River channel lateral mobility: metrics, time scales, and controls
Authors
Andrew D. Wickert,
Corresponding author
University of Colorado, Department of Geological Sciences and Institute of Arctic and Alpine Research, Colorado, USA
Corresponding author: A. D. Wickert, University of Colorado, Department of Geological Sciences and Institute of Arctic and Alpine Research, 1560 30th St., Boulder, CO 80303, USA. (wickert@colorado.edu)
[1] Alluvial river channels are intrinsically mobile. We mapped channel planform extent in a series of experiments to measure instantaneous rates of channel motion, loss of planform overlap with the original positions of the channels, and reworking of the fluvial surface over which the channels moved. These experiments comprise two aggrading deltas, one subsiding delta that underwent cyclical base level changes, and one braided channel system that was seeded with vegetation. We find that the amounts of channel planform overlap and remaining unreworked fluvial surface area both decay exponentially with time, and that these metrics and the instantaneouslymeasured rates of channel motion scale predictably with one another in spite of the different time scales of the processes they record. Rates of channel planform change increase with increasing sediment flux and bed and planform irregularity, and decrease with the establishment of riparian vegetation. Aggradation does not noticeably affect channel mobility, but induces avulsions that allow the channels to more rapidly rework the fluvial surface. Additional findings include that: (1) sediment flux in the braided experiment equals its rate of bar migration, (2) channel widths are normally distributed with time, and (3) we can use our channel mobility metrics to connect surface processes with the resultant fluvial stratigraphy.
[2] Alluvial river channels in nonincisional settings are highly mobile systems that interact dynamically with their beds, banks, and the fluvial surfaces such as braid plains, valleys, and alluvial fans, through which they flow. Their morphologic evolution is affected by sediment discharge, base level changes, riparian vegetation, and internallygenerated bed and planform irregularities such as barforms and sinuosity. We present novel analyses of a series of physical experiments to understand how these basic controls on morphodynamic change control overall rates of channel mobility across multiple time scales of measurement.
[3] We measure channel mobility over three time scales characteristic to fluvial systems. In the short term, we measure channel mobility as a continuous change in channel planform with time, which equates to a combination of lateral migration rate and avulsion magnitude times frequency. Over longer time scales, rivers gradually lose overlap with and “forget” the shape their previous planforms. Over even longer time scales, rivers move across and rework the sediments of their fluvial surfaces.
[4] We apply metrics for these three time scales of channel planform change to a set of analog material experiments [Kim et al., 2006a, 2006b; Martin, 2007; Sheets et al. 2007; Tal and Paola, 2007, 2010]. Although simplifications in the experimental conditions limit our ability to explore the full range of channel planforms and rates of landscape change in natural systems, these experiments offer the advantage of complete, highresolution data sets under controlled conditions, making them an attractive platform from which to develop this method. These systems were forced with different sediment discharges, base level changes, vegetation, and/or flow conditions. These differing inputs allow us to pick apart how external forcings control channel mobility, and analyses across the whole range of experiments show relationships among our measurements at different time scales.
2 Descriptions of Experimental Systems
[5] We examined river channel planform change in a series of experimental fluvial systems generated at the Saint Anthony Falls Laboratory at the University of Minnesota (Figure 1, Table 1). We observed three experimental alluvial fan deltas (depositional systems) [Kim et al., 2006a, 2006b; Martin, 2007; Sheets et al., 2007] and one braided river (transport system) to which riparian vegetation was added [Tal and Paola, 2007, 2010]. We observed these experiments during windows in which all of the inputs to each channel system remained approximately constant, and did not include any of the early time steps in which the channel had not yet developed a selfformed fluvial network. These experimental systems were predominantly braided owing to the relative ease of reproducing braiding versus meandering in noncohesive sediments in the laboratory [cf., Howard, 2009; Tal and Paola, 2010]. In this section we introduce the river systems studied and their basic characteristics (Table 2).
Table 2. Experimental Channel System Parameters and Channel Mobilities
Experiment
fw̄
ab̄
h̄
S
Q_{s}
Q
^{d}D_{50}
ζ̇
SLR
^{e}σ
Q_{s}
^{f}Ξ
[–]
[m]
[m]
[–]
[m^{3}/hr]
[m^{3}/hr]
[mm]
[m/hr]
[mm/hr]
[mm/hr]
[m/hr]
[–]
f_{w}: mean wetted fraction, b̄: mean channel system width, h̄ mean depth, S: downvalley slope, Q_{s}: sediment discharge, Q: water discharge, D_{50}: median sediment diameter, ζ̇: rate of channel lateral motion, SLR: rate of sealevel rise, σ: subsidence rate, Q_{s}: downstream sediment flux, Ξ: ratio of channel mobility to sediment flux (“efficiency” of crossstream or morphologicallyaltering sediment transport).
^{a}Channel system mean width b̄ equals fw̄ times the width of the fluvial surface.
^{b}Tal and Paola [2007, 2010] see a larger wetted width in BV1 than in BV2 because they used a lower dye intensity threshold to define a channel.
^{c}This is the highflow discharge, which the channel system experienced during our periods of observation.
^{d}Sediment in the delta experiments comprised both quartz and anthracite to simulate coarse and fineparticles, respectively, due to the lower density of anthracite. These D_{50} valuesare given for quartzdensityequivalent grain sizes.
^{e}XES02 was run with a linear hingetype subsidence [Kim et al., 2006a, 2006b]; the rate given here is the mean.
^{f}Ξ was calculated without accounting for channel mobility due to superelevationdriven avulsions (Equations (28) and (30)) because we see no relationship between superelevation and ζ̇ ( Sections 6.2 and 6.3; Figure 8.C).
