Morphology and dynamics of mountain rivers Mikaël ATTAL Marsyandi valley, Himalayas, Nepal Acknowledgements: Jérôme Lavé, Peter van der Beek and other scientists from LGCA (Grenoble) and CRPG (Nancy) Eroding landscapes: fluvial processes

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

RIVERS Borrego badlands, California ( Bryce Canyon, Utah ( 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

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

Rivers insure the transport of the erosion products to the sedimentary basin Dissolved load + suspended load + bed load

Hierarchical organization of fluvial network W S S = 0.46W Hovius, 1996, 2000

Hierarchical organization of fluvial network Hacks law (1957) Rigon et al., 1996 L = aA h L = length of stream a = constant h = constant in the range in natural rivers

Hierarchical organization of fluvial network Response to active tectonics Galy, 1999

Hierarchical organization of fluvial network Response to active tectonics Tectonic control on drainage development (Eliet & Gawthorpe, 1995) A B A B A: relay zone, large catchment, low subsidence rate B: fault wall, small catchments, large subsidence rate

JGR, 2002 Hierarchical organization of fluvial network Response to active tectonics

Hierarchical organization of fluvial network Response to active tectonics Humphrey and Konrad, 2000

Hovius, 2000 Development and evolution of river profiles

Uplift > Erosion Development and evolution of river profiles Rivers 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 < 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 profiles Rivers adjust their SLOPES to increase or reduce erosion rates

Sklar and Dietrich, 1998 Mountain bedrock rivers EROSION DEPOSITION

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 = K S A -θ where K S = 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 = K S A -θ K S 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 = cA b where b = 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 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 A 3/8 S -3/16 Development and evolution of river profiles Rivers adjust their SLOPES but also their WIDTH to increase or reduce erosion rates Channel steepening = cause of channel narrowing?

Fiamignano fault, Central Apennines, Italy Whittaker et al., 2007a W α A 3/8 S -3/16 Finnegan et al., 2005, supported by analytical work by Wobus et al., 2006b W α A 0.5 Rivers maximize their stream power in the zone of high uplift by steepening AND narrowing

Summary Steady-state bedrock rivers: hierarchical organization of the network (+ Hacks 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 II. 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 = ΔE p / ΔtΔx, where ΔE p = 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 α where 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 A a ; a 1 W A b ; 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 A m S n where: for E Ω m = n = 1 E ω m 0.5; n = 1 E τ m 0.3; n 0.7 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 A m S n E τ m 0.3; n 0.7 Demonstration: τ = ρgRS = ρgDS for large rivers. Using the same simplification, the Mannings law becomes: V = (1/N) D 2/3 S 1/2 (1). Also, V = Q/WD (2). (1) + (2) Q/WD = (1/N) D 2/3 S 1/2 D 5/3 = NQ/WS 1/2 = QNW -1 S -1/2 D = N 3/5 Q 3/5 W -3/5 S -3/10 τ = ρgDS = ρgN 3/5 Q 3/5 W -3/5 S 7/10 Q A and W A 1/2 τ A 3/5 A -3/10 S 7/10 τ A 3/10 S 7/10

River in steady-state: Thus: Power law between S and A Stream power law S = K S A -θ where K S = 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 - τ c ) 5. Transport-limited model ( Willgoose et al., 1991 ): 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) Role of sediment: the tools and cover effects L f can either be thought of as a length scale or as the ratio of transport capacity (Q c ) to detachment capacity [Cowie et al., 2006]. Erosion efficiency Q s /Q c 01 Meyer-Peter-Mueller transport equation (1948) Q c = k 1 (τ - τ c ) 3/2

7. « Tools and cover » effects model (Sklar & Dietrich, 1998, 2004) 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 = V i I r F e V i = volume of rock detached / particle impact, I r = rate of particle impacts per unit area per unit time, F e = fraction of the river bed made up of exposed bedrock. Erosion efficiency Q s /Q c 01

E = KA m S n.f(q s ) Stream Power Law(s) (laws 1, 2, 3): f(q s ) = 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(q s ) 1 General form: fluvial incision laws

(2002) Detachment-limited law (SPL, laws 1, 2, 3)Transport limited law (law 5) Transient response of fluvial systems

(2002) Transient response of fluvial systems Detachment-limited law (SPL, laws 1, 2, 3)Transport limited law (law 5) Erosion Specific Stream power (law 2): dz/dt = U – E = U - KA 0.5 S dz/dt = -KA 0.5 dz/dx + U Celerity of the wave in the x direction

(2002) Transient response of fluvial systems Detachment-limited law (SPL, laws 1, 2, 3)Transport limited law (law 5)

Field testing of fluvial incision laws (1) Basal shear stress: Fluvial erosion law: V D Excess shear stress model (law 4): Lavé & Avouac, 2001 τ = ρ g R S, where R = WD / (W+2D) τ = ρ g D S, if W >10D. E = K (τ - τ c )

Fluvial incision along Himalayan rivers MFT

Fluvial incision measured using terraces

[Lavé and Avouac, 2001] Comparison between fluvial incision (terraces) and excess shear stress (channel geometry)

E = K (τ* - τ c *) Independent measurements: E from terraces and τ from channel geometry. τ c * value used = 0.03 See Buffington and Montgomery, 1997, for extensive description of the critical shear stress concept. Important role of lithology

Modified from Lavé & Avouac, 2001 TSSHHCLH S Lavé & Avouac, 2001: maximum fluvial erosion rate in the HHC zone for 6 main Himalayan rivers

Modified from Lavé & Avouac, 2001 TSSHHCLH S Lavé & Avouac, 2001: maximum fluvial erosion rate in the HHC zone for 6 main Himalayan rivers

All laws predict similar steady-state topographies (concave-up profile, etc.), Predicted transient response to a disturbance depends on the fluvial incision law chosen. Field testing of fluvial incision laws (2) Using the transient response of the landscape

Fiamignano, Italy Xerias, Greece km Transient response to tectonic disturbance (Whittaker et al., 2007a, b, 2008; Cowie et al., 2008, Attal et al., 2008)

Fiamignano, Italy Xerias, Greece Transient response to tectonic disturbance (Whittaker et al., 2007a, b, 2008; Cowie et al., 2008, Attal et al., 2008)

Fiamignano, Italy Xerias, Greece Transient response to tectonic disturbance Italy closer to DL end-member, Greece closer to TL end-member (Cowie et al., 2008) SEDIMENTS DO MATTER! Erosion efficiency f(Q s ) Q s /Q c 01

Summary At least 7 different fluvial erosion laws. - 3 stream power laws (erosion = f (A, S)) - 4 laws including the role of sediment (f(Q s ) 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(Q s ) 1); - sediments could have tools and cover effects.

E = K A m S n E τ m 0.3; n 0.7 Demonstration: τ = ρgRS = ρgDS for large rivers. Using the same simplification, the Mannings law becomes: V = (1/N) D 2/3 S 1/2 (1). Also, V = Q/WD (2). (1) + (2) Q/WD = (1/N) D 2/3 S 1/2 D 5/3 = NQ/WS 1/2 = QNW -1 S -1/2 D = N 3/5 Q 3/5 W -3/5 S -3/10 τ = ρgDS = ρgN 3/5 Q 3/5 W -3/5 S 7/10 Q A and W A 1/2 τ A 3/5 A -3/10 S 7/10 τ A 3/10 S 7/10