Introduction
The retreat of knickpoints, i.e., localized steps in the river profile, is a
common process in erosion systems. Knickpoints are created in response to an
erosional perturbation and propagate information upstream into the landscape
as opposed to the downstream transport of sediments fed from hillslopes
(Whipple, 2004; Bishop, 2007; Allen, 2008). They are usually triggered by
relative fall of the river base level, whether by uplift of the river bed or
drop of the base level to which the river profile adjusts (e.g., a lake, a
dam, a fault offset or the sea level). Knickpoints distributed within a
landscape can thus be thought of as key signal carriers of external forcing
at play in the sediment routing system.
Through use of physical experiments, base-level falls can successfully
produce knickpoints over both alluvial/non-cohesive or bedrock/cohesive
substrates (for example: Brush and Wolman, 1960; Holland and Pickup, 1976;
Begin et al., 1981; Gardner, 1983; Bennett et al., 2000; Frankel et al.,
2007; Cantelli and Muto, 2014). Under supercritical flow conditions, the
shape of the knickpoints is well preserved (Bennett et al., 2000; Cantelli
and Muto, 2014). In some cases, upstream-migrating steps occur as a train of
closely spaced knickpoints bounded by hydraulic jumps, termed “cyclic
steps” by Parker (1996; Fig. 1). One might directly associate the presence
of single knickpoints or trains of cyclic steps along a river with an ongoing
or past external change, e.g., a relative base-level fall triggered by
climate change or tectonics. However, knickpoints may also form in response
to the reduction of sediment discharge along the river or can even be
autogenic, arising from natural variability within a drainage basin
(Hasbergen and Paola, 2000). Furthermore, dissipation is commonly observed as
knickpoint retreat, and so the height of a knickpoint face does not
necessarily reflect the initial base-level fall (Parker, 1977; Gardner, 1983;
Crosby and Whipple, 2006; Whipple, 2004; Bishop et al., 2005). Overall, there
is still much to be worked out about the specifics of how knickpoints encode
and carry erosional information.
Additionally, lithologic controls over river profiles and their knickpoints
have long been recognized (Hack, 1957; Bishop et al., 1985; Miller, 1991;
Pederson and Tressler, 2012). In recent field examples, Cook et al. (2013)
measured lower rates of knickpoint retreat above more resistant rock, while
Grimaud et al. (2014) documented the persistence of lithogenic knickzones
(e.g., > 30 km long steeper reaches) at continental scale.
Finally, Sklar and Dietrich (2001, 2004) highlighted bed lithology, i.e.,
variations in bedrock strength or alluvium thickness, as a major limiting
factor of river abrasion capacity, through, for example, boulder armoring
(Seidl at al., 1994), and therefore a control over the response timescale of
the sediment routing system (see also Gasparini et al., 2006).
Schematic longitudinal section of a river bed before
(a) and during (b) the propagation of a knickpoint
triggered by relative base-level fall. Blue arrows represent flow direction
and black arrows the motion of the bedload. The black and blue dashed lines
respectively represent the bedrock and water levels before knickpoint
propagation. (c) Idealized representation of a knickpoint
characterized by its velocity, Vkp, and the depth of associated
plunge pool, Hp.
In this study, we investigate experimentally the effect of bed lithology and
uplift style on knickpoint evolution. The experiments provide simple cases
of 1-D evolution that are relevant for comparison with individual river
segments. The results highlight the strong effect of bedrock lithology on
knickpoint characteristics and show how incision and knickpoint propagation
are influenced by transient deposits along streams. They also show a form of
self-organization in which multiple small base-level steps may be required
to produce a single knickpoint. This points to a new form of knickpoint
self-organization that controls the relative rate at which knickpoints are
generated as a function of the rate and magnitude of base-level fall. The
results suggest that knickpoint spacing, though not vertical magnitude
alone, is an indicator of base-level fall rate.
Experimental setup
Flume design and experiment sets
We carried out experiments on river incision at the St. Anthony Falls
Laboratory, University of Minnesota, Minneapolis. To minimize planform
complications such as bars, we constructed a small, narrow flume to test the
impact of base-level fall style and bed lithology on stream erosion. The
flume is 1.9 cm wide, about 100 cm long and 36 cm high (Fig. 2). We supplied
a constant water discharge (Qin=1250 mL h-1) over a
cohesive substrate, which eroded and formed a profile. The substrate is very
similar to the one used by Hasbargen and Paola (2000). It is composed of
silica sand (density = 2.65; d50=90 µm), kaolinite (density = 2.63; d50 < 4 µm) and water. The composition of the
substrate controls its erodibility, one of the key variables we wished to
study. This substrate is placed wet into the flume and its top surface
flattened as much as possible. The experiment starts immediately. Water
introduction causes the slow erosion of the first upstream 10 cm of the
flume that provides a constant minimum bedload (qs ∼ 3 g min-1). This bedload acts as an abrasion tool
throughout the experiments (Sklar and Dietrich, 2004; Fig. 1). The stream is
perturbed by lowering the downstream end of the flume using a sliding gate
(Fig. 2). In response to this perturbation, knickpoints develop and retreat
upstream (Figs. 3 and 4).
