Abstract

River channel erosion by plucking is poorly understood even though it is a dominant mechanism for bedrock river profile evolution. In an experimental flume with fractured slabs of plaster model bedrock, plucking from a bed lacking protrusion is produced by nonuniform flow, particularly in rapidly varied flow with hydraulic jumps and free-surface undulations. Model bedrock slides upstream toward water-surface depressions in regions lacking recirculation, groups of blocks bulge up when a trough of free-surface waves moves above, and bubbles and debris particles move in the bed crack network. The likelihood and completeness of plucking increases with average flow strength but relies on local flow properties for initiation of motion. Particle image velocimetry (PIV) analysis of flow during a plucking event suggests that flow structures smaller than the average size of the blocks may be important in the plucking process by generating velocity or pressure variations around the blocks. Because plucking typically occurred near free-surface undulations and we have observations consistent with crack network flow, we propose that the mechanism driving block lift starts in the static pressure gradients developed in the sub-bed crack network, which are locally and temporally enhanced by turbulent pressure fluctuations. Positive feedback occurs when plucked blocks alter flow character and allow other blocks to slide around the bed, promoting additional plucking. Negative feedback occurs where the deposition of plucked blocks downstream of nonuniform-flow reaches limits transport capacity by changing or damping the nonuniform flow upstream. Our experimental results are consistent with previous engineering studies of slab uplift under plunging jets and high-Froude-number hydraulic jumps in energy-dissipating spillways. Our results also point toward the ability of nonuniform flow in bedrock rivers with a low Froude number to generate lift of fractured bedrock below steps and constrictions, and suggest a need for further study of mechanisms that initiate block plucking in experimental and field settings.

INTRODUCTION

Fluvial bedrock erosion sculpts waterways and sets rates of geomorphic processes that shape landscapes (Whipple et al., 2013). Relying largely on the stream-power model of incision thus far, geomorphologists model evolution of river profiles or two-dimensional landscapes to: (1) understand geologic rates of landscape change and sediment delivery given tectonic, climatic, and geologic conditions (e.g., Cowie et al., 2006; Attal et al., 2011); and 2) invert profiles for information about past events such as uplift or base-level fall (e.g., Cook et al., 2009), ongoing tectonic deformation (Boulton and Whittaker, 2009; Miller et al., 2013), or river-basin capture (Willett et al., 2014). Although the stream-power model offers reasonable predictions of first-order response of a fluvial system to perturbations, it requires researchers to use simplified erodibility terms to quantify and express complex mechanisms that are studied at the river-reach or depth scale.

Removal of bedrock from river beds is accomplished only when it is exposed (Gilbert, 1877; Davis, 1889) and can be dissolved, attacked by material in transport, or removed by hydraulic forces (reviewed by Turowski, 2012). The primary mechanisms of erosion in most insoluble bedrock settings (Springer et al., 2003; Covington et al., 2015) are abrasion and plucking (Whipple et al., 2000a). Cavitation is deemed possible (Baker, 1988; Hancock et al., 1998) and has been found to be exceptionally minor in channels except during jökulhlaups (Carling et al., 2017), but remains largely unexplored in fluvial environments (Thompson and Wohl, 1998). Bedrock erosion by abrasion has been theoretically modeled by Sklar and Dietrich (1998), quantified using mill experiments (Sklar and Dietrich, 2001), studied through saltation experiments and modeling (e.g., Chatanantavet et al., 2013), and examined in field settings (e.g., Hobley et al., 2011).

Bedrock erosion by plucking is poorly documented in geological literature (Whipple et al., 2013), yet existing studies of bedrock river reaches suggest that plucking is the dominant erosional mechanism where bedding and joint spacing permit it and incision is rapid (Miller, 1991; Hancock et al., 1998; Tinkler and Wohl, 1998; Wende, 1999; Whipple et al., 2000b; Jansen, 2006; Anton et al., 2015). Field evidence from repeat surveys conducted by Beer et al. (2017) provides a case study wherein the potential for erosion by plucking is an order of magnitude larger than by abrasion. Baker’s (1978) characterization of plucking as erosion by hydrodynamic forces on fractured bedrock, such as forces arising from vortex motion and separated flow, indicated that plucking is a complex yet effective process. Flood events are capable of lifting or rotating blocks out of the bed (Hancock et al., 1998; Wende, 1999; Whipple et al., 2000a; Snyder et al., 2003), sliding or toppling blocks downstream when the downstream ends of the clasts are exposed (Whipple et al., 2000a; Lamb and Dietrich, 2009; Dubinski and Wohl, 2013), or rotating blocks when the upstream faces are exposed (Wende, 1999). Block submergence and position relative to downstream obstructions have influenced movement of exposed blocks that have slid and been lifted in model and field studies (Carling and Tinkler, 1998; Wende, 1999; Carling et al., 2002). Channel bed properties—including bedding-plane dip, weathering, and geometry of joints, fractures, and bedding planes—influence the threshold of motion in different channels and are important factors in frictional resistance to movement (Miller, 1991; Annandale, 1995; Wende, 1999; Dubinski and Wohl, 2013). Studies by Baker and Costa (1987) and Whipple et al. (2000a) suggest that block size, orientation, and friction against adjacent blocks, rather than the mechanical strength of bedrock, determine erosion thresholds. In the few geologic studies seeking to explore bed evolution through plucking, plucked blocks have been modeled as alluvial sediment using the assumption that the bedrock has been weathered or battered into a sufficiently loose condition to be entrained (Chatanantavet and Parker, 2009; Lamb and Fonstad, 2010; Chatanantavet and Parker, 2011). Given that plucking is shown to be a significant process in certain bedrock channel settings, there is a clear need for further experimental and field research into the conditions and hydraulic forces that initiate plucking in fractured bedrock.

