# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### General Lagrangian Theory of the Fluctuating Plume Model  
Let us consider a steady-state PBL of vertical extension \(z_{i}\), one point source at (\(x_{s}\),\(y_{s}\),\(z_{s}\)) with constant emission rate \(Q\) and a horizontally homogeneous spatial domain. In these conditions, the fluctuating plume model can be reformulated more realistically in a purely Lagrangian framework [49; 50; 51; 52]. The instantaneous concentration \(c(x\),\(y\),\(z)\), if considering a continuous point source (with characteristic physical dimension \(\sigma_{0}\)) with position (\(x_{s}\),\(y_{s}\),\(z_{s}\)), is statistically described by the probability density function \(p(x\),\(y\),\(z)\). Adopting the Fluctuat  
Figure 1: Graphical example of average plume (**a**) versus “single-disks” meandering plume (**b**). ing Plume philosophy, the dispersion can be considered as the combination (convolution) of two independent contributions (the floating part and the relative diffusion part). This distribution can be expressed by the convolution integral:  
\[p(x,y,z)\ =\ \int_{0}^{\infty}\int_{-\infty}^{+\infty}p_{cr}(c;x,y,z,y_{m},z_{m}) \cdot p_{m}(x,y_{m},z_{m})\cdot dy_{m}dz_{m} \tag{1}\]  
where \(p_{cr}\) (\(c\); \(x\), \(y\), \(z\), \(y_{m}\), \(z_{m}\)) is the probability density function describing at the downwind distance \(x\) the dispersion of the generic passive with respect to the centroid of an instantaneous plume located at (\(y_{m}\), \(z_{m}\)) (the relative dispersion) and \(p_{m}\) (\(x\), \(y_{m}\), \(z_{m}\)) is the probability density function that at \(x\) the centroid of the instantaneous plume has exactly those coordinates. Known the two probability density functions, we can calculate any \(n^{th}\) moment of the instantaneous concentration. As a matter of fact, in a Lagrangian framework, a generic (uncentered) moment of order \(n\) will be expressed as  
\[\overline{c^{n}}(x,y,z)\ =\ \int_{0}^{\infty}c^{n}\cdot p(c;x,y,z)\cdot dc \tag{2}\]  
For example, the average concentration and the variance are expressed as  
\[\overline{c}(x,y,z)\ =\ C(x,y,z)\ =\ \int_{0}^{\infty}c\cdot p(c;x,y,z)\cdot dc \tag{3}\]  
\[\sigma_{c}^{2}\ =\ \int_{0}^{\infty}c^{2}\cdot p(c;x,y,z)\cdot dc-C^{2} \tag{4}\]  
and the concentration intensity as  
\[i_{c}^{2}\ =\ \frac{\overline{c^{2}}-\overline{c^{2}}}{C^{2}}\ =\ \frac{\sigma_{c}^{2}}{C^{2}} \tag{5}\]  
Unlike [PERSON] (1959) and supported by experimental evidence [65; 70], here it is assumed that the fluctuations inside the instantaneous plume can be statistically described using a predefined PDF \(p_{cr}\) (\(c\); \(x\), \(y\), \(z\), \(y_{m}\), \(z_{m}\)). Although [PERSON] (2003) [51] assumes this distribution as [PERSON], [PERSON], and [PERSON] (2002), [PERSON] et al. (2000) and [PERSON] et al. (2009) [51; 52; 54] hypothesize more realistically that the PDF is a Gamma distribution of the type:  
\[p_{cr}(c;x,y,z,y_{m},z_{m})\ =\ \frac{\lambda^{\lambda}}{\bar{c}_{r}\Gamma( \lambda)}\cdot\left(\frac{c}{\bar{c}_{r}}\right)^{\lambda-1}exp\left[-\frac{ \lambda c}{\bar{c}_{r}}\right] \tag{6}\]  
where \(\bar{c}_{r}\) is the mean concentration relevant to the instantaneous plume centroid, \(c\) the instantaneous concentration, and \(\lambda\) is defined as \(\lambda\ =\ 1/i_{cr}^{2}\), where \(i_{cr}\) is the ratio of the standard deviation of the relative concentration \(\sigma_{r}\) to the mean relative concentration (see Section 2.6). On the other hand, the instantaneous plume centroid in \(x\) meanders stochastically [65] and this can be described by a probability density function \(p_{m}(x,y_{m},z_{m})\) associated to the coordinates \(y_{m}\) and \(z_{m}\).