Experimental setup. Base-level fall, of rate U, is produced by
lowering the sliding gate. Qin is the water discharge introduced the
flume using a constant head tank. Qout is the water discharge measured
at the outlet of the flume. Because of absorption by the substrate,
Qin (1250 mL min-1) is superior to Qout in every experiment
(see Table 1).
Illustration of a knickpoint observed along the flume during
experiment 10. (a) Overall view of the profile and
(b, c) details of the knickpoint. Note the
white color of the water due to suspended sediments.
Summary of the main characteristics for each experiments. τeq represents the equilibrium shear stress. NA stands for no
acquisition.
Experiment
1
2
3
5
6
7
8
9
10
11
Base-level fall rate, U (cm h-1)
2.5
5
1.25
0.5
50
2.5
5
2.5
2.5
2.5
Base-level drop, ΔZ (cm)
0.25
0.25
0.25
0.25
0.25
2.5
2.5
0.25
0.25
0.25
Kaolinite fraction, fk (%)
1
1
1
1
1
1
1
0
2
5
Discharge, Qout (mL min-1)
800
770
730
900
820
895
890
970
900
755
Flow depth, h (mm)
2.5
2
2.75
3.25
1.1
2
2.5
2.5
1.75
2
Flow velocity, Vf (m s-1)
0.28
0.34
0.23
0.24
0.65
0.39
0.31
0.34
0.45
0.33
Froude number
2.10
2.41
1.31
1.22
3.95
2.8
1.99
2.82
3.82
2.36
Reynolds number
2222
2232
1986
2353
2579
2594
2472
2694
2667
2188
Equilibrium slope
0.061
0.077
0.051
0.037
0.15
NA
NA
0.054
0.066
NA
τeq (Pa)
1.18 ± 0.14
1.28 ± 0.17
1.11 ± 0.11
0.88 ± 0.08
1.9 ± 0.33
NA
NA
0.91 ± 0.12
NA
NA
KP velocity, Vkp (cm min-1)
8.2
8.1
6.8
8.8
11.6
9.8
11.8
17
7
0.7
KP frequency (Hz)
0.0006
0.0008
0.0003
0.0001
0.0046
0.0003
0.0006
0.0009
0.0004
0.0003
Period between KP, Δt (min)
28.8
20.0
48.0
118.0
3.6
60.0
30.0
18.4
43.6
48.6
Plunge pool depth, Hp (cm)
1.23
1.19
0.97
1.13
1.31
NA
NA
1.25
1.82
3
Base case
Base-level fall variations
Base-level drop variations
Substrate variations
We carried out several experimental sets. Experiment 1 is the base case
to which other experiments can be compared (rate of base-level fall, U= 2.5 cm h-1; incremental base-level drops, ΔZ= 0.25 cm and
kaolinite fraction fk= 1 % by weight when dry; see Table 1). First,
we tested base-level fall scenarios. During experiments 2, 3, 5
and 6, U was set to 5, 1.25, 0.5 and 50 cm h-1, respectively, while
ΔZ and fk were kept similar to experiment 1. In other words,
the base level was dropped 0.25 cm every 30 min to get a 0.5 cm h-1
rate and every 3 min to get a 5 cm h-1 rate. During experiment 7, U and fk were similar to experiment 1 (2.5 cm h-1 and
1 %) but ΔZ was changed to 2.5 cm (Table 1). To keep the same
base-level fall rate, the base level was then dropped 2.5 cm every 60 min. Similarly, the base level was dropped 2.5 cm every 30 min in
experiment 8 so that it could be compared to experiment 2. Finally,
different substrate lithologies were tested. The kaolinite fraction,
fk, was changed to 0, 2 and 5 % during experiments 9, 10 and 11, respectively, while U and ΔZ were kept similar to experiment 1 (Table 1).
Evolution of two experiments with the same average rate of
base-level fall (U= 2.5 cm h-1), but different incremental base-level
drops, ΔZ. (a)–(d) For experiment 1 (ΔZ= 0.25 cm), a
knickpoint is propagating in between 96 and 103 min (a), leaving a
alluvial layer (b) that will be progressively removed as the base level of
the experiment is lowered between 105 and 130 min (c). A new knickpoint
starts retreating in between 132 and 140 min once the alluvium has
disappeared (d). (e)–(h) For experiment 7 (ΔZ= 2.5 cm), a
new knickpoint is generated each time the base level is dropped (i.e., in
between 0 and 8 min (e) and in between 60 and 69 min (g)). In
between these drops, the profile's slope is lowered by overall diffusion
((f) and (h); see also Fig. 7b). Blue and red colored lines correspond to
the successive elevation of the bedrock surface, while the light-blue and red
area corresponds to the alluvium. The position of the base level is tracked
on the left side of each frame. Vertical exaggeration is 1.375.
