Meandering channel dynamics in highly cohesive sediment on an intertidal mud flat in the Westerschelde estuary, the Netherlands

Geomorphology 105 (2009) 261–276

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Meandering channel dynamics in highly cohesive sediment on an intertidal mud flat in the Westerschelde estuary, the Netherlands Maarten G. Kleinhans ⁎, Filip Schuurman, Wiecher Bakx, Henk Markies Faculty of Geosciences, Department of Physical Geography, Utrecht University, PO-Box 80115, 3508 TC Utrecht, the Netherlands

a r t i c l e

i n f o

Article history: Received 5 May 2008 Received in revised form 2 October 2008 Accepted 3 October 2008 Available online 25 October 2008 Keywords: Meandering Cohesion Erosive step Bank erosion Separated bend flow

a b s t r a c t Small meandering channels of about 1 m wide on an intertidal mudflat in the Westerschelde estuary the Netherlands) were studied with the aim to improve understanding of the effect of highly cohesive bed and bank sediment on channel inception and meander geometry and dynamics. The study is supported by experiments and modelling. The estuarine meandering channels are less dynamical than alluvial meandering rivers, and the dynamics are more localised. Moreover, the high thresholds for bed sediment erosion and for bank failure lead to two processes, uncommon in larger rivers, that cause most of the morphological change. First, the beds of the channels are eroded by backward migrating steps under hydraulic jumps, while the remainder of the bed surface along the channel is hardly eroded. Second, channel banks erode i) where eroding steps locally cause undercutting of otherwise stable channel banks and ii) in very sharp bends where the flow separates from the inner-bend channel boundary and impinges directly on the bank on the opposite side of the channel. Further morphological change is probably induced by rainfall splash erosion and by storm waves that weaken the mud, and by large mud fluxes from the estuary. The steps were successfully reproduced in laboratory flume experiments. An existing model for step migration predicted celerities consistent with field and laboratory observations and demonstrated a strong dependence on the threshold for erosion. Bank stability models confirm that banks and steps only fail when undercut and weakened by waves, rain or excess pore pressure in agreement with observations. The effects of a high threshold for bank erosion was implemented in an existing meander simulation model that reproduced the observed locations of bank erosion somewhat better than without the threshold, but flow separation and its effect on meander bends remains poorly understood. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Unvegetated intertidal mudflats and vegetated tidal salt marshes commonly have small meandering channels (e.g., Bridges and Leeder, 1976; Gabet, 1998; Fagherazzi and Furbish, 2001; Temmerman et al., 2003b). Sediment on mudflats is generally very strong compared to the bank and bed sediment of terrestrial rivers even where the banks of the latter consist of cohesive sediment and are vegetated. Friedkin (1945) and Ferguson (1987) argued that the width–depth ratio and therefore channel pattern strongly depend on the strength of the banks. It is therefore insightful to compare meandering channels on mudflats to terrestrial rivers with weaker banks and noncohesive bed sediments. Various combinations of factors and processes have been put forward to explain the patterns and dynamics of channels on intertidal mudflats. The most important, detailed below, are i) the high strength and high threshold for erosion of the sediment, more so when salt marsh vegetation is abundant; ii) the shape and size of the drainage ⁎ Corresponding author. E-mail address: [email protected] (M.G. Kleinhans). URL: http://www.geog.uu.nl/fg/mkleinhans (M.G. Kleinhans). 0169-555X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2008.10.005

area; and iii) the action of rain or waves on the intertidal flat between the channels. The strength of intertidal mud depends on the water content and compaction of the mud and on the presence of salt marsh vegetation (van Eerdt, 1985; Gabet, 1998; Winterwerp and van Kesteren, 2004, chapters 2 and 8). The critical threshold for erosion and deposition determines the width and depth of the channels as demonstrated in a model for the cross-sectional shape of salt marsh channels (Fagherazzi and Furbish, 2001). The critical shear stress for erosion of mud from a channel boundary increases with depth because the mud consolidated under its own weight during sedimentation. Since the discharge increases in the downstream direction of tidal channel networks, both width and depth are expected to increase as well. However, the critical shear stress also increases with depth so that erosion increasingly takes place at the banks in downstream direction. Hence the width– depth ratio of tidal channels increases as discharge increases in downstream direction. Fagherazzi et al. (2004) demonstrated the effect of bank material strength under bidirectional bend flow on the shape of the meanders in the San Francisco Bay. They found that slump blocks correlated well with locations of the highest velocities near the banks predicted by a meander simulation model applied to

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both flood and ebb flow. Typical migration rates of bends were of the order of 1 × 10− 2 m/y, which, given widths of meters, is slow compared to terrestrial meandering rivers (see data of Van de Wiel in Kummu et al., 2008, Fig. 11). Thus, the high cohesion of the sediment and the salt marsh vegetation strengthen the banks, which renders the meander dynamics very slow. The large-scale shape and size of the tidal flat determine the flow capacity that causes morphological change in the channels, and are therefore important boundary conditions (the channels barely affect the large-scale morphology). Mudflat cross-shore profile shape and overall gradient are influenced by the hydrodynamics (tides, longshore current) and mud supply of the surrounding estuary, as well as by exposure of the site to waves (Roberts et al., 2000). Hence, the overall morphology of the flat is the result of the tidal water level

variation as well as the bidirectional flow properties. However, the current may be focussed through the channels when the estuarine stage is at the level of the mudflat. For very low-gradient flats the flow surges toward land (flood current) when the flat is just inundated and surges toward the estuary (ebb current) when the flat just emerges. Most of the morphological change of channels will take place during these surges only. The surges are delayed as time is needed for the flow to travel between shallow onshore areas and the mouth in either direction. The ebb surge is commonly stronger than the flood surge as the outflow of the upstream drainage area will be concentrated in the channels whereas the inflow takes place through the channels and over the entire mudflat (Fagherazzi et al., 2008). Splash erosion is the process where individual raindrops break down the cohesive structure of the (consolidated) clay during low

Fig. 1. Map of the study location. North is to the top. The top map is 11 km wide. (Produced by Geomedia based on topographical map and Google Earth accessed September 2008).

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tide, resulting in a more easily erodible surface layer (Mwamba and Torres, 2002; Pilditch et al., 2008; Tolhurst et al., 2008). Oscillatory shear stress exerted by waves also has a weakening effect. The water content of this surface layer may be so large that it becomes very weak mud or even liquefied mud, which is easily eroded as it has critical shear stresses and strengths orders of magnitude lower than selfconsolidated mud (Winterwerp and van Kesteren, 2004). Moreover, settling of mud becomes hindered for concentrations larger than a few tens of grams per litre (Winterwerp and van Kesteren, 2004). In the absence of currents, suspended and mud liquefied by waves is expected to be diffused by orbital motion and flow into the channels where it is transported in the following ebb surge. The weakening by waves of mud in the channel banks could destabilise and erode the bank slopes. We hypothesise that the seemingly stable meandering channels on mudflats are evolving very slowly because the thresholds for sediment displacement are very high compared to the capacity of the flow, while the duration of flows that may cause morphological change is limited to the short-duration surges that occur when tidal flats are just inundated or emptied. As such these channels may be on the very slow end of a continuum of river channels with varying stream power and bank strengths. Other relevant differences with terrestrial meandering rivers are the cohesive sediment in the bed, the increasing downstream gradient of the channel and the potentially reversing (tidal) current. The aim of this paper is to describe and explain the dynamics of meandering channels on an unvegetated intertidal mudflat. We will demonstrate that the high cohesion and critical shear stress of the