Engineering studies of plucking are considerably more developed. Under impinging jets in plunge pools, rock blocks are removed by “impulsive erosion” (Bollaert and Schleiss, 2005) wherein pressure fluctuations (e.g., Liu and Li, 2007; Pan et al., 2014; Li et al., 2016) weaken blocks by extending fractures (Annandale, 1995; Bollaert, 2002; Bollaert and Schleiss, 2003, 2005; Pan et al., 2014) and then amplify within the fractures (Bowers and Toso, 1988; Bollaert, 2016) to ultimately lift blocks (Liu and Li, 2007; Peiqing and Aihua, 2007). In the “accumulative plucking” model of Li et al. (2016), joint walls clamp back down on a block and allow repeated occurrences of minor uplift to incrementally lift the block out of place. Similarly in rivers, Hancock et al. (1998) defined hydraulic jacking, where sediment fills cracks as they widen, promoting incremental block uplift. In highly turbulent flows under energy-dissipating hydraulic jumps at spillways, pressure fluctuations (Narasimhan and Bhargava, 1976; Wang et al., 2015) are transmitted along with the fluid into the channel bed (Toso and Bowers, 1988; Fiorotto and Rinaldo, 1992; Bellin and Fiorotto, 1995; Fiorotto and Caroni, 2014; Fiorotto et al., 2016; Barjastehmaleki et al., 2016), which results in failure of the channel materials. Lastly, where plucking is experimentally caused solely by flow around or above a block, exposed blocks are moved by lift forces that co-vary with drag force (Carling et al., 2002; Lamb et al., 2015), or block protrusion is necessary to generate either stagnation pressure in the cavities around the block or wake flow above the block sufficient to cause lift (Reinius, 1986; Coleman et al., 2003; Annandale, 2005; Frizell, 2007; George et al., 2015).

This study is motivated by the need to understand erosion mechanisms of bedrock channels where rare plucking events may dominate erosion patterns and rate. In a small laboratory flume, we evaluate the hypothesis that irregularly fractured model bedrock can be plucked from an otherwise smooth bed lacking protrusion. In addition, we test whether plucking in nonuniform flow increases with greater stream power and with changes in bed geometry (Elium et al., 2014). We vary slope and downstream-facing steps to promote turbulence and enhance nonuniform flow features like hydraulic jumps. Our experimental results are supported by engineering work on rapidly varied flow, but rather than producing plucking phenomena like those of dam spillways and plunge pools, our model setting generates flow with Froude numbers closer to those of critical flow and thus illustrates phenomena likely to form in rivers away from waterfalls (e.g., Baynes et al., 2015a, 2015b). Our results have implications for understanding how nonuniform flow influences bedrock erosion and step evolution in natural settings, and for informing future modeling of bedrock erosion in channels.

METHODS

We conducted all experiments in a 245-cm-long × 14.3-cm-wide × 35-cm-deep flume (Fig. 1). The flume’s main compartment walls are 13-mm-thick clear acrylic. The main compartment sits above a 10-cm-high inflow channel, which is supported between two wooden beams. Water is recirculated at a rate of 3.5 ± 0.12 L/s from a 375 L tank by a Tsurumi HS2.45 pump. Water flows through the inflow channel into the main compartment, passing through a 4.7-cm-long soda-straw flow straightener. Water depth and velocity are controlled by a sluice gate mounted at the downstream end of the flume. The flume floor and test reach boundary are lined with immobile ceramic tiles that surround fractured plaster model bedrock.

The model bedrock in all cases was plaster of Paris (saturated density [ρ] of 1.40–1.58 g/cm3). This low-density material is appropriate for scaling the plucking of bedrock blocks in natural rivers (Lamb et al., 2015). For all experiments, except for a single particle image velocimetry (PIV) experiment, plaster mixed using standard instructions was poured into a 30-cm-long, 14-cm-wide, ∼1-cm-thick block set mold. We wrapped cured slabs in cloth and fractured each twice, once each on the long and short axes, using a rolling pin. We numbered each piece with permanent marker and photographed each set under a sheet of clear gridded plastic (Fig. 1). We rectified images in ArcGIS software to characterize block shape by planform area and average length and width. The planform area of the blocks used in our experiments averaged ∼18 cm2 and ranged between 10 and 25 cm2 (see the supplemental block set geometry data1). Before each experiment, we measured the dry and saturated weights of four to five blocks to characterize the properties of the entire block set using an electronic balance, and measured the thickness of the same blocks using a micrometer.

We conducted three styles of experiments to analyze the relationship between flow character and block removal: (1) 20 “slope-step” experiments that varied flume slope and the presence and height of a downstream-facing step at the upper side of the test reach (Figs. 1A–1C); (2) one “multilayer” experiment that comprised 12 plaster block sets (three layers, each consisting of four block sets laid end to end) (Fig. 1D); and (3) a single PIV experiment that aided flow visualization during plucking of one of three plaster blocks along the flume centerline.

Slope-Step Experiments

In the slope-step experiments, the 30-cm-long test zone extended from 78 to 108 cm upstream of the flume exit. The upstream extent of the test zone was 116 cm downstream of the flow straightener. In these experiments, we studied combinations of bed slope (0.493%, 0.800%, and 1.240%) and presence and height of an upstream step (no step, a one-tile step 0.8 cm high, and a two-tile step 1.6 cm high). We used each block set (A, B, C, etc.) in two experiments (Aa and Ab, Ba and Bb, etc.). For each experiment, we placed test blocks in the dry flume, covered them with a wire mesh, filled the flume to subcritical flow, removed the wire mesh, and agitated the blocks by hand to release trapped air bubbles from blocks and adjacent ceramic tiles.

