Geoscience Reference
In-Depth Information
as
r
, the scale of the averaging volume decreases; this has been supported by
measurements in a wide variety of flows. From
Eq. (7.44)
we have
r
μ
1
e
σ
2
e
ln
(/r)
μ
1
e
A
e
μ
1
ln
/r
=
∼
∼
.
(7.45)
If, for example,
r
∼
λ
, the Taylor microscale, then
λ
∼
uλ
ν
=
R
λ
,
(7.46)
and
Eq. (7.45)
becomes
e
σ
2
R
μ
1
∼
.
(7.47)
λ
Thus, the predictions of
Eq. (7.42)
become, for
r
∼
λ
,
S
∂u/∂x
∼
R
3
μ
1
/
8
∂u/∂x
∼
R
μ
λ
.
,
(7.48)
λ
The observations
(
Figure 7.5
)
imply that
μ
1
∼
1
/
3, somewhat greater than the
consensus measured value of
0
.
25 (
Sreenivasan and Kailasnath
,
1993
). This area
has attracted much attention and more sophisticated models have been proposed
(
Sreenivasan and Antonia
,
1997
).
Tennekes and Woods
(
1973
) argued that since collisions of cloud droplets are
fostered by high values of shear, the intermittency of fine structure in large-
R
t
cloud turbulence should significantly enhance the efficiency of turbulent coales-
cence.
Shaw
et al
.
(
1998
) discussed the resulting preferential concentration of
cloud droplets, which causes intermittency in the supersaturation field (
Shaw
,
2000
).
Shaw
(
2003
) sketched a mechanism for this preferential concentration.
†
The
turbulent velocity and number density
n
of droplets are governed by
∼
dv
i
1
τ
d
(u
i
−
∂n
∂t
+
v
i
∂n
n
∂v
i
dt
=
v
i
)
+
g
i
,
∂x
i
=−
∂x
i
,
(7.49)
with
τ
d
the droplet inertial time scale resulting from Stokes' solution and
v
i
the
droplet velocity. A negative droplet velocity divergence on the rhs of the second of
Eq. (7.49)
indicates local congregation of droplets.
For small
τ
d
we can write the solution of the first of
(7.49)
as
τ
d
∂u
i
u
j
∂u
i
O(τ
d
),
v
i
=
u
i
+
τ
d
g
i
−
+
+
(7.50)
∂t
∂x
j
†
The author is grateful to Raymond Shaw for this tailored discussion of this mechanism.