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intertidal flat sediment lead to localised dynamics with two channel processes that hitherto have only been identified in different geomorphic settings: erosive backward migrating steps that repeatedly excavate (incipient) channels, and separated flow in sharp bends which causes localised bank erosion and deposition that further sharpens the bends. Purely erosive, backward migrating steps under hydraulic jumps have been observed in various settings, including rills (e.g., Stefanovic and Bryan, 2007), streams with cohesive beds (Simon and Thomas, 2002) and as headcuts on mountain slopes in general (e.g., Parker and Izumi, 2000). They form in nonuniform subcritical flow with a downstream increasing Froude number, such as in the extreme case of a waterfall. The erosion then increases strongly downstream and in extreme cases a steep headcut undercut by a plunge pool may form. In our case, the steps initiate at the downstream edge of the mudflat in the falling tide. The steps excavate the bed and undercut the banks as they migrate upstream along the channels. The model of Izumi and Parker (2000) predicts how fast and how far the step migrate upstream depending on the Froude number far upstream of the step and on the threshold for erosion. In sharp channel bends the flow separates from the bank which leads to a bank erosion and deposition pattern that differs from gentler meander bends (Bridges and Leeder, 1976; Leeder and Bridges, 1975; Andrle, 1994; Hodskinson and Ferguson, 1998; Ferguson et al., 2003). Complex three-dimensional vortices form i) near the innerbend bank just upstream of the bend apex and ii) near the outer-bend bank just downstream of the bend apex. The main flow directly impinges on the outer bank at a high angle just downstream of the

Fig. 2. (A) Section of an aerial photograph taken on 24 January 2003, 13:35 h (MET) at a scale of 1:18,000. The photo shown here has been sharpened and contrast-stretched as a whole in Irfanview. (Topographical Survey, Kadaster, with permission.) (B) DTM (in m + sealevel, Dutch coordinate system) interpolated from DGPS measurements covering approximately the same area as (A). Black dots indicate DGPS measurement locations. Two contours indicate approximate transition from concave to convex (−1.3 m) and the transition from a gentle to a steep zone (−1.65 m).

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apex (Ferguson et al., 2003). In response, the banks erode locally; the bend sharpens and may develop a peculiar angular shape. We will adapt the model of Ikeda et al. (1981) and Johannesson and Parker (1989) by incorporating a high critical flow velocity for erosion to assess how this threshold for bank erosion changes the predicted locations of bend migration for cohesive banks. It is not possible yet to include flow separation effects. The bank erosion in such models is very simplistic so we will also study the effect of mud strength on the bank erosion with two more sophisticated channel bank stability models. The paper is organised as follows. First, the field site is described and methods of data collection and experimental setup are given. This is followed by the field observations and experimental results. Then we assess quantitatively the effects of cohesion with simple models for erosive step migration, bank failure, and sharp meander bend evolution. Finally, we present our interpretation of the morphodynamics on the intertidal flat followed by general discussion and conclusions. 2. Field site description The field site is located on the southern shore of the Westerschelde estuary (the Netherlands), on an unvegetated intertidal flat near the Paulinapolder (to the west of the site of Temmerman et al., 2003b) (coordinates 51°21′51.7′′ N. and 3°41′46.7′′ E.) (Fig. 1). The estuary is embanked, and the mudflat ranges from the dike to 220 m seawards (Fig. 2). Macrobenthos and microbial mats were not studied. Vegetation was absent at the channel sites that were studied. We collected data between 13 September and 29 October 2006, and additional observations were made during a field site visit on 22 March 2008 after an exceptionally heavy storm season. To prevent disturbances of the mud at the instrumented channel and its bank, we designated a small number of walking routes on the flat, which remained clearly recognisable during the entire field work period despite rain storms and wave action. Unfortunately this prevented us from collecting detailed DTM data of individual bends and channels. The mud is a slightly bimodal sediment with 20–50% silt and 2–4% clay with a mean grain size of about 50 µm (Fig. 3a). Sediment samples show no clear spatial sorting trends. For this analysis, sediment samples from several cross-sections along the instrumented channel were analysed in a Malvern particle sizer after treatment to deflocculate. Clay is defined as particle sizes smaller than 2 µm, sand is larger than 65 µm and silt has particle sizes between 2 and 65 µm.

3. Methods 3.1. Field data Data were collected at three spatial levels: first, the general morphology of the entire flat with differential GPS and aerial photographs collected by the Dutch Topographical Survey in 1999, 2003, and 2005, which had just enough resolution to identify the channels. The second spatial level was channel and meander geometry measured at least three locations along six channels as well as transect-averaged streamwise flow velocity measurements with dye tracer. Third, high-frequency flow and suspended sediment transport were measured during a full neap-spring tidal cycle in one channel with an instrumented frame. In addition, the morphology of the latter channel and two other channels of smaller size was mapped in detail through photography at a height of 4 m above the flat using a camera suspended on a fishing rod and by repeated observations and photographs at various locations. The development of backward migrating steps was followed during ebb surges and on subsequent days. Observations took place in a period of about half an hour from the moment when the channels became accessible, that is, when the flow became bankfull and the steps were not yet active (no hydraulic jumps). Migration distances were estimated from step locations relative to fixed poles on subsequent days. The migration of meander bends was followed during the entire field work period by noting bank failure locations, and between the field work period and a later site visit (March 2008, after a heavy storm season) by comparing photographs. The channel has a trapezoidal cross-section, asymmetrical in the bends, with an average top width W = 1.43 ± 0.31 m (two standard deviations), a bottom width W = 0.26 ± 0.07 m, and a depth h = 0.33 ± 0.06 m, and minimum bend radii R = 0.15 m. Bend radius was measured at the channel centre line by taper for a length of one channel width at the sharpest bend location. For the sharpest (‘hairpin’) bend of radius 0.15 m the width was 0.30 m so that the ratio of radius over width was R/W ≥ 0.5 for all data. The wetted perimeter is 1.73 m, and the wetted cross-sectional area at bankfull discharge is 0.27 m2. The slope of the outer-bend bank varies between 20 and 90°. With the instrumented frame, data was collected at 2 Hz for 6 weeks from neap tide to neap tide in one cross-section of a relatively large meandering channel. Flow velocity was measured with a calibrated electromagnetic flow sensor (EMF) at 0.08 m above the bed in the thalweg of the channel; depth was determined with a calibrated pressure sensor corrected for atmospheric pressure; and sediment concentration was measured at three heights above the bed (0.19–0.36 m) by daily cleaned optical backscatter sensors (OBS) that were calibrated on local sediment in the laboratory flume with a nearly linear second-order polynomial. The signals were despiked by a gradient threshold filter. Direct shear tests were done on samples from a typical bank in an outer bend at a depth of about 0.1 m below the surface to check whether the apparent cohesion and apparent angle of internal friction were in the same range as reported in Winterwerp and van Kesteren (2004), so that information about compaction of the sediment and effect of water content can be inferred from literature as well. 3.2. Experiments on erosion and backward steps

Fig. 3. Summary of cumulative particle size distributions: envelope (min, max, thin lines) and mean (bold line).

We did flume experiments in order to perform observations and measurements on two critical processes in greater detail than was possible in the field: incipient entrainment from a plane bed and backward step migration. We also aimed to obtain self-formed meandering channels in the laboratory, but this did not occur in a range of settings including unidirectional and tidal currents. Experiments with an initial carved meandering channel show that the

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Fig. 4. (A) Elevation above mean sea level measured with differential GPS is plotted against distance from the estuary. The thin black line at the top of Fig. 2B indicates the area over which this cross-shore profile has been averaged. (B) The gradient is calculated from the moving average over a 25 m window.

erosive power of the flow was too small in comparison to the cohesion of the sediment to initiate bend migration. A section 6 m long and 0.1 m deep of a flow-recirculating laboratory flume of 0.4 m wide, 0.5 m deep, and 17 m long was carefully filled with sediment collected from the intertidal flat. We collected dense sediment with large shovels so that it was mostly consolidated. The sediment was left to consolidate further for one week in still water. Consolidation experiments (not reported)

