Landscape Modeling
Nicole Gasparini
Arizona State University
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™ an d a
decompressor
are need ed to see this picture.
From
Greg
Tucker’s
Website
Attributes of Every Node:
z, elevation
nodes a, node area
edges A, drainage area = ai
Q, incoming fluvial discharge
Qs, incoming sediment load
Qsin, Qin from erosion upstream
S, downstream slope
z
Qsout, U E H
Qout t
Drainage Channel
Area, Profile,
increases slope
decreases
down-
stream down-
stream
Outlet
Outlet
Fluvial Erosion Model -
Detachment-limited model for incision into bedrock
E kb c
a
Shear Stress
Q “…force balance for steady,
S uniform flow in a wide channel”,
W Tucker, 2004
E = erosion rate (length/time)
= bed shear stress
c = threshold value
b = erodibility; f(lithology, process); stronger rock, smaller kb
k
a = positive constant
Q = fluvial discharge
W = channel width
, = positive constants, about equal
Discharge Relationship:
Q Pa i i or Q PA c Discharge-Area Relationship,
Hydrologic Steady State
i
Q = Fluvial Discharge
P = Effective Precipitation Rate = Rainfall - losses
positive constant 1
c =
Q = 0.0171*A0.9932 River basins
R2 = 0.9977 in Kentucky,
USA, from
Solyom and
Tucker, 2004
Channel Width:
b
W Q Hydraulic Geometry (e.g. Leopold &
Maddock 1953)
W Channel Width
Q = Fluvial Discharge PA c
b = positive constant ~ 0.5
Data from the
Q = 0.1335 * A0.9
Clearwater
W=4.2*A0.42 River,
Washington
State, from
Tomkin et al.,
2003.
Combining previous relationships with some parameter
value assumptions…
0.5
E kb (PA) S Functional form of erosion
equation in numerical
models, ignore thresholds
for now.
E Slope-Area relationship -
S 0.5
A0.5 Channel slopes (& relief) are
kb P 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
E 0.5
S 0.5
A
kb P
Uniform precipitation in space and time. Differences
between “more erosive (higher precipitation) and less
erosive climates”.
E 0.5
Lower Precip, S 0.5
A
more relief kb P
Higher Precip
less relief
Does topography influence local climate?
Spatially Variable Precipitation
Roe, 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…”
x
dA( x')
Q( x) p( x')
dx'
dx'
0
Spatially Variable Precipitation
Roe, Montgomery & Hallet, 2002
Precip Increases with Elevation
outlet
Precip Decreases with Elevation
Precip Increases with Elevation
Precip Decreases with Elevation
S A
Simple Examples with CHILD
Precipitation varies linearly with elevation (uniform uplift/erosion).
Total volume of rain is the same in both landscapes.
Precipitation increases with elevation Precipitation decreases with elevation
m m
80 km 80 km
20 km 20 km
Single outlet Single outlet
Precipitation varies with elevation.
S A
High
Low
Spatially Variable Precipitation,
Ellis, Densmore & Anderson, 1999
Precip
Distance
Time Variant Precipitation
(Tucker & Bras, 2000; Tucker 2004)
(see also Molnar 2001; Lague, Hovius and Davy, 2005)
Poisson Rainfall Model (Eagleson, 1978)
Rainfall Intensity
1 p
f p exp
Q p I A
P P
Storm duration
1 t
f tr exp r
Tr Tr
Interstorm period
1 t
f tb exp b
Tb Tb
Thresholds are important when modeling storm variation
Detachment-Limited Transport-Limited
E Qs
a
E k c
Qs kW c
p
E = detachment rate (L/T) 3
= shear stress Qs = sediment transport rate (L /T)
c = critical shear stess
k = transport coefficient
k, a = parameters; W = channel width
if shear stress formulation, a 1 p = exponent ~ 1.5
if unit stream power formulation, a 3/2
What does a threshold do to erosion rates under
conditions of stochastic storms?
F var= rainfall variability,
F var P / P larger implies more extreme events
P = mean storm rainfall
P = mean annual rainfall
From Tucker
(2004); F
var
Phoenix, AZ
calculated using
mean storm
intensity from
the month with
Astoria, OR
the greatest
mean intensity
P
What does a threshold do to erosion rates under
conditions of stochastic storms?
Transport-limited
Higher
threshold
“extreme events
become increasingly
important in
geomorphic systems
with large thresholds”
(Tucker & Bras 2000
and Baker 1977)
What does a threshold do to erosion rates under
conditions of stochastic storms?
Detachment-limited
Transport-limited
What does a threshold do to channel concavity?
Tucker (2004)
Detachment-limited Transport-limited
Slope
Slope
Higher
Higher threshold
threshold
Drainage area Drainage area
Simulations from Tucker (2004)
Transport-limited
*c 0 *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.
What else? Non-steady-state discharge -
Solyom and Tucker (2004)
Slope
Long Storm
Drainage Area
Slope
Drainage Area
Short Storm
Slope
Drainage Area
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.
Qc constant
I Kf (Qs ) p “tools” “cover”
f Qs
Qs
Qc
•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
tectonics show that many of our assumptions about how
climate influences erosion are wrong/too simplistic?