THE ORIGIN AND SIGNIFICANCE OF SINUOSITY
ALONG INCISING BEDROCK RIVERS
Jonathan Ross Barbour
Submitted in partial fulfillment of the requirements for
the degree of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2008
© 2008
Jonathan Ross Barbour
All Rights Reserved
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.
TABLE OF CONTENTS
Table of Contents…………………………………………………... i
Acknowledgments…………………………………………………..ii
Dedication…...............................................................……………... v
Preface................................................................................................vi
Chapter 1:
Migration and meandering of bedrock rivers…….....1
Chapter 2:
Typhoon-driven discharge variability and bedrock
river meandering…………………………………… 48
Chapter 3a:
Magnitude-frequency distributions of boundary
shear stress along a rapidly eroding bedrock
mountain river……………………………………… 92
Chapter 3b:
Monitoring the flow conditions and morphological
changes of a typhoon flood-prone bedrock river……111
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.
iv
To Oscar.
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
vi
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 floodprone 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
ix
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.
xi
CHAPTER 1 1
CHAPTER
Migration and meandering of bedrock rivers *
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 competition 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 evolution 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 interpretations 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 sediment, 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; Einstein 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 consistent 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
1
The terminology in the incised meander literature cited in this review has been inconsistent; 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; Jefferson 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 Powell 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 corresponds 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, meanders incised in Honduran highlands could reveal a regional isostatic response
to the detachment of a subducted slab (Rogers et al. 2002). Likewise, meanders 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 incision. For example, Missouri geologist Arthur Winslow disagreed with Davis’s
interpretation of the Osage since he knew of no field evidence for the peneplanation 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 impinges, 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 allowed 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
2
Unfortunately, we can no longer revisit Winslow’s field area to settle the DavisWinslow 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 construction 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 occur 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 crossvalley 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 associated 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 lacustrine 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 symmetric 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
(a)
T
de
s
(b)
122°E
120°E
122°E
24°N
22°N
22°N
24°N
la
nd
sli
I
120°E
(c)
100 m
N
22°55'0"N
N
120°41'0"E
120°42'0"E
22°55'0"N
Shetoushan
slip-off slope
landslid
22°54'0"N
22°54'0"N
cut bank
es
0
500 1,000 m
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 cutoff, resembles the head and neck of a snake; a famous example along the
Jukou River (Zhuókŏuxı̄) in southwestern Taiwan is even called Shetoushan
(Shétóushān), 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),
10
10
(a)
(b)
(c)
Shetoushan
0
0.5
1 Km
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 alluvium (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 Pennsylvania 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 different 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 lithologic, 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
Table 1: Examples of Incised Meanders
Incised Meander Location: References
Ozark Mountains of Missouri: Davis (1893a,b); Winslow (1893, 1894)
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)
Allegheny Plateau of New York: (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 velocity thread (e.g. Dietrich and Smith 1983), asymmetry of turbulent eddy
generation (e.g. Hey 1976), variations in the vertical flow velocity structure (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
direction varies sinusoidally downstream,
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 physicallybased 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, precurses 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). Furthermore, they require a mobile bed for bar formation and therefore neglect the
meanders that form in supraglacial meltwater streams, ocean currents, offshore 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 Andrews 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 profile 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 current in which the high-velocity thread is displaced from bank to opposite
bank along the channel; since the shear stress on the channel boundary increases 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. Blondeaux and Seminara 1985; Johannesson and Parker 1989), but it is not required 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 applicable 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
Special considerations for incised meanders
4.1
Bankfull discharge
Meander models tend to specify flow in terms of some combination of discharge, 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 concept 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 distributions 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 processes 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 buffering 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 extend 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 downcutting 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 downward 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 sediment 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 meander 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 contributes 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 numerical 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 direction. 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.
Channel planform reconstructions built through observations of fossil
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 migration (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
dz/dt
dx
=
,
dt
tan φ
(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≤
dz/dt
dx
≤
.
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 crossvalley 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 asymmetry in question does not have this sort of structural influence.
As a case study, we considered the Hsiukuluan (Xiùgūlúan) River (Figure 5), which drains the Central Mountains of Taiwan and flows north
26
26
a)
b)
|dxmax|
dt
T
|dx|
dt
T
dz
dt
T
c)
dz
dt
I
|dx1|
dt
|dx2,max|
dz1
dt
dt
dz2
dt
T
I
T
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 (Qı̆meı̆) 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 extremely sinuous and along its bends are several cutoff meander loops that
indicate this channel’s rapid planform evolution through orogenesis. Alluvial 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 intermeander 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 Equation 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 (Lavé and Avouac 2001), and a fault that brings hard metased-
28
28
122°E
24°N
25°N
120°E
22°N
23°N
I
A
n/a
C
K
M
~12.3–12.9
G
~22.7–27.3
>10–10.7
11.6–12.4
Q
~11.2–12.5
O
~15.1–19.1
E
~17.2–19.3
n/a
n/a
J
L
N
B
R
S
P
>17.5–18.3
D
F
H
T
0
0.5
1
2 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
Elevation above sea level [m]
196-251
mm yr
15.1-19.3
Elevation above sea level [m]
Elevation above sea level [m]
L
36.6˚
mm yr-1
K
10-12.9
4.7˚
í
í
C
í
216-276
N
33-42
mm yr-1
17˚
21.9˚
15.1-19.3
10-12.9
4˚
18.5˚
M
í
í
í
254-325
P
O
mm yr-1
20.2˚
20.9˚
n/a
15.1-19.3
E
F
10.9˚
3.4˚
n/a
í
í
í
mm yr
G
36.1˚
-1
Q
15.1-19.3
H
13.5˚
n/a
í
í
J
8.8˚
21.7˚
í
20.3˚
n/a
í
T
26.4˚
10-12.9
S
mm yr-1
I
í
65-83
n/a
í
R
13.8˚
13.6˚
í
62-80
D
mm yr-1
í
Elevation above sea level [m]
B
22.6˚
4.4˚
í
Elevation above sea level [m]
A
122-157
-1
n/a
í
Distance from thalweg [m]
í
Distance from thalweg [m]
Figure 6: Caption on next page
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 limited 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 landscape 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
= kAm S n ,
dt
32
32
(4)
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 profiles,
m
S = ks A− 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 process, 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 contributing 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 contributing 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°20'0"E
138°30'0"E
28°30'0"N
p
o
28°30'0"N
n
h
g
28°20'0"N
m
l
i
j
k
f
28°20'0"N
e
28°10'0"N
d
c
28°10'0"N
b
a
138°10'0"E
138°20'0"E
138°30'0"E
Drainage area [km2]
350
p
p
300
Elevation [m]
250
Cl z 1.64
o
o
o
m
j
k
100
h gf
e
h
gf e
0
150
d
d c
b
a
b
c
e b
d
d
a
0
Figure 7: Caption on next page.
35
c
e
ff a
b a
c
100
50
Distance from mouth [km]
35
i h
l
i
i
g
g
l
k
j
i
kj
n m
l
k
50
h
j
l
150
í
10
o
m
m
3
10
n
p
Cl z 1
n
n
200
2
10
p
í
10
*UDGLHQW>í@
1
10
Figure 7:
Top: Hillshaded Shuttle Radar Topography Mission (SRTM) 3arcsecond digital elevation model (DEM) of the Shimanto River with inset showing
location in Shikoku, Japan. The black network was derived through standard flowrouting 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 reducedsinuosity 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 mechanism 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 regions, 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 explore 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 evolution.
References
A. D. Abrahams. Lithologic control of bedrock meander dimensions in the
appalachian valley and ridge province; a comment. Earth Surface Processes and Landforms, 10(6):635–642, 1985.
D. Alexander. Leonardo da Vinci and fluvial geomorphology. American
Journal of Science, 282:735–755, 1982.
G. M. Ashley, W. H. Renwick, and G. H. Haag. Channel form and processes
in bedrock and alluvial reaches of the Raritan River, New Jersey. Geology,
16:436–439, 1988.
D. L. Baars. Controlling factors in the distribution and development of
incised meanders in the central Colorado Plateau: Discussion and reply.
Geological Society of America Bulletin, 102:233–242, 1990.
R. E. Bates. Geomorphic history of the Kickapoo region, Wisconsin. Geological Society of America Bulletin, 50:819–880, 1939.
Z. B. Begin. Stream curvature and bank erosion: A model based upon the
momentum equation. Journal of Geology, 89:497–504, 1981.
38
38
J. Blache. Le probleme des meandres encaisses et les rivieres Lorraines.
Journal Geomorphology, 2:201–212, 1939.
J. Blache. Le probleme des meandres encaisses et les rivieres Lorraines, II.
Journal Geomorphology, 3:311–331, 1940.
H. R. Blank. Incised meanders in Mazon County, Texas. Geological Society
of America Bulletin, 81:3135–3140, 1970.
P. Blondeaux and G. Seminara. A unified bar-bend theory of river meanders.
Journal of Fluid Mechanics, 157:449–470, 1985.
G. R. Brakenridge. Rate estimates for lateral bedrock erosion based on
radiocarbon ages, Duck River, Tennessee. Geology, 13:111–114, 1985.
D. D. Braun. Lithologic control of bedrock meander dimensions in the appalachian valley and ridge province. Earth Surface Processes and Landforms, 8(3):223–237, 1983.
R. A. Callander. Instability and river channels. Journal of Fluid Mechanics,
36:465–480, 1969.
C. W. Carlston. Free and incised meanders in the United States and their
geomorphic and paleoclimatic implications. In Abstracts of 1963, pages
29–29. Geological Society of America, 1964.
C. W. Carlston. The relation of free meander geometry to stream discharge
and its geomorphic implications. American Journal of Science, 263:864–
885, 1965.
M. A. Carson and M. F. Lapointe. The inherent asymmetry of river meander
planform. Journal of Geology, 91:41–55, 1983.
J. Challinor. The ”incised meanders” near Pont-erwyd, Cardiganshire. The
Geological Magazine, LXX:90–92, 1933.
W. S. Cole. The interpretation of intrenched meanders. Journal of Geology,
38:423–436, 1930.
W. S. Cole. Modification of incised meanders by floods. Journal of Geology,
45:648–654, 1937.
39
39
C. H. Crickmay. Lateral activity in a river of northwestern Canada. Journal
of Geology, 68(4):377–391, 1960.
W. M. Davis. The topographic maps of the United States Geological Survey.
Science, 21(534):225–227, 1893a.
W. M. Davis. The Osage River and the Ozark uplift. Science, 22(563):
276–279, 1893b.
W. M. Davis. Current notes on physiography. Science, 5(121):647–649, 1897.
W. M. Davis. Incised meandering valleys. The Bulletin of the Geographical
Society of Philadelphia, IV(4):182–192, 1906.
W. M. Davis. The Seine, the Meuse, and the Moselle, pages 587–616. Ginn
and Company, New York, 1909.
W. M. Davis. Meandering valleys and underfit rivers. Annals of the Association of American Geographers, III:3–28, 1913.
W. E. Dietrich and J. D. Smith. Influence of the point bar on flow through
curved channels. Water Resources Research, 19:1173–1192, 1983.
G. H. Dury. An approach to paleometeorology. Nature, 172:919, 1953.
G. H. Dury. Contribution to a general theory of meandering valleys. American Journal of Science, 252(4):193–224, 1954.
G. H. Dury. Bed-width and wavelength in meandering valleys. Nature, 176:
31–32, 1955.
G. H. Dury. Principles of underfit streams. Geological Survey Professional
Paper 452-A, pages A1–A67, 1964.
G. H. Dury. General theory of meandering valleys and underfit streams,
pages 264–275. Macmillan & Co., Little Essex Street, London, 1970.
G. H. Dury, R. G. Shepherd, and S. A. Schumm. Experimental study of river
incision: Discussion and reply. Geological Society of America Bulletin, 87:
319–320, 1976.
B. F. Edwards and D. H. Smith. River meandering dynamics. Physical
Review E, 65(046303):1–12, 2002.
40
40
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 (1954), pages
249–253. Crown Publishers, Inc., New York, 1926.
S. F. Emmons. The origin of the Green River. Science, 6(131):19–21, 1897.
F. Engelund and O. Skovgaard. On the origin of meandering and braiding
in alluvial streams. Journal of Fluid Mechanics, 57:289–302, 1973.
H. Flohn. Beitrage zur problematik der talmaander. Frankfurter Geographische Hefte, Hefte 1:1–96, 1935.
J. Fredsoe. Meandering and braiding of rivers. Journal of Fluid Mechanics,
84:609–624, 1978.
D. J. Furbish. River bend curvature and migration: How are they related?
Geology, 16:752–755, 1988.
T. W. Gardner. The history of part of the colorado river and its tributaries: An experimental study, pages 87–95. Canyonlands. Four Corners
Geological Society, eighth edition, 1975.
G. K. Gilbert. Report on the geology of the Henry Mountains. U.S. Geographical and Geological Survey of the Rocky Mountain Region, Government
Printing Office, Washington D.C., 1877.
J. Guo and P. Y. Julien. Shear stress in smooth rectangular openchannel flows. Journal of Hydraulic Engineering, 131(1):30–37, 2005.
doi:10.1061/(ASCE)0733-9429(2005)131:1(30).
J. T. Hack and R. S. Young. Intrenched meanders of the North Fork of the
Shenandoah River, Virginia. U.S.G.S Professional Paper, 354-A, 1959.
G. S. Hancock and R. S. Anderson. Numerical modeling of fluvial strathterrace formation in response to oscillating climate. Geological Society of
America Bulletin, 114(9):1131–1142, 2002.
D. J. Harbor. Dynamic equilibrium between an active uplift and the Sevier
River, Utah. Journal of Geology, 106:181–194, 1998.
D. R. Harden. Controlling factors in the distribution and development of
incised meanders in the central colorado plateau. Geological Society of
America Bulletin, 102:233–242, 1990.
41
41
R. D. Hey. Geometry of river meanders. Nature, 262:482–484, 1976.
E. J. Hickin and G. C. Nanson. The character of channel migration on the
beatton river, northeast british columbia, canada. Geological Society of
America Bulletin, 86:487–494, 1975.
J. B. L. Hol. Das problem der talmaander. Zeitscher. Geomorphologie, Band
10:169–195, 1938.
J. B. L. Hol. Meanders, hun beteekenis en onstaan. K. nederl. Aardrijksk.
Genoot. Tijdschr., ser. 2(Deel 56):161–177, 1939.
N. Hovius and C. P. Stark. Actively meandering bedrock rivers. In EOS,
Transactions AGU, page 506, 2001.
A. D. Howard. A detachment-limited model of drainage basin evolution.
Water Resources Research, 30(7):2261–2285, 1994.
M.-L. Hsieh, P.-M. Liew, and Y. Ota. The dynamic hualien-taitung coast,
eastern taiwan: a treasure for studying active tectonics and coastal evolution. Booklet, 2001. Field guide for the 2001 international meeting on
sea-level changes, coastal evolution and neotectonics (INQUA).
N. K. Huber. Amount and timing of Late Cenozoic uplift and tilt of the
Central Sierra Nevada, California - evidence from the upper San Joaquin
river basin. Geological Survey Professional Paper, 1197:1–28, 1981.
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. D. Jansen. Flood magnitude-frequency and lithologic control on bedrock
river incision in post-orogenic terrain. Geomorphology, 82:39–57, 2006.
M. S. W. Jefferson. The antecedent Colorado. Science, 6(138):293–295,
1897.
H. Johannesson and G. Parker. Linear theory of river meanders. In S. Ikeda
and G. Parker, editors, River meandering, Water Resources Monograph,
pages 181–214. American Geophysical Union, Washington, D.C., 1989.
S. Judson and G. W. Andrews. Pattern and form of some valleys in the
driftless area, Wisconsin. Journal of Geology, 63(4):328–340, 1955.
42
42
J. P. T. Kalkwijk and R. Booij. Adaptation of secondary flow in nearlyhorizontal flow. Journal of Hydraulic Research, 24(1):19–37, 1986.
E. Kirby, K. X. Whipple, W. Tang, and Z. Chen. Distribution of active rock
uplift along the eastern margin of the Tibetan Plateau: Inferences from
bedrock channel longitudinal profiles. Journal of Geophysical Research,
108(B4, 2217):16–1–16–24, 2003.
P. F. Kitanidis and J. F. Kennedy. Secondary current and river meander
formation. Journal Fluid Mechanics, 144:217–229, 1984.
J. S. Kobor and J. J. Roering. Using the stream power law and digital
topographic data to quantify lateral migration rates in bedrock channels,
Oregon Coast Range. In Cordilleran Section - 98th Annual Meeting. The
Geological Society of America, 2002.
S. T. Lancaster. A nonlinear river meandering model and its incorporation
in a landscape evolution model. PhD thesis, Massachusetts Institute of
Technology, Cambridge, MA, 1998.
S. T. Lancaster and R. L. Bras. A simple model of river meandering and
its comparison to natural channels. Hydrological Processes, 16:1–26, 2002.
doi:10.1002/hyp.273.
W. B. Langbein and L. B. Leopold. River meanders: Theory of minimum
variance. United States Geological Survey Professional Paper, (422-H),
1966. 15 p.
J. Lavé and J. P. Avouac. Fluvial incision and tectonic uplift across the
himalayas of central nepal. Journal of Geophysical Research, 106:26561–
26591, 2001.
L. B. Leopold and W. B. Langbein. River meanders. Scientific American,
214:60–70, 1966.
L. B. Leopold, M. G. Wolman, and J. P. Miller. Fluvial processes in geomorphology. W. H. Freeman, New York, 1964. 522 p.
W. R. Lowry. Dam politics: Restoring America’s rivers. Georgetown University Press, Washington, DC, 2003. ISBN:0878403906.
R. H. Mahard. The origin and significance of intrenched meanders. Journal
of Geomorphology, 5:32–44, 1942.
43
43
K. Masuch. Zur frave der talmaander. Berliner Geog. Arb., Heft 9, 1935.
T. S. McCarthy and S. Toth. Incised meanders along the mixed bedrockalluvial Orange River, Northern Cape Province, South Africa. Zeitschrift
Fur Geomorphologie, 48(3):273–292, 2004.
H. H. Mills and R. T. Mills. Evolution of undercut slopes on abandoned
incised meanders in the Eastern Highland Rim of Tennessee, usa. Geomorphology, 38:317–336, 2001.
R. C. Moore. Origin of inclosed meanders in the physiographic history of
the Colorado Plateau country. Journal of Geology, 34:29–57, 1926a.
R. C. Moore. Significance of enclosed meanders in the physiographic history
of the Colorado Plateau country. Journal of Geology, 34(2):97–130, 1926b.
G. C. Nanson and E. J. Hickin. Channel migration and incision on the
beatton river. jhe, 109:327–337, 1983.
G. Parker. On the cause and characteristic scales of meandering and braiding
in rivers. Journal of Fluid Mechanics, 76:457–480, 1976.
G. Parker and E. D. Andrews. On the time development of meander bends.
Journal of Fluid Mechanics, 162:139–156, 1986.
G. Parker, K. Sawai, and S. Ikeda. Bend theory of river meanders. part 2.
nonlinear deformation of finite amplitude bends. Journal of Fluid Mechanics, 115:303–314, 1982.
F. J. Pazzaglia and M. T. Brandon. A fluvial record of long-term steady-state
uplift and erosion across the Cascadia forearc high, Western Washington
state. American Journal of Science, 301:385–431, 2001.
F. J. Pazzaglia, T. H. Gardner, and D. J. Merritts. Bedrock fluvial incision
and longitudinal profile development over geologic time scales determined
by fluvial terraces. In E. E. Wohl and K. J. Tinkler, editors, Rivers Over
Rock: Fluvial Processes in Bedrock Channels, Geophysical Monograph Series 107, pages 207–235. American Geophysical Union, Washington, D.C.,
1998.
J. W. Powell. Exploration of the Colorado River of the West and its tributaries. Explored in 1869, 1870, 1871, and 1872 under the direction of
44
44
the secretary of the Smithsonian Institution. Government Printing Office,
Washington, 1875.
B. L. Rhoads and M. R. Welford. Initiation of river meandering. Progress
in Physical Geography, 15:127–156, 1991.
J. L. Rich. Certain types of stream valleys and their meaning. Journal of
Geology, 22:469–497, 1914.
R. D. Rogers, H. Karason, and R. D. van der Hilst. Epeirogenic uplift above
a detached slab in northern Central America. Geology, 30(11):1031–1034,
2002.
S. A. Schumm. Meander wavelength of alluvial rivers. Science, 157:1549–
1550, 1967.
S. A. Schumm and H. R. Khan. Experimental study of channel patterns.
Geological Society of America Bulletin, 83:1775–1770, 1972.
J. D. Sears. Relations of the Browns Park Formation and the Bishop Conglomerate, and their role in the origin of Green and Yampa Rivers. Bulletin of the Geological Society of America, 35:279–304, 1924.
M. A. Seidl and W. E. Dietrich. The problem of channel erosion into bedrock.
Catena Suppl., 23:101–124, 1992.
G. Seminara. Meanders. Journal of Fluid Mechanics, 554:271–297, 2006.
R. G. Shepherd. Incised river meanders: Evolution in simulated bedrock.
Science, 178:409–411, 1972.
R. G. Shepherd and K. A. Schumm. Experimental study of river incision.
Geological Society of America Bulletin, 85:257–268, 1974.
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, 2006.
L. Sklar and W. E. Dietrich. River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply. In E. E.
Wohl and K. J. Tinkler, editors, Rivers over Rock: Fluvial Processes in
Bedrock Channels, Geophysical Monograph Series, pages 237–260. American Geoph. Union, 1998.
45
45
C. P. Stark, J. R. Barbour, N. Hovius, H. Chen, C.-W. Lin, M.-J. Horng,
C. Huang, C.-H. Jen, J. K.-Q. Xu, J. M. Turowski, S. Dadson, C.-C. Wu,
and Y. Fukahata. Dependence of mountain river sinuosity on typhoon
flood magnitude and frequency. Nature, 2008. in prep.
A. N. Strahler. Elongate intrenched meanders of Conodoguinet Creek. American Journal of Science, 244:31–40, 1946.
W. A. Tarr. Intrenched and incised meanders of some streams on the northern slope of the Ozark Plateau in Missouri. Journal of Geology, 32:583–
600, 1924.
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.
K. J. Tinkler. The superimposition hypothesis for incised meanders – A
general rejection and specific test. Area, 4(2):86–91, 1972.
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.
C. R. Twidale. River patterns and their meaning. Earth Science Reviews,
67:159–218, 2004.
K. X. Whipple. Bedrock rivers and the geomorphology of active orogens.
Annual Review of Earth and Planetary Sciences, 32:151–185, 2004.
K. X. Whipple, G. S. Hancock, and R. S. Anderson. River incision into
bedrock: Mechanics and relative efficacy of plucking, abrasion, and cavitation. Geological Society of America Bulletin, 112:490–503, 2000.
G. P. Williams. Bank-full discharge of rivers. Water Resources Research, 14
(6):1141–1154, 1978.
G. P. Williams. River meanders and channel size. Journal of Hydrology, 88:
147–164, 1986.
46
46
A. Winslow. The Osage river and its meanders. Science, 22(546):32–32,
1893.
A. Winslow. Meandering rivers in Missouri. Science, 23(580):152–153, 1894.
C. Wobus, K. X. Wipple, E. Kirby, N. Snyder, J. Johnson, K. Spyropolon,
B. Crosby, and D. Sheehan. Tectonics from topography: Procedures,
promise, and pitfalls. In M. T. B. Sean D. Willett, Niels Hovius and
D. M. Fisher, editors, Tectonics, Climate, and Landscape Evolution, pages
55–74. Geological Society of America, special paper 398 edition, 2006a.
C. W. Wobus, G. E. Tucker, and R. S. Anderson. Self-formed bedrock
channels. Geophysical Research Letters, 33(L18408):1–6, 2006b.
R. J. Wright. Underfit meanders of the French Broad River, North Carolina.
Journal of Geomorphology, 5:183–190, 1942.
