# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development  
### Conceptual Bases of the Fluctuating Plume Model  
When a passive scalar, like a tracer or odorous gas, from a point source is emitted, in reality, many instantaneous plumes are generated in sequence, each different from the other and irregularly shaped. Only considering their average, they make up the usual regular plume characterized by a progressive enlargement in downwind direction, proportional to the turbulence present in the PBL. If we focus our attention on a single instantaneous plume, we can clearly see that in the first phase of dispersion, near the source, the plume is coherent, narrow and meandering, mainly horizontally and, to a lesser extent, also vertically. The plume meandering, more accentuated near the source, progressively reduces downwind until it disappears: the meandering extent is inversely proportional to the plume's characteristic dimension. From this phenomenological evidence, the famous Fluctuating Plume Model proposed by [PERSON] (1959) was inspired, which was originally formulated more as a conceptual tool than a quantitative model. It should be pointed out that, in this context, the term meandering refers to the slow oscillation of a plume due to the eddies characterized by a length scale greater than the plume length scale and not to the slow horizontal oscillations of the mesoscale motion (the \"submeso\" motions) practically always present in the PBL and very evident in high stability [57; 58; 59; 60; 61]. Let us put a point source in position (0, 0, \(z_{s}\)) and a coordinate system centered in it (\(x\) along the direction towards the mean wind is blowing, the \(y\) horizontal and transverse to \(x\) and with the vertical \(z\)-axis) [48; 62], and an averaging time of the order of the spectral gap (therefore of the order of the hour [8]). What we can see is a regular plume growing progressively with the distance downwind to the emission point, initially in linearly and subsequently in parabolic manner. Basically, in homogeneous and stationary turbulence, the average plume can be seen as a succession of slices (spreading disk), adjacent to each other in \(x\) direction, having flat surfaces in (\(y\), \(z\)) and thickness \(dx=Udt\), where \(U\) is the mean wind speed at the emission point and \(dt\) is an appropriate small time interval. The distribution of the average concentration in the plane (\(y\), \(z\)), at least in first approximation, can be considered Gaussian and centered at points \(y_{s}=0\) and \(z_{s}\) (plume centroid). If, instead, we subdivide the averaging time into a succession of shorter time steps, it is easy to imagine that the average plume is the result of the superimposition of ideally infinite instantaneous irregular plumes, whose barycenters no longer coincide with the \(x\)-axis, but meander around it. Further, in this case, the (instantaneous) plume can be seen as a series of slices placed one next to the other, but in each slice the concentration distribution, always Gaussian in the \(y\) and \(z\) directions, is centered around a new centroid placed now at the transversal distance \(y_{m}\) and the vertical distance \(z_{m}\). [PERSON] (1959) used this idea to propose a simple conceptual model of a stationary Gaussian Meandering Plume. In [PERSON]'s model, the concentration distribution within the instantaneous plume is strictly deterministic, being described in both \(y\) and \(z\) by a Gaussian distribution. This distribution is instead a stochastic variable, since the coordinates of the position of the centroid \(y_{m}\) and \(z_{m}\) are really stochastic variables due to the nature of the turbulent eddies in the PBL. However, [PERSON]'s model does not account, neither directly nor explicitly, for the action of turbulent eddies of smaller dimensions than the plume. This dispersion, not considered by the [PERSON]'s model, is named \"relative dispersion\" or \"internal plume dispersion\".