## 10.3. Gaussian dispersion models

### 10.3.1. Theory and limitations of Gaussian models

The turbulent diffusion equation (10.10) is a partial differential equation that can be solved with various numerical methods. Assuming a homogenous, steady-state flow and a steady-state point source, equation (10.10) can also be analytically integrated and results the well-known Gaussian plume distribution

 , (10.19)

where c is a concentration at a given position, Q is the source term, x is the downwind, y is the crosswind and z is the vertical direction and u is the wind speed at the h height of the release. The σy, σz deviations describe the crosswind and vertical mixing of the pollutant, thus they are constructed from the Kh, Kz values of equation (10.10). Equation (10.19) describes a mixing process that results a Gaussian concentration distribution both in crosswind and in vertical direction, centered at the line downwind from the source (Figure 10.3). The last term of equation (10.19) expresses a total reflection from the ground, therefore this formula does not count with dry and wet deposition. Adding a third vertical component to the equation, total reflection from an inversion layer can also be computed. Gravitational settling and chemical or radioactive decay are neglected.

One can observe that the x downwind distance from the source does not appear in equation (10.19). It originates from the obvious assumption that advection is more dominant than diffusion, that, however, can cause large error in situations with low wind speeds where a three-dimensional diffusion dominates. Unfortunately, these situations proved to have been the most dangerous ones in real-life atmospheric dispersion problems as they were often connected to stably stratified atmosphere or low-level inversions (Sharan and Gopalakrishnan, 1997).

Figure 10.3: Schematic figure of a Gaussian plume. The He effective stack height and the crosswind and vertical deviation of the profile are the key parameters of the model. (Source: Wikipedia)

Table 10.5: Features and assumptions of most Gaussian dispersion models

 Represented in most Gaussian models Not represented in most Gaussian models Advection Wind shear Horizontal turbulent diffusion Change of wind over time Vertical turbulent diffusion Change of source parameters over time Reflection from ground Wet and dry deposition Reflection from inversion layer Gravitational settling Elevated source Chemical reactions Buoyancy (effective stack height) Radioactive decay Multiple source points Complex terrain 3D diffusion (low-wind case)

Besides turbulence, the elevation of the source, often referred to as stack height is a key parameter of a Gaussian model, because ground concentrations are computed in an analytical way assuming the maximum concentration in the stack height (Figure 10.3). If buoyant pollutants are present, the horizontal advection starts from considerably higher than the stack’s top due to the buoyant rise of the released gas. It led to the definition of the effective stack height that is the stack height added to the buoyant plume rise. Both empirical and theoretical formulas exist to compute the plume rise using the temperature, specific heat capacity, release speed and flux of the material as input data, which can be successfully used even in extremely buoyant cases like pool fires. We note that in situations where buoyant pollutants and a low-level thermal inversion are present, more sophisticated simulations are required to estimate the penetration of the inversion layer by the plume.

### 10.3.2. History of development

The greatest advantage of Gaussian models is that they have an extremely fast, almost immediate response time. Their calculation is based only on solving a single formula (equation 10.11 or similar) for every receptor point, and the model’s computational cost mainly consists of meteorological data pre-processing and turbulence parameterization. Depending on the complexity of these submodules, the model’s runtime can be extremely reduced that enables its application in real-time GIS-based decision support software.

Gaussian dispersion models have become a uniquely efficient tool of air quality management for the past decades, especially in the early years when high performance computers had an unreachable price for environmental protection organizations and authorities. They have been successfully used for a wide range of studies of air quality in urban and industrial areas. However, industrial incidents like the ones in Seveso, Bhopal and Chernobyl showed out some critical weaknesses of Gaussian models, and strongly motivated the development of more advanced simulations to satisfy the scientific and public interest in the safety of atmospheric environment.

Although the toxic gas release in Seveso, Italy in 1976 happened during daytime in weakly unstable conditions, Gaussian models couldn’t perform well because of the strong horizontal wind shear and fast changing wind direction. As exact deposition maps became available, Cavallaro et al. (1982) managed to give a better computational result with a statistical method that estimated dispersion directions driven by measured wind vectors, which can be regarded as a Lagrangian approach. Eight years later, another serious accident happened in Bhopal, India. The high number of victims warned the world that release of toxic heavy gas in a situation where low-level night time inversion is present can cause catastrophic consequences. Because of the low wind speed, fast settling pollutant and strong temperature inversion connected with local scale terrain effects, it was impossible to obtain reliable results from Gaussian models, however, later simulations with advanced Lagrangian software showed a good agreement with measurements.

Inspired by the serious accidents and the more and more efficient computers, there were large efforts to develop Gaussian models in a way that they could provide more accurate air quality forecasts as well as to take into account some of the unrepresented physical processes (Table 10.5). It led to some respectable results like the more sophisticated treatment of vertical mixing in convective boundary layer or the parameterization of complex terrain effects. Due to the developments, today’s advanced Gaussian dispersion models like AERMOD, CTDM or ADMS still have a significant role in environmental modelling.

While the incidents in Seveso and Bhopal and other air pollution episodes were concentrated on a local scale, the Chernobyl accident in 1986 had serious consequences in several countries and the radioactive I-131 gas was measured globally (Pudykiewicz, 1988). It was clear that the steady-state assumption of Gaussian models couldn’t handle continental scale dispersion processes, however, existing Eulerian and Lagrangian models provided prescious information in the estimation of the impact of the accident (Pudykiewicz, 1988). The fast development of computers and NWP-s allowed researchers to create more and more efficient dispersion simulations using girded meteorological data. Eulerian and Lagrangian models are state-of-the-art tools of recent atmospheric dispersion simulations (Mészáros et al., 2010, Dacre et al., 2011).