Experiment timing began when subcritical flow stabilized above the test reach; we recorded water surface height at 30 cm intervals along the flume length. We raised the gate 1 mm every 5 min and recorded water height after the water surface profile stabilized following each gate change (10–20 s). As we raised the gate higher, allowing more water to pass through the gate, supercritical flow formed in the channel upstream of the test reach, creating an undular (Froude number [Fr] <1.7) to weak (Fr <2.5) hydraulic jump (Chow, 1959). The jump moved downstream and eventually across the test reach with each additional gate raise (Supplemental Movie A2). When the jump crest moved within 10 cm upstream of the test reach, we increased the time interval between gate height changes to 20 min to allow more time for plucking to occur. We continued this procedure until each experiment concluded at the first of two scenarios, either (1) all tiles were plucked from the test bed, or (2) the gate was fully open and plucking ceased. Experiments lasted 1–2 h. We measured water discharge three times after each experiment by filling a 20 L bucket for 3–4 s at the flume outlet. Discharge measurements varied by ∼3% and did not systematically change with gate height or slope; we thus assumed constant discharge for all experiments and used flow depth to estimate average channel velocity.

All experiments were recorded with a video camera. Short durations of some experiments were also recorded with a high-speed camera (500–1000 frames/s at 100 μs exposure time) for enhanced visualization (e.g., Supplemental Movie A [footnote 2], 00:58). Video footage for experiment Ba was lost; we thus conducted experiment Ja with identical flume setup as a substitute (Table 1). To investigate whether block size has an influence on bed erodibility, we conducted experiment Ka, an experiment with the same slope and step configuration as experiment Ja but with a plaster set containing more, smaller blocks. Including the repeated experiment and the trial investigating the importance of block size, the slope-step trials resulted in 20 experiments.

Multilayer Experiment

To evaluate the degree to which plucking of blocks creates or hinders conditions for further plucking, we performed one experiment with three layers of fractured bedrock along 120 cm of the flume floor (from ∼78 to 198 cm) such that plucking, transportation, and deposition of blocks created flow variability (Fig. 1D). Bed slope was 0.8%. The uppermost plaster set was flush with the stationary ceramic tiles upstream and downstream of the plaster reach, so that the bed was flush from the flow straightener to the outlet. Having multiple layers with small differences in plaster thickness led to minor protrusion (≤1 mm) at the upstream end of a block set. The flume was filled with standing water, and blocks were placed into position, allowed to saturate for 2 h, and were agitated to release air from between the blocks. We seeded the water with 150 μm urea particles to visualize flow. The experiment began when supercritical flow entered the upstream end of the plaster reach. Following plucking, some large blocks settled on the flume sidewalls at the downstream end of the plaster reach, creating a backwater such that plucking ceased. We removed these blocks to allow continued erosion; blocks resting on the bed were not disturbed. The multilayer experiment ran for 2 h.

Particle Image Velocimetry Experiment

We conducted one PIV experiment to measure velocity and visualize flow structures. A laser sheet was produced using a continuous-wave, optically pumped semiconductor laser (Coherent Genesis) with a power output of 100 mW. The sheet was formed using a series of spherical and cylindrical lenses (La Vision 1108405), producing an approximately uniform thickness of 1.0 mm throughout the image plane. Particle images were acquired with a high-speed digital camera at a frame rate of 1000 frames/s and an exposure time of 100 μs. The exposure time minimized particle blur within images, and the acquisition rate minimized particle dropout between sequential images. Images were rectified using a calibration target submerged in the flume to scale image size and account for refraction between the camera line of sight and the direction normal to the laser sheet. We processed images using commercial software (La Vision 7.1) with a constant window size of 32 × 32 pixels with 50% overlap and three passes. Vector filtering and smoothing were conservatively performed in post-processing.

We built a ceramic tile test bed to hold a 4.6-cm-wide rectangular set of three plaster blocks in the middle of the flume, starting 5 cm downstream of a 0.8-cm-tall step, with a flume slope of 0.8%. This special block arrangement ensured that if plucking occurred, it would occur in the plane of the laser sheet. The high-speed camera was triggered such that ∼2.5 s of flow was captured preceding the plucking event and ∼1.5 s following. We used unfiltered water and seeded the flow minimally with 150 μm urea particles, which produced sufficient particle density to calculate local velocity in the flow. The set of three plaster blocks was much less prone to plucking than the full plaster sets used in the other experiments, therefore we temporarily obstructed the flow well downstream of the plaster tiles to shift the jump position upstream through the test plaster reach, and then let it retreat downstream; plucking occurred ∼12 s after the obstruction was removed and the jump resettled downstream.

RESULTS

Slope-Step Experiments

We observed plucking of blocks from an initially flat plaster surface in all but three of the 19 slope-step experiments for which we had preserved video footage (i.e., excluding experiment Ba) (Table 1). At the lowest slope, blocks were not plucked without at least one tile step. At moderate slope without a tile step, plucking occurred in only one of the two experiments with this setup. Because discharge was constant and the one- or two-tile-thick step created a steep jump, results of these 20 experiments show that increased bed slope or step height created the conditions for successful plucking and yielded more complete plucking.