R. W. Young. The patterns of some meandering valleys in New South Wales.
Australian Geographer, 11(3):269–277, 1970.
S.-B. Yu and L.-C. Kuo. Present-day crustal motion along the Longitudinal
Valley Fault, eastern Taiwan. Tectonophysics, 333(1–2):199–217, 2001.
47
47
CHAPTER 2 2
CHAPTER
Typhoon-driven discharge variability and bedrock river meandering †
Typhoon-driven discharge variability
and bedrock river meandering†
†
This manuscript is in preparation for submission to the Journal of Geophysical Research with coauthors Colin P. Stark, Niels Hovius, Jens Turowski, Hongey Chen, Kaiqin Xu, Yukitoshi Fukahata,
Ming-Jame Horng, and Chingweei Lin.
48
Abstract.
The role of fluvial incision into bedrock and in the relationships and
feedbacks among landscape, climate, and tectonics have motivated geomorphology research for many years. Much of this work has considered channel
bed erosion and its responses to tectonically controlled slopes and climatically controlled discharges to be of primary importance, while considering
bank erosion to be of secondary or very little relevance. However, incising
rivers widen, migrate, and even meander through processes of horizontal
erosion that can outpace downcutting, that can have a strong effect on the
shape and evolution of mountain landscapes, and that respond to tectonic
and climatic forcing. Here we show that bedrock channel planforms, which
are the accumulated effect of bank erosion and its changes along stream
and through time, are strongly correlated to the variability of rainfall, the
relative flood intensity, and the frequency of extreme storm events in the
island mountains of the western North Pacific. These results indicate that a
quantitative signature of climatology is recorded in the planforms of incising
rivers.
49
49
1. Introduction
The assumption that bedrock streams maintain fixed planform geometries along which
all changes in slope result from the competing processes of vertical incision and rock
uplift implies that incising rivers do not migrate laterally, or if they do, it is of minor
consequence to the evolution of mountain landscapes. However, evidence for bank erosion
along bedrock rivers is ubiquitous, for example, in straths which form through channel
widening during pauses of thalweg lowering [e.g. Bucher , 1932; Smith, 1947; Merritts
et al., 1994; Pazzaglia and Brandon, 2001; Finnegan et al., 2005], and in incised meanders
erode laterally at alternating banks during incision, with rates that vary around each
bend. Active meandering along incising rivers leaves behind a wake of characteristic
landforms such as abandoned channel banks [e.g. Brakenridge, 1985], slip-off slopes [e.g.
Rich, 1914; Tarr , 1924; Moore, 1926], and cutoff meander loops [e.g. Mahard , 1942; Hovius
and Stark , 2001]. The loops of incised meanders can have many of the same features as
alluvial meanders, including compound or multibend shapes, neck cutoffs, and geometries
that skew with to up- or downstream bend migration.
A classic interpretation is that sinuosity along incising rivers is inherited from a prior
alluvial phase, locked in when incision rejuvenates during, for example, to regional uplift, sea level fall, or climate change [e.g. Powell , 1875; Davis, 1893; Sears, 1924; Bretz ,
1962]. This idea has led to inferences that use the present shape of incised meanders as
windows into tectonic and climatic pasts [e.g. Davis, 1893; Gardner , 1975; Huber , 1981;
Rogers et al., 2002]. However, field observations [e.g. Brakenridge, 1985; Hartshorn et al.,
2002], laboratory experiments [e.g. Shepherd, 1972; Shepherd and Schumm, 1974], and
50
50
physically-based theoretical results [eg Turowski et al., 2008; Stark , 2006; Stark et al.,
2007] prove the opposite, that lateral cutting can be active during incision. This indicates
that assumptions about fixed/inherited planforms and interpretations of the histories they
reveal are not only unfounded, but are also likely incorrect.
Some theories of meandering apply specifically to alluvial rivers, since they require
cohesive banks and fine-grained mobile bend material [e.g. Callander , 1969; Fredsoe, 1978].
Others focus instead on the influence of channel curvature on the interaction of the flow
with an erodible bank [e.g. Ikeda et al., 1981]. These meander bend theories do not consider
the composition of the channel boundaries; therefore, they apply equally to alluvial and
incising bedrock rivers. Positive feedbacks of bend growth which depend on planform
characteristics such as channel width, meander wavelength, and curvature [e.g. Begin,
1981; Parker et al., 1982; Furbish, 1988] should be active along incising channels, and
sinuosity is expected to grow through erosion at the outer bends and shrink through
neck and shoot cutoffs [e.g. Seminara, 2006]. However, there are a number of important
distinctions between incising and alluvial meanders; for example, bedrock rivers tend to
have larger bed sediment particles, less erodible channel boundaries, and greater alongchannel slopes. The concept of bankfull discharge does not apply to bedrock rivers which
are confined to narrow valleys with no floodplain, and this can influence the depth of
the extreme floods and therefore also the range of shear stresses driven by the discharge
distribution. The analog for alluvial bank collapse on an incising river is a landslide, and
since bank heights on incising rivers can extend from the channel thalweg all the way
to the ridge crest, landslides can produce significantly more sediment than an alluvial
bank collapses; this sediment can have a buffering effect on both the wall at the foot of
51
51
the slope failure and on the bed downstream. Incising meanders also differ from alluvial
ones in the composition of their point bars: on an alluvial meander, point bars are made
of sediment transported from upstream, but on incising rivers, they tend to have only
a thin veneer of coarse sediment over bedrock that is effectively supplied from below.
Both alluvial and incising rivers can reduce their sinuosity through cutoffs, but incising
rivers may also reduce sinuosity through phases of alluviation in which a temporary fill
of sediment supports a straighter, braided channel that traverses and eventually cuts
down across the elevated overlapping inter-meander spurs. Some of these differences
may contribute to negative feedbacks along incising rivers that work against the positive
feedbacks associated with curvature and bending; for example, as hillslopes lengthen with
incision, landsliding at the outer banks may produce more sediment causing greater wall
buffering and slowing bank erosion.
Changes in sinuosity are always accompanied by changes in longitudinal structure, since
during sinuosity growth, channels lengthen and slopes decrease. Although this can affect
slope-dependent measures such as bed shear stress and stream power, most recent inquiries
into the roles of bedrock erosion in landscape evolution tend not to consider bank erosion
as a mechanism for slope adjustment [e.g. Tinkler and Wohl , 1998; Whipple, 2004]. We
do not yet understand the roles of discharge, sediment load, channel slope, and bedrock
erodibility in controlling the pace or style of bank erosion, nor do we know what feedbacks
relate bank erosion to vertical incision. Answers to these questions will help to clarify the
range of ways that bank erosion affects catchment morphology.
52
52
2. Study area
An actively meandering river can, at any time, be straight along any of its reaches.
Measures of channel sinuosity at a given time along a single river should not necessarily
correlate to the parameters that control lateral cutting. However, the average rate of
lateral cutting along a river is likely to be a function of tectonics (slopes), climate (discharges), and lithology (erodibility), each of which acts regionally. Therefore, the effect of
these parameters on mountain river sinuosity should be understandable at a scale much
broader than a single reach. For landscapes evolving at a similar pace and over a similar
length of time, regions with tectonic, climatic, and/or lithologic conditions that promote
lateral cutting should tend to have a preponderance of sinuous mountain rivers, while
mountain regions with conditions that do not promote lateral cutting should tend to have
a network of incising rivers with straighter reaches.
For this reason, we have taken a regional approach, focusing our analysis on the islands of
Japan, Taiwan, the Philippines, Borneo, and New Guinea. These islands cross the western
North Pacific ocean, where climatology is dominated by the world’s most active tropical
cyclone basin. Storms are most frequent in the tropics (northern Philippines and Taiwan)
and decrease toward the mid-latitudes (along the axis of Japan) and toward the equator
(along the axis of the Philippines and southward to Borneo and New Guinea). Parts of
Taiwan, for example, which sits near the center of the cyclone basin, have been hit by an
average of nearly two typhoons per year over the past two decades, some of which drive
rain rates up to a meter per day [Wu and Kuo, 1999]; by comparison, Northern Honshu,
Japan, is affected by a typhoon once every two or three years at most. This variability
of tropical cyclone frequency across the cyclone basin provides a unique opportunity to
53
53
understand the role of storminess on river network geometry. Furthermore, our decision
to focus on tropical to extra-tropical active mountains avoids the complications of present
day and recent glaciation, since rapid evolution of these areas means that the fluvial and
hillslope processes are acting fast enough to have largely erased glacial and periglacial
features.
Differences in lithology and tectonics may confound the effect of typhoon climatology
on the landscape by promoting or restricting the rates of the erosion processes. We have
loosely accounted for both of these by limiting our analysis to a selection high relief areas within the accretionary subduction zone complexes of our selected islands along the
eastern margin of the Eurasian continent: the Japan Volcanic Arc and Accreted Terrain,
the Taiwan Thrust and Fold Belt, the Philippine Accretionary Prism, the Rajang-Crocker
Accretionary Prism in northeastern Borneo, and the New Guinea Foreland Mobile Belt
and Foreland Basin Fold Belt [names and areas as defined by Steinshouer et al., 1997].
Although each area is a different litho-stratigraphic terrane and is comprised of a different
distribution of lithologies and erodibilities, all have been forming by similar subduction
and accretion processes and therefore have similar characteristics; for example, these regions tend to consist mainly of accreted Cenozoic island arcs and arc fragments, ophiolites,
olistostrome mélange, turbidites, and Cretaceous volcanics [e.g. Brown et al., 1980; Benard et al., 1990; Faure and Natalin, 1992; Hutchison, 1992; Honza et al., 2000; Hall ,
2002; Belaguru and Nichols, 2004; Peucker-Ehrenbrink and Miller , 2004] and they have
similar age distributions [Steinshouer et al., 1997]. Our concentration on the high relief
parts of these regions further ensures that we have excluded anomalously soft substrates
and instead are considering only rivers flowing through bedrock.
54
54
It is possible that the same climatologically-controlled landscape features could result
from conditions that promote rapid evolution at work for a short time, or from conditions
that promote slow evolution at work for a long time. Since the same tropical cyclone
basin dominates the climatology across the entire study area, we can assume that its
effect has been felt by all of the landscapes for similar periods of time. Differences in
climatologically-controlled landscape features in this region are therefore likely to be the
result the strength, rather than duration, of the relevant climate parameters.
3. Data
3.1. Climatology data
The Joint Typhoon Warning Center provides an archive of approximately 60 years of
tropical storm data. We used a version of the dataset with 1676 storm tracks for events
that occurred in the western North Pacific cyclone basin between 1945 and 2002. Each
record includes the maximum wind speed and the coordinates of the storm’s center at
6-hour intervals. Recent storm records also list the storm type classification (tropical
depression, tropical storm, typhoon, or supertyphoon), the storm radius, and the eye
diameter. The Hotspots dataset includes global risk maps for several natural hazards,
including tropical cyclones, which are provided in raster form as the total number of
events that occurred over each pixel in the past 20 years, taking into account both the
track and the evolving storm radius.
Monthly rain gauge data is available from the NOAA Global Historical Climate Network
for thousands of stations world wide. We used data for 169 stations within or just outside
the mountains of the western North Pacific islands.
55
55
We acquired river discharge data from three sources. The Global Runoff Data Centre
(GRDC) provided data for rivers in the Philippines and Borneo, and for some rivers in
Japan. The Ministry of Land, Infrastructure and Transport of Japan provided data for
additional rivers in Japan. The Water Resources Agency (WRA) of Taiwan provided
data for rivers in Taiwan. Discharge is available from these sources as daily and monthly
means computed from continuous stage recorder measurements. The WRA also provided
a limited number of the hourly stage measurements from one gauge.
3.2. Optical remote sensing data
The University of Maryland’s Global Landcover Facility (http://glcf.umiacs.umd.edu/data)
is an archive of optical remote sensing imagery at a variety of processing levels and from
a variety of sensors. One product on offer is a nearly global collection of orthorectified
Landsat Thematic Mapper (TM) images. With 28.5 meter pixels, this imagery is able
to resolve the sinuosity of most mountain rivers, and is commonly sufficient to identify
channels that are actively meandering; abandoned loops tend to be optically distinct from
their surroundings because their flat fill of alluvium and colluvium makes them ideal sites
for villages and farms, and because these tend to be accentuated by the shadows cast by
the old valley walls (Figure 1).
GoogleEarth has provided a much more convenient platform for seamlessly surveying
surface characteristics in very high resolution DigitalGlobe imagery, making it a useful tool
for validation of Landsat observations. However, this dataset is a mosaic of fully but not
necessarily consistently processed images at different resolutions acquired on sometimes
unspecified dates, so it is not well-suited for unbiased regional comparisons.
56
56
1
2
km
Figure 1: Examples of active meanders visible in Landsat TM 28.5 m data displayed as RGB-351 color composites along
the Nakagawa River at 33.80 ◦ N, 134.45 ◦ E, Tokushima Prefecture, eastern Shikoku Island, Japan. Dotted circles mark
channel loops that have been cutoff and abandoned.
57
3.3. SRTM digital topography
The Shuttle Radar Topography Mission (SRTM) 3-arcsecond digital elevation model is
a snapshot of global topography from 56 ◦ S to 60 ◦ N acquired from a single instrument
during an 11-day mission in February of 2000. However, the data suffers from abundant
no-data cells associated with shadowing, layover, and backscatter problems, as well as
offshore cells with erroneous elevations due to radar returns from rough seas. Despite
these issues, some of which have already been addressed and corrected in recent releases
of the dataset, its consistency, resolution and extent make it ideal for a regional comparison
of landforms. We assembled SRTM Digital Elevation Models (DEMs) for the islands of
Japan, Taiwan, the Philippines, Borneo, and New Guinea. This included several steps:
projection of each DEM to minimize the distortion of distance; elimination of offshore data
values; interpolation across onshore voids; filling local topographic minima to generate
gridded topography on which all possible drainage paths end at the coastline.
4. Methods
4.1. Mapping tropical storm frequency and statistics of rainfall and discharge
We interpolated the 6-hour JTWC storm center locations to create tracks, and we projected the tracks onto a Lambert Azimuthal Equal Area projection centered over Taiwan
at 121 ◦ E and 23 ◦ N. Many storms in this area initiate over the tropical Pacific, move west
toward the Asian continent and then swing north toward Japan. Some travel more than
100 km into the continent and others die over the ocean before making landfall.
Because the storm type and storm diameter information is only available for the
youngest storms, we chose to convert each track in the archive to a common 200 km
wide swath gridded at 20 km representing the approximate region affected by an average
58
58
storm. We assigned a value of 1 inside the swath and 0 outside the swath. Summing
the set of 1676 swaths and dividing by the 58 years represented by the data yielded a
rasterized map showing the annual number of storm centers that pass within 200 km of
each pixel. Assuming that this average storm diameter applies to the entire dataset, the
resulting map pixels can be interpreted as measures of tropical cyclone strike frequency
(y−1 ).
We combined the JTWC storm tracks for the western North Pacific with a smoothed
version (200 km median filter) of the Hotspots gridded storm data for the western South
Pacific, which affects the south eastern tip of New Guinea. Since the Hotspots and JTWC
datasets represent different periods of time and were processed differently, we first applied
a minor correction to the entire Hotspots dataset that minimizes the difference between
the western North Pacific storm frequencies of the Hotspots and JTWC datasets.
We assessed the hydrologic effect of storms through a comparison to the gauge statistics.
For each rain and river gauge, we computed first and second order statistics including
the mean of rainfall Rμ and discharge Qμ , the standard deviations Rσ and Qσ , and the
coefficients of variation Rcv and Qcv , which are the standard deviations normalized by
the means. We computed a host of other statistics of the discharge data, including the
99th and 50th percentiles Q99 and Q50 , the skewness Qskew , mean runoff (Qμ normalized
by drainage area), runoff variability (Qσ normalized by drainage area). We also measured
the relative flood intensity as the 99th percentile discharge normalized by drainage area.
4.2. Mapping sinuosity
We surveyed GLCF Landsat TM imagery of all of the islands of the western North Pacific from New Guinea in the east to Myanmar in the west, and from Timor in the south
59
59
to Sakhalin Island in the north, as well as imagery on the continent in eastern China, the
Koreas, the Malay Peninsula and northern Australia. We evaluated the rivers in mountainous terrain, noting the abundance of sinuous rivers and evidence that the sinuosity is
actively evolving, such as cutoff meander loops, valley asymmetry, and meander skewness.
Our survey of Landsat images revealed that sinuous mountain rivers with evidence of
meander activity are most common on the islands of the western North Pacific cyclone
basin, such as Taiwan, Luzon, and Shikoku, and where bedrock lithology is especially
weak, such as the Loess Plateau of China and the Quaternary volcanics of North Korea.
We infer from these observations that storms enhance bank erosion while rock strength
limits it, which is consistent with the Montgomery [2004]’s observation of an inverse
relationship between lithologic strength and strath cutting, and with Hartshorn et al.
[2002]’s observations of bedrock channel bank erosion that occurred preferentially during
a supertyphoon flood.
To map these observations quantitatively from topography alone, we made two simplifying assumptions. The first is that rivers in high relief areas are bedrock channels
[Turowski et al., 2008]. The second is that the sinuosity of these rivers is a proxy measure for the rate of their meander development; faster meandering produces more sinuous
rivers. Although cutoffs complicate this assumption, they occur locally and at different
loops at different times, such that a rapidly meandering channel will tend to maintain
high sinuosity values along a series of loops.
With these assumptions, we developed a regionalized measure of mountain river sinuosity which is based on the ensemble of channel network segments for a specified area and
which can be obtained directly from the DEM. Its computation requires the extraction
60
60
of the channel network from digitized topography, which we accomplished with standard
GIS flow-routing techniques applied to the SRTM data for Japan, Taiwan, the Philippines, Borneo, and New Guinea: first we determined flow direction from each pixel, and
then accumulated flow downhill, and finally applied a flow-accumulation threshold to separate network and hillslope pixels. We used a threshold of 1000 pixels, or approximately 8
square kilometers, since a comparison with Landsat imagery indicated that SRTM-derived
channels with smaller support areas do not closely match the planform of the channels
they represent.
For each river network link between two tributary junctions, we computed the alongchannel length l, the straight line node-node down-valley length L, and the elevation
change dz. We used these measures to compute the along channel slope Sl =
straight line node-node down-valley slope SL =
dz
,
L
and the sinuosity s =
l
L
dz
,
l
the
of each
network segment.
We then restricted our analysis to steep terrain by eliminating all channel reaches flowing
through areas with less than 500 meters of elevation change within a 5 kilometer radius.
We further limited our study areas to the subduction zone complexes of Japan, Taiwan,
the Philippines, Borneo, and New Guinea as they are identified in vector GIS layers
of Steinshouer et al. [1997]. We also clipped away any channel reaches that crossed
interpolated SRTM data voids.
The sinuosity s and the total along-channel length of each network segment i that
remained after clipping l∗ (Figure 2b,c) contributed to the calculation of regionalized
61
61
145°E
45°N
130°E
s = 1.33
35°N
(b)
s = 1.10
L
(a)
l
s = 1.59
s = 1.16
0 1.25 2.5
L
5 Km
l*
l
(c)
0 1.25 2.5
5 Km
10°N
(d)
0°
5°N
(e)
15°N
Ȥ Ȥ Ȥ 110°E
120°E
1
Regionalized sinuosity Ȥ [-]
and Link sinuosity s [-]
1.5
Figure 2: GIS methodology for mapping sinuosity. a) Japan. b) Close-up view from Japan as indicated by the dotted
white box. c) Close-up view from the Philippines as indicated by the dotted black box. d) Borneo. e) The Philippines.
Tiles (a), (d), and (e) show the channel network colored by link sinuosity s = l/L over the gridded regionalized sinuosity
χ. Regionalized sinuosity is gridded at 20 km using a contributing area 100 km in diameter (shown as black circles around
each central χ pixel). The regionalized sinuosity χ for each central pixel (highlighted with a black square) is noted in the
lower corner of each tile. Tiles (b) and (c) show the lengths L, l, and l∗ used for the calculation of link sinuosity s and
regionalized sinuosity χ. Note that for each tile, only the portion of the channel network that contributes to the center
pixel χ calculation are colored; the rest of the network, shown in thin black lines, are excluded because they are either
outside the 100 km contributing area, or because they flow across areas of low relief, weak substrate, and/or SRTM data
voids.
62
1
TC = 1.05
F = 1.27
sP = 1.21
sV = 0.14
sCV= 0.12
s = 1.17
s = 1.74
35°N
45°N
10
p( s)
130°E
140°E
0
10
TC = 0
F = 1.22
sP = 1.17
sV = 0.12
sCV= 0.11
s = 1.15
s = 1.63
D = 2.67
E = 0.080
(a)
í
1
1.1
1.2
1.3
1.4
1.5
Link sinuosity s
1.6
1
1
10
0°
10°N
5°N
10
15°N
10
TC = 1.53
F = 1.44
sP = 1.32
sV = 0.24
sCV= 0.18
s = 1.26
s = 2.22
110°E
p( s)
p( s)
120°E
0
10
0
10
D = 2.64
E = 0.069
D = 2.42
E = 0.135
(b)
í
10
1
1.1
1.2
1.3
1.4
1.5
Link sinuosity s
1.6
(c)
í
10
1
1.1
1.2
1.3
1.4
1.5
Link sinuosity s
1.6
Figure 3: Probability distributions of link sinuosity s values for the three sample χ pixels shown in Figure 2. A parametric
Γ PDF is fit to each s distribution, with shape parameter α and scale parameter β values noted. Additional statistics of
each contributing river network are also listed.
63
2.6
99th percentile sinuosity s99>í@
Mean sinuosity sM>í@
(a)
1.3
1.25
1.2
KW r2 = 0.92
1.15
2.2
2
1.8
1.6
Adjusted mean sinuosity C*>í@
Sinuosity coefficient of variation scv>í@
1.2
1.3
1.4
Regionalized sinuosity C>í@
(c)
0.2
0.15
0.1
KW r2 = 0.65
1.35
(d)
1.3
1.25
1.2
1.15
KW r2 = 0.93
0.05
1.2
1.3
1.4
Regionalized sinuosity C>í@
0.18
0.16
1.2
1.3
1.4
Regionalized sinuosity C>í@
(e)
6
0.14
A>í@
0.1
0.08
0.06
0.04
(f )
5
0.12
B>í@
KW r2 = 0.65
1.4
1.2
1.3
1.4
Regionalized sinuosity C>í@
0.25
(b)
2.4
4
KW r2 = 0.17
3
KW r2 = 0.69
2
0.02
1.2
1.3
1.4
Regionalized sinuosity C>í@
1.2
1.3
1.4
Regionalized sinuosity C>í@
Figure 4: Comparison of each measurement of χ to the statistics of the contributing network link sinuosities s, including
the mean (a), the 99th percentile, the coefficient of variation (c), χ∗ (Equation 6) (d), and the shape and scale parameters
α (e) and β (f) of the parametric Γ distributions (Equations 2 through 5). Scatter points are colored and shaped by region:
Japan is shown as blue circles, Taiwan as purple triangles, the Philippines as green squares, Borneo as brown diamonds,
and New Guinea as cyan triangles. Kendall’s Kτ and Pearson’s r2 are also noted.
64
65
sinuosity χ,
(si × li∗ )
χ=
i
, i ∈ φ100 ,
li∗
(1)
i
which we computed across the islands of the western North Pacific using a 100 km diameter
moving window and gridding at 20 km spacing. Values of χ range from 1, where most
mountain rivers are straight, to 1.5, where many mountain rivers are sinuous. Three
example χ cells are shown in Figure 2a, d, and e, along with circles showing the extent of
the moving window and the contributing network segments. The value of χ is a function
of the distribution of sinuosities s of the contributing N network links, shown for the three
sample χ values in Figure 3 along with statistics of the ensemble of sinuosities. The p(s)
distribution for each χ window is fit relatively well by a Γ distribution,
p(s) = f (s|α, β) =
1
β α Γ(α)
s
sα−1 e β .
