Numerical models that predict channel evolution are an essential tool for investigating processes that occur over timescales which render field observation intractable. The current generation of morphodynamic models, however, either oversimplify the relevant physical processes or, in the case of more physically complete codes based on computational fluid dynamics (CFD), have computational overheads that severely restrict the space–time scope of their application. Here we present a new, open-source, hybrid approach that seeks to reconcile these modelling philosophies. This framework combines steady-state, two-dimensional CFD hydraulics with a rule-based sediment transport algorithm to predict particle mobility and transport paths which are used to route sediment and evolve the bed topography. Data from two contrasting natural braided rivers (Rees, New Zealand, and Feshie, United Kingdom) were used for model verification, incorporating reach-scale quantitative morphological change budgets and volumetric assessment of different braiding mechanisms. The model was able to simulate 8 of the 10 empirically observed braiding mechanisms from the parameterized bed erosion, sediment transport, and deposition. Representation of bank erosion and bar edge trimming necessitated the inclusion of a lateral channel migration algorithm. Comparisons between simulations based on steady effective discharge versus event hydrographs discretized into a series of model runs were found to only marginally increase the predicted volumetric change, with greater deposition offsetting erosion. A decadal-scale simulation indicates that accurate prediction of event-scale scour depth and subsequent deposition present a methodological challenge because the predicted pattern of deposition may never “catch up” to erosion if a simple path-length distribution is employed, thus resulting in channel over-scouring. It may thus be necessary to augment path-length distributions to preferentially deposit material in certain geomorphic units. We anticipate that the model presented here will be used as a modular framework to explore the effect of different process representations, and as a learning tool designed to reveal the relative importance of geomorphic transport processes in rivers at multiple timescales.
The dearth of morphodynamic models that resolve the bar-scale morphology of braided,
gravel-bed rivers remains a first-order weakness in fluvial geomorphology. Indeed, since
prediction is generally perceived to be the pinnacle of scientific enquiry, the
limitations associated with existing modelling frameworks that can be used to inform
management and natural hazard assessment of braided rivers restrict the contribution that
our science can make to addressing societal needs (Wilcock and Iverson, 2003). Focusing
model development on braided rivers is paramount since, in principle, if a numerical
framework can predict the evolution of multiple channels and bars during high-flow events
in these multi-thread rivers, then the same framework should be transferable to
single-thread rivers. Major progress has been made in applying contemporary measurement
technologies to quantify the morphodynamics of braided rivers at the timescale of
individual to sequences of high-flow events (10
A variety of spatially distributed morphodynamic modelling frameworks have been developed to address this problem and are reviewed in depth by Williams et al. (2016a). Current approaches have been broadly assigned into one of two philosophical categories. The first involves simplifying and abstracting physical processes using a set of rules or simplified algorithms, giving rise to so-called “reduced complexity” (RC) models, often in the form of cellular automata (CA) models (RC/CA; Murray and Paola, 1994; Coulthard et al., 2002; Thomas and Nicholas, 2002). These rule-based models offer high computational efficiency, allowing calculations over wide spatial and long temporal scales (e.g. Nicholas and Quine, 2007; Thomas et al., 2007; Ziliani et al., 2013), but often at the expense of morphological fidelity to any particular system, instead producing self-organization and generalized behaviours of a given channel type (e.g. Murray and Paola, 1994; Murray, 2003). RC/CA models are particularly well suited for investigating the generalized morphodynamic response of channels under shifting boundary conditions (Thomas et al., 2007). The alternative second subset of models is widely referred to as physics-based, and are driven by computational fluid dynamics schemes (CFD; Bates et al., 2005) typically involving two-dimensional approximations of solutions to the Navier–Stokes equations. Bed shear stress is used as a measure of the friction force imposed by flow to scale bedload transport. The direction and distance of bedload transport are calculated based upon the flow field and parameterizations are employed to represent gravity-driven sediment transport, particle settling and remobilization effects. The Exner equation (Paola and Voller, 2005) is then used to calculate bed elevation change from local sediment flux divergence. This physics-based approach to calculating morphodynamic evolution comes at the cost of significantly increased computational overheads, particularly for graded sediment. Nonetheless, physics-based models have been widely applied to simulate fluvial systems (Mosselman et al., 2000; Rinaldi et al., 2008; Kleinhans, 2010) and event-based, graded sediment simulations reproducing reach-scale morphodynamics of natural rivers are beginning to emerge (Williams et al., 2016b). Moreover, such models have been used to shed light on how the morphodynamics of braided rivers are influenced by sediment heterogeneity (Sun et al., 2015; Singh et al., 2017) and vegetation (Crosato and Saleh, 2011; Li and Millar, 2011; Nicholas et al., 2013a; Iwasaki et al., 2017). Despite this progress, mesoscale physics-based simulations require considerable computational resources and can diverge from predicting feasible morphology (Schuurman et al., 2015). Beyond this classic dichotomy, a third approach to morphodynamic simulation draws on particle-based methods from granular physics (Frey and Church, 2012). This approach is, however, computationally intensive and the absence of appropriate upscaling methods limits its application to patch-scale investigations of sediment entrainment, transport, and deposition (e.g. Escauriaza and Sotiropoulos, 2011; Nabi et al., 2013).
