CHAPTER 1ALONG INCISING BEDROCK RIVERS
the degree of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
ABSTRACT
Landscapes evolve through processes acting at the earth’s surface in response to
tectonics and climate. Rivers that cut into bedrock are particularly important since
they set the local baselevel and communicate changes in boundary conditions across
the landscape through erosion and deposition; the pace of topographic evolution
depends on both the rate of change of the boundary conditions and the speed of the
bedrock channel network response. Much of the work so far has considered the
effects of tectonically-controlled changes in slope and climatically-controlled
changes in discharges to the rate of channel bed erosion while considering bank
erosion, if active at all, to be of at best secondary importance to landscape evolution.
Sprinkled throughout the literature of the past century are studies that have
recognized lateral activity along incising rivers, but conflicting interpretations have
left many unanswered questions about how to identify and measure horizontal
erosion, what drives it, what effect it has on the landscape, and how it responds to
climate and tectonics. In this thesis, I begin to answer some of these questions by
focusing on bedrock river sinuosity and its evolution through horizontal erosion of
the channel banks. An analysis of synoptic scale topography and climatology of the
islands of eastern Asia reveals a quantitative signature of storm frequency in a
regional measure of mountain river sinuosity. This is partly explained through a
study of the hydro- and morphodynamics of a rapidly evolving bedrock river in
Taiwan which shows how the erosive forces vary along a river to influence the
spatiotemporal distribution of downcutting, sidecutting, and sediment transport.
Through these analyses, I also present evidence that suggests that the relative
frequency of erosive events is far more important than the absolute magnitude of
extreme events in setting the erosion rate, and I show that the horizontal erosion of
bedrock rivers is an important contributor to landscape evolution. This thesis
comprises a new look at the processes at work in bedrock rivers which suggests new
ideas about the ways that landscape and climate interact, new tools for interpreting
landscape morphology, and new insights into the processes that contribute to the
evolution of active orogens.
Chapter 2: Typhoon-driven discharge variability and bedrock
river meandering…………………………………… 48
shear stress along a rapidly eroding bedrock
mountain river……………………………………… 92
Conclusions………………………………………………………… 164
i
ACKNOWLEDGMENTS
Thanks first of all to the Department of Earth and Environmental Sciences
at Columbia University, NASA, the National Science Foundation, and the
taxpayers of the United States of America for funding the research summarized in
this dissertation.
Thanks to my advisory committee, Chris Scholz, Jeff Weissel, and
especially my primary research advisor Colin Stark, who guided me through each
project. Thanks to Art Lerner-Lam, Upmanu Lall, and Roger Buck for serving on
my examination committees, and thanks to everyone else at Lamont-Doherty Earth
Observatory, in particular: Mia Leo, Sally Odland, Regina Giacinto, Felicia Taylor,
Carol Mountain, Missy Pinkert, Bree Burns, Greg Yetman, Rob Kakascik, Bob
Arko, Doug Shearer, Art Lerner-Lam, Mark Cane, Nick Christie-Blick, Steven
Chillrud, Chris Small, Roger Anderson, Andrea Taramelli, Thorsten Nagel, Joe
Galewsky, Vicki Ferrini, Mladen Nedimovic, Kristina Czuchlewski, Dalia Bach,
Kori Newman, Janet Baran, Rob Bialas, Angela Slagle, David Grass, Mike Tischer,
Irina Gorodetskaya, Byrdie Renik, Tim Crone, Abby Spieler, Kevin Jones, Louisa
Bradtmiller, Katie Leonard, Julie Bonczkowski, Rose Anne Weissel, and
especially to Chadwick Holmes for saving me from countless Matlab pitfalls.
Thanks to my undergraduate advisor at Yale University, Mark Brandon,
and to Frank Pazzaglia at Lehigh University for encouraging me to pursue this
degree.
ii
Thanks to Simon Dadson and Jens Turowski, and to their advisor at
Cambridge University, Niels Hovius, for patiently answering my questions and
arguing with me over interpretations of landforms and surface processes in the
field and at meetings. Jens graciously brought me along for some of his PhD
fieldwork in Taiwan and accompanied me for some of mine; his curiosity,
enthusiasm, and positive outlook were always contagious. This dissertation has
greatly benefited from his suggestions, criticisms, and guidance.
Fieldwork in Taiwan was only possible through the support of Hongey
Chen in the Department of Geosciences at the National Taiwan University (NTU)
in Taipei, and of Ching-Weei Lin at the Disaster Prevention Research Center
(DPRC) of the National Cheng Kung University (NCKU) in Tainan. Chia-Hong
Jen and Meng-Long Hsieh, Chung (Yellow Clock) Huang, and others at NTU, and
Chin-Pin Ko, Te-Cheng (Xiao) Yi, Tsai-Tsung (Victor/Xiao Pang) Tsai,Wei-Shu
(Xiao Shu) Chang, Shin- Ping (Morris) Lee,Wei-Lin (William) Lee, Gong-Rey
He, Yun-Chung (Take-san) Tsang, Wen-Yi (Xiao) Lai, Wen-Chi (Da) Lai, and
others at DPRC/NCKU were also instrumental in providing logistical and
technical support, field assistance, data, insight, and…some very interesting meals.
I also owe a large debt of gratitude to the Maolin firefighters who risked their own
safety to help me cross a raging river to recover a notebook that was accidentally
left behind, just before a heavy downpour, on the wrong side of a river.
I am especially grateful for the collaborative efforts of Chung (Yellow
Clock) Huang at NTU, and Te-Cheng (Xiao) Yi at DPRC. Their resourcefulness,
iii
insightful suggestions, and technical expertise helped to shape much of the field
methodology in this thesis over several seasons in Taiwan.
Fieldwork in Japan benefited from the logistical support of Yasutaka Ikeda
at Tokyo University and the local knowledge and insights of Yukitoshi Fukahata,
also at Tokyo University, Masayoshi Tajikara at the Japan Atomic Energy Agency,
and Tsuyoshi Hattanji and Yuichi Hayakawa at Tsukuba University.
Ming-Jame Horng at the Water Resources Agency of Taiwan, Joe Xu at
Tsukuba University in Japan, and the Thomas Luellwitz at the Global Runoff Data
Centre generously provided discharge data.
And finally, thanks to my friends, and most of all, to my family for
supporting me and for putting up with my stress while I struggled to complete this
dissertation.
As I am getting older, my memory has increasingly betrayed me, so thanks
also to everyone I have forgotten. If you are my age, you probably understand. If
you’re younger, just wait.
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v
PREFACE
William Morris Davis sparked a debate in 1893 when he wrote a letter to
Science in which he described incised meanders along the Osage River of Missouri
(Davis 1893). He argued that because meandering is a state of alluvial rivers, the
incised meanders of the Osage indicate a series of erosion cycles in which the river
once meandered across a flat plain until regional uplift of the Ozark Mountains
rejuvenated incision along the stream, locking in the existing planform. This
interpretation that incised meanders are inherited from a past alluvial phase
continues to re-appear as a relatively common explanation for meanders in
bedrock. However, even Davis’s contemporaries recognized that not only is there
rarely evidence for planform inheritance, but there is also no need to assume an
alluvial origin for meanders (e.g. Winslow 1893). Indeed, active meandering is
ubiquitous along incising rivers (e.g. Mahard 1942, Tinkler 1970, Ikeda et al. 1981,
Seminara 2006, Shyu et al. 2006), and it can have a strong influence on the
morphology of mountain landscapes.
