Geoscience Reference
In-Depth Information
where
τ
L
is the Lagrangian integral time scale defined by
∞
τ
L
=
R(t) dt.
(4.24)
0
τ
L
is of the order of the Eulerian time integral scale
τ
(
Chapter 2
), but there is no
simple relation between them (
Corrsin
,
1963
).
We can also interpret these results in spatial terms. We can write
t
u
p
(t
)dt
,
x
p
(t)
=
(4.25)
0
and ensemble averaging this gives, using the equivalence of Eulerian and
Lagrangian statistics in homogeneous turbulence,
t
u
p
(t
)dt
=
x
p
(t)
=
Ut.
(4.26)
0
Equation (4.26)
says that the ensemble of particles moves downstream at velocity
U
. Thus, we canmake the conversion
x
=
=
Ut, d/dt
Ud/dx
and write
Eq. (4.16)
for
K
as
dσ
t
dz
p
dx
=
dz
p
dt
.
U
2
U
2
1
2
K
=
dx
=
(4.27)
In the short-time and long-time limits
(4.22)
and
(4.23)
, which we can now interpret
as
t
τ
L
and
t
τ
L
, respectively, we then have
w
2
U
w
2
t
w
2
τ
L
=
K(x)
∼
=
x, t
τ
L
;
K(x)
∼
constant
,t
τ
L
.
(4.28)
The behavior of the mean plume width
σ
t
is, from
(4.27)
,
w
2
τ
L
U
1
/
2
(w
2
)
1
/
2
U
x
1
/
2
,x
σ
t
∼
x, x
Uτ
L
;
σ
t
∼
Uτ
L
.
(4.29)
This indicates the mean plume grows linearly with distance downstream in the
initial stages, and slows to parabolic growth (as sketched in
Figure 4.2
)
aftera
distance of order
Uτ
L
.
In summary, the mean concentration profile
C(x,z)
in a slender plume diffusing
from a continuous, crosswind line source in homogeneous turbulence of mean
velocity
U
has the same form as the laminar solution
(4.8)
,
e
−
z
2
Q
√
2
πUσ
t
2
σ
t
,
C(x,z)
=