XES02SS
0.1711
0.39
0.005
0.024
0.01822
1.5012
0.12
0.9927
0
1.85
9.34
0.11
XES02SF
0.1746
0.44
0.009
0.036
0.01822
1.5012
0.12
1.0874
− 2.26
1.85
4.60
0.24
XES02SR
0.2327
0.5
0.008
0.022
0.01822
1.5012
0.12
1.1470
2.26
1.85
4.55
0.25
XES02RF
0.2341
0.29
0.017
0.064
0.01822
1.5012
0.12
0.6297
− 13.57
1.85
3.69
0.17
XES02RR
0.4643
0.52
0.004
0.036
0.01822
1.5012
0.12
0.9395
13.57
1.85
8.76
0.11
DB031
0.0632
0.14
0.006
0.05
0.03600
1.44
0.12
2.7114
5
0
42.86
0.06
DB032
0.1801
0.47
0.003
0.009
0.00018
1.44
0.12
0.0283
0.023
0
0.13
0.22
BV1
0.2763
^{b}0.34
^{b}0.012
0.015
0.00619
7.2
0.5
1.5554
0
0
1.52
1.02
BV2
0.2954
^{b}0.44
0.033
0.015
^{c}0.00334
^{c}7.2
0.5
0.0893
0
0
0.23
0.39
[6] Three of the four experiments in our study are deltaic systems. DB031 [Sheets et al. 2007] and DB032 [Martin, 2007] are alluvial fan delta experiments. Sediment discharge in DB032 was much lower than that in DB031; this resulted in DB032 having subcritical flow, a lower fan surface slope, and a much lower aggradation rate. Both of these deltas first prograded to a defined radius (2.5 m for DB031 and 2.8 m for DB032), after which base level (the water level in the basin) was raised at a rate that produced a purely aggradational delta of uniform size (i.e. depositional equilibrium). We sampled only the time period over which these deltas were in depositional equilibrium. XES02 [Kim et al., 2006a, 2006b] is a subsiding alluvial fan delta that underwent cyclical changes in base level. We sampled this experiment at five different intervals to obtain a picture of the effects of base level changes on rates of channel planform change and reworking of the fan surface: the stillstand (XES02SS), the slow cycle of base level rise and fall (XES02SR, XES02SF), and the rapid cycle of base level rise and fall (XES02RR, XES02RF). The final experiment that we analyzed is part of a set of experiments designed to study the interactions between braiding and riparian vegetation (referred to here as BV) [Tal and Paola, 2007, 2010]. The first stage of this experiment (BV1) was characterized by steady state braiding in noncohesive sand under uniform discharge. The second stage (BV2) consisted of repeated cycles of a 1hour high flow and a 6day low flow. Alfalfa seeds (Medicago sativa) were dispersed uniformly over the bed immediately after the end of each high flow. Plants colonized freshly deposited bars and areas of braidplain that were not inundated during low flow [Tal and Paola, 2007, 2010]. Cycling high and low discharge in conjunction with plant seeding led to a change in channel planform from braided to singlethread and the formation of a vegetated floodplain.
3 Data From Images of Experiments
[7] We separated the fluvial surface into binary maps of regions that were occupied by the river system (“wet”, 1) and those that were unoccupied (“dry”, 0). We measured planform change by observing how these “wet” and “dry” areas changed. The major advantage of this method is that it does not require a spatially coherent channel, and can therefore work in both multi and singlethread systems. This is essential for analyzing our largely braided experiments. Our methods are improved versions of those used by Wickert [2007], and are written in Matlab and Python.
3.1 Defining Fluvial Surface and River Channels
[8] We defined the fluvial surface (F) by removing regions that a river channel could not occupy (Figure 2). F = 1 across the fluvial surface, and F = 0 in regions that were excluded from our analysis. For all of the experimental deltas, we exclude all areas that are inundated by the ocean (XES02: Kim et al. [2006a, 2006b]; DB031: Sheets et al. [2007] DB032: Martin [2007]). In XES02 [Kim et al., 2006a, 2006b]. We also exclude abandoned terraces produced during the cycles of rapid base level change. These terrace surfaces were dyed for identification, and after the river channel incised into them, it occupied a limited and consistent valley over the duration of our measurements. For the braided and vegetated river systems (BV1 and BV2, respectively: Tal and Paola [2007, 2010]), we used the entire 10 meter study reach, which excludes a combined 6 meters at the upstream and downstream ends of the flume in order to avoid the effects of these boundary conditions.
[9] We separated the fluvial surface, F = 1, into binary matrices referred to as “channel maps”. These matrices, denoted K, consist of channel (“wet”, 1) and unoccupied fluvial surface (“dry”, 0). This “wet” vs. “dry” analysis has been performed extensively on fluvial system experiments [Cazanacli et al. 2002; Tal and Paola, 2007; Wickert, 2007; Martin, 2007; Martin et al. 2009; Kim et al. 2010; Tal and Paola, 2010]. In each experiment, colored dye was injected into the water (Figures 1 and 2), making it straightforward to use a set range of color values to separate the channel from dry sediment and/or vegetation. We processed the images to increase the contrast in color parameters (red, green, blue, hue, saturation, brightness) between the channel and the rest of the image, and then used a threshold to generate the binary channel maps. We applied a smoothing filter to remove smallscale (1–2pixel radius) noise due to small changes in lighting and dye intensity that otherwise would cause an inaccurately large difference to be reported between channel systems at adjacent time steps. While this filter did not solve all of the issues with noise in the image processing, it provided an improvement over earlier methods [e.g., Wickert, 2007; Martin et al., 2009].
3.2 Determining Channel Planform Change
[10] We used the channel maps (Section 3.1) to measure instantaneous rates of channel planform change, loss of channel system overlap with previous channel patterns, and reworking of the fluvial surface as the channels swept across it. These measures are summarized schematically in Figure 3.
3.2.1 Channel Overlap
[11] We measured the loss of planform overlap by comparing river channel patterns at two different times and varying the amount of time between these two observations. This method is a bulk variant of the method Sapozhnikov and FoufoulaGeorgiou [1997] used to measure the spatial structure of channel change in experimental braided streams. For each alluvial river system, we assembled a time series of channel maps K_{β}, in which K represents the channel map binary matrix and β is an integer that represents the time step. We then selected each channel map in turn to be the baseline channel map (K_{B}). We differenced each baseline channel map with a corresponding time series of “transient” channel maps, K_{T}, of all T ≥ B. The absolute values of these differences were summed across rows (m_{r}) and columns (n_{c}) to obtain the number of changed pixels, D, between each baseline–transient channel map pair.
D(B,T)=∑i=1mr∑j=1ncKB−KT(1)
[12] We then scaled this difference against the number of changed pixels that would be observed by randomly scattering the same proportions of wet (channel) and dry (unoccupied fluvial surface) pixels across the fluvial surface. This random scatter parameter, Φ, is:
Φ=(fw,B⋅fd,T+fd,B⋅fw,T)(2)
where F is “fraction”, subscript w is “wet”, and subscript D is “dry”. We used Φ to compute a normalized overlap (O_{Φ}) in channel position between time steps b and T. A value of 1 for O_{Φ} indicates that the two channel maps are identical, and a value of 0 indicates that the two channel maps have an amount of overlap indistinguishable from random noise. Negative values indicate that there is less overlap between the channel maps than would be expected for samples of random noise.
