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![Page 1: Morphology and dynamics of mountain rivers Mikaël ATTAL Marsyandi valley, Himalayas, Nepal Acknowledgements: Jérôme Lavé, Peter van der Beek and other.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5519268f5503462f428b4c7b/html5/thumbnails/1.jpg)
Morphology and Morphology and dynamics of dynamics of
mountain riversmountain riversMikaël ATTALMikaël ATTAL
Marsyandi valley, Himalayas, Nepal
Acknowledgements: Acknowledgements: Jérôme Lavé, Jérôme Lavé, Peter van der BeePeter van der Beek and other k and other
scientists from LGCA (Grenoble) scientists from LGCA (Grenoble) and CRPG (Nancy)and CRPG (Nancy)
Eroding landscapes: Eroding landscapes: fluvial processesfluvial processes
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Lecture overviewLecture overview
I. Morphology and geometry of mountain « bedrock » riversI. Morphology and geometry of mountain « bedrock » rivers
II. Fluvial erosion laws: models and attempts of calibrationII. Fluvial erosion laws: models and attempts of calibration
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Erosion in mountainsErosion in mountains
Glaciers and hillslope processes
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RIV
ER
S
Borrego badlands, California
(www.parkerlab.bio.eci.edu)
Bryce Canyon, Utah (www.smugmug.com)
Cascade Mountains, California
10 km
Salerno
Southern Apennines, Italy
Canyonlands National Park, Utah (Stu Gilfillan)
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Fluvial incision
In response to tectonic uplift, rivers incise into bedrock...
Uplift
http://projects.crustal.ucsb.edu/nepal/
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Uplift
Hillslope erosion
… and insure the progressive lowering of the base level for hillslope processes
http://projects.crustal.ucsb.edu/nepal/
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Rivers insure the transport of the erosion products to the sedimentary basin
Dissolved load + suspended load + bed load
http://projects.crustal.ucsb.edu/nepal/
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Hierarchical organization of fluvial network
W
S S = 0.46W
Hovius, 1996, 2000
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Hierarchical organization of fluvial network
Hack’s law (1957)
Rigon et al., 1996
L = aAh
L = length of stream
a = constant
h = constant in the range 0.5-0.6 in natural rivers
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Hierarchical organization of fluvial networkResponse to active tectonics
Galy, 1999
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Hierarchical organization of fluvial networkResponse to active tectonics
Tectonic control on drainage development(Eliet & Gawthorpe, 1995)
A
B
A
BA: relay zone, large catchment, low subsidence rate B: fault “wall”, small catchments, large subsidence rate
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JGR, 2002
Hierarchical organization of fluvial networkResponse to active tectonics
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Hierarchical organization of fluvial networkResponse to active tectonics
Humphrey and Konrad, 2000
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Hovius, 2000
Development and evolution of river profiles
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Uplift > ErosionUplift > Erosion
Development and evolution of river profilesRivers adjust their SLOPES to increase or reduce erosion rates
Slope increases erosion increases until U = E (Steady-State). Steady-State means: rate of rock uplift relative to some datum, such as mean sea level, equals the erosion rate at every point in the landscape, so that topography does not change.
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Uplift < ErosionUplift < Erosion
Slope decreases erosion decreases until U = E (Steady-State). Steady-State means: rate of rock uplift relative to some datum, such as mean sea level, equals the erosion rate at every point in the landscape, so that topography does not change.
Development and evolution of river profilesRivers adjust their SLOPES to increase or reduce erosion rates
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Sklar and Dietrich, 1998
Mountain “bedrock” rivers
ER
OSI
ON
DE
PO
SIT
ION
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Noyo River, California (Sklar and Dietrich, 1998)
Stream Power Law (SPL)Typical steady-state “concave-up” river profile: power law
between slope and drainage area
“Fluvial” bedrock channel
S = KSA-θ where KS = steepness index and θ = concavity index (0.5 ± 0.15)
θ
“Debris-flow-dominated” bedrock channel
Debris-flow-dominated reaches: S independent of A, S controlled mostly by rock mass strength (angle of repose)
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San Gabriel Mts, California (Wobus et al., 2006)
S = KSA-θ
KS is a function of uplift rate: high uplift high erosion rates needed to reach steady-state steep slopes needed.
For a given A, the slope of a channel experiencing a high uplift rate (black) is higher than the slope of a channel experiencing low uplift rate (grey). log S
log A
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log S
log A
NOTE: this applies to STEADY-STATE bedrock
channels experiencing uniform uplift !!!