Measurements and uncertainties
We define the knickpoint as the point where a river steepens, whereas the knickpoint face corresponds
to the steep reach starting at this knickpoint and ending at the bottom of
the plunge pool (e.g., Gardner, 1983; Figs. 1c and 3c). We measured
geometries of the profile and knickpoints using a camera placed along the
flume. Pictures were extracted every 24–30 s and corrected for lens
distortion and vertical stretching in order to measure the overall
experimental slope, knickpoint face slope, and knickpoint face length. Water
depth was measured using a point gauge, while water discharge (e.g.,
Qout; Fig. 2) was measured throughout experiments using a graduated
cylinder. The hydraulic parameters of each experiment were calculated using
these measures (Table 1). Reynolds numbers fall between 1900 and 2700, while
Froude numbers are all above 1, indicating that the flow regime is
respectively transitional to turbulent and supercritical (Table 1).
On the extracted pictures, no vertical or horizontal position could be
accurately measured below a two-pixel resolution, i.e., 1.33 mm. These
vertical and horizontal errors were combined in a simple propagation formula
based on variance (Ku, 1966) to assess uncertainties in the metrics used in
this study. A test evaluation calculated for experiment 3 showed that
variance of the overall experiment's slope was around 0.0017 (i.e.,
∼ 5 % equilibrium slope of experiment 3) and knickpoint
velocity variance was about 2 mm h-1 (i.e., ∼ 3 % of
average knickpoint velocity for experiment 3). Therefore, both overall
slope and knickpoint velocity do not vary significantly due to measurement.
On the other hand, measures of the variance of knickpoint face length and
slope have greater uncertainties. For instance, when the overall experiment
is steep (e.g., experiment 6; Table 1), the transition to the knickpoint
face along the profile is not sharp and a horizontal measurement error up to
15 mm is possible, especially approaching the plunge pool (Figs. 1 and 3).
The resulting knickpoint face slope variance, calculated for experiment 6 assuming a vertical error of 1.33 mm, is about 3∘. Therefore,
two knickpoint face slopes would be significantly different only if their
difference is greater than 3∘. Plunge pool depth was calculated from
knickpoint face slope and knickpoint face length and corrected for the overall
slope of experiments (e.g., Fig. 1c). Error on flow depth, h, is approximately
0.25 mm. This together with uncertainty in slope allowed us to estimate the
uncertainty in the shear stress, τeq, shown in Table 1.
Evolution of the entire experiment 3 (U=1.25 cm h-1;
ΔZ= 0.25 cm) showing alluvium thickness deposited in response to
the retreat of knickpoints (enumerated from 1 to 5). Blue and red colored
lines correspond to the elevation of the bedrock surface at the end of the
knickpoint retreat, while the blue and red colored dashed lines correspond to
the elevation of the bedrock before knickpoint propagation. Light-blue and
red areas represent the alluvium. A new knickpoint is generated only when
the alluvium is removed from the profile. Note the abortion of knickpoint 3
after 3 min of retreat (see text for explanations). Vertical
exaggeration is 1.375.
Results
Knickpoint generation and periodicity
We observe threshold behavior in the total base-level drop needed to
generate a knickpoint. In the case of ΔZ= 0.25 cm, two to eight drops
are needed to generate the first knickpoint. A small initial knickpoint
retreats about 30 average stream depths (7 cm) upstream and then remains
stationary for 1–2 min. During this period, the plunge pool at the
foot of the knickpoint face deepens and a hydraulic jump forms. This phase
is characterized by over-erosion, i.e., the bottom of the plunge pool becomes
lower than the newly imposed base level. After the plunge pool reaches a
depth of 1–3 cm (Fig. 4), the knickpoint begins to retreat at constant
speed. In the case of ΔZ= 2.5 cm, a knickpoint is generated for
each base-level drop and retreats uniformly (Fig. 4e). During knickpoint
retreat, the sand–kaolinite substrate is eroded and the kaolinite and sand
separate. The kaolinite is transported out of the system in suspension, while
the sand is deposited downstream of the knickpoint to form a layer
(alluvium; Figs. 3, 4a and e). Once a knickpoint reaches the upstream end of the
flume, the alluvium remains along the profile (Fig. 4b and f). This layer
is slowly removed as the river profile is smoothly lowered by overall
diffusion over both the alluvium and the bedrock substrate (Fig. 4b, c and f). This indicates that the sediment layer acts as a shield that prevents
erosion of the bedrock substrate (Sklar and Dietrich, 2004): no significant
knickpoint–hydraulic jump couple is observed during the diffusion phase.
