Understanding Lagrangian Stochastic Dispersion Models

1. Consiglio Nazionale delle Ricerche ISTITUTO DI SCIENZE DELL’ATMOSFERA E DEL CLIMA(ISAC) - Turin Section Corso Fiume 4 - 10133 TORINO - ITALY tel. +39 011 6306819 fax +39 011 6600364 e-mail anfossi@to.infn.it BASIC ASPECTS OF LAGRANGIAN STOCHASTIC DISPERSION MODELS Domenico Anfossi

3. Historical background Theoretical basis Codes implementation Applications to real cases in different conditions of terrain type thermodynamic stability with different aims: study control forecasting scenarios Basic Aspects Of Lagrangian Stochastic Dispersion Models

4. Around 1500 A.C. Leonardo da Vinci Second half of 19th century Brownian motion 1905 A. Einstein 1913 P. Langevin 1914 -1918 First world war /chemical war 1921 G.I. Taylor

5. 1959 A.M. Obukhov proposed that the evolution of the motion of an air particle in the atmosphere can be described as a Markov process 1968 F.B. Smith (1)   1979 S.R. Hanna experimentally verifies eq. (1) 1980 - 1987 empirical models based on eq. (1) 1982 F.A. Gifford identifies eq. (1) with the Langevin equation 1987 D.J. Thomson well-mixed condition /generalised Langevin equation 1980 - now Operative Lagrangian models

6. Lagrangian particle models are three-dimensional models • for the simulation of airborne pollutant dispersion, • able to account for flow and turbulence space-time variations  • Emissions in the atmosphere are simulated using a certain • number of fictitious particles named ”computer particle”. • Each particle represents a specified pollutant mass. • It is assumed that particles passively follow the turbulent motion of air masses in which they are, • thus it is possible to reconstruct the emitted mass concentration • from their space distribution at a particular time

7. We call particle a fluid portion containing the emitted substance, having dimensions appropriate to follow the motion of the smallest turbulence eddies present in the atmosphere, but containing a number of molecules large enough to allow disregarding the effect of each of them. Under the hypothesis, accurately demonstrated, that dispersion due to molecular motion is negligible compared to turbulent dispersion, it can be thought that these particles possess a concentration of their own that is preserved during the motion

8.  Particles motion in the computation domain, that simulates the airborne pollutant motion in the real domain (atmosphere), is prescribed by the local mean wind. Particle dispersion (operated by turbulent eddies) is obtained from random speeds. These last are the solutions of stochastic differential equations, reproducing the statistical characteristics of the local atmospheric turbulence.

9. In such a way, different parts of the plume “feel” different atmospheric conditions, thus allowing more realistic simulations in conditions difficult to be reproduced with traditional models.

10. Lagrangian Stochastic Models

11. In the single particle models: the trajectory of each particle represents an individual statistical realisation in a turbulent flow characterised by certain initial conditions and physical constraints. Thus the motion of any particle is independent of the other particles, and consequently the concentration field must be interpreted as an ensemble average. The basic relationship, for an instantaneous source located at (Csanady, 1973) is:  where: C is the concentration at time t and location , Q is the emitted mass at time t = 0 is the probability that a particle that was at at time arrives at x at time t. To compute it is necessary to release a large number of particles, to follow their trajectories and to calculate how many of them arrive in a small volume surrounding x at time t. !

12. particles move in the computational domain without any grid using as input the values of the first two or three (sometimes four) moments of the wind velocity probability density distribution (PDF) at the location of the particle. It is worth noting that This input information comes either from measurements or from parameterisations appropriate to the actual stability conditions (unstable, neutral, stable), to the type of site (flat or complex terrain, coast, etc.), and to the time and space scales considered.

13. Langevin equation with Where: x = particle position u = particle velocity fluctuation = mean wind velocity dW = stochastic fluctuation and

14. Lagrangian structure function Kolmogorov, 1941 from which where is a numerical constant b(x,u) can be obtained by the Kolmogorov theory of local isotropy in the inertial subrange

15. a(x,t) is obtained from the well-mixed condition PDF must be specified from the moments of measured turbulence velocities

16. In homogeneous turbulence the PDF of velocity fluctuations is assumed to be Gaussian, thus the resulting Langevin equation has the following form for each component: This assumption may also be made for inhomogeneous Gaussian turbulence in the neutral PBL, obtaining:

18. Particle trajectory PDF of vertical velocity fluctuations CONVECTIVE CONDITIONS

19. BI-GAUSSIAN PDF

20. and

21. closure Determination of the parameters of the BI-GAUSSIAN PDF

22. and GRAM-CHARLIER PDF

23. GRAM-CHARLIER PDF where and are the moments of x

24. 4rd ORDER GRAM-CHARLIER PDF since: and we obtain where

25. Meteo-diffusion parameters necessary for the Lagrangian Particle Models Surface layer parameters • Roughness length • Monin-Obhukov length • Friction velocity • Convection velocity • Scale temperature • 1) - from circulation models • 2) – from in situ measurements , using meteorological pre-processors

26. VERTICAL PROFILIES 1) - from circulation models (RAMS – MM5) 2) - from parameterisations (Degrazia et al., 2001; Hanna, 1982)

27. Concentration calculation Q = tracer emission rate (Kg/s) Dt = time step (s) Np = total number of particles emitted at each Dt Ni = number of particle in the i-th cell being finding concentration Ci is calculated dividing the mass of the i-th cell (where ) by the cell volume (Dx Dy Dz)

28. Calculation of the number of particle to be emitted to have a pre-fixed concentration precisionassociated to each particle Example: ; ; ; x x gives

29. Plume rise

30. PLUME RISE

31. PLUME RISE Anfossi et al., 1993; Anfossi 1985

32. Our Lagrangian Particle Model for the simulation of atmospheric dispersion S P R A Y designed and developed by our team in Turin (I) with ARIANET in Milan (I) and ARIA in Paris (F)

33. EXAMPLE OF COMPLEX TERRAIN (Carvalho et al., 2002) A vertical section, in the Rhein Valley (D), of wind field at two different hours: mid-night (left) and mid-day (right). In the valley wind reverses its direction, while aloft wind mantains its direction

34. EXAMPLE OF COMPLEX TERRAIN Sea, coast, plane, mountain

36. Industrial area of Venice (Italy)

38. SPRAY simulation 31/5/2001 00:00 - 1/6/2001 00:00 3-D particles and g.l. concentrations – hourly imagines

39. Lagrangian Particle Model for the simulation of “Long Range Dispersion” M I L O R D Model for the Investigation of LOng Range Dispersion designed and developed by our team in Turin (I)

40. MILORD simulation of Chernobyl accident air concentration of

41. MILORD simulation of Chernobyl accident radionuclides puff during 15 days