(2)
s, α, β > 0. The Γ shape parameter, α, is given by
α=
3−a+
(a − 3)2 + 24a)
,
12a
(3)
where
a = ln(
N
N
1 1 si ) −
ln(si ).
N i=1
N i=1
(4)
The Γ scale parameter, β, is the ratio of the mean sinuosity sμ and α,
β=
sμ
.
α
(5)
Across the western North Pacific, χ increases with Γ scale parameter β (Kendall’s rank
correlation Kτ = 0.64), reflecting the dependence of both χ and β on the weight of the tail
distribution. However, χ is also strongly correlated with several other statistics of the link
sinuosity distribution, including the mean sμ , the 99th percentile s99 , and the coefficient
65
66
66
of variation scv . The best correlation (Kendall’s Kτ = 0.86) is obtained by applying a
small correction to sμ , (Figure 4),
χ∗ = sμ + scv sσ ,
(6)
where the adjusted mean sinuosity χ∗ is approximately the same as χ for a given contributing area. In principle, any of these statistics is interchangeable with χ as a measure
of regionalized sinuosity. However, we tend to prefer χ since it accounts for the relative
lengths of the individual segments.
We also replaced sinuosity s in Equation 1 with channel slope Sl and with down-valley
slope SL to compute analogous regionalized measurements of slope.
5. Results
Because Taiwan and Luzon are near the center of the western North Pacific cyclone
basin, these islands tend to be hit more frequently by typhoons than the northern islands
of Japan or the southern islands of the Philippines. Over the time period covered by the
JTWC storm track archive, parts of Taiwan and Luzon were consistently hit by at least
one or two storms every year while a few years passed between storms in parts of Japan.
Much of Borneo and New Guinea is too close to the equator to ever see a tropical cyclone.
South of the equator, storms also pass through western South Pacific and occasionally
produce heavy rain in southeastern New Guinea. However, the southern cyclone basins
are much less active than the western North Pacific.
Sampling the tropical cyclone strike frequency map at the discharge and rain gage
station locations reveals a strong correlation between strike frequency and statistics such
as skewness, coefficient of variation, and relative flood intensity (99th percentile discharge
66
64
64
normalized by drainage area), but not between storm frequency and the means or standard
deviations (Figure 5, Table 1). These are statistics that express the relative peak intensity
of rainfall and discharge; their correlation with tropical cyclone frequency is therefore
not surprising. For example, frequent storms in Taiwan produce both high means and
high peaks of rainfall and discharge, while everwet but typhoon-free conditions in Borneo
produce similarly high means with much lower peaks.
Our qualitative Landsat survey and quantitative pattern of sinuosity χ indicate that
mountain rivers are most sinuous in Taiwan and Luzon. Sinuosity decreases to the north
and to the south, roughly matching the pattern of storm frequency (Figures 6 and 7).
Kendall’s Kτ for tropical cyclone frequency and regionalized sinuosity χ for areas of the
western North Pacific with at least one storm on record is Kτ = 0.5009, and Pearson’s r2
correlation coefficient reveals that more than 45% of the variance in regionalized sinuosity
can be explained by the frequency of tropical cyclones (Figure 8).
Sampling the regionalized sinuosity grid at the discharge and rain gauge station locations
also reveals strong correlations with statistics such as Rcv and Q99 /Area which reflect the
storm frequency (Figure 9). Also as expected, the mean Qμ and standard deviation Qσ of
discharge and rainfall do not correlate as well with regionalized sinuosity. However, the
other measures, such as runoff variability and the ratio of the 99th and 50th percentiles of
discharge, Qskew and the 99th percentile discharge normalized by area do correlate well with
sinuosity, each accounting for approximately 40% of the variability in sinuosity (Table 1).
Assuming a constant down-valley slope SL between network nodes, sinuosity growth
along a channel reach reduces the along-channel slope Sl . However, there tends to be no
relationship between the slope and sinuosity nearby channel segments, such as an ensemble
67
Table 1: Kendall’s Kτ and Pearson’s r correlation coefficients
independent
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
TC [y −1 ]
Qcv [-]
TC [y −1 ]
Rμ [mm mo−1 ]
Rσ [mm mo−1 ]
Rcv [-]
Qμ [m3 s−1 ]
Qμ /A [m s−1 ]
Qσ [m3 s−1 ]
Qσ /A [m s−1 ]
Qcv [-]
Qskew [-]
Q99 [m3 s−1 ]
Q99 /Q50 [-]
Q99 /A [m s−1 ]
(Q299 /g)1/5 [m]
(Q299 /g)1/5 /A1/2 [-]
(Q299 /g)2/5 /A [-]
dependent
Rμ [mm mo−1 ]
Rσ [mm mo−1 ]
Rcv [-]
Qμ [m3 s−1 ]
Qμ /A [m s−1 ]
Qσ [m3 s−1 ]
Qσ /A [m s−1 ]
Qcv [-]
Q99 [m3 s−1 ]
Q99 /A [m s−1 ]
Qskew [-]
Q99 /Q50 [-]
(Q299 /g)1/5 [m]
(Q299 /g)2/5 /A [-]
Qskew [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
χ [-]
Kτ
-0.103
0.206
0.510
-0.095
0.155
0.201
0.523
0.557
0.169
0.503
0.615
0.503
0.169
0.480
0.718
0.5009
-0.169
0.144
0.444
-0.038
0.206
0.238
0.508
0.442
0.496
0.232
0.411
0.477
0.232
0.440
0.440
68
PKτ
8.28E-02
5.47E-04
1.04E-17
4.13E-01
1.78E-01
8.02E-02
4.89E-06
1.11E-06
1.41E-01
1.12E-05
7.66E-08
1.12E-05
1.41E-01
2.77E-05
9.06E-13
6.4E-83
1.20E-03
5.65E-03
1.49E-17
7.45E-01
7.00E-02
3.63E-02
2.75E-06
5.71E-05
4.86E-06
4.11E-02
2.01E-04
1.26E-05
4.11E-02
6.43E-05
6.43E-05
r
-0.025
0.563
0.738
-0.389
0.312
0.105
0.671
0.652
0.080
0.643
0.745
0.509
0.145
0.610
0.915
0.6731
-0.123
0.412
0.629
-0.114
0.292
0.265
0.676
0.612
0.633
0.248
0.516
0.649
0.309
0.619
0.635
Pr
7.50E-01
1.52E-15
2.63E-30
1.57E-02
5.64E-02
5.30E-01
4.51E-01
9.04E-06
6.32E-01
1.34E-05
8.36E-08
1.11E-03
3.85E-01
4.80E-05
9.61E-16
7.3E-90
1.11E-01
2.51E-08
5.30E-20
4.95E-01
7.58E-02
1.07E-01
3.25E-06
4.52E-05
2.03E-05
1.34E-01
9.01E-04
1.07E-05
5.91E-02
3.44E-05
1.88E-05
r2
0.001
0.317
0.544
0.152
0.097
0.011
0.451
0.426
0.006
0.414
0.554
0.259
0.021
0.372
0.837
0.453
0.015
0.170
0.396
0.013
0.085
0.070
0.456
0.374
0.400
0.061
0.267
0.421
0.095
0.383
0.403
í
í
x 10
400
8
300
200
Q99/A [m sí]
10
QM/A [m sí]
RM [mm moí]
x 10
8
6
4
100
6
4
2
2
0
0.5
1
0
1.5
0.5
1
0
1.5
0.5
1
1.5
í
í
x Cyclones
10
per year [yrí]
Cyclones per year [yrí]
10
xCyclones
10
per year [yrí]
300
200
100
0
0.5
1
15
(Q299/g)2/5/A >í@
QS/A [m sí]
RS [mm moí]
400
10
5
6
4
2
0
1.5
0.5
1
0
1.5
0.5
1
1.5
í
15 Cyclones per year [yr ]
Cyclones per year [yrí]
í
1.5 Cyclones per year [yr ]
8
3
1
Qskew >í@
Qcv >í@
Rcv >í@
2.5
2
1.5
10
5
1
0.5
0.5
0
0.5
1
1.5
TCs perper
year
[y-1][yrí]
Cyclones
year
0
0
0.5
1
1.5
í
Cyclones
TCs perper
yearyear
[y-1] [yr ]
0
0.5
1
1.5
TCs per
year
[y-1[yr
] í]
Cyclones
per
year
Figure 5: Tropical cyclone strikes v. rain R and river Q gauge station statistics. Scatter points are colored and shaped by
region consistent with Figure 4. There is no relationship between strike frequency and the means and standard deviations
of rainfall and discharge, but there is a moderately strong positive correltation between storm frequency and the coefficients
of variation of rainfall and discharge, and between storm frequency and the ratio of the 99 th and 50th discharge percentiles.
69
110°E
120°E
130°E
140°E
150°E
(
!
(
!
40°N
E
0
(
!
River gauges
Rain gauges
5
0.2
TC frequency [y-1]
E
Japan
2
160°E
40°N
100°E
(
!
(
!
E
E
!
(
(
!
(
!
(
!
(
!
!
(
(
!
(
!
(!
!
(
!
(
E
E !(
30°N
(
!
(
!!
( (
!
E
(
!
(
!
(
!
(
!
(
!
25
1. 5
1.
Taiwan
20°N
E!(
E!( E
(
!
(
!
E
E !(!(E
E!(
E
(
!
30°N
E
1 Regionalized Sinuosity Ȥ [-] 1.5
0.5
Philippines
E
E !(
E
E
EE!(
E
E
E E
E
10°N
1.7
2
5
10°N
20°N
E
E
1.5
1
0.75
E
E!(
0.5
E
EE
E
EE
E
E E
E
0°
0
(
!
0°
(
!
Borneo
New Guinea
E
10°S
0
EE
E EE E
EE
EEEE
EEEEE
E EE
E
EE
EEE
EE
EE
E
EEE
E
EE EE
EE EEE
E
E
E
EE
E
E
E EE E E
E
E
EE
E
E
EE EEE
E
E
E
E EE
E
E
E
E
E
10°S
EE
0.25
0.25
0.5
110°E
120°E
130°E
140°E
150°E
Figure 6: Regionalized sinuosity of the island mountains of the western North Pacific cyclone basin displayed in a Lambert
Azimuthal equal area projection centered over Taiwan, at 23 ◦ N, 121 ◦ E. Rain- and river-gauge station locations are marked
by crosses and circles. Blue contours show the 58-year tropical cyclone (TC) frequency estimated from the JTWC best
track dataset north of the equator. TC frequency is the average number of TC storm eyes that passed within 100 km each
year between 1945 and 2002.
70
Figure 7: Variation of regionalized sinuosity χ (top) and tropical cyclone strike frequency (bottom) with latitute for the
islands of the western North Pacific. Mapped values are grouped per degree of latitude and plotted as the mean (black
line) with one-σ error bounds (gray lines).
71
Regionalized sinuosity χ [−]
1.45
1.4
1.35
1.3
1.25
1.2
1.15
0
0.5
1
1.5
2
Tropical cyclone strike frequency [yr−1]
Figure 8: Tropical cyclone strike frequency v. regionalized sinuosity χ for the islands of the western North Pacific.
Kendall’s Kτ for all areas within the cyclone basin (at least one storm made landfall during the 58-year period of record),
is 0.5009 (PKτ = 6.4e-83), and Pearson’s r is 0.6731 (Pr = 7.2e-90).
72
1.3
1.25
1.2
200
300
1.25
1.2
400
2
RM [mm moí@
Sinuosity C>í@
Sinuosity C>í@
1.25
1.2
100
200
300
1.2
5
10
1
Rcv >í@
1.3
1.25
1.2
15
2
í
Sinuosity C>í@
0.5
1.5
4
6
8
Q99/A >PV@ x 10í
1.35
QS/A [m sí@ x 10
4
6
8
10
(( Q299/g)2/5)/A >í@
x 10
í
1.35
1.35
1.2
1.2
2
1.25
400
1.35
1.25
1.25
10
1.3
RS [mm mo @
1.3
1.3
í
í
Sinuosity C>í@
8
1.35
1.3
1.15
6
QM/A [m sí@ x 10
1.35
1.15
4
Regionalized Sinuosity C>í@
100
1.3
Sinuosity C>í@
1.15
1.35
Sinuosity C>í@
Sinuosity C>í@
Sinuosity C>í@
1.35
1.35
1.3
1.25
1.2
1
2
Qcv >í@
3
1.3
1.25
1.2
0
5
10
Qskew >í@
Figure 9: Scatterplots of regionalized sinuosity χ against statistics of rainfall and discharge.
73
15
that lies within a 100 km χ viewing window (Figure 10a, b, and c), and across the western
North Pacific, our measure of regionalized slope has only a very weak inverse relationship
with regionalized sinuosity (Figure 10d, e, and f).
6. Discussion
6.1. Discharge statistics along dammed rivers
Damming upstream from a river gauge affects the distribution of discharges downstream
to a degree that depends on factors such as drainage area at the dam location, reservoir
capacity, and dam usage (e.g. hydropower, flood mitigation, irrigation, recreation). The
extent and usage of damming varies considerably across our study areas and is difficult to
quantify since time series data on reservoir inflow and outflow are not readily available. It
is a particular problem in Japan, where dams regulate flow through thousands of rivers.
We argue that our use of daily, rather than hourly statistics of discharge accounts for
much of the effect that dams have on river flow characteristics, such that the statistics we
report here are not significantly different than they would be with no damming in any of
our selected catchments. This is because the strongest effect of a dam on discharge is likely
to occur during the extreme floods. For our study area, these flows are driven by storms
that are hours to days in duration. The effect of dams on these extreme discharges is likely
to filter the peak inflow and release it at a reduced discharge over a longer period of time;
we hypothesize that this is analogous to averaging instantaneous hourly measurements of
discharge over 24 hour periods.
The Ministry of Land, Infrastructure and Transport of Japan publishes some annual
statistics for many of its dams, including the yearly maximum instantaneous inflow and
outflow discharges; the ratio of these should correspond to the maximum effect of dams
74
74
on downstream discharges. As a test we have compared the yearly inflow and outflow
statistics from the Midori River dam in Kyushu, Japan (359 km2 ) to hourly and daily
discharge data from undammed Zhuokou River in southwestern Taiwan (375 km2 ) (Figure 11). For these similarly sized catchments with similar tropical storm frequencies, the
effect of the Midori River dam on instantaneous discharge is similar to the effect of taking daily averages on the Zhuokou. Therefore, we infer that the daily mean discharge
timeseries that we have used in this study are likely to be largely unaffected by upstream
damming. However, we do so cautiously, since without inflow and outflow timeseries for
all of the reservoirs within the watershed of each of our selected river gauges, we cannot
know for sure how different our statistics would be under entirely natural flow conditions.
6.2. Flood discharge and channel mobility
Because it is a measure of an ensemble of channel segments, regionalized sinuosity χ
is essentially an indication of the likelihood that any given incising channel in that area
meanders. This likelihood increases with storm frequency through the effect of storms,
particularly tropical cyclones in our study area, on peak rainfall and discharge intensity.
If the relationship between tropical storm driven floods and sinuosity is causal, then there
must be enhanced lateral cutting within the channels during peak discharges.
Hartshorn et al. [2002] found along the Liwu River in northeastern Taiwan that horizontal erosion was preferentially accomplished during a peak discharge event. Turowski et al.
[2008] explained that in perfectly parabolic channels, across-channel shear stress profiles
are depth-dependent; in shallow flows the maximum shear stress is found at the center of
the channel, but as flows deepen, two shear stress maxima diverge from the center towards
the banks. However, they found that the complicated geometries and width-depth ratios
75
75
5HJLRQDOL]HGFKDQQHOVORSH>í@
45°N
(a)
35°N
Channel slope Sl,i>í@
0.2
0.15
0.1
130°E
140°E
0.05
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1.2
1.4
1.6
Sinuosity si>í@
1.8
1.2
1.3
1.4
Regionalized sinuosity C>í@
1.45
0.2
0°
(b)
Regionalized sinuosity C>í@
5°N
1
Channel slope Sl,i>í@
(d)
0.08
0.15
110°E
0.1
0.05
(e)
1.4
1.35
1.3
1.25
1.2
1.15
0
1
1.2
1.4
1.6
Sinuosity si>í@
1.8
20
40
60
80
100
,QYHUVHUHJLRQDOL]HGFKDQQHOVORSH>í@
5HJLRQDOL]HGFKDQQHOVORSH>í@
(c)
10°N
Channel slope Sl,i>í@
0.25
15°N
0.3
0.2
0.15
120°E
0.1
0.05
0.08
(f )
1:1
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
1.5
2
Sinuosity si>í@
0.02
0.04
0.06
0.08
0.1
5HJLRQDOL]HGGRZQíYDOOH\VORSH>í@
2.5
Figure 10: a, b and c) Slope Sl versus sinuosity s for channel segments within the 100 km diameter χ viewing windows
shown in Figure 2. (d): Regionalized sinuosity χ versus regionalized channel slope across the western North Pacific
islands. e) Inverse regionalized channel slope versus regionalized sinuosity χ across the western North Pacific islands. f)
Regionalized down-valley slope versus regionalized channel slope across the western North Pacific islands. The highlighted
circle (Japan), triangle (Borneo), and square (Philippines) in (d) and (f) represent the sample locations shown in (a), (b)
and (c). Scatter colors and shapes in (d), (e) and (f) are consistent with Figure 4.
76
3 -1
Annual max outflow Q (Midori) [m s ]
Daily mean Q (Zhuokou) [m3 s-1]
3500
Zhuokou River
3000
Midori River Dam
y=0.8406x
2500
y=0.7912x
1:1
2000
1500
1000
500
0
1:01
0
1000
2000
3000
3 -1
Max Hourly Q (Zhuokou) [m s ]
3 -1
Annual max inflow Q (Midori) [m s ]
Figure 11: A comparison to test the effect of damming on daily discharge statistics showing yearly maxima of instantaneous inflow versus outflow discharge at the Midori River Dam in Japan (black) compared to the daily maximum versus
mean of hourly discharge measured on the undammed Zhuokou River in Taiwan (grey). These rivers have similar drainage
areas and similar mean annual discharge. The close correspondence of points suggests that dams affect natural flow variability in a way that is similar to or less than the smoothing effect of taking daily averages of hourly instantaneous flow
measurements.
77
of real rivers like the Liwu make it exceedingly unlikely for shear stress distributions to
mimic the model behavior. They inferred that flow depth variation alone is insufficient
to explain trends in lateral cutting.
Stark et al. [2007], Stark and Barbour [2007a], and Stark and Barbour [2007b], however,
examined the depth-dependence of horizontal erosion rates with a dynamic channel model
designed to handle flow variability Qcv by considering two discharge values that are each
active for some percentage of time. They found that high stage widening rates increase
with Qcv , but mid-stage widening rates do not, implying that a positive relationship
between Qcv and horizontal cutting must largely be an effect of the high-stage flows.
They also found that channel base width, mean flow width, and mean flow width-depth
ratio, and an inverse meander timescale 1/T that derives from meander bend theory Ikeda
et al. [1981]; Johannesson and Parker [1989]; Edwards and Smith [2002] are all increasing
functions of Qcv , indicating that lateral cutting should increase with discharge variability.
Though the inverse timescale or meander rate 1/T may always be positively correlated
with discharge variability, the relationship with sinuosity may always be changing, since
sinuosity along a reach is reset at a cutoff. Other negative feedbacks associated with slope
reduction with sinuosity growth and sediment buffering with increasing valley relief may
reduce the meander rate to slow the effect of relative flood intensity on lateral cutting
Stark et al. [2008].
6.3. Sediment and channel mobility
Stark et al. [2007]; Stark and Barbour [2007a, b]’s results provide a sufficient explanation for how relative flood intensity alone can drive lateral cutting, but Turowski et al.
[2008] proposed that the sediment at the channel bed is more important. Erosion along
78
78
a river will increase with sediment only until there is enough sediment to protect the
channel bottom from further erosion; the implication for a meandering river is that banks
will remain exposed to the moving water and erosive impacts of the suspended load even
while the bed is covered, since channel banks are too steep to support a sediment cover
[Moore, 1926; Turowski, 2006; Turowski et al., 2008]. Sediment load tends to increase
with discharge in a typhoon-dominated catchment since the storms that drive the deepest
flows also trigger landslides by boosting pore fluid pressures and reducing shear resistance
in the valley walls, and by oversteepening the valley walls through channel erosion. The
most erosive flows should therefore occur while the bed is covered, leaving the banks to
preferentially feel the effect of these floods; horizontal erosion should be fastest and horizontal to vertical erosion rate ratios should be greatest along rivers that most frequently
channel extreme discharges.
Storm-triggered landslides are most likely to occur at the outer bends of a meandering
channel where flow impinges on the bank and where slopes tend to be steepest. Assuming
that subsequent flows will erode and/or move the sediment that is produced, these landslides effectively pull the channel centerline outward toward the failed slope, increasing
sinuosity. However, if subsequent flows are unable to transport the new sediment, then
landslides work against sinuosity growth by providing a buffer of sediment to protect the
channel bank from further erosion.
6.4. Time scales of channel evolution and climate change
Sinuosity and typhoon climatology can both evolve over a range of timescales. Correlations between measures of storminess and measures of channel planform only suggest a
causal relationship if these timescales match.
79
79
We can estimate the timescale of sinuosity development. For example, the Hsiukuluan
River drains the Central Mountains and flows north through Taiwan’s Longitudinal Valley
before making a sharp turn to cross the highly erodible flysch and turbidite sediments and
andesitic volcanic rocks of the Coastal Range [e.g. Yu and Kuo, 2001; Shyu et al., 2006].
The valley’s position across the range appears to be antecedent to the uplift of the coastal
mountains but the river is extremely sinuous and along its bends are several preserved
slip-off slopes and cutoff meander loops that indicate the evolution of channel’s shape
throughout orogenesis.
Concerned primarily with the Coastal Range uplift, Shyu et al. [2006] obtained 14C ages
across several sip-off slopes of the Hsiukuluan River and measured Holocene downcutting
rates ranging from 11.2 to 27.3 mm y−1 . The geometry of the asymmetric valley indicates
a ratio of horizontal to vertical cutting of approximately 10 to 1, from which we can
infer that the km-scale loops of the Hsiukuluan River have taken on the order of 10,000
years to grow. At a more typical rate of incision, such as 2 mm y−1 , and a slower ratio
of horizontal to vertical erosion of 1, kilometer-scale loops would take 500,000 to form.
In 500,000 years, climate can go through several cycles of warming and cooling, and the
frequency and intensity of tropical storms may vary dramatically.
We have little information about how storm tracks, intensities, and frequencies change
during periods of warming and cooling, or about how these changes affect the tropical
cyclone pattern across the western North Pacific; for example, we do not know if entire
cyclone basin contracts and expands, or if it migrates north and south. Nor do we know
how changes to the cyclone basin would affect river discharge distributions. However, the
correlation between a quantitative measurement of landforms and modern day climatology
80
80
is remarkable, and it may offer new clues, since it implies that storm paths and intensities
have not changed much over the past hundreds of thousands of years, or if they have, then
the long-term average pattern may be similar to what we observe today.
It may even be possible to extract information about changes in storm frequency from
the landscape. For example, the shapes of slip-off slopes along an individual channel reflect
the relative rates of horizontal and vertical erosion. As long as the slip-off slope remains
gravitationally stable, it will record the history of these relative rates. Therefore, a convex
slip-off slope could result from horizontal erosion that has decreased at a constant rate of
bed cutting, and could therefore signify a reduction in storm frequency. Such inferences
about climatology must be made with caution and with additional information about
incision rates and lithology, since the same morphology could also result from incision
that has increased at a constant rate of bank cutting, or from differences in erodibility
through the stratigraphic section.
6.5. Lithologic control of sinuosity development
Jefferson [1897] argued that the preservation of sinuosity during incision requires a
strong bedrock. In weaker bedrock, sinuosity will tend to be erased by the erosion of
the overlapping inter-meander spurs; that is, lateral erosion is more efficient in weaker
bedrock, exactly what we observed in our qualitative survey of optical remote sensing
imagery.
