Bed and width oscillations form coherent patterns in a partially confined, regulated gravel-cobble-bedded river adjusting to anthropogenic disturbances

Abstract

Understanding the spatial organization of river systems in light of natural and anthropogenic change is extremely important because it can provide information to assess, manage, and restore them to ameliorate worldwide freshwater fauna declines. For gravel- and cobble-bedded alluvial rivers studies spanning analytical, empirical and numerical domains suggest that at channel-forming flows there is a tendency towards covarying bankfull bed and width undulations amongst morphologic units such as pools and riffles, whereby relatively wide areas have relatively higher minimum bed elevations and relatively narrow areas have relatively lower minimum bed elevations. The goal of this study was to determine whether minimum bed elevation and flow-dependent channel top width are organized in a partially confined, incising gravel-cobbled bed river with multiple spatial scales of anthropogenic and natural landform heterogeneity across a range of discharges. A key result is that the test river exhibited covarying oscillations of minimum bed elevation and channel top width across all flows analyzed. These covarying oscillations were found to be quasiperiodic at channel-forming flows, scaling with the length scales of bars, pools and riffles. Thus, it appears that alluvial rivers organize their topography to have quasiperiodic, shallow and wide or narrow and deep cross section geometry, even despite ongoing, centennial-scale incision. Presumably these covarying oscillations are linked to hydrogeomorphic mechanisms associated with alluvial river channel maintenance. The biggest conclusion from this study is that alluvial rivers are defined more so by variability in topography and flow than mean conditions. Broader impacts of this study are that the methods provide a framework for characterizing longitudinal and flow-dependent variability in rivers for assessing geomorphic structure and aquatic habitat in space, and if repeated, through time.

Earth Surf. Dynam., 5, 120, 2017 www.earth-surf-dynam.net/5/1/2017/ doi:10.5194/esurf-5-1-2017 Author(s) 2017. CC Attribution 3.0 License.

Rocko A. Brown1,2 and Gregory B. Pasternack1

1University of California, Davis, 1 Shields Avenue, Davis, CA, USA

2Environmental Science Associates, 2600 Capitol Avenue, Suite 200, Sacramento, CA, USA

Correspondence to: Rocko A. Brown ([email protected])

Received: 5 December 2015 Published in Earth Surf. Dynam. Discuss.: 5 February 2016 Revised: 15 November 2016 Accepted: 6 December 2016 Published: 11 January 2017

Abstract. Understanding the spatial organization of river systems in light of natural and anthropogenic change is extremely important because it can provide information to assess, manage, and restore them to ameliorate worldwide freshwater fauna declines. For gravel- and cobble-bedded alluvial rivers studies spanning analytical, empirical and numerical domains suggest that at channel-forming ows there is a tendency towards covarying bankfull bed and width undulations amongst morphologic units such as pools and rifes, whereby relatively wide areas have relatively higher minimum bed elevations and relatively narrow areas have relatively lower minimum bed elevations. The goal of this study was to determine whether minimum bed elevation and ow-dependent channel top width are organized in a partially conned, incising gravelcobbled bed river with multiple spatial scales of anthropogenic and natural landform heterogeneity across a range of discharges. A key result is that the test river exhibited covarying oscillations of minimum bed elevation and channel top width across all ows analyzed. These covarying oscillations were found to be quasiperiodic at channel-forming ows, scaling with the length scales of bars, pools and rifes. Thus, it appears that alluvial rivers organize their topography to have quasiperiodic, shallow and wide or narrow and deep cross section geometry, even despite ongoing, centennial-scale incision. Presumably these covarying oscillations are linked to hydrogeomorphic mechanisms associated with alluvial river channel maintenance. The biggest conclusion from this study is that alluvial rivers are dened more so by variability in topography and ow than mean conditions. Broader impacts of this study are that the methods provide a framework for characterizing longitudinal and ow-dependent variability in rivers for assessing geomorphic structure and aquatic habitat in space, and if repeated, through time.

1 Introduction

Understanding the spatial organization of river systems in light of natural and anthropogenic change is extremely important because it can provide information to assess, manage and restore them to ameliorate worldwide freshwater fauna declines (Frissell et al., 1986; Richter et al., 1997).Alluvial rivers found in transitional uplandlowland environments with slopes < 0.02 and median diameter bed sediments ranging from 8 to 256 mm can exhibit scale-dependent organization of their bed sediments (Milne, 1982), bed elevation prole (Madej, 2001), cross section geometry (Rayberg

Bed and width oscillations form coherent patterns in a partially conned, regulated gravelcobble-bedded river adjusting to anthropogenic disturbances

and Neave, 2008), and morphological units (Keller and Mel-horn, 1978; Thomson et al., 2001). For these rivers a plethora of studies spanning analytical, empirical, and numerical domains suggest that at channel-forming ows, there is a tendency towards covarying bankfull bed and width undulations amongst morphologic units such as pools and rifes (Brown et al., 2016). That is, relatively wide areas have higher relative bed elevations and relatively narrow areas have lower relative bed elevations. While covarying bed and width undulations have been evaluated in eld studies using cross section data (Richards, 1976a, b), in models of sediment transport and water ow (Repetto and Tubino, 2001), ume studies

Published by Copernicus Publications on behalf of the European Geosciences Union.

2 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

(Nelson et al., 2015), and in theoretical treatments (Huang et al., 2004), this idea has never been evaluated in a morphologically dynamic river corridor for which a meter-scale digital elevation model is available across a wide range of discharges, from a fraction of to orders of magnitude more than bankfull. The goal of this study was to understand if and how bed elevation and ow-dependent channel width are organized in a partially conned, incising, regulated gravel cobble bed river with multiple spatial scales of landform heterogeneity across a range of discharges. The analysis of geometric organization was accomplished through a suite of spatial series analyses using a 9 km reach of the lower Yuba River (LYR) in California, USA, as a test bed. Our central hypothesis is that the test river reach will have covarying and quasiperiodic bed and width oscillations. Due to the test river corridors variability (White et al., 2010), past history (James et al., 2009), and a Mediterranean climate (Wolman and Gerson, 1978), these patterns may be dominant in a range of ows. Knowledge of spatial patterns is commonly used to infer the geomorphic processes that yielded those patterns (Davis, 1909; Thornbury, 1954) and/or what future processes will be driven by the current spatial structure of landforms (Leopold and Maddock, 1953; Schumm, 1971; Brown and Pasternack, 2014). However, such inferences rarely include transparent, objective spatial analysis of topographic structure, so this study demonstrates a new methodology accessible to most practitioners to substantiate the ideas behind the processmorphology linkages they envision to be driven by variability in topography. The results of the study contribute to basic knowledge by showing multiple layers of coherent structure between width and bed undulations, which alerts geomorphologists to the need to prioritize future research on the cause and consequences of structured channel variability as opposed to further work on the central tendency of morphological metrics.

1.1 Background

A multitude of numerical, eld, and theoretical studies have shown that gravel bed rivers have covarying oscillations between bed elevation and channel width related to rifepool maintenance processes. The joint periodicity in oscillating thalweg and bankfull width series for poolrife sequences in gravel bed rivers was identied by Richards (1976b), who noted that rifes have widths that are on average greater than those of pools, and he attributed this to ow deection over rifes into the channel banks. Since then, many studies related to processes that rejuvenate or maintain the relief between bars and pools (i.e., maintenance or self-maintenance) have implied a specic spatial correlation of width and depth between the pool and rife at the bankfull or channel-forming discharge (e.g., Wilkinson et al., 2004;MacWilliams et al., 2006; Caamano et al., 2009; Thompson, 2010). For example, Caamano et al. (2009) derived a criterion for the occurrence of a mean reversal in velocity

(Keller, 1971) that implies a specic correlation of the channel geometry of alluvial channels with undulating bed proles. Specically, for a reversal in mean velocity at the bank-full or channel-forming discharge (holding substrate composition constant), the rife must be wider than the pool and the width variation should be greater than the depth variation between the rife and residual pool depth. Milan et al. (2001) evaluated several rifepool couplets, from a base ow to just over the bankfull discharge. They found that convergence and reversals in section-averaged velocity and shear stress were complex and nonuniform, which suggests that different morphologic units may be maintained at different discharges.Wilkinson et al. (2004) explicitly showed that phase shifts in shear stress from the rife to the pool between high and low discharge required positively covarying bed and width undulations. White et al. (2010) showed how valley width oscillations inuence rife persistence despite larger channel-altering oods and interdecadal valley incision. Sawyer et al. (2010) used two-dimensional (2-D) hydrodynamic modeling and digital elevation model (DEM) differencing to illustrate how variations in wetted width and bed elevation can modulate regions of peak velocity and channel change at a poolriferun sequence across a range of discharges from 0.15 to 7.6 times bankfull discharge. DeAlmeida and Rodriguez (2012) used a 1-D morphodynamic model to explore the evolution of rifepool bedforms from an initially at bed while maintaining the channel width variability. The resulting simulations were in close agreement to the actual bed prole in their model. Thus, their study is another example that channel width can exert controls on the structure of the bed prole. The ows at which the above processes are modulated vary in the literature.