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confirmed that the sediment did not compact significantly after 60 h and that vertical gradients in density were limited. We assume that bulk density is similar to that at the field site although it is likely that bulk density strongly varies with depth and perhaps location in the field. The incipient entrainment of sediment was measured in the flume before all other experiments so that the mud was as pristine and flat as possible. The water depth was 0.27 m, and the flow velocity, measured with the field EMF at 0.12 m above the bed, was increased in small steps from 0 to 0.8 m/s. The water surface slope was measured with pitot tubes so that the shear stress could be calculated from the depthslope product. The sediment concentration was measured at 3.0, 6.5, and 15.5 cm above the bed using Optical Backscatter Sensors (OBS) at 2 Hz. The sediment concentration in the flume experiments was corrected here for the fact that the water volume in the flume was about fourfold the volume of water above the mud bed itself so that uncorrected concentrations were effectively diluted. The OBS-sensors, which were the same as used in the field, were calibrated to transverse suction tube concentration sampling. In the second set of experiments, straight or curved channels of initially 0.06–0.08 m wide and 0.005–0.01 m deep were carved in the sediment bed on initial bed gradients of 0.01–0.025 and a constant discharge of 0.06–0.24 m3/s. Bed incision was forced at the downstream boundary of the sediment by a base level at a few centimetres below the sediment bed. The incision initiated a step that migrated up-stream following the channel. Flow velocity was measured using dye tracer (which also highlighted recirculating flow zones in the sharper bends), and the depth was measured manually at several thalweg locations along the channel far upstream of the steps (more than one backwater adaptation length). The bed was monitored during night and day by time lapse photography with a 10 min interval. The migration rate of the steps was estimated from the timelapse photographs.

Fig. 5. Photo mosaic of the channel in which hydrodynamics were measured (from left to right is about 190 m). Location of the instrument bridge is indicated. Locations of bank erosion are indicated by lines along the eroding bank. The transition from gentle to steep slope occurs at the convergence of two main channels, where the largest and sharpest meanders are also observed. The ⁎ indicates the location of the sharp bend in Figs. 6B and 7 and is the location where sediment was collected for direct shear tests.

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4. Results 4.1. Meandering channels The differential GPS data show that the intertidal flat has a convex cross-shore bathymetry (Fig. 4) with a slightly concave zone on its landward side. The gradient of the low-sloping part of the flat (100– 200 m from the estuary) varies between 0.005 and 0.001 while the steep zone (0–100 m) has an increasing gradient up to 0.05 (hereafter called ‘zone of increasing gradient’). In the landward concave zone, poorly defined channels that meander around gentle hummocky topography are nearly regularly spaced along the mudflat at a distance of 30–80 m and have widths ranging between 0.2 and 1.3 m and depths of a few centimetres. All channels have at least some very

sharp bends with ratios of bend radius (R) and width (W) of R/W b 2 (calculated for the channel width measured at the bottom of the channels, so R/W is much smaller for the top width of bankfull discharge). Channel cross-sectional area decreases in the downstream direction (also see Fig. 5), particularly in the steep zone of the flat where the duration of near-bankfull discharge is much smaller despite the longer periods of flooding. Highly sinuous meandering channels were found all along the intertidal flat in the zone of increasing gradient at a more or less regular spacing of 300–600 m (Fig. 2A). The relative bend radius for the bend apices (R/W, where R = bend radius and W = channel width at the top of the banks) varies between 1 b R/W b 4. The sinuosity of the channels is on average 1.2 and 1.7 for the gentle and steep slope sections, respectively. The meanders in the steep section of the

Fig. 6. (A) Tracer in a sharp meander bend indicated in Fig. 5. The vortex on the right bank downstream of the corner rotates clockwise; the vortex in the left outer-bend bank corner (note the bank erosion) rotates counterclockwise. (B) Series of sharp bends (lower left bend is shown in A). Note that image is distorted by wide-angle lens. (C) Sharp bend with erosion on inner-bend bank just upstream of flow separation point. Note the shell lag. (D–F) Sharp bends in various channels with flow separation that increase in sharpness over time. Arrows indicate flow direction.

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the flow impinged on the bank. The separation point was identified in most bends at a sharp and pointed mud ridge (Fig. 6B–D). At the location of the inner-bank vortex the bank was very smooth, low sloping, and soft which suggests that this is a depositional zone (Fig. 6B–D). Only one case was found with erosion of the inner-bend bank just upstream of the flow separation zone (Fig. 6C). The only clear locations of bank erosion and inner-bend deposition are associated with sharp bends, whereas gentle bends were not observed to erode noticeably in the entire period. The morphology of several very sharp bends suggested that these bends sharpened over time (Fig. 6, particularly E and F). Apparently, a positive feedback emerges once a sharper bend is formed. The innerbend bank builds out as a sharp mud ridge (Fig. 6B,C) while the outerbend bank erodes to form a circular meander pool (as in Andrle (1994), also see Fig. 7). The combination leads to bend sharpening, which explains very sharp meanders commonly observed on cohesive intertidal flats or in meandering rivers in cohesive sediments (Leeder and Bridges, 1975; Bridges and Leeder, 1976; Andrle, 1994; Fagherazzi and Furbish, 2001; Ferguson et al., 2003). Fig. 7. Sketch (not to scale) of observed bend migration of the bends in Fig. 6B.

intertidal flat have relatively small width-to-depth ratios with the average W / h = 3.6 ± 1.6 (where the ± indicates the min–max range), whereas the gently sloped meanders have W / h = 38 ± 21. The best developed and largest bends occur in the zone of increasing gradient, starting near the confluence (near bottom in Fig. 5, also see Fig. 2) with vertical outer-bend banks, sometimes with tension cracks, and (on average) 45° inner-bend bank slope. Only two cases of neck cutoff were inferred from observed planform of two smaller channels on the mudflat. Comparison between the aerial photographs of 1999, 2003, and 2005 (resolution about 0.5 m) and our observations in 2006 clearly show that the meandering channels did not migrate more than about 1 m. Only the sharpest bends have migrated over distances of the order of the channel width over the entire period of 7 years. Our field observations in March 2008, after a heavy storm season, showed that the meanders had changed, particularly sharpened, by migration distances locally of the order of the channel width in a much shorter period (Fig. 7). This illustrates the importance of thresholds for incipient erosion and for bank failure in this system. The largest channel reaches with the highest-amplitude meanders occur near the transition in bed gradient. Farther downstream, on the steep slope section, the channel is much narrower and somewhat deeper. This downstream trend is probably caused by the large-scale geometry of the intertidal flat, which dictates how it empties during ebb-tide. The ebb-tidal currents will cut most efficiently at the slope break and on the lower high-sloping section of the flat, but the lower section is only exposed to the flow over a very short period when the low-sloping section has already mostly emptied. As a result, the discharge at the slope break is not only larger but also occurs over a longer time period each tide than on the high-sloping section further downstream. On the more landward part of the low-sloping section the meandering channels become narrower and much less deep as they are exposed to smaller currents fed by the most upstream tidal flat area only. In many sharp meander bends, horizontal circulation cells were present near the outer-bend bank just upstream of the apex and near the inner-bend bank just down-stream of the apex (Fig. 6A). The flow through the thalweg separated from the inner-bend and impinged on the outer-bend bank downstream of the apex, where bank erosion was focussed. The separated flow drove two vortices: one near the inner-bend bank starting just downstream of the flow separation zone, and one near the outer-bend bank upstream of the zone where

Fig. 8. Erosive backward migrating steps in a large meandering channel. The bird footprints are about 0.04 m wide. Note the remains of bank collapse by slumping, which was initiated by undercutting and tension cracks.