The processes of bedrock river incision have long been a focus of attention
among those who study landscape evolution, in part because the slopes of incising
channels and their variations downstream are thought to result primarily from a
competition between bed erosion and rock uplift (e.g. Mackin 1948, Whipple 2004)
which results in longitudinal profiles that encode information about tectonics and
climate (e.g. Roe et al. 2002, Whipple 2006). However, if meanders grow along
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these rivers, channel slopes will change, complicating the relationships between
climate, tectonics, erosion, and bedrock channel longitudinal profiles.
Theories of bedrock incision tend to treat bed erosion as an increasing
function of boundary shear stress (e.g. Howard and Kerby 1983, Whipple 2004),
which depends on channel slope and hydraulic geometry, while theories of
meandering tend to treat bank erosion as an increasing function of cross-channel
asymmetry in the flow speed around meander bends (e.g. Thomson 1876, Einstein
1926, Rhoads and Welford 1991, Seminara 2006). Since hydraulic geometry and
flow speed vary with discharge, erosion of the walls and bed of bedrock channels
should both increase as functions of some characteristic(s) of the discharge
distribution, slope, and erodibility. It is likely that the functional dependencies are
different for vertical bed and horizontal wall erosion, particularly because only a
portion of the flows through the channel inundate the walls and because bed
sediment cover (which may vary in thickness, extent, caliber, and persistence with
discharge and/or erodibility) tends to armor the channel bed while leaving the
banks exposed (Moore 1926, Turowski, 2008). However, data on how discharge,
channel geometry, shear stress, erodibility, and sediment supply affect horizontal
cutting rates and relative rates of horizontal and vertical erosion are sparse,
limiting our ability to understand how bedrock channels and their valleys evolve.
This thesis presents such empirical data on discharge, channel and
hydraulic geometry, shear stress, and sediment supply, and it assesses the role of
these parameters in the development of sinuosity along mountain rivers. It
vii
approaches the problem from a historical perspective, at the scale of a continent,
and at the scale of a relatively small mountain catchment. It has three chapters:
Chapter 1, “Migration and meandering of bedrock rivers,” introduces the
concepts addressed in this thesis through a survey of bedrock river meandering
that brings together classic literature on incised meanders, theories of river
meandering and bedrock river erosion, and original field observations from
bedrock rivers. This chapter makes a case for more attention to horizontal erosion
in studies of mountain landscapes based on observations that (1) despite
widespread misconceptions, actively growing incised meanders are common in
nature and have been recognized as such for more than a century, (2) meander
theories apply to and account for meandering through bedrock regardless of
whether or not the river is incising (3) meandering affects channel slope, which is
a key parameter in all studies of bedrock river erosion and landscape evolution, (4)
the relative rate of lateral erosion with respect to incision has a first-order effect on
valley and ridge morphologies, and (5) neglecting horizontal erosion can lead to
misinterpretations of the landscape and its boundary conditions.
Chapter 2, “Typhoon-driven discharge variability and bedrock river
meandering,” presents a mapping study that reveals a correlation between explicit
measurements of channel network morphology extracted quantitatively from
topographic data and measures of climate variability such as storm frequency and
statistics of rainfall and discharge. The analysis is restricted to areas of high relief
and to broadly similar lithologies across the islands of the western North Pacific
Ocean, where the tropical cyclones dominate regional trends in climate. The
viii
correlation suggests that the sinuosity of an ensemble of mountain river network
segments is an unambiguous quantitative signature of climate in the landscape.
Chapter 3 is an analysis of field observations and empirical data on
discharge, hydraulic geometry, shear stress, and friction along a meandering
bedrock river in the Central Mountains of Taiwan to account for semi-annual to
decadal changes along this channel and to provide some explanation for how the
channel has evolved to its present form over longer time scales. It is presented in
two manuscripts. The first, “Magnitude-frequency distributions of boundary shear
stress along a rapidly eroding bedrock mountain river,” is a study of discharge
records, high-resolution satellite imagery, and topographic surveys of the channel
to find the rating relationship of discharge and boundary shear stress at a transect.
This rating function is used to convert a long time-series of discharge to a
magnitude-frequency distribution of shear stress. The second manuscript,
“Monitoring the flow conditions and morphological changes of a typhoon flood-
prone bedrock river,” follows with similar empirical data from additional transects,
and also includes data on flow speeds, flow depths, friction, and incipient sediment
particle motion. These data indicate that although rare extreme floods have
significantly greater discharges than semi-annual floods, they produce shear
stresses that are only moderately more erosive and able to transport only
moderately larger sediment grain sizes. Comparison to a similarly-sized but much
more-slowly eroding catchment in the eastern United States indicates that erosion
rates of these channels is a stronger function of the frequency of erosive discharges
than it is of the absolute magnitude of the most extreme floods. Observations of
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changes in shear stress and hydraulic geometry for given flows through a series of
meander loops further indicate that changes channel geometry control the
downstream variation of wall inundation frequency which, if correlated to
planform curvature over the long term, could contribute to meandering.
The final section of the thesis shows how the results from each chapter
combine to reveal important ways that discharge magnitude-frequency
distributions contribute to bedrock river erosion processes and landscape evolution,
and identifies several outstanding questions that suggest directions for future study.
REFERENCES
W. M. Davis. The topographic maps of the United States Geological Survey, Science, 21 (534), 225–227, 1893.
A. Einstein. The cause of the formation of meanders in the courses of rivers and of the so-called Baer’s law. In Ideas and Opinions, pages 249–253. Crown Publishers, Inc., New York, 1926.
A. Howard, and G. Kerby. Channel changes in badlands, Geological Society of America Bulletin, 94, 739-752, 1983.
S. Ikeda, G. Parker, and K. Sawai. Bend theory of river meanders. Part 1. Linear development. Journal of Fluid Mechanics, 112, 363–377, 1981.
J. H. Mackin. Concept of the graded river: Geological Society of America Bulletin, 59, 463–512, 1948.
R. H. Mahard. The origin and significance of intrenched meanders. Journal of Geomorphology, 5, 32–44, 1942.
R. C. Moore. Origin of inclosed meanders in the physiographic history of the Colorado Plateau country. Journal of Geology, 34, 29–57, 1926a.
G. H. Roe, Montgomery D.H., and Hallet B. Effects of orographic precipitation variations on the concavity of steady-state river profiles, Geology, 30 (2), 143-146, 2002.
x
B. L. Rhoads and M. R. Welford. Initiation of river meandering. Progress in Physical Geography, 15, 127–156, 1991.
G. Seminara. Meanders. Journal of Fluid Mechanics, 554, 271–297, 2006.
J. B. Shyu, K. Sieh, J.-P. Avuoac, W.-S. Chen, and Y.-G. Chen. Millennial slip rate of the Longitudinal Valley Fault from river terraces: Implications for convergence across the active suture of eastern Taiwan. Journal of Geophysical Research, 111(B08403):doi:10.1029/2005JB003971, 2006.
J. Thomson. On the origin of windings of rivers in alluvial plains, with remarks on the flow of water round bends in pipes. Proceedings of the Royal Society, 16, 1876. Republished in Nature v.14 p.122, June 1976. Also in ”Collected papers in physics and engineering”, ch.16, p. 96-100, CUP, 1912.
K. J. Tinkler. Active valley meanders in South-Central Texas and their wider significance. Geological Society of America Bulletin, 82, 1783–1800, 1971.