The effect of lithology on sinuosity development is displayed clearly in Shikoku, Japan,
where the Median Tectonic Line (MTL) is a right-lateral strike slip fault that accommodates some of the oblique convergence between the Eurasian and Philippine Sea Plates
at the Nankai Trough [Tabei et al., 2002]. South of the MTL are recently exhumed weak
81
81
trench sediments which we excluded from our regional analysis. To the north are older,
harder rocks of a metamorphic belt [Steinshouer et al., 1997; Scholz , 2002], which we
included in our analysis. The MTL also marks a distinct change in river morphology.
Low-sinuosity rivers incise into the stronger metamorphic belt, and high-sinuosity rivers
cut into the weaker trench sediments (Figure 12). χ values are correspondingly low to
the north and high to the south of the MTL (Figure 12). Because there is no significant
difference in the storm frequency across the fault, we assume that the sharp change in
sinuosity, which likely reflects a difference in lateral cutting rates, is the result of the
difference in erodibility across the fault.
The effect of lithology on sinuosity development implies that erodibility may be mappable directly from river network morphology, but only in areas with comparable climates.
At a spatial scale over which climatology is relatively constant, regionalized sinuosity χ
values should be correlated to lithologic weakness, as they are in the Shikoku example.
While it is extremely unlikely that lithology alone could cause the variation in sinuosity
that we see across the western North Pacific islands (there is no evidence of a regional
variation in erodibility that peaks in Taiwan and Luzon and decreases to the north and
south), lithologic variation does occur locally everywhere; if our inference is correct that
χ is an increasing function of storm frequency, then a variation of lateral cutting with
lithology should be responsible for much of the scatter we see in Figure 8.
7. Conclusions
Quantification of the regional abundance of incised meanders is possible through standard flow-routing techniques applied to digital topography along with an assessment of
the sinuosity of channel network links within some specified area; the mean, the coeffi-
82
82
33°30'N
132°45'E
133°E
MB
L
33°15'N
MT
TS
1 Regionalized Sinuosity F[-] 1.5
0
10
20km
Figure 12: Regionalized sinuosity across the Median Tectonic Line (MTL) in Shikoku, Japan. To highlight the local effect
of lithology on sinuosity, here χ was computed separately for the areas north and south of the MTL. The mildly sinuous
rivers north of the MTL cut into stronger Paleozoic/Mesozoic, Jurassic, and Silurian/Devonian rocks of a metamorphic
belt (MB) which were included in the analyses above. The tightly meandering rivers south of the MTL cut into weak
Cretaceous accretionary wedge trench sediments (TS) which were excluded from the analyses in an attempt to minimize
lithologic variability across the study areas. Gray indicates low relief areas also excluded.
83
cient of variation, and the shape parameter of a Γ function fit to the distribution of link
sinuosities all could serve as a quantitative measure of regionalized sinuosity. However, an
equation such as Equation 1 is preferable since it weighs the contribution of each network
segment by its length.
A pattern in the regionalized sinuosity of mountain rivers across the islands of the
western North Pacific matches the patterns of tropical cyclones, rainfall variability, and
relative flood intensity: all peak in Luzon and Taiwan, and decrease to the north along the
axis of Japan, and to the south along the Philippines and into Borneo and New Guinea.
This relationship is reconcilable with model and empirical results that find faster horizontal erosion during extreme floods. However, sinuosity cannot grow indefinitely, because eventually adjacent bends intersect and reaches straighten. If lateral erosion rates
increase as a function of relative flood intensity and channel curvature, sinuosity will grow
increasingly fast until there is a cutoff, when sinuosity and lateral erosion rates will step
backwards, before beginning to increase again. In the simplest model, given enough time,
all rivers should eventually meander with periodic loop cutoffs. This should contribute to
relatively stable, high values of sinuosity everywhere.
This implies that the relationship we find between sinuosity and storminess across the
western North Pacific may indicate a state of transience; that is, the rivers of Borneo,
New Guinea, and Northern Japan are growing sinuosity slowly under relatively flood-free
flow conditions, and simply have not had enough time to fully meander yet. Meanwhile,
rapid bank erosion along the rivers of Taiwan and Luzon, has caused these areas to evolve
faster towards a state of universal meandering along all channels.
84
84
However, an alternative explanation is that negative feedbacks associated with horizontal and vertical erosion work against sinuosity growth and eventually shut down lateral
cutting, such that sinuosity evolves to a stable form. In this case, stronger climatological
forcing in Taiwan and Luzon has allowed rivers in these islands to develop very sinuous
planforms quickly, before lateral erosion has ceased. Rivers are straighter in Japan and
Borneo, where weaker forcing promotes planform evolution that has been too slow to
develop meanders before shutting down.
Whether the planforms of these rivers are in a state of transience or have reached
stability is a question that remains to be answered, but negative feedbacks are very likely
to be at work on incising rivers. For example, meander growth reduces channel slopes
which should reduce lateral cutting, and incision lengthens hillslopes which should lead
to greater wall buffering.
In either case, the correlations of regionalized sinuosity of mountain rivers and indicators of storminess (tropical cyclone frequency, rainfall variability, relative flood intensity)
suggest that incising mountain rivers have planforms that adjust in response to climatology. We infer that a regional measure of mountain river sinuosity is a signature of the
long-term climate. Along a single channel, changes in horizontal and vertical erosion rates
can result from changes in tectonics or changes in climate; since the relative magnitudes
of these rates are recorded in the shape of the valley, if uplift rate and its changes in time
are well-constrained, it may be possible to read the paleotempestology from the shape of
the valley. However, this will only be possible with better understanding of the negative
feedbacks of incised meandering.
85
85
Acknowledgments. This study was supported by the National Science Foundation
through SGER EAR 05-50087 and GLD EAR 06-17557, the National Aeronautics and
Space Administration through ESSF NGT5-30532 and NAG5-13772, and Lamont-Doherty
Earth Observatory.
References
Begin, Z. B., Stream curvature and bank erosion: A model based upon the momentum
equation, Journal of Geology, 89, 497–504, 1981.
Belaguru, A., and G. Nichols, Tertiary stratigraphy and basin evolution, southern Sabah
(Malaysian Borneo), Journal of Asian Earth Sciences, 23, 537–554, 2004.
Benard, F., C. Muller, J. Letouzey, C. Rangin, and S. Tahir, Evidence of multiphase
deformation in the Rajang-Crocker Range (northern Borneo) from Landsat imagery
interpretation: Geodynamic implications, Tectonophysics, 183, 321–339, 1990.
Brakenridge, G. R., Rate estimates for lateral bedrock erosion based on radiocarbon ages,
Duck River, Tennessee, Geology, 13, 111–114, 1985.
Bretz, J. H., Dynamic equilibrium and Ozark land forms, American Journal of Science,
260 (2), 427–438, 1962.
Brown, C. M., C. J. Pigram, and S. K. Skwarko, Mesozoic stratigraphy and geological
history of Papua New Guinea, Palaeogeography, Palaeoclimatology, Palaeoecology, 29,
301–322, 1980.
Bucher, W. H., ”strath” as a geomorphic term, Science, 75, 130–131, 1932.
Callander, R. A., Instability and river channels, Journal of Fluid Mechanics, 36, 465–480,
1969.
86
86
Davis, W. M., The topographic maps of the United States Geological Survey, Science,
21 (534), 225–227, 1893.
Edwards, B. F., and D. H. Smith, River meandering dynamics, Physical Review E,
65 (046303), 1–12, 2002.
Faure, M., and B. Natalin, The geodynamic evolution of the eastern Eurasian margin in
Mesozoic times, Tectonophysics, 208, 397–411, 1992.
Finnegan, N. J., G. Roe, D. R. Montgomery, and B. Hallet, Controls on the channel width
of rivers: Implications for modeling fluvial incision of bedrock, Geology, 33 (3), 229–232,
2005.
Fredsoe, J., Meandering and braiding of rivers, Journal of Fluid Mechanics, 84, 609–624,
1978.
Furbish, D. J., River bend curvature and migration: How are they related?, Geology, 16,
752–755, 1988.
Gardner, T. W., The history of part of the colorado river and its tributaries: An experimental study, pp. 87–95, Canyonlands, eighth ed., Four Corners Geological Society,
1975.
Hall, R., Cenozoic geological and plate tectonic evolution of SE Asia and the SW Pacific: Computer-based reconstructions, model and animations, Journal of Asian Earth
Sciences, 20, 353–431, 2002.
Hartshorn, K., N. Hovius, W. B. Dade, and R. L. Slingerland, Climate-driven bedrock
incision in an active mountain belt, Science, pp. 2036–2038, 2002.
Honza, E., J. John, and R. M. Banda, An imbrication model for the Rajang Accretionary
Complex in Sarawak, Borneo, Journal of Asian Earth Sciences, 18, 751–759, 2000.
87
87
Hovius, N., and C. P. Stark, Actively meandering bedrock rivers, in EOS, Transactions
AGU, p. 506, 2001.
Huber, N. K., Amount and timing of Late Cenozoic uplift and tilt of the Central Sierra
Nevada, California - evidence from the upper San Joaquin river basin, Geological Survey
Professional Paper, 1197, 1–28, 1981.
Hutchison, C. S., Evidence of multiphase deformation in the Rajang-Crocker range (northern Borneo) from Landsat imagery interpretation: Geodynamic implications-Comment
(1), Tectonophysics, 204, 175–177, 1992.
Ikeda, S., G. Parker, and K. Sawai, Bend theory of river meanders. Part 1. Linear development, Journal of Fluid Mechanics, 112, 363–377, 1981.
Jefferson, M. S. W., The antecedent Colorado, Science, 6 (138), 293–295, 1897.
Johannesson, H., and G. Parker, Linear theory of river meanders, in River meandering,
edited by S. Ikeda and G. Parker, Water Resources Monograph, pp. 181–214, American
Geophysical Union, Washington, D.C., 1989.
Mahard, R. H., The origin and significance of intrenched meanders, Journal of Geomorphology, 5, 32–44, 1942.
Merritts, D. J., K. R. Vincent, and E. E. Wohl, Long river profiles, tectonism, and eustasy: A guide to interpreting fluvial terraces, Journal of Geophysical Research, 99 (B7),
14,031–14,050, 1994.
Montgomery, D. R., Observations on the role of lithology in strath terrace formation and
bedrock channel width, American Journal of Science, 304, 454–476, 2004.
Moore, R. C., Origin of inclosed meanders in the physiographic history of the Colorado
Plateau country, Journal of Geology, 34, 29–57, 1926.
88
88
Parker, G., K. Sawai, and S. Ikeda, Bend theory of river meanders. part 2. nonlinear
deformation of finite amplitude bends, Journal of Fluid Mechanics, 115, 303–314, 1982.
Pazzaglia, F. J., and M. T. Brandon, A fluvial record of long-term steady-state uplift and
erosion across the Cascadia forearc high, Western Washington state, American Journal
of Science, 301, 385–431, 2001.
Peucker-Ehrenbrink, B., and M. W. Miller, Quantitative bedrock geology of east and
Southeast Asia (Brunei, Cambodia, eastern and southeastern China, East Timor, Indonesia, Japan, Laos, Malaysia, Myanmar, North Korea, Papua New Guinea, Philippines, far-eastern Russia, Singapore, South Korea, Taiwan, Thailand, Vietnam), Geochemistry Geophysics Geosystems, 5 (1), Q01B06, doi:10.1029/2003GC000619, 2004.
Powell, J. W., Exploration of the Colorado River of the West and its tributaries. Explored
in 1869, 1870, 1871, and 1872 under the direction of the secretary of the Smithsonian
Institution, Government Printing Office, Washington, 1875.
Rich, J. L., Certain types of stream valleys and their meaning, Journal of Geology, 22,
469–497, 1914.
Rogers, R. D., H. Karason, and R. D. van der Hilst, Epeirogenic uplift above a detached
slab in northern Central America, Geology, 30 (11), 1031–1034, 2002.
Scholz, C. H., The mechanics of earthquakes and faulting, second ed., Cambridge University Press, 471 pp., 2002.
Sears, J. D., Relations of the Browns Park Formation and the Bishop Conglomerate, and
their role in the origin of Green and Yampa Rivers, Bulletin of the Geological Society
of America, 35, 279–304, 1924.
Seminara, G., Meanders, Journal of Fluid Mechanics, 554, 271–297, 2006.
89
89
Shepherd, R. G., Incised river meanders: Evolution in simulated bedrock, Science, 178,
409–411, 1972.
Shepherd, R. G., and K. A. Schumm, Experimental study of river incision, Geological
Society of America Bulletin, 85, 257–268, 1974.
Shyu, J. B., 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, 2006.
Smith, H. T. U., Physical effects of Pleistocene climatic changes in nonglaciated areas:
eolian phenomena, frost action, and stream terracing, Geological Society of America
Bulletin, 60, 1485–1516, 1947.
Stark, C. P., A self-regulating model of bedrock river channel geometry, Geophysical
Research Letters, 32, doi:10.1029/2005GL023193, 2006.
Stark, C. P., and J. Barbour, Variable discharge in bedrock channels: 2. Sedimentbuffering and channel morphodynamics., in prep., to be submitted to JGR, 2007a.
Stark, C. P., and J. Barbour, Variable discharge in bedrock channels: 3. Exploration of
channel mobility., in prep., to be submitted to JGR, 2007b.
Stark, C. P., J. Barbour, C.-W. Lin, H. Chen, C. Huang, and M.-J. Horng, Variable
discharge in bedrock channels: 1. Channel width and flow width-depth ratios., in prep.,
to be submitted to JGR, 2007.
Stark, C. P., et al., Dependence of mountain river sinuosity on typhoon flood magnitude
and frequency, Nature, in prep., 2008.
Steinshouer, D. W., J. Qiang, P. J. McCabe, and R. T. Ryder, Maps showing the geology,
oil and gas fields, and geologic provinces of the Asia Pacific region, USGS Open File
90
90
Report 97-470-F, http://pubs.usgs.gov/of/1997/ofr-97-470/OF97-470F/, 1997.
Tabei, T., et al., Subsurface structure and faulting of the Median Tectonic Line, southwest
Japan inferred from GPS velocity field, Earth Planets Space, 54, 1065–1070, 2002.
Tarr, W. A., Intrenched and incised meanders of some streams on the northern slope of
the Ozark Plateau in Missouri, Journal of Geology, 32, 583–600, 1924.
Tinkler, K., and E. Wohl, A primer on bedrock channels, pp. 1–18, Geophysical Monograph
107, American Geophysical Union, 1998.
Turowski, J. M., Controls on bedrock channel morphology: Experimental and theoretical investigations and comparison with natural channels, Ph.D. thesis, University of
Cambridge, 2006.
Turowski, J. M., 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.
Whipple, K. X., Bedrock rivers and the geomorphology of active orogens, Annual Review
of Earth and Planetary Sciences, 32, 151–185, 2004.
Wu, C.-C., and Y.-H. Kuo, Typhoons affecting Taiwan: Current understanding and future
challenges, Bulletin of the American Meteorological Society, 80 (1), 67–80, 1999.
Yu, S.-B., and L.-C. Kuo, Present-day crustal motion along the Longitudinal Valley Fault,
eastern Taiwan, Tectonophysics, 333 (1–2), 199–217, 2001.
91
91
CHAPTER3a
3a
CHAPTER
Magnitude-frequency distributions of boundary shear stress along a rapidly eroding
Magnitude-frequency distributions of boundary
shear stress
bedrock mountain river ‡
along a rapidly eroding bedrock mountain river‡
‡
This manuscript has been submitted to Geophysical Research Letters with co-authors Colin P. Stark,
Chingweei Lin, Hongey Chen, Ming-Jame Horng, Chin-Pin Ko, Te-Cheng Yi, Tsai-Tsung Tsai,
Wei-Shu Chang, Shin-Ping Lee, and Chung Huang.
92
Abstract.
The magnitude-frequency distribution of boundary shear stress
frames erosion rates in bedrock rivers, but empirical constraints
are rare, particularly for extreme floods. Here we present measurements of mean stress τb and its frequency distribution along
a fast-eroding river in Taiwan. We construct rating functions of
discharge and hydraulic geometry from stage time-series, topographic surveys, and high-resolution satellite images and find the
derived PDF of τb has a steep power-law tail. Recorded floods
include two 50-year, 3000 m3 s−1 events driving τb ≥ 300 Pa, but
largely comprise semiannual events generating τb ≈ 100-200 Pa,
all capable of driving incision. Comparable-size channels in the
north-eastern US eroding 40 times slower experience 50-year
floods with stresses in the latter range. Therefore, unless incision
rates scale exponentially with maximum flood shear, the extreme
incision rates in Taiwan probably originate in the exceptional frequency of their erosive floods. We deduce that frequency rather
than magnitude primarily sets the pace of bedrock channel erosion.
93
93
1. Introduction
Few datasets exist that relate flows in mixed bedrock-alluvial channels to the velocities
and boundary shear stresses that determine their erosion and deposition [Molnar et al.,
2006]. Rarer still are data on the longitudinal variations in flow geometry at different
discharges, which limits our ability to understand the degree, nature, and consequences of
the spatiotemporal complexity of shear stresses along bedrock channels and to constrain
models of channel evolution [Stark , 2006; Turowski et al., 2008; Wobus et al., 2006].
Bedrock rivers lack the kind of empirical constraints on hydraulic geometry available for
alluvial rivers [Leopold and Maddock , 1953].
The deployment of stage recorders in large numbers along mountain rivers is not a
practical means for monitoring their along-stream hydraulic geometry. The advent of
high-resolution satellite imagery offers a solution, at least for mapping the planform flow
geometry over space and time. To date, remote-sensing studies have focused on very large
lowland rivers with broad floodplains [e.g. Smith, 1997; Bjerklie et al., 2003; Brakenridge
et al., 2005], using satellite images to monitor river stage [Koblinsky et al., 1993; Birkett,
1998; Birkett et al., 2002], changes in river stage [Alsdorf et al., 2000, 2001], flood inundation [Sippel et al., 1994, 1998; Townsend , 2001], and flow widths averaged over reaches
[Smith et al., 1995] or at transects [Xu et al., 2004; Zhang et al., 2004]. Although satellite
assessments of hydraulic geometry can be combined with ground-survey data to estimate
channel boundary shear stresses, none of the prior studies have taken this step, nor have
their methods been tested in narrow, O(100 m) wide river channels. Addressing these
issues is the main goal of our paper.
94
94
The application of remote gauging to small mountain rivers presents a very different
challenge to the assessment of large lowland rivers. Wetted channels 10-100 m wide require very high resolution, 1 m-5 m imagery, eliminating the option at present of stage
measurement using SAR data; mountain catchments are particularly prone to cloud cover
that reduces the number of usable images, especially during the important storm-stage
flows; high topographic relief necessitates and complicates accurate orthorectification for
image-series co-registration, requiring 10 m resolution or better DEM data and very good
ground control; ground validation of flow geometries is logistically and technically more
difficult; significant changes in alluvial bed geometry occur frequently, frustrating widthbased rating; and flows are typically shallow, rough, and sinuous, which complicates flow
modeling.
Remote gauging can nevertheless be performed in mountain rivers given (1) very
high resolution, multitemporal satellite imagery, (2) good ground control for image coregistration, and (3) a well-located river stage recorder providing a long time-series of
discharge estimates. These criteria are met in many mountain catchments in Taiwan, a
particularly compelling location for the spatiotemporal assessment of channel flows and
boundary shear stresses because of its exceptional rates of bedrock channel incision and
landscape evolution [Li , 1976; Hartshorn et al., 2002; Dadson et al., 2003; Turowski et al.,
2008]. The data presented here on hydraulic geometry in Taiwanese rivers will be a
welcome resource in the study of bedrock channel dynamics.
We focus on the Zhuókŏuxı̄ catchment, which is located on the south-western flank of
the Central Mountain Range (Fig. 1a) at a steep range front created by the oblique-thrust
95
95
WANS23
WANS21
WANS24
WANS20
SHTO20
WANS25
WANS28
WANS27
WANS29
WANS26
LUMS02
(a)
XINS03
0.8 Km
(b)
200
SHTO20
WANS28
WANS29
WANS27
WANS23
LUMS02
100
XINS03
150
DAJN20
Elevation [m]
250
0.4
22°N
DAJN20
0
122°E
24°N
120°E
Slope [%]
1.5
(c)
1
0.5
0
14
12 10 8
6
4
2
Downstream Distance [km]
0
Figure 1: (a) Shaded relief image of the Màolı́n Valley generated from a 5 m DEM showing the locations
of surveyed channel cross-sections. (b) Longitudinal channel topographic profile. (c) Smoothed downstream
gradient.
96
96
Cháozhōu (Chauchou) fault. The Zhuókŏuxı̄ (often rendered as the Jhuokou or Jukou Shi,
meaning the “Muddy Mouth River”) drains a 375 km2 catchment along the Màolı́n Valley;
our study reach extends 14 km upstream from its mouth at Dàjı̄n close to the confluence
with the Gāopı́ng (Kaoping) River. This steep (Fig. 1c), mixed bedrock-alluvial river cuts
rapidly [Dadson et al., 2003] through an intermittent veneer of mostly quartzite cobbles,
boulders, and argillaceous sand, into argillite bedrock. Peak flows occur during the onset
of the Méiyŭ in spring, and during typhoons in the summer and autumn, and can exceed
3000 m3 s−1 ; flood discharge usually lasts for no more than a few days before dropping to
typical low stages flows around ∼30 m3 s−1 . In winter, peak flows rarely exceed 100 m3 s−1 .
2. Mapping of mean boundary shear stress
We combine field and remote-sensing measurements along the Zhuókŏuxı̄ with a simple
theoretical analysis to estimate the mean boundary shear stress τb for a range of discharges.
For reasonably straight, constant-gradient reaches, and uniform steady flow, boundary
shear stress τb is
τb = ρg
A
S = ρgRS
P
(1)
where R is the hydraulic radius or ratio of cross-sectional area A to wetted perimeter P ,
ρ is water density, g is gravitational acceleration, S is the channel bed slope. Mean flow
speed is the discharge per unit cross-sectional area A,
U = Q/A .
(2)
Estimation of τb and U at a transect therefore requires measurement of the channel slope
and the flow cross-section. The latter is obtained by combining the transect topographic
profile with satellite observations of flow width at each discharge.
97
97
Channel topography
We surveyed topographic profiles of each channel transect and obtained the longitudinal
profile of the river by projecting DGPS surveys of the flow channel edges at low stage
onto the channel centerline. To reduce error, we used a 300 m median filter along the
longitudinal profile and computed longitudinal gradient from the best linear fit of the
filtered profile within each 300 m window (Fig. 1b,c). While there is some variation in
slope along the Zhuókŏuxı̄ channel below this scale, there are no significant knickpoints
present and the gradient varies on a O(1 km) length scale between 0.5% and 1%.
Discharge time series
The Water Resources Agency of Taiwan has maintained an automatic stage recorder
on the Zhuókŏuxı̄ since 1970, first at LUMS02 (1970–1983), then at DAJN20 (1989–
2005), and now at XINS03 (2006–present) (Figs. 1 and 2). Stage is measured hourly and
yields the daily statistics: mean Qμ = 39.5 m3 s−1 , standard deviation Qσ = 130 m3 s−1 ,
coefficient of variation Qcv = 3.31, 99th percentile Q99 = 567 m3 s−1 , and peak Qmax >
3000 m3 s−1 . For our rating study, we used the 10 am measurement for each F2 image date
(Table 1).
The empirical PDFs for hourly and daily discharge derived from records spanning 20042006 are indistinguishable (Fig. 2a); the longer daily discharge record spanning 1970-2006
has a PDF with a slightly lighter tail. All the empirical PDFs indicate a power-law decay
in the tail p(Q) ∼ Q−α−1 ∼ Q−2 , i.e., a Pareto exponent of α ≈ 1, which is indicative of
frequent extreme events [Turcotte and Green, 1993; Stark and Hovius, 2001; Lague et al.,
2005] relative to exponentially decaying PDFs.
