1. The shallow water equations in geomorphic modeling
Guy Simpson
Earth Science
University of Geneva - Switzerland

2. Water-induced erosion ….. in the absence of water!
(drainage area - water discharge proxy)
Stream-power type model
(detachment-limited)
Transport-limited model
A (drainage area)
Flux = c An
Introduce poorly constrained parameters
Neglect flow dynamics of water and other important physics

3. Model of coupled shallow water flow and erosion-sedimentation
Simpson and Castelltort (2006) Computers and Geosciences

4. Shallow water flow over a mobile bed
water mass balance
x- force balance
y- force balance
Sediment mass balance
Kinematic relationship between bed fluxes and bed elevation
c sediment concentration in fluid
z bed topography
E entrainment flux (between bed and water)
D deposition flux
f porosity
r density of water-sediment mixture
(rw(1-c)+rsc)
ro density of saturated bed
(rwf+rs(1-f))
rw density of water
rs density of sediment
Additional forces in x direction
(e.g., bed friction, bed slope)

5. Empirical functions for E and D (for cohesionless material)If If
Shields parameter
Particle Reynolds number
Friction velocity

6. Solution obtained with finite volume method using a Riemann solver

7. Dam break calculation in 1-D

8. Dam break calculation in 2-D

9. Computed water surface (in meters) after 60 seconds

10. Velocity field of water and water depth contours after 60 seconds

11. Dam break over mobile bed (1-D simulation)

12. Water reservoir
dam

13. Coastal sediment transport due to a Tsunami
Wave : period = 30 min.
amplitude = 5 m
Gently sloping coast
Ocean

15. What are the (dis/)advantages relative to alternative formulations
Improved (richer) physics
- wide range of applications
- Flexible non-capacity sediment transport formulation
Avoid ad hoc parameters
Readily testable (bench-marking)
Disadvantages
More complicated to program (development time)
Computationally more expensive (longer model run time)
Numerical stability issues (paricularly for small water depths)
Other issues
Well developed numerical methods to solve the SWE
Easy to parallelise
Unstructured meshes
Other numerical techniques (e.g., Lattice Boltzmann)

18. Vector formulation

19. Finite volume method

20. Riemann solver