Eroding landscapes:
fluvial processes
Morphology and
dynamics of
mountain rivers
Mikaël ATTAL
Acknowledgements: Jérôme
Lavé, Peter van der Beek and
other scientists from LGCA
(Grenoble) and CRPG (Nancy)
Marsyandi valley, Himalayas, Nepal
Lecture overview
I. Morphology and geometry of mountain « bedrock » rivers
II. Fluvial erosion laws: models and attempts of calibration
Erosion in mountains
Glaciers and hillslope processes
Borrego
badlands,
California
Southern Apennines, Italy
(www.parkerlab
Canyonlands National Park, Utah (Stu Gilfillan)
RIVERS
.bio.eci.edu)
Salerno
Bryce
Canyon, Utah (www.smugmug.com)
10 km
Cascade Mountains, California
In response to tectonic uplift, rivers
incise into bedrock...
http://projects.crustal.ucsb.edu/nepal/
Uplift
Fluvial incision
… and insure the progressive lowering of the
base level for hillslope processes
http://projects.crustal.ucsb.edu/nepal/
Uplift
Hillslope
erosion
Rivers insure the transport of the erosion
products to the sedimentary basin
http://projects.crustal.ucsb.edu/nepal/
Dissolved load + suspended load + bed load
Hierarchical organization of fluvial network
W
S
S = 0.46W
Hovius, 1996, 2000
Hierarchical organization of fluvial network
Hack’s law (1957)
L = aAh
L = length of stream
a = constant
h = constant in the range
0.5-0.6 in natural rivers
Rigon et al., 1996
Hierarchical organization of fluvial network
Response to active tectonics
Galy, 1999
Hierarchical organization of fluvial network
Response to active tectonics
A: relay zone, large catchment, low
subsidence rate
B: fault “wall”, small catchments,
large subsidence rate
B
A
B
A
Tectonic control on
drainage
development
(Eliet & Gawthorpe, 1995)
Hierarchical organization of fluvial network
Response to active tectonics
JGR, 2002
Hierarchical organization of fluvial network
Response to active tectonics
Humphrey and
Konrad, 2000
Development and evolution of river profiles
Hovius, 2000
Development and evolution of river profiles
Rivers adjust their SLOPES to increase or reduce erosion rates
Uplift > Erosion
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.
Development and evolution of river profiles
Rivers adjust their SLOPES to increase or reduce erosion rates
Uplift < 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.
DEPOSITION
EROSION
Mountain “bedrock” rivers
Sklar and Dietrich, 1998
Stream Power Law (SPL)
Typical steady-state “concave-up” river profile: power law
between slope and drainage area
θ
“Fluvial” bedrock
channel
“Debris-flowdominated” bedrock channel
S = KSA-θ
where KS = steepness index and θ = concavity index (0.5 ± 0.15)
Debris-flow-dominated reaches: S independent of A, S controlled mostly by rock
mass strength (angle of repose)
Noyo River, California (Sklar and Dietrich, 1998)
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
San Gabriel Mts, California (Wobus et al., 2006)
NOTE: this applies to
STEADY-STATE bedrock
channels experiencing
uniform uplift !!!
Humphrey and Konrad, 2000
log S
log A
If uplift is not uniform or landscape
is responding to a disturbance 
slopes adjust  local steepening +
profile convexities
Hydraulic scaling in
bedrock rivers
Channel width W:
W = cAb where b = 0.3-0.5.
In alluvial rivers, b ~ 0.5
[e.g. Leopold and Maddock, 1953]
NOTE: this applies
to STEADY-STATE
bedrock channels
experiencing
uniform uplift !!!
Montgomery and Gran, 2001
Development and evolution of river profiles
Rivers adjust their SLOPES but also their WIDTH to increase
or reduce erosion rates
Rivers cut across active fold  Zone of high uplift 
channel steepening + narrowing
New Zealand (Amos and Burbank, 2007)
Development and evolution of river profiles
Rivers adjust their SLOPES but also their WIDTH to increase
or reduce erosion rates
Yarlung Tsangpo, SE Tibet (Finnegan et al., 2005)
Zone of high uplift  channel steepening + narrowing
W α A3/8S-3/16
Channel steepening = cause of channel narrowing?
PAUSE
Summary
Steady-state bedrock rivers: hierarchical organization of the
network (+ Hack’s law), concave up profile, power law between
S and A, power law between W and A.
In response to variations in uplift rate in space or time, channels
adjust their slopes AND width. Channels steepen and narrow in
zones of high uplift to maximize their erosive « stream power ».