98
98
Table 1: FORMOSAT-2 image set
Date∗
Q†
Q% ‡
Cloud cover
Use
2006/3/17
5
34
10%
low-stage, reference
2006/9/28
41
82
0%
mean-stage
2006/7/29
126 93.9
0%
mid-stage
2006/7/11
200 96.3
0%
mid-stage
2005/8/07
801 99.5
30%
high-stage
2005/7/10
14
n/a
0%
pre-Hăitáng§
2005/8/01
1
n/a
0%
post-Hăitáng§
2003/12/30∗∗ 16
n/a
0%
pre-Mı̆ndūlı̀§
∗
Images acquired between 9:55 am and 10:05 am local time.
Hourly discharge in m3 s−1 was measured at Dàjı̄n in 2005 and at Xı̄nnóng in 2006.
Reported here is the discharge measured at 10 am, scaled by relative drainage area to the
Dàjı̄n gauge.
‡
Q percentile.
§
Discharge during Hăitáng reached 2894 m3 s−1 on 2005/07/20. The maximum discharge on record for this catchment, 3163 m3 s−1 occurred one year earlier, during Typhoon
Mı̆ndūlı̀.
∗∗
SPOT-5 image.
†
99
99
í
10
(a)
-D – 1 § -2
Probability density p(Q) [mí s]
í
10
í
10
í
10
í
10
Qhí
í
10
Qdí
Qdí
í
10
1
2
10
3
10
4
10
10
í
[m33 sí
]
Q [m s ]
Q
2
Probability density p(Tb) [Paí]
Tb [Pa]
10
(b)
300 (a)
10
6
200
4
100
2
0
0
í
10
(c)
(b)
-J – 1 § -4
í
10
í
10
Tb distribution
í
10
power law decay
exponential decay
30
Tb exceedance [days/yr]
4
10
U [m sí]
0
100
Tb [Pa]
300
300
(d)
(c)
100
30
10
3
1
0.1
Q99
50
100
150
Tb [Pa]
200
250
Figure 2: Hydraulic properties at WANS28: (a) PDFs of hourly (Qh ) and daily average (Qd ) discharge on
the Zhuókŏu River from the WRA archive along with schematic power law tangent to the tail portion of the
distribution (gray dashed line). (b) Observations and model regressions of shear stress τb ∼ Q1/3 (black circles)
and flow speed U ∼ Q2/5 (gray squares) versus discharge. (c) Distribution p(τb ) with power law and exponential
model regressions. (d) Complementary cumulative distribution of τb exceedance.
100
100
Image processing and hydraulic geometry estimation
To estimate hydraulic geometry at each transect and discharge, we mapped flow width in
a series of FORMOSAT-2 (F2) satellite images and supplemented this with field mapping
of extreme flood width and depth along the channel.
Since 2004, the F2 VNIR sensor has collected 2 m-resolution panchromatic images and
8 m-resolution multispectral images at approximately 10 am daily over Taiwan with a
swath width of 24 km. The return period for any given region of interest is irregular
because of scheduling priorities; image acquisition is often off-nadir, affecting both the
field of view and the amount of image distortion. Substantial cloud cover is common,
especially during the high-magnitude flood events that most alter the channel, which
further reduces the availability of the most useful imagery. Nevertheless, many near
cloud-free images of the Zhuókŏuxı̄ with good satellite geometry are available, most of
which are acquired during the dry season when the river is at low stage. On a few rare
passes, F2 has captured the river at high stage, providing a set of images that span much
of the range of recorded discharge (Table 1).
The accuracy of wetted channel width measurements depends on the precise coregistration of the image series, which we accomplished in two steps. We first orthorectified
the selected images to a 5m DEM using ground control points collected in the field and
from orthophotographs. We then warped each image to the 2006/03/17 low-stage reference image, which was acquired at our request during fieldwork. By deploying a set
of satellite-visible, precisely located targets on the river bed during the F2 overpass, we
achieved the best possible ground control for this image.
101
101
We mapped the wetted channel in the each F2 image using variety of processing techniques. The clarity of the wetted edge varied with shadows, haze, illumination angle,
channel orientation, and bed material: in some cases, the edge was sharply defined in the
panchromatic image; in others, it was clearer in the pan-sharpened multispectral image;
in yet other cases, it was most clear after applying a Sobel edge filter to the panchromatic
image.
Although the F2 satellite did not image the Zhuókŏuxı̄ during the two most extreme
flood events on record, we were able to map their approximate hydraulic geometry
by examining images from before and after the floods. Discharge reached 3163 m3 s−1
on 2004/07/04 during Typhoon Mı̆ndūlı̀ (Mindulle), the highest ever recorded on the
Zhuókŏu River. A year later, discharge during Typhoon Hăitáng reached 2894 m3 s−1 on
2005/07/20. Together, these floods left a clear high water mark along much of the channel which we mapped through a comparison of a SPOT-5 image acquired before Mı̆ndūlı̀
and F2 images acquired after each storm (Table 1). The mapping was validated at each
transect with observations of the high water mark measured in the field.
Mapping results
At each transect of the channel we find nonlinear relationships between R, τb , U and
Q which we approximate with power-law scaling functions. Since slope is fixed at each
transect, τb ∼ R (Equation 1). However, since channel geometry and boundary roughness
changes in a complex fashion along the channel, the scaling of τb with Q and therefore the
shape of p(τb ) is different at each transect. For example, at WANS28 (Fig. 2b), which is
102
102
approximately trapezoidal with relatively steep bank angles, we find
1
R ∼ τb ∼ Q 3
⇔
2
U ∼ Q5
⇔
5
τb ∼ U 6 .
(3)
At WANS29, located 650 m upstream, most flows are confined between gently dipping
banks; only the highest discharges are bounded by steep bedrock walls. Here we find
1
R ∼ τb ∼ Q 2
⇔
1
U ∼ Q5
⇔
5
τb ∼ U 2 .
(4)
The scaling of τb with Q has an exponent that ranges from 2/3 to 1/4 over the surveyed
reach, which points to a dynamic variability in channel roughness and shape that current
landscape evolution models, which hold the scaling constant, fail to capture.
PDFs of mean boundary shear stress
We calculate the shear stress distribution p(τb ) for each transect by applying each empirical scaling relation τb ∼ Q1/β to the discharge distribution p(Q). Fig. 2(c) illustrates
the result of the transformation at WANS28 where β = 3. The power-law tail of the
discharge PDF steepens under transformation from p(Q) ∼ Q−α−1 with α ≈ 1 to
p(τb ) ∼ τb −4 ∼ τb −γ−1
where γ = αβ = 3 .
(5)
Power-law distributions with such large exponents are relatively light-tailed, and in an
empirical PDF they are barely distinguishable from exponential in their decay (Fig. 2).
At WANS29, however, where β = 2, the τb tail decay is heavier,
p(τb ) ∼ τb −3 ∼ τb −γ−1
where γ = αβ = 2 .
(6)
Values of γ vary along our study reach between 1.5 and 4, but typically fall between 2
and 3.
103
103
3. Discussion & conclusion
We have shown how a combination of high-resolution remote-sensing, stream gauging,
and field surveying can be used to measure hydraulic geometry, mean boundary shear
stress τb , and its magnitude-frequency distribution at any transect along a mountain river
channel. Table 2 summarizes our estimates of width, depth, and shear stress at ten
selected transects for each of the discharges imaged by the F2 satellite. These estimates
show that τb can vary significantly along the channel during a single discharge event, as the
flow is squeezed and stretched through transects with different geometric and frictional
characteristics.
Our data illustrate the hydraulic geometry and minimum discharge Q∗ (and percentile
Q% ) required to begin inundation of the channel wall (right-hand columns of Table 2).
Field observation of flows below this critical stage, supplemented by simple Shields stress
calculations, show that they can move bedload boulders up to ≈ 0.5 m in diameter. We
conclude that a significant fraction of the moderate floods on the Zhuókŏuxı̄, and many
other bedrock rivers in Taiwan and beyond, can move bed sediment and drive bedrock
erosion, but cannot induce wear on the walls. Only more powerful floods, which in Taiwan
occur semiannually but in other regions far less often, can cause significant wall erosion
[Hartshorn et al., 2002; Turowski et al., 2008].
It is instructive to compare the magnitude-frequency distribution of mean boundary
shear stress (Fig. 2d) in this fast-eroding river with shear stress data from more slowly
eroding bedrock channels [e.g., Snyder et al., 2003]. We estimate that heavy semiannual
floods of about Q ≈ 300-500 m3 s−1 on the Zhuókŏuxı̄ drive shear stresses of around
104
104
105
105
0.47
0.42
0.66
W21 347.1
W29 347.1
W28 353.0
42, 1.2, 62
33, 1.2, 52
73, 1.7, 79
65, 1.7, 65
92, 3.8, 123
w, H, τb
163 m3 s−1
17, 1.1, 36
16, 0.8, 17
13, 1.0, 22
22, 1.4, 39
19, 0.8, 26
11, 0.5, 29
23, 1.5, 46
24, 1.2, 23
31, 1.5, 28
26, 1.6, 46
34, 1.7, 58
55, 1.8, 81
67, 3.6, 96
59, 2.7, 53
61, 2.5, 47
58, 3.4, 95
47, 2.3, 80
95, 2.8, 133
12, 0.3, 21 99, 2.5, 134 104, 2.6, 140
24, 0.8, 47
24, 1.1, 55
54, 1.9, 36
w, H, τb
w, H, τb
10, 0.6, 19
41 m3 s−1
5 m3 s−1
195, 8.0, 643
129, 5.6, 448
96, 4.6, 322
139, 6.1, 193
w, H, τb
3000 m3 s−1
90, 5.5, 191
97, 4.5, 94
66, 3.9, 92
153, 8.8, 249
69, 5.6, 236
109, 7.3, 261
134, 7.9, 189
148, 8.9, 226
161, 9.3, 264
157, 9.9, 315
150, 10.3, 782 155, 12.1, 925
183, 6.3, 480
110, 2.6, 135
91, 2.4, 104
126, 5.7, 188
w, H, τb
801 m3 s−1
H∗
τb ∗
Q̃∗
2.2
2.3
1.3
3.2
67
81
52
3.4
3.6
3.1
142 8.1
60
114 3.3
40
125 2.9
96
76
97
73
68
231
110
158
90
152
85
50
150
356
94
539
433
249
7
1665
666
76
[m] [m] [Pa] m3 s−1
w∗
†
Hydraulic geometry and minimum discharge at the initiation of channel wall inundation. Mean boundary shear stress τb is computed with equation 1 using w at which
wetting of the wall begins for the field-surveyed cross-section. Critical discharge Q̃ and
discharge percentile Q% derive from the correlation at each transect between w measured
in satellite images with Dàjı̄n Q scaled by relative drainage area.
‡
S20 ≡ SHTO20,W23 ≡ WANS23, etc.
0.60
W27 347.0
0.66
W20 347.0
1.26
W25 346.8
1.08
1.19
W24 346.8
W26 346.9
1.07
W23 346.8
[%]
[km2 ]
0.63
Slope
Area
344.6
S20†
ID
Table 2: Flow widths w [m], depths H [m], and boundary shear stresses τb [Pa] at channel transects.
95.2
98.2
92.3
99.1
98.6
97.4
45.0
99.9
99.4
90.5
[%]
Q% ∗
τb = 100-200 Pa. In contrast, Snyder et al. [2003] find that boundary shear stresses of
a similar magnitude occur on the Fall Creek (a bedrock river of comparable scale in the
Finger Lakes region of the north-eastern US) every 50 years or so during discharge around
Q ≈ 300-400 m3 s−1 . These flows can both move coarse bedload and cause significant wear
of the channel bedrock cross-section in both rivers.
The maximum flood on record on the Zhuókŏuxı̄ (during Typhoon Mı̆ndūlı̀) is an order
of magnitude greater in discharge (Q ≈ 3000 m3 s−1 ) than these floods, but produces
a shear stress that is only a factor of two greater (e.g., at WANS28, Fig. 2(b)), with
τb ≈ 300 Pa. Incision rates in each river are drastically different: Dadson et al. [2003]
and others estimate the rate of incision in the south-western Central Range of Taiwan at
around 6 mm y−1 ; Snyder et al. [2003] estimate a rate of 0.14 mm y−1 for the past 1400
years and infer τ ≈ 100-200 Pa for 50 year floods. The Zhuókŏuxı̄ is therefore eroding
about 40 times faster than Fall Creek.
Therefore, unless incision rates scale exponentially or very nonlinearly with maximum
flood shear, the extreme incision rates in Taiwan probably originate instead in the exceptional frequency of their moderately erosive floods. The much higher sediment fluxes in
Taiwan are an important cofactor, but it is not clear whether they boost (tool effects) or
diminish (cover effects) the rate of bedrock wear [Sklar and Dietrich, 2004]. We deduce
that the frequency of sufficiently high shear-stress floods, rather than the magnitude of
exceptional shear stresses during extreme floods, is the more important factor in setting
the pace of bedrock channel erosion. To test this assertion further will require long-term,
event-by-event monitoring of floods in fast-eroding rivers like the Zhuókŏuxı̄.
106
106
Acknowledgments. This study was supported by the National Science Foundation
through SGER EAR 05-50087 and GLD EAR 06-17557, the National Aeronautics and
Space Administration through ESSF NGT5-30532 and NAG5-13772, and Lamont-Doherty
Earth Observatory. The authors thank G.-R. He, W.-L. Lee, W.-Y. Lai and the Maolin
Municipal Fire Department for assistance in the field. FORMOSAT-2 images were used
under authorization of the National Space Organization through the Disaster Prevention
Research Center of the National Cheng Kung University of Taiwan.
References
Alsdorf, D., C. Birkett, T. Dunne, J. Melack, and L. Hess, Water level changes in a large
Amazon lake measured with spaceborne radar interferometry and altimetry, Geophysical
Research Letters, 28 (14), 2671–2674, 2001.
Alsdorf, D. E., J. M. Melack, T. Dunne, L. A. K. Mertes, L. L. Hess, and L. C. Smith,
Interferometric radar measurements of water level changes on the Amazon Flood Plain,
Nature, 404, 174–177, 2000.
Birkett, C. M., Contribution of the TOPEX NASA radar altimeter to the global monitoring of large rivers and wetlands, Water Resources Research, 34 (5), 1223–1239, 1998.
Birkett, C. M., L. A. K. Mertes, T. Dunne, M. H. Costa, and M. J. Jasinski, Surface
water dynamics in the Amazon basin: application of satellite radar altimetry, Journal
of Geophysical Research, 107 (D20), 8059, doi:10.1029/2001JD000609, 2002.
Bjerklie, D. M., S. L. Dingman, C. J. Vorosmarty, C. H. Bolster, and R. G. Congalton,
Evaluating the potential for measuring river discharge from space, Journal of Hydrology,
278, 17–38, 2003.
107
107
Brakenridge, G. R., S. V. Nghem, E. Anderson, and S. Chen, Space-based measurement
of river runoff, EOS, 19 (10), 185–188, 2005.
Dadson, S. J., et al., Links between erosion, runoff variability and seismicity in the Taiwan
orogen, Nature, 426, 648–651, 2003.
Hartshorn, K., N. Hovius, W. B. Dade, and R. L. Slingerland, Climate-driven bedrock
incision in an active mountain belt, Science, pp. 2036–2038, 2002.
Koblinsky, C. J., R. T. Clarke, A. C. Brenner, and H. Frey, Measurement of river level
variations with satellite altimetry, Water Resources Research, 29, 1839–1848, 1993.
Lague, D., N. Hovius, and P. Davy, Discharge, discharge variability, and the bedrock channel profile, Journal of Geophysical Research, 110, F04,006, doi:10.1029/2004JF000259,
2005.
Leopold, L. B., and T. Maddock, The hydraulic geometry of stream channels and some
physiographic implications, United States Geological Survey Professional Paper, 252,
1953.
Li, Y. H., Denudation of Taiwan island since the Pliocene epoch, Geology, 4, 105–107,
1976.
Molnar, P., R. S. Anderson, G. Kier, and J. Rose, Relationships among probability distributions of stream discharges in floods, climate, bed load transport, and river incision,
Journal of Geophysical Research, 111, F02,001, doi:10.1029/2005JF000310, 2006.
Sippel, S. J., S. K. Hamilton, J. M. Melack, and B. J. Choudhury, Determination of
inundation area in the Amazon River floodplain using the SMMR 37 GHz polarization
difference, Remote Sensing of Environment, 48, 70–76, 1994.
108
108
Sippel, S. J., S. K. Hamilton, J. M. Melack, and E. M. M. Novo, Passive microwave observations of inundation area and the area/stage relation in the Amazon River floodplain,
International Journal of Remote Sensing, 19 (16), 3055–3074, 1998.
Sklar,
L. S.,
and W. E. Dietrich,
A mechanistic model for river incision
into bedrock by saltating bed load, Water Resources Research, 40 (W06301),
doi:10.1029/2003WR002,496, 2004.
Smith, L. C., Satellite remote sensing of river inundation area, stage, and discharge: A
review, Hydrological Processes, 11, 1427–1439, 1997.
Smith, L. C., B. L. Isacks, R. R. Forster, A. L. Bloom, and I. Preuss, Estimation of
discharge from braided glacial rivers using ERS 1 synthetic aperture radar: first results,
Water Resources Research, 31 (5), 1325–1329, 1995.
Snyder, N. P., K. X. Whipple, G. E. Tucker, and D. M. Merritts, Importance of a stochastic
distribution of floods and erosion thresholds in the bedrock river incision problem,
Journal of Geophysical Research, 108 (B2), 2117, doi:10.1029/2001JB001655, 2003.
Stark, C. P., A self-regulating model of bedrock river channel geometry, Geophysical
Research Letters, 32, doi:10.1029/2005GL023193, 2006.
Stark, C. P., and N. Hovius, The characterization of landslide size-frequency distributions,
Geophysical Research Letters, 28 (6), 1091–1094, 2001.
Townsend, P. A., Mapping seasonal flooding in forested wetlands using multi-temporal
Radarsat SAR, Photogrammetric Engineering and Remote Sensing, 67 (7), 857–864,
2001.
109
109
Turcotte, D. L., and L. Green, A scale-invariant approach to flood-frequency analysis,
Stochastic Hydrology and Hydraulics, 7, 33–40, 1993.
Turowski, J. M., 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.
Wobus, C. W., G. E. Tucker, and R. S. Anderson, Self-formed bedrock channels, Geophysical Research Letters, 33 (L18408), 1–6, 2006.
Xu, K., J. Zhang, M. Watanabe, and C. Sun, Estimating river discharge from very highresolution satellite data: a case study in the Yangtze River, China, Hydrological Processes, 18, 1927–1939, 2004.
Zhang, J., K. Xu, M. Watanabe, Y. Yang, and X. Chen, Estimation of river discharge from
nontrapezoidal open channel using QuickBird-2 satellite imagery, Hydrological Sciences,
49 (2), 247–260, 2004.
110
110
CHAPTER 3b
3b
CHAPTER
Monitoring the flow conditions and morphological changes of a typhoon flood-
Monitoring the flow conditions
and morphological changes
prone bedrock river §
of a typhoon flood-prone bedrock river§
§
This manuscript is in preparation for submission to the Journal of Geophysical Research with coauthors Colin P. Stark, Chingweei Lin, Hongey Chen, Ming-Jame Horng, Chin-Pin Ko, Te-Cheng Yi,
Tsai-Tsung Tsai, Wei-Shu Chang, Shin-Ping Lee, and Chung Huang.
111
Abstract.
Quantitative constraints on the properties of bedrock chan-
nel flows are rare, particularly on their spatiotemporal variability and patterns of boundary shear stress during high magnitude/low frequency floods.
Here we present such data for a rapidly eroding high-relief catchment in
Taiwan, gathered through a combination of very high resolution, multitemporal satellite imagery, repeat pass topographic surveys, and long-term
discharge records. We estimate through empirical rating curves the changing patterns and distributions of channel hydraulic geometry, boundary
shear stresses, flow speeds, and friction coefficients along the stream and
through time. We find that discharge in our study river is distributed
with a heavy power-law tail characterized by typhoon-driven flood magnitudes that are up to a hundred times greater than average flow. However, the range of boundary shear stress is much smaller, with similar magnitudes (100-300 Pa) driven by intraannual (300 m3 s−1 ) to interdecadal
(3000 m3 s−1 ) floods. Our empirically-derived time series and probability
distributions of boundary shear stress serve to explain many of the morphological features and changes that we observe along this bedrock river.
112
112
1. Introduction
Bedrock rivers set the pace of mountain landscape evolution through the magnitudes
and frequencies of their flows. Morphological changes that occur during discharge events
of different sizes provide clues about how catchments evolve in the long term, but observations and descriptions that link real-time channel changes to quantitative analyses of their
flows along bedrock rivers tend to be limited to extreme events [e.g. Snyder et al., 2003a],
or to individual transects [e.g. Hartshorn et al., 2002], leaving unanswered questions about
the relative importance of flows of different magnitudes and frequencies and about how
their effects vary along stream. Here we describe changes we observed along a river in
Taiwan from 2002 to 2007, and show how these observations combine with measurements
of flow characteristics and their changes through time and along the river to explain some
aspects of the evolution of catchment morphology.
The long-term entrainment of sediment and erosion of bedrock depend on the magnitude frequency distribution of boundary shear stresses of the flow through the channel.
Measurement of boundary shear stress requires estimates of discharge, which are typically
made at gauges, where records of stage are converted to discharge through an empirically
derived and frequently recalibrated stage-to-discharge rating relationship. However, empirical data that relates discharges to their boundary shear stresses are rare. There is
generally no choice but to infer characteristics of shear stress distributions from distributions of discharge without empirical constraints [e.g. Molnar et al., 2006]. Even where
such data do exist, they typically only apply to the gauge location; along-stream variation
in flow characteristics would require large numbers of stage recorders, the deployment of
113
113
which is impractical. In the rare case that station data are available for a mountain river,
variations in flow geometry along stream therefore tend not to be recorded. This limits our ability to understand the degree, nature, and consequences of the spatiotemporal
complexity of shear stresses along bedrock channels and to constrain models of channel
evolution which would benefit from such data [e.g. Stark , 2006; Turowski et al., 2008;
Wobus et al., 2006].
Satellite remote sensing is a possible solution, since it provides a means for extending
hydraulic geometry data beyond a single station to an entire reach and for simultaneously
mapping the resulting morphological changes. To date, remote-gauging studies of have
focused on very large lowland rivers with broad floodplains [e.g. Smith, 1997; Bjerklie
et al., 2003; Brakenridge et al., 2005], applying altimetry data to monitor river stage
[Koblinsky et al., 1993; Birkett, 1998; Birkett et al., 2002], Synthetic Aperture Radar
(SAR) interferometry to monitor stage change [Alsdorf et al., 2000, 2001], SAR to map
flood inundation [Sippel et al., 1994, 1998; Townsend , 2001] or reach-averaged flow widths
[Smith et al., 1995], and Visible and Near Infra-Red (VNIR) imagery to measure flow
widths at channel transects with well-constrained cross-sectional geometries [Xu et al.,
2004; Zhang et al., 2004].
The potential exists to estimate discharge with only data collected from satellites [Bjerklie et al., 2003], which, combined with high-resolution topographic data could provide an
opportunity to explore the spatiotemporal patterns of shear stresses throughout the entire
wetted channel network. However, remote gauging efforts so far have also included data
from the ground, such as stage recorder discharge estimates for validation or calibration
of the remotely gauged discharge [e.g. Xu et al., 2004].