The lack of morphodynamic modelling approaches that are effectively optimized for use at the mesoscale presents the opportunity for an alternative framework. An attractive approach is to couple flow-routing predictions based upon the well-understood and robust shallow water wave equations (Nicholas et al., 2012; Williams et al., 2013) with a simplified, empirically derived rule set for sediment transport, which in turn is used to update the bed topography. This schema has the advantage of decoupling the fastest components of the system (the hydrodynamics) from the slowest (evolution of the bed). Furthermore, an approach to sediment transport modelling based on particle path lengths, the characteristic distance travelled by sediment particles during a flood (Davy and Lague, 2009; Furbish et al., 2016), has potential to simplify simulation of bedload transport representing the fundamental process that leads to bar formation and morphodynamics. Recent work indicates that morphology, bed structure, and texture play a key role in particle dispersion (Hassan and Bradley, 2017). Particle path-length distributions have been found to take several forms in braided gravel-bed rivers, including exponential decay, or heavy-tailed distributions, marked by a large number of mobilized particles short distances downstream, resulting from floods that do not generate sufficient shear stress for transport across the braidplain width (Pyrce and Ashmore, 2003b). During floods which are capable of transporting sediment (i.e., competent) across large areas of the braidplain, typical path-length distributions exhibit peaks which correspond to the location of likely depositional sites downstream. Kasprak et al. (2015) and Pyrce and Ashmore (2003a, b) both noted that these depositional sites were most frequently the locations of bar heads (e.g. flow diffluences); and as such, particle path-length distributions could be readily constructed using morphometric indices that reflect the characteristic confluence–diffluence spacing.
This paper presents a new hybrid morphodynamic modelling approach that employs two-dimensional CFD hydraulics resolved at the event scale and a rule set for sediment transport that leverages hydraulic predictions to determine particle path lengths along which to erode and route sediment and subsequently evolve topography. Previous research on particle path-length distributions has largely been conducted in braided gravel-bed rivers; and as such, the model described here, termed MoRPHED (Model of Riverine Physical Habitat and Ecogeomorphic Dynamics), has been developed for such environments. Characteristic particle travel distances have, however, also been documented for single-thread channels (Pyrce and Ashmore, 2003b), so it may therefore be possible to apply the underlying theory to a wide variety of channel forms. That said, the approach is only appropriate where the extremes of the path-length distribution are less than the length of the system to be modelled (i.e. for gravel- rather than sand-bed rivers).
Combining both dynamical (force-based hydraulics) and kinematic (motion-based prediction of particle transport) frameworks provides an effective compromise that incorporates the necessary physics to simulate the key driving forces and vectors of motion, but simultaneously offers a reduced complexity structure suitable for wide-area, long-term morphodynamic modelling. This paper explores the degree to which MoRPHED can (a) capture the emergent properties of natural fluvial environments; (b) maintain, but not necessarily form or produce, braided topography; and (c) exhibit sensitivity to contrasting process representations.
For transparency and ease of future development, the model presented here has been
packaged into an open-source code published at
As with many previously developed morphodynamic models, the model used here (MoRPHED v.1.1) simulates hydraulics and uses these calculations to predict bedload transport and morphological change. This section details the methods used in each of these components, along with ancillary routines such as the parameterization of model boundaries, sediment grain size, and bank erosion. Figure 1 shows a flow chart of model operation along with required and optional inputs and outputs; these components are discussed throughout this section.
MoRPHED model operation flow chart.
The model's hydraulic component is driven using the freely available open-source Delft3D
software (version 4.00.01). We employed the model in depth-averaged form (i.e.
two-dimensional) as this provided an ideal compromise between computational efficiency
and the ability to resolve hydraulics at the cellular scale of our digital elevation
models (DEMs; Lane et al.,
1999). Although Delft3D three-dimensional flow simulations have been applied in river
environments (e.g. Parsapour-Moghaddam and Rennie, 2017), the parameterization,
validation, and computational overhead associated with three-dimensional modelling
precludes their use for the development and assessment of MoRPHED (Lane et al., 1999;
Brasington and Richards, 2007). Delft3D solves the shallow-water form of the
Reynolds-averaged Navier–Stokes equations, which relate changes in momentum (left-hand
terms) in time and space to the cumulative surface and body forces acting on the fluid
(right-hand terms):
For all simulations, discharge was specified at the upstream boundary and a corresponding
water surface elevation was set to the downstream boundary, and was calculated using a
normal depth approximation based on reach-average slope and roughness. Horizontal eddy
viscosity (
Models were run with a steady upstream discharge, corresponding to the peak of a flood, and hydraulic predictions were extracted for further sediment transport calculations only once the reach had achieved a steady state (no observed change in depth, velocity, or inundation). From each hydraulic simulation, we exported (a) water depth, (b) flow velocity resolved into streamwise and lateral components, and (c) bed shear stress. Put simply, flow hydraulics were computed for the maximum discharge of a given flood, and these hydraulics were used to drive subsequent sediment mobilization and morphodynamic evolution. This approach was adopted because (a) the calculation of morphodynamics once per event allows for a greatly reduced computational overhead associated with the model, and (b) modelling at finer timescales – while allowing for the ability to capture rapid transient events such as prograding bedload sheets and bank retreat during the course of a single flood – is inherently difficult given that the most common observational data available to geomorphologists describe channel form before and after a single event (Bertoldi et al., 2010; Williams et al., 2011; Mueller et al., 2014), and sediment transport distances, or path lengths, resulting from that event (Pyrce and Ashmore, 2003a, b; Snyder et al., 2009; Kasprak et al., 2015). We acknowledge that a single-flood time step may necessarily oversimplify morphodynamic evolution resulting from floods that are of sufficient duration to produce multiple entrainment episodes for particles, or those floods that fundamentally alter the bar spacing, and thus the path-length distribution, for a reach of interest. While our motivation was to assess the validity of a purely event-based model time step, regardless of flood duration, we provide a nested experiment in which the hydrograph of a single flood event was schematized as a set of three steady discharges corresponding to the rising limb, peak, and falling limb of the event (Sect. 4.2). This schematization was used in an attempt to evaluate (a) the model's suitability for simulating floods of extended duration and (b) the need for quasi-dynamic flow predictions to capture distinct stage-specific morphodynamic processes, such as the dissection of bar-top chutes at falling stages (e.g. Wheaton et al., 2013).