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Table 1 Observations of step celerity and flow conditions upstream of the step (given as minimum–mean–maximum or min–max values when more observations are available) Field site

Flume experiments

Parameter

Hard mud

Soft mud

Hard mud

Sandy mud

Celerity

0.2–0.5– 0.8 m/day 0.03 0.10–0.27 0.03–0.10

0.25–0.5– 0.9 m/min

0.26 m/day

0.39 m/day

0.011–0.019 0.24–0.30 0.04–0.045

0.10–0.34 0.02–0.04

Gradient (m/m) Flow velocity (m/s) Water depth (m)

4.2. Backward steps In various places on the steep zone of the intertidal flat, backward steps were observed (Fig. 8) that migrated upstream. These steps had heights up to the order of the channel depth. Some of the largest steps were undercut by plunge pools. In a few cases, always on the most convex part of the intertidal flat, several steps were found in one channel with distances of the order of a few meters. We roughly measured their migration rate through observation of step locations on subsequent days. The steps migrated with a celerity of the order of 1 × 10− 5 m/s (or less than 1 m/day) (Table 1). The steps migrated by erosion of the bed as well as failure of the step, usually initiated at tension cracks a few centimetre upstream of the step. The latter agrees

with observations by Simon and Thomas (2002) who also found that migration was accelerated by the mass failures. After the 30 min bankfull discharge stage (see description of hydrodynamics below), a second important flow stage occurs for about an hour as observed throughout the measurement period. In this second stage, the channels are fed by base flow from the mudflat, resulting in a water depth of 0.01–0.02 m with flow velocities of up to 0.2 m/s as determined with tracer. Most of the incision occurred in small backward migrating steps, which created a secondary channel of about 0.1 m wide in the floor of the larger channel with vertical walls and a depth of a few centimetres. The secondary channel varied in cross-sectional position in the main channel but tended to focus in outer-bends. At the small step locations, the primary channel banks were occasionally undercut, which caused bank failure. The floor of the channels contained shell lags in many places (e.g., Fig. 6C), particularly where the banks had recently been eroded and where steps had just passed by. These shells originated from the channel wall and occurred at varying depths but in general about 0.1 m depth below the flat surface. No steps were observed in the shell lags indicating that the lags increased the threshold of the bed against erosion. During our field site visit in March 2008 following a heavy storm season, the channels were nearly entirely filled with fluid mud on the steep seaside part of the tidal flat, until the flow started to concentrate in the channels. (The mud was fluid in the sense that the surface

Fig. 9. Example time series of one day near spring tide (a–c) and neap tide (D–F) for water depth (A,D), flow velocity through the channel (B,E) and average concentration of the OBS sensors (C,F). Time axis is given as hours since 1 January 2006 0 h and is of equal length for spring and neap tide plots. The large dots indicate the times of the peak flood velocity, the peak water depth and the peak ebb velocity.

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4.3. Hydrodynamics Example data of a spring tide and neap tide water depth and concurrent velocity (Fig. 9) illustrate the extreme phase lag between water level and velocity. The low velocity during most of the submergence shows that the tidal flat filled and emptied (loosely like a bath tub) wherein the velocity was negligible except for the final emptying phase. The ebb flush occurred for about 30 min when the mudflat was nearly emerged (towards ebb) and the fastest flood-tidal flow occurred when the flat was just submerging. The recorded flow velocity was highest during the ebb flush when the channel was bankfull (Figs. 9,10B). The highest observed ebb flow velocity was −0.80 m/s while the maximum flood flow velocity was about 0.30 m/s. The data are summarised by identifying the water depth peak, the flood-tidal flow peak and the ebb-tidal flow peak (also illustrated in the example time series, Fig. 10). These points of water depth, velocity and sediment concentration as well as wave height and precipitation are presented for the entire period in Fig. 10.

Fig. 10. Time series of (A) peak water depth during slack tide, (B) peak flood velocity (dashed line), peak ebb velocity (continuous line), (C) Mean (RMS) wave height during peak water depth, (D) cumulative precipitation since burst 6800, and (E) sediment concentration during peak water depth (dashed line) and peak ebb velocity (continuous line). Time axis is indicative as each point represents a value for one tidal cycle. See Fig. 9 for identification of these values from the time series.

oscillated when perturbed by a stick, and lumps of harder mud submerged immediately when released on the surface.) From that time onward, several (quasi-cyclic) backward steps emerged in the fluid mud that rapidly cleaned the channels of the fluid mud. The celerity of these steps was of the order of 1 × 10− 2 m/s—a few magnitudes faster than in the consolidated mud. The large amounts of fluid mud must have diffused into these channels from the estuary adjacent to the flat by waves and perhaps tidal currents during the preceding peak water depth as most of the mud had disappeared because of the action of backward steps when the flat emerged. The overall mud concentration in the estuary likely was large, so that large amounts could settle during submergence of the flat and peak water depths to be subsequently eroded during emergence and then resettled again during the next submergence and peak water depth.

Fig. 11. Sediment concentrations during peak water depth and peak ebb velocities. (A) Correlation between peak water depth and the peak ebb velocity a few hours later. (B) Sediment concentration (from the estuary) during peak water depth. Wave and rain events are indicated by superimposed symbols. (C) Sediment concentration (from the intertidal flat) during peak flood and ebb velocity. Drawn line is from the flume experiment. Circles and stars same as in (B).

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The peak ebb velocity magnitude correlates well with the peak water depth (at slack tide) (Fig. 11A, circles) although the flow velocity is out of phase with the water depth (Fig. 9). On the other hand, the peak ebb velocity does not correlate with the water depth in the channel during the ebb surge (Fig. 11A, dots). This demonstrates that the amount of water stored on the mudflat determines the magnitude of the ebb surge in agreement with Fagherazzi et al. (2008). Furthermore it indicates that the ebb surge, which is likely the only phase with significant erosion, occurs for short fractions of time during spring tide only. Hence the meandering channels can be considered unidirectional flow channels. 4.4. Sediment transport Example time series (Fig. 9) show that sediment concentration increased during the peak water depth and the ebb surge. Sediment concentration for these conditions is plotted against water depth and velocity in Fig. 11B,C. During the measurement period the flood peak concentration was between 0.05 and 0.3 kg/m3 and was not correlated to the flood peak velocity (Fig. 11C), indicating that this is the concentration supplied at the mudflat boundary from the estuary. The sediment concentration in the estuarine flow itself, the likely source of the sediment suspended during flooding and slack tide, may obviously vary with storms, supply from the land (through the Scheldt river) and the sea and with season but no observations were available during the measurement period. The sediment concentration during peak water depth is correlated to the water depth (Fig. 11B), which suggests that sediment settles during slack tide. The settling velocity of 0.05 mm sediment in still water is about 1.4–2 mm/s (calculated with the Stokes law), which allows sediment to settle from the entire water depth during slack water. Thus sedimentation on the tidal flat, though limited by the supply from the estuary, potentially takes place during every flood. The sediment concentration during peak ebb velocity correlates weakly with the velocity (Fig. 11C). The source of this sediment is the intertidal flat itself, and indeed the concentration is often higher when rain storms and wave events mobilised the mud. This is supported by visual observations on the onshore half of the mudflat: we observed that the surface runoff during rain storms results in larger morphological changes and higher sediment concentrations. This is caused by rain-drop splash erosion, which breaks down the cohesive structure of the sediment, forming a fluid mud layer of a few centimetres thickness. Rainfall runoff easily entrained the fluid mud that was subsequently collected in the most upstream channels, which were incipient and poorly defined based on local observations and aerial photographs. These small channels combine into fewer larger channels in the first onshore 100 m. Furthermore Temmerman et al. (2003b) demonstrated that suspended sediment concentrations at a nearby mudflat are three to four times as high in winter as in summer, further indicating the role of waves and rain. To assess the importance of the channels in the sediment budget of the entire mudflat, the amount of sediment (mass per unit width) imported during flood from the estuary onto the flat landward of the instruments, and exported from the channel landward of the instruments by the ebb current was calculated for the measurement period of 27.66 d. The transport is the summed product of instantaneous velocity, water depth, and concentration. We assumed that the measured instantaneous velocity by the single sensor was representative for the entire channel cross-section; a simplification that will not affect our main conclusion from this analysis. It must be noted that the final phase of the ebb flush was not measured as the instruments emerged, but concentrations were negligible already before the instruments stopped recording. The ebb channels are only active for about 1 h/d whereas the flood phase duration is about tenfold. As a result, the ebb exported about 200 kg of mud per metre width of channel, while the floods imported about 1400 kg of mud per