OΦ=1−DA⋅Φ(3)
A is the area of the fluvial surface in pixels. O_{Φ} is calculated for each K_{B} − K_{T} pair because the proportions of observed “wet” and “dry” pixels were not constant through a data set, although they tended to be normally distributed about a mean (Figure 4).
[13] Each channel overlap time series is given relative to its baseline channel position occurring at t = 0. We superposed all of these time series to yield a better statistical picture of the time evolution of each channel system. We thus assumed that all of these systems are statistically stationary. This is reasonable so long as the mean external conditions (water and sediment discharge, bed and bank material, mean channel slope) remain the same in a timeaveraged sense. We tested for stationarity by applying an augmented Dickey–Fuller test [Dickey and Fuller, 1979, 1981] to our instantaneouslymeasured rates of channel planform change. With a single lag, all experiments exhibited statistical stationarity (p < 0.001). With multiple lags, the results of the augmented Dickey–Fuller tests were indistinguishable from the same tests performed on Gaussian white noise with time series lengths, means, and standard deviations derived from each of the experimental time series. To test for longrange trends, we attempted a linear fit to each instantaneous planform change time series. Here, all line slopes were near 0, the median R^{2} value was 0.0050, and the maximum R^{2} value was 0.0472; these values likewise indicate statistical stationarity.
3.2.2 Reworking of the Fluvial Surface
[14] To calculate a time scale over which the fluvial surface is reworked, we first picked a single baseline channel map (K_{B}) as above. Then we selected from the fluvial surface area matrix (F) only those pixels that were not occupied by channel at the baseline time step, as these constitute the region available for new channel occupation, and called this matrix F ′ .
F′=F−KB(4)
Because the initial channel area was subtracted from F to create F ′ , the total relevant pixel area decreases accordingly by a factor of f_{d,B}.
∑i=1mr∑j=1ncF′=A⋅fd,B(5)
[15] We summed the series of channel maps K ′ through K ′ for each time step β = T after the baseline time step. Here, the primes indicate that these channel maps have had the initial channel area removed from the analysis according to their respective modified fluvial surface mask, F ′ . This summed K ′ through K ′ matrix gives the total number of times each pixel has been occupied between time steps β = B + 1 and β = T. We then counted the number of zeros (unreworked pixels: N ′ ) in this matrix at each time step T corresponding to each baseline time step b:
N′=∑i=1mr∑j=1nc∑β=B+1TK′≡0(6)
We divided this sum by the observed area, A ⋅ f_{d,B}, to obtain the fraction of not initially occupied fluvial surface that has been reworked, f_{R}.
fR=1−N′A⋅fd,B(7)
We overlaid all of the f_{R} measurements onto a common relativetobaseline time axis by setting t = 0 at each baseline timestep as we do to measure loss of planform overlap (Section 3.2.1).
3.2.3 Instantaneous Channel Planform Change
[16] Our third measure of river channel mobility is the rate of channel planform change, measured as the amount of surface area reworked per unit time. For a singlethread stream, this is equivalent to the migration rate integrated down the length of the channel plus the avulsion magnitude (areal extent) times its frequency.
[17] While it would be simplest to difference channel maps at adjacent time steps, the colorbased technique of separating channels from unchannelized fluvial surface produces minor inconsistencies in the area that is classified as “channel” due to very small changes in (1) lighting and dye concentration (Section 3.1) and (2) channel depths in the braided systems that allow portions of them to fluctuate across the channel map dye intensity threshold. This noise would cause too much planform change to be measured between adjacent time steps, so we filter it out with a technique known as double differencing, which is commonly applied to remove noise from GPS data [e.g., Alber et al., 2000]. Noise in the channel maps occurs at a similar magnitude in all nearby time steps, so approximately the same amount of additional loss of overlap due to random noise, given by the independent and identically distributed random variables X_{1} and X_{2}, is observed between K_{B + 2} and K_{B}, and K_{B + 1} and K_{B}. In addition, the channel maps are recorded at a high enough temporal resolution that the channels move at very close to the same velocity during closelyspaced time steps. These two properties permit us to cancel out the component of observed change that is due to the noise in the channel maps by subtracting the two differences, (K_{B + 2} − K_{B}) and (K_{B + 1} − K_{B}), from each other.
Here, the subscript cc denotes the difference between the channel maps due only to actual river channel motion, and τ_{M} is a characteristic planform overlap time scale that is much greater than the time scale for which instantaneous measurements of channel motion must be made (see Sections 5, 7.3).
[18] In the planform overlap analysis we counted areas subject to both erosion and deposition. To obtain only a bank erosion rate for the experiments, we assumed that longterm lateral erosion and deposition were equal, which is consistent with the wellbehaved mean in channel widths with time (Figure 4). This allowed us to divide our double differenced result by 2 to obtain a rate of instantaneous channel motion in pixels per time step. We then converted these numbers into fractional areal change per unit time by dividing the observed pixel area reworked by the full fluvial surface area, A, and the time Δt_{B + 1,B + 2} that elapsed between the two time steps being studied. The resulting rate, ζ̇f, has units of [1/time]. Because it has no spatial units, it can be interpreted as a rate of fractional planform change over the whole fluvial surface or as an onaverage rate of lateral motion across a given crosssection.
ζ̇f=D(B+2)−D(B+1)2AΔtB+1,B+2(12)
We compiled this value for each double differenced time step pair.
4 Curve Fitting
[19] We fit curves to our observations of channel planform change to generate a minimal set of parameters that describe our measurements of these river systems and allow us to make some inferences about the relationships between them. Table 3 is a compilation of the parameters for the curve fits that are described in this section. These include loss of channel planform overlap (M), fluvial surface reworking (R), and rates of fractional planform change per unit time (ζ̇f). The curve fits are shown in Figures 5 and 6.
