66
Air Pollution and Turbulence: Modeling and Applications
solutions are practically coincident with the complete ones (Equations 3.116a and b)
and, thus, can be used in applications. In particular, if we compute the normalized
autocorrelation function of Equation 3.119, we fi nd:
(
)
− τ
τ =
τ +
τ
R( )
cos( ) sin( )
p
e
q
i
q
(3.120)
By taking the real part of Equation 3.120, the following EAF is obtained:
− τ
τ =
τ
R( )
cos( )
p
e
q
(3.121)
It is worth mentioning that the EAF presented in this last equation is exactly the one
proposed by Frenkiel (1953), which was verifi ed by Anfossi et al. (2005) as being the
correct EAF for LWS conditions.
According to the analytical derivations, there seems to be no need for a specifi c
mechanism, such as gravity waves (Olesen et al., 1984; Etling, 1990) and/or particular
stability conditions, to generate horizontal meandering of a fl ow in a LWS condition.
Furthermore, the present analysis shows that the appearance of wind meandering is
a phenomenon related to the structure of the NS equations; that is, a special condition (in this case the equilibrium between Coriolis force and the pressure gradient)
generates a solution that shows oscillatory characteristics. The solutions suggest that
an increase in the horizontal pressure gradient (i.e., an imbalance between Coriolis
force and the pressure gradient) prevents the meandering phenomenon appearance.
More generally, with increasing production terms on the right-hand side of the NS
equations (departure from geostrophic balance, increasing Reynolds-stress terms)
meandering is being damped and turbulence begins to play a key role. Finally, on the
basis of these results, a system of stochastic Langevin equations that could be used
for computing dispersion in LWS could be derived.
REFERENCES
Anfossi D., Oettl, D., Degrazia, G., and Goulart, A. 2005, An analysis of sonic anemometer
observations in low wind speed conditions, Bound. Lay. Meteorol., 114, 179–203.
Anfossi, D., Degrazia, G., Ferrero, E., Gryning, S.E., Morselli, M.G., and Trini C. 2000,
Estimation of the Lagrangian structure function constant C 0 from surface-layer wind
data, Bound. Lay. Meteorol., 95, 249–270.
Angell, J.K., Pack, D.H., Hoecker, W.H., and Delver, N. 1971, Lagrangian-Eulerian timescale estimated from constant volume balloon fl ights past a tall tower, Quart. J. Roy.
Meteorol. Soc., 97, 87–92.
Arya, S.P. 1995, Modeling and parameterization of near-source diffusion in weak winds,
J. Appl. Meteorol., 34, 1112–1122.
Batchelor, G.K. 1949, Diffusion in a fi eld of homogeneous turbulence, I: Eulerian analysis,
Aust. J. Sci. Res., 2, 437–450.
Caughey S.J. and Palmer, S.G. 1979, Some aspects of turbulence structure through the depth
of the convective boundary layer, Quart. J. Roy. Meteorol. Soc., 105, 811–827.
Champagne, F.H., Friche, C.A., Larve, J.C., and Wyngaard, J.C. 1977, Flux measurements,
fl ux estimation techniques, and fi ne scale turbulence measurements in the unstable
surface layer over land, J. Atmos. Sci., 34, 515–520.
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
solutions are practically coincident with the complete ones (Equations 3.116a and b)
and, thus, can be used in applications. In particular, if we compute the normalized
autocorrelation function of Equation 3.119, we fi nd:
(
)
− τ
τ =
τ +
τ
R( )
cos( ) sin( )
p
e
q
i
q
(3.120)
By taking the real part of Equation 3.120, the following EAF is obtained:
− τ
τ =
τ
R( )
cos( )
p
e
q
(3.121)
It is worth mentioning that the EAF presented in this last equation is exactly the one
proposed by Frenkiel (1953), which was verifi ed by Anfossi et al. (2005) as being the
correct EAF for LWS conditions.
According to the analytical derivations, there seems to be no need for a specifi c
mechanism, such as gravity waves (Olesen et al., 1984; Etling, 1990) and/or particular
stability conditions, to generate horizontal meandering of a fl ow in a LWS condition.
Furthermore, the present analysis shows that the appearance of wind meandering is
a phenomenon related to the structure of the NS equations; that is, a special condition (in this case the equilibrium between Coriolis force and the pressure gradient)
generates a solution that shows oscillatory characteristics. The solutions suggest that
an increase in the horizontal pressure gradient (i.e., an imbalance between Coriolis
force and the pressure gradient) prevents the meandering phenomenon appearance.
More generally, with increasing production terms on the right-hand side of the NS
equations (departure from geostrophic balance, increasing Reynolds-stress terms)
meandering is being damped and turbulence begins to play a key role. Finally, on the
basis of these results, a system of stochastic Langevin equations that could be used
for computing dispersion in LWS could be derived.
REFERENCES
Anfossi D., Oettl, D., Degrazia, G., and Goulart, A. 2005, An analysis of sonic anemometer
observations in low wind speed conditions, Bound. Lay. Meteorol., 114, 179–203.
Anfossi, D., Degrazia, G., Ferrero, E., Gryning, S.E., Morselli, M.G., and Trini C. 2000,
Estimation of the Lagrangian structure function constant C 0 from surface-layer wind
data, Bound. Lay. Meteorol., 95, 249–270.
Angell, J.K., Pack, D.H., Hoecker, W.H., and Delver, N. 1971, Lagrangian-Eulerian timescale estimated from constant volume balloon fl ights past a tall tower, Quart. J. Roy.
Meteorol. Soc., 97, 87–92.
Arya, S.P. 1995, Modeling and parameterization of near-source diffusion in weak winds,
J. Appl. Meteorol., 34, 1112–1122.
Batchelor, G.K. 1949, Diffusion in a fi eld of homogeneous turbulence, I: Eulerian analysis,
Aust. J. Sci. Res., 2, 437–450.
Caughey S.J. and Palmer, S.G. 1979, Some aspects of turbulence structure through the depth
of the convective boundary layer, Quart. J. Roy. Meteorol. Soc., 105, 811–827.
Champagne, F.H., Friche, C.A., Larve, J.C., and Wyngaard, J.C. 1977, Flux measurements,
fl ux estimation techniques, and fi ne scale turbulence measurements in the unstable
surface layer over land, J. Atmos. Sci., 34, 515–520.
© 2010 by Taylor and Francis Group, LLC