114
114
Application of remote gauging to small mountain rivers, as opposed to the broad alluvial ones that have so far been gauged remotely, is a different challenge mostly due to
smaller spatial scales and greater relief: wetted channels 10 m-100 m wide require data
with high spatial resolution to detect variations with flow, limiting us (at present) to the
use of optical data from only a few commercial imaging satellites; accurate image-series
coregistration requires imagery at or near nadir, 10 m resolution or better digital elevation
data, and very good ground control, which is sometimes impossible in narrow valleys with
limited Global Positioning System (GPS) reception; orographic clouds frequently obscure
flood-stage flows making it unlikely to capture the extreme events in any imagery, since
cloud cover and flood stage tend to be roughly correlated; access to the channel for ground
validation of flow geometries is logistically and technically more complicated; significant
changes in alluvial bed geometry occur frequently, frustrating width-based rating; flows
are typically shallow, rough, and sinuous, complicating any modeling component to the
gauging scheme.
We show here that remote gauging of mountain river discharge is possible with the right
data, which includes very high resolution, multitemporal satellite imagery, good ground
control for image coregistration, and a well-located river stage recorder that provides a
long time-series of discharge estimates. These criteria are met in many mountain catchments in Taiwan, a compelling location for the spatiotemporal assessment of channel flows
and boundary shear stresses because of its exceptional rates of bedrock channel incision
and landscape evolution [e.g. Li , 1976; Hartshorn et al., 2002; Dadson et al., 2003; Turowski et al., 2008]. We also show how remote gauging combines with topographic surveys
to provide estimates of boundary shear stresses and their magnitude frequency distri-
115
115
butions, something that none of the prior remote gauging studies have attempted, and
we demonstrate how our method of remote gauging can provide information to explain
observed channel changes and to inform studies of bedrock channel dynamics.
2. Study area
We have been monitoring the Zhuókŏuxı̄ (the name is often rendered as the Jhuokou
or Jukou Shi and means “Muddy Mouth River”) since 2002. It is a small mountain river
located on the south-western flank of the Central Mountain Range of Taiwan (Figure 1e)
at a steep range front created by the oblique-thrust Cháozhōu (Chauchou) fault. This
river drains a 375 km2 catchment along the Màolı́n Valley through a designated scenic area
with limited development along the channel. There are no dams in the catchment, but
there are some stone or concrete embankments that reinforce sections of the channel walls
(although most of these failed during recent floods). A modest tourist industry supports
three villages in the valley with sufficient infrastructure to provide very easy access to
many points along the channel; during periods of low flow, it is possible to walk along
and across at least the lower 15 km of the Zhuókŏuxı̄. This accessibility also encourages
open-pit mining of fresh gravel deposits after flood events.
Our study reach extends 14 km upstream from the mouth at Dàjı̄n where the Zhuókŏuxı̄
joins the Gāopı́ng (Kaoping) River. This steep (Figure 1c), and sinuous bedrock [sensu
Turowski et al., 2008] river cuts rapidly [Dadson et al., 2003] through an intermittent
veneer of mostly quartzite cobbles, boulders, and argillaceous sand, into argillite bedrock.
Abundant cutoff meander loops at a range of elevations along the valley, as well as alternating valley asymmetry along stream with preserved slip-off slopes on the inside of each
bend and active landsliding on the cutbank side (red polygons in Figure 1e) all indicate
116
116
WANS24
SHTO20
WANS20
WANS21
WANS29
LOTS04
WANS28
WANS23
(a)
~25m
150
1.5
Slope [%]
WANS03
SHTO20
WANS28
WANS29
WANS27
WANS23
LUMS02
(b)
XINS03
100
LUMS01
(c)
WANS07
LUMS10
LUMS02
1
(e)
0.5
14
12 10 8
6
4
2
Downstream Distance [km]
120°E
XINN21
XINN20
LUMS11
XINN22
0
XINS03
XINS02
122°E
(d)
XINS01
DAJN20
0
22°N
24°N
WANS10
WANS27
WANS22
WANS04
WANS06
WANS26
200
DAJN20
Elevation [m]
WANS25
250
0.5
1 Km
Figure 1: (e) A shaded relief image of the Màolı́n Valley generated from a 5 m digital elevation model showing
the locations of 28 surveyed channel cross-sections used as flow width measurement sites and as HEC-RAS
model input. Red polygons show new landslides mapped in a F2 image acquired 2006/03/17 that were not
present on December 31, 2003. (a) Photograph from camera-equipped remote controlled helicopter of landslides
at Shétóushān (location marked in (e) by curved red arrow). (b) Zhuókŏu River bed elevation and (c) downstream
gradient (smoothed). (d) Location in SW Taiwan.
117
117
that the river is actively meandering [e.g. Winslow , 1893; Rich, 1914; Tarr , 1924; Mahard ,
1942].
Discharge Q measured at Dàjı̄n by the Taiwan Water Resources Agency (WRA) rises
during the onset of the Méiyŭ rainy season in spring, and peaks during typhoons in
the summer and autumn. Twice in the past few years instantaneous Q has exceeded
3000 m3 s−1 , although extreme flood discharges usually persist for no more than a few
days with peaks that last only hours before dropping back to typical wet-season levels
below ∼100 m3 s−1 . Daily averages during the most extreme events therefore tend to
reach about 1000 to 2000 m3 s−1 . During the dry season from late fall through winter,
discharges drop close to zero.
The scale of the Màolı́n Valley (on the order of 100 m across) and its relief (up to
1 km from channel to ridge), and the rates of channel evolution (incision is approximately
5 mm yr− 1, and lateral migration is evident through field observations even on annual
to decadal timescales) make this reach an ideal choice for testing the feasibility limits of
remote-gauging, and for demonstrating how the use of remote gauging for monintoring a
mountain river can provide new insight into bedrock channel dynamics.
3. Observations of channel change
We have observed morphological changes along the Zhuókŏuxı̄ for the past six years
through repeat observations in the field and in high-resolution satellite imagery.
Significant changes resulted from two tropical cyclones that made landfall over this part
of Taiwan during the summers of 2004 and 2005: discharge reached 3163 m3 s−1 on July 4,
2004 during Typhoon Mı̆ndūlı̀ (Mindulle), the highest ever recorded on the Zhuókŏu River,
deep and powerful enough destroy almost all of the bridges that cross our study reach; a
118
118
year later on July 20, 2005, discharge during Typhoon Hăitáng reached 2894 m3 s−1 . The
Water Resources Agency (WRA) of Taiwan considers both of these discharge events to
match the 50 year flood magnitude.
For example, Figures 2 and 3 show subsets of a multi-spectral SPOT image (at 2.5 m
spatial resolution) acquired on December 31, 2003 alongside a pan-sharpened (8 m in
multispectral and 2 m in panchromatic) FORMOSAT-2 (F2) image acquired on March 17,
2006. Several differences are evident: an abandoned meander loop was reoccupied by the
deep flow and then abandoned again during the recession at Lóngtóushān (Figure 2a,b);
an outer and recently unused branch of the Shétóushān loop was re-established as the main
channel after an inner branch (itself incised at least a meter into bedrock) had contained
the flow for at least several years (Figure 2c,d); the inner channel and the vegetated hill
between these branches were replaced with a relatively thin and flat gravel and boulder
point bar; downstream at Upper (Figure 2a,b) and Lower (Figure 2c,d) Wànshān, the
deep flow eroded point bars and found a less sinuous path along a series of shoots that
are only wet during these extreme events; and in each of the F2 images of Figures 2 and
3, new landslides appear along the channel. These are also mapped and shown in red in
Figure 1e.
For the 3 years of our study after Hăitáng, discharge at the gauge remained below
1000 m3 s−1 . In 2006, it reached flood stage three times, peaking at 883 m3 s−1 on June
10, 673 m3 s−1 on July 14, 2006 and 385 m3 s−1 on July 25. These flows were sufficient to
erode into an embankment of 0.5-2 m boulders beneath the bridge at Lower Wànshān.
The embankment recession is clearly visible in precisely coregisterred high-resolution
FORMOSAT-2 images acquired on March 17 and September 28, 2006 (Figure 4a,b). Low-
119
119
(a)
(b)
Longtoushan
Longtoushan
SHTO20
SHTO20
(d)
(c)
SHTO20
SHTO20
Shetoushan
Shetoushan
WANS25
0
0.25
0.5
WANS25
0.75
1 Km
Figure 2: Channel changes along the Zhuókŏuxı̄. Right: SPOT image (December 30, 2003) acquired before two
extreme events. Left: F2 image (March 17, 2006) acquired after. Images are of Lóngtóushān (a,b) and Shétóushān
(c,d). The black lines mark the water’s edge mapped in the field with a DGPS during the week of March 17, 2006.
120
120
(a)
(b)
WANS25
Upper
Wanshan
WANS10
WANS10
Lower Wanshan
(c)
Lower Wanshan
(d)
WANS07
WANS07
WANS10
0
Figure 3:
0.25
WANS25
Upper
Wanshan
0.5
0.75
WANS10
1 Km
Channel changes along the Zhuókŏuxı̄. Right: SPOT image (December 30, 2003) acquired before
two extreme events. Left: F2 image (March 17, 2006) acquired after. Images are of Upper Wànshān (a,b) and
Lower Wànshān (c,d). The black lines mark the water’s edge mapped in the field with a DGPS during the week
of March 17, 2006.
121
121
March 17, 2006
September 28, 2006
(a)
(b)
N
N
0
50
100
200 m
March 17, 2006
July 27, 2007
(c)
(d)
N
15 m
Figure 4:
Channel comparison in F2 images from March 17, 2006 (a) and September 28, 2006 (b), and in
low-level aerial photographs taken with a camera mounted on a remote controlled helicopter on from March 17,
2006 (c) and July 27, 2007 (d). The footbridge near the bottom of (c) and (d) is marked by a diagonal green line
near the top of (a) and (b). Transect WANS03 is approximately 200 m upstream from the bridge, in the upper
half of the aerial photographs.
122
122
level aerial photographs taken March 17, 2006 and July 25, 2007 reveal the redistribution
of sediment, with well-sorted boulders confined to the embankment in the first image but
spread across the gravel bar in the second. That is, boulders at least 50 cm in size were
transported by discharges less that 1000 m3 s−1 . It is also clear in the aerial photographs
that these intraannual floods shifted the thalweg and modified it from a very straight to
a slightly curved planform through the thin alluvial cover.
These short term changes in channel morphology were all accomplished during interdecadal and intraannual floods. To relate them to the long term evolution of the channel,
we must understand both the physical properties of these floods and their relative magnitude and frequency of these within a long term discharge time series. This requires data
on the hydraulic geometry of the range of discharges that flow through the channel, data
we obtained through a combination of remote sensing, field surveys and discharge records.
4. Data
4.1. Optical satellite imagery
Since 2004, the Taiwanese FORMOSAT-2 (F2) satellite has collected 2 m-resolution
panchromatic images and 8 m-resolution multispectral images at approximately 10 am
daily over Taiwan along a swath 24 km wide. The return period for any given region of
interest is irregular because of scheduling priorities; image acquisition is often off-nadir, affecting both the field of view and the amount of image distortion, and makes many images,
particularly those in high relief areas, unsuitable for orthorectification and coregistration.
Substantial cloud cover is common, especially during high-magnitude discharges, since
these tend to occur contemporaneously with the storms that drive them. Nevertheless,
many near cloud-free images of the Zhuókŏuxı̄ with good satellite geometry are available.
123
123
Most of these are acquired during the dry season when the river is at low stage and the sky
is clear, but on a few rare passes, F2 has captured the river at higher stage. Because we
focus here on the way that the physical properties of different flows contribute to channel
evolution, we have strategically selected a series of images that capture the entire range of
discharges in the channel from the dry season base flow to the largest storm floods (Table 1
55
and Figure ??).
If our priority was instead to map the changes to channel morphology,
we would have selected instead a time series of images that capture the channel at low
stage between significant events.
We were able to take advantage of the F2 programming schedule to acquire an image of
the Zhuókŏuxı̄ at our request during fieldwork on March 17, 2006. Prior to the overpass,
we deployed a set of satellite-visible, precisely located targets on the river bed, each
consisting of two white reflective plastic sheets, 2 m by 6 m in size and oriented with the
cardinal directions in a cross on a dry part of the channel bed. We also mapped the edges
of the wetted channel along our entire study reach the week of March 17, 2006 with a
handheld differentially corrected GPS (DGPS). These points and vectors collected at the
time of image acquisition assure the best possible ground control for this image, which we
used as the reference for image series coregistration (Methods).
4.2. Topography
Precise coregistration of a high-resolution image series requires a high-resolution DEM
and ground control points for the orthorectification of each image. We used a 5m DEM of
our study area and a collection of ground control points that we acquired in the field with
a DGPS and in a set of large scale orthophotographs. Ground control points for all images
included the corners of buildings, bridge towers, large boulders, lone trees, street corners,
124
124
Table 1: FORMOSAT-2 image set
∗
Date
†
Q
Q%
‡
Cloud cover
Use
2006/3/17
5
34
10%
low-stage, reference
2006/9/28
41
82
0%
mean-stage
2006/7/29
126 93.9
0%
mid-stage
2006/7/11
200 96.3
0%
mid-stage
2005/8/07
801 99.5
30%
high-stage
2005/7/10
14
n/a
0%
pre-Hăitáng§
2005/8/01
1
n/a
0%
post-Hăitáng§
2003/12/30∗∗ 16
n/a
0%
pre-Mı̆ndūlı̀§
∗
Images acquired between 9:55 am and 10:05 am local time.
Hourly discharge in m3 s−1 was measured at Dàjı̄n in 2005 and at Xı̄nnóng in 2006.
Reported here is the discharge measured at 10 am, scaled by relative drainage area to the
Dàjı̄n gauge.
‡
Q percentile.
§
Discharge during Hăitáng reached 2894 m3 s−1 on 2005/07/20. The maximum discharge on record for this catchment, 3163 m3 s−1 occurred one year earlier, during Typhoon
Mı̆ndūlı̀.
∗∗
SPOT-5 image.
†
125
125
126
126
04
05
F2
801 m3s-1
2005/08/07
F2
5 m3s-1
2006/03/17
06
F2
126 m3s-1
2006/07/29
Figure 5. SPOT-5 and FORMOSAT-2 images of the Zhuókŏu River at a range of stages and the WRA
hydrograph of daily average discharge from 2004, 2005, and 2006. The dotted black line is the channel
centerline mapped in the field with a handheld GPS in March 2006 and July 2007. Colored vertical lines
in the hydrograph show the date of acquisition of satellite imagery.
SPOT-5
16 m3s-1
2003/12/30
F2
41 m3s-1
2006/09/28
and other objects clearly visible in the F2 images. The reference image from March 17,
2006 had additional ground control from the points and vectors that we collected in the
field during the image acquisition.
Estimation of boundary shear stress τb requires knowledge of the topography across and
along the channel, so we surveyed cross sections at 28 transects (Figure 1) in the field
in March of 2006 and July of 2007. Since these transects would be used with remote
measurements of wetted channel widths, their locations were selected stategically with
consideration of their geometry: we tended to avoided narrow gorges where widths vary
only slightly with discharge, and selected instead broad parts of the channel where at least
onebank dips gently, maximizing the variation of width with discharge. However, because
we sought to understand the along stream variation of flow properties, we also included a
few transects with steep banks on both sides.
We used a DGPS to map the channel edges in the field during the week of March 17,
2007. To reduce noise in the GPS measurements, we snapped both channel edges to
their centerline and converted them to evenly spaced coordinates of along-channel length
and elevation using a 300 m median filter. We derived the channel gradient from the
longitudinal profile by finding the slope of least-squares linear regressions along a 300 m
moving window. The field-mapped channel edges also serve as additional ground control
for the March 17, 2006 reference F2 image.
4.3. Discharge
The Water Resources Agency of Taiwan has maintained an automatic stage recorder on
the Zhuókŏuxı̄ since 1971, first at LUMS02 (1971–1983), then at DAJN20 (1989–2005),
and now at XINS03 (2006–present) (Figure 1). Stage is measured hourly and yields the
127
127
daily statistics: mean Qμ = 39.5 m3 s−1 , standard deviation Qσ = 130 m3 s−1 , coefficient
of variation Qcv = 3.31, 99th percentile Q99 = 567 m3 s−1 , and peak Qmax > 3000 m3 s−1 .
Since we have observed that even intraannual floods can move the intermittent bed
cover (Figure 4), the channel geometry may change significantly between annual surveys,
and changes in sediment cover at the gauge site can significantly affect the conversion
of water level or flow width measurements to discharge estimates. Maintenance of the
gauge therefore includes an annual survey of the channel cross section at the gauge and
approximately bi-weekly visits to the gauge location to measure flow velocities, depths,
and cross sectional areas used to construct and recalibration the rating curve.
The 36-year record of daily mean discharge along the Zhuókŏuxı̄ has a probability
distribution that is approximated in its tail by p(Q) ∼ Q−α−1 ∼ Q−2 , i.e., it has a Pareto
exponent of α ≈ 1 (Figure 6). Time series have probability distributions with this kind of
heavy tail when there is a significant number of events with extremely higher than average
magnitude [Turcotte and Green, 1993; Stark and Hovius, 2001; Lague et al., 2005]. On
the Zhuókŏuxı̄ these extreme floods are driven primarily by typhoons, which tend to make
landfall in this part of Taiwan at least a once or twice each year and can boost discharge
up to two orders of magnitude above average.
It is important to emphasize that the long-term record of discharge contains daily mean
values, but that the satellite imagery captures the instantaneous flow conditions. We
obtained hourly data from the WRA for the years covered by our image time series (20042006) and used the 10:00 am measurements for comparisons of discharge and aspects of the
hydraulic geometry. We also used the short time series of hourly data to evaluate the effect
of using the long daily time series for the construction and analysis of PDFs. PDFs of the
128
128
í
10
Probability density p(Q) [mí s]
-D – 1 § -2
í
10
í
10
í
10
í
10
í
10
í
10
Qhí
Qdí
Qdí
1
2
10
10
3
10
4
10
3 í
Q [m s ]
Figure 6: PDFs of hourly (Qh ) and daily average (Qd ) discharge on the Zhuókŏuxı̄ from the WRA archive along
with schematic power law tangent to the tail portion of the distribution (gray dashed line).
129
129
hourly data and their daily averages are shown in Figure 6; their very close correspondence
indicates that only minor, if any, error will result from combining instantaneous measures
of flow characteristics with the long term record of daily, rather than hourly, discharge.
5. Methods
5.1. Image series coregistration
Wetted channel widths vary with discharge as flow fills a channel with sloping banks
to different depths. However, in a series of imagery, wetted channel widths can also vary
between images due to distortion caused by different view angles or satellite positions
during acquisition. It is therefore necessary to precisely coregister the image series. We
accomplished this in two steps. First, we used the 5 m DEM and ground control points,
along with camera model information for the F2 sensors to orthorectify each image. Then,
we coregistered each orthorectified image to the March 17, 2006 reference image with
a rubber-sheet warping procedure that itself involved the identification of hundreds of
matching points in each subject-reference pair of images. For our purposes, precision of
coregistration of the image series is most important in the valley bottom which contains
the wetted channel. Poor registration in the ridges does not affect our measurements.
Therefore, we focused our selection of matching points in subject and reference image
pairs on features along the channel. The subject image was then stretched to match
precisely the reference image at these points, assuring that differences in wetted channel
width relate to the flow conditions in the channel and not the orbital conditions of the
satellite.
130
130
5.2. Measurement of wetted channel width and construction of widthdischarge rating curves
Remote gauging depends on the construction of a rating curve from measurements of
some aspect of hydraulic geometry, such as water level, channel width at transects [Xu
et al., 2004, e.g.], reach-averaged effective width [Smith et al., 1995, e.g.], or inundation
area, at a range of known discharges. Once a single rating curve for a transect or reach,
or a set of curves for several transects or reaches, is established from known discharges,
discharge can be estimated for hydraulic geometry measured in additional remotely sensed
datasets acquired at unknown discharge.
Our F2 image series captures the full range of intraannual flows on the Zhuókŏuxı̄, from
5 to more than 800 m3 s−1 . Although images were not acquired during the extreme floods
of Mı̆ndūlı̀ and Hăitáng, we were able to also measure the extent of these interdecadal
discharges, since together, they left a high water mark along much of the channel (Figure 8)
which we mapped through a comparison of a SPOT-5 image acquired before Mı̆ndūlı̀ and
F2 images acquired after each storm (Table 1). We validated this mapping at each transect
in the field with observations and surveys of the high water mark.
We used a variety of processing techniques to highlight the edges of the wetted channel,
since the clarity of this boundary depends on shadows, haze, illumination angle, channel
orientation, and bed material: in some cases, it is sharply defined in the panchromatic
image; in others, it is clearer in a pan-sharpened version of the multispectral image; in yet
other cases, it is most clear after applying a Sobel edge filter to the panchromatic image.
We measured the wetted channel width of the Zhuókŏuxı̄ at the transects shown in
Figure 1 in each F2 image. Since the satellite passes Taiwan at approximately 10 am
131
131
a
WANS26
WANS25
WANS24
WANS23
WANS20
WANS27
b
c
0
Figure 7:
50 100 m
FORMOSAT-2 image zooms of the Zhuókŏu River at Upper Wànshān at low, medium, and high
stages from (a) March 17, 2006, at Q = 5 m3 s−1 ; (b) July 11, 2006, at Q = 126 m3 s−1 ; (c) August 7, 2005, at
Q = 801 m3 s−1 . The colored polygons indicate the mapped inundation; the black-trimmed white lines are the
locations of channel transects.
132
132
5m
Figure 8: Photograph of the Mı̆ndūlı̀ and Hăitáng high water mark near WANS21.
133
133
each day, we matched each F2 image to the 10 am WRA discharge measurement fron its
image date. The collection of width measurements along with the associated discharges
22
comprise a unique width-discharge rating curve for each transect (Table ??).
5.3. Measurement of shear stress and flow speed
We combined the measurements of hydraulic geometry with a normal flow approximation for open channel flow to estimate the mean boundary shear stress τb for the discharges
imaged in the F2 imagery. We computed τb as
τb = ρg
Aw
S = ρgRS ,
P
(1)
where R is the hydraulic radius (the ratio of cross-sectional area Aw to wetted perimeter
P ), ρ is water density, g is gravitational acceleration, S is the channel bed slope which
is assumed to match the water surface slope [e.g. Turowski et al., 2008]. Equation 1 only
applies because we chose transect locations where the channel has a relatively uniform
gradient and is relatively straight.
Mean flow speed is defined as the discharge per unit cross-sectional area A,
U = Q/A .
(2)
Because τb and U depend on R and A respectively, their computation for a given flow at
a given transect requires knowledge of the hydraulic geometry.
Equation 1 indicates that shear stress τb ∼ R. Since the relationship between R and
Q can vary significantly downstream with changes in channel topography, friction, and
drainage area, the scaling of Q and τb , and therefore also the probility density of shear
stress p(τb ), varies along the river and must be computed empirically at each transect.
