Geoscience Reference
In-Depth Information
If we now write the velocity as a Fourier series in three dimensions, as in
Chapter 2
,
u
i
(
κ
,t)e
i
κ
·
x
,
u
i
(
x
,t)
=
ˆ
(6.24)
κ
then
u
i
u
j
is (suppressing the dependence on
t
)
u
i
(
κ
)e
i
κ
·
x
κ
u
j
(
κ
)e
i
κ
·
x
u
j
(
κ
)e
i((
κ
+
κ
)
·
x
)
.
u
i
u
j
=
ˆ
ˆ
=
u
i
(
κ
)
ˆ
ˆ
(6.25)
κ
κ
κ
Equation (6.25)
says that the wavenumber vectors of the Fourier components of the
product
u
i
u
j
are the sums of the wavenumber vectors of the Fourier components of
u
i
and
u
j
. The Fourier components of the filtered product
(u
i
u
j
)
r
havewavenumbers
of magnitude less than
κ
c
,
(u
i
u
j
)
r
u
j
(
κ
)e
i((
κ
+
κ
)
·
x
)
,
|
κ
+
κ
|
=
ˆ
ˆ
u
i
(
κ
)
<κ
c
.
(6.26)
κ
κ
But Fourier components of the velocity field having wavenumbers
κ
and
κ
of
magnitude larger than
κ
c
contribute to
(u
i
u
j
)
r
(
Figure 6.3
)
. We can see this by
writing
(u
i
u
j
)
r
as
(u
i
u
j
)
r
(u
i
+
u
i
)(u
j
+
u
j
)
r
(u
i
u
j
)
r
(u
i
u
j
)
r
(u
i
u
j
)
r
(u
i
u
j
)
r
.
=[
]
=
+
+
+
(6.27)
The last three terms in
(6.27)
involve
u
i
and, hence, involve Fourier components
of the velocity field having wavenumbers beyond the cutoff (
Figure 6.3
)
.
Figure 6.3 An illustration in
κ
1
,κ
2
space of the three types of interactions of
Fourier components of the velocity field that according to
Eq. (6.27)
contribute to
the resolvable-scale product
(u
i
u
j
)
r
. Resolvable (r) wavenumbers lie within the
circles; subfilter-scale (s) wavenumbers lie outside. The left panel shows wavenum-
ber vectors of Fourier components of
u
i
and
u
j
,
respectively. By
Eq. (6.26)
their
sum, which lies within the circle, is the wavenumber of a Fourier component of
the filtered product
(u
i
u
j
)
r
. Here the product of two r modes produces another r
mode, as in the first term on the far rhs of
Eq. (6.27)
. The second panel illustrates
the second and third terms in
(6.27)
, where the product of an r mode and an s mode
produces an r mode. The third panel illustrates the final term, where the product
of two s modes produces an r mode.
