The Impact of Truncated Lévy Jump Distribution on CTRW in Fluid Fields

连续时间随机漫步是布朗随机漫步的自然推广，它允许合并等待时间分布␺和一般跳跃分布函数。我们知道，要想进一步研究连续时间随机游可以通过研究跳跃步长概率密度函数以及等待时间概率密度函数的性质。将连续时间随机漫步推广到流动流体场中也是一大热点研究命题。这种推广可以导出在运动流体中连续时间随机漫步模型的宏观方程。在这里，我们考虑具有指数截断莱维跳跃分布的连续时间随机漫步，在这种情况下，流体极限导致具有指数截断分数阶导数的运输方程，该方程描述了中间渐进状态下记忆、跳跃和截断效应之间的相互作用。该模型在流体力学中具有重要应用，例如可描述污染物在湍流边界层的输运：高速流场主导平流输送，湍流脉动引发指数截断的莱维跳跃，而颗粒在涡旋结构的暂留形成幂律等待时间。通过调节参数，可量化评估海洋溢油扩散中早期快速扩展与后期受海流约束的竞争机制，为预测污染范围提供理论依据。Continuous-time random walk is a natural extension of the Brownian random walk, allowing for the combination of waiting time distribution and general jump distribution functions. We know that to further study continuous-time random walks, one can investigate the properties of the jump step length probability density function and the waiting time probability density function. Extending continuous-time random walks to fluid flow fields is also a major research topic. This extension can lead to the macroscopic equations of the continuous-time random walk model in moving fluid. Here, we consider continuous-time random walks with exponentially truncated Lévy jump distributions. In this case, the fluid limit leads to a transport equation with an exponentially truncated fractional derivative, which describes the interaction between memory, jumps, and truncation effects in the intermediate asymptotic state. This model has important applications in environmental fluid mechanics, such as describing the transport of pollutants in turbulent boundary layers: high-speed flow fields dominate advection transport, turbulent pulsations cause exponentially truncated Lévy jumps, and particles’ residence in vortex structures form power-law waiting times. By adjusting parameters, the competition mechanism between the early rapid expansion and the later constraint by ocean currents in the diffusion of marine oil spills can be quantitatively evaluated, providing a theoretical basis for predicting the pollution range.