Geoscience Reference
In-Depth Information
Figure 3.2 The evolution of the profile of mass flux
wc
in the turbulent diffusion
problem of
Figure 2.5
.
During its travel the diameter
d
of the filament increases due to molecular diffu-
sion of the scalar
c
into the fluid, causing the concentration averaged across its cross
c
ave
, to decrease with time. From the conserved scalar
equation (1.31)
we
estimate the magnitude of this rate of change of
section,
c
ave
as
˜
c
ave
d
2
c
ave
τ
molec
,
D
Dt
˜
γ
˜
˜
c
ave
2
c
ave
∼
∇
˜
∼
∼
γ
(3.16)
d
2
/γ
the time scale of this molecular diffusion process. The propor-
tionality of
τ
molec
to
d
2
indicates that only concentration anomalies of small spatial
scale are removed quickly by molecular diffusion. If
d
with
τ
molec
∼
10
−
3
m, for example,
and
γ
=10
−
5
m
2
s
−
1
(roughly the value for temperature and water vapor in air),
τ
molec
∼
=
10
5
s, about one day.
Under very stably stratified conditions
(Part II)
flow in the lower atmosphere can
be laminar not far above the surface. In such cases an effluent plume from a stack of
1m diameter in a wind of 10 m s
−
1
could extend hundreds of kilometers downwind
in nonturbulent air!
In turbulent flow the ensemble-averaged concentration field
C(x,y,z,t)
down-
stream of the source is formally defined by an average over many instantaneous
plumes:
10
−
1
s. But if
d
=
1mthen
τ
molec
∼
N
1
N
C(x,y,z,t)
=
lim
N
1
˜
c(x,y,z,t
;
α).
(3.17)
→∞
α
=
We'll refer to “ensemble-averaged” simply as “mean.”
Figure 3.3
shows a sketch
of the downstream development of the mean plume in the
x
−
z
plane. Its contrast
with the instantaneous plume is striking.
