Mathematical air quality models provide a powerful framework for understanding the dynamics of pollutants in the atmosphere and for assessing the impact emission sources have on pollutant concentrations. Two classes of models are commonly used. Empirical models provide an understanding of source impacts by statistically analyzing historical air quality data. Diagnostic models provide a comprehensive description of the detailed physics and chemistry of compounds in the atmosphere, following the evolution of pollutant emissions to their ultimate fate. The complexity and computational intensity of modern models have necessitated the development of algorithms for fast and accurate mathematical solution techniques. Air quality models are being applied to solving such problems as urban smog, acid deposition, regional ozone, haze in scenic regions, and the destruction of the protective stratospheric ozone layer. Mathematical models have grown increasingly detailed in descriptions of air pollutant dynamics. They are widely used in regulatory planning and analysis. This article focuses on the models used to understand air pollution dynamics. Development of a mathematical air quality model proceeds through the conceptual stage, proximating the physical system is derived to a formal description of the idealized system expressed as mathematical equations; third is the computational implementation of the model, including development and of the algorithms and computer code needed to solve the equations; and the final step is the application of the model. Models used in air pollution analysis fall into two classes: empirical–statistical and deterministic. Empirical–statistical models are based on establishing a relationship between historically observed air quality and the corresponding emissions. The linear rollback model is the simplest and easiest to use, and has been widely applied. A second empirical/statistical model is the receptor‐oriented model, used for estimating the contributions that distinguishable sources such as automobiles or municipal incinerators make to particulate matter concentrations. Receptor models are powerful tools for source apportionment of particulates because a vast amount of particulate species characterization data have been collected at many sampling sites worldwide. Deterministic air quality models describe the individual processes that affect the evolution of pollutant concentrations. These models are based on solving the atmospheric diffusion‐reaction equation. In the Lagrangian reference frame, the frame of reference moves with the flow of air, in effect maintaining the observer in contact with the same air parcel over extended periods of time. Because pollutants are carried by the wind, it is often convenient to follow pollutant evolution in a Lagrangian reference frame, and this perspective forms the basis of Lagrangian trajectory and Lagrangian marked‐particle or particle‐in‐cell models. One of the most basic and widely used transport models is the Gaussian plume model. Of the Eulerian models, the box model is the easiest to conceptualize. The atmosphere over the modeling region is envisioned as a well‐mixed box, and the evolution of pollutants in the box is calculated following conservation‐of‐mass principles including emissions, deposition, chemical reactions, and a changing mixing height (or inversion‐base). Eulerian “grid” models are the most complex, but potentially the most powerful, air quality models, involving the least‐restrictive assumptions. They are also the most computationally intensive. The temporal and spatial resolutions of models can vary from minutes to a year and from meters to hundreds of kilometers. Statistical models generally rely on several years' worth of measurements of hourly or daily pollutant concentrations. Pollutant removal processes, particularly dry deposition and scavenging by rain and clouds, are a primary factor in determining the dynamics and ultimate fate of pollutants in the atmosphere. Three different types of chemical mechanisms have evolved as attempts to simplify organic atmospheric chemistry: surrogate, lumped, and carbon bond. These mechanisms were developed primarily to study the formation of O
3
and NO
2
in photochemical smog, but can be extended to compute the concentrations of other pollutants, such as those leading to acid deposition. Inclusion of a description of aerosol dynamics within air quality models is of primary importance because of the health effects associated with fine particles in the atmosphere, visibility deterioration, and the acid deposition problem. Aerosol dynamics differ markedly from gaseous pollutant dynamics in that particles come in a continuous distribution of sizes and can coagulate, evaporate, grow in size by condensation, or be deposited by sedimentation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Historically, the computational intensity of the more complex chemically active models have limited their application. For example, modeling only the gas‐phase dynamics over an area like Los Angeles requires solving about 500,000 simultaneous, nonlinear equations. The development of parallel computers should allow for significantly more detailed modeling of large systems. Both receptor and analytical air quality models have proven to be powerful tools for understanding atmospheric pollutant dynamics. Receptor models, are effective in determining the contributions of various sources to particulate matter concentrations. Analytical air quality models have been used the most in modeling the dynamics of pollutants at local and urban scales.