The model employs a critical non-dimensional value of the bed shear stress (Shields
stress) to determine whether sediment can be entrained at a particular location. The
theory and threshold values of Shields stress for entrainment have been well studied in
gravel-bed river settings. Incipient motion for gravel occurs when the Shields stress
(
Most morphodynamic models compute bed elevation change using some form of the Exner
equation for sediment continuity (Paola and Voller, 2005). In this equation, bed
elevation (
To summarize, in lieu of a traditional continuity-based approach for estimating bed elevation change (Eq. 6), MoRPHED estimates bed sediment scour depth using an event-based approach (Eq. 7). This scour depth, in combination with the model grid (i.e. cell) size provides a volume of scoured sediment at any location within the model domain. This volume of entrained sediment is transported downstream along hydraulic flow lines and deposited according to a path-length distribution, each component of which is discussed in the subsections that follow. This routine was performed once per flood, regardless of flood duration; as our intent was to develop a purely event-scale morphodynamic model, we did not attempt to scale or correct estimates of bed sediment erosion as a function of flood duration in this study.
Readily erodible banks are a hallmark of braided rivers (Wheaton et al., 2013) yet continue to represent one of the most difficult geomorphic processes to numerically represent (Darby and Thorne, 1996; Simon et al., 2000; Rinaldi and Darby, 2007; Stecca et al., 2017). Most existing models rely on fine time steps and detailed predictions of the near-bank force balance to predict bank stability. As MoRPHED is a simplified event-scale model, here we estimate the lateral retreat distance by empirically scaling the distance of lateral retreat during a model run to near-bank shear stress and bank slope.
To begin, the model calculates the slope of all cells in the model domain; all
simulations presented herein used a cell resolution of 2 m (Rees River; Sect. 3.1) or
1 m (River Feshie; Sect. 3.2). The relatively coarser cell resolution on the Rees was
chosen for computational efficiency, particularly with regard to the hydraulic modelling
component of MoRPHED, as the Rees site encompassed a considerably larger spatial domain
than that of the Feshie. Cells that exceed a user-defined slope criterion, which was set
to 7 % by examining the average slope of cells that underwent bank retreat in field
surveys for all simulations presented herein (Fig. 2a), are then identified as candidate
cells for undergoing bank erosion. Whether bank erosion occurs by mass failure or lateral
channel migration, eroding banks are often marked by steep slopes; and as such, this
criterion was used as the first metric for computing bank sediment erosion. While below
the angle of repose for both consolidated and unconsolidated sediment, this 7 %
threshold simply served as a first approximation of those areas that may exhibit
sufficient slope to undergo bank retreat and was designed to be inclusive of areas that
might be below slope failure criteria, but may nevertheless undergo bank retreat as a
function of near-bank shear stress, discussed below. This slope delineation produces
groups of cells, which are removed from the selection if the group's area falls below a
user-specified threshold (Fig. 2b). This area cutoff was used for computational
efficiency, and was set to 30 cells (i.e. 30 m
Lateral migration algorithm schematic.
For each cell within these groups, the bed shear stress in the surrounding cells was
sampled using a
The cells that remained were those exceeding both the slope and near-bank shear stress
thresholds; and for each of these cells, the model then removed material from a number of
adjacent cells specified by Eq. (11) and shown in Fig. 2d1, which computes the lateral
extent of bank erosion (
Bank erosion and bank material transport and deposition are computed prior to bed morphodynamics (Fig. 1); and to conserve computational overhead, hydraulics are not recomputed between these steps; as a result, bed scour is not altered in areas proximal to eroding banks. Further, because candidate cells for bank erosion are identified simply on the basis of slope and shear stress, it is possible that cells away from the wet/dry boundary (i.e. the channel bed) can undergo erosion in this manner as well. We believe this approach is appropriate here as hydraulics and morphodynamics are computed only once per flood (at peak discharge); and as such, steeply sloping cells susceptible to erosion may be inundated within the channel at the event peak, but may still undergo subaerial failure on the rising or falling limb of the hydrograph. We note that Eq. (11) was used for simulations on both river systems studied here, which varied considerably in their constituent volumes of fine/cohesive sediment and vegetation extent (see Sect. 3). Thus, these values are likely applicable for a variety of rivers, but we caution that for systems with exceptional bank cohesion, whether via vegetation or fine sediment, adjustment of these parameters may be necessary. Finally, as opposed to a physically based relationship between shear stress, bank slope, and the extent of lateral erosion, Eq. (11) was derived through qualitative calibration; that is, rather than presenting a deterministic methodology for quantifying bank retreat, this approach is simply reflective of the best qualitative correspondence between modelling results and field observations of areas undergoing bank erosion.
Once entrained, bed or bank sediment is mobilized downstream along flow lines which are
delineated using velocity vectors from Delft3D. Although MoRPHED is inherently a
cell-based (i.e. raster) model, Delft3D-derived velocity vectors did not necessarily pass
through the centre of any given cell within the MoRPHED model grid. To account for this,
the nearest grid cell was computed along the Delft3D velocity vectors at downstream
intervals equal to the computational domain cell size (i.e. 1 m on the Feshie, 2 m on
the Rees; see Sect. 3). In the field, deposition of sediment consistently occurs in
diffuse patterns, such as “tear-drop” forms of lobate bars, prograding bedload sheets,
and thinly mantled overbank deposits (Ashmore, 1982; Ferguson and Werritty, 1983; Wheaton
et al., 2013). To mirror this diffuse deposition, the model distributes sediment within a
At each cell along the flowpath, the volume of sediment to deposit in the centre cell is
given by a path-length distribution (Fig. 4). In the simplest sense, this distribution
details the proportion of all eroded sediment which is deposited at a particular distance
downstream, and provides a representation of field-mapped source–sink pathways found in
braided rivers (Williams et al., 2015). These distributions have been studied by numerous
researchers and found to take several forms in braided rivers, as reviewed in Sect. 1.
