## 1. Introduction

[2] Sediment transport involves the movement of particles as suspended load or bed load by flows such as rivers, estuaries, or ocean currents. Extreme flow events, resulting from heavy storms, breach of dams, overflows from drainage ditches or large vortices associated with scour holes, increase the unsteadiness of flows and thus have major geomorphological effects. Accordingly, sediment transport because of extreme flow events could do great harm to water quality, structures, and land resources and thus cause immense economic and environmental damage. Despite the significant impacts of extreme flow events on sediment transport, movement of sediment particles in response to extreme flow events has not been fully understood and well characterized [e.g., *De Sutter et al.*, 2001].

[3] Existing sediment transport models can be classified into suspended load transport models and bed load transport models according to the transport mode of particles. Bed load is transported primarily near the bed and is in contact with the channel bed for a considerable fraction of the time. In contrast, particles of suspended load supported by the turbulence of the mean flow reside primarily in the water column [*Dade and Friend*, 1998]. In principle, these models have employed deterministic differential equations or algebraic equations appropriate to the particle transport mode to estimate sediment concentrations or fluxes [*Syvitski and Alcott*, 1995; *Arnold et al.*, 1998; *Moulin and Slama*, 1998]. The stochastic approach in the analysis of bed load sediment movement was initiated by *Einstein* [1937, 1950] and *Kalinske* [1947]. In recent years, several studies have also attempted stochastic approaches. Among others, most of studies used the Monte Carlo simulations to model sediment movement with morphological responses of the channels [*Singer and Dunne*, 2004; *Van Vuren et al.*, 2005]. *Lisle et al.* [1998] have proposed to simulate sediment particle movement using a two-state Markov process. *Cheong and Shen* [1976] and *Hung and Shen* [1976] have established stochastic descriptions of sediment particles due to turbulence using probability density functions. Nevertheless, to our knowledge, the physically-based stochastic particle transport modeling remains at the developmental stage due to the complexity of the stochastic model formulations based on stochastic differential equations (SDEs) or stochastic partial differential equations and a lack of appropriate data for model validation.

[4] Individual particle transport in water flow has recently been described by the particle tracking models (PTMs). PTMs typically employ a Lagrangian approach whose observation system moves with fluid particles. In groundwater modeling, the earlier PTMs such as those of *Ahlstrom et al.* [1977] and *Prickett et al.* [1981] have been applied to particle random motion due to turbulence considering mainly the mean flow velocity. Thereafter further work on the PTMs has been accomplished considering the drift term because of the variability of the dispersion coefficient [e.g., *Uffink*, 1985; *Tompson and Dougherty*, 1988; *Tompson and Gelhar*, 1990; *Bagtzoglou et al.*, 1992a, 1992b; *Tchelepi et al.*, 1993; *Fabriol et al.*, 1993; *Tompson*, 1993; *Kitanidis*, 1994]. More recently, *Dimou and Adams* [1993] applied the PTM to pollutant transport in coastal waters. In surface water modeling, *Prakash* [2004] and *Kuchikulla* [2006] applied PTMs to algal transport modeling in the Great Lakes. Recently, a few studies such as those of *Man and Tsai* [2007] and *Iso and Kamemoto* [2008] attempted to simulate solid particle trajectories in flows by the particle-tracking approach. The governing equations of more recent PTMs normally employ two terms including a drift term due to mean flow and a random term due to turbulence, which represent deterministic and stochastic properties of particle movement, respectively. The PTMs are Lagrangian based and focused on the movement of individual particles so that the stochastic concept can be easily applied to each particle.

[5] Recently, the measuring techniques using optical methods in multiphase flows have been advanced so that the velocity lag of sediment particles can be observed in laboratory [*Phillips and Sutherland*, 1990; *Muste and Patel*, 1997; *Best et al.*, 1997]. For instance, the discriminator laser-Doppler velocimetry can be used to separately measure both particle and fluid velocity. *Greimann et al.* [1999] in an earlier study presented a theoretical analysis associated with observed particle velocity lag in response to flows. *Cheng* [2004] and *Jiang et al.* [2004] evaluated the velocity lag due to the drag force in sediment-laden regular open channel flow by experiments and numerical simulations. Although it is assumed that the sediment particle velocity is the same as the flow velocity in the majority of sediment transport models, the lag of sediment velocity has been considered in some recent attempts of sediment transport modeling. For example, *Wu et al.* [2006] attempted to simulate sediment transport considering the lag effects with the suspended and bed sediment transport equations. To explain the temporal lag, *Wu et al.* [2006] introduced a correction factor to suspended load concentrations, i.e., the ratio of the sediment velocity to flow velocity, into the storage term of the depth-averaged advection-diffusion suspended load transport equation. As a result, they estimated the reduced concentration of the mixture flow using a correction factor not the temporal lag itself. If the temporal lag of the particle velocity can be evaluated, we can then consider the effect of larger particles that are more affected by the time lag in response to flow changes.

[6] In this study the proposed model takes the temporal lag of sediment particles in response to extreme flow perturbations into consideration. In this paper extreme flow perturbations are referred to as particular extreme flow events with a time scale that is comparable to that of flow turbulence. The proposed model is then able to account for the delayed response of a particle in sediment transport modeling and to test the validity of a commonly adopted assumption in suspended load transport that sediment particles move simultaneously with fluid. The particle relaxation time, i.e., a time lag between sediment and fluid particles due to the inertia effect of sediment particles, is often observed during an extreme flow event. Such a time lag results in a delayed instantaneous particle velocity and a decreased mean particle velocity in a given time period. Moreover, since the temporal lag is due to the inertia effect of particles, the extent of particle retardation is dependent on the particle size and density. However, there are limited studies to date on the delayed response of sediment particles due to extreme flow conditions (including extreme flow events and large flow perturbations). The proposed model has an added advantage to simulate the particle movement in extreme flows while accounting for the delayed response of particles.

[7] To describe the stochastic properties of entities, researchers in various fields have employed the stochastic differential equation (SDE) such as an Ito process or a Langevin equation. The Ito process or the Langevin equation is a mathematical expression of stochastic processes associated with random phenomena such as suspended particle movement in a fluid or stock price fluctuations. Since the movement of particles is stochastic under the influence of turbulence, the SDE is appropriate to depict the movement of particles. There are several studies on sediment transport from the stochastic viewpoint in the area of hydraulics and hydrology. Most of the studies have focused primarily on statistical properties of inputs [*Kleinhans and van Rijn*, 2002; *Sharma and Kavvas*, 2005]. On the other hand, *Man* [2008] has attempted to develop a stochastic differential equation-based particle-tracking model for trajectory and concentration of suspended sediment particles without consideration of the particle relaxation time. Some studies have profoundly advanced the particle stochastic model for turbulent two-phase flows using the probability density function of particle location or velocity [*Pope*, 1994; *Pope*, 2000; *Minier and Peirano*, 2001; *Peirano et al.*, 2006].

[8] This paper is one of the early efforts in developing a stochastic jump diffusion model for sediment particle movement in open channel flows. The proposed stochastic particle-tracking model is able to account for the delayed response of particles in response to extreme flow perturbations and to test the validity of a commonly adopted assumption that sediment moves along with flow in the case of extreme flow perturbations. The main objectives of this paper are to develop a physically based stochastic approach that better describes the essential stochastic characteristics of the sediment particle movement in response to probabilistic occurrences of extreme flows and to quantify the impact of particle relaxation time (temporal lag) on the sediment particle movement during extreme flows.