Collaborative Research: Geomorphic transport laws, landscape evolution, and fractional calculus

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If physical models are to achieve a realistic description of transport, they need to incorporate the breadth and richness of the multi-scale heterogeneity present in natural landscapes, but currently they fail to do so. We will develop the mathematical tools needed for this task by adapting and extending fractional calculus to the modeling of sediment motion. Fractional calculus allows us to write differential equations of particle fluxes that incorporate, simply and directly, broad-tailed probability distributions of grain size, single particle hops and storage times on the bed. It does so by providing differential operators of fractional order, which means that changes in model variables are calculated regionally or non-locally, rather than at a point in space or time as in classical calculus. It is this non-locality that allows a concise treatment of the multi-scale, spatiotemporal heterogeneities of particle sizes, velocities, and transport distances seen in natural rivers and hillslopes. We will develop the analytical treatments and numerical methods necessary to build and solve generalized geomorphic transport laws towards improved predictions and modeling of landscape dynamics. If physical models are to achieve a realistic description of transport, they need to incorporate the breadth and richness of the multi-scale heterogeneity present in natural landscapes, but currently they fail to do so. We will develop the mathematical tools needed for this task by adapting and extending fractional calculus to the modeling of sediment motion. Fractional calculus allows us to write differential equations of particle fluxes that incorporate, simply and directly, broad-tailed probability distributions of grain size, single particle hops and storage times on the bed. It does so by providing differential operators of fractional order, which means that changes in model variables are calculated regionally or non-locally, rather than at a point in space or time as in classical calculus. It is this non-locality that allows a concise treatment of the multi-scale, spatiotemporal heterogeneities of particle sizes, velocities, and transport distances seen in natural rivers and hillslopes. We will develop the analytical treatments and numerical methods necessary to build and solve generalized geomorphic transport laws towards improved predictions and modeling of landscape dynamics. Sediment transport is a key element of change in the natural environment, and one whose importance is increasing under the pressure of human population growth, land use and climate change. This project has the potential to gain new insights into the mechanisms of sediment transport, that will enable better predictions of the results of river flooding and aid in disaster management.