Only close observation of the bed indicates that smaller knickpoints (i.e.,
shallower than the stream depth) develop and propagate while the bed is
shielded by sediment.
Evolution of the profile's bed surface elevation as a function of
the base-level fall rate (see also Fig. 7a). The bed surface can be either
the bedrock or the alluvium surface. Note that the amount of knickpoint
increases with base-level fall rate.
Depending on the magnitude of base-level drop, ΔZ, the period between
knickpoints is not constant. In the case of ΔZ= 2.5 cm, and after
the alluvium is in place, the base-level drop is greater than the alluvium
thickness, allowing each drop to form a knickpoint (Fig. 4e and g). The
face of a new knickpoint is irregular, i.e., its slope changes at the
transition between the bedrock and the remaining bed sediments (Fig. 4g). In
that case, the average period between knickpoints corresponds to the time
between each base-level drop (e.g., 60 min for experiment 7 and 30 min
for experiment 8; Table 1). In the case of ΔZ= 0.25 cm, the
alluvium has to be removed before a new knickpoint can be generated and
retreat (Fig. 4c and d). In this regime, the average period between
knickpoints is therefore a function of the alluvium thickness to be eroded
in the flume (Table 1). A detailed sequence is shown in Fig. 5 for
experiment 3. Overall, the knickpoint period is about 70 min for most of
this experiment (e.g., the time needed to produce a base-level fall equal to
the alluvium thickness, 1.25 cm). However, the geometry of the bedrock
surface is irregular and hence the sediment thickness too. Accordingly, the
third knickpoint generated disappears upon reaching sediment deposits in the
flume (Fig. 5). First, the alluvial layer is rapidly removed along the upper
section of the knickpoint face. This produces a two-step knickpoint face
that is progressively smoothed. This smoothing disturbs the flow: the
hydraulic jump cannot be maintained and the knickpoint fades. As a
consequence, thinner alluvium is left along the flume and the next (fourth)
knickpoint starts after only 33 min (Fig. 5). This indicates that transient
alluvial deposits can disturb the flow and temporarily prevent knickpoint
formation or propagation.
(a)–(c) Evolution of mean slope of the experiments with time
for different sets of experiments. (a) Evolution with base-level fall rate.
(b) Evolution with different base-level fall styles. For experiments 5, 7 and 8 (respectively represented by the blue triangles, yellow
circles and orange circles), the minimum time between each base-level drop
is 30 min. (d) Evolution of the equilibrium shear stress as a function of
their base-level fall rate for experiments where ΔZ= 0.25 cm.
Exponential fit is shown with a dashed line.
Equilibrium slope and timescales
Figure 6 shows the overall evolution of experimental profiles as a function
of base-level fall rate (ΔZ=0.25 cm). These profiles correspond to
the bed surface and not to the bedrock surface. Each experiment starts with
a nearly flat profile whose slope increases (dashed lines; Fig. 6) until
stabilization (plain lines). As base-level fall rate increases, profiles
become steeper: Fig. 7a shows that profile slopes increase proportionally
to the rate of base-level fall. Each experiment reaches a quasi-equilibrium
slope that is proportional to the rate of base-level fall applied.
Knickpoint frequency also increases as a function of base-level fall rate
and more knickpoints are captured along the profiles from Fig. 6a to e
(see also Table 1). This configuration is enhanced for U= 50 cm h-1
(experiment 6), where several knickpoints can retreat simultaneously. In
this configuration, and similar to experiments 7 and 8,
knickpoints are propagating even though sediments are preserved along the
profile. However, the downstream reach (first 10 cm of the flume)
must be free of alluvium in order for a knickpoint to be generated.
Figure 7b shows the evolution of slope for experiments 7 and 8,
which have base-level fall rate similar to experiments 1 and 2,
respectively, but a ΔZ 10 times higher (e.g., 2.5 cm). Experiment 5 (U= 0.5 cm h-1; ΔZ= 0.2 5cm) is shown for comparison.
After 100 min, experiments 7 and 8 have a slope that is high but
lower than experiments 1 and 2, respectively. Furthermore, the
profiles of the former decrease and converge towards a low equilibrium
slope, which is close to the equilibrium slope in experiment 5. In all
these experiments (5, 7 and 8), a common characteristic is the
low frequency of base-level drops and the conversely long period in between
these drops (≥ 30 min). This suggests that these experiments are more
affected by smooth profile readjustment by diffusion during quiescent
periods and less by knickpoint retreat.