From a system perspective, bed and width undulations, both jointly and in isolation, are a means of self-adjustment in alluvial channels that minimize the time rate of potential energy expenditure per unit mass of water in accordance with the law of least time rate of energy expenditure (Langbein and Leopold, 1962; Yang, 1971; Cherkauer, 1973; Wohl et al., 1999). For bed proles, Yang (1971) and Cherkauer (1973) showed that undulating bed relief is a preferred conguration of alluvial channels that minimize the time rate of potential energy expenditure. Using eld, ume, and numerical methods, Wohl et al. (1999) showed that valley wall oscillations also act to regulate ow energy analogous to bedforms. In analyzing reach-scale energy constraints on river behavior, Huang et al. (2004) quantitatively showed that wide and shallow sections and deep and narrow sections are two end member cross-sectional congurations necessary for efciently expending excess energy for rivers, so these two types of cross sections imply covarying bed and width undulations as a means of expending excess energy.Therefore, the above studies suggest that both bed and width oscillations are a means to optimize channel geometry for the dissipation of excess ow energy. The question now is the extent to which this well-developed theory plays out in real

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R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 3

rivers, especially now that meter-scale river digital elevation models (DEMs) are available.

Flows that drive channel maintenance in western US rivers, such as the test river in this study (described in detail in Sect. 3 below), are thought to typically have recurrence intervals ranging from 1.2 to 5 years (Williams, 1978;Andrews, 1980; Nolan et al., 1987). Most of the literature investigating rifepool maintenance discussed above report bedform-sustaining ow reversals occurring at or near bank-full, often with no specicity to the frequency of these events (Lisle, 1979; Wilkinson et al., 2004). Studies that do report recurrence intervals have ranged from the 1.2- to 7.7-year recurrence ows (Keller, 1971; Sawyer et al., 2010). However, many rivers exhibit multiple scales of freely formed and forced landscape heterogeneity that should inuence u-vial geomorphology when the ow interacts with them, no matter the magnitude (Church, 2006; Gangodagamage et al., 2007). For example, Strom and Pasternack (2016) showed that the geomorphic setting can inuence the stage at which reversals in peak velocity occur. In their study an unconned anastomizing reach experienced velocity reversals at ows ranging from 1.5- to 2.5-year recurrence ows, compared to2.5- to 4.7-year recurrence ows for a valley-conned reach.

Given that river geometry can record memory from past oods (Yu and Wolman, 1987) and the presence of multiple layers of topographic variability (Brown and Pasternack, 2014), it is hypothesized that covarying bed and width undulations could also be present at discharges other than bank-full.

1.2 Study objectives

The primary objectives of this study were to determine if there are covarying bed and width oscillations in the test reach, if they exhibit any periodicity, and how they vary with discharge. Based on the literature review above, we hypothesize there will be covarying bed and width oscillations that form quasiperiodic patterns, with the strongest relationship occurring for a broad range of channel-forming ows. A secondary objective is to demonstrate how a geomorphic covariance structure (GCS) analysis of minimum bed elevation and wetted width, as dened below, can be generated from high-resolution topography and hydraulic models to assess ow-dependent spatial organization of river corridor topography.The study site was a 6.4 km section of the lower Yuba River (LYR), an incising and partially conned self-formed gravel cobble-bedded river (Fig. 1; described in Sect. 3). Several statistical tests were used on the serial correlation of minimum bed elevation, Z, channel top width, Wj , and their geomorphic covariance structure, C(ZWj ), where j indexes the ow discharge. The novelty of this study is that it provides the rst assessment of covarying bed and width oscillations in a partially conned, self-maintained alluvial river across a wide array of ows. The broader impact is that it provides a framework for analyzing the ow-dependent topographic

Table 1. Flows analyzed and their approximate recurrence intervals.

Q (m3 s1) Approximate recurrence interval

8.50 128.32 1.03 141.6 1.2 283.2 1.5 597.5 2.5 1195 4.7 2390 12.7 3126 20

variability of river corridors, without differentiating between discrete landforms such as rifes and pools. Further, an understanding of the ow-dependent spatial structure of bed and width GCS would be useful in assessing their utility in applied river corridor analysis and synthesis for river engineering, management, and restoration.

2 Experimental design

To evaluate covarying bed and width undulations, the concepts and methods of geomorphic covariance structures were used (Brown, 2014; Brown and Pasternack, 2014). A GCS is a bivariate spatial relationship amongst or between variables along a pathway in a river corridor. It is not a single metric as in statistical covariance, but a spatial series, and hence can capture spatially explicit geomorphic structure. Variables assessed can be ow-independent measures of topography (e.g., bed elevation, centerline curvature, and cross section asymmetry) and sediment size as well as ow-dependent hydraulics (e.g., top width, depth, velocity, and shear stress; Brown, 2014), topographic change, and biotic variables (e.g., biomass and habitat utilization). Calculation of a GCS from paired spatial series is straightforward by the product xstd,i ystd,i, where the subscript std refers to

standardized and possibly detrended values of two variables x and y at location i along the centerline, creating the serial data set C(XY ). Since this study is concerned with bedand ow-dependent top width undulations, the GCS series at each ow j is denoted as C(ZWj ). For simplication in this paper local values use similar notation and are differentiated by sentence context. More information on GCS theory is provided in Sect. 4.2 below. GCS series were generated for eight ows ranging from 8.50 to 3126 m3 s1, spanning a broad range of ow frequencies (Table 1). The range of selected ows spans a low-ow condition up to the ow of the last large ood in the river. These ows were selected to provide enough resolution to glean ow-dependent effects while not producing redundant results.

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4 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

(a)

(b)

River

Virginia

Ranch

Reservoir

Englebright

Lake

South Yuba

!

km

0 5 10 15 20

Timbuctoo Bend

Deer Creek

flow (m s )

3

1

8.5028.32 141.6 283.2 597.5 1195 2390 3126

Flow

0 330 660 m

Figure 1. Regional and vicinity map of the lower Yuba River (a) and extent of study segment showing inundation extents predicted by the 2-D model (b).

The rst question this study sought to answer was whether there was a tendency towards covarying Z and Wj and how it changed with discharge. If Z and Wj covary, then the sign of the residuals of both variables will be positive or negative, yielding a positive C ZWj

[parenrightbig]

> 0. Therefore, to determine if there are covarying bed and width oscillations, a histogram was generated for each ow-dependent series of C(ZWj ).

The second question was whether each ow-dependent series of C(ZWj ) was random, constant, periodic, or quasiperiodic. Quasiperiodicity in this setting is dened as a series with periodic and random components, as opposed to purely random or purely periodic (Richards, 1976a). Quasiperiodicity differs from periodic series in that there are elements of randomness blended in (Newland, 1993). To answer this question autocorrelation function (ACF) and power spectral density (PSD) analyses of each C(ZWj ) series were used to determine if there were statistically signicant quasiperiodic length scales (sensu Carling and Orr, 2002) at which C(ZWj ) covary and how that changes with discharge.

Based on the studies listed above (Sect. 1.1), we hypothesize that gravelcobble-bedded rivers capable of rejuvenating their rifepool relief should exhibit a topography (at any instant in time) with a tendency towards quasiperiodic and covarying bed and width oscillations. The basis for covary-

ing and quasiperiodic bed and width oscillations is founded on the idea that, on average, channel geometry is maintained during bankfull (e.g., geometric bankfull) discharge and that locally channels are shaped by rifepool maintenance mechanisms (Wilkinson et al., 2004; MacWilliams et al., 2006; Caamano et al., 2009; Thompson, 2010). Based on the literature reviewed in Sect. 1.1, we hypothesize that the C(ZWj ) GCS will, on average, become more positive with increasing ow until approximately the bankfull discharge, where the channel overtops its banks and non-alluvial oodplain features exert control on cross-sectional mean hydraulics. At that point there may not be a tendency towards positive or negative residuals if the topographic controls at that ood stage are not important enough to control channel morphology. For example, smaller events might occur frequently enough to erase the in-channel effects of the large infrequent events, especially in a temperate climate (Wolman and Gerson, 1978). On the other hand, if a system is dominated by the legacy of a massive historical ood and lacks the capability to recover under more frequent oods, then the C(ZWj ) GCS will continue to increase positively until the discharge that carved out the existent covarying bed and width oscillations for the current topography is revealed.Note that we do not expect a clear threshold where organiza-

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R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 5

tion in the C(ZWj ) GCS is a maximum but rather a range of ows near the bankfull discharge. The effect of a particular ow on a channel is dependent not just on that ow but on the history of ow conditions that led to the channels condition (Yu and Wolman, 1987). Therefore, it should not be expected that the observed patterns will be associated with a singular ow value. Also, this study looked at a river in a Mediterranean climate, and thus it may be more prone to exhibiting a wider range of positive C(ZWj ) GCS than a temperate or tropical river, as the number and frequency of recovery processes is reduced (Wolman and Gerson, 1978).With this logic, it is hypothesized that the C(ZWj ) GCS will be quasiperiodic for ows near the bankfull discharge, due to the presence of bar and pool topography, and that the ACF and PSD will yield length scales commensurate with the average spacing of these topographic features. For ows above the bankfull discharge, a river corridor has many local allu-vial landforms, bedrock outcrops, and articial structures on its oodplain and terraces. These features inuence bed adjustment during oods that engage them and hence impact the GCS. It is unknown how GCS length scales will change in response to the topographic steering these features induce, causing changes to bed elevation, but investigating that is a novel and important aspect of this study. In addition to performing these tests, we also present two 1.4 km sections

of the C(ZWj ) GCS, Z, W, and the detrended topography for three representative ows to discuss specic examples of how these patterns change with landforms in the river corridor across a wide array of discharges.

