Geoscience Reference
In-Depth Information
Following the steps in
Eq. (16.8)
yields
∂φ(
κ
)
∂t
d
κ
=−
iκ
j
dZ
∗
(
κ
)dF
j
(
κ
)
+
iκ
j
dZ(
κ
)dF
j
(
κ
)
γκ
2
dZ
∗
(
κ
)dZ(
κ
)
dZ(
κ
)dZ
∗
(
κ
)
(16.9)
−
+
dZ
∗
(
κ
)dS(
κ
)
dZ(
κ
)dS
∗
(
κ
)
.
+
+
We write the first pair of terms on the right side in the notation of
Chapter 15
,
dZ(
κ
)dF
j
(
κ
)
=
C
c,cu
j
(
κ
)d
κ
=
Co
c,cu
j
(
κ
)d
κ
−
iQ
c,cu
j
(
κ
)d
κ
,
dZ
∗
(
κ
)dF
j
(
κ
)
C
c,cu
j
(
κ
)d
κ
=
=
Co
c,cu
j
(
κ
)d
κ
+
iQ
c,cu
j
(
κ
)d
κ
,
(16.10)
with
C
c,cu
j
, the cross spectrum of
c
and
cu
j
, written in terms of its co- and
quadrature spectra. The first pair of terms on the rhs of
Eq. (16.9)
is then
−
iκ
j
dZ
∗
(
κ
)dF
j
(
κ
)
+
iκ
j
dZ(
κ
)dF
j
(
κ
)
=
2
κ
j
Q
c,cu
j
(
κ
)d
κ
.
(16.11)
Similarly, the diffusion and source terms in
Eq. (16.9)
yield
2
γκ
2
dZ(
κ
)dZ
∗
(
κ
)
2
γκ
2
φ(
κ
)d
κ
,
−
=−
dZ
∗
(
κ
)dS(
κ
)
dZ(
κ
)dS
∗
(
κ
)
+
=
2
Co
c,s
(
κ
)d
κ
,
(16.12)
with
Co
c,s
the cospectrum of
c
and
s
. The resulting spectral evolution equation is
∂φ(
κ
)
∂t
2
γκ
2
φ(
κ
).
=
2
Co
c,s
(
κ
)
+
2
κ
j
Q
c,cu
j
(
κ
)
−
(16.13)
The terms in
Eq. (16.13)
integrate over
κ
to the corresponding terms in
Eq. (16.3)
for variance. Since the turbulent transport term in
Eq. (16.3)
vanishes by
homogeneity, we see that the corresponding term in
Eq. (16.13)
integrates to zero:
2
κ
j
Q
c,cu
j
(
κ
)dκ
1
dκ
2
dκ
3
=
0
.
(16.14)
This implies that the term represents transfer within wavenumber space.
Under isotropy
(Chapter 14)
the quadrature spectrum
Q
c,cu
j
, being a vector
function of the vector
κ
,hastheform
Q
c,cu
j
(
κ
)
=
κ
j
F(κ),
(16.15)
2
κ
2
F(κ).
Under isotropy the other
spectra in
Eq. (16.13)
depend only on wavenumber magnitude:
with
κ
=|
κ
|
. It follows that 2
κ
j
Q
c,cu
j
(
κ
)
=
φ(
κ
)
=
φ(κ),
Co
c,s
(
κ
)
=
Co
c,s
(κ).
(16.16)
