## 2 Model Introduction and Simulation Design  
### Introduction to the Model  
FLEXPART is a Lagrangian particle dispersion model developed by the Norwegian Institute for Air Research (NILU) [15; 16; 17]. Initially designed to investigate the dispersion of long-range and mesoscale point-source air pollution, this model has evolved into an essential tool for simulating and analyzing atmospheric transport processes. It encompasses spatial scales ranging from tens of meters to global scales, and covers multiple fields including air pollution, atmospheric transport, greenhouse gases, water cycling, and volcanic ash. FLEXPART calculates the trajectories of particles released from point, line, area, or volume sources [18]. It describes the long-range and mesoscale processes involved in tracer transport, dispersion, dry and wet deposition, and radioactive decay in the atmosphere. The primary objective of the model is to study the source-receptor relationship of atmospheric pollutants: emissions act as the \"sources\", while observation stations serve as the receptors or \"sinks\". By examining the processes of horizontal transport, dispersion, convection, dry and wet deposition, radioactive decay, and first-order chemical reactions of pollutants, the model can provide grided pollution concentrations over time through forward simulation or grided residence times (also known as sensitivity coefficients or footprint functions) [19; 20; 21]. The core algorithm employed by FLEXPART utilizes a zero-acceleration scheme [22] to calculate particle trajectories, which can be expressed as follows:  
\[x(t+\Delta t)=x(t)+v(x,t)\times\Delta t\, \tag{1}\]  
where \(\Delta t\) is the time increment, \(x\) is the position vector, and \(v=\overline{v}+v_{t}+v_{m}\) indicates that the wind vector consists of a grid-scale component \(\overline{v}\), a turbulence component \(v_{t}\), and a mesoscale wind disturbance component \(v_{m}\).  
WRF-FLEXPART is established by coupling the numerical weather forecast model WRF with FLEXPART, enhancing the suitability of the model for simulation and forecasting over a mesoscale perspective. Particle position calculations of FLEXPART can use a grid that differs from that of the WRF model. The control file governs various parameters such as the initial positions, emission times, durations, number of particles, concentrations, grid resolution, and parameterization scheme of the physical processes. After transforming the source-sink relationship, the formula for calculating residence time at each grid point is expressed as:  
\[\Delta\tau_{i}=\frac{\Delta T}{NJ}\sum_{m=1}^{N}\sum_{j=1}^{J}f_{ijn} \tag{2}\]  
where \(\Delta T\) is the temporal resolution, \(N\) is the number of samples in the \(\Delta T\) time range, \(J\) is the total number of particles released, and \(f_{ijn}\) is a function that determines how many particles are \"contributing\" to a given grid point [23].  
Simply put, the fundamental principle of FLEXPART is to release a multitude of particles at the designated \"source\" point. Subsequently, by taking into account various atmospheric processes such as turbulent diffusion, advective transport, and dry and wet deposition of the particles, an assessment can be made regarding the number of particles that ultimately arrive at the \"sink\" point. This enables us to establish a comprehensive understanding of the relationship between the source and sink. Throughout this calculation process, detailed positional information of all particles at each output time is generated. By connecting these positions with those from previous time intervals, FLEXPART effectively captures trajectory information for all particles. Additionally, utilizing its built-in plume module, FLEXPART can analyze this extensive dataset comprising particle trajectories through clustering statistics in order to determine the most probable trajectory. This trajectory is the most representative trajectory after considering numerous uncertainties.