metre width of mudflat over the entire period, with most of the transport effected during spring tides. The channel is about 1 m wide and the interchannel spacing is of the order of 100 m, so that the amount of imported sediment is two orders of magnitude larger than the exported sediment in the period of observation (assuming a uniform supply of sediment onto the flat). Indeed the intertidal flat has a long-term sedimentation trend forced by a long-term change in tides (Temmerman et al., 2003a). 4.5. Experimental results The measured concentration increased rapidly after the flow velocity was increased in each step, and then became constant around a maximum concentration for that step (see line in Fig. 11). This behaviour is commonly observed in mud sediment transport (e.g., Winterwerp and van Kesteren, 2004, Fig. 9.3). The incipient entrainment of sediment was gradual with increasing flow velocity, and there is no clear threshold for motion (Fig. 11C). This may be because the bed was created by direct emplacement of mud from the field site, so that the surface was not entirely uniform and weak patches existed where the mud is more easily entrained. This may also explain why the experimental concentrations are on average higher than the concentrations measured in the field. Had the sediment been exposed to strong flows for several days without recirculating the water and sediment, then the concentrations for the stepwise-increased flow velocity would have been lower and perhaps a clearer critical shear stress would have been observed. A critical shear stress of 7 N/m2 is assumed here (in agreement with Winterwerp and van Kesteren, 2004) for the hard mud; half this value is estimated for a second set of experiments on much sandier mud and a value of 0.2 N/m2 is assumed for the very weak sandy mud (Lick and Gailani, 2004). Direct shear tests were done on six fully drained and submerged samples from near the location indicated in Fig. 2. The bulk unit weight of the sediment ρsg = 12 kPa. The samples compacted fully within a day in the direct shear box under the same normal pressure as used for the experiments, after which they were submitted to a 0.2 mm/h horizontal displacement. The resulting cohesion was C = 11 kPa, and the internal angle of friction was ϕ = 20°, which are typical values for this material (Winterwerp and van Kesteren, 2004). In the channel experiments, steps initiated in all cases at the downstream boundary. The steps migrated upstream at a more or less constant celerity following the straight or meandering channel (Fig. 12). The erosion by the migrating step resulted in a deepening and widening of the carved meandering channel. The flow characteristics upstream of the step were measured as well as the celerity of the steps in an experiment with hard mud and an experiment were about 50% of fine poorly sorted sand was mixed into the mud (Table 1). In experiments, the step celerity is of the same order of magnitude as in the field. The

Fig. 12. Erosive backward migrating step in the flume in strong mud. The step follows the initial meandering channel. Flume width is 0.4 m and length of shown section is 2 m. Time between time frames is 6 h.

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steps did not only migrate upstream by gradual erosion of the bed as Izumi and Parker (2000) assumed in their model, but also by repeated cantilever or planar failures at the step typical of cohesive sediments (van Eerdt, 1985; Simon and Thomas, 2002). Just downstream of the step, the deepening caused further failure of the banks as reported in Simon and Thomas (2002). In experiments without an initial channel, the steps also developed; but bifurcated in the upstream direction because the flow was not as focussed as it was by an initial channel, and then stalled because the divided flows could no longer erode sediment.

for Fr4/3 n − ψ N 0, for which steps migrate infinitely far upstream because the bed upstream of the step is eroding as well as the step itself, and cn =

ð1−ψÞγ 1−Frn2

ð3Þ

for Fr4/3 n − ψ ≤ 0, for which steps migrate only a finite distance upstream because the threshold for erosion is no longer exceeded at some distance upstream of the step. The resulting nondimensional celerity is transformed back to dimensional celerity as

5. Modelling c = cn 5.1. Description of model of step migration

271

α ðψÞ−γ Cf

ð4Þ

(see Izumi and Parker, 2000, for derivation and details). Izumi and Parker (2000) and Parker and Izumi (2000) described an analytical model for the stability of erosive steps. Starting from normal, subcritical flow upstream of the step, the flow becomes critical above the step leading to more intense erosion at the step than upstream of the step. This leads to the migration of the step as a function of the erosion rate. Izumi and Parker predicted the celerity of the step based on the backwater formulation and a formulation for the erosion of cohesive sediment or bedrock (generalised from literature). This erosion formulation is of the form: E = α ðτ − τc Þγ

ð1Þ

with erosion E in m2/s, α = erodibility coefficient, γ = power between 1 and 2, τ = bed shear stress exerted by the flow and τc = critical shear stress for entrainment. Here, α = 1 × 10− 7 τc for materials ranging from sand to clay (Hanson and Simon, 2001; Simon and Thomas, 2002). Note that α has a dimension depending on the power γ, e.g., Nm4/s for γ = 1. Further note that some authors (Hanson and Simon, 2001) cite α = 2 × 10− 7 τc for this relation and it is not clear which paper has the typographical error (Darby, pers. comm. 2008). Given the observed normal flow velocity u, depth h, and slope S far upstream of the steps [that is, one backwater adaptation length upstream, estimated as h / S (in m), where flow is no longer affected by the downstream step], the flow velocity above the step for Fr = 1 is calculated as u1 = (qg)1/3 with q = uh = specific flow discharge, and g = 9.81 m/s2 is the gravitational acceleration. Then Izumi and Parker (2000) defined a nondimensional velocity as un = u / u1 (with subscript n denoting nondimensionalisation), which is then used to calculate a Froude number Frn = u3/2 n (see Izumi and Parker, 2000, for the details of the derivation). An inverse nondimensional excess shear stress parameter, ψ, is defined as the ratio of critical and actual bed shear stress (far upstream of the step): ψ = τc / ρCfu21, where the flow friction is defined as Cf = ghS / u2. From the stability analysis, Izumi and Parker (2000) found that the step will migrate infinitely far upstream for ψ N 1. (Note that ψ N 1 is defined far upstream; at the step erosion can nevertheless take place because the local shear stress exceeds the critical shear stress at least at the step even though not farther upstream.). Note that we use bed slope S as surrogate for energy slope throughout the paper. This is specifically admissible in the Izumi and Parker (2000) model because there it refers to flow far upstream of the step which is steady and uniform. It is generally admissible in our tidal environment for the last moments of ebb drainage where tidal water level fluctuations do not directly affect the shallow channel flow which can therefore reasonably be assumed uniform. The nondimensional step celerity in the upstream direction is calculated by (Izumi and Parker, 2000):

cn =

 γ 4=3 ð1−ψÞγ − Frn −ψ 1−Frn2

ð2Þ

5.2. Predictions of step migration The observations of step migration are presented in four groups: field data, further subdivided into strong and weak mud, and experimental data, with the same strong mud as in the field and sandy mud (Fig. 13). The step celerities are predicted from all measured flow velocity and water depth values in the field and in the flume. The values plotted in Fig. 13 are mean and mean ± standard deviation of predicted celerity versus minimum, mean and maximum measured celerity (values presented in Table 1). Thus the different symbols for each subset represent the range of observed flow conditions and observed step celerities. The calculated dimensional step celerity is of the same order as the measured celerity (Fig. 13). Obtaining better results is possible by tuning the critical shear stress of each case but this was not done here. More accurate water depth measurements would likely also improve the predictions. Exact simultaneous measurements are not available because flow varied rapidly whereas step celerity had to be measured over a longer period, so we had to use representative values of the hydrodynamic parameters in the channel upstream of the steps. Despite the uncertainties, the present analysis suffices for the following observation. The three-order of magnitude difference between step celerity in weak and strong mud demonstrates the large effect of the threshold for erosion, which varies only one order of magnitude. A further consideration is that step migration is a more efficient erosion mechanism than simple surface erosion (also see Parker and Izumi,

Fig. 13. Predicted versus observed step celerity for field and flume data. The critical shear stress for erosion was τc = 7 Pa for the strong mud in the field and the flume, 3.5 Pa for the sandy mud in the flume and 0.2 Pa for the soft mud in the field during the 2008 site visit after a heavy storm season.