In all but one of the experiments in which plucking occurred, plucking initiated within 10 cm upstream or downstream of the hydraulic jump toe (Fig. 2; Supplemental Movie B3; supplemental movie transcripts4), or immediately downstream of the step where blocks were lifted in recirculating flow behind the step. In general, the first block was rotated from the downstream edge such that the upstream edge lifted into the flow, either on its own or due to being bowed up in contact with a neighboring block. In some experiments, several blocks bowed up together, with one eventually breaking free and the others dropping back to the bed. Pairs or groups of blocks bowed up on both the upstream-downstream and lateral edges. Blocks commonly oscillated in the bed before being plucked, particularly directly under the hydraulic jump (Supplemental Movie B [footnote 3] [experiment Cb], 05:45). In several experiments, small bubbles and debris particles beneath the plaster blocks and along the margins of the flume were observed to flow upstream away from deeper water downstream of the jump and toward the toe of the jump (Supplemental Movie B [footnote 3] [experiments Da and Fa], 06:34 and 08:20). Arriving at the toe of the jump, they would move up through cracks or voids where blocks had been removed. In the lee of a step, blocks in the first row commonly lifted at the upstream edge, but only in a few experiments did they lift far enough to be the first block to be plucked.

Following removal of the first block, other blocks commonly followed quickly, usually as a string of blocks downstream of the first (Fig. 3). Open space in the block set allowed remaining blocks to move longitudinally and laterally in the test reach. Many blocks slid downstream into open space until they encountered either another block or the end of the test section, and then were lifted from the upstream edge and plucked (Supplemental Movie B [footnote 3] [experiment Da], 06:24). Counterintuitively, blocks also slid upstream into sites of removed blocks and were then lifted at the upstream edge and removed (Supplemental Movie B [footnote 3] [experiment Ea], 07:30). Upstream sliding was commonly simultaneous among several blocks downstream of a void, with larger spaces growing between each block as they slid one by one up into the zone of plucking and were removed (Fig. 3; Supplemental Movies B [footnote 3] and C5). Upstream sliding generally caused blocks to slide from below the highest water surface of a hydraulic jump toward the toe. Even though upstream sliding did occur in the zone of recirculation in experiments with a step, it was common downstream of this zone and in experiments without a step as well.

Plucking commonly started shortly after a gate change that caused jumps to either change position or become stronger. Of the 16 experiments in which plucking occurred, half the experiments had plucking occur within 30 s of a gate change (Table 1), which is during the time that the nonuniform flow was quickly changing location and character. For example, in experiment Ab (moderate slope, one-tile step), the location of the jump toe, as determined by the water surface minimum, moved downstream 1.5 cm and dropped 1.5 mm in elevation over the 13 s prior to the first block lifting (Supplemental Movie B [footnote 3] [experiment Ab], 03:30). Jump transitions and adjustments were not smooth, with toe position and elevation oscillating over a distance of 1 and 0.1 cm respectively during the change. Plucking initiated directly underneath the deviation of the jump toe water surface that was lowest and farthest downstream following the gate change. During steady conditions (no gate changes), the height and slope of the hydraulic jump and position of its toe changed over a similar magnitude. This level of nonstationary behavior was not tracked for its influence on when or where plucking occurred.

Alteration to the test bed after blocks were plucked commonly caused changes in the location and character of the hydraulic jump. In cases where plucked blocks settled in a cluster downstream of the jump, the jump responded to the flow obstruction by moving upstream (Supplemental Movie B [footnote 3] [experiment Ha], 01:21). In other cases, the flow transported plucked blocks completely out of the flume, which caused the jump to migrate downstream. Jump migration often led to additional plucking near the new position of the jump toe.

In this study, block size or shape did not appear to influence the ability of the flow to lift and pluck a block, with one exception. Generally, the first blocks plucked by the flow were not consistently smaller, larger, longer, or wider than the average for all blocks in each respective set. The exception is that in three experiments, blocks much smaller than the block set mean and located at the upstream corners in the lee of a step were the first blocks to be plucked (Table 1; Supplemental Movie B [footnote 3] [experiments Hb, Ca, and Cb], 01:28, 04:23, and 05:02). These plucking events were not associated with the main plucking event, and no adjacent blocks eroded. The plucking events that resulted in removal of more than just one small block always initiated further downstream, close to the hydraulic jump. Following these findings, we conducted experiments Ja and Ka with identical flume setup to investigate whether block size or shape influences plucking. The two plaster sets had different numbers of blocks and therefore different sizes; set Ja had the fewest blocks (largest size) of any set used, and set Ka, the most blocks (smallest size). We found that plucking occurred in a similar location in both experiments, close to the trough of the hydraulic jump.

Multilayer Experiment

The multilayer experiment confirmed that plucking of layered model bedrock occurs in the presence of hydraulic jumps as well as free-surface oscillations (Supplemental Movie D6). Plucking of the initially smooth test surface began at two separate locations (Figs. 4A and 4B) via lift in subcritical flow with free-surface oscillations downstream of the entering supercritical flow. The upstream, and first, plucking event happened within several seconds of a gate change, causing the jump and free-surface waves to move (Supplemental Movie D [footnote 6], 00:22). The second plucking event happened just downstream of where the first block came to rest on the bed (Fig. 4B; Supplemental Movie D [footnote 6], 00:30). In both places, plucking began at the upstream end of a block set, where a single crack extended across the entire width of the bed. Later, after most of the top layer was plucked, a downstream-facing exposed edge on the upstream-most block set should have permitted the blocks in that set to slide downstream like in the experiments of Dubinski and Wohl (2013) (Fig. 4C; Supplemental Movie D [footnote 6], 02:20). However, in this example, the exposed edge acted as a step and changed the flow to create free-surface undulations downstream of the step. Plucking below the step occurred before the blocks slid off the step itself. Counterintuitively, when the blocks in the step-forming set did erode, they were plucked by lifting rather than sliding (Fig. 4C; Supplemental Movie D [footnote 6], 02:30).

