Analytical Solution for Anomalous Diffusion of Bedload Tracers Gradually Undergoing Burial
Analytical Solution for Anomalous Diffusion of Bedload Tracers Gradually Undergoing Burial
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DOI：
10.1029/2018jf004654
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发表时间：
2019-01
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通讯作者：
Zi Wu;E. Foufoula‐Georgiou;G. Parker;Arvind Singh;Xu-dong Fu;Guangqian Wang
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文献类型：
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作者：
Zi Wu;E. Foufoula‐Georgiou;G. Parker;Arvind Singh;Xu-dong Fu;Guangqian Wang
考虑推移质输沙过程中示踪剂颗粒的埋藏是河流示踪剂扩散公式的重要组成部分。在这里，我们提出了一个修正的活动层公式，它考虑了埋藏的影响，并提供了解析解，使得能够在中间时间尺度上对推移质示踪的超扩散现象进行深入的探索。这一现象已经在最近使用基于二维Exner的主方程的数值结果中观察到。通过假设活化层中的示踪剂可以与基准层中的非示踪剂粒子交换以保持质量，并且进入基准层的示踪剂在分析时间尺度内被永久捕获，我们能够推导出两层中示踪剂浓度的控制方程。活化层示踪剂浓度受具有汇项的平流-扩散方程控制，衬底层示踪剂的增加受相应源项的驱动。示踪群方差的解是解析确定的，可以用中间时间尺度上的扩散诱导标度项(∝T1)和平流诱导标度项(∝T3)之和来近似，这解释了超扩散现象。通过与现有的数值模拟结果的比较，表明所提出的公式能够捕捉到示踪剂传输的关键特征。
Accounting for the burial of tracer particles during bedload transport is an important component in the formulation of tracer dispersal in rivers. Herein we propose a modified active layer formulation, which accounts for the effect of burial and admits analytical solutions, enabling insightful exploration of the phenomenon of superdiffusion of bedload tracers at the intermediate timescale. This phenomenon has been observed in recent numerical results using the 2‐D Exner‐Based Master Equation. By assuming that tracers in the active layer can exchange with nontracer particles in the substrate layer to preserve mass, and that tracers entering the substrate layer get permanently trapped during the timescale of analysis, we are able to deduce governing equations for the tracer concentration in both layers. The active layer tracer concentration is shown to be governed by an advection‐diffusion equation with a sink term, and the increase of tracers in the substrate layer is driven by a corresponding source term. The solution for the variance of tracer population is analytically determined and can be approximated by the sum of a diffusion‐induced scaling (∝t1) and an advection‐induced scaling (∝t3) terms at the intermediate timescale, which explains the phenomenon of superdiffusion. The proposed formulation is shown to be able to capture the key characteristics of tracer transport as inferred by comparison with available results of numerical simulations.