J. M. Turowski, N. Hovius, M.-L. Hsieh, D. Lague, and M.-C. Chen, Distribution of erosion across bedrock channels, Earth Surface Processes and Landforms, 33 (3), 353–363, 2008.
K. X. Whipple, Bedrock rivers and the geomorphology of active orogens, Annual Review 625 of Earth and Planetary Sciences, 32, 151–185, 2004.
A. Winslow, The Osage river and its meanders. Science, 22 (546), 32–32, 1893.
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Migration and meandering of bedrock rivers*
* This manuscript is in preparation for submission to Earth Surface Processes and Landforms with co-authors Colin P. Stark, Chingweei Lin, and Hongey Chen.
1
Abstract
Studies of bedrock rivers and their role in landscape evolution tend to focus on channel slope and its changes in response to a com- petition between climate-driven erosion and relative baselevel fall; a common simplifying assumption is that incising rivers do not shift laterally. However, theories of meandering that predict planform evo- lution along any channel with deformable boundaries are supported by abundant field evidence of active meandering along incising rivers, even ones in strong bedrock. Here we review a century of interpreta- tions of incised meanders and discuss how meander theory, developed primarily to explain alluvial meanders, also applies to meanders in bedrock. However, we also present our own observations to show that incised meanders have several important differences from meanders in alluvium that are related to the channel geometry, the role of bed sed- iment, and the erodibility and heterogeneity of bank material, and we discuss how these differences may induce feedback mechanisms that are unique to incising meanders. If neglected, the effect of sinuosity growth on channel slope at constant drainage area and in the absence of tectonic influence may lead to misinterpretations of tectonics from morphology.
1 Introduction
River meanders have sparked curiosity for centuries (e.g. Thomson 1876; Ein-
stein 1926; Alexander 1982), and continue to inspire a great deal of scientific
research (e.g. Rhoads and Welford 1991; Seminara 2006); the literature is
replete with theories to account for their origin and evolution, descriptions
of their regular planforms, and empirical measurements that reveal consis-
tent relationships among aspects of their geometry and hydrology. Much
of the attention has focused on meanders through cohesive sediment of flat
alluvial landscapes. However, some have examined how and why meanders
2 2
also exist incised into the bedrock of plateaus, uplifting mountains, offshore
canyons, and Martian highlands.
Meanders are actually quite common along the incising bedrock rivers of
some areas of high relief, and the processes of active meandering, which is
the result of horizontal bank erosion, must be important to the evolution of
these landscapes. However, these processes tend to be neglected in models
of landscape evolution which consider bedrock erosion to be a key parameter
while fixing the river network planform. A long overdue review of meander
theory and its application to bedrock rivers will inform interpretations and
models of landscape evolution.
2 The Osage River debate
William Morris Davis sparked a debate in 1893 with letter to Science in
which he presented a theory of the origin of incised meanders along the Osage
River of Missouri (Davis 1893a,b; Winslow 1893, 1894)1. He believed in
cycles of landscape evolution, in which young topography has high relief with
incising rivers while old landscapes are flat and mantled with fine grained
1The terminology in the incised meander literature cited in this review has been in- consistent; words like intrenched (Mahard 1942), entrenched (Rich 1914), incised (Tarr 1924), inclosed (Moore 1926a), and in-grown (Rich 1914), have taken turns as labels for all incised meanders and for each type. We prefer to use the unambiguous words active and passive meandering, which refer specifically the state of bank erosion processes in the channel. Passive meandering may occur through planforms that were inherited from an alluvial phase, or through channels that were shaped by active meandering in higher stratigraphy. We further distinguish between the inheritance of channel planform and the antecedence of valley position (e.g. Powell 1875; Davis 1897; Emmons 1897; Jeffer- son 1897; Sears 1924), since, for example, an actively meandering river may traverse and incise into an anticline along a valley axis that is antecedent to the uplift.
3 3
sediment through which rivers meander. The Osage seemed to defy this
paradigm, since it is “extremely tortuous in a steep-sided valley, trenched
two or three hundred feet below the level of the surrounding upland” (Davis
1893a). Davis deduced that the curious morphology of the Osage must
indicate a series of landscape evolution cycles; some time in the past the
Osage had “worn down its basin to a surface of faint relief,” such that “its
slope became gentle and its current had taken to a deviating path, peculiar
to old streams, which so generally meander on their flat flood plains.” As
the plateau began to rise, “the faithful stream once more turned the task
of cutting down its channel...but in doing so, it retained in the new cycle
of its life the meandering course that it had attained in its old age in the
previous cycle.” That is, renewed uplift of old topography causes a switch
from horizontal to vertical erosion; whatever planform happens to exist when
this occurs is locked in place and is retained through the young stages of the
next erosion cycle.
This was not an entirely new idea; twenty years earlier, John Wesley Pow-
ell arrived at a similar sequence of events to explain the sinuous canyons he
found in expeditions to Colorado Territory, arguing that along some reaches,
“the drainage was established antecedent to the corrugation or displacement
of the beds by folding and faulting” (Powell 1875). This is similar to Davis’s
idea, but not quite the same, since the antecedence of valley position that
Powell described does not require an inherited channel planform.
If sinuosity, defined as the ratio of the curved along-channel length l
4 4
to the straight down-valley length L, does not change during incision, and
if channel widths adjust quickly to changes in discharge, then empirical
scaling relationships of alluvial channel width and wavelength imply that
inherited meanders are windows into the past conditions under which the
sinuosity formed (e.g. Dury 1953; Carlston 1964, 1965; Schumm 1967). That
is, meander wavelengths correspond to particular channel widths, so incised
meander wavelengths indicate the width of the channel when meandering
was active. The variation of channel width with discharge further indicates
that the past width inferred from an inherited meander wavelength corre-
sponds to the past discharge conditions, which, if different from the current
conditions, may reveal a change in climate or network structure (Davis 1913;
Wright 1942; Dury 1954, 1955, 1964, 1970; Williams 1986).
Since planform inheritance requires baselevel to fall, it also signifies a
physiographic history that may include either eustatic change in mean sea
level, isostatic or tectonic uplift, or the breach of an ice or landslide dam.
Gardner (1975)’s experiments indicated that sinuosity inheritance occurs
when baselevel drops without regional tilting, such that, for example, mean-
ders incised in Honduran highlands could reveal a regional isostatic response
to the detachment of a subducted slab (Rogers et al. 2002). Likewise, mean-
ders incised into the Sierra Foothills may have resulted from the upstream
propagation of a knickpoint into a lava plateau over which the rivers had
meandered through alluvium (Huber 1981).
Implicit in the concept of inheritance is an assumption that incising
5 5
rivers either do not cut their banks, or they do so symmetrically. Therefore
inherited incised meanders also reveal properties of the flow and load that
would support either zero or symmetric bank erosion and that must have
persisted through the time of incision.
However, these inferences are invalid if meandering occurs during inci-
sion. For example, Missouri geologist Arthur Winslow disagreed with Davis’s
interpretation of the Osage since he knew of no field evidence for the pene-
planation required by inheritance (Winslow 1893). Instead, Winslow said
that vertical incision along the Osage was always accompanied by “lateral
degradation and movement.” He explained that “where the current im-
pinges, sapping will increase the convexity and the sinuosity will become
more pronounced,” even as “the channel sinks vertically at the same time.”