Similar to the plucking produced in the slope-step experiments, plucking was much more likely to occur during transient flow conditions. During changes in flow, we observed multiple blocks in a layer bridge up under water-surface undulations and then drop back down to the bed (Supplemental Movie E7). Plucking of blocks created a hole in the uppermost layer, and multiple blocks commonly resettled on the bed downstream to create a localized jump in a further downstream reach (Supplemental Movie D [footnote 6], 00:30). Thus, changes in bed topography created flow feedbacks like those observed in the slope-step experiments. New steps, as well as the holes left behind by plucked blocks, caused jumps and free-surface undulations to move upstream or downstream, in some cases enhancing continued plucking and other cases restricting it.

PIV Analysis

Flow velocity calculated from PIV measurements identifies the development and downstream movement of clockwise-rotating (in a left-to-right–oriented flow) coherent flow structures in the lee of a tile step (Fig. 5; Supplemental Movie F8). The middle block was the first to lift by rotating up at its front edge. The front edge of this block was ∼7.5 cm downstream of the step, a position nearly coinciding with the water-surface elevation minimum near the jump toe and just downstream of the expected reattachment point of flow separation behind the step (Schmeeckle, 2015). Mean velocity determined from discharge in the 3-cm-deep flow was 0.8 m/s with a Froude number of 1.4; PIV-measured horizontal velocity above the upstream crack just above the bed averaged 0.62 m/s for the 2.4 s immediately before the block lifted, which then disturbed the flow field and blocked the laser beam. Velocity varied strongly over the 3-cm-deep flow (Figs. 5C and 5D). Downstream of the middle block (downstream of 100 mm in Fig. 5), the thread of higher velocity moved upward into the jump. The regions of concentrated clockwise vorticity were ∼0.3–0.5 cm in diameter and moved downstream at ∼0.35 m/s, somewhat slower than the average horizontal flow velocity of 0.66 m/s at the height of the vortex centers. The flow structures were “uncorrelated” (sensuBollaert, 2002) because they were an order of magnitude smaller than the 3.8-cm-long block and did not deliver similar conditions to all cracks around a block simultaneously. During the 2.4 s prior to the plucking event, multiple flow structures crossed the surface of each block. At the onset of plucking, the degree, location, and size of rotational flow was not anomalous. However, at the initiation of upward rotation of the block (2.4 s into the 4 s of recording), the region above the crack at the front of the plucked block underwent an anomalous period of slowed downstream and increased vertical flow (Fig. 6; Supplemental Movie F [footnote 8], 01:29). We characterize these flow changes as “ejections” based on the quadrant analysis of Kline et al. (1967). This excursion from the mean is made more apparent by comparison to the downstream crack, where the opposite flow conditions existed in the 0.1–0.2 s prior to the beginning of protrusion (Fig. 6).

DISCUSSION

Initiation of Plucking

The experimental results highlight a potentially important phenomenon wherein a fractured but otherwise smooth channel bed lacking protrusion can be plucked by nonuniform, unsteady flow. In nearly every experiment, areas of low water-surface elevation in free-surface undulations and hydraulic jumps are the sites of initial motion. Several observations suggest that plucking is initiated by fluid flow and associated pressure disturbances in cracks surrounding the blocks: upstream sliding of blocks near the jump, movement of sub-bed particles and bubbles through the crack network toward the site of plucking, plucking most commonly occurring near the toe of the jump, and plucking commonly initiating during transient flow. We propose three contributions to the pressure distribution acting on the lower surface of a block (Fig. 7) that combine to initiate plucking: (1) flow in the crack network driven by the gradient in the elevation head of nonuniform flow (Fig. 7A), (2) pressure fluctuations transmitted to the crack network from unsteady, turbulent flow structure within the hydraulic jump (Fig. 7B), and (3) pressure fluctuations transmitted to the crack network by turbulent flow structures generated upstream of the site of plucking (Fig. 7C). These mechanisms align with both our observations and extensive engineering results. In the following discussion, we first take up the issue of protrusion. We then estimate the contributions of our proposed mechanisms using the example of plucking in experiment Eb. Lastly, we examine feedbacks and the impact of flow transience, and consider how these mechanisms scale to flows in rivers.

Our proposed mechanisms explain block uplift from a cracked layer that lacks protruding edges, often viewed as necessary for the initiation of plucking (Reinius, 1986; Coleman et al., 2003; Lamb and Dietrich, 2009; Lamb et al., 2015). It has been observed that stagnation pressure can lead to plucking when relative block protrusion (protrusion height normalized to block length) is >0.1 (Reinius, 1986; Coleman et al., 2003) or, if relative protrusion is less, in flows with a Froude number much greater than 3 (Reinius, 1986; Frizell, 2007; George et al., 2015). In this study, however, plaster blocks with an average length of 34 mm protruded substantially less than 1 mm, and generally <0.1 mm. The Froude number of flows was generally <2. Instantaneously following initiation of plucking from a smooth bed, block protrusion arose locally (Fig. 3; Supplemental Movie A [footnote 2]). Where it did exist, protrusion appeared to render blocks easier to pluck, as shown during the multilayer experiment where some block corners had 1 mm of protrusion; the first two areas to pluck had this minor protrusion. However, most of the plucking events that we observed initiated from a set of blocks lacking protrusion in low-Froude-number flows.

To initiate plucking, the pressure distribution beneath the block must overcome the pressure distribution on top of block plus the submerged weight of the block, which includes the buoyancy force arising from uniform pressure distributions around the block. As such, we focus on the pressure distribution below the block as the cause of plucking. We use experiment Eb (Fig. 3; Supplemental Movie A [footnote 2], 00:58) as an example to estimate the pressure acting on a block at the time of plucking. The first block to pluck (block 13) has a saturated density of 1.44 g/cm3, a thickness of ∼5.75 mm, and top-surface dimensions of 4 cm × 8.8 cm. Dividing the saturated, submerged weight by the top-surface area results in a downward stress of 25 Pa.

