Abstract

Locations of landslides, bank failures, and floodplain deposition during recent intense flooding in Vermont and Colorado (USA) were spatially nonuniform, indicating that some reaches are more prone to these types of geologic hazards. These three key flood effects signal redistribution of sediment across landscapes, reflecting hillslope-channel coupling and the sources and/or sinks of material. We show that spatial gradients in total stream power (Ω) provide critical additional information beyond at-a-point Ω magnitudes for predicting which reaches are likely to be susceptible to these hazards during floods. Field tests in four rivers (watershed areas 0.8–180 km2) indicate that downstream increases in Ω coincide with erosion and mass wasting into channels, and downstream decreases in Ω are associated with floodplain deposition. Our analytical approach, supported by field evidence, predicts geologic hazards and, more broadly, sources and sinks of material along rivers. These are critical concerns from a practical and theoretical standpoint.

INTRODUCTION

Catastrophic floods cause billions of dollars (U.S.) in damages and reshape landscapes; some locations are increasingly susceptible to these events due to changes in climate and land use. The 2011 Vermont and 2013 Colorado (USA) floods demonstrated the societal and geologic relevance of these floods and the need for quantitative tools to predict where different types of geomorphic changes will occur in these mountainous regions. These were record-breaking events: peak discharges throughout Vermont greatly exceeded the 1-in-100-yr flood magnitude (Magilligan et al., 2015), while in Colorado precipitation totals exceeded the 1-in-1000 yr 24 h rain event (Gochis et al., 2014). Spatially variable erosion occurred in the form of landslides, debris flows, bank failures, and channel incision. Equally widespread but variable deposition occurred in the form of overbank and floodplain accumulation of fresh sediment with transport of sediment ranging from silt to >1-m-diameter boulders (Buraas et al., 2014; Coe et al., 2014; Yellen et al., 2014; Anderson et al., 2015; Magilligan et al., 2015). The dramatic and variable geomorphic response provided field evidence to test sediment transport theory on hillslope-channel coupling along rivers.

Geologic boundary conditions modify long profiles of rivers, creating discrepancies from archetypal concave-upward profiles (Gilbert, 1877) such that rivers exhibit relatively steep reaches with high stream power (Ω) interspersed by less steep, more tranquil reaches with low Ω. Stream power is defined as 
graphic
where Q, S, ρ, and g are stream discharge, local channel slope, water density, and gravity, respectively (Bagnold, 1977). Hack (1957, 1973) linked gradients in channel slope, lithologic controls, and erosion by showing that downstream changes in the stream-gradient index, which scales approximately with Ω, correspond to differences in bedrock type and caliber of delivered sediment load. Others have used gradients in Ω over broad scales to reveal differential uplift rates (McKeown, 1988; Kirby and Whipple, 2001) and rainfall (Schlunegger et al., 2011) across mountain ranges. These studies show that a longitudinal profile can be a dependent variable, reflecting influences over long time scales. However, on short time scales, a longitudinal profile can be essentially invariant and thus act as an independent variable that constrains gradients in Ω and regulates sediment transport.

Exner (1920) showed that downstream gradients in velocity, a proxy for sediment transport, cause vertical bed erosion or deposition at the scale of stream-bed ripples; more recent work has coupled Exner’s approach with continuity equations for sediment transport (Paola and Voller, 2005). However, these one-dimensional models are constrained to vertical erosion and deposition on the bed and, for understandable simplicity, neglect lateral sources and sinks on banks, floodplains, and hillslopes. Furthermore, direct field evidence of Exner’s (1920) theory has been elusive, especially beyond the scale of the reach of a stream and over short time scales. Most work has relied on reconstructing vertical aggradation and downcutting over century to millennial time scales, for example, in studies that have sought to explain arroyo formation (Graf, 1983), terrace development (Hancock and Anderson, 2002), and classification of confined channel types (Bizzi and Lerner, 2015). Other studies implicitly reference Exner (1920) without drawing direct, quantitative relationships to his model (Miller, 1995).

This paper has two goals. First, we aim to expand the Exner (1920) model to include lateral exchange of sediment between stream channels, banks, floodplains, and hillslopes. We develop a model using conservation of mass to predict locations of erosion and deposition based on downstream gradients in sediment transport. Erosion and deposition are equivalent to sources and sinks of sediment and associated material (e.g., carbon, other nutrients, and pollutants) along river profiles. Second, we utilize field evidence to test this model. Taking advantage of the abundant and readily delineated landslides and floodplain deposits in the 2011 Vermont and 2013 Colorado floods, we compare locations of predicted and observed erosion and deposition. We hypothesize that locations of increases and decreases in Ω inferred from digital elevation model (DEM) analysis correspond with locations of predominant inputs and outputs, respectively, of stream sediment.

