Geoscience Reference
In-Depth Information
15.6.2.3 A conserved scalar
From
(15.102)
the spectral density of a conserved scalar is related to its three-
dimensional spectrum through
E
c
(κ)
φ(κ)
=
4
πκ
2
.
(15.120)
The spectrum in the horizontal plane is
∞
E
c
(κ)
φ
(
2
)
(κ
h
)
=
4
πκ
2
dκ
3
.
(15.121)
−∞
Since this is again axisymmetric, the two-dimensional scalar spectrum
E
(
2
)
is
c
∞
κ
h
E
c
(κ)
2
κ
2
E
(
2
)
2
πκ
h
φ
(
2
)
=
=
dκ
3
.
(15.122)
c
−∞
Using the geometry of
Figure 15.5
and the inertial subrange form
βχ
c
−
1
/
3
κ
−
5
/
3
,
E
c
(κ)
=
(
7
.
9
)
this yields
π/
2
βχ
c
−
1
/
3
κ
−
5
/
3
E
(
2
)
cos
5
/
3
θdθ,
=
(15.123)
c
h
0
which through
(15.112)
becomes
0
.
84
βχ
c
−
1
/
3
κ
−
5
/
3
E
(
2
)
=
,
(15.124)
c
h
the two-dimensional spectral constant here being 0
.
84
β
.
Questions on key concepts
15.1 Outline the notion of an
autocorrelation function.
15.2 Explain why we say the autocorrelation function is an indicator of the
“memory” of a stochastic function.
15.3 How do the integral scale and the microscale of a stochastic process differ?
15.4 Explain in simple terms the meaning of the power spectral density of a
function. How does randomness enter, and how is it accommodated?
15.5 Interpret and explain
Eq. (15.34)
physically.
15.6 Explain physically the notion of a
three-dimensional
spectrum in turbulence
and why it is attractive.
15.7 Explain how in relatively low-
R
t
flows the Taylor microscale might be
misinterpreted as the scale of the viscous eddies.
