Overview of atmospheric transport and dispersion

• atmosphere is highly dispersive
• amount of dispersion will depend on the state of turbulence
• turbulence is determined by the atmosphere's stratification / stability, surface roughness, and wind speed
• traditionally transport and dispersion for regulatory purposes, is modelled using variants of a ``Gaussian Plume Model'' which have been modified to account for inversions, terrain, etc.
• mean concentrations of pollutants downwind of a stack will closely follow a normal distribution
• the Gaussian model equation:

  $$\chi {\frac{X}{2 \pi \sigma_{y} \sigma_{z} \overline{u}}} \left[\vphantom{-\frac{y^{2}}{2 \sigma_{y}^{2}}}\right. {\frac{y^{2}}{2 \sigma_{y}^{2}}} \left.\vphantom{-\frac{y^{2}}{2 \sigma_{y}^{2}}}\right] \left[\vphantom{\exp{\left(-\frac{(z-H)^{2}}{2 \sigma_{z}^{2}}\right)} + \exp{\left(-\frac{(z+H)^{2}}{2 \sigma_{z}^{2}}\right)}}\right. \left(\vphantom{-\frac{(z-H)^{2}}{2 \sigma_{z}^{2}}}\right. {\frac{(z-H)^{2}}{2 \sigma_{z}^{2}}} \left.\vphantom{-\frac{(z-H)^{2}}{2 \sigma_{z}^{2}}}\right) \left(\vphantom{-\frac{(z+H)^{2}}{2 \sigma_{z}^{2}}}\right. {\frac{(z+H)^{2}}{2 \sigma_{z}^{2}}} \left.\vphantom{-\frac{(z+H)^{2}}{2 \sigma_{z}^{2}}}\right) \left.\vphantom{\exp{\left(-\frac{(z-H)^{2}}{2 \sigma_{z}^{2}}\right)} + \exp{\left(-\frac{(z+H)^{2}}{2 \sigma_{z}^{2}}\right)}}\right]$$ (1)

• where X is the rate of emission from the source (kg s^-1)
• $\sigma_{y}^{}$, $\sigma_{z}^{}$ are the horizontal and vertical standard deviations of the pollutant distribution in the y and z directions (m)
• $\overline{u}$ is the mean horizontal wind speed through the depth of the plume direction (m s^-1)
• H is the effective stack height (m)
• $\chi$ is the pollutant concentration (kg m^-3), and is a function space, and the nature of turbulence.
• if only the ground level concentrations are required, this equation simplifies somewhat:

  $$\chi {\frac{X}{\pi \sigma_{y} \sigma_{z} \overline{u}}} \left[\vphantom{-\left(\frac{y^{2}}{2 \sigma_{y}^{2}} + \frac{H^{2}}{2 \sigma_{z}^{2}}\right)}\right. \left(\vphantom{\frac{y^{2}}{2 \sigma_{y}^{2}} + \frac{H^{2}}{2 \sigma_{z}^{2}}}\right. {\frac{y^{2}}{2 \sigma_{y}^{2}}} {\frac{H^{2}}{2 \sigma_{z}^{2}}} \left.\vphantom{\frac{y^{2}}{2 \sigma_{y}^{2}} + \frac{H^{2}}{2 \sigma_{z}^{2}}}\right) \left.\vphantom{-\left(\frac{y^{2}}{2 \sigma_{y}^{2}} + \frac{H^{2}}{2 \sigma_{z}^{2}}\right)}\right]$$ (2)

• the $\sigma$'s are a measure of the vertical and horizontal spreading of a plume and thus represent the amount of atmospheric dispersion which depends on the state of turbulence
• they will be a function of x, as well as atmospheric stability and surface roughness
• the trick in much of this kind of dispersion modelling is in accurate calculation / estimation of the $\sigma$'s
• in the absence of accurate turbulence data, it is possible to crudely categorize the stability of the atmosphere based on routine atmospheric observations (table 1).
  

  Table 1: Note: A, extremely unstable; B, moderately unstable; C, slightly unstable; D, neutral (heavy overcast day or night); E, slightly stable; F, moderately stable.

  Surface	Daytime solar radiation			Nighttime conditions
  wind m s-1	Strong	Moderate	Slight	$\geq$ 4/8 clouds	$\le$ 3/8 clouds
  < 2	A	A-B	B	-	-
  2-3	A-B	B	C	E	F
  3-4	B	B-C	C	D	E
  4-6	C	C-D	D	D	D
  > 6	C	D	D	D	D

  
  

  Briggs (1973) proposed a series of empirical formulae for the determination of $\sigma_{z}^{}$ and $\sigma_{y}^{}$:

  
  

  Table 2: Briggs' $\sigma_{y}^{}$, $\sigma_{z}^{}$ formulae for elevated small releases, where 10² < x < 10⁴ m.

  Stability	$\sigma_{y}^{}$ (m)	$\sigma_{z}^{}$ (m)
  Class
  Open country conditions
  A	0.22x(1 + 0.0001x)- .5	.20x
  B	0.16x(1 + 0.0001x)- .5	.12x
  C	0.11x(1 + 0.0001x)- .5	.08x(1 + 0.0002x)- .5
  D	0.08x(1 + 0.0001x)- .5	.06x(1 + 0.0015x)- .5
  E	0.06x(1 + 0.0001x)- .5	.03x(1 + 0.0003x)-1
  F	0.04x(1 + 0.0001x)- .5	.016x(1 + 0.0003x)-1
  Urban conditions
  A-B	0.32x(1 + 0.0004x)- .5	.24x(1 + 0.001x).5
  C	0.22x(1 + 0.0004x)- .5	.20x
  D	0.16x(1 + 0.0004x)- .5	.14x(1 + .0003x)- .5
  E-F	0.11x(1 + 0.0004x)- .5	.08x(1 + .00015x)- .5

  
  
• [OH Oke 9.8, 9.11]
• the above model does not include many very important effects
  • terrain effects on impingement and steering the plume
  • vertical variations in stability
  • horizontal or vertical variations in the wind field
  • local circulations
• modern implementations of the GPM have parameterizations to try and remedy these shortcomings
• other methods of modelling transport and dispersion include:
  • Puff models - treat emissions as a series of 'puffs' which are individually transported and dispersed (variant of GPM)
  • Box models - based upon conservation of pollutant mass within a volume (often used in chemical transformation models, e.g. photochemical models)
  • Gradient transport models - based upon flux/gradient (K) turbulent transport (GPM is a special case of this in which the eddy diffussivities are constant)
  • Trajectory models - move a vertical column at the mean wind speed with pollutants added at the bottom as they are generated
  • Mesoscale models / Lagrangian Particle Dispersion Models - mesoscale atmospheric models produce gridded mean wind fields, many particles are inserted into the wind field and are transported by the mean wind and diffused by a random element representing turbulence