Enviroware - LAPMOD

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