MoRPHED deposits a proportion of the path-length specified volume of sediment in each
wetted cell of the
Particle transport and deposition routine. Panel
Path-length distributions used in MoRPHED modelling on
For each simulated event, the model tracks the volume of sediment passing the downstream or lateral reach boundaries. In effect, export of sediment occurs when the length of the user-specified path-length distribution exceeds the downstream or lateral boundaries of the model domain. When this occurs, the remaining volume of sediment (the amount of eroded sediment not yet deposited along the flow line) is recorded as having been exported from the reach. Sediment import is user-specified and can be (a) set equal to the volume of sediment export during the preceding event (e.g. sediment equilibrium; Grams and Schmidt, 2005), (b) specified as a percent of sediment export during the preceding event, or (c) specified via a text file detailing absolute volumetric sediment import during each event (e.g. sedigraph time series). Algorithmically, the model computes flowpaths from each wetted cell at the upstream reach boundary and distributes the total volume of imported sediment to each cell of each flowpath as specified in the user-input path-length distribution. Sediment is introduced to the reach at the end of each event (Fig. 1; i.e. following hydraulic and bed or bank morphodynamic modelling) and prior to initiation of the subsequent event, as this allows for the import of sediment once exported sediment volumes are known. Sediment leaving the lateral reach boundaries was included in the total export volume, along with sediment leaving via the downstream reach boundary.
To compare the outputs of the model with field-based surveys of channel evolution, we derived several morphometric parameters along with comparing DEMs of difference (DoDs) and contributions of individual braiding mechanisms to total geomorphic change. Each of these three verification components are discussed below.
We manually quantified the braiding index (
We differenced DEMs of initial and final field surveys and model simulations using
Geomorphic Change Detection software (GCD, version 6;
From DoDs, we extracted elevation-change distributions (ECDs, a histogram of all volumetric changes), along with deriving the net sediment imbalance during each survey and model period (the percent departure of the sediment budget from equilibrium conditions).
Using Geomorphic Change Detection (GCD) software, we mechanistically segregated both field- and model-derived DoDs by delineating the processes responsible for each area of geomorphic change (Ashmore, 1991; Wheaton et al., 2013). These processes can be separated into 4 “braiding mechanisms” described by Ashmore (1991): central bar development, lobe dissection, transverse bar conversion, and chute cutoff. To these, Wheaton et al. (2013) added an additional 6 mechanisms that are not unique to braided rivers: bank erosion, channel incision (i.e. bed erosion), overbank sheets, confluence pool scour, bar trimming, and lateral bar development. Hereafter, these 10 processes are referred to simply as “braiding mechanisms”. We mapped the areas in DoDs where each of these mechanisms occurred; areas where we could not confidently assign a mechanism of change were classified as questionable or unresolved change. While this approach is inherently subjective, the delineation of braiding mechanisms, and more broadly mechanisms of geomorphic change, has been completed for numerous study sites (Wheaton et al., 2013; Kasprak et al., 2015; Sankey et al., 2018a, b), and efforts aimed at the automated mechanistic segregation of geomorphic change have also been presented in the literature (Kasprak et al., 2017). This classification allowed us to compute and compare the volumetric contribution of each braiding mechanism to total geomorphic change in both field and model-derived DoDs.
The braided gravel-bedded Rees (Fig. 5) drains a 402 km
The weakly braided gravel-bedded Feshie (Fig. 5) is a tributary of the Spey River and
drains 231 km
Morphodynamic modelling sites. Overview maps of Rees River
Because we modelled the same reach of the Rees previously analysed by
Williams et al. (2013), we simply used the values of
Delft3D hydraulic modelling parameters.
In contrast to the Rees, no comprehensive validation and verification of Delft3D exists on the Feshie; as such, we leveraged existing surveys of wetted areas from 2003 to 2007 in concert with surveyed water depth in those years to examine the performance of Delft3D. Because field surveys were conducted at low flows to facilitate rapid measurement of braidplain topography, here we are only able to verify the results of Delft3D at these low flows. However, Delft3D has been employed and verified on braided gravel-bed rivers at the flood stage (Javernick, 2013), demonstrating that the model can accurately reproduce flood-stage hydraulic features and can be used to drive morphodynamic evolution at the event-scale. For modelling on the Feshie, we estimated discharge by downscaling the average observed flow for the relevant survey period at the nearest gauging station (Scottish Environment Protection Agency no. 8013, Feshie at Feshiebridge) located approximately 11 km downstream using an empirically derived discharge-area coefficient of 0.71 (Wheaton et al., 2013). The value of discharge taken at Feshiebridge was the average flow during each year's survey period (average of 2 weeks). We estimated the downstream water elevation using surveyed inundation extent in combination with the DEM for each year modelled. Downstream water surface elevation estimated from the spatial data were cross-checked using a reach-scale conveyance calculation (Williams et al., 2013).
Results of our validation of Delft3D on the Feshie at low flow are shown in Fig. 6. Here
we report (a) the mean of depth differences between modelled and observed values
(
River Feshie hydraulic modelling. Surveyed water surface extent in each of the five years (2003–2007) was compared to modelled inundation extent in the same years using discharge levels approximated using data from the gauge at Feshiebridge. Areas observed (but not predicted) to be inundated shown in blue, areas predicted (but not observed) to be inundated shown in green. Areas which were correctly predicted as being inundated shown in red. Base layer is a hill-shaded 1 m DEM.
We modelled a single flood event on the Rees River that occurred between
8–16 December 2009 as the result of heavy rainfall in the upstream watershed (Fig. 7).
