Probability densities and distributions - Turbulence in the Atmosphere - page 312

13.9 Explain why the characteristic function, Eq. (13.12) , can be interpreted as
the expected value of e iκu .
13.10 Prove Eqs. (13.14 )and (13.15) .
13.11 Prove Eq. (13.16) .
13.12 Explain the interpretation of Eq. (13.19) that is given just below it.
13.13 Interpret the second of Eqs. (13.8 ) physically.
13.14 Sketch the pdf of the top-down scalar, using its joint pdf with vertical velocity
shown in Figure 1 3.5 .
13.15 Show that when u u) n in the integran d of Eq. (13.9) yields

central moments (moments about the mean),

= 0, using (u −

n .

[ u − u ]

References
Batchelor, G. K., 1960: The Theory of Homogeneous Turbulence. Cambridge University
Press.
Deardorff, J. W., and G. E. Willis, 1975: A parameterization of diffusion into the mixed
layer. J. Appl. Meteor. , 14 , 1451-1458.
Deardorff, J. W., and G. E. Willis, 1985: Further results from a laboratory model of the
convective boundary layer. Bound.-Layer Meteor. , 32 , 205-236.
Kolmogorov, A. N., 1941: The local structure of turbulence in incompressible viscous
fluid for very large Reynolds numbers. Doklady ANSSSR , 30 , 301-305.
Lamb, R., 1978: A numerical simulation of dispersion from an elevated point source in the
convective boundary layer. Atmos. Environ. , 12 , 1297-1304.
Lamb, R., 1982: Diffusion in the convective boundary layer. In Atmospheric Turbulence
and Air Pollution Modelling , F. Nieuwstadt and H. van Dop, Eds., Dordrecht: Reidel,
pp. 159-229.
Lumley, J. L., and H. A. Panofsky, 1964: The Structure of Atmospheric Turbulence .
New York: Interscience.
Lundgren, T. S., 1967: Distribution functions in the statistical theory of turbulence. Phys.
Fluids , 10 , 969-975.
Taylor, G. I., 1938: The spectrum of turbulence. Part I . Proc. R. Soc. A , 164 , 476-490.
Willis, G. E., and J. W. Deardorff, 1974: A laboratory model of the unstable planetary
boundary layer. J. Atmos. Sci. , 31 , 1297-1307.
Willis, G. E., and J. W. Deardorff, 1978: A laboratory study of dispersion from an elevated
source within a modeled convective planetary boundary layer. Atmos. Environ. ,
12 , 1305-1311.
Wyngaard, J. C., and C.-H. Moeng, 1992: Parameterizing turbulent diffusion through the
joint probability density. Bound.-Layer Meteor. , 60 , 1-13.
Wyngaard, J. C., and H. Tennekes, 1970: Measurements of the small-scale structure of
turbulence at moderate Reynolds numbers. Phys. Fluids , 13 , 1962-1969.