Humphrey and Konrad, 2000
If uplift is not uniform or landscape is responding to a disturbance slopes adjust local steepening + profile convexities
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Channel width W:
W = cAb where b = 0.3-0.5.
In alluvial rivers, b ~ 0.5 [e.g. Leopold and Maddock, 1953]
Montgomery and Gran, 2001
Hydraulic scaling in bedrock rivers
NOTE: this applies to STEADY-STATE
bedrock channels experiencing
uniform uplift !!!
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New Zealand (Amos and Burbank, 2007)
Rivers cut across active fold Zone of high uplift channel steepening + narrowing
Development and evolution of river profilesRivers adjust their SLOPES but also their WIDTH to increase
or reduce erosion rates
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Yarlung Tsangpo, SE Tibet (Finnegan et al., 2005)
Zone of high uplift channel steepening + narrowing
W α A3/8S-3/16
Development and evolution of river profilesRivers adjust their SLOPES but also their WIDTH to increase
or reduce erosion rates
Channel steepening = cause of channel narrowing?
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SummarySummary
Steady-state bedrock rivers: hierarchical organization of the Steady-state bedrock rivers: hierarchical organization of the network (+ Hack’s law), concave up profile, power law between network (+ Hack’s law), concave up profile, power law between S and A, power law between W and A. S and A, power law between W and A.
In response to variations in uplift rate in space or time, channels In response to variations in uplift rate in space or time, channels adjust their slopes AND width. Channels steepen and narrow in adjust their slopes AND width. Channels steepen and narrow in zones of high uplift to maximize their erosive « stream power ».zones of high uplift to maximize their erosive « stream power ».
Remark: this can also result from variations in rock type. What Remark: this can also result from variations in rock type. What about climatic variations?about climatic variations?
II. Fluvial erosion laws: models and attempts of calibrationII. Fluvial erosion laws: models and attempts of calibration
PAUSE
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Stream power per unit length (Ω) = amount of energy available to do work over a given length of stream bed during a given time interval. Ω = ΔEp / ΔtΔx, where ΔEp = potential energy loss = mgΔz, and m = mass of the body of water.
As m/Δt = ρQ Ω = mgΔz/ΔtΔx = ρQgΔz/Δx
Ω = ρ g Q S
ρ = density of water,g = acceleration of gravity ,W = channel width,D = channel depth,z = elevation,S = channel slope,V = flow velocity,Q = discharge.
Stream power: theory
Acknowledgement: Peter van der Beek
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Shear force exerted by the body of water moving downstream (F):F = ρgWDX.sin αwhere X is the length of the reach. For low angle α, sin α ~ tan α F = ρgWDXS. Shear stress τ = shear force / wetted area of the channel: τ = F / ((W+2D)X) = ρgSWD / (W+2D)
τ = ρ g R S, where R = hydraulic radius = WD/(W+2D)
τ = ρ g D S, if W >10D.
Stream power: theory
ρ = density of water,g = acceleration of gravity ,W = channel width,D = channel depth,z = elevation,S = channel slope,V = flow velocity,Q = discharge.
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1. Incision Stream power / unit length (Ω)(Seidl et al., 1992; Seidl & Dietrich, 1994)
E Ω E = K Q S
2. Incision Specific Stream power (ω) (Bagnold, 1977)
E Ω / W E = K Q S / W
3. Incision basal shear stress (τ) (Howard & Kerby, 1983; Howard et al., 1994)
E τ E = K Q S / W V,as τ ~ ρgDS and Q = WDV.
Fluvial incision laws, part 1 Fluvial incision = f (hydrodynamic variables)
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• Simplification - hydrology and hydraulic geometry:
Q Aa ; a 1W Ab; b 0,5
• Expression for flow velocity (e.g. Manning equation):
The 3 fluvial erosion laws can be written in the same general form: STREAM POWER LAW (SPL) – Detachment-limited model:
E = K Am Sn
where: for E Ω m = n = 1 E ω m 0.5; n = 1 E τ m 0.3; n 0.7
213
2
2
1S
DW
WD
NV
Fluvial incision = f (hydrodynamic variables)
The influence of rock strength, rainfall, sediment supply, grain size, discharge variability, etc., are lumped together into the K parameter!
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E = K Am Sn
E τ m 0.3; n 0.7
Demonstration:τ = ρgRS = ρgDS for large rivers. Using the same simplification,
the Manning’s law becomes: V = (1/N) D2/3S1/2 (1).Also, V = Q/WD (2).