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2000), so that the presence of steps determines much of the channel evolution on the steeper sloping part of the flat. 5.3. Bank stability modelling Bank failure is often triggered by undercutting of the banks through fluvial erosion and by large groundwater pore pressures during falling stages (Simon et al., 2000; Simon and Collinson, 2002; Darby et al., 2007). In our case, banks are also undercut by erosive bend flow and by the passage of backward steps in a similar manner as in rills described by Stefanovic and Bryan (2007). Afterward, the failed sediment mass deposits on the channel bed and locally reduces the shear stress and the immediate probability of renewed bank erosion, until this sediment is removed by the flow. When such failed blocks contain salt marsh vegetation, they remain in place for a long time (van Eerdt, 1985); but for the present field case, vegetation is absent and the block of mud is removed in at most a day. The most important parameters determining bank stability are the effective cohesion C′(in kPa) of the sediment and the effective angle of internal friction ϕ' (in degrees). We can exclude roots from our analysis since the study area is not vegetated. The apparent ϕ and C are determined from direct shear tests on well-drained saturated sediment and the Mohr–Coulomb equation: σ = C + tanð/Þ

ð5Þ

where σ = peak strength (in kPa) of the material at the point of failure. For reference, typical apparent cohesion and apparent friction angle for well-drained clay are C = 1–10 kPa and ϕ = 10–25°. However, these values will vary strongly in nature. Notably, the cohesion depends strongly on water content, and mud need not be well drained when the water draws down rapidly as in our tidal conditions. On the other hand, during flooding the matric suction of partially saturated banks increases the strength. Hence, the effective cohesion and effective friction angle may differ very much from the apparent cohesion and apparent friction angle. For water content (by weight) of less than, say, 25%, the undrained shear strength is about 10 kPa, while for 100% it decreases to 0.1 kPa (Fig. 3.21a in Winterwerp and van Kesteren, 2004). The apparent cohesion that will be determined in the direct shear test with drained sediment can therefore be interpreted as a maximum possible value. Our observations show that the failure surface of bank failures is a plane passing through the toe of the slope. Hence, the Culmann method can be applied. To assess the effect of cohesion, the order of magnitude will be estimated of the critical height of the bank at which failure takes place (e.g., Carson, 1971, p. 100): Hc =

2C sinSb ρs g sinðSb −βÞðsin β− cos β tan /VÞ

Simon et al. (2000) and Simon and Collinson (2002) developed an advanced bank stability and toe erosion model that was used here to calculate safety factors as a function of water table, cohesion, and bank angle. The model combines three failure mechanisms including with tension cracks and cantilever, allows specification of C′ and ϕ' for five vertical layers, effect of positive and negative pore water pressure, rooting and added-weight effects of vegetation and includes a bank toe erosion model. Their model is available at http://www.ars.usda. gov/Research/docs.htm?docid=5044. This model will be used to calculate safety factors to explore effects of tension cracks, oversteepened banks of varying angles, depth-dependent cohesion following Fagherazzi and Furbish (2001) and varying flow depth in the channel, in our case related to tidal water level fluctuation. 5.4. Predictions of bank stability The Culmann model is applied to a reasonable range of conditions and bank slopes to assess the necessary conditions for bank failure. A bank is predicted to fail when the critical bank height for failure is smaller than the actual bank height. The critical bank height at failure is of the order of 10 m for the highest cohesion values (15 kPa) (Fig. 14A). This is also the case for banks with tension cracks (Fig. 14B). Banks of about 0.1 m, that occur at the field site, only fail when the apparent cohesion is 0.1–1 kPa (Fig.14A,B), which only is the case for higher water content (Fig. 3.21a in Winterwerp and van Kesteren, 2004). This implies that the banks only fail when they are saturated. The combination of unconsolidated sediment, saturated banks and an empty channel occurs when the tidal flat has just emerged. At that point in time self-weight

ð6Þ

where Hc = critical bank height, C = apparent cohesion, Sb = bank slope, β = failure slope (here assumed to be β = (Sb + ϕ') / 2), ϕ' = effective angle of internal friction, ρs = density of sediment, g = gravitational acceleration. The effective angle of internal friction here includes the effect of pore pressure: ϕ' = (1 − ru)ϕ, where the pore pressure ratio is ru =

ρgHw ρs gHb

ð7Þ

where ρ = density of water, Hw = water height above the bank toe, and Hb = bank height. The presence of tension cracks and a vertical bank allows a simplification of the Culmann method (Carson, 1971): Hc = 2:67

  C π /V + tan ρs g 4 2

ð8Þ

This equation will be compared with the former (applied to vertical banks as well) to illustrate the effect of tension cracks.

Fig. 14. Critical height of the banks at failure for various combinations of bank slope and cohesion as predicted with the Culmann model. Dots (bottom, ‘tc’ in legend) indicate critical height for banks with tension cracks of 0.1 m depth in the 90° slope. As a rule of thumb, the critical height is about 30% lower when the bank is still saturated but the water level has dropped to the bank toe.

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consolidation of freshly deposited sediment did not yet increase the strength above critical values for bank failure. The only mechanism that significantly weakens the banks after consolidation is waves. In support of this hypothesis, we observed that most of the meander migration took place during a heavy storm season, whereas no significant change occurred in the preceding years. The more advanced model of Simon et al. (2000) and Simon and Collinson (2002) is applied to some similar scenarios as the Culmann model for comparison. The safety factors are plotted for a range of realistic parameters (Fig. 15). For safety factors larger than 1–1.3 the bank will fail. The input bank height was 0.4 m at an angle of 90°, effective friction angle 30°, failure plane angle 15°, and a density of 1200 kg/m3. The flow depth was varied, but the groundwater table is at the top of the bank to simulate the conditions immediately after the ebb peak flow. The calculations indicate that banks fail when they are still submerged (Fig. 15A), as emergent banks have decreasing water pressure to support part of the weight. Interestingly, the presence of tension cracks increases the stability of the bank (Fig. 15B). Furthermore,

273

extremely oversteepened banks with angles between 90 and 170° (where 90° is already vertical) and a large value of cohesion (C = 10 kPa) only fail at unrealistically high bank angles (Fig. 15C). Additional more complicated and perhaps more realistic scenarios were run (not plotted) as follows with the Simon et al. (2000) and Simon and Collinson (2002) model. When the groundwater table was lowered (representing drained, unsaturated banks), the safety factor increased because of matric suction. When the cohesion was increased with depth from C = 0.1 to 10 kPa to simulate the effect of self-weight compaction (as in Fagherazzi and Furbish, 2001), the safety factor decreased about fourfold compared to the case with 10 kPa at all depths. This further supports the outcome that banks fail when they are saturated and the mud is weakened. Both the simple Culmann model and the advanced Simon et al. (2000) and Simon and Collinson (2002) bank stability model indicate that a step or bank with a height of 0.1–0.4 m may just fail under a combination of rather unlikely conditions: when it is vertical or even oversteepened, contains considerable tension cracks or has a lower cohesion caused perhaps by waves, rain when the tidal flat is emergent and probably only in combination with excess pore pressure immediately after the tidal flat emerged. These conditions do not occur often which agrees with the rare occurrence of bank failure locations in the field. The models clearly indicate that banks are much more likely to fail when they are undercut, which emphasises the importance of bank and step undercutting in the morphodynamics of the mudflat channels. 5.5. Migration simulation of meanders with sharp bends Many meander simulation models have been developed in the past. Most account for the detailed interaction between flow and bars (e.g., Camporeale et al., 2005; Crosato, 2007) based on theory by Struiksma et al. (1985) and others (see Camporeale et al., 2005, for references). In contrast to the sophisticated treatment of the channel bed, the bank erosion in these models is simplistically related to flow velocity and/or depth in the outer bend, so that bank erosion is highest in the deepest pools. A major assumption of these models is that the bed consists of mobile sediment, such as sand or gravel, which is not the case in the channels on the intertidal mudflat. Bars and pools and concurrent overdeepening effects were not observed in the mudflat meanders. Therefore, the meander migration model of Ikeda et al. (1981) and Johannesson and Parker (1989), where bank erosion is related only to the flow velocity at the outer-bend bank, is more appropriate. Note that recent work has included more sophisticated treatments of bank erosion in 2D flow models (Darby et al., 2002; Duan and Julien, 2005) but this is beyond the scope of the present paper. The model of Ikeda et al. (1981) is based on the shallow flow equation and the momentum redistribution A' by secondary flow in bends as introduced by Johannesson and Parker (1989), which we will use. The velocity ub near the eroding bank is related to reach-averaged velocity u and local channel curvature K: u

Fig. 15. Safety factor calculated with the model of Simon et al. (2000) and Simon and Collinson (2002) plotted against flow depth (for cohesion C = 0.1,1 kPa, tension crack depth (for C = 10 kPa) and bank angle (0.5–0.94π radians or 90–170° for C = 10 kPa).

