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

如果物理模型要实现运输的现实描述，它们需要纳入自然景观中存在的多尺度异质性的广度和丰富性，但目前它们未能做到这一点。 我们将开发这项任务所需的数学工具，调整和扩展分数阶微积分泥沙运动的建模。 分数阶微积分使我们能够编写微分方程的颗粒通量，合并，简单，直接，宽尾概率分布的粒度，单颗粒跳和存储时间的床上。 它通过提供分数阶的微分算子来实现，这意味着模型变量的变化是在区域或非局部计算的，而不是像经典微积分那样在空间或时间中的一点上计算。 正是这种非局部性，允许一个简洁的治疗的多尺度，时空异质性的颗粒大小，速度和运输距离在自然河流和山坡。 我们将开发必要的分析处理和数值方法，以建立和解决广义地貌输运规律，以改善景观动态的预测和建模。 如果物理模型要实现运输的现实描述，它们需要纳入自然景观中存在的多尺度异质性的广度和丰富性，但目前它们未能做到这一点。 我们将开发这项任务所需的数学工具，调整和扩展分数阶微积分泥沙运动的建模。 分数阶微积分使我们能够编写微分方程的颗粒通量，合并，简单，直接，宽尾概率分布的粒度，单颗粒跳和存储时间的床上。 它通过提供分数阶的微分算子来实现，这意味着模型变量的变化是在区域或非局部计算的，而不是像经典微积分那样在空间或时间中的一点上计算。 正是这种非局部性，允许一个简洁的治疗的多尺度，时空异质性的颗粒大小，速度和运输距离在自然河流和山坡。 我们将开发必要的分析处理和数值方法，以建立和解决广义地貌输运规律，以改善景观动态的预测和建模。沉积物迁移是自然环境变化的一个关键因素，在人口增长、土地使用和气候变化的压力下，其重要性日益增加。该项目有可能获得对沉积物输运机制的新见解，这将有助于更好地预测河流洪水的结果，并有助于灾害管理。

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.