Peak flows reached a maximum instantaneous discharge of 259 m
Rees River event morphodynamic modelling results. Continuous hydrograph and
modelled discharge shown in
Results of morphodynamic modelling on the Rees for this event are shown in Fig. 7. This
event-scale model was run using a steady-state discharge of 259 m
Rees River event modelling: geomorphic change detection
results; results of discretized modelling shown in
The choice of model time step is one of the more important considerations in
morphodynamic modelling, whereby the user must strike the optimal balance between a model
time step fine enough to preserve computational stability and which is coarse enough to
allow computation over meaningful timescales (Brasington et al., 2007). To investigate
these questions using MoRPHED, we discretized the hydrograph used in event-scale
modelling on the Rees so as to model three discrete points over the course of the
modelled flood (Fig. 8). These discharges were 75, 259, and 75 m
Rees River discretized hydrograph case study. DoD from
MoRPHED modelling shown in
We modelled morphodynamics during the 1-year period from July 2003 to July 2004 on the
Feshie. The estimated hydrograph for the study reach, based on the empirical downscaling
coefficient applied to the Feshiebridge gauge (Sect. 3.2) during this period, is shown in
Fig. 9a. This survey epoch contained 16 flood peaks above competence (20 m
River Feshie annual morphodynamic modelling results. Refer to the caption of Fig. 7 for details.
Wheaton et al. (2013) noted that the most volumetrically significant braiding mechanisms
during the 2003–2004 period were chute cutoff (29 %), bank erosion (16 %), and
channel incision (15 %). Overall, geomorphic change was primarily confined to a main
channel bisecting the braidplain longitudinally, with one anabranch on the left side of
the braidplain undergoing bank erosion and central bar development. Braiding index
(
River Feshie annual modelling, geomorphic change detection results.
Transport and deposition of eroded bed or bank sediment in MoRPHED is a function of the
path-length distribution used in the model. While field or laboratory data describing
particle transport distances can be used to produce a path-length distribution,
parameterizing such a distribution for sites where tracer data are not available is not
straightforward. Here we employed average confluence–diffluence spacing to estimate the
peak of the distribution (Fig. 3; Pyrce and Ashmore, 2003a, b; Kasprak et al., 2015). To
further understand the effect of contrasting path-length-distribution shapes and
distances, we used MoRPHED to model the annual hydrograph on the River Feshie in a manner
identical to Sect. 4.3, except we varied the characteristics of the specified path-length
distribution (Fig. 10). We employed a compressed Gaussian distribution (Fig. 10a), a
flattened Gaussian distribution (Fig. 10b), and an exponential-decay-type distribution
(Fig. 10c; Pyrce and Ashmore, 2003b); each of these distributions had a total length
identical to our original distribution on the Feshie (Fig. 10a). We also modelled two
shortened distributions (length
River Feshie variable path length case study. Path-length distributions used are
compressed Gaussian
Overall, the geomorphic changes predicted by modelling differing path-length
distributions were similar at the reach scale, and areas of scour and deposition
generally aligned between all five of the simulations (Fig. 10a–e).
Analysis of the ECD produced using each
path-length distribution revealed that while the compressed and stretched Gaussian
distributions were marked by more laterally extensive deposition at higher magnitudes
(e.g.
We modelled morphodynamics during the 10-year period between July 2003 and June 2013
along the River Feshie. The estimated hydrograph at Glen Feshie during this period is
shown in Fig. 11a; and using an equilibrium sediment budget, we modelled all peaks above
20 m
Differencing DEMs from survey data at the beginning and end of the analysis period
reveals elevation changes ranging from
Results of morphodynamic modelling from 2003 to 2014 are shown in Fig. 11e. Total model
runtime for the 185-flood series was approximately 72 h. Geomorphic change ranged from
River Feshie decadal morphodynamic modelling results. Refer to the caption of
Fig. 7 for details. Note variable axes limits in
River Feshie decadal modelling, geomorphic change detection results.
We developed a morphodynamic model that computes sediment transport according to user-specified path-length distributions, and subsequently employed this model to predict channel evolution at two braided river reaches across timescales ranging from a single event to a decade. We observed that the model reproduced many of the geomorphic changes observed in the field, although the magnitude and mechanisms of those changes often diverged from observations using field data. At the same time, the modular design of the modelling framework may hold promise for exploration of braided channel evolution, and also raises questions regarding the way processes are algorithmically represented and the model's sensitivity to those process representations.
Braided rivers undergo geomorphic change as a result of numerous morphodynamic processes or braiding mechanisms (Ashmore, 1991; Wheaton et al., 2013; Kasprak et al., 2015). Given its highly simplified nature, the degree to which these braiding mechanisms must be explicitly represented as algorithms in morphodynamic modelling deserves exploration. Of the 10 braiding mechanisms discussed in Sect. 2.7.3, the model produced 8 simply as a result of the bed erosion, transport, and deposition functions included in the model. Only bank erosion and bar edge trimming required the inclusion of a lateral channel migration component. As the surfaces undergoing bank erosion and bar edge trimming are not necessarily submerged during a flood, these processes cannot be simply captured by an excess shear stress scour approach (Eq. 7; Sect. 2.2).
The geomorphic changes that the model most commonly produce are those that result from focused scour and longitudinally continuous deposition, given the nature of the scour and deposition functions used in the model. In particular, channel incision, bar edge trimming, bank erosion, and lateral or central bar development are common processes produced by the model (Figs. 7, 9, 11). Additionally, given the single-peaked Gaussian distributions used herein, those braiding mechanisms which involve deposition immediately downstream of an erosional source are difficult to reproduce. For example, Wheaton et al. (2013) demonstrated the importance of chute cutoff as a braiding mechanism on the Feshie, noting that chute development across point bars not only manifested as erosion, but that the scoured material was often deposited immediately downstream of the chute. Similarly, scoured bank material (e.g. mass failures) may often be deposited at the bank toe rather than transported downstream. In both cases, the model makes no differentiation in transporting the eroded sediment, mobilizing the material according to the user-specified path-length distribution; as such, proximal couplets of erosion and deposition are difficult to reproduce in the model. Finally, we note that chute cutoff in the model always occurred in locations of pre-existing chutes across point bars. Because chute cutoff often occurs at the falling stage of floods, when braidplain-inundating flows are first being confined into anabranches, our model may not properly reproduce chute cutoff as a result of only computing peak flood hydraulics, and averaging shear stress across a range of model cells. Although headward erosion of these pre-existing chutes typically occurred in the model, thus increasing their extent, we did not observe any instances where chute cutoff was initiated in the model without an existing chute or channel head being present on a bar surface. As such, chute cutoff may represent a braiding mechanism that must be explicitly included in the model's code in order to be properly represented in the future.