Table 3. Measured Channel Mobility Parameters
Experiment
Channel Planform Overlap
Fluvial Surface Reworking
Instantaneous Mobility
a_{M}
M
p_{M}
R^{2}
a_{R}
R
p_{R}
R^{2}
ζ̇f

[1/hr]



[1/hr]


[1/hr]
XES02SS
0.7534
3.3261
0.0759
0.5287
0.8916
0.6482
0.1274
0.899
0.4355
XES02SF
0.8246
5.2873
0.1119
0.5653
0.9232
0.9214
0.1299
0.9484
0.5338
XES02SR
0.7608
4.5664
0.0504
0.5858
0.9006
1.3327
0.0712
0.8969
0.4315
XES02RF
0.7611
3.7436
0.0887
0.3809
0.8746
0.9209
0.0725
0.8964
0.8389
XES02RR
0.8823
6.2162
0.1008
0.5988
0.8955
2.7496
0.079
0.8609
0.5083
DB031
–
–
–
0.3182
0.889
1.2103
0.0904
0.9446
1.224
DB032
0.8664
0.1965
0.1601
0.5799
0.9704
0.0395
0.003
0.9634
0.01085
BV1
0.8493
1.7537
0.2069
0.8294
0.9218
0.6197
0.2067
0.9708
1.264
BV2
0.988
0.6474
0.6713
0.9211
0.9918
0.5293
0.8287
0.8267
0.05992
4.1 Planform Overlap
[20] The change in overlap with the original planform over time takes the rough form of an exponential decay (Figure 5), which can be described by the equation
OΦ=(aM−pM)e−Mt+pM(13)
where the channel mobility parameter M is the decay constant. M describes how quickly the channel system reconfigures itself. The channel preference parameter, p_{M}, is the asymptote that gives the fraction of the observed area that the channel never touches. The third parameter, a_{M}, is the yintercept. It is made a free parameter here to be a correction factor due to the artificiallyhigh measured loss of overlap between the first and second time steps, as described in Section 3.2.3.
[21] The parameter p_{M} is zero when the river channel(s) move freely across the observed area, and is greater than zero when the channel(s) appear to prefer one part of the observed area more than another. In our study, this term is generally near zero but positive. It is near zero because we define the observed area to be the fluvial surface, across which the braided channels in noncohesive sediment that dominate all but the vegetated BV2 experiment freely moved. It is positive because in the delta experiments [Kim et al., 2006a, 2006b; Sheets et al., 2007; Martin, 2007], a channel always existed immediately downstream of the sediment and water point source, and in the braidingvegetation experiment [Tal and Paola, 2007, 2010] the channel system remained close to the wall along some portions of the fluvial surface for long periods of time. Even with careful experimental design, boundary conditions typically exert some influence on channel form.
4.2 Reworking of the Fluvial Surface
[22] The amount of unreworked fluvial surface decays exponentially with time (Figure 6). This exponential decay can be described in a form identical to that for loss of planform overlap
1−fR=(aR−pR)e−Rt+pR(14)
where f_{R} is the fraction of the total surface area that has been visited by the flow, not including the area initially occupied by the channel, and R is the decay constant for fluvial surface reworking (also called the reworking rate). The terms a_{R} and p_{R} are analogous to a_{M} and p_{M}, respectively, in Section 4.1. p_{M} is positive if there are areas on the fluvial surface that the channel preferentially does not visit. The term a_{R} is a necessary yintercept correction factor due to the image analysis. Sudden increases in observed channel area, for example due to changes in lighting, cause a small but sudden spike in the amount of area observed to be reworked, especially around t = 0 when the entire fluvial surface area is open to this artificial reworking.
[23] The fluvial surface reworking data (Figure 6) display less scatter than the channel planform overlap data (Figure 5). The scatter in the overlap data reflect the fact that even if a channel system “forgets” its initial channel pattern, it will at times overlap it more than at others, resulting in large spikes in the measured O_{Φ}. Because we cycle through baseline time steps, these data form a large point cloud around the asymptote. In our reworking analysis, oncereworked regions cannot be reworked again, resulting in a significantly tighter point cloud about the asymptote.
4.3 Instantaneous Channel Planform Change
[24] The simplest way to calculate channel planform change per unit time would be to take an average of the instantaneous planform change (defined in Section 3.2.3) divided by the time elapsed between each pair of observations. As a few outliers exist (caused, for example, in BV by moving the calibration dye trays and not by river dynamics), we unweighted these outliers to calculate rates of real channel motion (Table 3). These direct measurements of fractional planform change ζ̇f were then multiplied by mean fluvial surface widths to obtain rates of average channel lateral mobility across a crosssection in meters per hour (ζ̇, Table 2).
5 Linking Measurements Across Time Scales
[25] All three of our measurement methods are fundamentally linked to channel mobility. By the definition of a derivative, ζ̇f is the change in channel position as a fraction of the fluvial surface area as Δt, the time between measurements, → 0. Loss of planform overlap and fluvial surface reworking are described by exponential functions, so their derivatives at t = 0 are linear functions of the decay constants M and R, respectively. This convergence at t = 0 means that ζ̇f, M, and R, must be linearly related to one another.
[26] We investigated this relationship by comparing measured time scales for loss of planform overlap and fluvial surface reworking with characteristic time scales that use the instantaneous rate of channel position fractional change, ζ̇f. We defined an idealized channel time scale, τ_{ch,M} [modified from Jerolmack and Mohrig, 2007], to compare with measured loss of channel system overlap, and an idealized fluvial surface time scale, τ_{fs,R}, to compare with measured fluvial surface reworking.
τch,M=fw(1−(Φ̄∕2))(1−OΦ)ζ̇f(15)
τfs,R=fd̄fRζ̇f(16)
Here, Φ̄ is the timeaveraged value of Φ for the experiment. The channel time scale is that required for a channel to move monotonically in one direction a distance that would cause it to have a spatial correlation of O_{Φ}. The fluvial surface time scale likewise is that required for a channel to rework a fraction f_{R} of fd̄, the average fraction of the fluvial surface that is not initially occupied by the channel, without reoccupying any alreadyreworked area (i.e. with perfect efficiency). These time scales therefore represent the shortest amount of time it could possibly take a channel moving at a known rate to lose overlap with itself or rework the fluvial surface.
