A general geomorphological recession flow model for river basins

Authors

  • Basudev Biswal,

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    1. Department of Civil Engineering, Indian Institute of Science, Bangalore, India
    2. Department of Civil Engineering, Indian Institute of Technology Hyderabad, India
    • Corresponding author: B. Biswal, Department of Civil Engineering, Indian Institute of Technology Hyderabad, Andhra Pradesh 502205, India. (basudev02@gmail.com)

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  • D. Nagesh Kumar

    1. Department of Civil Engineering, Indian Institute of Science, Bangalore, India
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Abstract

[1] Recession flows in a basin are controlled by the temporal evolution of its active drainage network (ADN). The geomorphological recession flow model (GRFM) assumes that both the rate of flow generation per unit ADN length (q) and the speed at which ADN heads move downstream (c) remain constant during a recession event. Thereby, it connects the power law exponent of –dQ/dt versus Q (discharge at the outlet at time t) curve, α, with the structure of the drainage network, a fixed entity. In this study, we first reformulate the GRFM for Horton-Strahler networks and show that the geomorphic α (αg) is equal to inline image, where D is the fractal dimension of the drainage network. We then propose a more general recession flow model by expressing both q and c as functions of Horton-Strahler stream order. We show that it is possible to have α =α g for a recession event even when q and c do not remain constant. The modified GRFM suggests that α is controlled by the spatial distribution of subsurface storage within the basin. By analyzing streamflow data from 39 U.S. Geological Survey basins, we show that α is having a power law relationship with recession curve peak, which indicates that the spatial distribution of subsurface storage varies across recession events.

1. Introduction

[2] Recession flows are characterized by continuously decreasing streamflow over time occurring during dry or no-rain periods. Drainage basins, irrespective of their location and size, follow an interesting recession flow pattern that –dQ/dt and Q (Q being discharge at the outlet at time t) exhibit a power law relationship [Brutsaert and Nieber, 1977]:

display math(1)

[3] Biswal and Marani [2010] suggested that recession flow in a basin is controlled by the dynamics of its active drainage network or ADN (the part of the drainage network that is actively draining at time t). In particular, they hypothesized that the gradual shrinking of the ADN [see, e.g., Gregory and Walling, 1968; Blyth and Rodda, 1973; Day, 1983] originates the power law relationship between –dQ/dt and Q (equation (1)). In order to prove their hypothesis, they proposed a conceptual model, the geomorphological recession flow model (GRFM), by making two simple assumptions that during a recession event: (i) the rate of flow generation per unit ADN length (q) remains constant in both space and time and (ii) the speed at which the heads of the ADN configuration move downstream (c) also remains constant in both space and time. With these two assumptions, one can find that the power law exponent α in equation (1) is equal to the power law exponent of the N(t) versus G(t) curve [Biswal and Marani, 2010], where N(t) and G(t) are, respectively, the number of heads and the total length of the ADN configuration at time t. Note that as c is constant, t = l/c, l being the distance of any ADN head from its farthest source or channel head at time t. Because of the linear relationship between l and t, both the N(t) versus G(t) curve and the N(l) versus G(l) curve give the same power law exponent, the geomorphic α (αg). Thus, αg depends only on the structure of the channel network, an entity that remains unchanged over time.

[4] Biswal and Marani [2010] observed that –dQ/dt versus Q curves from a basin can be different for different recession events, which implies that α must be computed individually for the available recession curves. They then considered the median of the distribution of the α values as the representative α (αr) of the basin, as the distribution is often skewed. Considering streamflow data from 67 basins situated across the United States, they found that αr is nearly equal to 2, particularly when the basins are steep and free from significant human interventions (see also Shaw and Riha [2012]). Then they used digital elevation models (DEMs) for the same set of basins and found that their αg values are nearly equal to 2.1. The observation αr being nearly equal to αg, or the error ϵr − αg) being nearly equal to zero, implies that the assumptions made by the GRFM (both q and c remain constant during a recession event) are adequate for the majority of recession events in a basin. However, the GRFM does not explain why some recession events do not give α equal to αg. Also, it was found that some natural basins display large ϵ values (see the inset of Figure 4, Biswal and Marani [2010]). We propose that, when observational errors are negligible, α deviates from αg because either q or c, or both vary during the recession event.