Finally, it is worth noting that we did not explicitly include avulsion as 1 of the 10 braiding mechanisms examined here for consistency with the mechanisms analysed by Ashmore (1991) and Wheaton et al. (2013), and also because avulsion, by nature, arises from a combination of the examined braiding mechanisms; that being said, avulsion is certainly essential to the development and maintenance of braided planform rivers (Ferguson, 1993). While the model is algorithmically capable of producing avulsion through a combination of lateral bank retreat and sediment scour or deposition, these did not occur at the same frequency as was seen in the field; this is particularly evident in the decadal-scale simulation's incised and simplified channel planform, along with the reduction in channel nodes as compared to field observations (Fig. 11). At these extended timescales, the model appears to exhibit a positive feedback loop where bed sediment scour drives increased flow capture and bed shear stress, resulting in further downcutting of the channel. Adjustments in bed sediment scour, sediment deposition (in particular, more focused deposition of material to offset erosion and drive flow deflection), along with increased rates of lateral bank erosion all have the potential to drive more frequent avulsion, prevent over-scouring of the bed, and provide greater fidelity to field observations of channel morphodynamics when modelling at decadal and longer timescales.
In Sect. 4.2.1, we modelled a single event on the Rees as three discrete discharges on
the hydrograph (Fig. 8). Because MoRPHED under-predicted the volume of change,
particularly due to the absence of low-magnitude scour, during the single-event
simulation (Fig. 7), we sought to understand whether discretizing the hydrograph would
allow for an improved prediction of overall volumetric change, and low-magnitude
erosional change in particular. Overall, discretizing the hydrograph into three modelling
time steps only marginally increased predictions of volumetric change across the study
reach: total volumetric change in the discretized run (Fig. 8) was 39 771 m
Discretizing the hydrograph also increased the amount of low-magnitude scour predicted by the model (Fig. 8b), more accurately reflecting the field-derived ECD (Fig. 7b). It is likely that this is the result of the 0.1 m elevation threshold used in our change detection (Sect. 2.7.2), whereby additive changes due to erosion largely did not exceed 0.1 m depth after a single flood event, but did exceed this threshold when three discrete hydrograph points were measured. Whereas erosional processes that lead to high-magnitude scour, such as bank erosion, dominated the ECD in the single-event model run, processes such as channel incision and lobe dissection were more prevalent in the discretized hydrograph model run. Additionally, several areas of high-magnitude bank and bar trimming were largely offset by deposition of imported or scoured material during the discretized hydrograph run, thereby decreasing the overall magnitude of scour in those areas (Fig. 8b).
We modelled annual-scale morphodynamics on the Feshie using five different path-length distributions (Sect. 4.3.1). There is overall similarity between the DEMs produced by the model (and hence the DoDs shown in Fig. 10) using these distributions, yet subtle differences may reflect the distinct nature of the spatial arrangement of erosion and deposition. Overall, the similarity between the modelled distributions may also be the result of the smoothing algorithms used in the model to ensure computationally stable output surfaces, such as the along-flow averaging of shear stress and neighbourhood windows used for deposition (Sect. 2.4), both of which may act to reduce the variability introduced by the choice of a particular path-length distribution.
In fluvial settings, erosional processes typically operate over small spatial scales
(e.g. bank erosion, bar trimming, pool scour), and the magnitude of scour in these
focused areas is typically higher than diffuse, broad-scale depositional processes such
as overbank sheets or accretion of mid-channel or lateral bar material (Wheaton et al.,
2013). As such, we suggest that the overall similarities, as well as the differences,
between our model's outputs using these contrasting path-length distributions reflects
the ability of deposition to counterbalance elevation changes due to scour of material,
and the fact that the diffuse nature of the path-length distributions used here make this
counterbalancing difficult. For example, the compressed and stretched Gaussian
distributions (Fig. 10a, b) are both marked by broad areas of erosion and deposition
typically falling between
While a full benchmarking comparison of our approach with other CA or CFD schemes is
beyond the scope of this investigation, which solely seeks to demonstrate a novel hybrid
approach to morphodynamic modelling, we can nevertheless draw general conclusions
regarding the performance of this approach as contrasted with existing modelling efforts.
We note that Williams et al. (2016a, b) performed both CA hydraulic modelling using an
approach similar to Murray and Paola (1994), and 2-D CFD morphodynamic modelling (using
Delft3D) on the same reach of the Rees River that was investigated using MoRPHED. At both
low and high flows (7.3 and 54.7 m
With regard to morphodynamics, we can compare the general results of Williams et
al. (2016b), who simulated the event-scale evolution of the same Rees River study reach
used here. During a flood with a peak discharge of 227 m
In our modelling approach, the parameterization of time as equal to a single event, or several points within one event, in the case of the hydrograph discretization case study (Sect. 4.2.1), stands in contrast to traditional CFD approaches, where time steps are represented by common units (i.e. seconds, minutes, hours, etc.). We recognize that this approach requires several simplifications, and perhaps chief among these is the notion that event-scale erosion is largely independent of event duration (see Sect. 2.2). For example, using the approach described here, a flood of several days duration would result in much less erosion than several short floods of the same magnitude, separated by periods of low flow, as the former would be modelled as a single event and the latter as multiple discrete events. We made this simplification for two main reasons. First, laboratory and field research into particle travel distances during floods has developed robust relationships between channel morphology and event-scale deposition patterns (Kasprak et al., 2015; Pyrce and Ashmore, 2003a, b) and similarly computing event-scale erosion magnitudes, while inherently uncertain, across an area of interest provides a computationally efficient way to utilize our growing knowledge of event-scale deposition dynamics. Second, in lieu of continuous intra-flood topographic surveys and/or intra-flood sediment monitoring at many points within a river, it is much more commonplace for geomorphologists to instead rely on information collected prior and subsequent to a competent flow. In effect, the modelling technique developed here operates on the same temporal scale as geomorphologists often work at: before and after a geomorphically effective flow. If such an approach is to prove useful for predicting fluvial morphodynamics, improvements in our ability to forecast event-scale erosion will be required, and this is an area deserving of further research. Similarly, this approach would benefit from an improved understanding of the timescales at which event-scale scour may be largely independent of event duration, versus those longer-duration events where scour is intrinsically linked to event duration. Certainly, for those floods of sufficient duration and/or magnitude to fundamentally alter bar spacing and thus invalidate the initial path-length relationship used for modelling, adjustment of the modelling approach, specifically the user-defined path-length distribution, is required at intra-flood timescales.