(Winslow 1893).2 Only a single cycle of erosion is necessary for the channel
to shape itself this way.
Although Davis continued to argue his case for the Osage, he soon al-
lowed that active planform evolution can occur during incision, even with
lateral migration rates that far outpace downcutting (Davis 1906, 1913).
Winslow, meanwhile, maintained his argument for the Osage based mostly
on his knowledge of the geological history of the Ozark region (Winslow
2Unfortunately, we can no longer revisit Winslow’s field area to settle the Davis- Winslow debate over the nature of the Osage planform; the meanders in question were submerged dozens of meters below the surface of the Lake of the Ozarks by the con- struction of Bagnell Dam, completed in 1931. Likewise, the rivers Powell described were flooded by the construction of Glen Canyon Dam, completed in 1966, which created the lake that takes his name. These reservoirs remain popular vacation destinations, and together their hydroelectric plants have the capacity to generate nearly 1500 megawatts of power (Lowry 2003; http://www.usbr.gov).
6 6
1894); however, he did not refute the possibility that inheritance can oc-
cur elsewhere. The question therefore became one of classification: what
features on a sinuous incised channel are indicative of inherited sinuosity
versus active meandering? Most subsequent studies focused on the cross-
valley shape, since downcutting with no centerline migration should produce
symmetric valleys while downcutting with simultaneous centerline migration
should produce valley asymmetry that, along a sinuous river, will alternate
polarity downstream (e.g. Davis 1906, 1913; Rich 1914; Tarr 1924; Moore
1926a,b; Hol 1938, 1939; Masuch 1935). A corollary set of features is associ-
ated with the topography of the inter-meander spurs: on an inherited train
of meanders, the spurs have co-planar surfaces that overlap at high elevation
and, in a plateau, are continuous with the surrounding peneplain (e.g. Tarr
1924); active meanders, however, have slip-off slope spurs that dip into each
bend (Mahard 1942) (Figure 1).
Valley shape alone could be misleading, since the channel curvature
should always induce some lateral cutting, and may lead to asymmetry even
along inherited meanders (Mahard 1942). Instead, the best discriminators
are meander cutoffs, which occur when adjacent bends intersect and pinch
off a section of the channel. In an alluvial system, the abandoned channel
segment becomes an oxbow lake which eventually fills with fine-grain lacus-
trine sediment to form a plug within the alluvial plain. Erosional resistance
of such plugs may promote the development of planform bend asymmetry
and complex or multiple bends which exhibit reversals of curvature within
7 7
Figure 1: Left: Passive meandering on an incised channel. The steep and sym- metric valley walls indicate that this river has not migrated much while it incises. The overlapping inter meander spurs are flat and coplanar with the surrounding plateau, on which there are alluvial indicators of an abandoned loop (gray dashed lines) Right: Active meandering on an incised channel. The asymmetric valley preserves slip-off slopes at each bend. Bend growth has been sufficient to cut off and abandon a meander loop. On both rivers, the dotted red line shows the minimum sinuosity required at the onset of incision (after Mahard 1942)
8 8
120°E
slip-off slope
cut bank
Figure 2: Low-level aerial photographs of the Jukuo River in southwestern Taiwan (a) and location maps (b and c). This reach exhibits the characteristic landforms of incised meandering including valley asymmetry with relatively gentle slip-off slopes φ on the inner bends and steep hillslopes θ on cutbanks, cutbank landsliding, and necking. The meander loop around Shetoushan will become abandoned some time in the near future; high stage flows are now attacking both sides of the intermeander spur (red arrows) causing slopes to fail and reducing the ridge height to leave a narrow, sharp, saddle-shaped spur neck. As the ridge continues to fall, deep flows will overtop it and eventually cut off the loop that surrounds Shetoushan.
9 9
a single loop. In an incising system the cutoff process leads to a suite of
characteristic landforms. As adjacent bends come together, they attack the
elevated intervening intermeander spur from both sides; undercutting in the
channel induces slope failure, reducing both the width and elevation of the
spur to form a narrow saddle-shaped neck behind a rounded hill at the
end of the spur. The morphology of the spur at this point, prior to cut-
off, resembles the head and neck of a snake; a famous example along the
Jukou River (Zhuokoux) in southwestern Taiwan is even called Shetoushan
(Shetoushan), which translates as snake head mountain. After cutoff, the
head portion of the spur is surrounded by the abandoned channel loop which
tends to fill with alluvium and colluvium to form a flat field in otherwise high
relief terrain, well-suited for villages, farms, soccer fields, etc. (Figure 3).
The position of cutoffs in the landscape and their associated features
therefore correspond to the river’s position at the time of active meandering.
Mahard (1942) recognized in particular that remnants of an alluvial system,
such as oxbow lakes, preserved in the peneplain would be the strongest
evidence of inheritance, although we know of no examples of these in nature
or literature. Cutoff and abandoned meander loops at a variety of elevations
above the channel, however, are very common.
Since active meanders form in situ, they obviate the need to invoke a
past alluvial phase for explaining existing sinuosity; however, this does not
preclude the possibility that the channel used to meander through alluvium
(e.g. Winslow 1894; Hack and Young 1959; Leopold et al. 1964; Blank 1970),
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Shetoushan
Figure 3: The Jukou River river (22.88 N, 120.65 E) in Taiwan in a 5 m DEM and SPOT (December 31, 2003), and Formosat-2 (March 17, 2006) images, left to right. This river shows all of the characteristics of active meandering, including cutoff meander loops (purple polygons in the DEM), alternating valley asymmetry with gently-dipping slip-off slopes, and cutoffs. Two extreme floods occured in this basin between the SPOT and Formosat-2 image dates. Landslides that occured during these floods (marked by red polygons in the DEM and arrows in the Formosat image) were limited to the outer bends, indicating that the slopes were destabilized by focused outward erosion around each meander loop.
11 11
or even that the existing planform may retain some features of a geometry
that formed in higher stratigraphy (e.g. Harden 1990), perhaps even in al-
luvium (Twidale 2004). For example, variation of lateral erosion rates with
erodibility, or a negative feedback that slows the meandering process with
incision (Stark et al. 2008) could each produce an existing planform that is
not evolving today but that formed during incision through overlying layers.
Whether or not these processes occur remains to be tested.
There is also a class of incised meanders which form consequent to
bedrock features (e.g. Powell 1875; Jefferson 1897; Strahler 1946) or other
external influences such as emerging topography or glacial obstructions (e.g.
Challinor 1933). Well-known examples on the Conodoguinet Creek in Penn-
sylvania and the North Fork of the Shenandoah River in Virginia both have
a series of tight loops connected by long, straight, parallel reaches that run
perpendicular to a strong cleavage in their shaley bedrock (Strahler 1946;
Hack and Young 1959). Lateral cutting occurs on these rivers only where
the flow direction matches the cleavage and the force required to erode is
minimized. This sort of externally controlled sinuosity is distinctly differ-
ent from sinuosity that is a function of the interaction of the flow and its
boundaries.