The first proposed contribution to the pressure under the block [Ph(x), Fig. 7A] is an established mechanism due to flow and pressure transmission in the presence of a water surface gradient (Bowers and Toso, 1988). The resulting interstitial hydraulic gradient creates a nonuniform pressure distribution below the block, and subsequent flow generates shear stress on the block lower surface. Flow beneath the bed aligns with models of a dam-failure case where fluid and pressure under a hydraulic jump was transmitted upstream through the sub-bed drain and caused uplift of blocks on the dam spillway (Bowers and Toso, 1988). In other settings, seepage and transient flow under blocks has been modeled as part of spillway and plunge-pool block uplift (e.g.,Liu and Li, 2007; Pan et al., 2014). In experiment Eb, the neighboring jump crest is 4 cm higher than the toe at the time of plucking, yielding 400 Pa of static head, which can drive crack flow, create sub-block shear, and potentially create block uplift by increasing sub-block pressure. If we approximate the pressure distribution in the crack network as decreasing linearly over the 15 cm between the jump crest and the site of plucking, the 400 Pa of head could induce a pressure gradient of 2.6 kPa/m within the crack network. Assuming that plane Poiseuille flow applies in the thin (0.5–1 mm) horizontal space below the plaster block, average flow velocity (forumla) is given by (Fay, 1994): 
graphic
where h is the thickness of the crack (in m), μ is the viscosity (in Pa·s), and dP/dx is the pressure gradient (in Pa/m) ignoring the slope of the flume. The average sub-block velocity in experiment Eb at the time of plucking is estimated to be on the order of 0.1 m/s. This sub-block flow velocity gives rise to a shear stress of 1–2 Pa, about one-third of the macroscopic shear stress in the supercritical flow over the block. If the pressure gradient of 2.6 kPa/m is transferred under the block, the gradient across a 4-cm-long block (block 13) yields an area-averaged uplift pressure of ∼50 Pa. Flow and shear below the block helps explain upstream movement of blocks in a constant downstream flow and the tendency of plucking to initiate near water-surface minima.

For the second proposed mechanism [Pp(x, t), Fig. 7B], the static pressure distribution under blocks is modified by pressure variations with time caused by turbulent, unsteady behavior in the overlying nonuniform flow. Variations include unsteady behavior of hydraulic jump location and height (e.g., Toso and Bowers, 1988; Wang et al., 2015), highly turbulent flow in jumps (Fiorotto and Rinaldo, 1992), and dynamic pressures from vertical flow at constrictions (Venditti et al., 2014). Disturbances generated by these flow variations are transmitted into the crack network and temporally modify the pressure distribution below the block surface. If the disturbances are of a spatio-temporal scale capable of pressurizing a sufficient portion of the block lower surface, lift can occur (Pan et al., 2014). For experiment Eb, pressure disturbances should peak between the toe and crest of the jump (Toso and Bowers, 1988; Farhoudi and Narayanan, 1991; Fiorotto and Rinaldo, 1992; Fiorotto and Caroni, 2014; Wang et al., 2015) and vary the local pressure by ±400 Pa, which is nearly equivalent to the velocity head of the flow entering the jump (Bellin and Fiorotto, 1995). Although the distribution of pressure is unknown for our flows, this doubling of pressure that influences the crack network occurs close to the jump toe (Bellin and Fiorotto, 1995), which is where we observed plucking to initiate most commonly in our experiments. Certainly, jump pressure disturbances modify the pressure distribution on both the top and bottom surfaces of the block, but because the disturbances must be transmitted through the crack network to reach the bottom surface, we expect the disturbances to the top and bottom to be uncorrelated (sensuBollaert, 2002). Hence, large increases in pressure in the crack network might occur when a reduction in pressure occurs over the top of the block.

For the third mechanism [Pd(x, t), Fig. 7C], turbulent flow structures passing above the blocks produce pressure disturbances or flow in the cracks that could trigger a plucking event (Whipple et al., 2000a). Experiment Eb compares well in scale with a flume study of sediment transport behind a step (Nelson et al., 1995) that was computationally modeled by Schmeeckle (2015). His results from modeling subcritical flow over a step without strong water-surface topography suggest that the turbulent structure of flow downstream of a step can generate local pressures fluctuations at the bed on a scale of ±10–12 Pa, which is a value nearly half of the submerged downward stress exerted by block 13 in our experiment Eb. The addition of steps increased the occurrence of plucking in our experiments, which indicates that this extra source of turbulence helps promote plucking. The PIV-imaged turbulent structures we measured are much smaller in diameter than the length of the block surface and therefore provide localized pressure disturbances at a specific crack location. Thus, blocks are pressurized nonuniformly, which could help to explain why we observed plucking occur via rotating up from one edge or bowing up together at a crack (e.g., Supplemental Movie B [footnote 3], 04:50) in most of our experiments.

