Geoscience Reference
In-Depth Information
scalar variance come fromsmaller wavenumbers (larger scales), those in the energy-
containing range. By contrast,
Eq. (5.32)
shows that the density of contributions to
the scalar derivative variance grows as
κ
1
/
3
in the inertial range, meaning that the
contributions to the scalar derivative variance come mainly from wavenumbers in
the dissipative range.
We'll denote the intensity scale of the dissipative-range
c
-fluctuations as
s
d
and
their length scale as
η
, the Kolmogorov microscale. (We are assuming here that
γ
ν
. We discuss the general case when
γ
ν
in
Chapter 7
.) Then
χ
c
scales as
γ
s
d
∂c
∂x
j
∂c
∂x
j
∼
χ
c
=
2
γ
η
2
.
(5.33)
(ν
3
/)
1
/
4
,
Solving for
s
d
yields, using
η
=
χ
1
/
2
γ
1
/
4
1
/
4
c
s
d
∼
.
(5.34)
s
2
u/
. With
u
3
/
this gives from
(5.34)
We saw earlier that
χ
c
∼
∼
u
ν
1
/
4
s
s
d
∼
R
1
/
4
∼
.
(5.35)
t
Since
R
t
is large, this confirms that
s
d
s
, meaning that the dissipative eddies
of the scalar field are of low intensity. This is the same relation that connects the
velocity scales of the dissipative and energy-containing eddies,
Eq. (1.35)
.
If we write
Eq. (5.33)
for
χ
c
as
s
d
s
d
χ
c
∼
η
2
/γ
=
τ
r
,
(5.36)
η
2
/γ
is the
time scale for their removal bymolecular diffusion. The time required for molecular
diffusion through distance
η
should depend only on
η
and
γ
, which indeed yield
a time of order
η
2
/γ
. We conclude that
χ
c
does represent the diffusive removal of
η
-sized
c
fluctuations.
then since the
η
-sized
c
fluctuations have amplitude
s
d
, we see that
τ
r
∼
5.3.5 Simple limiting cases
As we shall discuss in
Part II
, near the surface in a quasi-steady, locally homoge-
neous turbulent boundary layer the turbulent transport term in the variance budget