[27] We compared these characteristic time scales with measured nefolding times τ_{M} and τ_{R}, where n is the number of exponential decay times (“efolds”) measured for the channel system:
τM=nM(17)
τR=nR(18)
When substituting n efolds for the amount of lost planform overlap (1 − O_{Φ}) in Equation (15) and the fraction of reworked fluvial surface (f_{R}) in Equation (16), we obtain:
τch,M=fw(1−(Φ̄∕2))(aM−pM)(1−e−n)ζ̇f(19)
τfs,R=fd(aR−pR)(1−e−n)ζ̇f(20)
The ratios between the measured times and the characteristic time scales are:
τMτch,M=ζ̇ffwM(1−(Φ̄∕2))(aM−pM)n1−e−n(21)
τRτfs,R=ζ̇ffd̄R(aR−pR)n1−e−n(22)
The first term on the right hand side of Equation (22) combines the important quantities in determining the ratio of the measured fluvial surface reworking time to the shortest possible reworking time. Because the measured time scale (τ_{R}, Equation (18)) increases linearly with n while the efolding time scale (τ_{fs,R} Equation (20)) increases as − e^{ − n}, the ratio between the two is a function of the chosen time scale. This satisfies the expectations of the behavior of the equations in the limits: As f_{R} → p_{R}, n ∕ (1 − e^{ − n}) → ∞ , and therefore τ_{R} ∕ τ_{fs,R} → ∞ , following the asymptote in Equation (14). As f_{R} → 0, n ∕ (1 − e^{ − n}) → 1 and therefore τR∕τfs,R→ζ̇f∕(fd̄R(aR−pR)). In theory, τR∕(fd̄R(aR−pR)) should be unity, because over very short time periods, the river channel will be able only to move away from its original position and will not be able to rework previouslyreworked fluvial surface area, thereby making the instantaneous and longterm reworking rates be equal. This can be shown rigorously by solving the time derivative of Equation (14) at t = 0 while knowing that at t = 0, any channel motion (ζ̇f) can only rework previouslyunvisited parts of the fluvial surface, or more simply by taking n = 0 in Equation (20):
R=ζ̇ffd(aR−pR)(23)
The same logic applies for loss of planform overlap (Equation (19)):
M=ζ̇ffw(1−(Φ̄∕2))(aM−pM)(24)
Equations (23) and (24) allow us to remove the first terms from the right hand sides of Equations (21) and (22); this sets them equal both to each other and to the ratio of linear and exponential time scales in the second term on the right hand side:
τMτch,M=τRτfs,R=n1−e−n(25)
As a reasonable approximation of complete loss of planform overlap and fluvial surface reworking, we choose a 3efolding time, which corresponds to 95% loss of overlap and reworking.
τRτfs,Rn=3=τMτfs,Mn=3=3.16(26)
Figure 7 shows that most of the data points lie near the theoretical τ_{R} ∕ τ_{fs,R} = 3.16 line, in accordance with expectations. There are, however, two marked deviations. First, channels in the braided system, BV1, move rapidly but do not lose overlap with their original positions or rework their fluvial surface as quickly as would be predicted. This suggests that while BV1 rapidly moves bars, it maintains a more persistent and longerlived selfsimilarity. Second, the measured reworking rate in DB032 is barely any slower than the most rapid possible (Equation (16)) reworking rate. Its subcritical flow causes ripples to develop pervasively throughout the channels (Martin, 2007), roughening them and pushing the channels more rapidly across the fluvial surface (see Section 6.5). The grouping of all of the other data around 3.16 implies that it may be possible to use shortterm measurements of channel motion (ζ̇f) to predict longerterm rates of channel planform change and fluvial surface reworking.
6 Measured Channel Mobility and Fluvial System Forcings
6.1 Theory
[28] River channel lateral motion in a given crosssection (ζ̇ [m/hr]) is a combination of channel migration rate (ζ̇mig) and avulsion magnitude times frequency (ζ̇av).
ζ̇=ζ̇mig+ζ̇av(27)
[29] Because the experimental systems exhibited only depositional avulsions (i.e. those avulsions that are caused by aggradation of the bed and superelevation of the channel), we chose a channel superelevation criterion for the frequency of avulsion, following Jerolmack and Mohrig (2007) [see also Bryant et al., 1995, Mohrig et al., 2000]. This avulsion frequency is the reciprocal of the time required for the channel to aggrade (aggradation rate η̇) to a full channel depth (mean channel depth h̄). The magnitude of an avulsion is equal to the total channel width (i.e. all threads) in a crosssection; for this, we use the mean width, b̄.
ζ̇av=b̄η̇h̄H(0)(28)
H(0) is the Heaviside function, defined to be 1 for η̇≥0 and to be 0 otherwise; this is to exclude the case of an “antiavulsion” with negative channel planform change as the channel incises. When calculating η̇ on the deltas, we assumed that delta surface aggradation rates matched the respective rates of base level change (i.e. sea level rise plus subsidence). For rivers in the field, direct measurements of inchannel aggradation rates would be used instead.
[30] Channel lateral migration is the morphologic equivalent to a flux of sediment in a crossstream (y) direction. Whatever the cause of the lateral movement of sediment, we hypothesize that the lateral sediment flux (i.e. channel migration) should increase with increasing downstream sediment transport. More formally, the downstream sediment flux, q_{s,x} = q_{s} [m/s] (the x direction of transport being generally implied), should be proportional to the crossstream sediment flux q_{s,y}, which is the channel lateral migration rate ζ̇ times one minus the porosity: qs,y=ζ̇mig(1−λp). In order to be dimensionally consistent, this relationship must be linear by a constant of proportionality, Ξ. We measured downstream sediment discharge, Q_{s} [m^{3}/s], as the sediment input to the deltas and the sediment output from the braidedvegetated systems. This difference is because the deltas are depositional systems for which we use sediment input as a characteristic discharge, while sediment discharge in an inland system is best measured as the throughput, not the input. We used mean channel width, b̄ [m], and mean flow depth, h̄ [m], to calculate sediment flux as qs=Qs∕(b̄h̄):
ζ̇mig=Ξqs(1−λp)=Ξ1(1−λp)Qsb̄h̄(29)
Beard and Weyl [1973] showed experimentally that λ_{p} = 0.39 for wellsorted wetpacked sand at 120 and 500 microns (i.e. the sizes that were used in the experiments; see Table 2). Since the bed and banks were made of the same material in these experiments, we do not need to worry about the complications of grain size or material property heterogeneity. All of the material moves as bed load, also freeing us from complications resulting from multiple modes of sediment transport. As such, these experiments provide a datarich and simplified testing ground for our hypothesis that sediment flux controls channel lateral mobility.
[31] Equation (29) is a morphodynamic sediment transport relationship [e.g., Popov, 1962a, 1962b; Neill, 1971; Church, 2006] that relates channel planform change to bed material transport as observed via the evolution of the channel. Analogous relations relating sediment transport to bedform evolution are presented by McElroy and Mohrig [2009].
[32] With this morphodynamic relationship for channel lateral migration, we rewrite Equation (27) in more concrete terms:
ζ̇=Ξ1(1−λp)Qsb̄h̄+bη̇h̄H(0)(30)
Our constant of proportionality, Ξ, captures other factors that affect channel migration rates. Bank stabilization by vegetation or consolidated bank materials should inhibit channel migration and decrease Ξ [e.g., Simon and Collison, 2002; Pollen and Simon, 2005; Constantine et al., 2009; Güneralp and Rhoads, 2011]. Channel curvature, on the other hand, increases the lateral variability of channel velocities, bed shear stress, and therefore sediment transport capacity [Hooke, 1975]; this should enhance channel migration rates [e.g., Furbish, 1988; Güneralp and Rhoads, 2010] and therefore also increase Ξ. This parameterization is consistent with the work of Howard and Knutson [1984], who posited that lateral migration rates are controlled by a combination of bank erodibility and sediment discharge.
