Geoscience Reference
In-Depth Information
so to leading order the droplet velocity divergence is
4
s
ij
s
ij
−
r
ij
r
ij
,
∂v
i
τ
d
∂u
i
∂x
j
∂u
j
τ
d
∂x
i
=−
∂x
i
=−
(7.51)
where we have used
u
i,j
r
ij
,
the sum of strain- and rotation-rate tensors,
Eq. (2.73)
.
Equation (7.51)
says that strong local vorticity causes positive cloud
droplet divergence; weak local vorticity causes negative divergence, droplet con-
gregation, and therefore higher droplet collision rates. The variance of this droplet
velocity divergence is
=
s
ij
+
∂u
∂x
2
2
τ
d
F
∂u/∂x
2
∂v
i
∂x
i
∂v
j
∂x
j
=
∂u
i
∂x
j
∂u
j
∂x
i
∂u
k
∂x
m
∂u
m
∂x
k
∼
τ
d
τ
d
F
∂u/∂x
∼
,
ν
(7.52)
and so is very much larger in cumulus clouds than in laboratory experiments, say.
7.5 Two-dimensional turbulence
Two-dimensional turbulence, flow confined to a plane, has been used as a model
for the largest scales of motion in the atmosphere. As we shall see it shares features
with three-dimensional turbulence and is much more accessible computationally.
7.5.1 Vorticity and conserved scalars
We'll consider motion in an
x,y
plane:
u
i
=
u(x,y,t), v(x,y,t),
0
.
Vorticity
can have only a component normal to the plane:
ω
3
(x,y,t)
.
As a
result the vortex-stretching term in the vorticity
equation (1.28)
is identically zero
and the equation reads
=
0
,
0
,
ω
i
˜
˜
∂
2
D ω
3
∂ ω
3
u
i
∂ ω
3
ω
3
Dt
=
∂t
+˜
∂x
i
=
ν
∂x
i
∂x
i
,
(7.53)
which is identical to
Eq. (1.31)
for a conserved scalar. The absence of a vortex-
stretching term in
Eq. (7.53)
makes the velocity field in two-dimensional turbulence
differ in some striking ways from that in three-dimensional turbulence (
Tennekes
,
1978
).
The gradient
g
i
=
c
,i
of a conserved scalar
c
satisfies
D
g
i
Dt
=˜
˜
g
i,t
+˜
u
j
˜
g
i,j
=−˜
u
j,i
˜
g
j
+
ν
g
i,jj
.
˜
(7.54)