Limitations to this study (but not the GCS approach) for worldwide generalization include not considering other variables relevant to how alluvial rivers adjust their shape, such as grain size, channel curvature, and vegetation, to name a few. Some of these limitations were not study oversights but reected the reality that the study reach used had relatively homogenous sediments (Jackson et al., 2013), low sinuosity, and limited vegetation (Abu-Aly et al., 2014). This yielded an ideal setting to determine how much order was present for just bed elevation and channel width but does not disregard the importance of these other controls, which can be addressed in future studies at suitable sites. Also, this study is not a direct test of the response to or drivers of morpho-dynamic change. The extent to which GCS can be used as an indicator of change to simplify geomorphic analysis instead of doing morphodynamic modeling remains unknown, but nding metrics that link landforms, the agent that shape them, and the responses they induce has always been the goal of geomorphology (Davis, 1909).

3 Study area

3.1 River context

The study area was the 6.4 km Timbuctoo Bend reach of the lower Yuba River (LYR) in northeastern California, USA.

The reach begins at the outlet of a bedrock canyon that is dammed 3 km upstream, and the watershed above the dam

drains 3480 km2 of dry summer subtropical mountains. Little is known about the pre-European Yuba River, but in this reach it is conned by valley hillsides and bedrock outcrops, and these are evident in some photos from early European settlers panning the river for gold in the late 1840s. During the mid to late 19th century there was a period of extensive hydraulic gold mining of hillside alluvial deposits in the upper Yuba watershed that delivered an overwhelming load of heterogeneous sediment to the lowland river valley (James et al., 2009). Geomorphologist G. K. Gilbert photo-documented the LYR around the time of its worst condition in the early 20th century and provided foundational thinking related to how the river would evolve in time (Gilbert, 1917). In 1941 Englebright Dam was built to hold back further sediment export from the mountains, and that allowed the river valley to begin a process of natural recovery, which was reviewed by Adler (1980) and more recently by Ghoshal et al. (2010). However, this process was interfered with by widespread dredger mining in the early to mid 20th century. In two locations of the study reach there are wide relict dredger tailings piles on the inside of the two uppermost meander bends that the river has been gradually eroding.

The hydrology of the regulated LYR is complex and quite different from the usual story of signicantly curtailed ows below a large dam. Englebright Dam primarily serves as a sediment barrier, and it is kept nearly full. As a result, it is operated to overtop when outow is > 127.4 m3 s1, long enough to ll its small remaining capacity, so ood hydrology is still seasonal and driven by rainfall and snowmelt in the watershed. Two of three sub-catchments do not have large dams, so winter oods and spring snowmelt commonly cause spill over Englebright sufcient to exceed the bankfull channel in Timbuctoo Bend. The one regulated sub-catchment does have a large dam, New Bullards Bar (closed in 1970), and this reduces the frequency and duration of oodplain inundation compared to the pre-dam record (Escobar-Arias and Pasternack, 2011; Cienciala and Pasternack, 2016), but it is not like other rivers where the entire upstream watershed is regulated. Sawyer et al. (2010) reported the 1.5-year recurrence interval for the post-Englebright, pre-New Bullards Bar period as 328.5 m3 s1 and then for post-New Bullards

Bar as 159.2 m3 s1. California has long been known to exhibit a roughly decadal return period for societally important major oods that change river courses (Guinn, 1890), though the magnitude of those oods is not necessarily a 10-year recurrence interval scientically. Since major ow regulation in 1970, the three largest peak annual daily oods came roughly 10 years apart, in the 1986, 1997, and 2006 water years. The ood of 1997 was the largest of the post-dam record. The 2006 peak ood event had a recorded peak 15 min discharge of 3126.2 m3 s1 entering the study reach.

Wyrick and Pasternack (2012) analyzed LYR inundation patterns in a high-resolution DEM of the river produced af-

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6 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

ter the 2006 wet season, and they considered how channel and oodplain shapes change dramatically through the study reach. Their ndings apply to the Timbuctoo Bend reach. Different locations exhibited spillage out of the channel into low-lying peripheral swales and onto lateral and point bars at ows from 84.95141.6 m3 s1. When the

water stage rises to 141.6 m3 s1, relatively at active bar tops become inundated and the wetted extents line up with the base of willows along steeper banks anking the channel. These and other eld indicators led to the consideration of 141.6 m3 s1 as representative of the bankfull discharge adjusted to the modern regulated ow regime since 1970. By a ow of 198.2 m3 s1, banks are all submerged and water is spilling out to various degrees onto the oodplain. The oodplain is considered fully inundated when the discharge reaches 597.5 m3 s1. Above that ow stage exist some terraces, bedrock outcrops, and soil-mantled hillsides that become inundated. Regarding the two relict dredger tailings piles mentioned earlier, they interact with the ows ranging from 597.51,195 m3 s1. Apart from these piles, the ow width interacts predominately with the valley walls for discharges at 1195 m3 s1 and above. Given the estimate of bankfull discharge for the LYR, the instantaneous peak ow during the 2006 ood was 23 times that, so quite substan

tial compared to those commonly investigated in modern geomorphic studies.

3.2 Timbuctoo Bend details

A lot is known about the geomorphology of Timbuctoo Bend, and this information helps inform this study to substantiate the possibility that the rivers topography is organized in response to differential topographic steering as a function of ow stage. According to Wyrick and Pasternack (2012), the reach has a mean bed slope of 0.002, a thalweg length of 6337 m, a mean bankfull width of 84 m, a mean oodway width of 134 m, an entrenchment ratio of 2.1 (dened per Rosgen, 1996), and a weighted mean substrate size of 164 mm. Using the system of Rosgen (1996), it classies as a B3c stream, indicating moderate entrenchment and bed slope with cobble channel material. A study of morphological units revealed that its base ow channel area consists of 20 % pool, 18 % rife, and then a mix of six other landform types. More than half of the area of the riverbank ecotone inundated between base ow and bankfull ow is composed of lateral bars, with the remaining area containing roughly similar areas of point bars, medial bars, and swales (Wyrick and Pasternack, 2012). A study of bankfull channel substrates found that they are differentiated by morphological unit type, but the median size of all units is in the cobble range (Jackson et al., 2013), even depositional bars that are often thought of as relatively ne in other contexts. Vegetated cover of the river corridor ranged from 0.8 to 8.1 % of the total wetted area at each ow, with more inundated vegetation at higher ows.

White et al. (2010) used a sequence of historical aerial photos, wetted channel polygons, repeat long proles from 1999 and 2006, and a valley width series to conclude that even though Timbuctoo Bend has incised signicantly since 1942 in response to many oods, there are several rifes and pools that persist in the same wide and constricted valley locations, suggesting that valley width oscillations maintain those positions and drive morphodynamic response. This suggests that it may not matter exactly which instant topography one might analyze to look at the effect of topographic variability in controlling or responding to large ood processes, as they all should reect the same topographic steering regime induced by the valley walls.

Two studies have been done to look at the hydraulic processes associated with different ood stages in Timbuctoo Bend. Sawyer et al. (2010) found that one of the poolrife run units in this reach experienced ow convergence routing between baseow, bankfull ow, and a ow of roughly 8 times bankfull discharge that maintained rife relief. Strom et al. (2016) assessed the hydraulics of the whole reach over the same range of ows in this study, and they reported that the reach exhibits a diversity of stage-dependent shifts in the locations and sizes of patches of peak velocity. The spatial persistence of such patches decreased with discharge until ows exceeded 1000 m3 s1, at which point valley

walls sustained their location for ows up to the peak of 3126 m3 s1. Also, peak velocity patches resided preferentially over chute and rife landforms at within-bank ows, several morphological unit types landforms for small oods, and pools for oods > 1000 m3 s1. These studies corroborate the process inferences made by White et al. (2010) in that hydraulics were found to be stage-dependent in ways that were consistent with the mechanism of ow convergence routing.