6.2 Sediment Flux Controls Channel Mobility
[33] Figures 8.A and 8.B show a positive correlation between channel mobility and sediment flux on both dimensional (Figure 8.A) and dimensionless (Figure 8.B) axes. If all of these experiments were identical, and therefore had the same value of Ξ, we predict that the relationship between their dimensional forms would be linear. While these data are from very disparate systems, they do approximate an increasing trend because the 2.5 order of magnitude range in sediment flux outweighs the one order of magnitude variability in Ξ. Independently from but simultaneously with our laboratory investigations, Dunne et al. (2010) used field observations and numerical modeling to show that increased sediment supply forces more rapid channel motion, and future laboratory, computer, and field work should help to better understand the mechanics behind these observations. Figure 8.B is a dimensionless version of this relationship between sediment transport and channel mobility that uses common input variables for experimental studies: grain size (D) and sedimenttowater ratio (Q_{s} ∕ Q). A combination of this graph and the calculated values of Ξ for each experiment (Table 2) therefore constitute a set of predictions of channel mobilities that can be tested and expanded experimentally.
[34] Figure 8.C shows that there is little relationship between channel mobility and our aggradational avulsion criterion in the experimental deltas, even with a wide range of aggradation rates. That XES02 (circles) shows no relationship between lateral mobility and aggradation rate is particularly important to this point: in this experiment, aggradation rate was controlled primarily by base level fluctuations and did not covary with sediment input (as it did for DB031 and DB032). Further, XES02 follows the trend of the other experiments in Figures 8.A and 8.B, supporting a stronger sediment flux control. This contradicts expectations of increased channel mobility with increased superelevationdriven avulsion frequency (Equation (29)) (Mohrig et al. 2000; Jerolmack and Mohrig, 2007) (Section 6.2), and is likely due to the noncohesive sediment that allows the channels to migrate much more rapidly than they move by avulsion.
6.3 Base Level Rise and Avulsion Promote Fluvial Surface Reworking
[35] While sediment flux controlled the rate at which channels moved laterally, avulsions helped to facilitate more rapid fluvial surface reworking. Aggradation, often linked to deltaic lobe deposition, lead to channel backfilling and caused channels to superelevate and avulse (Bryant et al. 1995; Mohrig et al. 2000) to unoccupied regions of the delta surfaces. This and analogous situations have been observed in field studies [e.g., Frazier, 1967; Roberts, 1998], laboratory experiments [e.g., Sheets et al., 2007; Martin et al., 2009; Powell et al., 2012; Reitz and Jerolmack, 2012], and numerical models [e.g., Overeem et al., 2005] of deltas.
[36] Figure 8.D shows fluvial surface reworking rate (R) vs. avulsion magnitude times frequency (ζ̇av). Some variability in reworking rate occurred with little or no depositional avulsion forcing. This scales with increasing sediment flux: in the absence of avulsions, sedimentfluxinduced channel mobility lead to more rapid reworking. Where avulsions did occur, they increased the rate of fluvial surface reworking by rapidly moving the channel to unoccupied portions of the fluvial surface, thereby reworking the the new channel footprint and facilitating channel migration into hithertountouched regions.
[37] XES02 experienced base level changes that we compare in Figure 9 with our channel overlap (M) and fluvial surface reworking (R) decay constants. These data show that rates of fluvial surface reworking and loss of planform overlap increase with base level rise. While rates of fluvial surface reworking are significantly higher for the base level rise phases of XES02 than they are for the base level fall phases, rates of loss of planform overlap are roughly constant throughout, as these are tied more closely to local channel migration. A “relative reworking efficiency”, calculated as a ratio of the reworking and overlap rates, shows that with increasing base level rise the reworking rate increases much faster than the rate of loss of channel planform overlap. This provides further evidence that rivers that move laterally by avulsive channel redistribution rework their fluvial surface more rapidly than systems that move laterally at the same rate (i.e. the same ζ̇f) but by lateral migration.
6.4 Vegetation
[38] Lateral migration rates for the steady state braided and vegetated channel systems were 1.56 and 0.09 meters per hour, respectively: the addition of vegetation caused a 17fold reduction in channel migration rate. Our measured migration rate for the vegetated case is lower than that reported by Tal and Paola (2010) because we smoothed the data to remove noise (Section 3.2.3), which exists on the fringes of these channels where the dye intensity is close to our threshold. In spite of this 17fold decrease in channel migration rate, Ξ decreased only by a factor of 2.6. This disparity indicates that the majority of the reduction in channel migration rate was due to a combination of deeper channels and lower sediment transport capacity—likely from the mechanical strength and additional roughness of the vegetation. The 2.6fold change in Ξ can be attributed to a combination of changes in channel geometry and flow fields as the system became singlethread, and the greater difficulty of detaching sediment from the vegetated banks. This shows that vegetation and vegetationinduced bank cohesion produce a combined influence on channel lateral mobility by changing channel geometry, channel roughness, and grain detachment potential.
[39] The fraction of the fluvial surface that was actively reworked, 1 − p_{R}, dropped from 79% to 17% as the addition of vegetation confined the flow [consistent with the observations of Tal and Paola, 2007, 2010]. Within these differentlysized areas, the fluvial surface reworking rate (R) and rate of loss of channel planform overlap (M) were approximately the same for both the vegetated and unvegetated stages of the experiment. This shows that while vegetation confined BV2 to a narrower corridor, both braided and singlethread rivers share a common bar reconfiguration time scale.
6.5 Channel Patterns and Bed Topography
[40] In BV1, Ξ is unity, meaning that the barforms moved at the same rate as the bed material. This implies that all of the sediment passing through this system interacted with the bars, corroborating the work of Pyrce and Ashmore [2003], who show that sediment path lengths in experiments are given by the poolbar spacing. This lends additional support for the oftenused morphological (or morphodynamic) method of estimating sediment transport in braided streams in the field [e.g., Church, 2006], in which the observed motion of barforms is inverted to obtain rates of bed material transport.
