## 1 Introduction
#### 2.2.3 Coupling a Dispersion Model  
Lagrangian models simulate random pathways through the turbulent flows that fluid elements carve (78), which makes them especially adept at simulating tracer paths through atmospheric turbulent flow. When used in the context of atmospheric dispersion models, trajectories are created for a large number (often tens of thousands) of particles over the course of hours, which can trace out the atmosphere's mean flow and turbulence as well as account for subgrid-scale transport processes. By contrast, in Eulerian simulations a point source is instantly averaged out into its corresponding grid box, resulting in loss of resolution. In Lagrangian models, each air parcel can be assigned a mass and a loss process to match the limitations of the particular investigation. Although their stochastic differential equations may lead to some numerical errors (79), in general Lagrangian models have less numerical diffusion than their Eulerian or semi-Lagrangian counterparts (80; 81). When factoring in that Lagrangian models can simulate backwards in time, allowing them to act as reliable tracers to source regions, they become especially useful in investigating emission regions and making comparisons against in situ measurements (82).  
Here, we couple the Lagrangian Particle Dispersion Model (LPDM) defined in (83) as an adjoint to the WRF forward model to create influence functions from current CO\({}_{2}\) measurement sites in the Aix-Marseille region. The influence functions are used to understand the footprint of the airmass before its arrival at the observation site in order to understand the fluxes that influence atmospheric CO\({}_{2}\) concentrations at the location. The LPDM is provided with the u, v, and w wind components, the temperature, and the Turbulent Kinetic Energy (TKE) from the WRF output files for our innermost (1 km \(\times\) 1 km) domain at hourly resolution. Then, at the desired receptor spots and altitudes corresponding to the aforementioned real or potential future measurement sites (see Table 1), 35 particles are released backwards in time every 30 s, advecting through the WRF-generated wind fields. Files are saved every 2 min; these files are then used to create hourly influence function files by integrating along all particles within the lowest 400 m, following the procedure laid out in (84). This height was chosen to account for the difficulties LPDM has with sudden terrain height changes, which are present within our domain. Following the procedure set out in (50), 12 hours of influence function files are used to create a single influence function  
Figure 3: Comparison of the CO\({}_{2}\) flux amounts from the different components for the different time periods of the model in the 1 km domain.  
(footprint) for a receptor site of interest at a given time of interest in order to allow enough time for all particles to sufficiently traverse the domain.  
In our study, influence functions were generated from the three receptor positions outlined in Table 1: Observatoire Haute-Provence (OHP), Marseille Longchamp (CAV), and End-dome (SME). These were the sites for which measurements were available during our period of interest. Their respective positions in the 1 km domain can be seen in Figure 1. These influence functions are used in Section 3.4 to showcase the differences in potential source regions during the different seasons as well as during particular distinct wind condition events, as demonstrated with a case study of 24 July 2016.