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.
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 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
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