for sloping basins, are 0.48 and 0.53 respectively, to be compared with values of h in the range 0.52 – 0.60, with mean h = 0.55, obtained for a different set of basins by Rigon et al. [1996]).

5. Conclusions

[15] Recession curves in sloping basins bear the signature of the geomorphic structure of the channel network which produces them. Such a signature can be identified through a simple model of the ADN dynamics under the following assumptions: i) Q is proportional to the total length, G(l), of the ADN, ii) the discharge per unit length of the ADN, q, is constant in space, iii) the rate, c, at which the ADN contracts is constant in space and time, and iv) the rate of change of the total length of the active drainage network dominates over the rate of change of q, such that dQ/dt is proportional to the number, N(l), of the terminal links of the ADN.

[16] N(l) is likely to become small during prolonged drought periods, even in sloping basins. In this regime the ADN is limited to the main and most stably saturated links of the channel network, such that the term containing dq/dt in the expression for dQ/dt(2) can no longer be neglected. In the case of prolonged recession periods one must thus also account for the term dq/dt, as in Brutsaert and Nieber [1977]. The theoretical interpretation developed here, however, does capture the properties of observed recession curves for the large number of sloping basins considered, suggesting that indeed q · dG/dtdq/dt · G(t) in a relatively frequent dynamical regime. In fact, the power-law exponents obtained from observational −dQ/dt vs. Q curves agree well with the exponents obtained from the corresponding N(l) vs. G(l) curves when basins are relatively steep. This result provides a valuable conceptual interpretation of recession hydrographs and possibly a way to infer their characteristics solely from DTM's.

[17] Our analyses show that the power-law form of −dQ/dt vs. Q curves does not stem from a one-to-one relationship between Q(t) and the volume of water stored within the basin, but rather from the underlying morphological structure of the channel network. We also show that, in order to avoid a severe underestimation of the scaling exponent, recession analysis should be performed separately for each single event, rather than by binning all recession data together.

[18] Finally, our results identify some general connections between recession curves and Hack's law, which characterizes the organizational structure of channel networks. Values of Hack's exponent retrieved from observed recession curves and from N(l) vs. G(l) curves are in good general agreement with those obtained from traditional DTM analyses, in further support of the geomorphic roots of recession curves properties.

[19] In closing, it is interesting to note that the good agreement between the predictions from the model of ADN dynamics and the observed exponents of −dQ/dt vs. Q curves is a confirmation that the hypotheses of the model indeed hold for the large set of sloping basins and events considered. In particular, the specific discharge q appears to be approximately spatially constant. One is thus led to argue that the incision of the channel network may be due, to a significant degree, to subsurface flow, in such a way as to produce an approximately uniform drainage of the local groundwater system and thus a unform distribution of q.


[20] The authors gratefully acknowledge the CARIPARO foundation for funding the PhD scholarship awarded to BB, and for financial support through the research project ‘Transport phenomena in river basins: theory and hydrologic and geophysical observations’. Funding from the University of Padova ‘Strategic Research Project’ on ‘Geological and Hydrological Processes: Monitoring, Modelling and Impacts in North-east Italy’ is also acknowledged.