Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 35
deviation of the wind direction increases, which makes it more diffi cult to defi ne a
mean plume direction. Even when the stability (during nighttime) reduces the vertical dispersion and the instantaneous plume may be thin, meandering disperses the
plume over a rather wide angular sector. As a consequence, horizontal diffusion
is enhanced because of the meander. Since the meandering phenomenon occurs
frequently, their description is of fundamental importance in air dispersion modeling
in the atmospheric boundary layer.
Based on fact that the Navier–Stokes (N–S) equations describe the multitude of
phenomena associated to the turbulence processes, the investigation of both cases
(decaying turbulence in the CBL and the low-frequency horizontal wind oscillations) will be accomplished employing these conservation equations. In the decaying
turbulence case, the N–S equations will be used to derive eddy diffusivities. In the
LWS case, the N–S equations will be utilized to explain the low-frequency horizontal wind oscillations.
3.1.1 TAYLOR’S MODEL
By considering Taylor’s classical paper (Taylor, 1921) on diffusion by continuous
movements, we assume the motion of fl uid particles (massless) in a turbulent fl ow
fi eld by velocity fl uctuations.
We take the simplest case of dispersion in one direction only, where X i corresponds to an arbitrary direction associated to the i-component of the velocity of the
fl uid particle (i = u, v, w). If the fl uid particle leaves the origin at t = 0, its position X i
at time t is given by
=
′ ′
∫
0
( )
( )d
t
i
i
X t
v t t
(3.1)
An eddy diffusivity can be obtained by multiplying Equation 3.1 by v i (t)
⎛
⎞
=
=
=
′ ′
⎜
⎟
⎝
⎠ ∫
2
0
d
d 1
( ) ( )
( )
( ) ( )d
d
d 2
t
i
i
i
i
i
i
i
X
X t v t
X t
X
v t v t t
t
t
(3.2)
and taking an ensemble-mean over many realizations (i.e., consider a large number
of fl uid particles that are assumed to start in succession from a fi xed time t = 0), we
obtain
⎛
⎞ =
′ ′
⎜
⎟
⎝
⎠ ∫
2
d 1
( ) ( )d
d 2
t
i
i
i
o
X
v tv t t
t
(3.3)
In the above equation, both sides have dimensions of an eddy diffusivity (m 2 s −1 ).
Taylor’s theory applies to dispersion in a fi eld of homogeneous and stationary
turbulence, that is, turbulence whose statistical properties have quantitatively the
same structure in all parts of the fl ow fi eld and the statistical properties of variables
© 2010 by Taylor and Francis Group, LLC
deviation of the wind direction increases, which makes it more diffi cult to defi ne a
mean plume direction. Even when the stability (during nighttime) reduces the vertical dispersion and the instantaneous plume may be thin, meandering disperses the
plume over a rather wide angular sector. As a consequence, horizontal diffusion
is enhanced because of the meander. Since the meandering phenomenon occurs
frequently, their description is of fundamental importance in air dispersion modeling
in the atmospheric boundary layer.
Based on fact that the Navier–Stokes (N–S) equations describe the multitude of
phenomena associated to the turbulence processes, the investigation of both cases
(decaying turbulence in the CBL and the low-frequency horizontal wind oscillations) will be accomplished employing these conservation equations. In the decaying
turbulence case, the N–S equations will be used to derive eddy diffusivities. In the
LWS case, the N–S equations will be utilized to explain the low-frequency horizontal wind oscillations.
3.1.1 TAYLOR’S MODEL
By considering Taylor’s classical paper (Taylor, 1921) on diffusion by continuous
movements, we assume the motion of fl uid particles (massless) in a turbulent fl ow
fi eld by velocity fl uctuations.
We take the simplest case of dispersion in one direction only, where X i corresponds to an arbitrary direction associated to the i-component of the velocity of the
fl uid particle (i = u, v, w). If the fl uid particle leaves the origin at t = 0, its position X i
at time t is given by
=
′ ′
∫
0
( )
( )d
t
i
i
X t
v t t
(3.1)
An eddy diffusivity can be obtained by multiplying Equation 3.1 by v i (t)
⎛
⎞
=
=
=
′ ′
⎜
⎟
⎝
⎠ ∫
2
0
d
d 1
( ) ( )
( )
( ) ( )d
d
d 2
t
i
i
i
i
i
i
i
X
X t v t
X t
X
v t v t t
t
t
(3.2)
and taking an ensemble-mean over many realizations (i.e., consider a large number
of fl uid particles that are assumed to start in succession from a fi xed time t = 0), we
obtain
⎛
⎞ =
′ ′
⎜
⎟
⎝
⎠ ∫
2
d 1
( ) ( )d
d 2
t
i
i
i
o
X
v tv t t
t
(3.3)
In the above equation, both sides have dimensions of an eddy diffusivity (m 2 s −1 ).
Taylor’s theory applies to dispersion in a fi eld of homogeneous and stationary
turbulence, that is, turbulence whose statistical properties have quantitatively the
same structure in all parts of the fl ow fi eld and the statistical properties of variables
© 2010 by Taylor and Francis Group, LLC