Atmospheric dispersion modeling

Discussion of the layers in the Earth's atmosphere is needed to understand where airborne pollutants disperse in the atmosphere. The layer closest to the Earth's surface is known as the troposphere. It extends from sea-level to a height of about 18 km (11 mi) and contains about 80 percent of the mass of the overall atmosphere. The stratosphere is the next layer and extends from 18 km (11 mi) to about 50 km (31 mi). The third layer is the mesosphere which extends from 50 km (31 mi) to about 80 km (50 mi). There are other layers above 80 km, but they are insignificant with respect to atmospheric dispersion modeling.

The lowest part of the troposphere is called the atmospheric boundary layer (ABL) or the planetary boundary layer (PBL) . The air temperature of the atmosphere decreases with increasing altitude until it reaches what is called an inversion layer (where the temperature increases with increasing altitude) that caps the Convective Boundary Layer, typically to about 1.5 to 2 km (0.93 to 1.24 mi) in height. The upper part of the troposphere (i.e., above the inversion layer) is called the free troposphere and it extends up to the tropopause (the boundary in the Earth's atmosphere between the troposphere and the stratosphere). In tropical and mid-latitudes during daytime, the Free convective layer can comprise the entire troposphere, which is up to 10 to 18 km (6.2 to 11.2 mi) in the Intertropical convergence zone.

The ABL is of the most important with respect to the emission, transport and dispersion of airborne pollutants. The part of the ABL between the Earth's surface and the bottom of the inversion layer is known as the mixing layer. Almost all of the airborne pollutants emitted into the ambient atmosphere are transported and dispersed within the mixing layer. Some of the emissions penetrate the inversion layer and enter the free troposphere above the ABL.

In summary, the layers of the Earth's atmosphere from the surface of the ground upwards are: the ABL made up of the mixing layer capped by the inversion layer; the free troposphere; the stratosphere; the mesosphere and others. Many atmospheric dispersion models are referred to as boundary layer models because they mainly model air pollutant dispersion within the ABL. To avoid confusion, models referred to as mesoscale models have dispersion modeling capabilities that extend horizontally up to a few hundred kilometres. It does not mean that they model dispersion in the mesosphere.

The technical literature on air pollution dispersion is quite extensive and dates back to the 1930s and earlier. One of the early air pollutant plume dispersion equations was derived by Bosanquet and Pearson.[2] Their equation did not assume Gaussian distribution nor did it include the effect of ground reflection of the pollutant plume.

Sir Graham Sutton derived an air pollutant plume dispersion equation in 1947[3] which did include the assumption of Gaussian distribution for the vertical and crosswind dispersion of the plume and also included the effect of ground reflection of the plume.

Under the stimulus provided by the advent of stringent environmental control regulations, there was an immense growth in the use of air pollutant plume dispersion calculations between the late 1960s and today. A great many computer programs for calculating the dispersion of air pollutant emissions were developed during that period of time and they were called "air dispersion models". The basis for most of those models was the Complete Equation For Gaussian Dispersion Modeling Of Continuous, Buoyant Air Pollution Plumes shown below:[4][5]

${\displaystyle C={\frac {\;Q}{u}}\cdot {\frac {\;f}{\sigma _{y}{\sqrt {2\pi }}}}\;\cdot {\frac {\;g_{1}+g_{2}+g_{3}}{\sigma _{z}{\sqrt {2\pi }}}}}$

where:
${\displaystyle f}$	= crosswind dispersion parameter
	= ${\displaystyle \exp \;[-\,y^{2}/\,(2\;\sigma _{y}^{2}\;)\;]}$
${\displaystyle g}$	= vertical dispersion parameter = ${\displaystyle \,g_{1}+g_{2}+g_{3}}$
${\displaystyle g_{1}}$	= vertical dispersion with no reflections
	= ${\displaystyle \;\exp \;[-\,(z-H)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]}$
${\displaystyle g_{2}}$	= vertical dispersion for reflection from the ground
	= ${\displaystyle \;\exp \;[-\,(z+H)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]}$
${\displaystyle g_{3}}$	= vertical dispersion for reflection from an inversion aloft
	= ${\displaystyle \sum _{m=1}^{\infty }\;{\big \{}\exp \;[-\,(z-H-2mL)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]}$
	${\displaystyle +\,\exp \;[-\,(z+H+2mL)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]}$
	${\displaystyle +\,\exp \;[-\,(z+H-2mL)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]}$
	${\displaystyle +\,\exp \;[-\,(z-H+2mL)^{2}/\,(2\;\sigma _{z}^{2}\;)\;]{\big \}}}$
${\displaystyle C}$	= concentration of emissions, in g/m³, at any receptor located:
	x meters downwind from the emission source point
	y meters crosswind from the emission plume centerline
	z meters above ground level
${\displaystyle Q_{}}$	= source pollutant emission rate, in g/s
${\displaystyle u}$	= horizontal wind velocity along the plume centerline, m/s
${\displaystyle H}$	= height of emission plume centerline above ground level, in m
${\displaystyle \sigma _{z}}$	= vertical standard deviation of the emission distribution, in m
${\displaystyle \sigma _{y}}$	= horizontal standard deviation of the emission distribution, in m
${\displaystyle L_{}}$	= height from ground level to bottom of the inversion aloft, in m
${\displaystyle \exp }$	= the exponential function

The above equation not only includes upward reflection from the ground, it also includes downward reflection from the bottom of any inversion lid present in the atmosphere.