Finally, Carley et al. (2012), Wyrick and Pasternack (2015), and Pasternack and Wyrick (2016) used DEM differencing, uncertainty analysis, scale-stratied sediment budgeting, and topographic change classication to analyze how the LYR changed from 1999 to 2008, including Timbuctoo Bend. These studies took advantage of the repeated mapping of the LYR in 1999 and 20062008, with Timbuctoo Bend mapped entirely in 2006. They found large amounts of erosion and deposition, strong differential rates of change among different landforms on three spatial scales, and topographic changes driven by 19 different geomorphic processes. For Timbuctoo Bend, the dominant topographic change processes found were in-channel downcutting (including knickpoint migration) and overbank (i.e., oodplain) scour, with noncohesive bank migration a distant third. Thus, the river appears to change through adjustments to its bed elevation far more than changes to its width in this reach. This nding will come into play in interpreting the results of this study later on.

In summary, even with modern technology it is impossible to monitor the hydrogeomorphic mechanics of uvial change

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dependent sample pathways using 2-D model hydraulic outputs of depth, velocity, and wetted area. The effect of having different sample pathways for each ow is that it accounts for ow steering by topographic features in the river corridor.

For each ow a grid of kinetic ow energy (di v2i) was generated in ARCGIS, where di is the depth and vi is the velocity at node i in the 2-D model hydraulics rasters. Then a sample pathway was manually digitized using the momentum grid, following the path of greatest kinetic energy. For ow splits around islands, if the magnitude of energy in one channel was more than twice as great as the other, it was chosen as the main pathway. If they were approximately equal, then the pathway was centered between the split. Once a sample pathway was developed, it was then smoothed using a Bezier curve approach over a range of 100 m or approximately a bankfull channel width to help further minimize section overlaps. For each sample pathway, cross sections were generated at 5 m intervals and clipped to the wetted extent of each ow, with any partially disconnected backwater or non-downstream-oriented areas manually removed.

Despite smoothing there were areas of the river where the river has relatively high curvature in the sample pathway, causing sample section overlaps to occur. These were manually edited by visually comparing the sample sections with the kinetic ow energy grid and removing overlapped sections that did not follow the downstream ow of water. This was more prevalent at the lower discharges than the higher ones due to the effects topographic steering creating more variable sample pathways.

To provide a constant frame of spatial reference for comparison of results between ows, while preserving ow-dependent widths, sections were mapped to the lowest ows sample pathway using the spatial join function in ARCGIS.

The lowest ow was used because that had the longest path.

This ensures no multiple-to-one averaging of data would happen, as that would otherwise occur if data were mapped from longer paths to shorter ones. To create evenly spaced spatial series the data were linearly interpolated to match the original sampling frequency of 5 m. Minimum bed elevation along each section was sampled from the DEM using the same sections for measuring width for the lowest-ow sample pathway.

4.2 Developing geomorphic covariance structures

To generate GCS series for bed- and ow-dependent width undulations, the two variables, Z and Wj , were rst detrended and standardized. Detrending is not always needed for width in GCS analysis, but some analyses in this study did require it. A linear model was used for Z, (Table 2) as is common in many studies that analyze reach-scale bed variations (Melton, 1962; Richards, 1976a; McKean et al., 2008). Similarly, each Wj series was linearly detrended, but the trends were extremely small, with a consistent slope of just 0.002 (Table 2). Finally, each series was standardized by

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R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 7

in a large river for ows up to 22 times bankfull discharge, so recent studies have tried to get at the mechanisms during such events with a range of strategies. Historical river analysis, hydrodynamic modeling, and topographic change detection and analysis have been used together to reveal a picture of a river that is changing in response to multiple scales of landform heterogeneity that drive topographic steering. Even though the river has changed through time, there has been a persistence of nested landforms, and thus it would be useful to understand how topographic features are organized purely through an analysis of the DEM per the methods developed in this study. This study exclusively uses the 2006 map made during the dry season that followed the dramatic 2006 wet season, which included the large ood, two other notable peaks, and a total of 18 days of oodplain lling ow. Thus, it addresses the topography as it existed after that river-altering wet season and how it will in turn inuence the dynamics of the next one.

4 Methods

The meter-scale topographic map of Timbuctoo Bend produced from echosounder and robotic total station ground surveys was used for extraction of Z (Carley et al., 2012; see Supplement), while a corresponding meter-scale 2-D hydrodynamic model was used to generate data sets for Wj for each discharge. Details about the 2-D model are documented in the Supplement and previous publications (Abu-Aly et al., 2013; Wyrick and Pasternack, 2014; Pasternack et al., 2014); it was thoroughly validated for velocity vector and water surface elevation metrics, yielding outcomes on par with or better than other publications using 2-D models.

4.1 Data extraction

A rst step was to extract Z and Wj spatial series from the digital elevation model and 2-D model outputs. This required having a sample pathway along which bed elevation could be extracted from the DEM and top width from the wetted extents from the 2-D model. Sampling river widths was done using cross sections generated at even intervals perpendicular to the sample pathway and then clipped to the 2-D model-derived wetted extent for each ow. Because of this, the pathway selected can have a signicant bearing on whether or not sample sections represent downstream-oriented ow or overlap where pathway curvature is high. There are several options in developing an appropriate pathway for sampling the river corridor. The thalweg is commonly used in ow-independent geomorphic studies, but the thalweg is too tortuous within the channel to adhere to a reasonable denition of top width. Further, as ow increases, central ow pathway deviates from the deepest part of the channel due to higher ow momentum and topographic steering from submerged and partially submerged topography (Abu-Aly et al., 2014). Therefore, in this study we manually developed ow-

8 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

75

600

(a)

(b)

500

70

Elevation (m)

400

65

Width (m)

300

200

60

100

55 0 2000 4000 6000

0 0 2000 4000 6000

Station (m)

Station (m)

6

GCS/ residual magnitude

(c)

C(W,Z) Z W

4

2

0

-2

-4

-6 0 1000 2000 3000 4000 5000 6000

Station (m)

. We did not assess the statistical signicance of coherent landform patterns, but one could do so following Brown and Pasternack (2014).

4.3 Data analysis

Before any statistical tests were performed, we rst visually assessed the data in two approximately 1.4 km long sections to illustrate how C(ZWj ) is affected by ow responses to landforms. For these two examples only three discharges were selected to illustrate ow-dependent changes in Z, Wj ,

and C(ZWj ) with uvial landforms. The lowest and highest ows, i.e., 8.50 and 3126 m3 s1, were selected to bracket the range of ows investigated. The intermediate ow selected was 283.2 m3 s1 based on the shifts in C(ZWj ) observed in the histogram, ACF, and PSD tests as shown below in the results. For these examples the exact magnitudes of C(ZWj )

are not as important as the patterns and how they relate to visually discernible landforms.

A MannWhitney U test was performed between each C(ZWj ) data set to determine if they were statistically different at the 95 % level. Histograms were then computed for each C(ZWj ) data set to evaluate whether there was a tendency for the data to be positively covarying and how that changes with discharge. Two histograms were developed, one based on the quadrant classication of C(ZWj ) for each ow and another showing the C(ZWj ) magnitude. This was done so that the distribution of both the type of C(ZWj )

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Figure 2. Raw bed prole (a) and ow width (b) series for 283.2 m3 s1. After detrending and standardizing, values of Z (black line in c) and W (blue line in c) are multiplied together to compute C(ZWj ) (red line in c). The whole series of C(ZWj ) is the GCS.

the mean and variance of the entire detrended series (Salas et al., 1980) to achieve second-order stationarity, which is a prerequisite for spectral analysis (described in the following section). Second-order stationarity of a series means that the mean and variance across the domain of analysis are constant (Newland, 1983). Removal of the lowest frequency of a signal, which can often be visually assessed, has little impact upon subsequent spectral analyses (Richards, 1979). A linear trend was used over other options such as a polynomial because a linear trend preserves the most amount of information in the bed series, while a polynomial can lter out potential oscillations. After detrended and standardized series of Z and Wj were generated, then the GCS between them was computed by taking the product of the two at each centerline station, yielding a spatially explicit measure of how the two covary (Fig. 2). The GCS is the whole series of C(ZWj )

values and not a single metric such as the traditional statistical denition of covariance. Interpretation of a GCS is based on the sign, which in turn is driven by the signs of contributing terms. For C(ZWj ), if both Z and Wj are positive or negative, then C ZWj

[parenrightbig]

> 0, but if only one is nega-

tive then C ZWj

[parenrightbig]

>0. These considerations yield four sub-reach-scale landform end members that deviate from normative conditions (Fig. 3). Normal conditions in this context refer to areas where both variables are close to the mean and thus C ZWj

[parenrightbig]

0. Note that the signs of Z and Wj are not

only important, but the magnitude is, too. Since C ZWj

[parenrightbig]

is

generated by multiplication, if either Z or Wj is within the

range of 1 to 1, then it serves to discount the other. If Z or

Wj is > 1 or < 1, it ampliesC ZWj

R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 9

Table 2. Linear trend models and R2 for Z and Wj used in detrending each series.