An analysis of the stream slope according to lithology is shown in Fig. 7c.
Lithology or substrate strength is represented as the kaolinite percentage
within the substrate, fk. For similar uplift rates, the experiment
without kaolinite has a lower equilibrium slope than the experiment with 1 % kaolinite. However, the equilibrium slopes of experiments 1 and 10 (with respectively 1 and 2 % of kaolinite) are similar. Therefore,
despite their different bedrock strengths, these two cases are at
equilibrium with the alluvium and not the substrate. Indeed, shear stress
calculated at the equilibrium slope for experiments 1, 2, 3, 5 and 6 goes as the base-level fall rate (Fig. 7d). A tentative
exponential fit suggests that the shear stress for U= 0 cm h-1
(0.91 ± 0.5) would be above the shear stress of motion (i.e., ∼ 0.13 Pa for d50= 0.1 mm; Julien, 1998) and that the evolution of
these slopes is controlled by alluvium removal. The comparison between Fig. 7a and c further suggests that the overall equilibrium slope varies more
strongly with base-level fall rate than lithology. When fk= 5 %,
no equilibrium is attained and the quasi-equilibrium state has a strong
sinusoidal shape (Fig. 7c): a maximum value is reached about every 100 min.
Given a typical knickpoint velocity of about 0.7 cm min-1 (experiment 11; Table 1) and the flume experimental section length 75 cm, 100 min corresponds to the time required for a knickpoint to reach the
upstream part of the flume. This indicates that low knickpoint velocity
lengthens the readjustment timescale of the overall profile as higher relief
can be maintained until knickpoints pass through the system.
Knickpoint characteristics as a function of base-level fall rate
and substrate. (a)–(d) Illustrations of the knickpoint shapes as a
function of the kaolinite content (fk) in the substrate. Note that the
plunge pool depth could not be measured from photographs for experiment 11 ((d); fk=5 %): the substrate was so cohesive that it stuck
on the walls and the bottom of the plunge pool was not accessible. Hp
was, however, estimated to be ca. 3 cm on the flume during experiment 11. In
this experiment, the geometry of the bed was more heterogeneous and the
channel narrowed to incise the bedrock. The dashed line corresponds to the
approximate bottom on the plunge pool. (e) Variations in knickpoint slope
and plunge pool depth as a function of fk. (f) Variations in knickpoint
slope and plunge pool depth as a function of the base-level fall rate, U.
(g)
Mean knickpoint retreat velocity shown as a function of fk. The
exponential fit is represented with a dashed line. (h) Mean knickpoint
retreat velocity shown as a function of U.
Controls on knickpoint characteristics
In Fig. 8, we investigate knickpoint properties in relation to U and
fk. Figure 8a to d show that the knickpoint face slope and plunge pool
depth increase linearly as a function of fk (Fig. 8e). These
characteristics do not vary significantly as a function of the uplift rate:
only a slight increase in knickpoint slope and plunge pool depth are
suggested as functions of U (Fig. 8f). This shows that these knickpoint
properties are primary controlled by lithology. The same statement applies
for knickpoint retreat velocity: while variations in U from 0.5 to 50 cm h-1 do not show a statistically significant increase in knickpoint
velocity (Fig. 8h), an increase from 0 to 5 % kaolinite is responsible
for a knickpoint velocity decrease from 17 to 0.7 cm h-1 (Fig. 8g). The
effect of kaolinite fraction on knickpoint velocity can be fit by an
equation of the form
Vkp=Vmaxe-α⋅fk,
where Vmax is the maximum velocity attained over sand (e.g., fk= 0) and α is a dimensionless fitting parameter. Less dramatically,
the increase in ΔZ from 0.25 to 2.5 cm increases knickpoint
retreat velocity by 20 % (i.e., comparison between experiments 1 and 7 and experiments 2 and 8 in Table 1). This indicates that
knickpoint velocity may still be partially influenced by base-level fall
velocity. Finally, while Bennett et al. (2000) showed that plunge pool depth
increases with water discharge, our results suggest that this depth also
goes with the kaolinite fraction (Fig. 8e):
Hp∼fk.
Discussion
Knickpoint self-organization
The experiments presented in this study were carried out in a small 1-D flume
with very simple conditions compared to natural systems: constant discharge,
constant lithology per experiment, no interfluve processes (debris-flow,
pedimentation, etc.) and no possibility for the channel to widen (although
channel narrowing has been observed in experiment 11; see caption of Fig. 8). The first and most striking result of this study is that, even under
these simple conditions, knickpoint dynamics remain surprisingly complex and
exhibit strong autogenic (self-organized) variability mediated by alluvium
dynamics and associated bed sheltering, and by the erosional threshold for
the bedrock substrate. Indeed, the interaction between bed lithology and
base-level fall style (i.e., overall rate and distribution of vertical
offsets) provides a variety of configurations that strongly affects the
evolution of river profiles.
