Nicole GaspariniArizona State University

Landscape Modeling

What is the point of numerical landscape evolution models?

•Use landscape evolution models to understand the behavior of different erosion processes and theories.

•What details matter?•Under what circumstances?•How do different processes interact?

•Use numerical models to understand how sensitive the landscape is to variability in forcing (climate, tectonics).

“Document the state of the art and identify the rate-limiting challenges…”

• Intro to fluvial incision.• How is precipitation included in a landscape

evolution model?– Uniform (space and time) but varies between

experiments– Uniform in time, varies in space– Uniform in space, varies in time - intensity, duration,

interstorm duration

Landscape Evolution Models(Use CHILD as example; Tucker et al, 2001)

• Water falls onto the landscape, aggregates downstream, and can entrain, transport, and deposit sediment and incise into bedrock.

• Hillslopes deliver sediment to fluvial channels. Hillslope processes are not usually modeled as a function of soil water content or overland flow.

• Glaciers? Debris Flows?

QuickTime™ and a decompressor

are needed to see this picture.

From Greg Tucker’s Website

Attributes of Every Node:

z, elevationa, node areaA, drainage area = ai

Q, incoming fluvial discharge

Qs, incoming sediment load from erosion upstream

S, downstream slope

nodes

edges

Qsin, Qin

Qsout,

Qout

OutletOutlet

Drainage Area, increases down-stream

Channel Profile, slope decreases down-stream

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∂z

∂t=U − E − H

Fluvial Erosion Model - Detachment-limited model for incision into bedrock

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E = kb τ − τ c( )a Shear Stress

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E = erosion rate (length/time)τ = bed shear stressτ c = threshold valuekb = erodibility; f(lithology, process); stronger rock, smaller kb

a = positive constantQ = fluvial dischargeW = channel widthα ,β = positive constants, about equal

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τ ∝ Q

W

⎛

⎝ ⎜

⎞

⎠ ⎟

α

Sβ“…force balance for steady, uniform flow in a wide channel”, Tucker, 2004

Discharge Relationship:

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Q∝ PA c Discharge-Area Relationship,Hydrologic Steady State

River basins in Kentucky, USA, from Solyom and Tucker, 2004

Q = 0.0171*A0.9932

R2 = 0.9977€

Q = Fluvial Discharge P = Effective Precipitation Rate = Rainfall - lossesc = positive constant ≤ 1

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Q = Piai

i

∑ or

Channel Width:

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W ∝Qb

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W = Channel WidthQ = Fluvial Discharge ∝ PA c

b = positive constant ~ 0.5

Hydraulic Geometry (e.g. Leopold & Maddock 1953)

Data from the Clearwater River, Washington State, from Tomkin et al., 2003.

Q = 0.1335 * A0.9

W=4.2*A0.42

Combining previous relationships with some parameter value assumptions…

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E = kb (PA )0.5 S Functional form of erosion equation in numerical models, ignore thresholds for now.

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S =E

kbP0.5A−0.5 Slope-Area relationship -

Channel slopes (& relief) are inversely proportional to precipitation.

Major issues already! Spatial patterns of precipitation, temporal patterns of precipitation - This just assumes an effective precipitation rate and steady-state flow.

Uniform precipitation in space and time. Differences between “more erosive (higher precipitation) and less

erosive climates” Whipple, Kirby & Brocklehurst (1999).

Less erosive climate shown in gray, and more erosive climate, in black lines

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S =E

kbP0.5A−0.5

Uniform precipitation in space and time. Differences between “more erosive (higher precipitation) and less

erosive climates”.

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S =E

kbP0.5A−0.5Lower Precip,

more relief

Higher Precipless relief

Does topography influence local climate?

Spatially Variable PrecipitationRoe, Montgomery & Hallet, 2002

“where winds are forced upslope, the air column cools and saturates … and rains out” ; “Conversely, prevailing downslope winds dry out the air column, and precipitation is suppressed…”

€

Q(x) = p(x' )dA(x' )

dx'dx'

0

x

∫

Spatially Variable PrecipitationRoe, Montgomery & Hallet, 2002

Precip Increases with Elevation

Precip Decreases with Elevation

outlet

Precip Increases with Elevation

Precip Decreases with Elevation

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S∝ A−θ

Simple Examples with CHILDPrecipitation varies linearly with elevation (uniform uplift/erosion).