[41] This full sediment interchange with the bed was not observed in the experimental deltas. Here, values of Ξ are much lower, and much of the sediment passed directly through these systems to build deltaic lobes. We attribute this largely to the shorter transport distance on the deltas: these were 1–4 m long, compared with the 16 m long BV1 and BV2, and this distance is too short for flow divergence and reconvergence around barforms (2–3 m long in BV1) and significant barform migration to occur.
[42] DB032 was built with a very low surface slope to permit Fr < 1 flow. This subcritical flow over a sandy bed formed ripples with heights that were a large fraction of the flow depth and acted as “bars” (at a smaller scale than those in BV1) to steer the channel [Martin, 2007]. This combination of low downstream gradients and selfforming inchannel topographic variability and roughness caused the channel to be selfrepelling and very rapidly rework its whole fluvial surface: its 3efolding reworking time is barely longer than it would take an ideally efficient channel to wipe across the fluvial surface from one side to the other (Figure 7; see Equation (16)). Ξ for DB032 is almost four times Ξ for its sister experiment, DB031, that had supercritical flow. This represents a fourfold increase in sediment exchange with the fan surface.
[43] Increased sinuosity along a reach of BV2 correlates with more rapid planform change (Figure 10). This is consistent with research on meander bends [e.g., Hooke, 1975; Furbish, 1988, 1991; Güneralp and Rhoads, 2009, 2010; Parker et al., 2011], which shows that channel curvature sets up a situation in which sediment deposits on the inner bank and erodes from the outer bank, thereby further enhancing the sinuosity of the channel. This is not a runaway process, however, and increases in sinuosity beyond a radius of curvature to width ratio of ∼ 3 [Hickin and Nanson, 1975; Nanson and Hickin, 1983]) lead to decreased meander migration rates.
7 Discussion
7.1 Normally Distributed Channel Widths
[44] Channel widths, as obtained from wetted area ( f_{w}) in our channel maps, are normally distributed about the mean (Figure 4). These normal distributions support the concept of an equilibrium alluvial channel width [Parker, 1979; Parker et al. 2011], in which deviations from the mean width are allowed, but are unstable and brought back to equilibrium by erosion of and/or deposition on the banks due to increased wall shear stresses in narrower channels and decreased wall shear stresses in wider channels.
[45] Figure 4 shows histograms of channel width normalized to mean width. We observe the greatest standard deviations in normalized channel width for the deltas, which experienced repeating cycles of (1) channelization and lobe building and (2) sheetflow and searching for a new region to channelize. In the braidedvegetated dynamics experiment, this standard deviation dropped significantly with the addition of vegetation. This indicates that the width of the deeper singlethread channel in BV2 is more stable than the width of the braided channels, as expected, and suggests that there exists a strong coupling between point bar deposition and cut bank erosion that maintains this enhanced stability [cf. Parker et al., 2011].
7.2 Exponential Decay in Reworking
[46] The exponential decay functions used to model fluvial surface reworking imply that channel positions are independent of their previous positions. This is physically reasonable for highlyavulsive aggrading fan delta experiments, in which the fans maintain a symmetrical shape with subtle surface topography. Prograding deltas, on the other hand, have been shown to persistently reoccupy certain positions [Reitz et al. 2010, Figure 3] because topographic signatures of past channel occupation are not erased by channel burial as they are in our oftenaggradational experiments.
[47] While our exponential decay curves provide good fits to our data with few free parameters, they are not perfect and the complex feedbacks between flow, sediment transport, and landscape morphology may preclude the ability to understand any but the simplest landscapes with a single function. Near the maximum runtimes of many of the experiments, our exponential curves for fluvial surface reworking approach their asymptotes, p_{R}, and start to underpredict the amount of reworked fluvial surface (Figure 6). Although the data become sparse at long time intervals, making interpretations more tenuous, this deviation could indicate a slower process of river channel planform change that is superimposed on the exponential function. We hypothesize that the exponential functions are good indicators of fluvial surface reworking due to bar and bank migration within the river corridor and avulsions that move the location of the river corridor. The remaining unreworked areas are often high terraces, which can be reworked only when the a channel comes into contact with their edges—a much lower probability occurrence than the river channel coming into contact with one of its bars. Our longestrunning experiment, DB032, had a surface with no terrace regions due to its rippleforced rapid channel motion (Section 6.5), and almost completely reworked its surface (asymptote p_{R} = 0.3% unreworked). The exponential function for DB032 fits its data without the overpredictive discrepancy that characterizes all of our other experiments that approach their reworking asymptotes. Additional longterm experiments would help us to further characterize the longterm statistics of fluvial surface reworking.
7.3 Measurement Frequency
[48] Our measured loss of channel planform overlap with time can be viewed as a measurement of rates of channel planform change over various time scales of averaging. When measurements are spaced closely relative to a channel migration rate, there is little risk of error in interpolating between the channel positions at two different times. Over longer measurement intervals, part of a channel may leave its old position and then return, reducing the measured amount of river channel planform change. In the longest term, we see the measured loss of planform overlap reach an asymptote (Figure 5) that is often near the amount of overlap expected for two random sets of channel and unchannelized fluvial surface. This saturation in loss of planform overlap has also been noted in experiments by Sapozhnikov and FoufoulaGeorgiou [1997]. The observation that apparent rates of change decrease with measurement time scale may help to resolve the discrepancy that Hooke [1980] noticed in that shortterm field measurements showed more rapid alluvial channel migration than did measurements from historical maps spanning 135 years (1840 to 1975).
7.4 Channel Mobility and Fluvial Stratigraphy
[49] The relationship between channel mobility (as avulsions) and preserved channel strata was first developed by Leeder [1978]. Our approach to measuring channel mobility can be used to formulate a simple relationship between fluvial surface reworking rates and bulk volumes of channel and overbank deposits (i.e. the net to gross ratio for reservoir rock) preserved in the stratigraphic record:
fch=fw̄+fd̄(1−pR)1−exp−Rh̄η̇ob(31)
Here, f_{ch} is the fraction of the deposit volume that is channel deposit, f_{w} is the wetted (channel) fraction, f_{d} = (1 − f_{w}) is the dry (not channel) fraction, p_{R} is the reworking asymptote, h̄ the mean channel depth, and η̇ob is the mean overbank aggradation rate, which we approximate as sea level rise plus subsidence (a reasonable estimate for the aggradational effects of shallow flow in these experiments). This equation states that the volume of channel deposit is given by the planview fractional area of the channel, plus the fractional area reworked over the time required for overbank deposition to raise the mean surface level by one scour depth such that these deposits are protected from further erosion and reworking by the channel. Because of our spatiallydimensionless (fluvial surface normalized) approach in calculating fluvial surface reworking (R [1/hr]) and channel area (f_{w} [–]), our deposit “volumes” can apply equally to 3D deposits, crosssections, or single cores; we find a strike section to be easiest to visualize. We assume that the stratigraphy is composed of a bimodal distribution of channel and overbank deposit. Therefore
fob=1−fch(32)
where f_{ob} is the overbank deposit fraction. The endmembers of the set of Equations (31) and (32) are that no overbank deposit is produced when η̇ob→0 and/or when R → ∞ , and that one channelwidth of channel deposit remains as η̇ob→∞ and/or R → 0.
