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From Watershed Hydrology to Landscape Evolution: A New Semi-Discrete 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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Example:

How do surface processes self-organize into such different landscapes?

Time & space scales: 102 -106 yrs

Badlands NW of Interior, South Dakota; photo by Louis J. Maher, Jr.

Colorado River tidal flats; National Geographic

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So what is the modeling problem?

RMB

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, but we need to…. 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

RMB

Catchment Scale 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 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) Use GIS tools to decompose horizontal projection of the study area into Delauney triangles (i.e., a TIN)

2) Project each triangle vertically to span the ‘‘active flow volume’’ forming a prismatic volume

3) Subdivide prism into layers to account for various physical process equations and materials

4) Use adaptive gridding

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A new strategy for integrated hydrologic and landscape modeling

4) Write down equations describing hillslope and channel surface processes

5) 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 semi-distributed approach, but with full coupling of all elements

Example: Conservation of Mass

Becomes….

V Stc

c¶

+ Ñ × =¶

2 3

1 1k i

k i

dQ Q

dtc

= =

= -å å

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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”

Solve global system by SUNDIALS (SUite of Nonlinear

and DIfferential/ALgebraic equation Solvers) or

PETSc (Portable, Extensible Toolkit for Scientific computation)

' ( , , , )M f x y tc c=

II o

i

dSP E P

dt

æ ö÷ç ÷= - -ç ÷ç ÷çè ø

3

1

( ) /ijo s oc

j i

dhP Q Q A

dt =

æ ö÷ç ÷ç = + - ÷ç ÷ç ÷çè øå

etc.

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Advantages

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 Canopy-interception Snowmelt runoff Evapotranspiration

Bucket Model II o

i

dSP E P

dt

æ ö÷ç ÷= - -ç ÷ç ÷çè ø

Temperature Index Model snowsnow

i

dSP E w

dt

æ ö÷ç ÷= - -ç ÷ç ÷çè ø&

Evapotranspiration: Pennman-Monteith Equation

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One Possible Realization: PIHMSed Subsurface

unsaturated flow Subsurface saturated

flow

0

30

1

Richard Equation

( ) /

si

ijg l gc

j i

dI q ET

dt

dq Q Q Q A

dt

x

z

=

æ ö÷ç = - - ÷ç ÷ç ÷è øæ ö÷ç ÷ç = + - + ÷ç ÷ç ÷çè ø

å

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One Possible Realization: PIHMSed Surface overland and

channel flows

( ) ( ) ( ) 2

1k

k

h uh vhq

t x y

r r rr

=

¶ ¶ ¶+ + =

¶ ¶ ¶å

( ) ( )( ) ( )( )

2 2 / 2ox fx

u h ghuh uvhgh S S

t x y

rr rr

¶ +¶ ¶+ + = - +

¶ ¶ ¶

( ) ( ) ( )( )( )

2 2 / 2oy fy

v h ghvh uvhgh S S

t x y

rr rr

¶ +¶ ¶+ + = - +

¶ ¶ ¶{

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One Possible Realization: PIHMSed Sediment transport and

bed evolution equations for non-cohesive

sediment from Cao et al. [2002] for illustration:

( ) ( ) ( )ch cuh cvhE D

t x y

¶ ¶ ¶+ + = -

¶ ¶ ¶

1

z D E

t l

¶ -=

¶ -

( )0 1m

D c cw= -

( ) ( )0.8

1160 c

c

dUE

hR

l q q

q¥- -

={

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One Possible Realization: PIHMSed Sediment transport

Detachment rate by rain-splash

Bed armoring Concept of active layer

( ) 2R rD c k h i=

Other processes? Downslope flux by tree-

throw 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 Domain decomposition

566 elements

Precipitation forcing Daily precipitation from

2004 repeated for 100 years

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Example: Definition of erosion “hotspots” in the Shale Hills CZO Initial conditions

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 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 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 physically-based, 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