There are two main frameworks for modeling air pollution: Eulerian and Lagrangian. Eurlerian models resolve analytically (simple cases) or numerically the conservation equations. **Lagrangian models follow the motion of the mass of pollutant in the atmosphere**.

## Basic concepts

- The mass of pollutant emitted is divided in a given number of
**computational particles**
- Each of these particles moves in the atmospheric fluid with the same velocity of the fluid (i.e.
**mean wind** + **turbulent fluctuations of wind**)
- the mean field (`u`) is estimated using a meteorological model or from measurements
- the turbulent fluctuations (`u'`) are computed using the
**Langevin equation** with coefficients depending from local atmospheric stability conditions through the PBL scaling variables (`u_star, w_star, L, H`)

- The
**trajectory** of each particle is reconstructed by estimating its **position at discrete time intervals**
- The
**concentration** in a given point at a given time is computed by summing the **contribution of all particles** close to the point

## The motion of the particles

The trajectory of each particle is calculated with the equation

`x_i (t+Delta t) = x_i (t) + Delta t (u_i + u'_i)`

`u_i` is the mean wind component along the i-th direction and `u'_i` represents the turbulent velocity fluctuation along the same i-th direction

The time evolution of the velocity fluctuation is described by the non-linear Langevin equation

`d u_i = a_i(x,u'_i,t) dt + b_(ij) (x,u'_i,t) d xi_j(t) `

Both `a_i` and `b_(ij)` are linked to the structure of turbulence through functional relations with the meteorological variables.

## The calculation of concentrations

The concentration in a given point at a given time is obtained by summing the contribution of all particles close to the point:

`C(x_0,y_0,z_0) = sum_p c_p K_p (x_0,y_0,z_0) `

The concentration associated to each particle (p) is a function of the mass associated to the particle and the relative diffusion of the mass since the release. The K function is a *smoothing kernel*

## Advantages of Lagrangian particle models

Source attribution: each particle belongs to an emitting source

Arbitrary resolution: the mass of pollutant is not distributed in a computational cell. This is particularly important close to the source.

Fast computation: high space resolution does not limit numerical stability and integration time steps

Read more