There are published descriptions of incised meanders in a variety of litho-
logic, climatic, and tectonic environments (Table 1), and many of them show
clear evidence of active meandering. Countless more actively meandering
bedrock rivers are visible with spatially variable abundance across the planet
12 12
Incised Meander Location: References
Rich (1914); Tarr (1924); Lancaster (1998)
Colorado Plateau: Powell (1875); Sears (1924); Moore (1926a,b)
Huber (1981); Harden (1990); Baars (1990)
French Lorraine: Davis (1906, 1909); Blache (1939, 1940)
Eifel region of Germany: Davis (1906); Flohn (1935)
Belgian Ardennes: Davis (1906)
Appalachian Mountains: Davis (1906); Bates (1939); Wright (1942)
Strahler (1946); Hack and Young (1959); Braun (1983); Brakenridge (1985)
Mills and Mills (2001); Twidale (2004)
Edwards Plateau of Texas: Blank (1970); Tinkler (1971, 1972)
Eastern Highlands of New South Wales, Australia: Young (1970)
California Sierra Nevada: Huber (1981)
Southern Sierra Nevada of Spain: Jansen (2006)
Central American Highlands of Honduras: Rogers et al. (2002)
Coast Range of Oregon: Kobor and Roering (2002)
Northern Cape Province of South Africa: McCarthy and Toth (2004)
Watchung Hills of New Jersey: Ashley et al. (1988)
Foothills of the Canadian Rockies: Crickmay (1960)
Central and Coastal Mountains of Taiwan: Hovius and Stark (2001)
Hsieh et al. (2001); Shyu et al. (2006)
13 13
in any web-based virtual globe program with high resolution imagery. It is
clear that the processes of meandering are common not only in alluvial rivers
but in incising ones, as well; indeed many theories of meandering suggest
that this is entirely expected.
3 Meander theories and bedrock rivers
Rivers meander if their boundary shear stresses are erosive along each bank
with offset periodicity. Theories of what brings about such a condition
have evolved over time, with discussions that have focused on the effects
of, for example, centripetal acceleration around bends (e.g. Thomson 1876),
boundary shear friction (e.g. Einstein 1926), Coriolis-induced secondary flow
currents (e.g. Kalkwijk and Booij 1986), topographic steering of the high ve-
locity thread (e.g. Dietrich and Smith 1983), asymmetry of turbulent eddy
generation (e.g. Hey 1976), variations in the vertical flow velocity struc-
ture (Engelund and Skovgaard 1973), and periodic sediment bedforms (e.g.
Callander 1969).
In a classic pair of papers, Langbein and Leopold (1966) and Leopold and
Langbein (1966) wrote that river meanders have a shape that simultaneously
minimizes the energy required to flow through the bent path and minimizes
the variance in direction along that path. Although the curve itself is not
a typical sine waveform, the angle Θ that it makes with the down-valley
14 14
Θ = ω sin l, (1)
where l is the distance along the stream and ω is a coefficient that varies
with the angle of the stream when it crosses the valley axis; larger values of
ω correspond to fatter loops. This concept of a sine-generated curve proved
useful as a mathematical description of meander planform for subsequent
modelers (e.g. Parker et al. 1982; Johannesson and Parker 1989; Edwards
and Smith 2002; Seminara 2006). However, the shape of anything growing in
a periodic curvilinear geometry can be described at any point in its evolution
by a sine-generated curve; therefore, while these curves do approximate the
shape of many alluvial and incised meanders, they do not provide insight
into the processes that cause meanders to form and evolve.
Others (e.g. Callander 1969; Parker 1976; Fredsoe 1978) found a physically-
based explanation for meander initiation in an instability that develops as
water flows over a mobile bed. Submerged sediment bars tend to grow on
alternate sides of the channel; shoaling over these elevated bars and into the
adjacent deep pools generates thalweg sinuosity which, they argued, pre-
curses a meandering channel characterized by a wavelength that matches
the periodicity of the original bars. This theory was supported by flume
experiments in which alternate bars formed along a straight channel prior
to the initiation of meandering (Schumm and Khan 1972). However, these
bar theories fall short in their attempts to explain meandering dynamics
15 15
for several reasons; most important, they do not include descriptions of
bank erosion, which means they can explain sinuosity of the thalweg but
not meandering of the channel (e.g. Rhoads and Welford 1991). Further-
more, they require a mobile bed for bar formation and therefore neglect the
meanders that form in supraglacial meltwater streams, ocean currents, off-
shore canyons, and incising bedrock rivers which have at most limited bed
sediment.
Ikeda et al. (1981) and others (e.g. Parker et al. 1982; Parker and An-
drews 1986; Edwards and Smith 2002) have addressed these shortcomings
by considering the stability of the channel planform to perturbations of its
centerline. These bend theories focus on asymmetry in the flow velocity pro-
file which results from such a perturbation through an interaction of the
flow with channel curvature and boundary friction; friction at the walls and
bed reduce the flow speed near the boundary, allowing this thread to be
more effectively turned around the bends than the faster flow at the free
surface (Thomson 1876; Einstein 1926). This causes a secondary flow cur-
rent in which the high-velocity thread is displaced from bank to opposite
bank along the channel; since the shear stress on the channel boundary in-
creases with flow speed (e.g. Guo and Julien 2005), fastest horizontal erosion
will tend to occur where the fastest flow thread most closely approaches the
channel walls. It is common for the periodic displacement of the fastest flow
to be offset with respect to the curvature of the channel centerline; if erosion
is maximized downstream from the bend apex, the entire meander train will
16 16
migrate down the valley.
At a transect on a bend, the flow asymmetry is characterized in its
simplest form by an excess velocity at the concave bank relative to the
convex bank; lateral migration of the channel centerline is then explained
with an erosion rule that is based on the magnitude of this across-channel
velocity variation. Since the redirection of a flow around a curved channel
requires a force applied by the outer bank on the flow, a corresponding force
is also applied by the flow on the outer bank (e.g. Begin 1981). The likelihood
and rate of bank erosion is expected to increase with this force. Increasing
curvature boosts the centrifugal acceleration around a bend, enhancing the
divergence of near-boundary and near-free surface flow threads. At constant
channel width and arc length, this increase in flow asymmetry increases
the rate of outer bank erosion (Furbish 1988) in a positive feedback which
operates until a stable form is achieved (e.g. Stark et al. 2008), for example,
through negative feedbacks between bend growth and its associated slope
reduction, or through wall buffering that increases as faster bank erosion
produces greater volumes of sediment. On incising rivers, bank erosion may
likewise slow with incision if wall buffering increases as hillslopes lengthen.
If these negative feedbacks do not occur, erosion rates will increase with
curvature until adjacent bends intersect and bypass the loop entirely, cutting
it off and resetting the reach sinuosity to 1.
Factors that influence the production of secondary currents and the rate
of bank erosion include curvature and friction as well as flow speed, dis-
17 17
charge, water surface slope, flow width, and arc length. Bar formation can
also play an important role in establishing the initial perturbation (e.g. Blon-
deaux and Seminara 1985; Johannesson and Parker 1989), but it is not re-
quired by bend theories, since curvature and friction alone are sufficient to
produce the required flow asymmetry. Importantly, bend theories do not
consider the role of sediment, but instead assume that erosion of the outer
bank of an alluvial river is matched by bar formation at the other side of the
channel. This assumption is not necessary for meandering of incising rivers,
since the combination of vertical and horizontal erosion allow the channel to
maintain its width passively. In this way, bend theory is arguably more ap-
plicable to incising rivers than it is to alluvial ones; in fact incised meanders
appear explicitly in several studies (e.g. Ikeda et al. 1981; Parker et al. 1982;
Carson and Lapointe 1983; Kitanidis and Kennedy 1984; Blondeaux and
Seminara 1985), indicating that for those developing meander theory, there
is nothing particularly surprising or unusual about the active meandering of
bedrock rivers. However, there are critical differences between meandering
rivers that incise and those that do not, and these considerations are the
focus of the following section.