GRADIENTS IN SEDIMENT TRANSPORT

Previous studies use a conservation of mass approach to show that the divergence in downstream sediment transport is offset by deposition or erosion on the stream bed, inducing bed elevation changes (Exner, 1920; Paola and Voller, 2005). We extend this concept to accommodate lateral inputs and outputs of sediment, thereby equating downstream changes in sediment flux to lateral and vertical exchanges of sediment between the water column, banks, and bed: 
graphic
where Qsx is volumetric sediment flux transported in the downstream direction x; qsy is the magnitude of cross-stream, y direction volumetric input of sediment from channel-adjacent banks, floodplains, and hillslopes per unit length of channel; As is the cross-sectional area of sediment mantling the bed per unit length of channel; t is time; and ε ≡ 1 – ϕ, where ϕ is porosity of the bed (Fig. DR1 in the GSA Data Repository1).

The essence of Equation 2 is that reaches with downstream increases in sediment transport (dQsx/dx > 0) convey all the sediment delivered from upstream plus additional inputs from the bed, banks, and adjacent hillslopes. This is associated with net erosion from the channel bed, banks, and possibly adjoining hillslopes, thus inducing landslides and bank failures. In contrast, reaches with downstream decreases in sediment transport (dQsx/dx < 0) do not convey all the material supplied from upstream, and thus lateral or vertical deposition in the form of floodplain and bed deposition must result. A necessary condition for either scenario is that sediment is in transport, Qsx ≠ 0, with available sediment and thresholds for sediment transport exceeded at the location of analysis or upstream. Equation 2 can be used to predict locations of erosional and depositional reaches, given an appropriate index of sediment flux, Qsx.

METHODS

We examined four watersheds with some of the highest rainfall amounts during the 2011 Vermont and 2013 Colorado floods: Saxtons River (Vermont), West Branch of the White River (Vermont), Fourmile Canyon Creek (Colorado), and an unnamed creek on Mount Sanitas (Colorado) (Fig. 1; Figs. DR2–DR8). These are alluvial rivers in humid and semi-arid mountainous terrain with variable slopes, typically 10% to 0.2%. Bed caliber ranges from silt to boulders.

To test locations of erosion and deposition predicted by gradients in Qsx, we used Ω as a proxy for Qsx in Equation 2, because Qsx scales with Ω (Bagnold, 1977; Dade, 2012). Reaches of downstream increases in Ω reflect locations where dQsx/dx > 0 and downstream decreases in Ω reflect locations where dQsx/dx < 0. The value of Ω was computed based on analysis of 10 m DEMs (Gesch et al., 2002; Finlayson and Montgomery, 2003). We used a linear relationship for Q as a function of flow accumulation area, referenced to the peak flow measured at nearby U.S. Geological Survey (USGS) gages (01154000, 01144000, and 6727500). Stream power was smoothed over a distance of 200–1000 m upstream using a least-squares best-fit of point elevation measurements along the longitudinal profile, consistent with other basin-scale studies that smooth over a scale ∼1/10 the square root of drainage area (Kasprak et al., 2012). Mapped gradients in Ω are approximations of gradients in sediment transport (given that transport thresholds are exceeded), subject to the smoothing scale and artifacts of the resolution, accuracy, and processing of DEMs.

Near-channel erosion and deposition were measured along a total of 52 river kilometers, examined in pre- and post-satellite imagery (Fig. DR2) and verified by field measurements. We measured fresh flood deposits and erosion scars on banks and hillslopes using a real time kinematic GPS, meter tape, and metal probe while walking the channels and floodplains. Erosion was considered an input to the channel. Lateral inputs, termed collectively mass wasting, consisted primarily of landslides extending up hillslopes and bank failures to the tops of channels, both triggered by stream undercutting, and to a lesser extent gully erosion and debris flows. Floodplain and near-channel deposits were considered outputs from the channel. Measurements of channel bed incision and deposition were feasible based on field evidence in the smaller Sanitas watershed, but not practical on the other larger rivers given the lack of high-resolution pre-flood long profiles. Erosion and deposition reflect primarily lateral inputs and outputs along the three larger waterways, and reflect lateral plus vertical inputs and outputs along the Mount Sanitas channel.

RESULTS

Broad regions of increases and decreases in Ω are evident in each river (Fig. 2). Decreases in Ω occur where downstream decreases in slope offset the normal downstream increases in flow accumulation area. Increases in Ω occur where increases in flow accumulation area are coupled with increases in slope or exceed decreases in slope. Field observations and geologic maps (Colton, 1978; Ratcliffe et al., 2011) suggest that slope variations are partly controlled by varying resistance of underlying bedrock and relict glacial features (consistent with Hack, 1957, 1973); however, some changes in geologic boundary conditions are too subtle to be evident at the resolution of geologic maps.