[50] We test this model with DB031 and DB032 by comparing predicted and observed fractions of channel deposit. Martin [2007] measured ∼ 60% channel deposit in DB031 and ∼ 95% channel deposit in DB032 by slicing the experimental deltas and measuring stratigraphy. Equation (31) gives 72% channel deposit in DB031 and 99% channel deposit in DB032, reproducing the observed trend. The fact that we overshoot in both cases is likely because only a portion of the measured channel area that we use to calculate reworking rates actually produces the deep scours preserved in the stratigraphic record. With additional work and refinement, and especially with the application of crosssectional and threedimensional topographic data, these principles may help to translate between channel mobilities and channel deposit fractions in fluvial stratigraphy.
8 Conclusions
[51] We create an areabased methodology to measure instantaneous rates of channel planform change, loss of channel system planform overlap, and fluvial surface reworking. We develop these metrics using a variety of laboratory experiments that in turn provide new insights into (1) the relationships between time scales of fluvial system evolution and (2) response of channel systems to external forcings. Our major contributions are that:
[52] We have developed functional relationships that connect river channel lateral mobility from shortterm measurements to longterm loss of planform overlap and fluvial surface reworking (Section 5). Fluvial surface reworking rates can be used in turn to connect surface processes to fluvial stratigraphy (Section 7.4). In alluvial systems, the present and the past are the keys to one another.
[53] Channel mobility is driven primarily by sediment flux, Bank stabilization by vegetation decreases channel mobility by decreasing sediment flux and adding bank stability, and irregularities in channel bed and planform geometry (ripples and sinuosity, respectively) increase channel mobility. Aggradationallyforced depositional avulsions correlate poorly with channel mobility, but well and positively with fluvial surface reworking rates, indicating that they act to distribute the channel system more efficiently across the whole fluvial surface. All of the sediment passing through BV1, the braided experiment, is incorporated into migrating barforms. This means that its sediment flux and barform migration rate are identical, and lends laboratory support for morphodynamic methods of measuring sediment transport rates.
[54] In addition to these, we show that:
[55] Channel widths are normally distributed with time, implying that they are stable with constant forcings on the fluvial system.
[56] Saturation in loss of planform overlap with time can explain reduced rates of observed channel mobility with longer intervals between observations.
[57] We are able to explain channel response to forcings (vegetation, sediment flux, aggradation, and planform geometry) and relationships between rates of channel planform change as measured across multiple time scales in these experimental settings. This bodes well for our ability to better understand the evolution of alluvial systems through time.
Notation
[58]
F
fluvial surface area matrix (1 = fluvial surface; 0 = regions outside the fluvial surface).
β
time step [hr].
K_{β}
channel map binary matrix at time step β (1 = “wet”; 0 = “dry”).
B
baseline time step; plotted at time t = 0.
T
transient time step; these go from each baseline time step to the end of each experiment.
t
time since baseline [hr].
D
number of pixels that changed from “wet” to “dry” or “dry” to “wet” between two channel maps.
m_{r}
number of rows in matrix.
n_{c}
number of columns in matrix.
f_{w,β}
fraction of fluvial surface that is “wet” at time step β.
f_{d,β}
fraction of fluvial surface that is “dry” at time step β.
Φ
fraction of overlap observed between channel maps of randomlydistributed “wet” and “dry” pixels.
OΦ
the amount of overlap between two channel maps, scaled to 1 for perfect overlap and 0 when overlap equals Φ (random noise).
A
fluvial surface area [pixels].
′
primes indicate reference to a baseline channel map in which the initial channel position is excluded from the analysis.
N′_{B,T}
number of unreworked pixels between time steps b and T.
f_{R}
reworked fraction of the fluvial surface.
Δ_{tβ,β}
time elapsed; two time steps (β) optionally specified [hr].
ζ˙f
rate of channel planform change as fraction of total fluvial surface area [1/hr].
M
decay constant for rate of loss of channel planform overlap (M → “mobility" ) [1/hr].
R
decay constant for rate of fluvial surface reworking; also called “reworking rate” [1/time].
a_{M}
yintercept correction factor for loss of planform overlap.
a_{R}
yintercept correction factor for fluvial surface reworking.
p_{M}
channel preference parameter: asymptote for the overlap analysis.
p_{R}
asymptote for the reworking analysis (i.e. fraction of fluvial surface never touched by channel).
n
number of efolds.
τ_{ch,M}
ideally efficient time scale for loss of channel planform overlap.
τ_{fs,R}
ideally efficient time scale for fluvial surface reworking.
τ_{M}
measured time scale for loss of channel planform overlap.
τ_{R}
measured time scale for fluvial surface reworking.
H(0)
Heaviside function.
Θ
Ratio of channel mobility to sediment flux: efficiency of sediment flux in driving channel planform change.
Fr
Froude number.
ζ˙
rate of channel planform change [m/hr].
ζ˙mig
rate of channel planform change due to lateral migration [m/hr].
ζ˙av
rate of channel planform change due to avulsion [m/hr].
b
channel width [m].
h
channel depth [m].
η˙
aggradation rate [m/hr].
λ_{p}
bed material porosity.
Q_{s}
sediment discharge [m^{3}/hr].
Q
water discharge [m^{3}/hr].
S
downvalley slope.
D_{50}
median grain size [mm].
SLR
rate of sea level rise [mm/hr].
σ
subsidence rate [mm/hr].
f_{ch}
channel deposit fraction.
f_{ob}
overbank deposit fraction.
Acknowledgments
[59] ADW thanks Matt Wolinsky, John Southard, Mariela Perignon, Peter Ashmore, and Bob Anderson for their insights. Reviews by Alex Densmore, Stephen Lancaster, and four anonymous reviewers helped us to improve the manuscript. This work was supported by the National Science Foundation (NSF) Research Experiences for Undergraduates program, the MIT UROP program through the Paul Gray UROP fund, and the Science and Technology Center program of the NSF via the National Center for EarthSurface Dynamics under Agreement Number EAR0120914.