Remark: this can also result from variations in rock type. What
about climatic variations?
 II. Fluvial erosion laws: models and attempts of calibration
Stream power: theory
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.
ρ = density of water,
g = acceleration of gravity ,
W = channel width,
D = channel depth,
z = elevation,
S = channel slope,
V = flow velocity,
Q = discharge.
Acknowledgement: Peter van der Beek
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= ρgQS
Stream power: theory
Shear force exerted by the
body of water moving
downstream (F):
F = ρgWDX.sin α
where X is the length of the reach.
For low angle α, sin α ~ tan α
 F = ρgWDXS.
ρ = density of water,
g = acceleration of gravity ,
W = channel width,
D = channel depth,
z = elevation,
S = channel slope,

V = flow velocity,

Q = discharge.
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.
Fluvial incision laws, part 1
Fluvial incision = f (hydrodynamic variables)
1. Incision  Stream power / unit length (Ω)
(Seidl et al., 1992; Seidl & Dietrich, 1994)
EΩE=KQS
2. Incision  Specific Stream power (ω) (Bagnold, 1977)
EΩ/WE=KQS/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 = f (hydrodynamic variables)
•
•
Simplification - hydrology and hydraulic geometry:
Q  Aa ;
W  Ab;
a1
b  0,5
2
Expression for flow velocity
(e.g. Manning equation):
1  WD  3 12
V 
 S
N  W  2D 
 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 
Eω 
Eτ 
m=n=1
m  0.5; n = 1
m  0.3; n  0.7
The influence of rock strength, rainfall, sediment supply, grain size, discharge
variability, etc., are lumped together into the K parameter!
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
Stream power law
dz
U  E  0
River in steady-state:
dt
Thus:
U  KA S
m
n
Looks familiar?
1
 Power law between S and A
θ
 U  n m n
S   A
K
S = KSA-θ
where KS =
steepness index and θ = concavity
index (0.5 ± 0.15)
SPL: Fluvial incision = f (hydrodynamic variables)
 Simplistic model! Threshold for erosion? Role of sediments?
Fluvial incision laws, part 2: beyond the SPL…
4. Excess shear stress model (Densmore et al., 1998; Lavé & Avouac, 2001):
E = K (τ - τc)
5. Transport-limited model (Willgoose et al., 1991):
1 Qs
E
 ;
W x
Qs  K t Amt S nt
Sediment transport continuity equation
(non-linear diffusion equation)
Role of sediment: the “tools and cover” effects
(Gilbert, 1877)
Cover
Tools
Experimental study of bedrock abrasion by saltating particles
Sklar & Dietrich, 2001
Role of sediment: the “tools and cover” effects
6. Under-capacity model: cover effect (sediment needs to be moved for
erosion to occur). CASCADE uses this model (Kooi & Beaumont, 1994)
1
Qc  Qs 
E
W Lf
Meyer-Peter-Mueller transport
equation (1948)
Qc= k1 (τ - τc)3/2
Erosion efficiency
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].
Qs/Qc
0
1
Role of sediment: the “tools and cover” effects
Q
E s
W Lf
 Qs 
1  
 Qc 
1998: theoretical
E = ViIrFe
2004: mechanistic
(Sklar & Dietrich, 1998, 2004)
Erosion efficiency
7. « Tools and cover » effects model
Qs/Qc
0
1
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.
 At least 7 different fluvial incision models!
+ Low amount of field testing.
General form: fluvial incision laws
m n
E = KA S .f(qs)
 Stream Power Law(s) (laws 1, 2, 3): f(qs) = 1
 Laws including the role of the sediments: 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.
Transient response of fluvial systems
Detachment-limited law (SPL, laws 1, 2, 3)
(2002)
Transport limited law (law 5)
(2002)
Transient response of fluvial systems
Detachment-limited law (SPL, laws 1, 2, 3)
Transport limited law (law 5)
(2002)
(2002)
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
Transient response of fluvial systems
Detachment-limited law (SPL, laws 1, 2, 3)
(2002)
Transport limited law (law 5)
(2002)
Summary
At least 7 different fluvial erosion laws.
- 3 “stream power laws” (erosion = f (A, S))
- 4 laws including the role of sediment (f(Qs) ≠ 1)
Low amount of field testing but recent work strongly
support that:
- sediments exert a strong control on rates and processes of
bedrock erosion (f(Qs) ≠ 1);
- sediments could have “tools and cover effects”.
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