Geoscience Reference
In-Depth Information
Equation (5.15) indicates that the mean-gradient production and molecular
destruction terms in Eq. (5.7) are of the same order, which by the scaling
guidelines is
s 2 u
u j c ∂C
us s
∂x j
=
.
(5.18)
The turbulent transport term is also of this order.
The molecular diffusion term scales as
γ
u
γ
ν
R 1
2 c 2
∂x j ∂x j
γ s 2
2
s 2 u
s 2 u
s 2 u
γ
=
,
(5.19)
t
so it is negligible.
With these scaling results the scalar-variance conservation equation reduces to
(retaining the time-change and mean-advection terms for now)
∂c 2
∂t =−
U j ∂c 2
∂c 2 u j
∂x j
2 u j c ∂C
∂x j
∂x j
χ c .
(5.20)
Its terms (except for time change and mean advection) are of order s 2 u/ .
We have inferred through Eq. (5.15) that the molecular destruction term χ c in
the scalar variance budget is of leading order, s 2 u/ . Thus we can write
s 2 u
s 2
/u ,
∂c
∂x j
∂c
∂x j
χ c =
2 γ
=
(5.21)
which indicates that the time scale of the removal of c 2 by molecular destruction
is /u , the large-eddy turnover time. This says that if their production mechanism
were suddenly shut off the scalar fluctuations would disappear within a time of the
order of a large-eddy turnover time. It is difficult to imagine how this could happen
more quickly. This reflects the strongly dissipative nature of turbulence.
5.3.3 Quasi-steadiness, local homogeneity
As Guideline 6 indicates, the mean-advection and time-change terms in a turbu-
lence budget such as Eq. (5.20) can involve additional, externally imposed scales.
For example, conditions in the atmospheric boundary layer change in the stream-
wise direction due to variations in the surface conditions (variable temperature or
roughness, for example), and change in time due to evolving synoptic conditions
and the diurnal cycle. Thus, for mean advection we i nt roduce a scale U of the mean
velocity and a scale L x of streamwise variations in c 2 ,
U j ∂c 2
U s 2
∂x j
L x ,
(5.22)
 
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