18 18
4.1 Bankfull discharge
Meander models tend to specify flow in terms of some combination of dis-
charge, velocity, or flow depth corresponding to a characteristic event (e.g.
Blondeaux and Seminara 1985). For alluvial rivers, the bankfull discharge
(e.g. Williams 1978) is a convenient and natural choice. However, the con-
cept of a bankfull discharge has no meaning in incising bedrock channels
(Tinkler 1971, 1972), since these are commonly bounded by valley walls
with no floodplain; there is no way for an incising river to dynamically adjust
the bankfull depth through the construction of natural levees. A statistical
choice, such as the mean annual maximum or the 99th percentile discharge,
could be an alternative. However, it is unlikely that a single choice would
apply to both alluvial and incising rivers, since all floods are confined to an
incising bedrock channel, whereas only those that are less than bankfull are
confined to an alluvial one. This could significantly affect the relationships
of discharge, shear stress, and erosion in bedrock versus alluvial channels;
the same long term distribution of discharges should produce different dis-
tributions of shear stress in channels with and without floodplains.
The best option would be to avoid generalization and consider all of
the discharges that occur in a channel, and combine them with empirical
data on the hydraulic geometry of each flow. However, this is impractical,
since hydraulic geometry depends on channel slope, bed roughness, and
19 19
cross sectional geometry, and is therefore unique at every transect. Instead,
empirical data on the spatiotemporal variation of hydraulic geometry and
shear stress along alluvial and incising rivers may help to identify a robust
statistic or set of statistics to use for generalizing discharge.
4.2 Bank height
The erosion of channel banks can occur gradually through shear-related pro-
cesses like plucking and abrasion, or incrementally through collapses and
landslides that result when undercutting in the channel destabilizes the
overriding slope. Collapses and landslides produce significant volumes of
sediment which buffers the bank, protecting it from further erosion until
subsequent flows are sufficiently powerful to erode or remove it; the long
term ability of flows to eliminate this sediment may limit the long term rate
of horizontal erosion (Seminara 2006). Since the amount of sediment buffer-
ing the banks is an increasing function bank height and collapse frequency,
an inverse relationship may exist between channel relief and lateral erosion
rate (Hickin and Nanson 1975; Nanson and Hickin 1983).
Along alluvial rivers, the bank height subject to failure is simply the
bankfull depth, but slope failures along incising bedrock channels can ex-
tend from the thalweg to the drainage divide. We expect this to result in
more buffering along incised channels than alluvial ones, and may result
in more buffering with incision if hillslopes are simultaneously lengthening.
However, the valley shape of incising rivers may compensate for this by con-
20 20
fining all flows to the channel and maximizing the stresses available to erode
the buffering boulders and carry them away; however, the details of such
tradeoffs have yet to be addressed in theory, experiments, or observations.
4.3 Bed buffering
Bed sediment in an incising channel can be both an agent of erosion and
a protective covering against it, causing the rate of downcutting to be a
nonlinear function of sediment supply with a positive correlation at small
loads and a negative one at higher loads (Gilbert 1877). Moore (1926a)
applied this idea to a curved channel and wrote that “the effectiveness of
sideward cutting in proportion to downcutting seems to be controlled by the
relative loading of the stream...if the load is relatively large, and especially
if it consists in part of coarse material which is rolled and slid along the
stream bed, there is a blanketing effect which partly protects the bed...and
the effect of the erosion on the unprotected side walls which is thus produced
is relatively very important.”
Similarly, (Tinkler 1971, 1972) argued that to accomplish any downcut-
ting in a mixed bedrock-alluvial channel, there must be a recurrent flood of
great enough magnitude to clear all of the sediment and expose the bed to
erosion. Otherwise, only lateral erosion would be possible. This is supported
by flume experiments in which flow through an initially sinuous channel in
a simulated isotropic bedrock of sand, silt, and kaolinite clay incises down-
ward only when all of the available sediment is entrained; while there is bed
21 21
sediment, erosion occurs at the outer bank of the bends (Shepherd 1972;
Shepherd and Schumm 1974; Dury et al. 1976). Studies of natural channels
have also shown that bedcover may induce widening (e.g. Pazzaglia et al.
1998; Pazzaglia and Brandon 2001; Hancock and Anderson 2002), which,
unless perfectly symmetric, results in centerline migration.
However, the effect of bed sediment is likely complicated by shape of
the channel and the relative scale of the discharge variation and the sedi-
ment extent, thickness, quality, and caliber. For example, flows of a certain
magnitude may be required to wet the banks, and even greater flows may
be required to accomplish bank erosion. These extreme floods that erode
the channel walls may also carry enough sediment to protect and cover the
bed, leaving the steeper banks exposed (Turowski et al. 2008). Meanwhile,
discharges at lower stage than is required to wet the banks may be sufficient
to move enough bed sediment to attack the channel bed. Thus the most
powerful events may carve the channel walls while the bed is protected by
bed sediment, while significantly less extreme discharges may be required to
erode the bed while the banks remain dry.
Bed sediment on incising rivers is typically thin and intermittent, but it is
commonly extensive enough to potentially form the alternate bars which me-
ander bar theories require for the initiation of thalweg sinuosity, and which
can produce the perturbations required by bend theories of meandering to
initiate channel curvature (e.g. Ikeda et al. 1981). Thus bedload need not
be eliminated from theories or models of meanders along incising bedrock
22 22
rivers, although differences with respect to alluvial sediment in grain size
and cohesion should be considered.
4.4 Erodibility
In general, incising rivers have stronger, less erodible banks than alluvial
ones, which, all else equal, will result in slower meandering (Davis 1913;
Tinkler 1971); along incising rivers, the meandering rate should increase with
bedrock erodibility. Likewise, harder rocks should be more likely to preserve
existing sinuosity against both loop enlargement and downstream migration,
while weaker rocks would be more vulnerable to lateral erosion (Jefferson
1897; Davis 1906). In soft or weak enough bedrock, sinuosity could even
disappear through downstream migration or erosion of the overlapping spurs
(Moore 1926a; Cole 1930, 1937). For example, in the Virginia Appalachians,
meanders cut into weak shaley rocks tend to have amplitudes two to three
times the size of nearby meanders with similar wavelength and drainage
area that are cut into harder, massive carbonates; if these rivers have been
evolving for the same amount of time, lateral erosion is at least twice as fast
in the weaker rocks (Braun 1983; Abrahams 1985).
How erodibility influences the relative rates of centerline migration and
downcutting is not immediately obvious, since a range processes may be in
competition. For example, if erodibility promotes lateral erosion, it may lead
to greater sediment loads which may act to protect the weaker bed, thereby
slowing down vertical incision. Or, that same increased lateral erosion may
23 23
be accomplished through undercutting and slope destabilization, that con-
tributes to a negative feedback due to increased wall buffering. Investigations
of rock strength and meandering rates may help to clarify whether or not
these feedbacks exist and how they work.
4.5 Bank heterogeneity
Meander bend theories do not yet explicitly account for the formation of
multiple loops, along which curvature changes more frequently than the
dominant wavelength of sinuosity. However, Lancaster and Bras (2002)’s nu-
merical experiments indicate that topographic steering (Dietrich and Smith
1983) of the bed may be sufficient, but they noted that heterogeneity of
bank erodibility is also likely to be important. Bank erodibility varies along
alluvial meanders due to the lacustrine plugs that form in oxbow lakes, and
along bedrock rivers due to changes in lithology and lithologic structure, or
to anisotropy of structures that causes erodibility to vary with flow direc-
tion. If heterogeneity is to be included in meander models, its sources, which
differ in incising and alluvial meanders, should also be considered.