Top width Bed elevation

Discharge (m3 s1) Linear trend model R2 Linear trend model R28.50 y = 0.0016x + 193.03 0.0231 y = 0.002x + 194.2 0.8727

28.32 y = 0.0025x + 234.27 0.0429 y = 0.002x + 194.26 0.8713

141.6 y = 0.003x + 301.61 0.0423 y = 0.0021x + 194.04 0.8731

283.2 y = 0.0002x + 332.87 0.0002 y = 0.0021x + 194.23 0.8710

597.5 y = 0.0101x + 528.6 0.2286 y = 0.0021x + 194.16 0.8711

1195 y = 0.0133x + 665.02 0.3037 y = 0.0021x + 194.29 0.8703

2390 y = 0.012x + 710.57 0.2420 y = 0.0022x + 193.92 0.8736

3126 y = 0.0121x + 733.12 0.2437 y = 0.0022x + 193.94 0.8733

and magnitudes could be assessed. Additionally, the bivariate Pearsons correlation coefcients (r) were computed between Z and Wj to assess their potential interdependence.Bivariate Pearsons correlation coefcients were also computed each series of Wj . Statistical signicance was assessed for (r) using a white-noise null hypothesis at the 95 % level.

Next, ACF and PSD analyses were used to determine if C(ZWj ) was quasiperiodic or random, as it was visually evident that it was not constant or strictly periodic. If a series is quasiperiodic, this will be reected in statistically signi-cant periodicity in the ACF (Newland, 1993; Carling and Orr, 2000). Because the PSD is derived from the ACF the two tests show the same information but in different domains, with the ACF in the space domain and the PSD in the frequency domain. So while the ACF analysis reveals periodicity in the signal (if present), the PSD analysis presents the associated frequencies. Both are shown to visually reinforce the results of the PSD analysis. This is helpful because spectral analysis can be very sensitive to the algorithm used and associated parameters such as window type and size. Showing the ACF allows a visual check of dominant length scales that may have quasiperiodicity (e.g., as in Carling and Orr, 2000).The ACF analysis was performed for each ow-dependent series of C(ZWj ), and then these were compared among ows to characterize stage-dependent variability and to analyze how spatial structure changed with discharge. This test essentially determines the distances over which C(ZWj ) are

similar. An unbiased estimate of autocorrelation for lags was used:

Rk =

pn (Salas et al., 1980). For red noise, a rst-order autoregressive (AR1) model was t to the standardized residuals for each spatial series of bed elevation and channel width. For comparison, rst-order autoregressive (AR1) models were produced for 100 random spatial series (each with the same number of points as the ow width spatial series) and averaged. Each averaged AR1 ow width series was then multiplied with the AR1 bed elevation series to create an AR1 model for each C(ZWj ). The red-noise estimate was then taken as the average of all AR1 models of C(ZWj ). The ACF plots were made so that values not exceeding the white-noise signicance are not shown, along with a reference contour for the AR1 estimate. Frequencies can be gleaned from the ACF analysis by taking the inverse of the lag-distance-associated repeating peaks following Carling and Orr (2002).

Power spectral density was estimated for each C(ZWj )

series using a modied periodogram method (Carter et al., 1973). The periodogram is the Fourier transform of the biased estimate of the autocorrelation sequence. The periodogram is dened as

P (f ) =

nk

Pi=1(xi x)(xi+k x) nk

Pi=1(xi x)2

n

n k

, (1)

where xi is a value of a GCS series at location i, x is the

mean value of the GCS (zero due to standardization process), and the terms 1

nk and 1n account for sample bias (Cox, 1983;

Shumway and Stoffer, 2006). Each Rk vs. lag series was plotted against discharge for a maximum of 640 lags (3.2 km, or approximately half the study length), creating a surface that

shows how ACF evolves with ow. Lag intervals are equal to sample interval for the data sets (e.g., 5 m). Statistical signicance was assessed relative to both white- and red-noise autocorrelations. White noise is associated with random processes that are uncorrelated in space, while red noise is associated with data that have properties of rst-order autocorrelation (Newland, 1993). The benet of this approach is that(i) many uvial geomorphic spatial series display autoregressive properties (Melton, 1962; Rendell and Alexander, 1979; Knighton, 1983; Madej, 2001) and (ii) it provides further context for interpreting results beyond assuming white-noise properties. The 95 % condence limits for white noise are given by 1n

2

[Delta1]x

N

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

N1

Xn=0 hnxnei2fn

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2

, (2)

where P (f ) is the power spectral density of x, hn is the window, [Delta1]x is the sample rate, and N is the number of data points (Trauth et al., 2006). While the raw periodogram can exhibit spectral leakage, a window can reduce this effect.A hamming window was used with a length equal to each

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10 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

- C(Z, Wj)

5 Results

5.1 Relating C(ZW j) patterns to landforms

The rst example is located at the lower end of the study area and transitions from a valley meander to a straighter valley section with several valley corridor oscillations (Fig. 4). Starting upstream there is a large point bar on river left with a pool (i.e., Z) that transitions to a broad rife with a 200 m

long zone with Z > 1. Downstream, the river channel impinges on the valley walls, creating two forced pools with localized negative spikes in Z (Fig. 4a, b). Downstream of this, the low-ow channel is steered to the left of the valley, being bounded by two bars. In this zone Z values are positive and 1. Past this there is an inset anabranch that transitions

to a constricted pool with a broad terrace on river left. In this lower zone, Z uctuates between 0 and 1.

Given that bed elevation is held xed for this type of analysis, changes in Wj act to modulate the sign and magnitude of the C ZWj

[parenrightbig]

(a)

Relative bed elevation

-Z

+Z

+ Wj

Relative width

+C(Z, Wj)

+ C(Z, Wj) - C(Z, Wj)

- Wj

(b) (c)

High Low

GCS with increasing ow. In particular, when Z is near a value of 1, the relative ow W modulates the sign and strength of the GCS signal, with several possible changes including persistence, shifting, reversal, and emergence. For example, a persistent positive W oscillation occurs near station 1500, where this zone is always relatively wide regardless of ow. The anabranch zone, however, shows the positive peak in Wj shift downstream from station 900 to 600 from 8.5 to 283.2 m3 s1. Two reversals in Wj occur from low to high ow near stations 350 and 1100, which also create reversals in the GCS, but with different signs. Near station 400, Z and Wj are negative at 8.5 and 283.2 m3 s1, creating a positive GCS. However, Wj increases with ow discharge with an emergent positive peak in W at 3126 m3 s1, which yields a negative GCS.

The other example area occurs at a transition from a valley bend to a straighter section where the river transitions from a broad point bar on river left and eventually crosses over between two smaller inset point bars (Fig. 5a, b). Starting at the upstream extent a large point bar is located on river left with two forced pools in the channel at approximately 3500 and 3600 that have the strongest negative spikes in Z (Fig. 5c, d). Downstream where the point bar ends, the bed prole increases over a broad rife with Z > 1 located above station 3000. As mentioned above in Sect. 3, this poolriferun sequence was studied in great detail by Sawyer et al. (2010), who conrmed the occurrence of naturally rejuvenating rifepool topography. Immediately below the broad rife is a localized zone where Z>1 adjacent to a small bedrock outcrop. Within the alternate bars the bed prole is between 0 and 1 for 300 m, followed by a localized

negative peak in Z around station 2300.

For the rst 200 m, Wj is < 0 for all three ows but gradually increases downstream with increasing ow (Fig. 5c).Since the two deep pools in this initial zone have Z>1, the GCS is > 1 for all ows but reaches a maximum magni-

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2.0

2.0

GCS magnitude

1.0

1.0

0.0

0.0

-1.0

-1.0

-2.0

0 100 200 300 400

Distance upstream (m)

-2.0

0 50 100 150 Distance upstream (m)

Figure 3. Conceptual key for interpreting C

ZWj

[parenrightBig]

geomorphic

covariance structures (a). For quadrant 1, Z and Wj are both relatively high, so that implies wide and shallow areas associated with deposition. Conversely, in quadrant 2, Z is relatively low, but Wj is relatively high, which implies deep and wide cross areas, which in turn implies that these areas may have been scoured at larger ows. In quadrant 3, Z and Wj are both relatively low, so that implies narrow and deep areas associated with erosion. Finally, in quadrant 4, Z is relatively high and Wj is relatively low, so that implies narrow and topographically high areas. Prototypical channels and GCS with positive (b) and negative (C) C

ZWj

[parenrightBig]

colored according to

(a).

data set. Since samples were taken every 5 m, this resulted in a sampling frequency of 0.2 cycles m1 and a Nyquist frequency, or cutoff, of 0.1 cycles m1. The number of data points used for the analysis was roughly half the largest data set, resulting in a bandwidth of 0.00016 cycles m1. For

PSD estimates a modied LombScargle condence limit for white noise at the 95 % level was used as recommended by Hernandez (1996). Since this study was concerned with changes in PSD with ow, estimates were plotted relative to the standard deviation of all PSD results for all series. This was done instead of using the standard deviation of each series because that inates power within a series without context for the variance of adjacent ows.

