Direction and Non Linearity in Non-local Diffusion Transport Models

1. Direction and Non Linearity in Non-local Diffusion Transport Models F. Falcini , V. Ganti V.1,2, R. Garra , E. Foufoula Georgiou, C. Paola, V.R. Voller University of Minnesota

2. Hydrological Examples of interest: Sediment Depositional Deltas Water and sediment input Sediment Fans 1km Main characteristic: Channels (at multiple scales) transporting and depositing sediment through and on system

3. ~3m An experimental system: Sediment input into standing water, in a subsiding basin “Jurasic Tank” Experiment system for building deltas Multi-scaled channelized surface with heterogeneities A “Two-D Porous Media” ? Will first look at non-local and non-linear effects in this system

4. A first order mass balance model: Geometry and Governing Equation

5. A first order mass balance model: Geometry and Governing Equation

6. A first order mass balance model: Geometry and Governing Equation q –input sed-flux balances subsidence Piston subsidence

7. A first order mass balance model: Geometry and Governing Equation q –input sed-flux balances subsidence Assume a diffusion model Sediment elevation above datum

8. ~3m Compare with experiment Diffusion solution “too-curved” When compared to experiment

9. One proposed Improvement is a non-linear model Posma et al., 2008 q –input sed-flux balances subsidence One solution use a non-linear diffusion model

10. Better Comparison with Experiment ------BUT Required value of betamuch smaller than expected theoretical value

11. Another solution is a non-local model Voller and Paola, 2010 q –input sed-flux balances subsidence use a non-local model Note: same form as non-linear Where: The right hand Caputo derivative is Interpret as weighted sum of down-- stream local slopes Why RIGHT HAND ?

12. Get the identical Comparison with Experiment ------BUT Difficult to know how to obtain value for alpha But non-locality is clearly in system so expected to be less than 1

13. Motivates development of a non-local non-linear NLNL model Current work with Fede Falcini and others q –input sed-flux balances subsidence use a NLNL model Note: same form as non-linear The weighted sum of down-stream non-linear slopes down The same form again ! But now The non-local “dilutes” the non-linearity--- To obtain same fit the non-local allows for a weaker non-linearity

14. But now The non-local “dilutes” the non-linearity--- To obtain same fit the non-local allows for a weaker non-linearity A range of alpha and beta values can achieve fit Including values of beta in theoretical range

15. What about direction in Non-local models? Consider a simple source to sink sediment transport model coolgeology.uk.com The Sediment Cycle weathering-erosion hill-slope by-pass transport deposition-burial delta subsidence uplift

16. 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. 0 1 divergence of flux Exner mass-balance normalize domain deposit thickness above datum Expected profile shape

17. Now model this combined erosion-depositional system with a fractional model coolgeology.uk.com use a general non-local model for flux And exam role of for fixed alpha (0.7) direction

18. Now model this combined erosion-depositional system with a fractional model And us a general non-local model for flux First gamma = 1—only upstream non-locality Control-information from upstream Correct shape and max location for fluvial surface In erosional domain

19. And us a general non-local model for flux First gamma = 1—only upstream non-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 !

20. And us a general non-local model for flux Now try gamma = -1—only down-str. non locality Control-information from downstream Correct shape and min location for fluvial surface In depositional domain

21. And us a general non-local model for flux Now try gamma = -1—only down-str. non locality Control-information from downstream Correct shape and max location for fluvial surface In depositional domain But incorrect shape in erosional domain maximum elevation not at continental divide !

22. 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 Voller et al, GRL 2012

23. 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

24. Direction and Non Linearity in Non-local Diffusion Transport Models Although difficult to quantify there is sufficient Physical evidence to suggest that Non-locality is present in Sediment transport systems “toy” models presented here have shown that Non-linear Non-local Non-locality Dilutes apparent Non-linearity NLNL The Direction of flow of information matters in non-local systems