5 Incised meander valley morphology and
mesurement of meander rate
Changes in channel width w and centerline migration dx/dt both depend
on how each bank moves through erosion and deposition with respect to
24 24
some fixed datum. If we consider the sign on the bank migration to indicate
erosion (positive) or deposition (negative), then w at a given transect will
always change unless the sum of the migrations of each bank is zero. Such a
condition could occur at a transect where both banks have zero migration,
or, on an alluvial river, from erosion on one bank and equal-magnitude
deposition on the other. Along an incising river, no deposition is required,
since erosion of the bed and one bank passively sets the position of the
other bank according to the flow depth. Migration of the centerline will
always occur unless the bank migrations are equal magnitude and oppositely
directed.
banks, levees, swales, and paleosols, combined with radiocarbon ages can
reveal rates and histories of lateral channel migration (e.g. Brakenridge
1985). Time series imagery can also provide information on channel mi-
gration (Crickmay 1960), although this is of limited use unless migration
rates are exceedingly high and/or the temporal baseline of the image series
is sufficiently long.
Where such data is not available, we can at least infer the relative rates of
centerline migration and vertical incision from the shape of the valley. Along
a channel transect with dx/dt = 0, the slope of the valley walls is set by a
lithologically or structurally controlled hillslope angle θ. Where dx/dt = 0, a
slip-off-slope angle φ is defined by the arctangent of the ratio of bed erosion
dz/dt to dx/dt. The active or cut-bank side of a such a channel’s valley will
25 25
have a slope set by θ, but the passive slip off slope side will have a slope set
by θ or by φ, whichever is smaller (Figure 4).
If there is a fluvially set slip-off slope angle φ, and if the rate of bed
erosion dz/dt is greater than zero, φ alone provides an estimate of the relative
rates of horizontal and vertical erosion; estimation of the absolute dx/dt also
requires a knowledge of dz/dt, since
dx
dt =
dz/dt
tanφ , (2)
for φ > 0. Although valley symmetry has long been considered an indicator
of zero dx/dt, we expect any transect where dx/dt is relatively too small
to produce a stable φ given the local value of θ to have a symmetric cross
section. Thus the most we can say for a symmetric valley is
0 ≤ dx
dt ≤ dz/dt
tan θ . (3)
Since hillslope angle θ is strongly influenced by bedding, cleavage, and joints,
it may be spatially non-uniform, resulting in structurally-controlled cross-
valley asymmetry (e.g. Judson and Andrews 1955). Inferences about dx/dt
made qualitatively through observations of valley geometry or quantitatively
with Equation 2 should always be supported by field evidence that the asym-
metry in question does not have this sort of structural influence.
As a case study, we considered the Hsiukuluan (Xiuguluan) River (Fig-
ure 5), which drains the Central Mountains of Taiwan and flows north
26 26
|dx| dt
dz dt
Figure 4: Schematic valley shapes. a) A symmetric valley with walls that have slopes set by the hillslope angle θ on both sides. This valley shape does not indicate the direction or magnitude of channel centerline migration since the rate of horizontal erosion is too small relative to downcutting to preserve slip-off slopes; however, lateral erosion and centerline migration may still occur in such a valley, with a rate described by Equation 3. b) An asymmetric valley with a stable slip-off slope angle φ on one side and the hillslope angle θ on the other. Here horizontal erosion is relatively fast enough to preserve a record of the relative rates of horizontal and vertical erosion in the slip-off-slope, while the cutbank slope is maintained by slope failure. c) A valley in which the ratio of horizontal to vertical erosion decreased, either through a reduction of horizontal erosion or through an increase in downcutting. The dotted line marks the break in topographic slope, but it does not necessarily denote the point in time when the transition in erosion rate ratios occured. This is because the relative increase in vertical erosion causes hillslopes on both sides of the valley to fail: if the relative rates of horizontal and vertical erosion match the hillslope angle, and horizontal erosion continues to operate on the left side of this schematic valley, then the slope break marks the point in time when a change in erosion rate ratios occured; otherwise, slope failure on the right side of the valley will cut into the slip-off slope, eventually eliminating it entirely and forming a symmetric valley like the one in (a).
27 27
through the Longitudinal Valley. It then makes a sharp turn to cross highly
erodible flysch and turbidite sediments of the western flank of the Coastal
Range. Near the range axis, it crosses the Chimei (Qme) fault and crosses
harder andesitic volcanic rocks before reaching the Pacific Ocean (e.g. Yu
and Kuo 2001; Shyu et al. 2006) (Figure 5). The valley position appears
to be antecedent to the uplift of the coastal mountains, but the river is ex-
tremely sinuous and along its bends are several cutoff meander loops that
indicate this channel’s rapid planform evolution through orogenesis. Allu-
vial deposits within the cutoff loops are proof that these are abandoned
channel segments. Patches of similar deposits all along the intermeander
slip-off slope spurs indicate that these are fluvial features formed through
simultaneous incision and centerline migration. Shyu et al. (2006) obtained
14C ages from several of the spurs (Figure 5) and found incision rates to
vary along the channel and through time between 11.2 and 27.3 mm y−1.
We measured the slip-off slope angles along the ridge of each intermean-
der spur of the Hsiukuluan River from a 40m DEM of Taiwan. Using these,
along with Shyu et al. (2006)’s estimates of dz/dt, we deduced through Equa-
tion 2 the outward horizontal erosion rate at each of the bends (Figure 6).
Since valley geometry is a function of the relative rates of dx/dt and dz/dt,
its variation can result from along stream changes in either dx/dt or dz/dt.
This is complicated along the Coastal Range Hsiukuluan since the river
crosses both a growth anticline which has uplift rates that are fastest near
the ridge axis (Lave and Avouac 2001), and a fault that brings hard metased-
28 28
0 1 20.5 Km
Figure 5: Hillshaded 40m DEM of the Hsiukuluan River with inset location map of Taiwan. Cross-valley profiles are drawn down the ridge line of minimum slope on each meander spur and are shown in figure 6. The approximate location of the Chimei Fault, a lithologic boundary between hard andesitic volcanics to the east and weaker flysch and turbidite sediments to the west, is also shown. Stars indicate Shyu et al. (2006)’s sample locations along with their inferred incision rates in mm yr−1. Contour interval is 100 m.
29 29
A B
El ev
at io
n a
b ov
e se
a le
ve l [
30 30
Figure 6: Valley topography looking downstream across the Hsiukuluan River at locations shown in Figure 5. Transects AB, CD, EF, and IJ are all almost entirely within the weaker flysh and turbidite sediments. Transects OP, QR, and ST are entirely within the stronger volcanics. KL, and MN are bisected by the fault, with the cutbank of KL in harder rocks and the cutbank of MN in softer rocks. In general, channel geometry has alternating asymemtry west of the fault, and is symmetric to the east, indicating that the relative rate of horizontal erosion corresponds to the erodibility of the bedrock. Application of Equation 2 to Shyu et al. (2006)’s incision rate estimates and the slope of a linear regression of the inside bank’s valley wall (segments used for fitting are in thick grey) yields rates of horizontal erosion. In this case, we applied the range of incision rates for the upstream cluster of samples to transects AB through GH, and the range of dates for the downstream cluster of samples to transects IJ through MN. Without constraint on the incision rates east of the Chimei fault, we are unable to estimate horizontal erosion, but we know from the hillslope angles the maximum erosion rate ratios. Note that the geometry of EF, and MN are also complicated by a cutoff loop.
