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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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