The sum of the four exponential terms in ${\displaystyle g_{3}}$ converges to a final value quite rapidly. For most cases, the summation of the series with m = 1, m = 2 and m = 3 will provide an adequate solution.

${\displaystyle \sigma _{z}}$ and ${\displaystyle \sigma _{y}}$ are functions of the atmospheric stability class (i.e., a measure of the turbulence in the ambient atmosphere) and of the downwind distance to the receptor. The two most important variables affecting the degree of pollutant emission dispersion obtained are the height of the emission source point and the degree of atmospheric turbulence. The more turbulence, the better the degree of dispersion.

Equations[6][7] for ${\displaystyle \sigma _{y}}$ and ${\displaystyle \sigma _{z}}$ are:

${\displaystyle \sigma _{y}}$(x) = exp(I_y + J_yln(x) + K_y[ln(x)]²)

${\displaystyle \sigma _{z}}$(x) = exp(I_z + J_zln(x) + K_z[ln(x)]²)

(units of ${\displaystyle \sigma _{z}}$, and ${\displaystyle \sigma _{y}}$, and x are in meters)

The classification of stability class is proposed by F. Pasquill.[8] The six stability classes are referred to: A-extremely unstable B-moderately unstable C-slightly unstable D-neutral E-slightly stable F-moderately stable

The resulting calculations for air pollutant concentrations are often expressed as an air pollutant concentration contour map in order to show the spatial variation in contaminant levels over a wide area under study. In this way the contour lines can overlay sensitive receptor locations and reveal the spatial relationship of air pollutants to areas of interest.

Whereas older models rely on stability classes (see air pollution dispersion terminology) for the determination of ${\displaystyle \sigma _{y}}$ and ${\displaystyle \sigma _{z}}$, more recent models increasingly rely on the Monin-Obukhov similarity theory to derive these parameters.

The Gaussian air pollutant dispersion equation (discussed above) requires the input of H which is the pollutant plume's centerline height above ground level—and H is the sum of H_s (the actual physical height of the pollutant plume's emission source point) plus ΔH (the plume rise due to the plume's buoyancy).

Visualization of a buoyant Gaussian air pollutant dispersion plume

To determine ΔH, many if not most of the air dispersion models developed between the late 1960s and the early 2000s used what are known as the Briggs equations. G.A. Briggs first published his plume rise observations and comparisons in 1965.[9] In 1968, at a symposium sponsored by CONCAWE (a Dutch organization), he compared many of the plume rise models then available in the literature.[10] In that same year, Briggs also wrote the section of the publication edited by Slade[11] dealing with the comparative analyses of plume rise models. That was followed in 1969 by his classical critical review of the entire plume rise literature,[12] in which he proposed a set of plume rise equations which have become widely known as "the Briggs equations". Subsequently, Briggs modified his 1969 plume rise equations in 1971 and in 1972.[13][14]

Briggs divided air pollution plumes into these four general categories:

• Cold jet plumes in calm ambient air conditions
• Cold jet plumes in windy ambient air conditions
• Hot, buoyant plumes in calm ambient air conditions
• Hot, buoyant plumes in windy ambient air conditions

Briggs considered the trajectory of cold jet plumes to be dominated by their initial velocity momentum, and the trajectory of hot, buoyant plumes to be dominated by their buoyant momentum to the extent that their initial velocity momentum was relatively unimportant. Although Briggs proposed plume rise equations for each of the above plume categories, it is important to emphasize that "the Briggs equations" which become widely used are those that he proposed for bent-over, hot buoyant plumes.

In general, Briggs's equations for bent-over, hot buoyant plumes are based on observations and data involving plumes from typical combustion sources such as the flue gas stacks from steam-generating boilers burning fossil fuels in large power plants. Therefore, the stack exit velocities were probably in the range of 20 to 100 ft/s (6 to 30 m/s) with exit temperatures ranging from 250 to 500 °F (120 to 260 °C).

A logic diagram for using the Briggs equations[4] to obtain the plume rise trajectory of bent-over buoyant plumes is presented below:

where:
Δh	= plume rise, in m
F	= buoyancy factor, in m4s−3
x	= downwind distance from plume source, in m
xf	= downwind distance from plume source to point of maximum plume rise, in m
u	= windspeed at actual stack height, in m/s
s	= stability parameter, in s−2

The above parameters used in the Briggs' equations are discussed in Beychok's book.[4]

• EPA's Support Center for Regulatory Atmospheric Modeling
• EPA's Air Quality Modeling Group (AQMG)
• NOAA's Air Resources Laboratory (ARL)
• UK Atmospheric Dispersion Modelling Liaison Committee web site
• UK Dispersion Modelling Bureau web site
• Atmospheric Chemistry transport model LOTOS-EUROS
• The Operational Priority Substances model OPS (in Dutch)
• HAMS-GPS Dispersion modelling
• Wiki on Atmospheric Dispersion Modelling. Addresses the international community of atmospheric dispersion modellers - primarily researchers, but also users of models. Its purpose is to pool experiences gained by dispersion modellers during their work.