Models in the Earth sciences are necessarily imperfect (Oreskes et al., 1994), and the highly simplified nature of MoRPHED, combined with the highly dynamic and non-linear nature of braided river morphodynamics (Ashmore, 1991; Bristow and Best, 1993), implies that our model will necessarily fail to achieve perfect replication of field-observed geomorphic dynamics. However, even imperfect models can provide meaningful insight into the processes behind morphologic evolution of fluvial systems (Paola et al., 2009; Paola and Voller, 2005). MoRPHED is designed to facilitate experimentation, particularly with regard to process inclusion or the particular aspects of process representation of bed and bank erosion, transport, deposition, and import dynamics (Fig. 1). For example, in Sect. 4.3.1, we explored the implications of altering the path-length distributions for bed and bank sediment transport or deposition, along with seeking to understand the advantages and drawbacks of discretizing hydrographs during model runs. The notion of morphodynamic modelling that employs sediment transport routines based on particle path-length distributions is in its infancy, and we have built the model as an exploratory tool that can be used to investigate the utility of this approach toward predictive modelling of braided river evolution. Several components of the model may be employed to investigate longstanding questions in our understanding of braided river dynamics, starting with the path-length approach itself. Recent field-based research on braided rivers has confirmed the coupled nature of sediment sources and sinks and the influence of sediment pathways on braiding maintenance (Williams et al., 2015). While it has long been hypothesized, and field and laboratory data have often confirmed, that mobilized particles in braided rivers are preferentially deposited in association with regularly occurring channel bars (e.g. Pyrce and Ashmore, 2003a, b; Kasprak et al., 2015), the form of the path-length distribution, and its relationship to geomorphic unit spacing, is deserving of further study across braided systems. As such, the choice of path-length distribution and subsequent comparison of model results with field observations may provide insight into the applicability of path-length distributions on a system-by-system basis (Hassan et al., 2013).
MoRPHED may also be used to investigate the utility of event-scale monitoring. Existing morphodynamic models typically employ a sediment continuity approach (e.g. Exner; Eq. 6), operating at very fine temporal scales, typically seconds to minutes. This approach produces results consistent with field observations (Bates et al., 2005), but comes at the expense of computational overhead, thereby restricting the timescales that can be modelled (Brasington et al., 2007). Because the time step of the model is, by default, a single event, computational resources are conserved, allowing for extended simulations at annual and decadal timescales. However, it is unclear whether processes that occur over the course of a competent flow (e.g. avulsion, bank failures) can be adequately captured using an event-scale modelling approach. In lieu of a continuity-based approach for computing morphodynamic change, here we employed a simplified event-scale estimation of bed scour depth as a function of flow hydraulics and bed sediment characteristics (Eq. 7, Sect. 2.2). At present, it is unclear whether this approach is valid across a wide range of river styles and/or independent of event duration; one potential way forward in model calibration may be to scale Eq. (7) such that field-observed changes, particularly those seen in ECDs at the event scale (Fig. 7), are matched as closely as possible, and then proceeding with modelling over longer timescales. Such a model calibration approach was not attempted here, but may improve event-scale modelling fidelity in future work. Similarly, the degree to which a hydrograph may need to be discretized and its constituent parts modelled (Sect. 4.2.1) to capture stage-dependent processes such as the development of chute cutoffs or bar edge trimming (Wheaton et al., 2013) is deserving of further investigation.
Finally, we note that the sequencing of individual floods may have implications for the geomorphic evolution of modelled reaches using this approach. For example, a large flood event may result in widespread scour of the reach and, under equilibrium sediment import parameters, may thus drive a large amount of bedload import and deposition at the upstream end of the reach prior to the next event being modelled. The choice of path-length distribution used for imported sediment may further affect the degree to which large amounts of deposition could potentially occur at the upstream model boundary, and the implications of these choices on model fidelity, although not explicitly evaluated here, deserve further study. Finally, although periodic sediment-supplying floods have been surmised to maintain the braided planform in lieu of continuous sediment supply, and thus may allow braided rivers to persist under conditions of sediment deficit at annual scales (Wheaton et al., 2015), the influence of flood sequencing on reach-scale morphodynamics has received relatively little attention in the literature and is deserving of further work in the field as well as via numerical modelling.