Observations of near-channel erosion and deposition support the prediction that river channels are vulnerable to erosion where dΩ/dx > 0 and likely to be loci of deposition where dΩ/dx < 0. Reaches generally exhibit either mass wasting or floodplain deposition, but not both in the same reaches. Furthermore, zones of abundant sediment inputs to channels, with mass wasting up to and exceeding 10,000 m3/km, align approximately with regions where dΩ/dx > 0, and zones of abundant sediment outputs from channels, with floodplain deposition as much as 26,000 m3 km–1, align roughly with regions where dΩ/dx < 0. Some regions do not match our predictions, for example, ∼7 and ∼8.2 km in Figure 2D, where erosion occurs in a zone of dΩ/dx > 0. Overall the model succeeds in 34 of 39 reaches, with predominant erosion where dΩ/dx > 0, predominant deposition where dΩ/dx < 0, or no effect.

Locations of equal Ω show different geomorphic responses depending on the downstream gradient in Ω. To isolate the effect of Ω gradients from at-a-point magnitudes, we highlight the 6 locations on the Saxtons River where Ω is roughly equivalent, ∼30,000 W/m, but increases at some locations and decreases at others (Fig. 2A). Deposition is observed in each of the three locations where deposition is predicted by dΩ/dx < 0 (∼18, ∼23, and ∼26 km). In contrast, where dΩ/dx > 0, abundant mass wasting is observed in two of three locations (∼15 and ∼24 km) and little to no deposition or erosion are observed at ∼20 km.

The data are consistent with the stipulation that at-a-point thresholds for sediment transport must be exceeded within the watershed for gradients in Ω to favor erosion or deposition. We observed a general lack of abundant near-channel erosion or deposition in the first 4 km of the Saxtons and 5 km of the West Branch, where Ω was consistently low. Where higher and also variable Ω prevailed in downstream reaches, Ω gradients, rather than at-a-point Ω magnitudes, better predicted if a reach of high activity was prone to erosion or deposition.

DISCUSSION

This study augments long-established theory on downstream spatial gradients in sediment transport by demonstrating the influence of downstream gradients on hillslope-channel coupling, both mathematically and with field evidence, and by providing an initial, quantitative means to predict areas prone to be lateral sources or sinks of material. Improved predictions might be obtained from more detailed analysis of local sediment transport dynamics, for example, bed resistance to erosion and the hydraulics of flow expansion and contraction (Miller, 1995). Our analysis shows success with a relatively generalized but efficient DEM-based derivation of Ω.

We stress that Equation 2 incorporates longitudinal changes in total sediment transport; thus we examine Ω rather than unit stream power, ω, where ω ≡ Ω/w and w is channel width. A growing body of work shows the importance of ω in single events and at geologic time scales. Without examining gradients in forces, studies have shown that at-a-point magnitudes of ω predict locations of extreme geomorphic changes in floods (Magilligan, 1992; Buraas et al., 2014), stream bed grain size (Snyder et al., 2013), abundance of landslides (Larsen and Montgomery, 2012), and bedrock incision rates (Dietrich et al., 2003; Ouimet et al., 2009). Our analysis builds on this work by showing that additional information can be extracted from the spatial gradients in Ω. Furthermore, our analysis, combined with other work related to the Exner model, raises the question of whether geomorphic effects attributed to at-a-point Ω magnitudes in some studies are also modulated by local changes in Ω.

This study has further implications for river management and ecology. Because nutrients and contaminants often sorb to sediment (Klotz, 1985; Brigham et al., 2009; Landis et al., 2012), downstream gradients in Qsx may predict the sources and sinks of nutrients and contaminants in riparian ecosystems in small and large floods. Following the Vermont and Colorado events, considerable effort was directed toward removing floodplain deposits and stabilizing banks. Predictions of areas prone to erosion or deposition can guide decisions to protect structures or simply allow natural riverine processes where possible.

CONCLUSIONS

Geologic constraints can impose downstream variations in Ω that create reaches where sediment transport predictably increases or decreases with respect to distance downstream. When thresholds for sediment mobility are exceeded, reaches of downstream increases in sediment transport favor erosion of the bed and banks, and reaches of downstream decreases in sediment transport are prone to bed and floodplain deposition. In two large flood events, remotely mapped gradients in Ω predicted from DEMs are spatially associated with field-surveyed stream sediment sources, in the form of bank erosion and landslides, and sinks, in the form of floodplain deposition. Overall, the effect of downstream gradients in sediment transport may shed light on the lateral flux of material in basic geomorphology studies as well as applied issues in river protection and restoration. The physical controls on lateral fluxes of material between channels and adjacent areas are critical concerns, especially as the potential for floods may be increasing with climate change and urbanization.

James Pizzuto, Evan Dethier, and Helen Doyle reviewed earlier drafts. We also thank by Andrew Miller, Fritz Schlunegger, and an anonymous reviewer for thoughtful reviews. The National Science Foundation (NSF, grants BCS-1103172, BCS-1160301, and BCS-1222531) and the Geological Society of America (Graduate Student Research Grant supported by NSF 1354519) supported this research.

1GSA Data Repository item 2015331, discussion of the sites, flood events, modified Exner equation, total stream power, channel width, and field measurements with photographs, is available online at www.geosociety.org/pubs/ft2015.htm, or on request from editing@geosociety.org or Documents Secretary, GSA, P.O. Box 9140, Boulder, CO 80301, USA.