Boundary-Layer Meteorology

, 129:371 | Cite as

A K-Theory of Dispersion, Settling and Deposition in the Atmospheric Surface Layer

Original Paper

Abstract

Dispersion estimates with a Gaussian plume model are often incorrect because of particle settling (β), deposition (γ) or the vertical gradient in diffusivity (Kv(z) = K0μz). These “non-Gaussian” effects, and the interaction between them, can be evaluated with a new Hankel/Fourier method. Due to the deepening of the plume downwind and reduced vertical concentration gradients, these effects become more important at greater distance from the source. They dominate when distance from the source exceeds Lβ = K0U/β2, Lγ = K0U/γ2 and Lμ = K0U/μ2 respectively. In this case, the ratio β/μ plays a central role and when β/μ = 1/2 the effects of settling and K gradient exactly cancel. A general computational method and several specific closed form solutions are given, including a new dispersion relation for the case when all three non-Gaussian effects are strong. A more general result is that surface concentration scales as C(x) ~ γ−2 whenever deposition is strong. Categorization of dispersion problems using β/μ, Lγ and Lμ is proposed.

Keywords

Air pollution Deposition Diffusion Dispersion Settling Surface layer Transport 

List of Symbols

U, V

Horizontal wind components (m s−1)

K0

Vertical diffusivity at z = 0 (ground or canopy top) (m2 s−1)

Kv(z)

Vertical diffusivity (m2 s−1)

KH

Horizontal diffusivity (m2 s−1)

x, y

Horizontal coordinates (m)

z

Distance above ground or canopy top (m)

\({\hat{z}}\)

Offset vertical distance (m)

h

Scale factor for vertical distance (m−1/2)

η

Scaled offset vertical distance

η0

η at the ground or canopy top

G

Logarithmic derivative of Ĉat z  =  0 (m−1)

k, l

Horizontal wavenumber components (m−1)

C(x, y, z)

Concentration (kg m−3)

Ĉ(klz)

Fourier Transform of C(x, y, z) (kg m−1)

S(x, y)

Source strength (kg m−2 s−1)

a

Horizontal dimension of source (m)

σ

Intrinsic frequency (s−1)

α

in situ decay rate (s−1)

β

Particle settling speed (m s−1)

γ

Deposition velocity (m s−1)

μ

Vertical K gradient (m s−1)

d

Coefficient in asymptotic formulae

L

Various horizontal length scales (m)

\({H_p^1}\)

Hankel function of the first kind of order p

\({\hat{x}}\)

