Non-local Transport in Channel Networks Vaughan Voller Civil Engineering University of Minnesota

1. Non-local Transport in Channel Networks Vaughan Voller Civil Engineering University of Minnesota source coolgeology.uk.com sink and help from his “collective” Work with Tetsuji Muto, Wonsuck Kim, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man Liang Matt Wolinsky, Colin Stark, Andrew Fowler, Doug Jerolmack, Dan Zielinski Vamsi Ganti, Chris Paola, Efi Foufoula

2. It is a truth universally acknowledged1 that the transport of sediment in the landscape can have a significant effect on the safety and wellbeing of humankind hhwa.dot.gov Wonsuck Kim, EOS 2010 Upland Debris Flows Ocean Delta Building Let us first look at modeling deltas J. Austen 1813

3. Examples of Sediment Deltas Water and sediment input Sediment Fans 1km Main characteristic: Channels (at multiple scales) transporting material through system

5. Land profile view Water advancing shore-line A simple model for sediment transport in delta sediment flux land water bed-rock

6. Sediment Transport and Diffusion 101 x Driver--sediment flux [volume/length-time] A linear diffusion model good place to start (Paola 92) Balance of flux across A LOCAL model -local slope land Bed-rock subsidence water bed-rock Exner balance

7. How well does this model work ? ~3m “Jurasic Tank” Experiment at close to steady state diffusion Diffusion solution “too-curved” subsidence

8. But in Experiment Assumption in Model Heterogeneity occurs at all scales Up to an including the domain. REV can not be identified --scales are power-law distributed ~3m Volume over which average properties can be applied globally. Is this equation valid Clear separation between scale of heterogeneity and domain. An REV can be identified Suggests a “non-local” Model

9. An aside: A simple example of non-local transport: A block sliding on an oil film down an inclined At equilibrium: down-slope weigh balanced by up-slope shear-stress velocity LOCAL slope

10. Now consider a series of 3 rigidly connected blocks on different slopes At equilibrium: velocity of system 3 2 1 Then Velocity of Block 1 Or A WEIGHTED SUM OF UPSTREAM SLOPES VELCOITY DEPENDS ON NON-LOCAL SLOPES

11. But in Experiment Assumption in Model Heterogeneity occurs at all scales Up to an including the domain. REV can not be identified --scales are power-law distributed ~3m Volume over which average properties can be applied globally. Is this equation valid Clear separation between scale of heterogeneity and domain. An REV can be identified Suggests a “non-local” Model

12. How can we conceptualize a non-local model? Cannels arriving at X-X have a distribution of up stream lengths Flux in a given channel is controlled by slope at channel head Or in limit X X One possible set of power law weights gives

13. A second aside: Fractional Derivatives Basic Calculus Integral of second derivative is first derivative Cauchy Repeated integral nth integral of n-1th is 1st derivative Generalize to non-integer case The 1-alpha integral of the first derivative is the alpha order fractional derivative Definition by analogy

14. How can we conceptualize a non-local model? Cannels arriving at X-X have a distribution of up stream lengths Flux in a given channel is controlled by slope at channel head Or in limit X X One possible set of power law weights gives

15. But in reality Heterogeneity occurs at all scales Up to an including the domain. REV can not be identified Non-local sediment transport measure of locality Model (local) 0 1 suggests we need a non-local model, e.g., A “weighted” sum of upstream slopes flux ~ local slope Flux at a point depends on slopes at up-stream locations --information is forwardly propagated Can do the opposite –have a downstream dependence--backward propagation Weighted sum of downstream slopes

16. So for general non-local treatment we model flux as 0 1 Locality direction (all up-stream) upstream dependence and/or downstream dependence Introduce some nomenclature The Caputo fractional Derivative of order alpha Can be interpreted as the integral of the 1st derivative Then non-local governing transport equation has the form Note In scaled domain

17. Application a source to sink sediment transport model coolgeology.uk.com The Sediment Cycle weathering-erosion Key variable sediment flux hill-slope by-pass transport deposition-burial delta subsidence uplift

18. A first order model Mass Balance Model (divergence of flux) Eliminate by-pass -region Model with Exner Equation erosion-uplift deposition/ subsidence erosion/ uplift depo.-sub. divergence of flux Exner mass-balance deposit thickness above datum 0 1 normalize domain

19. Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta) i.e., sediment flux ~ slope Easy-solution Uplift: Sub: Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1 Consistent with field and lab But---- Surfaces may be too-”curved” erosion deposition

20. Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta) i.e., sediment flux ~ slope At this point I can go one of two ways: Easy-solution • I could add more physics, and • features to provide a better • match with reality but with • more parameters n • --OR 2. Explore the model space of this simple construct and see how much it might be able to inform about possible system behaviors Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1 I am in the modeling school that believes Understanding Parameters Consistent with field and lab But---- Surfaces may be too-”curved” So I will do 2 erosion deposition

21. Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta) i.e., sediment flux ~ slope Easy-solution Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1 Consistent with field and lab But---- Surfaces may be too-”curved” erosion deposition

22. Consider—non-local depositional system with down-stream dependence beta=-1 —sub. rate Can be fit to observations Voller and Paola JGR 2010 Before After

23. But the BIG question remains Is this non-local model physically meaningful ? Some good evidence— Channels scales are known to be fractal (power-law scaling) pdf’s --e.g., measured sed. transport at a point over time is thick tailed But no direct measure of locality value alpha Also--Can we extend the treatment to the hillslope? (YES-- Vamsi Ganti et al. JGR 2010) And what is the effect of the Locality direction (beta)?

24. To answer last question let us return to our combined erosion-depositional system coolgeology.uk.com use a general non-local model for flux And exam role of Beta for fixed alpha (0.7)

25. To answer last question let us return to our combined erosion-depositional system And us a general non-local model for flux First Beta = 1—only upstream locality Control-information from upstream Correct shape and max location for fluvial surface In erosional domain

26. To answer last question let us return to our combined erosion-depositional system And us a general non-local model for flux Beta = 1—only upstream locality Control-information from upstream Correct shape and max location for fluvial surface In erosional (hillslope) domain But incorrect shape in depositional domain minimum elevation not at sea-level !

27. To answer last question let us return to our combined erosion-depositional system And us a general non-local model for flux Now try Beta = -1—only downstream locality Control-information from downstream Correct shape and min location for fluvial surface In depositional domain

28. To answer last question let us return to our combined erosion-depositional system And us a general non-local model for flux Now try Beta = -1—only downstream locality Control-information from downstream Correct shape and mix location for fluvial surface In depositional domain But incorrect shape in erosional domain maximum elevation not at continental divide !

29. To answer last question let us return to our combined erosion-depositional system coolgeology.uk.com IN fact Only physically reasonable solutions UNDER FRAC. DER. MODEL OF NON-LOCALITY Require that locality points upstream in The erosional domain but needs to point Downstream in the depositional domain. Transport controlled by upstream features in erosional regime but controlled by downstream features in depositional domain

30. Is there a distinguishing feature between these regimes that may explain this switch in The direction of transport (flow of information) ---- Depositional domain Diverges information down-stream Erosional domain Converges information down-stream A win-win The MATH is suggesting something interesting about nature If confirmed this could have important consequences for our understanding earth-surface Dynamics and transport in channel systems-- If invalidated might require rethinking and assessment of current non-local transport models Direction matters in non-local systems

31. To end of a Philosophical note--- Math Modeling can be used in two ways validation Data Driven validation mathematical construct physical description/hypothesis  observation  e.g., laminar-turbulent transition Theory Driven mathematical  construct observation ? physical description/hypothesis  e.g., relativity Both approaches offer valid methods for advancement of our understanding

32. Delta growth with Channel formation Sediment transport rules Coupled to simplified Shallow water solver. Man Liang— With Paola and Voller

33. Sediment Transport and Diffusion 101 Diffusive Flux x Divergence of flux across This divergence of flux can be balanced by or surface rise subsidence land bed-rock water bed-rock Diffusive Exner Equation

34. A One D Experiment mimicking building of delta profile, Tetsuji Muto and Wonsuck Kim Sediment and Water Mix introduced into a slot flume (2cm thick) with a fixed Sloping bottom and fixed water depth shore-line and sediment/rock boundary moves in response to sediment input h(t) Can model with a diffusion equation (in terms of sediment height h) between two moving boundaries—the shoreline Ssh and the sediment/rock Sba Exhibits Closed Form Solution !

35. Jorge Lorenzo Trueba, et al J. Fluid Mech. (2009), vol. 628, pp. 427–443 Experiment vs. Analytical: VALIDATION experimental analytical Time s Position mm