[5] In this study, we propose a broader theoretical framework to explain the deviation of observed α from αg by allowing both q and c to vary along stream channels. First, we reformulate the GRFM in the context of Horton-Strahler tree networks. We then generalize the model by expressing both q and c as functions of Horton-Strahler stream order. We show that the constant q and constant c assumptions, as adopted by the GRFM, are not the necessary conditions for having α =α g. We then analyzed observed recession curves from 39 U.S. Geological Survey (USGS) basins and found that there exists a power law relationship between α and the recession flow curve peak. Our analysis attempts to explain how the variation in subsurface storage distribution across streams of different orders controls the value of α.

2. Horton-Strahler Tree and the GRFM

[6] River networks are classic examples of a binary tree. They observe some deep structural regularities that can be expressed quantitatively by means of the well-known Horton-Strahler ordering scheme [Horton, 1945; Strahler, 1952] (see Figures 1a and 1b). In the Horton-Strahler ordering scheme, a stream that does not receive flow from any other stream is called a first-order stream. Two streams of order ω join to form a stream of order ω + 1. If two streams of different orders join, then the resulting stream will have the order that of the higher order stream. Horton [1945] found that in a typical river network, the ratio of average length of streams of order ω + 1 ( inline image) to that of streams of order ω ( inline image), RL, and the ratio of number of streams of order ω ( inline image) to that of order ω + 1 ( inline image), RB, are approximately constant for any chosen ω. That means, inline image and Nω for a river network with its highest order stream having order Ω can be expressed as:

display math(2)
display math(3)

where LΩ is the length of the Ω order stream or the main stream of the channel network. Note that the number of streams of order Ω is always equal to 1 for a real river network.

Figure 1.

(a) A graphical illustration of a group of stream reaches with different lengths (o) joining one another to form a drainage network (Ω = 2). Closed circles represent active ends (ends that are receiving flow) and open circles represent inactive ends (ends that are not receiving flow). (b) The network has been modified following the rule that a stream of order ω can only drain into a stream or order ω + 1 (Ω is 3 now). In either case, the application of the assumption that both q and c are constant gives the same N(l) versus G(l) curve, implying that the modification scheme preserves recession curve characteristics. (c) The N(l) versus G(l) curve for the two networks when both q and c are 1.

[7] In a channel network, a channel reach can only drain into its downstream channel reach; thus during a recession event, the channel reach will dry or stop flowing before its downstream channel reach. It also means that the drying pattern of the channel reach will be determined by its upstream channel reach and not by its downstream channel reach. This property of recession dynamics suggests that the channel reach can be relocated to any other place in the channel network as long as the drying pattern of its upstream channel reach remains the same. That means it is possible to restructure a channel network such that its drying patterns remain the same as those of the original network. The method of restructuring a drainage network according to a specific use is also known as a “dynamic tree” approach [Zaliapin et al., 2010]. Our objective here is to restructure a drainage network such that it gives the same N(t) versus G(t) curve or N(l) versus G(l) curve as that of the original network. In the new structure, we allow a stream of order ω to drain only into a stream of order ω + 1 (see Figure 1b). We also assume that streams of a certain order have equal length; that means, inline image is equal to Lω, length of any chosen stream of order ω. So if the constant c assumption is applied, all streams of a certain order will take the same amount of time to dry up. Considering that at time t, only the streams of order greater than or equal to ω are contributing (i.e., they are actively draining), application of the constant q assumption gives the expression for Q as:

display math(4)

where G(ω) is the length of the ADN at time t or the total length of the streams with order greater than or equal to ω. –dQ/dt is then defined as:

display math(5)

where inline image is the time taken by a stream of order w to dry up, which is also equal to inline image (equation (2)), owing to the constant c assumption. Thus, by combining equations (2)(5), we obtain:

display math(6)

[8] where inline image.

[9] If the constant c and the constant q assumptions remain valid, then the exponent of the inline image versus inline image curve should be the same as that of the –dQ/dt versus Q curve as inline image (equation (4)) and inline image (equation (6)). Figure 2 shows inline image versus G(ω) curve for a Horton-Strahler tree network with RB = 4 and RL = 2.

Figure 2.

The N(ω) versus G(ω) curve for a modified Horton-Strahler network with RB = 4 and RL = 2 when LΩ, q, and c are equal to 1. The slope of the curve in log-log plot, the geomorphic α (αg), decreases as ω approaches Ω. Thus, αg was computed by excluding the last two points (shown as lighter olive green dots). The value of αg for the network is equal to 1.95, which is close to the value obtained by the analytical expression in this study, 2.