We accomplished this by using the hydraulic radius R that comes from the intersection of
134
134
135
135
Area
km2
343.5
344.6
346.8
346.8
346.8
346.9
347.0
347.0
347.1
347.1
353.0
353.0
353.3
353.3
355.2
355.2
356.1
358.3
358.5
364.8
365.5
365.6
365.7
365.8
365.9
366.4
372.1
375.1
ID
LOTS04
SHTO20
WANS23
WANS24
WANS25
WANS26
WANS20
WANS27
WANS21
WANS29
WANS28
WANS10
WANS06
WANS22
WANS04
WANS03
WANS07
LUMS02
LUMS01
LUMS10
LUMS11
XINN22
XINN21
XINN20
XINS02
XINS03
XINS01
DAJN20
1.05
0.63
1.07
1.19
1.26
1.08
0.66
0.60
0.47
0.42
0.66
0.48
0.71
0.65
0.76
0.66
0.77
0.89
0.80
0.50
0.61
0.46
0.48
0.61
0.66
0.52
0.66
0.27
Slope
%
519
521
524
524
524
524
525
525
525
525
534
534
534
534
537
537
538
542
542
551
552
553
553
553
553
554
562
567
Q99
m3 s−1
0.3
0.6
1.1
0.8
0.3
0.5
0.8
1.4
1.0
0.8
1.1
0.7
1.0
1.5
0.5
0.5
0.5
0.7
0.5
0.3
0.5
0.4
0.7
0.3
0.4
0.9
31
11
15
25
1.9
1.4
0.4
0.8
2.4
1.6
0.6
0.3
0.7
0.7
0.5
0.6
0.8
0.3
1.4
1.1
1.3
1.4
1.3
1.8
1.1
1.3
0.4
3.3
2.4
0.8
21
9
7
16
15
19
55
47
21
29
26
39
22
17
36
11
19
45
25
18
22
28
14
10
14
10
5 m3 s−1
H
U τb
18
10
24
24
12
11
19
22
13
16
17
33
21
23
10
15
13
10
21
13
19
17
w
4
1
2
5
2
3
10
8
4
5
5
7
4
3
6
3
6
8
3
3
3
5
3
1
2
2
D
0.5
0.9
1.0
1.0
45
20
30
31
0.5
1.9
1.2
1.2
2.5
1.8
1.7
1.6
1.5
1.2
1.5
0.8
1.1
2.6
1.6
1.5
1.0
1.3
0.9
1.0
4.3
1.2
2.3
1.7
0.4
0.9
1.2
1.8
2.1
2.9
2.4
4.5
5.0
0.7
1.4
1.1
2.3
3.5
2.1
2.4
2.0
7.7
5.2
4.6
13
17
7
20
32
36
52
62
134
81
58
46
28
23
46
10
21
79
44
42
31
40
28
22
41 m3 s−1
H
U
τb
28
54
33
42
99
55
34
26
31
24
23
39
26
44
45
49
43
24
53
35
w
2
5
2
4
5
10
11
13
27
17
10
8
6
4
8
3
6
15
11
9
6
10
6
4
D
36
55
81
75
63
40
92
65
73
104
95
47
58
61
59
67
67
57
72
89
75
71
62
80
72
55
67
w
0.8
0.9
1.0
1.5
2.0
1.8
3.8
1.7
1.7
2.6
2.8
2.3
3.4
2.5
2.7
3.6
2.1
2.6
4.2
4.6
1.7
1.6
1.6
1.2
2.4
1.4
1.1
2.7
0.8
3.7
3.0
1.3
1.3
2.6
1.6
2.4
2.0
1.5
1.9
2.0
1.1
0.6
3.5
2.9
5.9
3.7
1.7
4.4
6.6
10.7
5.9
5.2
4.0
2.6
25
31
19
35
25
138
123
65
79
140
133
80
95
47
53
96
54
91
127
209
39
57
36
41
61
38
17
163 m3 s−1
H
U
τb
4
5
4
9
5
17
21
16
17
29
27
13
18
10
10
19
9
16
24
30
10
10
13
9
10
7
4
D
150
87
126
91
110
183
150
69
153
66
97
90
w
5.7 1.9
2.4 7.7
2.6 5.6
6.3 1.0
10.3 0.6
5.6 2.9
8.8 1.1
3.9 4.2
4.5 3.3
5.5 2.8
2.7 4.2
3.0 6.0
81
40
188
104
135
480
782
236
249
92
94
191
801 m3 s−1
H
U
τb
16
7
31
22
26
69
97
32
45
16
17
31
D
5.5
6.9
5.7
4.4
3.8
8.9
4.1
4.8
8.3
9.5
9.6
7.1
6.1
4.7
6.9
152
124
117
149
121
235
163
91
114
163
218
6.1
4.6
5.6
8.0
12.1
9.9
9.3
8.9
7.9
7.3
6.3
8.9
5.4
2.7
2.0
3.5
3.8
3.6
4.4
6.3
10.3
7.0
7.8
10.2
10
3.6
10.6
7.1
3.6
1.8
2.4
6.0
5.5
5.9
4.1
153
536
112
165
286
324
434
328
231
198
87
243
231
253
179
193
322
448
643
925
315
264
226
189
261
3000 m3 s−1
H
U
τb
74
111
103
97
139
96
129
195
155
157
161
148
134
109
w
29
62
18
25
33
39
51
41
27
27
16
34
39
38
25
33
43
58
87
113
56
48
36
29
41
D
Table 2. Flow widths w [m], maximum depths H [m], average flow speeds U [ms− 1], boundary shear stresses τb [Pa], and particle diameter D (for τ ∗ = 0.07) [cm] at channel
transects. Note that reported here for H and D are the maximum flow depths Hmax and the diameter of the largest mobile particle Dmax
measured flow widths and surveyed channel transects to obtain estimates of τb for the set
of imaged and mapped flows at each transect. We compared Q and τb of the measured
flows at transect to find their empirical scaling. This scaling can be applied to the to
the time series of Q to infer the time series of τb , which can be evaluated to find the τb
probability distribution.
In general the scaling of Q and τb tends to approximately follow a power-law model
τb ∼ Q1/β . Therefore, an alternative approach for finding the τb PDF is to apply this
power-law model to the discharge PDF to transform it directly to a τb PDF.
For example, at WANS28, which is approximately trapezoidal with relatively steep bank
angles, we find (Fig. 18
??)(a)
1
R ∼ τb ∼ Q 3
⇔
2
U ∼ Q5
⇔
5
τb ∼ U 6 .
(3)
The power-law tail of the discharge PDF steepens under the transformation from p(Q) ∼
Q−α−1 with α ≈ 1 to
p(τb ) ∼ τb −4 ∼ τb −γ−1
where γ = αβ = 3 .
(4)
At WANS29, located 650 m upstream, most flows are confined between gently dipping
banks; only the highest discharges are bounded by steep bedrock walls. Here we find
1
R ∼ τb ∼ Q 2
⇔
1
U ∼ Q5
⇔
5
τb ∼ U 2 .
(5)
Since β = 2, the τb tail decay here is somewhat heavier,
p(τb ) ∼ τb −3 ∼ τb −γ−1
where γ = αβ = 2 .
(6)
Many transects have more complicated geometry than WANS28 and WANS29, making
the regression of a power law a poor choice to approximate the scaling of Q and τb . In these
136
136
cases, a piecewise interpolation of the Q and τb relationship is more appropriate. However,
despite differences along the river in cross-sectional geometry and friction, which combine
to affect the scaling relationship of Q and τb , and despite being driven by discharge
with a heavy-tailed probability distribution, the τb probability distributions for all of
our transects decay sharply in the higher magnitudes. Power-law distributions with this
kind of semi-heavy tail decay are barely distinguishable from exponential distributions.
This reflects a fundamental difference between the magnitude frequency distribution of
discharge and that of shear stress: the discharge time series has interdecadal floods that are
two orders of magnitude greater than average flow conditions and one order of magnitude
greater than the intraannual floods; the shear stress of extreme floods, however, is similar
in scale to the shear stress of yearly floods and only one order of magnitude greater than
average flow conditions.
5.4. Incipient motion of bed sediment
Erosion of the channel bed requires transport of the overlying bedcover, so it is important to establish what flows are required to move the boulders lining the channel
bed. To assess the size of boulders that flows can move, we use the Shields number τ ∗
[Shields, 1936], a ratio of the boundary shear stress of the moving fluid to the weight of
a cohesionless particle submerged in the flow. It is defined by
τ∗ =
τb
(ρs − ρ)gD
(7)
where ρs is the density of the sediment particle, ρ is the denisty of water, and D is the
particle diameter. Particles are suspended in the flow when τ ∗ = 1, but there is an
empirically determined critical level τ ∗ << 1 which marks incipient motion. Particles in
137
137
gravel-bedded rivers typically moving when 0.03 < τ ∗ < 0.07, although isolated particles
have been observed to move at τ ∗ = 0.01, and imbricated clusters may require τ ∗ > 0.2
[e.g. Fenton and Abbott, 1977; Buffington and Montgomery, 1997; Church and Hassan,
2002].
Substituting for τb in Equation 1 yields
τ∗ =
ρgRS
,
(ρs − ρ)gD
(8)
recalling that we have defined R as hydraulic radius and S as slope. Note that the symbol
R is sometimes used in Shields stress equations for the submerged specific gravity. Instead
we use ρ∗ , which is defined as
ρ∗ =
ρs − ρ
.
ρ
(9)
Simplifying Equation 8 and combining with Equation 9 gives
τ∗ =
RS
.
ρ∗ D
(10)
We rearrange Equation 10 so that it is solved for particle diameter D, and use the
maximum flow depth Hmax instead of hydraulic radius R to find the maximum grain size
Dmax that can be moved by a particular flow,
Dmax =
Hmax S
.
ρ∗ τ ∗
(11)
Using a conservative τ ∗ = 0.07 for incipient particle motion, an assumption of ρ∗ ≈ 1.65,
our GPS-derived S, and an H that comes from the intersection of our F2-measured widths
and surveyed channel cross sections, we used Equation 11 to find a conservative estimate
22
for the maximum partical size Dmax that each measured flow can move (Table ??).
Much
larger particles may move if they are isolated and protruding above their surroundings,
since the shields number for these could be as low as 0.01 [Fenton and Abbott, 1977].
138
138
Dmax, 3000= 33 cm
5
Dmax, 99 = 31 cm
(a)
0
50
100
Distance from thalweg [m]
0
Max Depth [m]
Elevation [m]
10
150
Q [m3 sí]
0
10
(b)
0.1
U
Cf
0
Tb [Pa]
10
Cf>í@
U [m sí]
10
0
(c)
100
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
30
100
Tb exceedance [days/yr]
Tb [Pa]
(f)
100
30
10
Q99
3
1
dry
walls wet walls
0.1
0
50
100
150
Tb [Pa]
Figure 9: Cross section topography with the 3000 m3 s−1 and the 99th percentile flow depths indicated along
with the diameter of the largest particle that each flow can move (a). Comparison of flow speed U and Chezy
friction Cf with discharge Q (b). Comparison of boundary shear stress τb with discharge Q (c). Comparison of
maximum particle diameter Dmax (d). Boundary shear stress probability density with γ ≈ 4 (e). Boundary shear
stress complimentary cumulative distribution (f), all at transect SHTO20.
139
139
6
Dmax, 3000 cm
D,max, 99 = 20 cm
(a)
í
0
60
Distance from thalweg [m]
0
Max Depth [m]
Elevation [m]
80
Q [m3 sí]
0
10
Cf
8
6
(b)
U
0
300
Tb [Pa]
10
Cf>í@
U [m sí]
10
0
100
(c)
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
Tb [Pa]
100
300
(f)
100
30
10
Q99
3
1
0.1 dry walls wet walls
0
100
Tb [Pa]
300
Figure 10: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 1.5 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS23.
140
140
Dmax, 3000= 58 cm
Dmax, 99 = 25 cm
(a)
0
50
Distance from thalweg [m]
0
Max Depth [m]
Elevation [m]
100
Q [m3 sí]
0
10
10
Cf
0.1
(b)
U
0
Tb [Pa]
Cf>í@
U [m sí]
10
0
(c)
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
Tb [Pa]
100
300
(f)
100
30
10
Q99
3
1
dry walls wet walls
0.1
0
100
300
Tb [Pa]
Figure 11: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 1.5 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS24.
141
141
0
Dmax, 99 = 47 cm
(a)
í
Max Depth [m]
Elevation [m]
Dmax, 3000 = 87 cm
í
í
0
Distance from thalweg [m]
Q [m3 sí]
0
10
3
10
U [m sí]
1
Cf>í@
1
U
(b)
Cf
0
Tb [Pa]
10
0
(c)
D [cm]
0
(d)
0
0
10
10
10
Probability density p(Tb) [Paí]
Q [m3 sí]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
100
Tb [Pa]
300
(f)
100
30
10
Q99
3
1
dry
0.1 walls wet walls
0
Tb [Pa]
Figure 12: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 2.5 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS25.
142
142
Dmax, 3000 = 113 cm
10
Dmax, 99 = 64cm
(a)
í
0
Max Depth [m]
Elevation [m]
0
100
Distance from thalweg [m]
Q [m3 sí]
0
10
U
1
1
Cf>í@
U [m sí]
Tb [Pa]
10
(b)
Cf
0
1000
0
(c)
0
100
D [cm]
10
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
100
Tb [Pa]
300
(f)
100
30
10
Q99
3
1
dry
0.1 walls wet walls
0
600
Tb [Pa]
800
1000
Figure 13: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 2.25 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS26.
143
143
Dmax, 3000 = 56 cm
10
Dmax, 99 = 23 cm
(a)
í
0
í
0
Distance from thalweg [m]
Max Depth [m]
Elevation [m]
Q [m3 sí]
0
10
(b)
0.1
Cf
U
0
300
Tb [Pa]
10
Cf>í@
U [m sí]
10
0
(c)
100
D [cm]
0
60
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
30
100
300
Tb exceedance [days/yr]
Tb [Pa]
(f)
100
30
10
Q99
3
1
dry walls wet walls
0.1
0
100
Tb [Pa]
300
Figure 14: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 2.5 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS20.
144
144
10
Dmax, 99 = 41 cm
5
(a)
0
50
100
Distance from thalweg [m]
0
Max Depth [m]
Elevation [m]
Dmax, 3000 = 48 cm
150
Q [m3 sí]
U [m sí]
10
10
10
Cf
(b)
U
0
300
Tb [Pa]
Cf>í@
0
0
100
(c)
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
100
Tb [Pa]
300
100
30
10
dry walls wet walls
3
1
Q99
0.1
(f)
0
100
300
Tb [Pa]
Figure 15: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 3 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS27.
145
145
Dmax, 3000= 36 cm
10
Dmax, 99 = 13 cm
(a)
í
0
Max Depth [m]
Elevation [m]
0
Distance from thalweg [m]
Q [m3 sí]
0
10
(b)
0.08
0.06
Cf
U
0
Tb [Pa]
10
Cf>í@
U [m sí]
10
0
(c)
100
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
Tb [Pa]
100
(f)
100
30
10
Q99
3
1
0.1
dry walls wet walls
0
100
Tb [Pa]
Figure 16: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 1.8 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS21.
146
146
10
Dmax, 3000 = 29 cm
Dmax, 99 = 15 cm
(a)
í
0
Max Depth [m]
Elevation [m]
í
0
Distance from thalweg [m]
Q [m3 sí]
U [m sí]
10
10
(b)
Cf
U
Tb [Pa]
0
Cf>í@
0
10
0
(c)
100
D [cm]
0
(d)
30
10
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
Tb [Pa]
100
(f)
100
30
10
Q99
3
1
dry walls wet walls
0.1
0
100
Tb [Pa]
Figure 17: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 2 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS29.
147
147
10
Dmax, 3000 = 41 cm
Dmax, 99 = 29 cm
5
(a)
195
0
Max Depth [m]
Elevation [m]
í
0
50
Distance from thalweg [m]
Q [m3 sí]
0
10
10
10
Cf
0.15
0.1
(b)
U
Tb [Pa]
0
Cf>í@
U [m sí]
6
0.05
0
100
(c)
D [cm]
0
(d)
0
0
10
10
10
3 í
Probability density p(Tb) [Paí]
Q [m s ]
í
10
í
10
í
10
í
10
(e)
Tb exceedance [days/yr]
30
100
Tb [Pa]
300
(f)
100
30
10
Q99
3
1
0.1
dry walls wet walls
0
50
100
150
Tb [Pa]
Figure 18: Cross section topography (a), flow characteristics (b-d), boundary shear stress probability density
with γ ≈ 3 (e), and boundary shear stress complimentary cumulative distribution (f) at transect WANS28.
148
148
6. Results
22 and Figures 9 through 18.
Our results are summarized in Table ??
2 shows the characteristics of each measured flow at each transect. Widths w
Table 2??
are those measured directly in F2 imagery except for the 3000 m3 s−1 floods which was
measured indirectly by mapping the differences between before-and-after imagery and by
surveying high water marks in the field. Maximum depths Hmax for each flow are derived
from the intersection of the measured widths with the surveyed transects. Flow speeds U
for each flow at each transect come from the gauged discharge Q scaled by drainage area
to the transect location and divided by the cross sectional area of the flow. Boundary
shear stresses τb and grain sizes Dmax come from Equations 1 and 11 respectively. The
grain size estimation is for the maximum flow depth and therefore represents the largest
grain sizes that can move in the flow. However,
it is a conservative estimate since we set
2
the Shields parameter τ ∗ equal to 0.07. Since the range of τ ∗ values typically quoted for
incipient motion is from 0.03 to 0.07, the flows may be able to move particles more than
twice the diamater that appears in the table.
In some images, clouds or shadows obscured the view of the channel such that width
measurement was impossible at one or more transects. For such cases, the table entries
remain empty.
Each of Figures 9 through 18 shows flow characteristics at a single transect. Box
(a) of each figure is the surveyed channel cross section shown with five times vertical
exaggeration. The dotted horizontal line represents the water level of the 99th percentile
flow. The solid horizontal line represents the water level of the 3000 m3 s−1 flood. Noted
149
149
on each of these lines is our conservative estimate of the largest boulder diameter Dmax
that can move in each of these flows using Equation 11.
Boxes (b) through (d) of each figure show the variation of flow characteristics with
discharge. Flow speeds U , shear stresses τb , and particle sizes Dmax match the values
22 Chezy friction coefficients Cf are defined as the ratio of τb and the
presented in Table ??.
product of ρ and U2 . Two curves are shown to relate τb and Q in box (c): in grey is a
power law regression and in black is a piecewise cubic hermite interpolating polynomial
(PCHIP), which makes no assumption of any functional behavior.
Box (e) is the probability density function of τb derived from the conversion of the Q
timeseries to a τb timeseries through the empirical relation of Q and τb from Box C. The
grey PDF comes from the power law regression of the Q and τb points. The black PDF
comes from the PCHIP interpolation.
Box (f) in each figure is the complimentary cumulative distribution function, scaled
to show the number of days each year that each value of τb is exceeded, with the grey
distribution from the power law regression and the black from the PCHIP interpolation.
The 99th percentile discharge is identified with a dotted horizontal line. A vertical dashed
line shows the point at which the flow begins to inundate the channel walls.
7. Discussion
7.1. The utility of remote gauging
The accuracy of a width-discharge rating curve, and of any discharge estimates that
come from additional width measurements fit into such a curve, depends on a channel cross section that does not significantly change. However, we have shown that in a
bedrock river with intermittent alluvial cover, the bed sediment may be reorganized by
150
150
intraannual floods. It is best, therefore, to collect rating information from an image series
that spans a short period time, and to apply the resulting rating curve only to additional
imagery from the same period so that the extent of change to the channel cross section is
minimized. This is problem is not unique to remote gauging; traditional gauges in Taiwan are recalibrated and resurveyed at least annually to account for topographic change.
Indeed, remote gauging is advantageous in this regard, since it provides an opportunity
to construct a more robust rating curve by averaging over reaches [e.g. Smith et al., 1995].
By smoothing out changes in alluvial bed morphology, measures of inundation area per
unit distance downstream (or mean inundation width) should remain a reliable indicator of discharge through longer periods of time than wetted channel widths at individual
transects. A practical application of remote gauging of discharge includes the ability to
fill gaps in a hydrograph from a traditional gauge, such as the gaps that appear in the
discharge time series for our study river which occured when the gauge was out of service for several months following Typhoon Mı̆ndūlı̀ (shaded parts of the hydrograph in
55
Figure ??).
The strength of remote gauging for mountain rivers, however, is its ability to provide
empirical data on the spatiotemporal patterns of flow characteristics, such as estimates of
hydraulic geometry, friction, flow speeds, and shear stress. Methods for reach-averaging
of channel hydraulic properties have been developed [e.g. Wiele and Smith, 1996; Griffin
et al., 2005] and could be used with our remote gauging data to collect reach-averaged
boundary shear stresses, flow speeds, and frictions. However, since we are interested in
understanding the variation of these properties, we focused on their values at individual
transects and found, for example, flow speeds that tend to increase with discharge from
151
151
less than 1 m s−1 at low stage to more than 5 m s−1 during interdecadal floods. We also find
Chezy friction coefficients Cf vary with discharge. However unlike flow speeds, friction
coefficients vary much less predictably. We attribute this to the way the bed character
changes along the Zhuókŏuxı̄, with patches of fine grained sediment, exposures of hummocky argillite bedrock, and piles of rounded quartzite boulders. In some parts of the
channel, grain sizes are well-sorted. In other parts, they are mixed. Roughness character
therfore varies both downstream and across each transect such that each flow encounters
different roughnesses along the stream, and different flows feels different roughnesses at
each transect.
7.2. Relative importance of flood magnitude and flood frequency
The semi-heavy decay of the p(τb ) distribution suggests that differences in erosion rates
may be a stronger function of flood frequency than flood magnitude; i.e., the absolute
frequency of erosive events is likely the key rate factor rather than the relative frequency of
extreme events. Snyder et al. [2003b], for example, present an estimate of boundary shear
stress for a ≈ 50 year flood of Q ≈ 335 m3 s−1 in Fall Creek, a river in the Finger Lakes that
has been eroding at approximately 0.14 mm y−1 for the past 1400 years, about 50 times
slower than the similarly sized Zhuókŏuxı̄. With an assumption of Manning’s n = 0.05,
they infer a boundary shear stress of τ ≈ 100 − 200 Pa during the Fall Creek 50 year flood.
The similarly rare floods Zhuókŏuxı̄ have an order of magnitude greater discharge than
Fall Creek; however, they tend to produce a τb only a factor of 2 greater than the Fall
22 which is insufficient to explain the large difference erosion rates
Creek flood (Table ??),
of these two rivers. However, the semi-heavy shear stress PDF tail distributions reflect
the fact that along the Zhuókŏuxı̄, shear stresses similar to those driven by the Fall Creek
152
152
50 year flood are attained at least annually at many transects. That is, a difference in
the frequency of erosively significant floods approximately matches, and may explain, the
difference in erosion rates.
7.3. Width and planform evolution
The ability to measure the boundary shear stress distribution at any transect for which
the cross-sectional geometry is known enables comparisons not just between catchments
22 illustrates this point, since widths, depths, velocbut also along a single reach. Table ??
ities, and shear stresses all vary along stream during each flow as water is stretched and
squeezed through parts of the channel with different geometric and frictional characteristics.
It is of particular interest that the Zhuókŏuxı̄ actively meanders, indicative of a longterm along-stream variation of bank erosion rates. For insight into how this may be
accomplished, we focus on transects SHTO20 through WANS28, which are located around
a series of meander bends with channel geometry that varies significantly around each
point of inflection (Figures 9 through 18). Lateral erosion requires a flow which is at
least deep enough to inundate the channel walls. With this in mind, when surveying
the transect cross sections, we noted the point at which the flow begins to inundate the
channel wall, commonly clearly marked in the topography as a sharp break in slope.
Table 3 summarizes the hydraulic geometry at each transect for the flow which reaches
this point, that is, the flow which marks the initiation of wall inundation.
It is apparent from the discharge percentiles in Table 3 that each transect begins to
experience wall inundation during flows of different magnitudes and frequencies; flows
which do not even touch the walls at one transect may do a significant amount of work
153
153
Table 3: Flow characteristics at the initiation of channel wall inundation. Q, w, H, and τb are associated with
the minimum flow required for wall inundation.
ID
w
H
τb §
Q†
Q% †
[m] [m] [Pa] m3 s−1 [%]
SHTO20
76 2.3
50
76
90.5
WANS23 96 2.2
85
666
99.4
WANS24 125 2.9 152
WANS25 40 1.3
§
90
1665 99.9
7
45.0
WANS26 114 3.3 158
249
97.4
WANS20 60 3.2 110
433
98.6
WANS27 142 8.1 231
539
99.1
WANS21 52 3.1
68
94
92.3
WANS29 81 3.6
73
356
98.2
WANS28 67 3.4
97
150
95.2
τb is computed with equation 1 using the w and H of the cross section that correspond
to the field-surveyed point where wall inundation initiates.
†
Q̃ and Q% come from the rating of width measured in satellite images and Dajin Q
scaled by relative drainage area to each transect.
