It is well known that knowledge about the mean concentration field is not sufficient in a wide range of cases, such as the dispersion of toxic, flammable and chemical reacting gases. Usually, further information about concentration fluctuations, e.g. by determining the concentration variance, is needed. In particular, it is necessary to determine the 90th percentile of the hourly concentration distribution when dealing with odour-hour modelling, whereby an odour-hour is defined by at least 6 minutes of perceivable odour concentrations within one hour. For convenience, we define R90 as the 90th percentile normalised by the mean concentration. It can be determined by assuming a PDF (Probability Density Function) that is generally completely defined by the concentration fluctuation intensity *i* and the mean concentration. While the latter is provided by many types of dispersion models, even very simple ones, obtaining an estimation of *i* requires a more sophisticated approach, that should allow estimating the concentration variance dispersion. In the frame of the Lagrangian stochastic models several approaches can be found for estimating this quantity: Thomson (1990), Borgas and Sawford (1994), Pope (1985); Cassiani et al. (2005), Yee et al. (1994); Yee and Wilson (2000); Luhar et al. (2000); Franzese (2003); Mortarini et al. (2009); Ferrero et al. (2013), Manor (2014), Ferrero et al. (2017). Lagrangian stochastic models for the assessment of odour were proposed and tested by Oettl and Ferrero (2017), and Ferrero and Oettl (2019). Extensive review can be also available: Ferrero et al. (2020), Cassiani et al. (2020).

- Borgas MS and Sawford BL (1994) A family of stochastic models for particle dispersion in isotropic homogeneous stationary turbulence. J Fluid Mech 279:69–99
- Cassiani M, Franzese P, Giostra U (2005) A PDF micromixing model of dispersion for atmospheric flow. Part I: development of the model, application to homogeneous turbulence and to neutral boundary layer. Atmos Environ 39:1457–1469
- Cassiani, M., Bertagni, M.B., Marro, M., Salizzoni, P. (2020) Concentration Fluctuations from Localized Atmospheric Releases, Boundary-Layer Meteorology, 177(2-3), pp. 461–510
- Ferrero E, Mortarini L, Alessandrini S, Lacagnina C (2013) Application of a bivariate gamma distribution for a chemically reacting plume in the atmosphere. Boundary-Layer Meteorol 147(1): 123–137
- Ferrero E., Mortarini L., Purghè F. (2017), A simple parameterization for the concentration variance dissipation in a Lagrangian single-particle model, Boundary-Layer Meteorol., 163 (1), pp 91–101
- Ferrero E., Oettl D., (2019) An evaluation of a Lagrangian stochastic model for the assessment of odours, Atmospheric Environment, Volume 206, Pages 237-246
- Ferrero, E.; Manor, A.; Mortarini, L.; Oettl, D. (2020) Concentration Fluctuations and Odor Dispersion in Lagrangian Models. Atmosphere, 11, 27
- Franzese P (2003) Lagrangian stochastic modeling of a fluctuating plume in the convective boundary layer. Atmos Environ 37:1691–1701
- Luhar A, Hibberd M, Borgas M (2000) A skewed meandering-plume model for concentration statistics in the convective boundary layer. Atmos Environ 34:3599–3616
- Manor A (2014) A stochastic single particle lagrangian model for the concentration fluctuation in a plume dispersing inside an urban canopy. Boundary-Layer Meteorol 150:327–340
- Mortarini L, Franzese P, Ferrero E (2009) A fluctuating plume model for concentration fluctuations in a plant canopy. Atmos Environ 43:921–927
- Oettl D. and Ferrero E. (2017) A simple model to assess odour hours for regulatory purposes, Atmos Environ, 155, 162-173
- Pope S (1985) PDF methods for turbulent reactive flows. Prog Energy Combust Sci 11:119–192
- Thomson DJ (1990) A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. J Fluid Mech 210:113–153
- Yee E, Chan R, Kosteniuk PR, Chandler GM, Biltoft CA, Bowers JF (1994) Incorporation of internal fluctuations in a meandering plume model of concentration fluctuations. Boundary-Layer Meteorol 67:11–38
- Yee E, Wilson DJ (2000) A comparison of the detailed structure in dispersing tracer plumes measured in grid-generated turbulence with a meandering plume model incorporating internal fluctuations. Boundary- Layer Meteorol 94:253–296

Note prepared by E. Ferrero (11/2023). For corrections or expansions please contact us.