2.1. An Analytical Expression

[10] Because LΩ is constant for a basin and q and c are assumed to be constant during a recession event, equation (4) gives the expression for Q as:

display math(7)

[11] When inline image (which is true for real river basins) and inline image [Rodriguez-Iturbe and Rinaldo, 1997]:

display math(8)

[12] Manipulation of equation (2) gives

display math(9)

as LΩ is a constant. Now using the expression for inline image (equation (9)) in equation (8) and taking logarithms of both the sides:

display math(10)

[13] The term inline image is known as the fractal dimension of the drainage network, inline image [La Barbera and Rosso, 1989]. Exponentiating both sides of equation (10), it can be found that

display math(11)

[14] Noting that inline image and inline image, equations (5) and (11) give

display math(12)

[15] Then combining equations (11) and (12):

display math(13)

because RL is a constant. Thus, the expression for the geomorphic α (αg) is:

display math(14)

[16] Equation (14) suggests that the fractal geometry of the drainage network [e.g., Rinaldo et al., 2006; Mantilla et al., 2010] gives rise to the power law relationship between –dQ/dt and Q. From Figure 2, αg obtained for the drainage network having RB = 4 and RL = 2 is 1.95, which is close to the value predicted by equation (14), 2. The discrepancy, although small, is introduced during the transformation of equation (7) to equation (8).

[17] According to de Vries et al. [1994], inline image, where β is defined as inline image, with inline image being the probability of a randomly chosen stream pixel having contributing area A greater than or equal to a [Rodriguez-Iturbe et al., 1992]. From Rigon et al. [1996], inline image, where h is the Hack's exponent [Hack, 1957]. So one can now find that αg for the Horton-Strahler stream network is equal to inline image, a relationship which was also obtained by Biswal and Marani [2010]. This proves that the modified Horton-Strahler network mathematically preserves recession flow characteristics of the original network. For real basins, inline image [Rodriguez-Iturbe and Rinaldo, 1997], which gives inline image, a value consistent with the earlier observation [Biswal and Marani, 2010].

3. A General GRFM: q and c Following Horton's Laws

[18] Geomorphological and ecological properties vary gradually along a stream channel in a drainage basin, suggesting that, in a relatively homogeneous hydro-geomorphological region, channel reaches located at equal distance from their respective channel heads are likely to have the same geometrical and physical properties [Vannote et al., 1980]. Often, the variation of physical properties along stream channels is characterized by power law relationships, e.g., between drainage area, distance l, channel slope, and channel cross section [Leopold and Miller, 1956; Hack, 1957; Montgomery and Foufoula-Georgiou, 1993; Rodriguez-Iturbe and Rinaldo, 1997]. The existence of power law relationships suggest that river basins exhibit self-similar property manifested in terms of Horton's laws [Peckham and Gupta, 1999]. Horton's laws can be expressed in a general form:

display math(15)

valid for any ω, where inline image is the average value of the random variable inline image for streams of order ω. Equation (15) can also be written as inline image or inline image, because inline image is a constant for the basin. It is found that equation (15) is valid not only for stream length and stream number, as originally suggested by Horton [1945], but also for many other physical variables like drainage area, channel cross section, discharge, channel slope, and vegetation indices [Schumm, 1956; Leopold et al., 1964; Dunn et al., 2011]. In this context, we assume that inline image and inline image, where Θ and inline image are constants; that means, inline image and inline image. From equation (4), the expression for Q is:

display math(16)

[19] The term inline image can be considered as the summation of lengths of the streams with order greater than or equal to ω in a Horton-Strahler network having either bifurcation ratio equal to inline image or length ratio equal to inline image, denoted here as inline image. Similar to the expression for G(ω) (equation (7)), when inline image and inline image can be expressed as:

display math(17)

[20] Combining equations (9) and (17), we find:

display math(18)

where inline image. The expression for –dQ/dt similarly will be

display math(19)

where inline image is the time taken by a stream of order ω to dry up when inline image or inline image (obtained by using equation (2)). Thus, the expression for inline image becomes (recalling that inline image and inline image):

display math(20)

where inline image. As RL is a constant, the combination of equations (17) and (20) gives the expression for the generalized GRFM as:

display math(21)

which gives the expression for geomorphic α for the modified Horton-Strahler tree as:

display math(22)

[21] D will remain constant for a basin. Figure 3 shows inline image versus inline image curves for different combinations of Θ and inline image. It can be noticed that inline image increases with decreasing inline image and/or increasing inline image. Increasing inline image and increasing inline image imply increasing Θ and increasing inline image, respectively, and vice versa.

