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cross-section in the large eddy to that in the turbulent eddy =
W
/
w
*
. The magnitude
of
W
is smaller than that of
w
*
(Eq. 1.1). The ratio
W
/
w
*
is equal to the upward mass
flux of surface air/unit time/unit cross-section in the rising large eddy. The volume
fraction of turbulent eddy across unit cross-section of large eddy envelope is equal
to
r
/
R
. Turbulence scale upward mass flux of surface air equal to
W
/
w
*
occurs in
the fractional volume
r
/
R
of the large eddy. Therefore, the net upward mass flux
f
of surface air/unit time/unit cross-section in the large eddy environment is equal to
W
w
r
R
f
=
.
(1.5)
*
The large eddy circulation speed
W
and the corresponding temperature perturbation
θ may be expressed in terms of
f
and
z
as follows:
Wwfz
fz
=
=
*
(1.6)
θθ
.
*
In Eq. (1.6), θ
*
is the temperature perturbation corresponding to the primary turbu-
lent eddy circulation speed
w
*
.
The corresponding moisture content
q
at height
z
is related to the moisture con-
tent
q
*
at the surface and is given as follows:
qqfz
=
.
(1.7)
*
Substituting from Eqs. (1.1), (1.2), and (1.4), the net upward flux
f
of surface air at
level
z
is now obtained from Eq. (1.5) as follows:
1
WR
wrz
2
f
=
ln
z
=
ln
z
=
ln .
z
(1.8)
kz
π
z
*
In Eq. (1.8),
f
represents the steady-state fractional volume of surface air at any level
z
. A fraction
f
of surface aerosol concentration
N
*
is carried upward to normalized
height
z
. The aerosol number concentration
N
at level
z
is then given as follows:
NNf
=
*
.
(1.9)
Since atmospheric aerosols originate from surface, the vertical profile of mass and
number concentration of aerosols follow the
f
distribution. The vertical mass ex-
change mechanism predicts the
f
distribution for the steady-state vertical transport
of aerosols at higher levels. The vertical variation of atmospheric aerosol number
concentration given by the
f
distribution is shown in Fig.
1.5
. The vertical variation
of large eddy circulation speed
W
, the corresponding temperature θ and the moisture
content
q
are shown in Fig.
1.6
.
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