  4  Aub u W AK u u2 −u2 + Cf K + 2 Cf ub = + ð A + A V−1Þ 2 h 2 As As h gh

ð9Þ

where u = reach-averaged flow velocity, ub = flow velocity at the eroding bank, Cf = flow friction, K = local curvature calculated as K =∂χ/∂s where s = stream-line coordinate, χ = angle between two segments (with spacing Δs) of the channel, and A = parameter for transverse bed slope (specified later) and A' = 16.7 calculated from the original equation in Johannesson and Parker (1989). Given the insignificance of the flood peak flow velocities at our field site, only peak flow velocities during ebb surges will cause morphological change. In the original model of Ikeda et al. (1981), and in fact in most meander simulation models, the erosion of the bank, equal to the

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5.6. Simulation results of meander migration in highly cohesive sediment

displacement of a node on the channel, is calculated with E = kðub Þ

ð10Þ

where k is the erodibility coefficient. This coefficient determines the rate of meander displacement and hence a sensible time step of the model, but it is strongly affected by numerical aspects such as distance between nodes and method and level of smoothing between time steps and should therefore be calibrated on meandering history (see Crosato, 2007, for extensive discussion). Hence meander simulation models are not reliable predictors of channel migration. Therefore, the model is here only used heuristically to determine the locations of bank erosion. To account for the large strength of the banks and bed, we adapt the model to include a critical shear stress for bank erosion: E = kðub −uc Þ

ð11Þ

where uc is the critical flow velocity for bank erosion, here calculated from the critical shear stress determined in the experiments with similar water depth. We will compare model predictions with and without this critical flow velocity to assess its effect on meander shape. The model of Johannesson and Parker (1989) and many similar models assumes steady bankfull flow and gentle bends. We will apply the model to bankfull flow conditions but we do not need to specify how long this flow condition occurs in the tidal cycle because we will not predict the amount of bank erosion but only its locations. The gentle bend assumption is of course violated in the sharp bends with flow separation but at present there is no formulation available for such bend flows. In fact, it is unknown when flow separates and what parameters to use for describing the sharpness of bends. In defence of our approach we note that Fagherazzi et al. (2004) successfully predicted the locations of bank erosion with the same model except for the threshold of motion.

The model is applied to a simplified representation of the tidal channel shown in Fig. 5. Various model settings were applied which all have similar results. Here, results are presented where values of the model constants are width W = 1.0 m, depth h = 0.35 m, u = 0.7 m/s, A = 6 (following Fagherazzi et al. (2004)), A' = 16.7, Cf = 0.0981, and k = 7 × 10− 9. The critical excess near-bank velocity is 1 m/s. The sensitivity of the meander migration rate to the various parameters was determined to be very large. However, the locations of erosion as well as the meander shape were fairly similar. The main aim of this model application is to demonstrate the effect of the erosion threshold on the locations of bank erosion and hence the meander shape. Thus we use the model heuristically as in Fagherazzi et al. (2004). The meander shape depends strongly on the critical near-bank velocity compared to the depth–width-averaged velocity. With the model that includes the erosion threshold, bank erosion becomes more localised so that the bends become quite sharp in comparison to the original model (Fig. 16). Bank erosion and bend migration occurs only in the sharpest bends in agreement with the observations (Figs. 7,6B). Clearly the observed bends are even sharper than modelled here, which is probably because of the separated bend flow that impinges on the banks in a narrow zone. The agreement between predicted and observed channel migration is weak, which is common for meander simulation models (Camporeale et al., 2005; Crosato, 2007). Several simple parameterisations for the effect of separated flow on bank erosion were tried in the model but did not improve the results. This subject deserves much further study. 6. Discussion The following suite of processes and phenomena explains the dynamics of meandering channels on the intertidal mudflat as inferred from our study combined with the literature. Boundary conditions. The large-scale mudflat morphology is determined by tidal range, the wave climate, and the suspended sediment supply (given the physical boundaries of the estuary and the dikes). This is inferred from our data of hydrodynamics and sediment transport and on Roberts et al. (2000) and Temmerman et al. (2003a). Hydrodynamic regime. The size, gradients and convexity of the mudflat determine the magnitude of the ebb and flood surges through the tidal channels and hence the potential incision and dynamics of the channels. The channels themselves probably do not modify the large-scale flat morphology. This is based on the observations of the extremely asymmetrical tides (with the ebb surge) and limited channel change and is in agreement with Roberts et al. (2000) and Fagherazzi et al. (2008). Landward channel inception. Incipient channels emerge on the upstream gentle flat because ebb flow concentrates in local lows. The tidal flat surface is somewhat irregular (‘hummocky’), partly because of the action by diatoms and macrobenthos. Waves and rain storms weaken the mud at the bed surface, and the concentrated flows transport this sediment and carve poorly connected, very shallow and wide (sometimes braided) channels. For a convex flat, the highest channel incision is found on the transition from gentle to steeper slope. This is based on our observations.

Fig. 16. Initial (dashed lines) and final (drawn lines) step of the meander model. (A) Original model. (B) Model adapted with a critical erosion threshold. Compare bank erosion locations to Fig. 5.

Seaward channel inception and deepening by steps. On the steep, seaward part of the flat, the focussed ebb flows initiate multiple backward steps in cohesive sediment (the steps are deeper than the weakened top layer). These steps migrate upstream at a rate depending on the threshold for erosion and excavate the incipient channels. Following an ebb surge, a longer phase of low-discharge

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outflow (base flow) initiates secondary steps on the bottom of the large channel, which, in places, create secondary inner channels and undercut main channel banks. This is based on our observations and in agreement with Izumi and Parker (2000). Channel widening. The sediment is so cohesive and the critical threshold for erosion is so high that channel banks erode only locally. Banks commonly fail after they are oversteepened after the passage of a step. This coincides with the draw-down of water during and after the ebb surge, so that pore pressure in the bank sediment further destabilises the banks. Storm waves and rain may also liquefy the bank surfaces leading to erosion. The failure blocks are rapidly eroded, and the finest sediment flows out into the estuary while the sand collects downstream in the channel so that the channels coarsen downstream. This is based on our observations of eroding banks and inferred from our data of waves and rain combined with Winterwerp and van Kesteren (2004, wave-initiated liquefaction) and Mwamba and Torres (2002); Tolhurst et al. (2008, effects of rain). Meandering. The banks are directly attacked in very sharp bends where flow separates from the channel boundary. The direct flow attack leads to undercutting, after which the bank fails locally. As a result, the meander sharpens while the channel just upstream and downstream of this location remains stable. Meander neck cutoff rarely occurs by downstream migration of such sharp bends. In general, meander dynamics are very slow because of the high thresholds. Stream capture and bifurcation occurs mostly through backward migrating steps that arrive at a hitherto unconnected channel. This is based on our observations and in agreement with Andrle (1994). Stabilisation of channels. The cohesion increases with depth from autocompaction, which limits the effective depth of erosion by backward steps. Furthermore, shells (bivalves) collect on the channel bed where they form an effective armour layer that protects the bed against erosion by the flow (but not against backward steps). These shells originate from the eroded banks. This is inferred from our observations in combination with Fagherazzi and Furbish (2001, for autocompaction). In-channel deposition. Separated flow in sharp meander bends leads to a low-velocity zone in the inner bend where mud is deposited and autocompacted. This deposition rate apparently mirrors the outer-bend bank erosion as the channels are not much wider in sharp migrating bends than elsewhere. More episodically, storms entrain so much sediment into the estuary that deposition rates during one slack tide increase dramatically. A deposit from one slack may nearly fill up a channel, but this sediment is so weak that it is nearly entirely removed again in the ebb surge by rapid migrating backward steps. This is based on our observations and in agreement with Bridges and Leeder (1976). The armouring effect of the shells at our field site has been inferred from limited observations. The effects of shells on the erosion of mud has barely been investigated but may be large as it provides a high threshold against erosion. Da S. Quaresma et al. (2007) found that shell and shell hash was transported landward by waves as bedload over an intertidal flat and was so abundant that it formed extensive cheniers. Neumeier et al. (2006) found that live cockles locally weakened the bed through bioturbation and acted as large roughness elements on the bed, causing disturbance of the flow, so that the effective critical shear stress was reduced up to a factor of two. The density of cockles in their experiments was so low that the surface was far from covered entirely by dead shells. Our field site takes an intermediate position between these extremes of shell coverage. The channel bed was locally covered entirely in dead bivalves that shielded the bed entirely from erosion; but in general, the conditions did not

