Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 39
the velocity autocorrelation coeffi cients ρ τ
L ( )
i
. A second important utilization of
Equation 3.7 concerns the derivation, from the actually measured data, of the turbulent velocity fi eld. This kind of application has already been partly initiated by
Frenkiel (1953) early in the fi fties. More recently Taylor’s diffusion theorem has been
used in the limits of small and large diffusion times, Equations 3.14 and 3.10 respectively, to derive by employing Equation 3.17 an analytical formula for the lateral
dispersion parameter σ =
2
y
v
X adjusted for a specifi c experiment (in that case, the
Hanford-67 experiment, Degrazia, 1991). This dispersion parameter admits a much
better representation when renormalized (rescaled) by the Lagrangian velocity and
Lagrangian timescales. The closer packing of the experimental data obtained by this
normalization allows an analytical fi tting of the results. This in turn helps to determine both the values of relevant parameters like σ = σ
2
v
v and the timescale T Lv ,
besides allowing for a derivation of the form of the lateral velocity autocorrelation
coeffi cient ρ Lv . We fi nally stress that this derivation was applied to the Hanford-67
data series but should be equally valid for other experiments as long as the underlying
hypotheses of Taylor’s theorem are applicable. This refers mainly to the homogeneity
and stationarity character of the turbulence. Whenever these features are warranted,
the same procedure should lead to the local values of the Lagrangian quantities
(σ
2
i and L i
T ) and to a functional form of the velocity autocorrelation coeffi cient. The
appropriated adjusting constants should be obtained by fi tting an analytical curve to
the experimental data.
3.2 THE WIENER–KHINCHIN THEOREM: SELECTING
ENERGY-CONTAINING EDDIES
It is said that turbulent fl ow consists of a superposition of eddies. All these varioussized eddies of which a turbulent motion is composed have a certain kinetic energy,
quantifi ed by their vorticity or by the magnitude of the velocity fl uctuation of the
corresponding frequency. These eddies interact continuously with the turbulent forcing mechanism, from which they extract their energy, and also with each other. An
interesting question is how the kinetic energy of turbulence is distributed according
to various scales (frequencies) of eddy motion. From a practical point of view, it is
still fundamental to identify the scales (frequencies) associated to the large energycontaining eddies. These contain most of the kinetic energy and are responsible for
the turbulent transport in the PBL. In this context, it seems appropriate to introduce
the concept of Lagrangian energy spectrum.
Following Sorbjan (1989), the spectrum measures the distribution of the variance of a variable over frequencies or wavelengths. If the variable is a fl uid particle
turbulent velocity component, the spectrum also describes the distribution of kinetic
energy over frequencies or wavelengths.
Using the Fourier transform, we can defi ne (Panofsky and Dutton, 1984)
+∞
ωτ
−∞
Φ ω =
τ
τ
π ∫
L
L
1
( )
( ) d
i
i
i
R
e
(3.18)
© 2010 by Taylor and Francis Group, LLC
the velocity autocorrelation coeffi cients ρ τ
L ( )
i
. A second important utilization of
Equation 3.7 concerns the derivation, from the actually measured data, of the turbulent velocity fi eld. This kind of application has already been partly initiated by
Frenkiel (1953) early in the fi fties. More recently Taylor’s diffusion theorem has been
used in the limits of small and large diffusion times, Equations 3.14 and 3.10 respectively, to derive by employing Equation 3.17 an analytical formula for the lateral
dispersion parameter σ =
2
y
v
X adjusted for a specifi c experiment (in that case, the
Hanford-67 experiment, Degrazia, 1991). This dispersion parameter admits a much
better representation when renormalized (rescaled) by the Lagrangian velocity and
Lagrangian timescales. The closer packing of the experimental data obtained by this
normalization allows an analytical fi tting of the results. This in turn helps to determine both the values of relevant parameters like σ = σ
2
v
v and the timescale T Lv ,
besides allowing for a derivation of the form of the lateral velocity autocorrelation
coeffi cient ρ Lv . We fi nally stress that this derivation was applied to the Hanford-67
data series but should be equally valid for other experiments as long as the underlying
hypotheses of Taylor’s theorem are applicable. This refers mainly to the homogeneity
and stationarity character of the turbulence. Whenever these features are warranted,
the same procedure should lead to the local values of the Lagrangian quantities
(σ
2
i and L i
T ) and to a functional form of the velocity autocorrelation coeffi cient. The
appropriated adjusting constants should be obtained by fi tting an analytical curve to
the experimental data.
3.2 THE WIENER–KHINCHIN THEOREM: SELECTING
ENERGY-CONTAINING EDDIES
It is said that turbulent fl ow consists of a superposition of eddies. All these varioussized eddies of which a turbulent motion is composed have a certain kinetic energy,
quantifi ed by their vorticity or by the magnitude of the velocity fl uctuation of the
corresponding frequency. These eddies interact continuously with the turbulent forcing mechanism, from which they extract their energy, and also with each other. An
interesting question is how the kinetic energy of turbulence is distributed according
to various scales (frequencies) of eddy motion. From a practical point of view, it is
still fundamental to identify the scales (frequencies) associated to the large energycontaining eddies. These contain most of the kinetic energy and are responsible for
the turbulent transport in the PBL. In this context, it seems appropriate to introduce
the concept of Lagrangian energy spectrum.
Following Sorbjan (1989), the spectrum measures the distribution of the variance of a variable over frequencies or wavelengths. If the variable is a fl uid particle
turbulent velocity component, the spectrum also describes the distribution of kinetic
energy over frequencies or wavelengths.
Using the Fourier transform, we can defi ne (Panofsky and Dutton, 1984)
+∞
ωτ
−∞
Φ ω =
τ
τ
π ∫
L
L
1
( )
( ) d
i
i
i
R
e
(3.18)
© 2010 by Taylor and Francis Group, LLC