R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 11

FLOW

(a)

4

3

2

1

Z

0

-1

-2

-3

(b)

-4

4

3

2

1

WJ

0

-1

-2

-3

(c)

8.5

8.50 m

s

3

1

283.2

283.2 m s

3

1

3126

3126 m

3

s

1

-4

4

3

2

GCS magnitude

1

0

-1

-2

-3

(d)

-4 200 400 600 800 1000 1200 1400 1600 1800

Station (m)

Figure 4. Example section in the middle of the study area showing inundation extents (a). Below are plots of minimum bed elevation (b), ow widths for 8.50, 283.2, and 3126 m3 s1 (C), and C

ZWj

[parenrightBig]

for the same ows. The aerial image is for a ow of 21.29 m3 s1 on

28 September 2006.

tude of 6 at 283.2 m3 s1. Beyond this area Wj increases for all ows, but the relative peak broadens and shifts downstream with increasing discharge. At 8.5 m3 s1 the peak is centered near station 3000, where it appears that a back

water increases ow widths upstream of station 2900. For 283.2 m3 s1 the peak shifts downstream 150 m as the

anabranch becomes activated and begins to spread water out. At 3126 m3 s1 the peak is shifted another 300 m down

stream as the bounding point bars are inundated. These shifts in relative Wj act with the bed prole to create a sharper positive peak in C(ZWj ) near the rife at low ows, but then this peak dampens and shifts downstream with increasing ow. There is a similar phase shifting reported for a mixed alluvialbedrock rifepool unit reported by Brown and

Pasternack (2014), associated with a corresponding phasing of peak velocity from the rife to the pool with increased ow. Given that the lower 500 m of this example area have

Z 0, the C ZWj

, GCS is also 0.

Overall, both examples show that zones where Z was either > 1 or < 1 were associated with large pools and rif

es in the study area and were characterized by strong peaks (e.g., > 1) in C(ZWj ). Patterns of Wj can work with Z to create a variety of ow-dependent responses including emergence, reversals, amplication, and shifting. An interesting result is that most of the locations where Z < 1 were short in length, whereas areas where Z > 1 tended to be broader in length.

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12 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

(a)

4

3

(b)

2

1

Z

0

-1

-2

-3

-4

-5

4

3

(c)

8.5

8.50 m

s

3

1

283.2

283.2 m s

3

1

3126

3126 m

s

3

1

2

1

WJ

0

-1

-2

-3

-4

6

(d)

-2 2300 2500 2700 2900 3100 3300 3500 3700 3900

5

4

GCS magnitude

3

2

1

0

-1

Station (m)

Figure 5. Example section at the lower extent of the study area showing inundation extents (a). Below are plots of minimum bed elevation(b), ow widths for 8.50, 283.2, and 3126 m3 s1 (C), and C

ZWj

[parenrightBig]

for the same ows. The aerial image is for a ow of 21.29 m3 s1 on

28 September 2006.

5.2 Is there a tendency towards positively covarying bed and width oscillations?

The histogram of C(ZWj ) showed that regardless of discharge, there was a tendency towards positive values (e.g., where both Z and Wj covary) and that this changed with stage (Fig. 6a). At least 55 % of the data always had C ZWj

[parenrightbig]

data sets, where some ranges of ows were signicantly different from each other and others were similar (Table 3). For example, the8.50 m3 s1 C ZWj

[parenrightbig]

> 0, increasing to 68 % at 283.2 m3 s1 and then slightly declining beyond this ow and stabilizing around 60 % (Fig. 6). There were at most 5 % of values < 1, with

an average and standard deviation of 3 and 2 %, respectively. Contrasting this, values > 1 peaked at 35 % at 141.6 m3 s1 and declined with increasing discharge. So out of the two ex-

tremes, the data exhibited a tendency towards positive values, with negative values < 1 being very rare.

The MannWhitney U test showed interesting ow-dependent aspects of the C ZWj

[parenrightbig]

had p values that were all signicant at the 95% level for each other ow, indicating differences in their distributions. For ows between 28.32597.5 m3 s1, the p values indicated that the series were statistically similar but not for higher ows. The p values for 1195, 2390,

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R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 13

(a)

80 %

70 %

60 %

50 %

Percent

40 %

30 %

20 %

10 %

0 %

8.50 28.32 141.6 283.2 597.5 1195 2390 3126

Discharge (m s )

3

1

< -1 Negative Positive > 1

40 %

(b)

35 %

30 %

Percent

25 %

20 %

15 %

10 %

5 %

0 %

8.50 28.32 141.6 283.2 597.5 1195 2390 3126

Discharge (m s )

3

1

+W,-Z -W, +Z -W,-Z +W,+Z

Figure 6. Histogram of C(ZWj ) classied by positive and negative values as well as > 1 and < 1 (a). Also shown is a histogram classied by quadrant (b). The top histogram (a) illustrates the dominance of values C(ZWj ) > 0 regardless of discharge and also illustrates an increasing tendency towards positive values of C(ZWj ) > 1 up until 141.6 m3 s1, after which it declines. The bottom histogram (b) illustrates the distribution of the four quadrant types shown in Fig. 3, with shallow and wide (e.g., +W, +Z) and deep and narrow (e.g., W, Z) zones

being most dominant at 283.2 3 s1. Colors represent bin-centered values.

and 3126 m3 s1 were statistically similar at the 95 % level

but not for lower ows.

The quadrant-based histogram reveals further insight into the distribution of river geometry with ow (Fig. 6b). The average percentage of C(ZWj ) for each quadrant across all ows was 30 % +W, +Z; 14 % +W, Z; 25 % W, +Z; and 31 % W, Z, with standard deviations ranging

from 2 to 3 %. Percentages of positive C ZWj

[parenrightbig]

Further insights into the positive nature of C ZWj

can be

inferred from bivariate Pearsons correlation coefcients of Z and Wj (Fig. 7). Similar to C ZWj

[parenrightbig]

the ow-dependent response was that the correlation between Z and Wj increased with ow until 283.2 m3 s1 and then subsequently declined.

To further reinforce these results, one can also inspect the plot of Z,Wj , and C ZWj

[parenrightbig]

[parenrightbig]

were rel

atively evenly distributed between +W, +Z and W, Z,

although the latter was slightly more prevalent. The percent of the data in the +W, +Z quadrant increased from 26 %

at 8.50 m3 s1, peaked at 34 % at 597.5 m3 s1, decreased to 30 % at 1195 m3 s1 and stabilized near this value for higher ows. Meanwhile, the percent of the data in the W, Z

quadrant increased from 29 % at 8.50 m3 s1 and peaked at 35 % at 141.6283.2 m3 s1 ow and then decreased to 30 % at 597.5 m3 s1. After that it increased to 33 % and stabilized at and beyond 1195 m3 s1. Both the +W, Z and +W, Z

quadrants followed a similar but opposite trend, reaching a minimum at 283.2 m3 s1.

for 283.2 m3 s1, visually showing the synchronous nature of Z and Wj (Fig. 2) The correlations between combinations of Wj show that each series is signicantly correlated to the next highest ow, but there is an interesting ow-dependent pattern (Fig. 8). Correlations between series decrease with increasing ow, reaching a minimum between 597.5 and 1195 m3 s1 and then increasing again.

5.3 Are bed and width oscillations quasiperiodic?

The ACF of C(ZWj ) also showed similar changes with discharge as the above analyses, with increases in the presence and magnitude of autocorrelation from 8.50 to 597.5 m3 s1 and then a subsequent decline with increasing ow (Fig. 9a).

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14 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

Table 3. MannWhitney U test p values amongst all combinations of Z and Wj ; bold values are at the 95 % level.