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favour chenier formation, and the shell supply was not sufficient to form extensive deposits. It is insightful to compare our meandering channels to larger meandering rivers. Terrestrial meandering rivers differ in two important aspects from meandering channels on an intertidal mudflat. First, the bed sediment generally is mobile, and second, bank erosion occurs along most of the outer-bend bank and elsewhere. In the present intertidal mud channels, the threshold against erosion is so high that bank erosion only occurs in the sharpest bends where flow separates and at backward steps that undercut banks significantly—and even then only in exceptional storm seasons. The threshold for bank erosion in terrestrial rivers is lower but cannot be ignored because it determines the width–depth ratio (Ferguson, 1987; Simon and Thomas, 2002; Darby et al., 2007; Parker et al., 2007, and many earlier papers). Hence the intertidal mudflat meanders are situated near the least dynamic end of the continuum. Bar pattern (alternating bars versus braiding bars) strongly depends on the width–depth ratio (e.g., Struiksma et al., 1985), while bar pattern in turn determines the locations where near-bank velocity and water depth are the highest. The effective fluvial erosion generally is the largest in deeper parts of the rivers, notably the pools near the outer banks between bars. Hence, in meander simulation models, the bank erosion process has been simplified through a simple linear relation between pool depth, velocity, and bank erosion while assuming that the inner-bank deposition mirrors the outerbend bank retreat (e.g., Ikeda et al., 1981; Camporeale et al., 2005; Crosato, 2007). Fagherazzi et al. (2004) showed that such models are also applicable to meandering channels on mudflats with bidirectional flow. However, meander simulation models describe the interaction between a mobile bed and the flow in great detail, while the interaction with the banks is described very simplistically. In mudflat channels, the bed often consists of cohesive and mostly immobile sediment, while the processes of bank erosion and deposition are complicated by the high strength of bank material. Future work on meander simulation models should therefore focus on floodplain formation as well as bank erosion. Meandering channels on intertidal mudflats present an interesting test case for our understanding of meandering as they are on the very cohesive side of the continuum of rivers with weak to very strong banks relative to the flow shear stress. 7. Conclusions This study reports observations, experiments, and modelling of the morphodynamics of small meandering channels in highly cohesive sediment on an intertidal mudflat dominated by ebb-tidal flow. The high geomorphic threshold for mud erosion and for bank failure leads to two processes that are uncommon in terrestrial meandering channels: backward eroding steps and formation of very sharp meander bends. As such these channels are on the least dynamic end of the continuum from wide and shallow rivers with weak banks to narrow and deep rivers with strong banks. On the steep seaward side (the estuary) channels incept by the backward migrating steps in ebb surges. The backward eroding steps migrate at a predictable rate, depending strongly on the threshold of erosion. Repeated passage of steps leads to widening and deepening of meandering channels. Second, inner channels one magnitude smaller in size were frequently observed within the larger channels, which were created by a prolonged base flow with small backward steps after the main tidal flow ceased. On the landward, gentler sloping flat, the meandering channels are only eroded in very sharp bends where the flow separates so that it impinges the outer banks nearly perpendicularly. A meander simulation was moderately successful in predicting the locations of bank erosion after a threshold for erosion was introduced. Bank

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stability modelling indicates that bank failure is most likely during the end of the ebb surge when banks have been undercut and the channel has just emptied but the pore pressure is still high. The collapsed bank material is rapidly removed by the flow, but a lag layer of shells remains that protects the bed locally against further erosion. Deposition and consolidation on the inner bank occurs at such a rate that the channel maintains an approximately constant width. Acknowledgements The advice of Theo van Asch and Rens van Beek on slope stability and direct shear measurements is gratefully acknowledged. We are sincerely thankful for no less than five thorough and constructive reviews by S. Darby, R. Ferguson and three anonymous reviewers. MK is supported by the Netherlands Earth and Life sciences Foundation (ALW) with financial aid from the Netherlands Organisation for Scientific Research (NWO) (grant ALW-VENI-863.04.016). This paper is partly based on the M.Sc. studies by FS and WB. The authors contributed in the following proportions to conception and design, data collection, data analysis and conclusions, and manuscript preparation, respectively: MK(70,20,50,80%), FS(10,30,20,10%), WB (10,30,20,10%) and HM(10,20,10,0%). References Andrle, R., 1994. Flow structure and development of circular meander pools. Geomorphology 9, 261–270. Bridges, P., Leeder, M., 1976. Sedimentary model for intertidal mudflat channels, with examples from the Solway Firth, Scotland. Sedimentology 23, 533–552. Camporeale, C., Perona, P., Porporato, A., Ridolfi, L., 2005. On the long-term behavior of meandering rivers. Water Resources Research 41, W12403. Carson, M., 1971. The Mechanics of Erosion. Pion Limited, London, UK. Crosato, A., 2007. Effects of smoothing and regridding in numerical meander migration models. Water Resources Research 43, W01401. Da, S., Quaresma, V., Bastos, A., Amos, C., 2007. Sedimentary processes over an intertidal flat: a field investigation at Hythe flats, Southampton Water (UK). Marine Geology 241, 117–136. Darby, S., Alabyan, A., Van de Wiel, M., 2002. Numerical simulation of bank erosion and channel migration in meandering rivers. Water Resources Research 38 (9), 1163. Darby, S., Rinaldi, M., Dapporto, S., 2007. Coupled simulations of fluvial erosion and mass wasting for cohesive river banks. Journal of Geophysical Research 112, F03022. Duan, J., Julien, P., 2005. Numerical simulation of the inception of channel meandering. Earth Surface Processes and Landforms 30, 1093–1110. Fagherazzi, S., Furbish, D., 2001. On the shape and widening of salt marsh creeks. Journal Geophysical Research 106 (C1), 991–1003. Fagherazzi, S., Gabet, E., Furbish, D., 2004. The effect of bidirectional flow on tidal channel planforms. Earth Surface Processes and Landforms 29, 295–309. Fagherazzi, S., Hannion, M., D’Odorico, P., 2008. Geomorphic structure of tidal hydrodynamics in salt marsh creeks. Water Resources Research 44, W02419. Ferguson, R., 1987. Hydraulic and sedimentary controls of channel pattern. Inst. British Geographers Special Publication 18. Blackwell, Oxford, UK, Ch. 6, pp. 129–158. Ferguson, R., Parsons, D., Lane, S., Hardy, R., 2003. Flow in meander bends with recirculation at the inner bank. Water Resources Research 39, 1322–1333. Friedkin, J., 1945. A laboratory study of the meandering of alluvial rivers U.S. Waterways Experiment Station.

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