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

合作研究：地貌传输定律、景观演化和分数阶微积分

• 批准号: 0823953
• 负责人: Colin Stark
• 金额: $ 9.97万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0823953&HistoricalAwards=false
• 关键词: Collaborative; Research; Geomorphic; transport; laws

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.

如果物理模型要实现对交通的现实描述，则需要考虑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。 我们将通过调整和扩展分数阶微积分来模拟沉积物运动来开发这项任务所需的数学工具。 分数阶微积分使我们能够编写粒子通量的微分方程，该方程简单而直接地包含了粒度、单粒子跳跃和床上存储时间的宽尾概率分布。 它通过提供分数阶微分算子来实现这一点，这意味着模型变量的变化是区域性或非局部性计算的，而不是像经典微积分中那样在空间或时间点上计算。 正是这种非局域性使得我们能够对自然河流和山坡中所见的颗粒大小、速度和传输距离的多尺度、时空异质性进行简洁的处理。 我们将开发建立和解决广义地貌传输定律所需的分析处理和数值方法，以改进景观动力学的预测和建模。 如果物理模型要实现对交通的现实描述，则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。 我们将通过调整和扩展分数阶微积分来模拟沉积物运动来开发这项任务所需的数学工具。 分数阶微积分使我们能够编写粒子通量的微分方程，该方程简单而直接地包含了粒度、单粒子跳跃和床上存储时间的宽尾概率分布。 它通过提供分数阶微分算子来实现这一点，这意味着模型变量的变化是区域性或非局部性计算的，而不是像经典微积分中那样在空间或时间点上计算。 正是这种非局域性使得我们能够对自然河流和山坡中所见的颗粒大小、速度和传输距离的多尺度、时空异质性进行简洁的处理。 我们将开发建立和解决广义地貌传输定律所需的分析处理和数值方法，以改进景观动力学的预测和建模。泥沙输送是自然环境变化的一个关键因素，在人口增长、土地利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泥沙输送机制的新见解，从而更好地预测河流洪水的后果并帮助灾害管理。