Geoscience Reference
In-Depth Information
with
κ
n
=
2
πn/L
.Since
u
i
is random its Fourier coefficients are different in each
realization.
Let us apply a wave-cutoff filter to
u
i
, with the cutoff wavenumber set at 2
πN
c
/L
,
with
N
c
N
. Fourier coefficients of wavenumber less than or equal to 2
πN
c
/L
are passed without change; those of wavenumber greater than 2
πN
c
/L
are made
zero. Then we can write the resolvable and subfilter-scale parts of
f
as
≤
N
N
c
n
=−
N
ˆ
n
=−
N
c
ˆ
u
i
(x
α)T (κ
n
)e
iκ
n
x
α)e
iκ
n
x
,
;
α)
=
u
i
(κ
n
;
=
u
i
(κ
n
;
u
i
(x
u
i
=
α)e
iκ
n
x
;
α)
=
u
i
−
>N
c
ˆ
u
i
(κ
n
;
(6.84)
|
n
|
−
N
N
α)e
iκ
n
x
α)e
iκ
n
x
.
=
1
ˆ
u
i
(κ
n
;
+
1
ˆ
u
i
(κ
n
;
n
=−
N
c
−
n
=
N
c
+
Then the covariance
u
i
u
i
is
N
c
−
N
N
c
N
m
=−
N
c
−
1
ˆ
m
=
N
c
+
1
ˆ
u
i
u
i
=
u
i
(κ
n
)
u
i
(κ
m
)
ˆ
+
u
i
(κ
n
)
u
i
(κ
m
)
ˆ
=
0
,
n
=−
N
c
n
=−
N
c
(6.85)
because the wave-cutoff filter ensures that the r and s components have no Fourier
modes in common so there are no contributions to
the s
um.
Thi
s h
olds
in three
spatial dimensions, so the wave-cutoff filter yields
u
i
u
i
u
i
u
i
+
u
i
u
i
,
neatly
=
separating the TKE into
resolvable-
and
subfilter-scale
components.
Appendix 6.2
Interscale transfer
In our homogeneous turbulence “thought problem” of
Section 6.
3
we deduced that
the resolvable-TKE budget
(6.30)
contains a term
u
i
(u
i
u
j
)
,j
−
and the subfilter-
u
i
(u
i
u
j
)
,j
. We deduced that these
represent “interscale transfer” - transfer of energy, without loss, from the r scales
to the s scales.
This term appears also in the horizontally homogeneous boundary-layer problem
in
Section 6.4
. In that problem we applied the filter in the horizontal plane, rather
than in three dimensions. For that reason we would like to analyze this termwithout
requiring three-dimensional homogeneity.
TKE budget
(6.36)
contains an analogous term
−