Figure 3.

inline image versus inline image curves for different values of Θ and inline image: when (a) inline image and (b) inline image. In all cases, RB and RL were 4 and 2, respectively, and inline image, q, and c are equal to 1. It can be observed that the power law exponent inline image is increasing with inline image but decreasing with Θ.

Figure 4.

(a) A hypothetical single stream channel that connects a source (open circle) and the basin outlet (closed circle) undergoing desaturation (in the downstream direction) during a recession event. (b) Storage gradient (ds/dl) versus l curves when: both Θ and inline image are 1 (ds/dl is constant along the stream channel, blue line), either inline image or inline image (ds/dl increases with l, red line), and either inline image or inline image (ds/dl decreases with l, green line).

3.1. The Condition of “Pseudoequilibrium”

[22] We define “equilibrium” as the state of a basin in which different subsurface storage systems interact with each other such that both q and c become constant during a recession event. In this case, the power law exponent of the –dQ/dt versus Q curve is αg, which is equal to inline image. However, one can envision a scenario where both q and c vary during a recession event but still give α =α g, i.e., a scenario in which inline image. Thus,

display math(23)

[23] Simplifying equation (23), we obtain

display math(24)

[24] This means that the basin is in a “pseudoequilibrium” condition when inline image. Therefore, constant q and constant c, as assumed by Biswal and Marani [2010], are not the necessary conditions for having α =α g for a recession event. When inline image must be equal to inline image, which also means that Θ must be equal to inline image.

4. Analysis of Observed Recession Curves

[25] Flow of water beneath the earth surface is largely unknown due to technological limitations, e.g., it is not yet fully understood why “old water” dominates streamflows during flood events [e.g., Botter et al., 2010]. As recession flows occur during dry periods, they provide key information about the basin's subsurface storage systems [e.g., Krakauer and Temimi, 2011; Biswal and Nagesh Kumar, 2013]. The knowledge of the spatial distribution and movement of subsurface water is essential to efficiently manage water resources as well as to study the transport of solutes [Cardenas, 2007; Welch and Allen, 2012]. Groundwater may follow short (local) flow paths or long (regional) flow paths to reach stream channels [Toth, 1963]. Thus, the portion of water infiltrating into the subsurface of a hillslope adjacent to a lower order stream may follow longer flow paths and reach a higher order stream channel. As a consequence, base flow generation per unit channel length, which is also active subsurface storage per unit length (s) will increase in the downstream direction, a hypothesis supported by experimental evidence [Ophori and Toth, 1990]. For a channel reach of length dl situated at a distance l from its farthest source (Figure 4a), s is equal to the product of its q and the time period for which the channel reach drains, t: inline image. The expression for the gradient of s at the channel reach, ds/dl, can then be found as:

display math(25)

[26] Thus, for a channel reach of order ω, inline image. When both Θ and inline image are 1 (i.e., when both q and c are constant), ds/dl remains constant along a stream channel (see Figure 4b) or s increases linearly with l (as inline image if c remains constant). In this case, the power law exponent inline image according to equation (22). If either inline image and/or inline image, ds/dl increases in the downstream direction. In this case, inline image. Similarly when inline image and/or inline image, ds/dl decreases in the downstream direction, which gives inline image. The spatial distribution of subsurface storage in a basin depends on the topography and the distribution of hydraulic conductivity in the subsurface zones [e.g., Toth, 1963; Sophocleous, 2002; Haitjema and Mitchell-Bruker, 2005; Cardenas, 2007]. Some subsurface storage zones may store more water during a rainfall event than others. Therefore, with increase in effective rainfall volume characterized by the peak discharge (QP), which is also the peak of the associated recession curve, ds/dl may either decrease or increase with l depending on the distribution of hydraulic conductivity.

