# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### New Operational Gaussian Fluctuating Plume Model
\right)\ =\ 1\ \ \text{for $k=0$, 1 the relation (\ref{eq:15}) is reduced to:}\)  
\[C\ =\ \frac{Q}{2\pi\sigma_{y}\sigma_{z}U_{m}}\cdot exp\left[-\frac{ \left(y-y_{s}\right)^{2}}{2\sigma_{y}^{2}}\right]\cdot\left[exp\left[-\frac{ \left(z-z_{s}\right)^{2}}{2\sigma_{z}^{2}}\right]+exp\left[-\frac{\left(z+z_{s }\right)^{2}}{2\sigma_{z}^{2}}\right]\right] \tag{16}\]  
i.e., the usual steady-state Gaussian relation for the mean concentration when only the ground reflection of a steady-state plume is considered. In this relation, the parameters \(\sigma_{y}\) and \(\sigma_{z}\) of the absolute dispersion are set equal to \(\sigma^{2}{}_{y}=\sigma^{2}{}_{ym}+\sigma^{2}{}_{yr}\) and \(\sigma^{2}{}_{z}=\sigma^{2}{}_{zm}+\sigma^{2}{}_{zr}\), respectively. The \(\lambda\) parameter is not present in this equation, so \(C\) does not depend on the PDF: it depends only on parameters of the absolute dispersion. Proceeding the analysis and setting \(n=2\), it is then possible to obtain the spatial distribution for the Gaussian plume uncentered second concentration moment:  
\[\overline{c^{2}}\ =\ \Big{(}\frac{Q}{2\pi\sigma_{y}\sigma_{z}U_{m}} \Big{)}^{2}\frac{\sigma_{yr}}{(2\sigma_{y}^{2}+\sigma_{y}^{2})^{0.5}}\frac{ \sigma_{yr}}{(2\sigma_{z}^{2}+\sigma_{y}^{2})^{0.5}}\frac{1}{\lambda^{\lambda }}\ \frac{\Gamma(2+\lambda)}{\Gamma(\lambda)}.\] \[\cdot exp\left[-\frac{2\left(y-y_{s}\right)^{2}}{2\left(2\sigma_{ zm}^{2}+\sigma_{yr}^{2}\right)}\right]\cdot\sum\limits_{k=0}^{2}\left\{\left( \begin{array}{c}2\\ k\end{array} \right)\cdot exp\left[-\frac{\left(2-k\right)\cdot\left(z-z_{s} \right)^{2}}{2\left(2\sigma_{zm}^{2}+\sigma_{yr}^{2}\right)}\right]. \tag{17}\] \[\cdot exp\left[-\frac{k\left(z+z_{s}\right)^{2}}{2\left(2\sigma_{ zm}^{2}+\sigma_{yr}^{2}\right)}\right]\cdot exp\left[-\frac{2\left(z\right)^{2}k \left(2-k\right)}{2\left(2\sigma_{zm}^{2}+\sigma_{yr}^{2}\right)}\frac{ \sigma_{yr}^{2}}{\sigma_{y}^{2}}\right]\right\}\]  
But, \(\frac{1}{\lambda^{2}}\cdot\frac{\Gamma(2+\lambda)}{\Gamma(\lambda)}\ =\ \frac{1}{ \lambda^{2}}\cdot\lambda\cdot(1+\lambda)\ =\ \frac{1+\lambda}{\lambda}\), and setting \(S_{y}^{2}\ =\ 2\sigma_{ym}^{2}+\sigma_{yr}^{2}\) and \(S_{z}^{2}\ =\ 2\sigma_{zm}^{2}+\sigma_{zr}^{2}\), the second uncentered moment assumes the following analytical form:  
\[\overline{c^{2}}\ =\ \frac{1+\lambda}{\lambda}\frac{Q^{2}}{\left(2\pi U_{m} \right)^{2}\sigma_{yr}\sigma_{z}}\frac{1}{S_{y}S_{z}}\cdot exp\left[-\frac{ \left(y-y_{s}\right)^{2}}{S_{y}^{2}}\right]\cdot\] \[\cdot\left\{exp\left[-\frac{\left(z-z_{s}\right)^{2}}{S_{z}^{2}} \right]+exp\left[-\frac{\left(z+z_{s}\right)^{2}}{S_{z}^{2}}\right]+\right. \tag{18}\] \[+\left.2\cdot exp\left[-\frac{\left(z-z_{s}\right)^{2}}{2S_{z}^{2 }}\right]\cdot exp\left[-\frac{\left(z+z_{s}\right)^{2}}{2S_{z}^{2}}\right] \cdot exp\left[-\frac{1}{2}\frac{\left(2z\right)^{2}}{S_{z}^{2}}\frac{\sigma_{yr }^{2}}{\sigma_{yr}^{2}}\right]\right\}\]  
Differently from the mean concentration, the second uncentered moment explicitly depends on the PDF parameter \(\lambda\), which statistically describes the fluctuations of the instantaneous plume, and depends both explicitly and separately on the horizontal and vertical parameters that characterize the relative dispersion and meandering of the plume centroid. Finally, the steady-state Gaussian Fluctuating Plume Model thus obtained consists of the following two equations: Equation (16) for the mean concentration and Equation (18) for its second moment. The concentration variance is equal to the second uncentered order moment minus the square of the mean concentration (see Equation (4)). Then, the standard deviation of the concentration is the root square of the concentration variance. This is the general form of the new steady-state Fluctuating Gaussian Plume Model used in the following.