The estimated magnitude of each proposed mechanism acting in experiment Eb is significant with respect to the downward stress imposed by block 13 and could therefore initiate plucking. Yet, plucking also occurred in our experiments where water-surface undulations were less extreme (Fig. 4; Supplemental Movie B [footnote 3]) and in conditions where turbulence would be expected to be lower (moderate-slope, no-step experiments). Hence, rather than one cause, we believe that combinations of these mechanisms initiate plucking in unsteady, nonuniform flow. If flow or pressure variation in the cracks is slower to respond than in the overlying flow, an unsteady water surface profile will augment the disequilibrium pressure distribution that facilitates plucking. In the slope-step experiments, plucking was most likely to occur soon after a gate height change when the water surface was still adjusting (Table 1; Supplemental Movie B [footnote 3]). In the multilayer experiment, plucking typically occurred when hydraulic features such as water-surface undulations moved due to gate changes or erosion and deposition in adjacent reaches. George et al. (2015) also observed plucking in many of their experiments during transitional flow conditions. The importance of unsteady behavior in generating flow in the subsurface is supported by bulging bed layers that appeared to follow a moving water-surface trough during the multilayer experiment (Supplemental Movie E [footnote 7]).

Implications for Plucking in Rivers

Although our observations of plucking take place in a narrow flume using low-density “bedrock,” the proposed mechanisms involving transmission of pressure variations to the site of plucking likely apply to conditions common in bedrock rivers. Plucking is common in flows reaching critical flow (Fr = 1) conditions (Tinkler, 1997a, 1997b). The oblique waves, water-surface undulations, standing waves, and jumps and/or jets at bedrock steps and constrictions observed by Jansen (2006) produce temporal and spatial changes of water-surface height and velocity (Kieffer, 1985). Similar to observations from model studies, we expect pressure to fluctuate due to turbulent structures under these unsteady waves (Kieffer, 1987). As another example, Tinkler (1997a) documented strong water-surface topography in bedrock streams where plucking occurs.

To arrive at river-scaled estimates for the proposed pressure mechanisms, we offer the following scaling example using Froude scaling of experiment Eb (Fig. 3). This scaling effort is motivated by a study by Tinkler and Parish (1998), who observed erosion of bedrock slabs of 10 cm or greater in thickness. Based on experiment Eb, a 40:1 depth ratio (natural channel to model scenario) scales to a 1-m-deep supercritical flow in a natural channel with a Froude number of 2.1 and a velocity of 6.6 m/s at the entrance to the jump. This velocity falls within the range considered by Tinkler and Parish (1998), who used the methods of Reinius (1986) to validate the erosion of bedrock blocks. Assuming a locally flat bed slope, the jump height resulting from scaling of experiment Eb is 0.66 m (Roberson and Crowe, 1985). A jump or standing wave with a height of 0.66 m and a streamwise distance of 4.2 m generates a static head of 6.5 kPa in the bed crack network with a hydraulic gradient of 1.5 kPa/m. These conditions lead to a flow velocity of 1.4 m/s in a 1-mm-wide bedrock crack. With a bedrock density of 2.6 g/cm3, a 10-cm-thick submerged block exerts a downward stress of ∼1.6 kPa. If the hydraulic pressure distribution Ph(x) across the bottom of the block is not sufficient to drive uplift of the block itself or induce crack network flow around the block that drags the block up, modifications to the pressure distribution by flow structure downstream [Pp(x, t)] or upstream [Pd(x, t)] of the block could combine to exceed the pressure caused by the submerged weight of the block. If the hydraulic jump is 0.66 m high, the pressure fluctuations under the jump itself are on the order of ±6 kPa, which is ∼4× the downward stress exerted by a 10-cm-thick block, and suggests that significant vertical motion of the bedrock blocks is possible. Given the common occurrence of 0.5- to 1-m-high waves in rivers during near-critical, high-discharge events, our experiments suggest a potential block extraction mechanism that needs further exploration in rivers.

In our experiments, discharge and hence reach-scale stream power remained constant for each slope, but the location of flow features and fate of plucked blocks controlled the energy-loss concentration. These observations suggest that “unexceptional” floods (sensuAnton et al., 2015) with conditions that generate local flow structures such as jumps and rollers could be more effective flows for plucking than deeper flows in which the amplitude of surface topography is negligible when compared to the depth of the flow. Nonetheless, our results do support findings of previous studies that initiation and rate of plucking scales to the imposed flow through some measure of channel-averaged shear stress or energy loss (Carling and Tinkler, 1998; Wende, 1999; Dubinski and Wohl, 2013; Lamb et al., 2015) because plucking occurred and was more complete as slope and step height increased (Table 1).

Feedback between plucking and deposition of eroded blocks is an important component of long-term maintenance of channel profiles. In flows over reverse steps, Wende (1999) observed the development of imbricated, transverse bars of plucked blocks that limited further plucking. Both Carling and Grodek (1994) and Carling (1995) documented cases where material eroded from or transported through hydraulic jumps was unable to leave the tranquil flow region below the jump where flow velocity and turbulence intensity decline. The facts that blocks erode from one section of the flow but cannot be re-entrained in others, and that in our multilayer experiment the erosion-deposition zones moved in both time and space, reinforce the notion that reach-averaged energy loss or stream power, on their own merit, do not effectively characterize the types of flow producing erosion (Tinkler, 1997a; Venditti et al., 2014). As such, three-dimensional models are required to describe bedrock erosion in rapidly varied flow common to bedrock rivers and gorges (Liao et al., 2014). The feedback between flow and transport of the plucked blocks in our study lends support to Howard’s (1998) idea that either the plucking of new blocks or their continued transport can control the rate of erosion by plucking. Downstream transport of plucked bocks in our experiments does limit further plucking in some cases, which resumes only when the previously plucked blocks are eroded and nonuniform flow conditions return to a more powerful configuration.

