1 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

2 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

3 Example: Dynamic Hydrology How does hydrologic system change if weathering of bedrock and erosion of sediment are dramatically increased? Time & space scales: yrs

4 Example: How do surface processes self- organize into such different landscapes? Time & space scales: yrs Badlands NW of Interior, South Dakota; photo by Louis J. Maher, Jr. Colorado River tidal flats; National Geographic

5 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).

6 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.

7 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

8 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….

9 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) etc.

10 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

11 One Possible Realization: PIHMSed Canopy-interception Snowmelt runoff Evapotranspiration

12 One Possible Realization: PIHMSed Subsurface unsaturated flow Subsurface saturated flow

13 One Possible Realization: PIHMSed Surface overland and channel flows

14 One Possible Realization: PIHMSed Sediment transport and bed evolution equations for non-cohesive sediment from Cao et al. [2002] for illustration:

15 One Possible Realization: PIHMSed Sediment transport Detachment rate by rain- splash Bed armoring Concept of active layer Other processes? Downslope flux by tree- throw Etc.

16 Example: Definition of erosion “hotspots” in the Shale Hills CZO from Henry Lin

17 Example: Definition of erosion “hotspots” in the Shale Hills CZO Domain decomposition 566 elements Precipitation forcing Daily precipitation from 2004 repeated for 100 years

18 Example: Definition of erosion “hotspots” in the Shale Hills CZO Initial conditions Lower third of regolith is saturated overland flow and stream flow depth = m sediment load = 0 Sediment sizes: , 0.002, and 0.02 m Boundary conditions No-flow around the watershed perimeter Weir condition at stream outlet

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

20 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