Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-19T06:44:09.111Z Has data issue: false hasContentIssue false
  • English
  • Français

On the spectral turbulent diffusivity theory for homogeneous turbulence

Published online by Cambridge University Press:  19 April 2006

Ruwim Berkowicz  and
Lars P. Prahm
Ruwim Berkowicz
Affiliation:
National Agency of Environmental Protection, Air Pollution Laboratory, Risø National Laboratory, DK-4000 Roskilde, Denmark
Lars P. Prahm
Affiliation:
National Agency of Environmental Protection, Air Pollution Laboratory, Risø National Laboratory, DK-4000 Roskilde, Denmark
Get access Rights & Permissions [Opens in a new window]

Abstract

The spectral turbulent diffusivity (STD) theory, originally deduced from a spectral generalization of the gradient-transfer theory (Berkowicz & Prahm 1979), is here derived from a basic concept of turbulent mixing for the case of homogeneous turbulence. The turbulent mixing is treated in a way similar to Prandtl's mixing-length concept. The contribution to the turbulent flux from eddies of different length is represented by a linear superposition. The spatial variation of the concentration distribution is described in terms of Fourier series. This procedure results in the spectral diffusivity formulation, which is Eulerian and scale dependent. If the concentration distribution is approximated by a truncated Taylor expansion instead of an exact representation by the Fourier series, the gradient-transfer approximation is retrieved.

The turbulent energy density, as function of the eddy length, is related to the eddy transport velocity and a probability of the occurrence of the eddies. The eddy transport velocity, derived from the relation between the energy spectrum and the Lagrangian correlation function, is used for computation of the spectral turbulent diffusivity. The turbulent energy spectrum is approximated by the inertial sub-range form $(-\frac{5}{3}$ law). The STD coefficient obtained here has, for large wavenumbers, a slope of $k^{-\frac{4}{3}}$ as predicted previously.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 100 , Issue 2 , 25 September 1980 , pp. 433 - 448
Copyright
© 1980 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. & Townsend, A. A. 1956 Turbulent diffusion. Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 352–399. Cambridge University Press.
Berkowicz, R. & Prahm, L. P. 1978 Pseudospectral simulation of dry deposition from a point source. Atmos. Environ. 12, 379–387.Google Scholar
Berkowicz, R. & Prahm, L. P. 1979 Generalization of K-theory for turbulent diffusion. Part I. Spectral turbulent diffusivity concept. J. Appl. Met. 18, 266–272.Google Scholar
Berkowicz, R., Prahm, L. P. & Louis, J.-F. 1979 Global 2-D spectral dispersion model. Proc. WMO Symp. on the long-range transport of pollutants and its relation to general circulation including stratospheric-tropospheric exchange processes, Sofia. WMO Rep. no. 538.Google Scholar
Christensen, O. & Prahm, L. P. 1976 A pseudospectral model for dispersion of atmospheric pollutants. J. Appl. Met. 15, 1284–1294.Google Scholar
Hanna, S. R. 1968 A method of estimating vertical eddy transport in the planetary boundary layer using characteristics of the vertical velocity spectrum. J. Atmos. Sci. 25, 1026–1032.Google Scholar
Heisenberg, W. 1948 Zur statistischen Theorie der Turbulenz. Z. Phys. 124, 628–657.Google Scholar
Hinze, J. O. 1975 Turbulence. 2nd edn. McGraw-Hill.
Inoue, E. 1950 On the turbulent diffusion in the atmosphere (I). J. Met. Soc. Japan 28, 444–455.Google Scholar
Lewellyn, W. S. & Teske, M. E. 1976 Second-order closure modelling of diffusion in the atmospheric boundary layer. Boundary-Layer Met. 10, 69–90.Google Scholar
Markee, E. H. 1963 On the relationships of range to standard deviation of wind fluctuations. Mon. Weath. Rev. 91, 83–87.Google Scholar
Monin, A. S. 1955 Equation of turbulent diffusion. Dokl. Akad. Nauk S.S.S.R. 105, 256–259.Google Scholar
Monin, A. S. 1956 Horizontal mixing in the atmosphere. Izv. Akad. Nauk S.S.S.R., Ser. Geofiz. 3, 327–345.Google Scholar
Monin, A. S. & Yaglom, A. M. 1965 Statistical Fluid Mechanics, Mechanics of Turbulence. Part I. Washington: Trans-Joint Publication Research Service.
Pasquill, F. 1974 Atmospheric Diffusion. 2nd edn Chichester: Ellis Norwood. Wiley.
Phythian, R. & Curtis, W. D. 1978 The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow. J. Fluid Mech. 89, 241–250.Google Scholar
Prahm, L. P., Berkowicz, R. & Christensen, O. 1979 Generalization of K-theory for turbulent diffusion. Part II. Spectral diffusivity model for plume dispersion. J. Appl. Met. 18, 273–282.Google Scholar
Prahm, L. P. & Christensen, O. 1977 Long-range transmission of pollutants simulated by the 2-D pseudospectral dispersion model. J. Appl. Met. 16, 896–910.Google Scholar
Prandtl, L. 1925 Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. angew. Math. Mech. 5, 136–139.Google Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. Roy. Soc. A 110, 709–737.Google Scholar
Roberts, P. M. 1961 Analytical theory of turbulent diffusion. J. Fluid Mech. 11, 257–283.Google Scholar
Schmidt, W. 1925 Der Massenhaustausch in freier Luft und Verwandte Erscheinungen. In Probleme der Kosmischen Physic. Hamburg: Von Henri Grand.
Schönfeld, J. C. 1962 Integral diffusivity. J. Geophys. Res. 67, 3187–3199.Google Scholar
Smith, F. B. 1977 Application from field programmes to estimation of K-profiles and vertical dispersion. British Met. O. 14, Turbulence and Diffusion Note no. 86 (unpublished).Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. London Math. Soc. Ser. 2, 20, 196–211.Google Scholar