31 31
iments and volcanics next to weak turbidites. This lithologic change occurs
a few kilometers from the coast, with the harder bedrock downstream, and
seems to have a clear effect on the valley shape across the fault. Although
there has yet to be a thorough investigation of the relative response of dx/dt
and dz/dt to changes in uplift rate and erodibility, and although there is lim-
ited constraint on dz/dt within the harder volcanics east of the Chimei fault,
we found that the transects with fastest dz/dt tend also to have the fastest
dx/dt within the flysch and turbidite sediments. Marked valley asymmetry
in the weaker substrates has slip-off slopes on the inside of each bend with
φ produced by the relative rates of centerline migration and downcutting;
east of the fault, in the harder rocks, the valley is symmetric, with lateral
erosion is relatively too slow to preserve stable slip-off slopes and with valley
slopes on both sides set by θ.
6 Implications for inferences on tectonics
Fluvial incision into bedrock channels communicates local changes in base
level throughout the channel network and ultimately sets the pace of land-
scape evolution. The vertical component is commonly considered to be a
function of stream power (e.g. Seidl and Dietrich 1992), or boundary shear
stress (e.g. Howard 1994) through a stream power law,
dz
32 32
where dz/dt is the rate of vertical erosion, A is drainage area which serves
as a surrogate for a characteristic discharge, and S is channel slope. The
coefficient k may depend on anything other than slope and area that may
affect incision, such as erodibility, climate, cross sectional channel geometry,
flow hydraulics, roughness, and the volume, grain size, and cover extent of
sediment load (e.g. Whipple et al. 2000; Wobus et al. 2006a).
Rearranged, Equation 4 gives a description of channel longitudinal pro-
files,
n , (5)
where ks, the steepness index, is ((dz/dt)/k)1/n (e.g. Whipple 2004). The
exponents m and n also vary with several influences, including erosion pro-
cess, channel hydraulic geometry, debris flow frequency, and alluviation (e.g.
Whipple 2004). Their ratio m/n describes the concavity of the channel long
profile and is commonly called the concavity index (e.g. Sklar and Dietrich
1998; Whipple 2004; Wobus et al. 2006a).
A river which incises at a rate that matches rock uplift U has widths
and slopes that are adjusted to maintain this balance; narrower and steeper
channels counter faster uplift rates. If widths have simple scaling with con-
tributing drainage area (e.g. Whipple 2004), the form of Equation 5 for
adjusted channels should vary with U , since ks is a function of dz/dt. In
this case, plots of slope against area should provide information about the
relative rates of uplift among nearby catchments; the scaling of channel slope
versus upstream area has been shown to increase with uplift rate (e.g. Kirby
33 33
et al. 2003; Wobus et al. 2006a), so that relative uplift rates may be inferred
from the slopes of such plots.
However, absent tectonic complications, any increase in sinuosity occurs
with an increase in reach length and results in a reduction of slope, which
means meandering rivers have slopes that can change irrespective of con-
tributing drainage area. This will reduce the scaling of slope and area in
much the same way as we expect from a reduction of uplift rate (Figure 7).
Therefore, inferences about tectonics drawn from a relationship of slope to
area may be wrong unless the development of sinuosity is also considered.
There is a limit to the magnitude of this effect, since cutoffs prevent reach
sinuosities from growing indefinitely; however, it implies that a condition of
equilibrium between uplift and incision rates may require channels to narrow
in response to sinuosity-driven slope reduction, against the assumptions of
how channels widen predictably downstream. More important, this type of
narrowing is exactly opposite what we expect to occur based on the increased
flux of sediment to the channel that should also result from sinuosity growth,
since increased sediment supply tends to drive widening (e.g. Pazzaglia et al.
1998; Pazzaglia and Brandon 2001; Hancock and Anderson 2002).
Alternatively, meandering rivers may be able to restore their adjusted
slopes and maintain slope-area relationships that reveal U by allowing the
entire catchment to tilt, something that has been shown to occur in alluvial
channels (Harbor 1998). If it occurs in incising rivers as well, it implies that
active meandering indicates a state of transience, and that passive mean-
34 34
138°10'0"E
138°10'0"E
138°20'0"E
138°20'0"E
138°30'0"E
138°30'0"E
28 °1
0' 0"
b
p
35 35
Figure 7: Top: Hillshaded Shuttle Radar Topography Mission (SRTM) 3- arcsecond digital elevation model (DEM) of the Shimanto River with inset showing location in Shikoku, Japan. The black network was derived through standard flow- routing over the SRTM topography using a drainage-hillslope threshold of 1000 pixels (≈ 8 km2). The heavy black line is the trunk channel of the catchment, selected as the path of maximum upstream area at each junction. A reduced- sinuosity version of the trunk stream path was drawn between nodes of the channel network and is shown in red. The mean sinuosity of the SRTM-derived channel is 1.64, but at the reach scale it varies from 1.1 to 2.6. The elevations at each node were extracted from the DEM and combined with the path lengths between nodes to determine the long profiles and the slopes of the channel and its straightened analog. Bottom: Long profile and slope-area plots of the Shimanto River’s trunk channel and its straightened analog. Sinuosity growth lengthens the channel and reduces the slope-area scaling in a way that is similar to what Wobus et al. (2006b) showed for slow versus fast zones of uplift. That is, the effect on a slope-area plot from increasing sinuosity is the same as the effect from reducing uplift.
36 36
dering should take over upon restoration of steady state. It also implies
that catchment relief may increase through sinuosity development. Since
meander models suggest that that bank erosion is strongly dependent on
discharge, incised meandering may provide a previously unrecognized mech-
anism through which climate can produce topographic relief.
At the very least, slope adjustments that occur with meandering should
introduce considerable noise to a slope-area plot and could even obscure the
differences in uplift rate these curves seek to reveal (Figure 7). It may be
possible to infer tectonics from topography through Equation 5 for some re-
gions, but doing so requires an assumption that any changes in slope, width,
roughness, and sediment load character that are due to active meandering
will be captured by the steepness index ks. A better option would be to ex-
plore in detail the factors that contribute to and suppress meandering along
incising rivers and to incorporate these into a new model of bedrock erosion
that includes these considerations; until then, models of landscape evolution
are unlikely to reproduce the landforms associated with meandering.
7 Conclusions
The contribution of bedrock river erosion to landscape evolution involves
processes that modify both the bed and the banks, so it is important to
consider horizontal processes with as much attention as has been afforded to
vertical incision. This requires the identification and analysis of the forces
that drive bank erosion and the conditions that affect it in incising rivers,
37 37
instead of ignoring them or simply collapsing them into a catch-all prefactor
like the steepness index ks. It also requires a better understanding of the
differences between meanders in bedrock and in alluvium so that meander
theories can better address and account for these differences. In particular,
theoretical, experimental, and empiricle observations on the role of feedback
processes that may be unique to incising meanders are currently lacking.
Greater attention to the planform evolution of bedrock rivers will yield a
better physical understanding of bedrock river erosion and landscape evolu-
tion.
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