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运输的现实描述，则需要将自然景观中存在的多尺度异质性的广度和丰富性结合在一起，但目前却没有这样做。 我们将通过调整和扩展分数计算到沉积物运动的建模来开发此任务所需的数学工具。 分数演算使我们能够编写粒子通量的微分方程，这些方程简单而直接地，直接，直接地，谷物尺寸，单个粒子啤酒花和床上存储时间的宽尾概率分布。 它通过提供分数顺序的差分运算符来做到这一点，这意味着模型变量的变化是在区域或非局部计算的，而不是在时空或时间点，如经典的微积分。 正是这种非本地性允许对粒度，速度和自然河流和山坡上的粒径，速度和运输距离的多尺度，时空异质性进行简洁处理。 我们将开发建立和解决广义的地貌运输定律所需的分析处理和数值方法，以改善景观动态的预测和建模。 如果物理模型要实现对运输的现实描述，则需要将自然景观中存在的多尺度异质性的广度和丰富性结合在一起，但目前却没有这样做。 我们将通过调整和扩展分数计算到沉积物运动的建模来开发此任务所需的数学工具。 分数演算使我们能够编写粒子通量的微分方程，这些方程简单而直接地，直接，直接地，谷物尺寸，单个粒子啤酒花和床上存储时间的宽尾概率分布。 它通过提供分数顺序的差分运算符来做到这一点，这意味着模型变量的变化是在区域或非局部计算的，而不是在时空或时间点，如经典的微积分。 正是这种非本地性允许对粒度，速度和自然河流和山坡上的粒径，速度和运输距离的多尺度，时空异质性进行简洁处理。 我们将开发建立和解决广义的地貌运输定律所需的分析处理和数值方法，以改善景观动态的预测和建模。沉积物运输是自然环境变化的关键要素，其重要性在人口增长，土地使用和气候变化的压力下增加。该项目有可能获得对沉积物运输机制的新见解，这将更好地预测河流洪水的结果并帮助灾难管理。