As observed in other geomorphic physical experiments (Paola et al., 2009),
the transient storage and release of sediments along the flume is
responsible for self-organized dynamics that in the problem at hand delay
knickpoint propagation in response to base-level fall (Figs. 4 and 5). This
behavior is particularly observed when ΔZ is on the order of or lower
than the flow depth (i.e., 0.25 cm; Table 1). As described for
alluvial-bedrock rivers (Sklar and Dietrich, 2004), the alluvium acts as a
shield for incision by knickpoint retreat and the river profile is
characterized by overall diffusive removal of the sediments until it becomes
too thin to shield the bedrock. However, when the incremental or cumulated
base-level fall is large enough, i.e., larger than the sediment thickness,
the effect of transient alluvium is less prominent, suggesting that high-magnitude external forcing is still likely to produce knickpoints (Fig. 4;
Jerolmack and Paola, 2010). Hence, one directly testable outcome of this work
is that offset can generate a knickpoint only when its magnitude exceeds the
thickness of any alluvial layer present on the bed. The thickness of the
alluvial layer sets an offset threshold for knickpoint generation. In an
environment in which uplift is generated by earthquakes, we expect ()
knickpoint propagation in response to fault displacement if the offset
exceeds the thickness of piedmont/alluvial deposits but () overall
diffusion (no knickpoint) for offset is lower than the alluvial thickness. The
latter therefore points to the ability of alluvial covers to filter
small-scale base-level variations that may not be recorded by knickpoint
propagation.
While the rate of base-level fall (or uplift) primarily controls overall
slope (Figs. 6, 7a and c; Bonnet and Crave, 2003), knickpoint
characteristics are dominated by bedrock strength, which in the experiments
increases with kaolinite content (Fig. 8). Earlier work has demonstrated
that the critical shear stress of sand/clay mixtures increases with their
clay content (Mitchener and Torfs, 1996). Hence, similar to field
measurements (Cook et al., 2013), the velocity of knickpoint retreat is
inversely proportional to substrate strength in our experiments. This
militates against assuming that the retreat rate of knickpoints is constant
over varying bedrock lithologies. Future studies investigating uplift
history through inverse modeling should therefore integrate a lithological
term (see Wilson et al., 2014) to simulate knickpoint or knickzone retreat
rate.
Surprisingly, our 1-D experiments show that base-level variation, a key
parameter studied in erosion /deposition systems, is not encoded by
knickpoint height, i.e., Hp. Instead, Hp mostly goes with water
discharge and bedrock strength (Bennett et al., 2000; this study).
Specifically, our experiments show that, for base-level fall created by
offsets, the sum of the offsets must reach a threshold (> sediment thickness) to trigger a knickpoint. The experiments of Cantelli and
Muto (2014) give insight into the complementary case: if the offset is too
large, a series of knickpoints rather than just one is generated. Together,
these findings suggest that, similar to drainage basins that tend to be
regularly spaced in mountain belts (Hovius, 1996), knickpoints tend toward
an optimal knickpoint shape – a kind of “unit knickpoint”. This unit
knickpoint is a function of water discharge and lithology (Eq. ), and
presumably could be strongly influenced by, for example, layering in the
substrate (e.g., Holland and Pickup, 1976), which is not present in our
experiments and those of Cantelli and Muto. To summarize, there is no one-to-one correlation between knickpoints along river profiles and base-level
events: one base-level drop can generate multiple knickpoints, but one
knickpoint can also result from multiple events.
At this point, we are not able to predict theoretically the properties of
unit knickpoints. Overall, plunge pool depth goes inversely with knickpoint
velocity (Table 1), although there is more scatter when the lithology is
constant and base-level fall rate varies (e.g., experiments 2, 3, 5 and 6). This suggests that slow retreat of a knickpoint and
associated plunge pool results in more vertical erosion of the bed by
scouring and increases the plunge pool depth (see Stein and Julien, 1993). A
second useful limit is the cyclic steps described by Parker (1996), which
can be thought of as a train of linked unit knickpoints, and are what we
observe in our experiments under rapid base-level fall (Fig. 6e). However,
while Parker described these features as self-formed, the ones presented in
this study are forced externally. The connection between individual
knickpoints and trains of cyclic steps deserves further study; however, we note
that in terms of local hydraulics and sediment motion, the knickpoints we
generated function similarly to Parker's steps, despite being solitary
except in the limiting case of rapid base-level fall. Hence, the geometry of
cyclic steps may provide a constraint on that of a unit knickpoint and hence
a means of predicting the characteristics of knickpoints generated by
specific scenarios of base-level fall. Another limit is that unit
knickpoints may not be generated or preserved in the case of catastrophic
base-level fall. This is suggested by the evolution of the Rhone Valley in
response to the 1500 m drop associated with the salinity crisis in the
Mediterranean Sea (Loget et al., 2006) and also in the case of a
catastrophic drop simulated experimentally (A. Cantelli, personal
communication, 2015).