Total volume of rain is the same in both landscapes.

Single outlet

Precipitation increases with elevation

20 km

80 km

m

Single outlet

Precipitation decreases with elevation

20 km

80 km

m

Precipitation varies with elevation.

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S∝ A−θ

High

Low

Spatially Variable Precipitation, Ellis, Densmore & Anderson, 1999

Pre

cip

Distance

Time Variant Precipitation(Tucker & Bras, 2000; Tucker 2004)

(see also Molnar 2001; Lague, Hovius and Davy, 2005)

Poisson Rainfall Model (Eagleson, 1978)

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f p( ) =1

Pexp −

p

P

⎛

⎝ ⎜

⎞

⎠ ⎟

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f tr( ) =1

Tr

exp −tr

Tr

⎛

⎝ ⎜

⎞

⎠ ⎟

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f tb( ) =1

Tb

exp −tb

Tb

⎛

⎝ ⎜

⎞

⎠ ⎟

Rainfall Intensity

Storm duration

Interstorm period

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Q = p − I( )A

Thresholds are important when modeling storm variation

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E = detachment rate (L/T)τ = shear stressτ c = critical shear stessk , a = parameters;if shear stress formulation, a =1if unit stream power formulation, a = 3/2

Detachment-Limited

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E = k τ − τ c( )a

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€

Qs = kW τ − τ c( )p

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E ∝∇Qs

Transport-Limited

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Qs = sediment transport rate (L3 /T)k = transport coefficientW = channel widthp = exponent ~ 1.5

What does a threshold do to erosion rates under conditions of stochastic storms?

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F var= rainfall variability, larger implies more extreme eventsP = mean storm rainfallP = mean annual rainfall

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F var= P / P

From Tucker (2004); calculated using mean storm intensity from the month with the greatest mean intensity

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P€

F varPhoenix, AZ

Astoria, OR

“extreme events become increasingly important in geomorphic systems with large thresholds” (Tucker & Bras 2000 and Baker 1977)

Transport-limited

What does a threshold do to erosion rates under conditions of stochastic storms?

Higher threshold

What does a threshold do to erosion rates under conditions of stochastic storms?

Transport-limitedDetachment-limited

What does a threshold do to channel concavity?Tucker (2004)

Detachment-limited

Higher thresholdS

lop

e

Drainage area

Transport-limited

Higher thresholdS

lop

e

Drainage area

Simulations from Tucker (2004)

Transport-limited

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τ *c = 0

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τ *c = 0.1

Storm variability may explain other mysteries about landscapes…

•Snyder et al (2003), Northern California - When stochastic rainfall was not considered, model could only reproduce slope characteristics of landscape using unrealistic erosion parameters. However, a stochastic rainfal model with an erosion threshold fit slope data quite nicely.

•Also, Baldwin et al (2003) found that the inclusion of stochastic storms with a transport-limited erosion model could produce longer lived topography in decaying landscapes, such as Appalachians.

Long Storm

Short StormS

lop

e

Drainage Area

Slo

pe

Slo

pe

Drainage Area

Drainage Area

What else? Non-steady-state discharge - Solyom and Tucker (2004)

Where do we go from here?

• Geomorphologists add more and more detail to fluvial erosion models. Sediment delivery, both from upstream and from hillslopes is a critical parameter to model.

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f Qs( )

€

QsQc

“tools” “cover”

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Qc constant

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I = Kf (Qs)τp

•Channel Width too.

But is the weakest link (climate, tectonics) already limiting what more we can learn from more detailed erosion models?

Where do we go from here?• Variation in storm intensity appears to be critical for

capturing extreme events.– What are we getting right/wrong about modeling storm

variability?– How will this effect landscape evolution with more

sophisticated erosion models (hillslopes, rivers, glaciers)?

• How important is spatial variability in rainfall?– Does spatially variable climate just mean spatially variable

rainfall intensity?– Sediment delivery to different parts of the landscape could

have profound affects on local erosion rates.

• Mapping precipitation to discharge - how far off are we?

• CAVEAT - Will coupled models of surface processes and CAVEAT - Will coupled models of surface processes and tectonics show that many of our assumptions about how tectonics show that many of our assumptions about how climate influences erosion are wrong/too simplistic?climate influences erosion are wrong/too simplistic?