Transcript From Watershed Hydrology to Landscape Evolution: A New Semi- Intermediate Complexity

From Watershed Hydrology to
Landscape Evolution: A New SemiDiscrete Finite Volume Model of
Intermediate Complexity
Chris Duffy, Shuangcai Li, Mukesh Kumar,
Yizhong Qu, and Rudy Slingerland
Departments of Civil and Environmental
Engineering & Geosciences
The Pennsylvania State University
July 2008
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What is our Objective?

To understand how Earth surface systems form
spontaneously in response to their internal
dynamics

To understand
how Earth
surface systems
couple across
large time and
space scales
Albert Bierstadt; Rocky Mountains
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Example: Dynamic Hydrology

How does
hydrologic system
change if
weathering of
bedrock and erosion
of sediment are
dramatically
increased?

Time & space
scales: 101 -104 yrs
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Badlands NW of Interior, South Dakota; photo by Louis J. Maher, Jr.
Example:

How do surface
processes selforganize into such
different
landscapes?

Time & space
scales: 102 -106 yrs
Colorado River tidal flats; National Geographic
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So what is the modeling
problem?

We want a physically-based, spatially-distributed, hydrologic & sediment
routing model that is morphodynamic, captures all relevant processes at the
precipitation event time-scale….. and simulates thousands of years

Being mindful that….
“….attempting to extract the dynamics at higher levels from
comprehensive modeling of everything going on at lower levels is……like
analyzing the creation of La Boheme as a neurochemistry problem.”
--Chris Paola (2000)

We think a continuum approach is going to work,
RMB but we need to….
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improve representations of morphodynamic processes,
correctly and efficiently treat strongly coupled effects spanning wide ranges of spatial and
temporal scales,
acknowledge that the defect rate for large communal codes is about seven faults per 1000
lines of FORTRAN (Hatten and Roberts, 1994).
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Earlier Approaches

Catchment Scale
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ANSWERS –Bierly et al.
CREAMS – Alonso, Knisel, et al.
SHESED – Wicks & Bathurst
KINEROS – Woolhiser et al.
EUROSEM – Morgan et al.
InHM – Heppner et al.
Landscape Scale




RMB
SIBERIA –Willgoose et al.
GOLEM/CHILD – Tucker
CASCADE – Braun et al.
CAESAR -- Coulthard et al.
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A new strategy for integrated
hydrologic and landscape modeling
1)
2)
3)
4)
Use GIS tools to decompose
horizontal projection of the
study area into Delauney
triangles (i.e., a TIN)
Project each triangle
vertically to span the ‘‘active
flow volume’’ forming a
prismatic volume
Subdivide prism into layers to
account for various physical
process equations and
materials
Use adaptive gridding
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A new strategy for integrated
hydrologic and landscape modeling
4)
5)
Write down equations
describing hillslope and
channel surface
processes
Use semi-discrete finite
volume method to
transform the PDEs into
ODEs


For small-scale numerical
grids, FVM yields contiuum
constitutive relationships
For larger grids the method
reflects assumptions of semidistributed approach, but with
full coupling of all elements
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Example:

Conservation of Mass
¶c
+ Ñ ×cV = S
¶t
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Becomes….
dc
=
dt
2
å
3
Qk -
k= 1
å
Qi
i= 1
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A new strategy for integrated
hydrologic and landscape modeling
6)
Assemble all ODEs within a
prism, each associated with its
appropriate layer(s)
1)

“local system”
Combine the local system over
the domain of interest into a
“global system”
ædS I
ö
÷
çç
÷
=
P
E
P
I
o÷
ççè dt
÷
ø
i
3
ædh
ö
÷
çç
ij
÷
=
P
+
(
Q
Q
)
/
A
÷
å s
çç
o
oc
÷
÷
è dt
øi
j=1
etc.

Solve global system by
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SUNDIALS (SUite of Nonlinear
and DIfferential/ALgebraic
equation Solvers) or
PETSc (Portable, Extensible
M c ' = f ( c , x , y, t )
Toolkit for Scientific
computation)
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Advantages


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
Mass conservation at all elements
All major hydrologic and sediment transport
processes fully coupled into one ODE
system
Interactions treated as internal terms on the
right hand side of ODE system
Flexible model kernel
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One Possible Realization:
PIHMSed
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Canopy-interception
Snowmelt runoff
Evapotranspiration
Bucket Model
T emperat ure Index Model
Evapot ranspirat ion:
ædS I
ö
÷
çç
÷
÷
ççè dt = P - E I - Po ø
÷
i
ædS snow
ö÷
çç
&÷
ççè dt = P - E snow - w ø÷
÷
i
P ennman-Mont eit h Equat ion
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One Possible Realization:
PIHMSed


Subsurface
unsaturated flow
Subsurface saturated
flow
Richard Equat ion
æd x
ö
÷
çç = I - q 0 - ET ÷
s÷
çè dt
÷
øi
3
æd z
ö
÷
çç
0
ij
= q + ( å Qg - Ql + Qgc ) / A ÷
÷
çç
÷
÷
è dt
øi
j=1
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One Possible Realization:
PIHMSed

Surface overland and
channel flows
¶ (r h )
¶t
+
¶ (r uh )
¶t
¶ (r vh )
¶t
+
+
¶ (r uh )
¶x
+
¶ (r vh )
¶y
2
=
å
k= 1
¶ (r (u 2h + gh 2 / 2 ))
¶x
¶ (r uvh )
¶x
+
r qk
+
¶ (r uvh )
¶y
¶ (r (v 2h + gh 2 / 2 ))
¶y
= - r gh (S ox + S fx
)
= - r gh (S oy + S fy )
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One Possible Realization:
PIHMSed

Sediment transport and
bed evolution
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equations for non-cohesive
sediment from Cao et al.
[2002] for illustration:
¶ (ch )
¶t
+
¶ (cuh )
¶x
+
¶ (cvh )
¶y
= E - D
¶z
D- E
=
¶t
1- l
m
D = w0 (1 - c ) c
160 (1 - l
E =
qc
R 0.8
)(q -
qc )dU ¥
h
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One Possible Realization:
PIHMSed

Sediment transport

Detachment rate by rainsplash
DR = cr k (h )i 2
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Bed armoring
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Concept of active layer
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Other processes?

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Downslope flux by treethrow
Etc.
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Example: Definition of erosion
“hotspots” in the Shale Hills CZO
from Henry Lin
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Example: Definition of erosion
“hotspots” in the Shale Hills CZO
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Domain decomposition
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566 elements
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Precipitation forcing
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Daily precipitation from
2004 repeated for 100
years
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Example: Definition of erosion
“hotspots” in the Shale Hills CZO
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Initial conditions
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Lower third of regolith is saturated
overland flow and stream flow depth = 10-6 m
sediment load = 0
Sediment sizes: 0.0004, 0.002, and 0.02 m
Boundary conditions

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No-flow around the watershed perimeter
Weir condition at stream outlet
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Example: Definition of erosion
“hotspots” in the Shale Hills CZO
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Conclusions

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A rich class of problems requires knowing how
Earth surface systems form spontaneously in
response to their internal dynamics
To solve these problems we need a physicallybased, spatially-distributed, morphodynamic
water & sediment routing model of catchment to
river basin scale
The mathematical know-how already exists; it is
the process laws that require work
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