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

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

Erosion in mountainsErosion in mountains

Glaciers and hillslope processes

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)

Fluvial incision

In response to tectonic uplift, rivers incise into bedrock...

Uplift

http://projects.crustal.ucsb.edu/nepal/

Uplift

Hillslope erosion

… and insure the progressive lowering of the base level for hillslope processes

http://projects.crustal.ucsb.edu/nepal/

Rivers insure the transport of the erosion products to the sedimentary basin

Dissolved load + suspended load + bed load

http://projects.crustal.ucsb.edu/nepal/

Hierarchical organization of fluvial network

W

S S = 0.46W

Hovius, 1996, 2000

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

Hierarchical organization of fluvial networkResponse to active tectonics

Galy, 1999

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

JGR, 2002

Hierarchical organization of fluvial networkResponse to active tectonics

Hierarchical organization of fluvial networkResponse to active tectonics

Humphrey and Konrad, 2000

Hovius, 2000

Development and evolution of river profiles

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.

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

Sklar and Dietrich, 1998

Mountain “bedrock” rivers

ER

OSI

ON

DE

PO

SIT

ION

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)

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

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

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 !!!

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

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?

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

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

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.

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)

• 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!

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

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?

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)

Sklar & Dietrich, 2001

Role of sediment: the “tools and cover” effects(Gilbert, 1877)

Experimental study of bedrock abrasion by saltating particles

Tools

Cover

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

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

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

(2002) (2002)

Detachment-limited law (SPL, laws 1, 2, 3) Transport limited law (law 5)

Transient response of fluvial systems

(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

(2002) (2002)

Transient response of fluvial systemsDetachment-limited law (SPL, laws 1, 2, 3) Transport limited law (law 5)

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”.

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