154
154
on the walls at another. If the frequency of wall inundation correlates with channel
curvature over long timescales, then the longitudinal variation of channel cross section
could contribute to the development of sinuosity. On this reach, such correlation now
exists only around some bends, but it is important to consider that the conditions we
observe only apply to the short timescale over which we conducted our study. We do
not yet have theory or observations to relate these modern longitudinal variations in wall
inundation frequency to the long term pattern. Nonetheless, channel geometry clearly
plays an important role in setting the along stream variation of horizontal erosion at least
in the short term. This observation and our measurments of the range of flows that do
and do not affect the channel walls are important empirical constraints for recent models
of channel erosion that assess the ways that vertical and horizontal erosion contribute to
meandering processes in bedrock rivers [e.g. Stark et al., 2008].
7.4. Effects of intraannual and interdecadal floods
Landslides can occur during a typhoon as heavy rain infiltrates the valley walls, increasing pore fluid pressures and reducing shear resistance to failure. Alternatively, flow
in the channel driven by the storm may be powerful enough to undercut the valley walls,
causing them to oversteepen and fail. Incising rivers that actively meander tend to have
asymmetric valleys, the result of outward migration at each bend which drives slope instability and failure of the outer bank and leaves behind a slip-off slope on the inner bank
[e.g. Mahard , 1942]. The Zhuókŏuxı̄ is a typical example, with cutbanks that are steep
and prone to failure and slip-off slopes that are very gentle, indicating rapid outward migration at the meander bends relative to incision. However, steep slopes also occur on this
river at sinuosity nodes and at the outside of abandoned loops. Infiltration of heavy rain
155
155
sufficient to cause landslides should affect all of these steep slopes equally, irrespective
of where they lie along the channel, provided consistent lithological, soil, and vegetation
conditions; however the landslides of typhoons Mı̆ndūlı̀ and Hăitáng occurred exclusively
at the outer bends of the active channel, precisely where lateral erosion is expected to be
fastest on a meandering river. This indicates that the storm-induced slope failures were
not caused by infiltration of the rain alone, but instead by undercutting of the deep flow.
We interpret this as direct evidence of concentrated lateral erosion and meander growth
during the interdecadal floods.
At Shétóushān, for example, flow-induced landsliding was accomplished with a significant change in the position of the main thread of the channel. Prior to Mı̆ndūlı̀ and
Hăitáng, the main thread of the channel at this bend was in the middle of a 200 m wide
channel (Figure 2c,d). A second thread along the outer wall had been abandoned, and
an elevated bar between this abandoned outer thread and the active inner thread was
fully vegetated, indicating its immunity from intraannual floods. During the interdecadal
typhoon floods, the vegetated bar was eliminated and its sediment redistributed and possibly transported downstream, the inner thread was abandoned and filled with alluvium,
and the outer thread was reoccupied and re-established as the main thread. Only in this
position could the river once again attack the outer wall and produce landslides.
Bedrock was exposed in the main thread prior to the interdecadal typhoon floods and
is visible in the SPOT image (Figure 2c,d). This supports our results which indicate an
ability of intraannual floods to produce shear stresses sufficient to move bed sediment and
attack bedrock. In the time after the two interdecadal typhoon floods, we have continued
to observe shifts in the position of the main thread all along the study reach, which also
156
156
indicates this ability of intraannual discharges to move bed sediment. At Lower Wànshān,
for example, our air photos show gradual changes in channel planform in the absence of
extreme discharge (Figure 4c,d). As this continues, bedrock will be exposed and covered
along different threads at different times, such even without completely clearing the bed
sediment, the river will eventually attack and lower all parts of its bed under intraannual
conditions.
These observations of bed erosion ocurring during normal to intraannual flows and bank
erosion during the interdecadal floods are fully consistent with Hartshorn et al. [2002],
who monitored the erosion of bedrock at a transect of the Lı̀wù river in northeastern
Taiwan. They measured with submillimeter precision erosion of the bed during normal
flow conditions but not during the interdecadal flood associated with Typhoon Bilis in
2000. During that event, only erosion of the walls occurred. Turowski et al. [2008] argued
that alluviation of the bed, which is nearly complete during extreme events, is likely
responsible for this, since the cover of sediment protects the bed but not the banks. Our
results confirm that lower stage flows do indeed drive boundary shear stresses sufficient
to excavate this sediment and accomplish the observed bed erosion, and that interdecadal
floods are both powerful enough and deep enough to cut into the channel banks.
8. Conclusions
Wall erosion is evident in a series of landslides that were inuduced at the outer meander
bends during interdecadal scale floods, and vertical incision is likely to occur as the main
flow thread migrates across the valley exposing bedrock during moderate to intraannual
flows. The variation in lateral erosion rates along stream is also a function of channel
morphology, which influences the depth of flow required to wet the banks and therefore
157
157
also the frequency with which the banks are wet, the depth to which they are underwater,
and the resulting shear stresses acting on the wall. Thus our first conclusion is that the
relative rates of horizontal and vertical erosion depends on the frequency of extreme floods
and the channel geometry.
The second conclusion is that remote gauging is possible in small upland catchments
through a combination of high resolution VNIR satellite imagery, topographic data, stage
records, and limited field mapping, and it provides empirical data on the spatiotemporal
variation of flow characteristics. The heavy-tailed distribution of discharge, which comes
from the huge variability of flows from nearly zero during the dry season to deep floods
during my monsoon and typhoon storms, produces a boundary shear stress PDF with
only a semi-heavy tail; although intraannual floods have significanly smaller discharges,
they have shear stresses that are similar to those driven by extreme floods. The boundary
shear stresses of intraannual floods are even sufficient to move much of the bed sediment
and likely also to erode bedrock. Flow speeds U vary with discharge and exceed 5 m s−1
during the extreme floods at many transects of our study river in Taiwan, and they vary
along the river during any given flow due to changes in slope and cross sectional geometry.
Chezy friction coefficients Cf also vary with discharge, but irregularly due to the different
roughnesses felt by flows of different depths over the thin and intermittent alluvial cover
with variable sorting along the channel and through time.
In general these findings indicate that Taiwan’s unusually rapid rates of erosion and
disproportionate contribution of sediment to the ocean result from processes that are
unusual in their frequency, with the magnitude of intraannual floods in a Taiwanese river
matching that of interdecadal floods in a slowly eroding but similarly sized river in the
158
158
northeastern United States. However, the processes in Taiwan are not unusual in their
magnitudes; the interdecadal discharges in Taiwan have boundary shear stresses that are
not much greater than the intraannual floods.
This study helps to address the paucity of empirical data on bedrock channels by providing such data for a rapidly evolving river in Taiwan, and by describing a methodology
for its acquisition. Future application of our methodology to rivers in other environments
will provide insights into and better constraints on bedrock channel dynamics by delivering the data needed to understand the spatiotemporal distribution of erosion along a
mountain river and to constrain models of bedrock channel dynamics.
Acknowledgments. This study was supported by the National Science Foundation
through SGER EAR 05-50087 and GLD EAR 06-17557, the National Aeronautics and
Space Administration through ESSF NGT5-30532 and NAG5-13772, and Lamont-Doherty
Earth Observatory. The authors thank G.-R. He, W.-L. Lee, W.-Y. Lai and the Maolin
Municipal Fire Department for assistance in the field, and J. Turowski and N. Hovius
for engaging discussions and thoughtful suggestions. FORMOSAT-2 images were used
under authorization of the National Space Organization of Taiwan through the Disaster
Prevention Research Center of the National Cheng Kung University of Taiwan.
References
Alsdorf, D., C. Birkett, T. Dunne, J. Melack, and L. Hess, Water level changes in a large
Amazon lake measured with spaceborne radar interferometry and altimetry, Geophysical
Research Letters, 28 (14), 2671–2674, 2001.
159
159
Alsdorf, D. E., J. M. Melack, T. Dunne, L. A. K. Mertes, L. L. Hess, and L. C. Smith,
Interferometric radar measurements of water level changes on the Amazon Flood Plain,
Nature, 404, 174–177, 2000.
Birkett, C. M., Contribution of the TOPEX NASA radar altimeter to the global monitoring of large rivers and wetlands, Water Resources Research, 34 (5), 1223–1239, 1998.
Birkett, C. M., L. A. K. Mertes, T. Dunne, M. H. Costa, and M. J. Jasinski, Surface
water dynamics in the Amazon basin: application of satellite radar altimetry, Journal
of Geophysical Research, 107 (D20), 8059, doi:10.1029/2001JD000609, 2002.
Bjerklie, D. M., S. L. Dingman, C. J. Vorosmarty, C. H. Bolster, and R. G. Congalton,
Evaluating the potential for measuring river discharge from space, Journal of Hydrology,
278, 17–38, 2003.
Brakenridge, G. R., S. V. Nghem, E. Anderson, and S. Chen, Space-based measurement
of river runoff, EOS, 19 (10), 185–188, 2005.
Buffington, J. M., and D. R. Montgomery, A systematic analysis of eight decades of
incipient motion studies, with special reference to gravel-bedded rivers, Water Resources
Research, 33 (8), 1993–2029, 1997.
Church, M., and M. A. Hassan, Mobility of bed material in Harris Creek, Water Resources
Research, 38 (11), 1237, doi:10.1029/2001WR000753, 2002.
Dadson, S. J., et al., Links between erosion, runoff variability and seismicity in the Taiwan
orogen, Nature, 426, 648–651, 2003.
Fenton, J. D., and J. E. Abbott, Initial movement of grains on a stream bed: the effect of
relative protrusion, Proceedings of the Royal Society of London. Series A, Mathematical
and Physical Sciences, 352 (1671), 523–537, 1977.
160
160
Griffin, E. R., J. W. Kean, R. Vincent, J. D. Smith, and J. M. Friedman, Modeling effects of bank friction and woody bank vegetation on channel flow and boundary shear stress in Rio Puerco, New Mexico, Journal of Geophysical Research, 110,
doi:10.1029/2005f000322, 2005.
Hartshorn, K., N. Hovius, W. B. Dade, and R. L. Slingerland, Climate-driven bedrock
incision in an active mountain belt, Science, pp. 2036–2038, 2002.
Koblinsky, C. J., R. T. Clarke, A. C. Brenner, and H. Frey, Measurement of river level
variations with satellite altimetry, Water Resources Research, 29, 1839–1848, 1993.
Lague, D., N. Hovius, and P. Davy, Discharge, discharge variability, and the bedrock channel profile, Journal of Geophysical Research, 110, F04,006, doi:10.1029/2004JF000259,
2005.
Li, Y. H., Denudation of Taiwan island since the Pliocene epoch, Geology, 4, 105–107,
1976.
Mahard, R. H., The origin and significance of intrenched meanders, Journal of Geomorphology, 5, 32–44, 1942.
Molnar, P., R. S. Anderson, G. Kier, and J. Rose, Relationships among probability distributions of stream discharges in floods, climate, bed load transport, and river incision,
Journal of Geophysical Research, 111, F02,001, doi:10.1029/2005JF000310, 2006.
Rich, J. L., Certain types of stream valleys and their meaning, Journal of Geology, 22,
469–497, 1914.
Shields, A., Application of similarity principles and turbulence research to bed-load movement, Mitteilunger der Preussischen Versuchsanstalt für Wasserbau und Schiffbau, 26,
5–24, 1936.
161
161
Sippel, S. J., S. K. Hamilton, J. M. Melack, and B. J. Choudhury, Determination of
inundation area in the Amazon River floodplain using the SMMR 37 GHz polarization
difference, Remote Sensing of Environment, 48, 70–76, 1994.
Sippel, S. J., S. K. Hamilton, J. M. Melack, and E. M. M. Novo, Passive microwave observations of inundation area and the area/stage relation in the Amazon River floodplain,
International Journal of Remote Sensing, 19 (16), 3055–3074, 1998.
Smith, L. C., Satellite remote sensing of river inundation area, stage, and discharge: A
review, Hydrological Processes, 11, 1427–1439, 1997.
Smith, L. C., B. L. Isacks, R. R. Forster, A. L. Bloom, and I. Preuss, Estimation of
discharge from braided glacial rivers using ERS 1 synthetic aperture radar: first results,
Water Resources Research, 31 (5), 1325–1329, 1995.
Snyder, N. P., K. X. Whipple, G. E. Tucker, and D. M. Merritts, Channel response to
tectonic forcing: analysis of stream morphology and hydrology in the Mendocino triple
junction region, northern California, Geomorphology, 97–127, 2003a.
Snyder, N. P., K. X. Whipple, G. E. Tucker, and D. M. Merritts, Importance of a stochastic
distribution of floods and erosion thresholds in the bedrock river incision problem,
Journal of Geophysical Research, 108 (B2), 2117, doi:10.1029/2001JB001655, 2003b.
Stark, C. P., A self-regulating model of bedrock river channel geometry, Geophysical
Research Letters, 32, doi:10.1029/2005GL023193, 2006.
Stark, C. P., and N. Hovius, The characterization of landslide size-frequency distributions,
Geophysical Research Letters, 28 (6), 1091–1094, 2001.
Stark, C. P., et al., Dependence of mountain river sinuosity on typhoon flood magnitude
and frequency, Nature, in prep., 2008.
162
162
Tarr, W. A., Intrenched and incised meanders of some streams on the northern slope of
the Ozark Plateau in Missouri, Journal of Geology, 32, 583–600, 1924.
Townsend, P. A., Mapping seasonal flooding in forested wetlands using multi-temporal
Radarsat SAR, Photogrammetric Engineering and Remote Sensing, 67 (7), 857–864,
2001.
Turcotte, D. L., and L. Green, A scale-invariant approach to flood-frequency analysis,
Stochastic Hydrology and Hydraulics, 7, 33–40, 1993.
Turowski, J. M., 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.
Wiele, S. M., and J. D. Smith, A reach-averaged model of diurnal discharge wave propagation down the Colorado River through the Grand Canyon, Water Resources Research,
32 (5), 1375–1386, 1996.
Winslow, A., The Osage river and its meanders, Science, 22 (546), 32–32, 1893.
Wobus, C. W., G. E. Tucker, and R. S. Anderson, Self-formed bedrock channels, Geophysical Research Letters, 33 (L18408), 1–6, 2006.
Xu, K., J. Zhang, M. Watanabe, and C. Sun, Estimating river discharge from very highresolution satellite data: a case study in the Yangtze River, China, Hydrological Processes, 18, 1927–1939, 2004.
Zhang, J., K. Xu, M. Watanabe, Y. Yang, and X. Chen, Estimation of river discharge from
nontrapezoidal open channel using QuickBird-2 satellite imagery, Hydrological Sciences,
49 (2), 247–260, 2004.
163
163
CONCLUSIONS
CONCLUSIONS
164
thesis
comprises
a collection
of observations
analyses
of the
ThisThis
thesis
comprises
a collection
of observations
and and
analyses
of the
morphology
of of
bedrock
channels
andand
their
flood
flows.
It focuses
on on
thethe
rivers
morphology
bedrock
channels
their
flood
flows.
It focuses
rivers
that
that drain
drainthe
theisland
islandmountains
mountainsofofthe
thewestern
westernNorth
NorthPacific
PacificOcean,
Ocean,where
wheretropical
tropical
cyclones dominate cyclones
regional trends
in climate.
dominate
regional trends in climate.
The first chapter reviews literature on incised meanders and meander
theory to show that meandering is common along bedrock rivers and has
consequences for landscape morphology and its interpretation. Meandering occurs
through secondary flow currents which advect momentum to the outside of each
bend, enhancing outwardly directed erosion and driving bend growth. Meander
growth reduces the channel slope, and it can influence incision rates, distort the
scaling of channel slope with upstream drainage area, and feed back on the
meandering process. If the planform geometry is mistakenly assumed to be static,
meandering can invalidate inferences of tectonics and climate from channel and
catchment morphology.
The second chapter shows that regionalized measurements of sinuosity for
ensembles of rivers, incising through broadly similar lithologies, correlate with the
statistics of rainfall and discharge that describe the frequency and relative intensity
of floods; for the study areas considered here, the relative intensity and frequency
of floods are controlled primarily by the pattern of tropical cyclone strikes. The
correlation of sinuosity and measures of relative flood statistics implies that if loop
cutoffs are the only mechanism that resets sinuosity, then areas with straighter
channels and no cutoffs, like Borneo, must have slower rates of lateral cutting, but
that given sufficient time, continued sinuosity growth will proceed to the cutoff
165
limit everywhere. A problem with this interpretation, however, is that the
landscapes in question have been evolving for millions of years and have
undergone kilometers of exhumation; if meanders are limited only by cutoffs, all
of the rivers in these areas should have sinuous planforms and many abandoned
loops by now. Most rivers, particularly in the low sinuosity areas like Borneo and
New Guinea, exhibit no cutoffs at all, which suggests that loop cutoffs are not the
only way sinuosity or its rate of development can be reduced. Instead, another
limit to meandering may be imposed by the timescale of drainage network
reorganization which could serve to reset (or “anneal”) sinuosity periodically; in
this case, the sinuosity today reflects the rate of meander development relative to
the frequency of resetting. A second interpretation is that meandering is a selflimited process, slowing down asymptotically as slopes decrease with bend growth
and reduce wall erosion rates. If so, then the sinuosity we see now along matured
channels is that which developed prior to reaching the asymptotic limit, and scatter
in the correlations of sinuosity and statistics of rainfall, discharge, and storm
frequency may be due in part to a mismatch between the trends of present-day
climate, and those that existed when the meanders formed.
In any case, the strength of the correlation and the timescale of meander
development (tens to hundreds of thousands of years) together suggest that the
pattern of tropical cyclones in the western North Pacific remains relatively
consistent through climate cycles; the cyclone basin may swell and contract, or it
may shift along lines of latitude toward and away from the Eurasian continental
166
166
margin, but a trend is maintained in which the frequency of storms peaks in
Taiwan and Luzon and decreases toward the pole and equator.
The third chapter looks in detail at the dynamics of these river channels. It
demonstrates how to remotely gauge narrow incising channels of uplifting
mountains with the right combination of river gauge data, channel topography, and
high-resolution satellite imagery to acquire measurements of variables that are
important for understanding bedrock river erosion processes such as hydraulic
geometry, mean boundary shear stress, flow speed, friction, and incipient particle
motion, as well as how each of these variables changes downstream. Comparisons
of discharge and boundary shear stress at transects of the ZhuókǂuxƯ, a rapidly
eroding river in Taiwan, reveal a consistent relationship that has approximately
power-law scaling, which transforms heavy-tailed magnitude-frequency
distributions of discharge into steeper, lighter-tailed distributions of boundary
shear stress; discharge during extreme storms on this river peak two orders of
magnitude above mean flow but drive shear stresses only one order of magnitude
greater than average. We gain further insight through a comparison of the
Taiwanese river to a similar-size river in the northeastern United States that is
eroding 50 times more slowly: the extreme flood shear stress magnitudes differ by
only a factor 2, but the shear stresses driven by a 50-year event on the slowly
eroding US river occur semiannually in the Taiwanese channel. Very strong
nonlinearity in the relationship of boundary shear stress and erosion is required if
extreme flood magnitudes are to explain the difference in erosion rates of these
167
167
two rivers; instead, differences in the frequency of erosive floods, which does not
require such strong nonlinearity, is a more likely explanation.
The third chapter also presents empirical data on how longitudinal
variation in channel geometry interacts with the distributions of discharge and
shear stress to modulate the frequency of wall inundation: with simple crosssectional geometry, constant slope, and uniform friction, the frequency of wall
inundation and depth of wall-inundating flows both decrease with channel width.
For a fixed slope, uniform friction, and a stationary distribution of discharge,
narrow reaches should therefore erode laterally more quickly than wider reaches in
a feedback which tends to maintain an equilibrium channel width over long
timescales. Cross-sectional variation may contribute to meandering if over the
timescales of meander development, channels tend to narrow around the bends
where lateral erosion is presumably fastest. In general, however, we see the
opposite, with meander bends occupied by the widest transects. An alternative
explanation is that cross-sectional asymmetry can increase the outer wall
inundation frequency, regardless of channel width. The asymmetry required to
boost outer wall inundation frequency this way is consistent with the asymmetry
we tend to see along incised meanders.
Together these conclusions suggest that the frequency and magnitude of
erosive flood discharges have important consequences for the morphology of
bedrock channels and mountain landscapes: predictions from meander theory
(Chapter 1) and observations across the western North Pacific (Chapter 2) indicate
that lateral cutting and meandering is a function of the relative magnitude of the
168
168
extreme floods, and is modulated by channel morphology (Chapter 3), while
observations from a river in Taiwan indicate that the catchment-wide erosion rate
is controlled by the relative frequency of erosive discharges (Chapter 3). These
conclusions suggest that catchment-wide erosion requires sufficiently high shear
stress flows to move coarse bedload and cause wear on the channel boundary;
however, the discharges that do work on the bed do not necessarily contribute to
lateral cutting along meanders, since sinuosity growth requires sufficiently deep
flows to establish secondary flow currents that are erosive at the outer bends.
The results and conclusions of this thesis suggest several directions for
future work by illuminating a range of open questions about the processes active in
bedrock channels. For example:
1. We lack empirical constraints and model-based statistics on flow
asymmetry around meander bends and its variation with discharge. Such
data would be useful for testing the assumption that meandering in bedrock
has the same hydrodynamic control as meandering in alluvium predicted
by meander bend theories, and it would provide additional evidence to
evaluate the functional dependence of lateral cutting on discharge.
2. If secondary currents are the origin of faster downstream flow speeds at the
outside of meander bends, and therefore the root cause of meandering, then
any disruption of the secondary flow will diminish the rate of meandering.
One possibility is that roughness in the channel bed, for example from
169
169
boulders or bed topography, could affect the establishment of secondary
flow currents. However, we lack observations and theory of how the
roughness relative to the flow depth relates to the planform evolution of
incising rivers.
3. We also lack theory to explain the dependence of meandering on lithology
and rock strength. Faster meandering in weaker rocks relative to thalweg
lowering rates indicates that vertical and horizontal erosion rates have
different dependencies on erodibility, but how and why remain unclear.
Boundary roughness could be a key connection between lithology and
meander activity if meandering slows with relative roughness, and if the
roughness height-scale is lithologically controlled. Observations of the
roughness height-scale in channels through a range of rock types will help
to establish whether or not lithologically or structurally controlled
roughness explains the qualitative correlation of sinuosity development and
substrate weakness.
4. The conclusion in the second chapter that processes other than loop cutoffs
limit meandering is inferred indirectly from the correlation of regionalscale sinuosity and rainfall, discharge, and storm track statistics. Without
any direct observations of these other limiting processes, we are currently
unable to identify with certainty the source of the limitation. Comparisons
of channel slope and indicators of meander activity for a given discharge
170
170
distribution, for example, may help to determine whether or not slope
reduction slows meandering, but only in combination with observations of
roughness, erodibility, and channel cross section, and a better
understanding of how each of these contributes to the rate of wall erosion.
Furthermore, if the sinuosity limitation is an effect of slope reduction, then
rivers may continue to meander if the meander-driven slope reduction is
accommodated by channel narrowing, enhancement of transect asymmetry,
or steepening within the catchment, each of which has yet to be tested with
observations or theory.
5. The inference that erosion rate is a greater function of erosive flood
frequency than it is of the extreme flood discharge also requires further
testing, which can be accomplished through the evaluation of available
stage records, channel slopes, and channel cross-sections at gauge locations
along bedrock rivers with known (or measurable) erosion rates. However,
results will be more robust following the methodology outlined in the
manuscripts of Chapter 3 using gauged discharge, satellite imagery, and
topographic surveys (or, very high resolution digital topography) to assess
conditions and their changes, since a single transect at the gauge may not
be representative of the rest of the reach.
6. The results presented in Chapter 3 show clearly that wall inundation
frequency varies downstream with channel geometry. A correlation of
171
171
inundation frequency and planform curvature would support the hypothesis
that longitudinal variation in transect geometry contributes to meandering.
However, for the limited number of meander bends included in the analysis,
the correlation is weak at best. Similar measurements of wall inundation
frequency along other actively meandering rivers will help to clarify the
importance of this effect in setting the longitudinal variation in wall erosion
rates that meandering requires.
172
172