Analysis of knickpoint distribution
The evolution of river bed and knickpoint retreat is commonly simulated
numerically using a combined advection–diffusion equation (Howard and
Kerby, 1983; Rosenbloom and Anderson, 1994; Whipple and Tucker, 1999; see
Bressan et al., 2014). In this study, advection is observed through
knickpoint generation every 3–120 min (Table 1). As a comparison, the
diffusion response timescale T of the experiments can be approximated in the
same way than alluvial systems, using the system (flume) length L and width
W (m), the sediment discharge qs (m3 min-1), and the overall
equilibrium slope S (Métivier and Gaudemer, 1999; Allen, 2008).
T=L2WSqs
This timescale is 300–1400 min, i.e., longer that the period in between
knickpoints. This indicates that most experiments presented in this study
are dominated by knickpoint advection (except experiments 5, 7 and 8; Sect. 3.2): despite their relatively fast migration, knickpoints
are generated too often to allow the stream to entirely relax by diffusion.
Erosion of the bed is usually modulated by a threshold that must be
surpassed in order for the river to erode (van der Beek and Bishop, 2003;
Snyder et al., 2003; Sklar and Dietrich, 2004). However, many simulations of
knickpoint retreat assume that each base-level drop can generate a new
knickpoint and that the initial geometry of knickpoints is offset by the
base-level drop. As pointed out before, this is not reasonable if
knickpoints tend to a unit form, independent of the magnitude of base-level
fall. Our analysis has shown that unit knickpoints are generated when the
alluvium is removed from the river bed, i.e., every time the base level
reaches the bottom of the plunge pool,Hp (Figs. 4 and 5). The period
between knickpoints, Δt, can then be simply approximated as a function
of the base-level fall rate:
Δt=HpU.
This is supported by the comparison between knickpoint period measured from
the experiments and estimated after Eq. () (e.g., for experiments 1, 2, 3, 5, 6, 9, 10 and 11; Fig. 9). Equation ()
can then be derived to estimate the spacing between knickpoints:
Δx=ΔtVkp=HpUVkp.
and a dimensionless spacing is obtained when divided by the flow depth.
Δx∗=HpU⋅hVkp
These equations can be derived to simulate knickpoint generation and retreat
using a rule-based model (Fig. 10). The upstream distance and elevation of
the nth knickpoint, with migration velocity Vkp are then
respectively
xn=Vkp⋅[t-n-1⋅Δt],yn=-Hpn-1⋅Δt.
In all simulations with a constant lithology, the upstream distance of the
first knickpoint is similar, independent of the base-level fall rate (Fig. 10). Hence, rather than giving information about base-level fall rate, the
position of this knickpoint allows assessment of the incipiency of
base-level fall within the model. In the field, this would correspond to
when the base-level fall or uplift had first exceeded the thickness of
alluvium within the channel.
Comparison of the measured period between knickpoints (Δt) to the calculated period between knickpoints using Eq. (). Linear fit of
the data is shown in black.
Snapshots of knickpoint migration calculated using
Eqs. (), () and (). Each snapshot represents a simulation with a different set
of parameters (U, Vkp, Hp) stopped after 6 min of runtime. The
bedrock surface (red line) is simulated by tracking the positions of the
knickpoint (white squares) and the bottom of their associated plunge pool
(white circles). The alluvium surface (blue line) is shown for comparison
with the experiments. The bedrock surface initial elevation is set to zero.
The first knickpoint is assumed to retreat instantaneously at a velocity
Vkp. The base-level falls at a rate U. A new knickpoint is generated each
time the base level (shown by the black dashed line) reaches the
depth of the plunge pool (Hp) associated with the previous retreating
knickpoint. For the sake of simplicity, no diffusive processes are
considered in the simulations. The water discharge and horizontal distance
between knickpoints and their plunge pool bottom (2 cm) are assumed
constant, while the velocity and height of unit knickpoints vary according to the main
trend observed in the experiments (Table 1). The simulations are varying
vertically as a function of base-level fall rate and horizontally as a
function of substrate strength. This controls two parameters: when it is
high, Vkp is low and Hp is deep, while when it is low,
Vkp is high and Hp is shallower (Table 1).