(1) + (2) Q/WD = (1/N) D2/3S1/2
D5/3 = NQ/WS1/2 = QNW-1S-1/2
D = N3/5Q3/5W-3/5S-3/10
τ = ρgDS = ρgN3/5Q3/5W-3/5S7/10
Q A and W A1/2 τ A3/5A-3/10S7/10
τ A3/10S7/10
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River in steady-state:
Thus:
Power law between S and A
0 EUdt
dz
nmn
AK
US
1
Stream power law
nmSAKU
S = KSA-θ where KS =
steepness index and θ = concavity index (0.5 ± 0.15)
θ
Looks familiar?
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Simplistic model! Threshold for erosion? Role of sediments?
4. Excess shear stress model (Densmore et al., 1998; Lavé & Avouac, 2001):
E = K (τ - τc)
5. Transport-limited model (Willgoose et al., 1991):
tt nmts
s SAKQx
Q
WE
;1
SPL: Fluvial incision = f (hydrodynamic variables)
Fluvial incision laws, part 2: beyond the SPL…
Sediment transport continuity equation(non-linear diffusion equation)
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Sklar & Dietrich, 2001
Role of sediment: the “tools and cover” effects(Gilbert, 1877)
Experimental study of bedrock abrasion by saltating particles
Tools
Cover
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6. Under-capacity model: cover effect (sediment needs to be moved for erosion to occur). CASCADE uses this model (Kooi & Beaumont, 1994)
scf
QQLW
E 1
Role of sediment: the “tools and cover” effects
Lf can either be thought of as a length scale or as the ratio of transport capacity (Qc) to detachment capacity [Cowie et al., 2006].
Ero
sion
eff
icie
ncy
Qs/Qc
0 1
Meyer-Peter-Mueller transport equation (1948)
Qc= k1 (τ - τc)3/2
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7. « Tools and cover » effects model (Sklar & Dietrich, 1998, 2004)
c
s
f
s
Q
Q
LW
QE 1
Role of sediment: the “tools and cover” effects
At least 7 different fluvial incision models! + Low amount of field testing.
2004: mechanistic
1998: theoretical
E = ViIrFe
Vi = volume of rock detached / particle impact,Ir = rate of particle impacts per unit area per unit time,Fe = fraction of the river bed made up of exposed bedrock.
Ero
sion
eff
icie
ncy
Qs/Qc
0 1
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E = KAmSn.f(qs) Stream Power Law(s) (laws 1, 2, 3): f(qs) = 1
Threshold for erosion (law 4), slope set by necessity for river to transport sediment downstream (law 5), cover effect (law 6), tools + cover effects (law 7).
Similar predictions at SS: concave up profile with power relationship between S and A.
Different predictions in terms of transient response of the landscape to perturbation.
Laws including the role of the sediments: f(qs) ≠ 1
General form: fluvial incision laws
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(2002) (2002)
Detachment-limited law (SPL, laws 1, 2, 3) Transport limited law (law 5)
Transient response of fluvial systems
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(2002) (2002)
Transient response of fluvial systemsDetachment-limited law (SPL, laws 1, 2, 3) Transport limited law (law 5)
Erosion Specific Stream power (law 2):
dz/dt = U – E = U - KA0.5S
dz/dt = -KA0.5 dz/dx + U
Celerity of the “wave” in the x direction
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(2002) (2002)
Transient response of fluvial systemsDetachment-limited law (SPL, laws 1, 2, 3) Transport limited law (law 5)
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SummarySummary
At least 7 different fluvial erosion laws.At least 7 different fluvial erosion laws.- 3 “stream power laws” (erosion = - 3 “stream power laws” (erosion = f f ((AA, , SS))))- 4 laws including the role of sediment (- 4 laws including the role of sediment (ff((QQss) ≠ 1)) ≠ 1)
Low amount of field testing but recent work strongly Low amount of field testing but recent work strongly support that:support that:- - sediments exertsediments exert a strong a strong control on rates and processes of control on rates and processes of bedrock erosion (bedrock erosion (ff((QQss) ) ≠ 1);≠ 1);
- sediments could have “tools and cover effects”.- sediments could have “tools and cover effects”.
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E = K Am Sn
E τ m 0.3; n 0.7
Demonstration:τ = ρgRS = ρgDS for large rivers. Using the same simplification,
the Manning’s law becomes: V = (1/N) D2/3S1/2 (1).Also, V = Q/WD (2).
(1) + (2) Q/WD = (1/N) D2/3S1/2
D5/3 = NQ/WS1/2 = QNW-1S-1/2
D = N3/5Q3/5W-3/5S-3/10
τ = ρgDS = ρgN3/5Q3/5W-3/5S7/10
Q A and W A1/2 τ A3/5A-3/10S7/10
τ A3/10S7/10