485 - Fractional Stochastic Particle Tracking Model with Long-term Memory : A Numerical Investigation on Differentiable Probabilistic Trajectories Achieved by Malliavin-calculus-based Fractional Brownian Motion (fBm)

This study aims to develop a fractional stochastic particle tracking model (FSD-PTM) to establish probabilistic particle trajectories by solving the Ito-type Langevin equation. The study considers the position of suspended sediment particles, which is described as time-dependent and represented as random variables based on the solution of the stochastic differential equation. When immersed in fully developed turbulent flow, particles exhibit chaotic behaviors due to being carried by turbulent coherent structures simultaneously. To account for this, the study incorporates fractional Brownian motion (fBm), a long-term dependent stochastic process, to simulate the random variable with a memory effect. In other words, particles are assumed to retain the influence caused by eddies rather than moving randomly. This approach allows for investigating sediment resuspension mechanisms from a novel perspective without introducing additional mechanisms.
The study utilizes smooth fBm achieved through the Malliavin calculus-based method with Wiener-Ito expansion to accurately differentiate the position random variable. This Malliavin calculus-based method smoothens the random variable, enabling the extension of the study's results from particle position to other variables such as velocity and particle shear stress. The study validates the position using the concentration method proposed by Noguchi and Nezu (2009) and verifies the velocity using the method suggested by Muste et al. (2009). These validations ensure the applicability of the Malliavin calculus approach. For future work, the study plans to simulate the velocity Langevin equation and integrate the velocity random variables to obtain the position, allowing for the linkage between velocity and position Langevin equations.