Morpho-geologic map showing two tributaries of the St. Louis
River, close to Lake Superior shore, Duluth, Minnesota (a), and their
associated long profiles: the Mission Creek (b) and Kingsbury Creek
(c)
rivers. Note that while the Kingsbury Creek watershed substrate is resistant
gabbro, the substrate of the Mission Creek watershed is composed of loose
sedimentary rocks (mainly sillstone, shale, mudstone and sandstone). The white
area represents unmapped bedrock, the black line the watershed limit and the dashed line
the Minnesota–Wisconsin border. Rivers are in blue. After Fitzpatrick et al. (2006). Vertical exaggeration is 20.
Equation () and Fig. 10 also show that an increase in base-level fall rate leads to the creation of more knickpoints and that the spacing between
knickpoints, Δx, is inversely proportional to base-level fall rate
(e.g., Fig. 10; Eq. ). Equation () therefore provides an alternative
relationship for interpreting uplift or base-level fall rate from knickpoint
distribution/spacing on the field. Knickpoint size (e.g., plunge pool
depth) is the other critical parameter of this equation; it is strongly
dependent on water discharge and substrate strength. In environments with
poorly consolidated material, i.e., alluvial rivers, where substrate is
strengthened only by a weak compaction or vegetation, base-level falls are
quickly compensated for by the migration of close, shallow knickpoints (e.g.,
right side of Fig. 10). In the case of bedrock rivers (e.g., left side of
Fig. 10), where the substrate is more resistant and more widely spaced,
deeper knickpoints are observed indicating that the response timescale of
the sediment routing system is increasingly longer. Interestingly, this
behavior is the opposite of the one predicted by the analysis of Whipple (2001) that the advection response time (i.e., the time for a knickpoint to
pass through a river system) is longer for alluvial (low-slope) rivers than
for steeper bedrock rivers. To the extent that low-slope rivers are
associated with weaker substrates, these strength variations act oppositely
to the effect of slope on knickpoint propagation. At this point, without
further information, the overall outcome of this competition cannot be
determined.
Overall the experimental results suggest promising approaches for analyzing
knickpoint dynamics as well as their spatial distribution in landscapes in
relation to relative base-level fall. Figure 11 exemplifies how bedrock
lithology affects knickpoint distribution on the field based on two
neighboring watersheds of similar size (25 ± 2 km2) near Duluth,
Minnesota. In both watersheds, base-level history is controlled by the
evolution of the level of Lake Superior during glaciation–deglaciation
cycles (Wright, 1973). The major difference between the two watersheds is
their bedrock lithology (Fig 11a; Fitzpatrick et al., 2006). While the
stream flowing above a loose sedimentary bedrock shows a small knickpoint
located 10 km upstream (Fig. 11b), the stream flowing over a resistant
gabbroic bedrock displays a big knickpoint located closer to the watershed
outlet (4 km; Fig. 11c). These first-order observations are consistent with
our experimental results that the increasing rock strength is favorable to
the creation of bigger knickpoints whose upstream propagation is slower.
Knickpoints and waterfalls: erosion processes
Our experiments highlight the effects of sediment
transport and lithology on knickpoint dynamics; a remaining challenge is to effectively link these
laboratory observations to theoretical, empirical and field data. To achieve
this, the mechanics and process of erosion in play must be understood and
characterized. In our experiment, two erosion regimes can be observed: a
background/“clear water” regime where erosion of the bed is triggered by
sediment abrasion through saltation (e.g., erosion rate ∼ 0.2 mm min-1; Sklar and Dietrich, 2004; Fig. 4c) and (ii) a waterfall
regime where measured erosion rate is 10 times higher (∼ 1.5 mm min-1; Fig. 4a and d). The turbidity observed within the plunge pool
suggests that most sediments may be in suspension there, uncovering the
bottom of the pool (Lamb et al., 2007) and perhaps providing abrasive tools
for erosion. The steep knickpoint face is furthermore conducive to erosion
rates higher than the background rate. A more accurate quantification of
erosion through abrasion would, however, require detailed tracking of sediment
and flow dynamics than we were able to do, particularly to identify what
fraction of the sediment is transported in suspension as opposed to bedload.
Our observations are indeed limited by the size of the experiment, but
detailed study using advanced particle- and flow-tracking techniques such as
laser holography (Toloui and Hong, 2015) in a larger facility would be a
logical next step in this line of research.
Finally, we observe undercutting and collapse of the knickpoint face in the
case of more resistant bedrock (2–5 % kaolinite), similar to natural
examples (Seidl et al., 1994; Lamb et al., 2007). In this case, we
hypothesize that sediment-laden flows in the pool are able to erode backward
compared to the overall flow sense due to vorticity in the pool and,
potentially, the angle of incidence of the flow, which is set by the
knickpoint slope. The conditions necessary for undercutting would be worth
investigation in the future, for example combining physical experiments and
high-resolution numerical simulations of flow and sediment transport.