Geoscience Reference
In-Depth Information
We also showed that
N
N
f(κ
n
) f
∗
(κ
n
)
κ
f(κ
n
) f
∗
(κ
n
)
f
2
ff
∗
=
=
=
κ
n
=−
N
n
=−
N
N
=
φ(κ
n
)κ,
(5.28)
n
=−
N
with
φ
the power spectral density of
f
.From
Eq. (5.27)
the derivative of
f
is
N
df
dx
=
f(κ
n
;
α)e
iκ
n
x
,
iκ
n
(5.29)
n
=−
N
so that the derivative variance is
df
dx
2
N
N
κ
n
f(κ
n
) f
∗
(κ
n
)
κ
n
φ(κ
n
)κ.
=
=
(5.30)
n
=−
N
n
=−
N
In the limit of large
N
and
L
this becomes
df
dx
2
∞
κ
2
φ(κ)dκ.
=
(5.31)
−∞
In the three-dimensional case we showed that the variance of a conserved scalar
fluctuation
c(
x
,α)
is given by
∞
c
2
=
E
c
(κ) dκ,
(2.58)
0
where
E
c
is the three-dimensional spectrum for the scalar,
κ
i
κ
i
=
κ
2
φ(κ
1
,κ
2
,κ
3
)dσ.
E
c
(κ)
=
(2.57)
The corresponding expression for the derivative variance is
∞
∂c
∂x
i
∂c
∂x
i
=
κ
2
E
c
(κ) dκ.
(5.32)
0
1
/η)
it is found that
E
c
(κ)
∼
κ
−
5
/
3
(
Chapter 7
). Thus,
Eq. (2.58)
confirms that the principal contributions to the
In the inertial range of wavenumbers
(
1
/
κ