Our exploratory research into decadal-scale modelling on the River Feshie (Sect. 4.4) indicates that both the accurate prediction of event-scale scour depth and subsequent deposition location present methodological hurdles in the development of valid morphodynamic models that operate at the timescale of competent flows. In our model, as in the field, erosion occurs in discrete, focused areas of high-magnitude scour (e.g. pool scour, bar trimming, bank retreat; Ashmore, 1991; Bristow and Best, 2003; Wheaton et al., 2013). Deposition occurs thinly over more broad spatial areas (e.g. bedload sheets, overbank sheets, bar formation), a result of the path-length distributions employed here and the smoothing required to avoid the generation of rough topography that would lead to hydraulic instability. One result of the differences in the nature of erosion and deposition is that deposition may never “catch up” to erosion if a simple path-length distribution is employed, thus resulting in over-scouring of channels (Fig. 11). In practice, this emergent behaviour can be seen in the large-scale downcutting seen in the central anabranch on the Feshie over the decadal run from 2003 to 2013 (Fig. 11, Sect. 4.4). It is possible that this artefact may simply be the result of the differing algorithmic nature of erosional processes versus depositional ones as represented in the model (e.g. discrete versus dispersed). On the other hand, it is well known that erosion and deposition carry divergent spatial signatures in the field (Wheaton et al., 2013), and it is also possible that the over-scouring of existing channels may arise from a lack of bed sediment armouring in the model; improper representation of bank erosion magnitudes, as discussed below; over-dispersion of deposited sediment as described by path-length distributions; or a combination thereof. As opposed to scour, the morphodynamic signature of deposition generally mirrors that seen in the field (see ECDs in Figs. 7, 9, 11), with a large contribution of total change coming from areas of shallow deposition. However, in future event-scale morphodynamic models, it may be necessary to augment path-length distributions so as to preferentially deposit material in certain geomorphic units (e.g. confluence pools, deep channels adjacent to banks) in order to develop lateral flow, bank material removal, and channel migration or avulsion (Ashmore, 1991).
Another process that is difficult to capture algorithmically is the lateral retreat of banks. Highly erodible banks are a hallmark of braided rivers and lead to the development of central bars and multiple anabranches (Ashmore, 1991). However, the Cartesian grid employed in the model, along with the event-scale time step of the model, makes the generation of smooth bank features difficult. While approaches are available that compute bank stability based on a factor-of-safety approach that balances downslope gravitational forces with the ability of bank material to provide cohesive resistance to failure (Darby and Thorne, 1996; Simon et al., 2000; Rinaldi and Darby, 2007), parameterization of these models, especially at the reach scale, is quite difficult. The simplified approach employed in the model averages the slope of bank cells and the near-bank shear stress of the flow to predict bank retreat distances (Sect. 2.3). The threshold slope and shear stress, and their effect on lateral retreat distance, have been empirically adjusted to emulate field-observed bank dynamics. We have observed that the simple treatment of lateral erosion in the model produces bank erosion and bar edge trimming. However, whether this simplified approach will provide computational stability over longer-term (e.g. centennial) simulations, when bank retreat is only computed once per flood, is unknown. Additionally, further investigation is needed to determine whether the inclusion of bank toe deposition, as opposed to immediate downstream transport according to a user-specified path-length distribution, is necessary in the model, along with whether bank material should be transported and deposited according to the same path-length distribution as was used for bed material.
The morphodynamic model developed here simulates braided river evolution at a variety of
timescales via a path-length-based approach, and here we applied the model to two braided
river reaches at the event, annual, and decadal scales. The premise of MoRPHED is that
sediment transport can be approximated using a steady flow set to the event peak
discharge and path-length distributions (Pyrce and Ashmore, 2003a, b; Kasprak et al.,
2015), which, when coupled with two-dimensional hydraulic simulations at peak flood
discharge, results in decreased computational overhead, thus enabling longer simulations
at improved spatial resolution. We observed overall correspondence between model outputs
and field observations when comparing planform changes, geomorphic change described by
elevation-change distributions (ECDs), and morphometric indices such as sinuosity,
braiding index, and channel node counts (Sect. 2.7.1). At the same time, divergence
between field and model datasets was evident, suggesting that a purely path-length-based
approach may oversimplify the highly dynamic nature of braided systems (Bristow and Best,
1993). Specifically, at event scales on the Rees River, New Zealand, we observed
under-predictions of areal and volumetric changes (total areal extent of change in the
field was 287 184 m
While we did observe reproduction of all field-observed braiding mechanisms (Wheaton et al., 2013), the relative contribution of these mechanisms often varied from values seen in the field. In contrast to all other braiding mechanisms, neither bank erosion nor bar edge trimming emerged simply as a result of bed scour and deposition, and instead needed to be explicitly parameterized in the model. In addition, chute cutoff only occurred at areas where pre-existing chutes or channel heads were observed, and did not appear to emerge across previously flat bar tops. While this model represents a first step in event-scale path-length-based morphodynamic modelling, further testing is needed to evaluate the feasibility of the approach for longer-term modelling runs and/or for the determination of whether process representation that accounts for inter-flood geomorphic change, such as avulsion or bank mass failure (Leddy et al., 1993; Ashworth et al., 1994), will require explicit parameterization. We have designed the model to be a modular framework for exploring the effect of various process representations, and as a learning tool designed to reveal the relative importance of geomorphic transport processes in braided river dynamics at multiple timescales.
All model code used in this
manuscript can be found at
All authors contributed to model conceptualization and formulation, collection and analysis of field data, and manuscript preparation.
The authors declare that they have no conflict of interest.
This research was supported by a grant from the National Science Foundation (no. 1147942). We thank Philip Bailey (North Arrow Research) for extensive assistance with model code and algorithm development. Field work on the Feshie was completed with the generous permission of Thomas MacDowell and the Glenfeshie Estate, with the assistance of Niall Lehane (Queen Mary University of London), Mark Smith (University of Leeds), Julian Leyland (Southampton University), and Damiá Vericat (Forest Technology Institute of Catalonia). Rees River field data were acquired during the Natural Environment Research Council project (NE/G005427/1). Our modelling efforts benefited from substantial discussion with Sara Bangen, Nate Hough-Snee, Wally MacFarlane, Eric Wall, and Peter Wilcock (Utah State University), along with Rebecca Hodge (Durham University) and David Sear (Southampton University). The comments of Murray Hicks and two anonymous reviewers greatly improved the quality of this manuscript, and we thank them for their contribution. Edited by: Jens Turowski Reviewed by: Murray Hicks and two anonymous referees