[27] Biswal and Nagesh Kumar [2013] defined a recession curve as a continuously decreasing streamflow time series lasting for at least 5 days and found no appreciable correlation between α and QP. The reason might be that observational and other errors can significantly affect the value of α. Therefore, we followed a more stringent criterion in this study to select recession flow curves by defining a recession curve as a discharge time series lasting at least for 5 days during which both Q and –dQ/dt decrease continuously [Shaw and Riha, 2012]. We then computed Q and –dQ/dt following Brutsaert and Nieber [1977] as: inline image and inline image, where inline image is the time step. We used daily discharge data for 39 relatively steep USGS basins (see Table 1 of the online supporting information; discharge data were obtained from http://waterwatch.usgs.gov/) and computed α for each recession event of a basin using the least square regression method. Basins with significant human interventions were avoided as activities like the presence of cities or dams can considerably alter recession curve properties [e.g., Wang and Cai, 2009; Biswal and Marani, 2010]. Although αr obtained by considering the new definition is not very different from that by considering the earlier definition (R2 = 0.80), by following the new definition, we found that α and Qp exhibit a power law relationship: inline image (see Figure 5). The R2 correlation of α versus Qp curve was found to be less than 0.1 for 21 study basins. The weak value of correlation may indicate that the distribution of subsurface storage along stream channels is unaffected by the amount of effective rainfall volume. Remarkably, the value of σ is positive for 31 basins, i.e., α decreases with increase in Qp, which may imply that high-intensity rainfall events cause Θ to increase and/or inline image to decrease. Similarly, negative values of σ for the remaining eight basins may imply that high-intensity rainfall events are causing Θ to decrease and/or inline image to increase. The modified GRFM can thus be used to obtain information on subsurface storage distribution, potentially for many practical applications like stream restoration [e.g., Bukaveckas, 2007].

Figure 5.

The power law exponent (α) of a recession curve versus its recession peak (Qp) curve ( inline image) for: (a) Bear (USGS id: 03076600; area = 126.65 sq km), (b) Tunkhannock (USGS id: 01534000; area = 991.75 sq km), and (c) Paluxy (USGS id: 08091500; area = 1061.90 sq km). Positive value of sigma indicates ds / dl increasing with l, and vice versa.

5. Summary

[28] The interplay between various subsurface storage units within a basin determines the drainage of water during no-rain or recession periods. Possibly, the channel network reflects the distribution of subsurface storage. The GRFM suggests that the exponent of the power law relationship between –dQ/dt and Q, α, has links with the channel network structure. In particular, this model assumes that both q and c remain constant in a basin during individual recession events. For most steep and natural basins, the median of the observed α values or the representative α, αr, is nearly equal to the power law exponent of the modeled recession curve (N(l) versus G(l)), αg, which indicates that the GRFM is generally able to capture the real recession flow characteristics. However, the GRFM cannot explain the discrepancy (if any), either between α, the power exponent of an individual recession event, and αg or between αr, the representative power law exponent, and αg.

[29] In this study, we reformulated the GRFM for Horton-Strahler tree networks. Particularly, we restructured Horton-Strahler trees such that a stream of order ω can only drain into a stream of order ω + 1. This scheme simplifies the computations while preserving the original recession flow characteristics. We found that the geomorphic α of a basin, αg, is related to its fractal dimension inline image (which is equal to inline image) as: inline image. We showed that this expression for αg also leads to the relationship: inline image (h being Hack's exponent), a relationship which was previously obtained and experimentally confirmed by Biswal and Marani [2010]. Hence, our network modification scheme preserves the recession characteristics of a drainage basin.

[30] We then proposed a broader conceptual framework to study the exponent –dQ/dt versus Q curve by allowing both q and c to vary across streams of different Horton-Strahler stream order (ω). Particularly, we assumed that both q and c follow the generalized Horton's law as: inline image and inline image, which also means inline image and inline image, where Θ and inline image are constants for the drainage network. The modified model gives geomorphic (modeled) α, inline image, equal to inline image. We showed that it is not necessary that both q and c remain constant (“equilibrium” condition) during a recession event for α to be equal to αg. There can be a “pseudoequilibrium” condition in which inline image is equal to αg when inline image.

[31] The modified GRFM suggests that the value α depends on the distribution of subsurface storage along the stream channels. If both Θ and inline image are 1, i.e., when both q and c remain constant, the subsurface storage gradient (ds/dl) remains constant along a stream channel, giving inline image. If inline image and inline image, ds/dl increases with l and inline image. Similarly, if inline image and inline image, ds/dl decreases with l, which gives inline image. We found that α and recession curve peak (Qp) exhibit a power law relationship: inline image, which possibly indicates that ds/dl is sensitive to effective rainfall intensity. Results obtained in this study are indicative of the possibility that information on the subsurface storage distribution of a basin can be obtained by analyzing its recession flow curves.

Acknowledgments

[32] The authors are grateful to the three anonymous reviewers for their helpful comments and suggestions.