CONCLUSION

Our experiments suggest that plucking of blocks can occur from a fractured but otherwise initially smooth experimental channel bed where locally nonuniform flow generates hydraulic jumps and water-surface undulations. Increased stream power and the presence of steps are shown to increase the likelihood and completeness of experimental plucking. We observed blocks sliding upstream in downstream flow, particles and bubbles flowing through the experimental bed crack network, and PIV-imaged flow structures that caused changes in velocity above the blocks and crack network. These observations suggest that plucking is driven by three potential mechanisms: flow in the crack network driven by the difference in elevation head of nonuniform flow; pressure fluctuations transmitted to the crack network from unsteady, turbulent flow structures within the hydraulic jump; and pressure fluctuations transmitted to the crack network by turbulent flow structures generated upstream of the site of plucking. Collectively, these mechanisms are of sufficient magnitude to generate block uplift. Plucking occurs in the absence of protrusion and results in feedback that either enhances or restricts continued plucking. We find support for these mechanisms in the data and models of previous investigations, which potentially offer insight to how plucking starts in low-Froude-number environments common in bedrock rivers. The mechanisms we propose are compatible with conditions in rivers, and render both flume and field studies of plucking worthy of further exploration.

ACKNOWLEDGMENTS

Funding for the study was provided solely by Washington and Lee University, as faculty Lenfest grants to Harbor, undergraduate student Summer Research Scholar stipends to Wilkinson and Helgans, and miscellaneous personnel and equipment support by the Washington and Lee Geology Department. These same sources funded design, testing, and construction of the flume by Liz Elium, Murtaza Kapasi, Jack Wilbur, and James Freeman during the summer of 2014 (Elium et al., 2014). Preliminary flume investigations began with Liz Elium’s unpublished geology thesis in 2015. The use of plaster of Paris as a simulated bedrock was first tested by James Biemiller in his 2015 thesis in geology (Biemiller, 2015). Emily Falls provided mechanical assistance with flume construction, operation and maintenance. David Pfaff of the Washington and Lee IQ Center provided the high-speed camera and other videography equipment, and assisted with video creation and editing. Previous drafts were substantially improved following reviews by an anonymous reviewer, Paul Carling, Ellen Wohl, Geosphere Science Editor Shanaka de Silva, and Geosphere Associate Editor Brandon McElroy. Support for the publication of this article was provided by the class of 1956 Provost’s Faculty Development Endowment at Washington and Lee University.

1Supplemental block set geometry data. Images of each block set, table of the average size of blocks compared to the first blocks that are plucked in the two experiments using the set, and box-and-whisker plots comparing the set geometry to the first block for area, length, width, and length-width ratio. Please visit https://doi.org/10.1130/GES01623.S1 or access the full-text article on www.gsapubs.org to view the supplemental data.
2 Supplemental Movie A. Example of the experimental setup procedure using slope-step experiment Eb. The plucking event shows plucking by rotation and upstream sliding of blocks, captured with a high-speed camera. Incoming flow is right to left. To view Supplemental Movie A, click https://doi.org/10.1130/GES01623.ma to download the movie file.
3 Supplemental Movie B. Initial plucking events during each experiment, shown one after another and grouped from lowest to highest slope. For each set of experiments with the same slope, steps are built then removed in the following sequence: none, one tile, two tiles, two tiles, one tile, none. Incoming flow is right to left. To view Supplemental Movie B, click https://doi.org/10.1130/GES01623.mb to download the movie file.
4Supplemental movie transcripts. Time stamps and a written account of important events in Supplemental Movies A–F (text footnotes 2, 3, 58). Please visit https://doi.org/10.1130/GES01623.S2 or access the full-text article on www.gsapubs.org to view the movie transcripts.
5 Supplemental Movie C. High-speed video of experiment Db (high slope, one-tile step) showing bowing up of a group of blocks and plucking, followed by upstream sliding of overturned blocks back into the vacancy created by the plucking. Incoming flow is right to left. To view Supplemental Movie C, click https://doi.org/10.1130/GES01623.mc to download the movie file.
6 Supplemental Movie D. Plucking phenomena observed during the multilayer experiment. Incoming flow is right to left. To view Supplemental Movie D, click https://doi.org/10.1130/GES01623.md to download the movie file.
7 Supplemental Movie E. Video sequence from the multilayer experiment showing the movement of a water-surface wave and the simultaneous bulging of the plaster set under the wave trough. The movement is best observed by “scrubbing” or sliding the time bar of the video back and forth. Incoming flow is right to left. To view Supplemental Movie E, click https://doi.org/10.1130/GES01623.me to download the movie file.
8 Supplemental Movie F. Results of the particle image velocimetry (PIV) experiment. This video has been flipped from the original recording so that flow is shown conventionally, left to right, to match the data shown in Figure 5 of the manuscript. The first segment of the video shows the vertical sheet laser crossing the experimental flow during plucking the middle of three plaster blocks. In next two video segments, three panels show the PIV results with the video of the flow. The lower panel is the high-speed video. The middle panel shows clockwise vorticity (counterclockwise is black). The upper panel shows either instantaneous velocity direction and magnitude, or velocity difference (forumla), with color indicating absolute velocity magnitude in both. forumla is the instantaneous velocity vector at a given location in the current vector image; forumla is the average velocity vector of any current vector image. Subtracting forumla provides the difference vector, which represents deviation from the mean. forumla is a function of (x,y,t) and forumla is a function of (t). The time “t” is the moment of time represented by the current vector image. The final video segment is a repeat with a slower frame rate near the 2.4 s mark (2400 ms) when the middle block begins to lift at the upstream edge. This almost imperceptible movement is best observed by “scrubbing” or sliding the time bar of the video back and forth near 2400 ms. To view Supplemental Movie F, click https://doi.org/10.1130/GES01623.mf to download the movie file.
Science Editor: Shanaka de Silva
Gold Open Access: This paper is published under the terms of the CC-BY-NC license.

Supplementary data