# Lagrangian Modeling of Marine Microplastics Fate and Transport: The State of the Science
## 1 Introduction
Furthermore, numerical models can simulate some key physical processes using fine grid resolution, given the ever-increasing computational capacity and the ongoing development of numerical techniques. Moreover, these models are capable of simulating different vertical and horizontal flow patterns encompassing microplastics with good accuracy. Numerical simulations of microplastics' fate and transport in water are generally carried out using Eulerian and Lagrangian models. Unlike Lagrangian trajectory models that use a movable coordinate system in space, Eulerian grid models typically employ a non-movable frame of reference [58]. In other words, a Lagrangian framework, known as the particle-tracking method, tracks only individual particles, while an Eulerian approach addresses advection and diffusion for given locations (computational nodes) [78]. Numerical Eulerian hydrodynamic models can be coupled with Lagrangian models. Accordingly, hydrodynamic modeling provides the ambient hydrological parameters as inputs for particle-tracking models (PTM), which are known as promising, cost-effective tools for interpreting the fate and transport of microplastics and providing better insights into microplastics' behavior in dynamic flow systems [79]. Although some studies have used an Eulerian approach for the mixing and dispersion processes in dynamic water systems [80], some other studies have used Lagrangian models for tracking particles to investigate plastic particles' fate and transport [58]. Lagrangian models solve for a lower number of equations per node so that they do not need to solve the equations for each location in the mesh. However, given the pros and cons of both Eulerian and Lagrangian approaches, they can be employed to complement each other. The particle-tracking model coupled with the hydrodynamic model can provide invaluable details about the potential trajectories of plastic litter from local [81; 82; 83] to global scales [70] that lead to the assessment of microplastics' transportation, distribution, residence time, and fate under various conditions [58; 67; 70; 84; 85; 86; 87]. Particle-tracking methods simulate the trajectory of particles and their transport in water bodies. They are applicable to a wide range of plastic particles' transport in their surrounding environment [78; 88]. PTM has been employed in many studies as a reliable alternative for field surveys. This is due to the logistic challenges involved in field surveys [70; 72; 89; 90; 91; 92]. Most Lagrangian particle-tracking models do not simulate the flow hydrodynamics. They can be considered as an important tool to simulate physical processes and transport mechanisms that microplastics undergo. Unlike the structured mesh, the unstructured mesh enables numerical models not only to mesh complex geometries more easily but also to address arbitrary positions, even though it should be noted that greater memory is needed for executing the simulation for unstructured mesh. Hydrodynamic values in each element of the Eulerian mesh are interpolated in the same location of the particle (as the host element) in PTM. On the other hand, the spatio-temporal interpolation of the flow characteristics (mainly including the velocity, diffusivities, salinity, sea surface height, temperature, etc.) in both horizontal and vertical directions may be necessary, especially when the duration between successive hydrodynamic outputs is different in comparison with the time step of particles' motion. Thus, the hydrodynamic information provided on an Eulerian grid, as an input to the PTM model, should be interpolated for the spatio-temporal location of the particles before being read by the PTM [61]. For instance, the particle's velocity at each location is found using the spatial and temporal interpolation of the hydrodynamic model's velocity. Most available numerical models have limitations in considering all effective factors relevant to microplastics' fate and transport.