3.1 Introduction: From Physical Principles to Quantitative Prediction

While understanding the physical principles of transport and dispersion is essential, it is not sufficient. To make actionable decisions, we must translate this qualitative knowledge into quantitative predictions. We need to be able to estimate the specific concentration of a pollutant at a specific location downwind from a source. This requires the use of mathematical models.

3.2 The Gaussian Plume Model: An Empirical Approach

Early attempts to model atmospheric dispersion tried to solve the classical diffusion equation:

dχ/dt = ∂(Kx ∂χ/∂x)/∂x + ∂(Ky ∂χ/∂y)/∂y + ∂(Kz ∂χ/∂z)/∂z

However, this approach has proven to be of limited use in atmospheric science. The diffusion coefficients (the K values) are not constant; they represent the effects of turbulent eddies, which vary dramatically with height, distance from the source, and overall atmospheric stability. Determining their correct values is exceptionally difficult.

A more successful and widely used approach has been empirical. The most common of these is the Gaussian Plume Model. This model assumes that the concentration of a pollutant within a plume has a normal (or Gaussian) distribution in both the horizontal and vertical directions, relative to the plume’s centerline. The concentration (χ) at any point (x, y, z) is given by:

χ = (Q / (2πVσyσz)) * exp[-(y²/2σy²)] * {exp[-((z-H)²/2σz²)] + exp[-((z+H)²/2σz²)]}

Let’s break down the components of this crucial equation:

• χ (chi): The predicted concentration of the pollutant (e.g., in micrograms per cubic meter).
• Q: The source strength, or the rate at which the pollutant is emitted (e.g., in grams per second).
• V: The mean wind speed. It appears in the denominator, reflecting that higher wind speeds lead to greater dilution and lower concentrations.
• σy and σz (sigma-y and sigma-z): These are the standard deviations of the concentration distribution in the horizontal (y) and vertical (z) directions, respectively. They represent the plume’s width and height and are the key parameters that describe the effects of dispersion.
• H: The effective source height, as defined in Module 2.

3.3 Determining Dispersion Parameters (σy and σz)

The entire predictive power of the Gaussian model hinges on our ability to accurately determine the dispersion parameters, σy and σz, as they change with downwind distance and meteorological conditions.

The most popular method for this is the Pasquill-Gifford method. This system provides a practical way to estimate the atmosphere’s dispersive capability without needing sophisticated instruments to measure the Richardson number directly. Instead, it classifies atmospheric stability into six “stability categories,” from A (very unstable) to F (moderately stable), based on simple observations like wind speed, time of day, and cloud cover.

Once the stability category is determined, σy and σz can be found using a set of empirical graphs that relate them to the downwind distance from the source. For example, under category ‘A’ conditions (strong convection), the graphs show that σy and σz grow very rapidly with distance, indicating strong dispersion. Under category ‘F’ conditions (stable), they grow much more slowly.

Despite its widespread use, the Pasquill-Gifford method has a primary weakness: its empirical curves were developed from experiments conducted over smooth, open terrain. Consequently, the method tends to underestimate dispersion when applied to rougher surfaces like forested areas or, most importantly, cities. This can lead to an over-prediction of pollutant concentrations in urban environments.

3.4 Model Refinements and Modern Developments

The basic Gaussian model must be refined to account for real-world constraints. A critical factor is the presence of an elevated inversion layer, which acts as a “cap” on the atmosphere. As a plume spreads vertically, it will eventually be limited by this inversion. Once the pollutant becomes well-mixed between the ground and the inversion lid, its vertical distribution becomes uniform. In this situation, the concentration becomes inversely proportional to the “ventilation factor” (VD), where V is the average wind speed in the mixed layer and D is the depth of that mixed layer (i.e., the height of the inversion). This simple relationship is a powerful tool for understanding daily and seasonal variations in pollution.

Topography can also impose severe constraints. A pollutant released in a narrow valley may be unable to disperse laterally. This trapping effect can maintain high concentrations over very long distances down the valley.

More recent research, particularly for the important case of light-wind, sunny conditions, has shown that vertical distributions are far from Gaussian. A modern framework has been developed that uses more physically direct predictors. The two key parameters are:

1. zi: The height of the planetary boundary layer (the mixed layer).
2. w* (w-star): The convective vertical velocity scale, which is a measure of the strength of thermal updrafts.

In this framework, the vertical distribution is definitely not Gaussian and is described by more complex formulas that have been validated through numerical models, laboratory experiments, and full-scale field observations.

3.5 City Models: Integrating Multiple Sources

The models discussed so far are for single sources. To understand the air pollution landscape of an entire urban area, we need “city models.” These are essential for developing comprehensive control strategies, as they calculate the composite pollution field resulting from the combined effect of all sources in the area.

Creating a complete city model is a computationally complex and expensive undertaking. It requires summing the contributions from thousands of individual sources—from large power plants to small neighborhood businesses—across a wide spectrum of meteorological conditions (different wind speeds, directions, and stability classes). To make this problem tractable, simplifications are often employed. For example, numerous small sources within a given area might be grouped together and represented as a single “line source” to reduce the computational burden. These models, though complex, are the only way to truly understand and manage urban air quality.

Having explored the theoretical models, we will now examine the real-world, observable pollution patterns that these models are designed to predict.