8.50 28.32 141.6 283.2 597.5 1195 2390 3126

8.50 0.0002 0.0000 0.0000 0.0000 0.0008 0.0498 0.040328.32 0.0126 0.0001 0.0262 0.6152 0.0865 0.1009 141.6 0.125 0.7627 0.0015 0.0000 0.0000 283.2 0.0859 0.0000 0.0000 0.0000 597.5 0.0033 0.0000 0.0001 1195 0.2673 0.3129 2390 0.9487 3126

-0.05

0.55

Similar to ACF analysis, PSD analysis showed quasiperiodic components of C(ZWj ) exhibiting ow-dependent behavior (Fig. 9b). For 8.50283.2 m3 s1 there is a high power band (e.g., PSD/ 1216) centered on 0.0014 cycles m1,

which is conrmed from the ACF analysis above. For 8.50 141.6 m3 s1 there are also smaller magnitude peaks ranging from 3 to 8, spread out over several frequencies. There is also a high magnitude component at the lowest frequency band that emerges at 28.32 and declines by 283.2 m3 s1. These low-frequency components are commonly associated with rst-order auto-regressive behavior in the data (Shumway and Stoffer, 2010). At 597.5 m3 s1 power is still associated on 0.0014 cycles m1, albeit with a 50 % reduction in

magnitude. Beyond this ow the frequency range and magnitude of statistically signicant values decline with discharge.Overall, both ACF and PSD results show that C(ZWj ) is

quasiperiodic from 8.50 to 283.2 m3 s1 but then decreases in strength as ow increases. Further, the PSD results show that the C(ZWj ) GCS is ow dependent and multiscalar, being characterized by a range of statistically signicant frequencies.

6 Discussion

6.1 Coherent undulations in cobblegravel bed river topography

The primary result of this study is that in an incising, partly conned, regulated cobblegravel river whose ow regime is dynamic enough to afford it the capability to rejuvenate its landforms, there was a tendency towards positive C(ZWj )

and thus covarying Z and Wj amongst all ows analyzed.

Based on the ACF and PSD analyses, the C(ZWj ) GCS

undulations are quasiperiodic. The results of this study associated channel organization across a range of recurrence interval frequencies within the range of commonly reported channel-forming discharges for western US rivers (e.g., 1.22.5 years) as well as substantially larger ows. These conclusions are obviously limited to the study reach, but this should not prohibit discussing possible mechanisms that could lead to these observed patterns, as well as the role of variable ows and incision.

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0.45

0.35

Correlation

0.25

0.15

0.05

8.50

8 80 800

28.32

141.6

283.2

597.5

1

1195

2.390

3126

Discharge (m3 s )

Figure 7. Pearsons correlation coefcient for Z and Wj between each ow.

At the lowest discharge there are approximately two broad bands of positive autocorrelation that exceeded both the white noise and AR1 threshold at lag distances of 1400 and 2100 m. At 28.32 m3 s1 these three peaks broaden, and the highest correlation was found at lag distance 1400 m, which increased from 0.4 to 0.7. At the bankfull dis

charge of 141.6 m3 s1, the peak at 1400 m diminishes, while the peak near 2100 m increased in strength (e.g., correlation magnitude). At 283.2 m3 s1 there are still peaks near 1400 and 2100 m that exceed both white noise and the AR1 threshold, but two other signicant peaks emerge near 700 and 2800 m. Similar statistically signicant correlations are found at 596.5 m3 s1, albeit with narrower bands of correlation. The correlation distances at 283.2 and 596.5 m3 s1 average 700 m, and this average would have a frequency

of approximately 0.0014 cycles m1. Beyond 596.5 m3 s1 the ACF diminishes rapidly with no peaks that are statistically signicant compared to red noise. Overall, the ACF results show that C(ZWj ) is quasiperiodic from 8.50 m3 s1 to 141.6597.5 m3 s1, but then the periodicity decreases in strength as ow increases.

R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 15

0.00

1.00

0.80

Correlation

0.60

0.40

0.20

8.5 to28.32

28.32 to 141.6

141.6 to

283.2

283.2 to 597.5

597.5 to 1195

1195 to

2390

2390 to

3126

Discharge pair (m s )

3 1

Figure 8. Pearsons correlation coefcient for sequential pairs of ow-dependent wetted width series.

Most notably, the test river exhibited a dominance of covarying values of Z and Wj across all ows, being characterized by a quasiperiodic pattern of wide and shallow or narrow and deep cross sections. This supports the idea that alluvial river reaches have a tendency to adopt wide and shallow or narrow and deep cross sections to convey water ow (Huang et al., 2004). Rather than select a single type of cross section to maximize energy dissipation to create a uniform cross section geometry at a single channel-maintaining ow, commonly referred to as bankfull, it appears that alluvial rivers adjust their channel topography to have cross sections that roughly alternate between those that are, on the one hand, wide and shallow and, on the other, narrow and deep (Fig. 6b; Huang et al., 2004), with some locations having a prismatic channel form indicative of normative conditions, particularly in transition zones. Whether this is attributed to minimizing the time rate of potential energy expenditure per unit mass within a reach (Langbein and Leopold, 1962; Yang, 1971; Cherkauer, 1973; Wohl et al., 1999) or channel unit scale mechanisms associated with rifepool maintenance (Wilkinson et al. 2004; MacWilliams et al., 2006; Caamano et al., 2009; Thompson, 2010) remains to be determined. Given that extremal hypotheses and rifepool maintenance act at different, yet interdependent scales, it is likely that both play an intertwined and inseparable role in channel form. That said, extremal theories are limited to predicting mean channel conditions within a reach (Huang et al., 2014), with no models that can yet fully predict sub-reach-scale alluvial river topography, so we turn our attention to more tractable hydrogeomorphic processes related to the maintenance of rife and pool topography.

Presumably, the quasi-oscillatory C(ZWj ) GCS pattern is also linked to ow-dependent patterns of convective acceleration and deceleration zones (Marquis and Roy, 2011; MacVicar and Rennie, 2012), as the length scales of the GCS were aligned with the spacing of erosional and depositional landforms such as bars and pools. This aspect is supported by ACF and PSD results as well as two other studies on the test reach. First, it appears that the quasiperiodicity of the C(ZWj ) GCS is related to the poolrife oscilla-

Figure 9. Autocorrelation (a) and PSD (b) of C(ZWj ) with increasing ow. For the ACF plot (a), only values exceeding white noise at the 95 % level are shown and the red contour demarcates the 95 % level for an AR1 process (red noise). For the PSD plot (b)

only values exceeding white noise at the 95 % level are shown.

tion in the river corridor. The PSD analysis showed that the dominant frequency of C(ZWj ) was 0.0014 cycles m1,

which equates to a length scale of 700 m (Fig. 9). Three of

the morphologic units (MUs) studied by Wyrick and Pasternack (2014) can be used for context, including pools, rifes, and point bars. In their results for the Timbuctoo Bend reach, pools, rifes, and point bars had an average frequency of 0.0029, 0.0028, and 0.001 cycles m1. Considering that pools and rifes are dened as two end-members of positive C(ZWj ), then the frequency of rifes and pools should be twice that of the C(ZWj ) GCS found herein. That is, a single oscillation of C(ZWj ) GCS would include both a narrow and deep (e.g., pool) and a wide and shallow (e.g., rife) cross section geometry, although transitional forms are possible within a cycle, too (Fig. 3). Therefore, it appears that the quasiperiodicity of the C(ZWj ) GCS is related to the poolrife oscillation in the river corridor. This is in agreement with studies based on eld investigations and numerical models that relate this observation to quasiperiodic bed and width variations associated with barpool topography (Richards, 1976b; Repetto and Tubino, 2001; Carling and Orr, 2002).

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16 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

Second, Sawyer et al. (2010) showed that stage-dependent ow convergence maintained bed relief by topographically mediated changes in peak velocity and shear stress at the central rife in a second example (Fig. 5). Interestingly, the ow width series phases relative to bed elevations in accordance with theory (Wilkinson et al., 2004) and eld and numerical studies (Brown and Pasternack, 2014). This supports an already reported relationship between the C(ZWj ) GCS and the process of ow convergence routing (Brown and Pasternack, 2014; Brown et al., 2016).

Lastly, Strom and Pasternack (2016) showed that peak zones of velocity undergo variable changes in their location with discharge, with most velocity reversals occurring after 597.5 m3 s1. In this case the zones of peak velocity patches underwent complex changes from being associated with narrow topographic high points at base ows (Wj ,+Z) to to

pographic low points where ow width is constricted at high ows (Wj ,Z). Overall, the presence of oscillating, on

the one hand, wide and shallow and, on the other, narrow and deep cross sections appears to be linked to hydrogeomorphic processes of rifepool maintenance.

