Theoretical models implemented in the SLAM software

The SLAM software relies on a meteorological pre-processor capable of describing the atmospheric surface boundary layer by determining the friction velocity \(u_\star\) and the Monin-Obukhov length \(Lmo\). To do this, the user must provide data such as wind speed \(u(z)\), cloud cover \(Cld\), ground temperature, etc.

SLAM is capable of handling very different meteorological conditions:

• Uniform weather.
• 3D weather using one or more wind fields pre-calculated by a CFD-type tool.

It is important to note that the pre-calculated wind fields are sorted according to the stability condition under investigation. The characteristic quantities used to define a condition of atmospheric stability are the friction velocity \(u_\star\) and the Monin-Obukhov length \(Lmo\). The parameters calculated by the meteorological pre-processor therefore allow \(u_\star\) and \(Lmo\) to be determined, so that the cases to be considered in the database can be identified.

From the site roughness \(z_0\), the wind speed at altitude \(z\) \(u(z)\), the ground temperature and the cloud cover, \(Lmo\) and \(u_\star\) can be determined by iteratively solving the system of equations below. \[L_{MO}= \frac{-\rho \, C_p \, u_\star^3}{\kappa \, (\frac{g}{T_0}) \, H_0} \] where:

• \(L_{MO}\) is the Monin-Obukhov length.
• \(u_\star\) is the friction velocity.
• \(\rho\) is the friction velocity.
• \(\kappa\) is the Karman constant.
• \(C_p\) is the specific heat capacity.
• \(g\) is the acceleration due to gravity.
• \(T_0\) is the ground temperature.
• \(H_0\) is the sensible heat flux at ground level, determined by calculating the ground energy balance.

\[u(z)=\frac{u_\star}{\kappa} \left [ ln \left (\frac{z-d+z_0}{z_0} \right ) - \left ( \psi_m \left ( \frac{z-d+z_0}{L_{MO}} \right ) - \psi_m \left ( \frac{z_0}{L_{MO}} \right ) \right ) \right ] \] where:

• \(u(z)\) is the wind speed at height z.
• \(z\) is the height above ground level.
• \(u*\) is the friction velocity.
• \(\kappa\) is the Karman constant.
• \(z_0\) is the aerodynamic roughness of the site.
• \(d\) is the displacement thickness of the site.
• \(\psi_m\) is the universal function.
• \(L_{MO}\) is the Monin-Obukhov length.

Lagrangian modelling approaches focus on the trajectories of pollutant particles. Thus, the principle underlying the SLAM software is to simulate the trajectories of a large number of particles based on an average velocity field, to which a random component representing turbulent fluctuations is added at every point and at each iteration.

To determine the random velocity of a particle, SLAM solves the Langevin equation at each time step: \[ \left \{ \begin{array}{r c l} dX \, = \, U_x(t) \, dt \, = \, u_x(x,y,z,t) \,dt \\ dY \, = \, U_y(t) \, dt \, = \, u_y(x,y,z,t) \, dt \\ dZ \, = \, U_z(t) \, dt \, = \, u_z(x,y,z,t) \, dt \end{array} \right . \] where:

• \(x,y\) and \(z\) are the Lagrangian coordinates of the particle.
• \(U_x,U_y\) and \(U_z\) are the Lagrangian components of the velocity field.
• \(u_x,u_y\) and \(u_z\) are the Eulerian components of the velocity field.

SLAM also utilises numerous theoretical models such as:

• Micro-mixing model (micro-mixing) for determining instantaneous concentrations.
• Interpolation method and Kernel approach.
• Overhead model.
• etc.