Non-dimensional downstream distance

A, B, C

Regime symbols

References

  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New York, p 1046Google Scholar
  2. Aylor DE (1975) Deposition of particles in a plant canopy. J Appl Meteorol 14: 52–57 doi:10.1175/1520-0450(1975)014<0052:DOPIAP>2.0.CO;2CrossRefGoogle Scholar
  3. Bowman F (1958) Introduction to Bessel functions. Dover, New York, p 135Google Scholar
  4. Calder KL (1949) Eddy diffusion and evaporation in flow over aerodynamically smooth and rough surfaces: a treatment based on laboratory laws of turbulent flow with special reference to conditions in the lower atmosphere. Q J Mech Appl Math 2: 182–197. doi:10.1093/qjmam/2.2.153 CrossRefGoogle Scholar
  5. Calder KL (1961) Atmospheric diffusion of particulate material, considered as a boundary value problem. J Meteorol 18: 413–416Google Scholar
  6. Chamberlain AC (1967) Transport of lycopodium spores and other small particles to rough surfaces. Proc R Soc Lond A Math Phys Sci 296: 45–70. doi:10.1098/rspa.1967.0005 CrossRefGoogle Scholar
  7. Chrysikopoulos CV, Hildemann LM, Roberts PV (1992) A three-dimensional steady state atmospheric dispersion-deposition model for emissions from a ground-level area source. Atmos Environ 26A: 747–757Google Scholar
  8. Corrsin J (1974) Limitations of gradient transport models in random walks and in turbulence. Adv Geophys 18A: 25–29Google Scholar
  9. Csanady GT (1963) Turbulent diffusion of heavy particles in the atmosphere. J Atmos Sci 20: 201–208 doi:10.1175/1520-0469(1963)020<0201:TDOHPI>2.0.CO;2CrossRefGoogle Scholar
  10. Deacon EL (1949) Vertical diffusion in the lowest layers of the atmosphere. Q J R Meteorol Soc 75: 89–103. doi:10.1002/qj.49707532312 CrossRefGoogle Scholar
  11. Deardorff J (1966) The counter-gradient heat flux in the lower atmosphere and in the laboratory. J Atmos Sci 23: 503–506 doi:10.1175/1520-0469(1966)023<0503:TCGHFI>2.0.CO;2CrossRefGoogle Scholar
  12. Denmead OT, Bradley EF (1987) On scalar transport in plant canopies. Irrig Sci 8: 131–149. doi:10.1007/BF00259477 CrossRefGoogle Scholar
  13. DiGiovanni F, Beckett P (1990) On the mathematical modeling of pollen dispersal and deposition. J Appl Meteorol 29: 1352–1357 doi:10.1175/1520-0450(1990)029<1352:OTMMOP>2.0.CO;2CrossRefGoogle Scholar
  14. Eltayeb IA, Al Hassan MH (1993) Two-dimensional transport of dust from an infinite line source at ground level: non-zero roughness height. Geophys J Int 115: 211–214. doi:10.1111/j.1365-246X.1993.tb05599.x CrossRefGoogle Scholar
  15. Hage KD (1961) On the dispersion of large particles from a 15-m Source in the Atmosphere. J Atmos Sci 18: 534–539 doi:10.1175/1520-0469(1961)018<0534:OTDOLP>2.0.CO;2CrossRefGoogle Scholar
  16. Horst TW (1984) The modification of plume models to account for dry deposition. Boundary-Layer Meteorol 30: 413–430. doi:10.1007/BF00121964 CrossRefGoogle Scholar
  17. Lin J-S, Hildemann LM (1996) Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities. Atmos Environ 30: 239–254. doi:10.1016/1352-2310(95)00287-9 CrossRefGoogle Scholar
  18. Overcamp TJ (1976) A general gaussian diffusion-deposition model for elevated point sources. J Appl Meteorol 15: 1167–1171 doi:10.1175/1520-0450(1976)015<1167:AGGDDM>2.0.CO;2CrossRefGoogle Scholar
  19. Pasquill F (1962) Atmospheric diffusion. Van Nostrand, London, p 297Google Scholar
  20. Pasquill F, Smith FB (1983) Atmospheric diffusion. Ellis Harwood, Chichester, England, p 437Google Scholar
  21. Qiu G, Warland JS (2007) A Lagrangian solution to the relationship between source strength and concentration profile under conditions of local advection. Boundary-Layer Meteorol 122: 189–204. doi:10.1007/s10546-006-9098-9 CrossRefGoogle Scholar
  22. Raupach MR (1989) A practical Lagrangian method for relating scalar concentrations to source distributions in vegetation canopies. Q J Roy Meteorol Soc 115: 609–632. doi:10.1002/qj.49711548710 CrossRefGoogle Scholar
  23. Rounds W (1955) Solutions of the two-dimensional diffusion equation. Trans Am Geophys Union 36: 395–405Google Scholar
  24. Smith FB (1962) The problem of deposition in atmospheric diffusion of particulate matter. J Atmos Sci 19: 429–434 doi:10.1175/1520-0469(1962)019<0429:TPODIA>2.0.CO;2CrossRefGoogle Scholar
  25. Smith RB (2003) Advection, diffusion, and deposition from distributed sources. Boundary-Layer Meteorol 107: 273–287. doi:10.1023/A:1022174410745 CrossRefGoogle Scholar
  26. Smith RB (2005) Analytical approach to shear diffusion and tracer age. Boundary-Layer Meteorol 117: 383–415. doi:10.1007/s10546-005-1446-7 CrossRefGoogle Scholar
  27. Sutton OG (1947) The theoretical distribution of airborne pollution from factory chimneys. Q J Roy Meteorol Soc 73: 426–436CrossRefGoogle Scholar
  28. Wilson JD, Sawford BL (1996) Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Boundary-Layer Meteorol 78: 191–210. doi:10.1007/BF00122492 CrossRefGoogle Scholar
  29. Wyngaard JC, Brost RA (1984) Top-down and bottom-up diffusion of a scalar in the convective boundary layer. J Atmos Sci 41: 102–112 doi:10.1175/1520-0469(1984)041<0102:TDABUD>2.0.CO;2 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Geology and GeophysicsYale UniversityNew HavenUSA

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