6.2 Hierarchical nesting, variable ows, and the role of incision

This study quantitatively supports the idea that river morphology in partially conned valleys is hierarchically nested with broader exogenic constraints such as the bedrock valley walls, as well as channel-width-scale alluvial controls such as point bars and islands. Our study quantitatively characterized interesting shifts in the amount of correlation amongst ow width series and in the presence of quasiperiodic oscillations in C(ZWj ) with changes in ow. Each series of Wj were signicantly correlated with the next highest ow, but this was lowest between 597.5 and 1195 m3 s1, where the valley walls begin to be engaged (Fig. 7). Further, both the ACF and PSD show that quasiperiodicity in C(ZWj )

declines after 597.5 m3 s1 (Fig. 9). In addition, Strom and Pasternack (2016) showed that reversals in peak velocity occur when ows exceed 597.5 m3 s1. While results show that statistically signicant correlations between Z and Wj occur for a range of ows, the greatest magnitude is not when the valley walls are inundated, but it is for the 283.2 m3 s1 channel and incipient oodplain. Given that correlations were still signicant for the ows that inundate the valley walls, this does not refute the role of valley width oscillations in potentially controlling rife persistence (White et al., 2010) but rather adds new insight to the morphodynamics of rivers incising in partially conned valleys. This suggests that the incision process may be decoupling the organization of the riverbed away from being controlled by the valley walls and instead phased towards reshaping channel topography within the inset bars that are nested within the valley walls. As the riverbed incises further down through knickpoint migration (Carley et al., 2012), this may act to shift zones of high and

low wetted width upstream unless lateral erosion can keep pace.

6.3 Broader implications

This study quantied relationships between ow width and minimum bed elevation in a partly conned and incising gravelcobble-bedded river, as well as for the rst time how they change with stage. While study results are currently limited to rivers similar to the study reach, there are several key results of this study that may have broader relevance to river restoration and management.

First, a key result of this study was that channel geometry was organized into covarying Z and Wj undulations across all ows analyzed, alternating between, on the one hand, wide and shallow and, on the other, narrow and deep cross sections. This is a very different view from the classical denition of singular and modal bankfull channel geometry often used to guide river and stream restoration (Shields et al., 2003). Instead, our study found that channel geometry at all ows had a relatively even mixture of, on the one hand, wide and shallow and, on the other, narrow and deep cross sections. Studies that deconstruct the complexity of river channel geometry to modal ranges of channel width and depth have always shown scatter, which has mostly been attributed to measurement uncertainty and/or local conditions (Park, 1977; Philips and Harman, 1984; Harman et al., 2008;Surian et al., 2009). Our study suggests that this variability is a fundamental component of alluvial river geometry. While this concept was proposed by Hey and Thorne (1983) over 2 decades ago, few studies have integrated these ideas into river engineering and design (e.g., see Simon et al., 2007).Thus, this study further supports a necessary shift away from designing rivers with modal conditions to designing rivers with quasi-oscillatory and structured variations in channel topography. An example of this is the formprocess synthesis of channel topography that experiences ow reversals using GCS theory (Brown et al., 2016)

Second, this study has implications for restoration design and ow reregulation in that a wide array of discharges beyond a single channel-forming ow are presumably needed for alluvial channel maintenance (Parker et al., 2003). Commonly singular values of channel-forming discharge, usually either bankfull or effective discharge, are used in stream and river restoration designs (Shields et al., 2007; Doyle et al., 2007). This study refutes this concept for rivers such as those studied herein, as supported by the results that show gradual changes in channel organization within a band of discharges with recurrence intervals ranging from 1.2 to 5 years, and 4-fold range in absolute discharges. Instead, stream and river restoration practitioners should analyze ranges of ow discharges and the potential topographic features (existing or designed) that could invoke stage-dependent hydrodynamic and geomorphic processes associated with complex, self-maintaining natural rivers.

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R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers 17

Third, while the length scales of covarying Z and Wj undulations are approximate to the spacing of bars and pools in the study area, they are quite complex and lack explicit cutoffs that illustrate power in a singular frequency band.Thus, river restoration efforts that specify modal values of bedforms may overly simplify the physical structure of rivers with unknown consequences to ecological communities and key functions that are the focus of such efforts. River restoration designs need to mimic the multiscalar nature of self-formed topography by incorporating GCS into river engineering (Brown et al., 2014) or somehow ensure that simpler uniscalar designs will actually evolve into multiscalar ones given available ows and anthropogenic boundary constraints.

Fourth, this study has potential implications for analyzing the effect of ow-dependent responses to topography and physical habitat in river corridors. Valley and channel widths have been shown to be very useful in predicting the intrinsic potential of salmon habitat (Burnett et al., 2007). Further, the role of covarying bed and width undulations in modulating velocity signals and topographic change has implications for the maintenance of geomorphic domains used by aquatic organisms. As one example, consider that adult salmonids use positively covarying zones such as rifes (e.g., +Wj ,

+Z) for spawning and pools (e.g., Wj , Z) for holding

(Bjorn and Reiser, 1991). In the study reach Pasternack et al. (2014) showed that 77 % of spawning occurred in rifes and chute morphologic units, which are at or adjacent to areas where C(ZWj ) > 1 (Figs. 4, 5), supporting this idea.The presence and structure of covarying bed and width undulations is also thought to be important indirectly for juvenile salmonids that require shallow and low-velocity zones as refugia during large oods. For example, the expansions that occur at the head of rifes would presumably provide lateral zones of shallow depths and moderate velocities needed for ood refugia. In the absence of positive bed relief, and zones of +W,+Z, ow refugia zones would be hydrologically dis

connected from overbank areas, impacting the ability of juvenile salmon to utilize these areas as refugia during oods and potentially leading to population level declines (Nick-elson et al., 1992). Future work should better constrain the utility of GCS concepts in assessing aquatic habitat.

Lastly, it is possible that the C(ZWj ) GCS could be used as a comparative proxy in remote-sensing applications to determine how the topographic structure of rivers changes with ow and how that may also change though time. The zoomed examples of C(ZWj ) and the detrended river topography highlight how this type of GCS can be used to characterize the topographic inuence on wetted width and bed elevation variability in river corridors. The C(ZWj ) GCS may be used diagnostically to assess riverine structure and hydraulic function in a continuous manner within a river across an array of ows. While not studied herein, prior work (Brown and Pasternack, 2014) showed that the magnitude ofC(ZWj ) can

also be related to ow velocity, though lagged effects do oc-

cur. Since the magnitudes can be linked to both unique land-forms and ow velocity, they may have utility in assessing topographic and hydraulic controls in river corridors.

Lidar and analytical methods for developing bed topography in rivers have improved considerably (McKean et al, 2009). For example, Gessese et al. (2011) derived an analytical expression for determining bed topography from water surface elevations, which can be obtained from lidar (Ma-girl et al, 2005). Assuming one has an adequate topographic data set, whether numerical ow modeling is needed to generate wetted width data sets places a considerable constraint on performing this type of analysis. This could potentially be relaxed, especially at ows above bankfull, using a constant water slope approximation for various ow stages. At smaller discharges in rivers, there are typically defects in the water surface elevation, where the bed topography exerts a strong control on bed elevations (e.g., Brown and Pasternack, 2008). However, many studies suggest that on large alluvial rivers, bankfull and ood proles show that they generally atten and smoothen once bed forms and large roughness elements such as gravel bars are effectively submerged. In this case, one can then detrend the river corridor and take serial width measurements associated with various heights above the riverbed (Gangodagamage et al., 2007). The height above the river can then be related to estimates of ow discharge and frequency, so that the changed GCS structure can be related to watershed hydrology (Jones, 2006). There is also the obvious option of using paired aerial photography with known river ows by correlating discharge with imagery dates and widths. Future work should establish whether similar conclusions can be reached using eld- and model-derived estimates of wetted width as opposed to modeled solutions.

7 Conclusions

A key conclusion is that the test river exhibited covarying oscillations of minimum bed elevation and channel top width across all ows analyzed. These covarying oscillations were found to be quasiperiodic at channel-forming ows, scaling with the length scales of pools and rifes. Thus, it appears that alluvial rivers organize their topography to have oscillating, on the one hand, shallow and wide and, on the other, narrow and deep cross section geometry, even despite ongoing incision. Presumably these covarying oscillations are linked to hydrogeomorphic mechanisms associated with alluvial river channel maintenance. As an analytical tool, the GCS concepts here treat the topography of river corridors as system, which is thought of as an essential view in linking physical and ecological processes in river corridors on multiple scales (Fausch et al., 2002; Carbonneau et al., 2012).While much research is needed to validate the utility of these ideas for these broader concepts and applications in ecology

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18 R. A. Brown and G. B. Pasternack: Bed and width oscillations form coherent patterns in alluvial rivers

and geomorphology, the idea of GCSs, especially for width and bed elevation, holds promise.

8 Data availability

Each C(ZWj ) data set is available from either author by request.

The Supplement related to this article is available online at http://dx.doi.org/10.5194/esurf-5-1-2017-supplement

Web End =doi:10.5194/esurf-5-1-2017-supplement .

Acknowledgements. Although not directly funded by any source, this study used data and models from studies previously sponsored by Pacic Gas & Electric Company, the US Fish and Wildlife Service Anadromous Fish Restoration Program, Yuba County Water Agency, and the Yuba Accord River Management Team. Co-author Gregory B. Pasternack received support from the USDA National Institute of Food and Agriculture, Hatch project number #CA-D-LAW-7034-H.

Edited by: J. K. HillierReviewed by: C. J. Legleiter, D. Thompson, and J. M. Turowski

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