Geoscience Reference
In-Depth Information
6.5 The physical mechanisms of interscale transfer
Interscale transfer plays a key role in the balances
(6.40)
,
(6.41)
for TKE and
(6.46)
,
(6.48)
for scalar variance in our “thought problem” involving three-dimensional,
homogeneous turbulence. The interscale-transfer terms in these equations are third
moments (mean values of triple products) that involve both r and s components. We
showed in
Chapter 2
that second moments can be represented as a sum of contribu-
tions from different wavenumbers; we'll now extend that analysis to third moments.
We'll begin with a one-dimensional example, an ensemble of one-dimensional,
homogeneous, random functions
a(x
;
α)
,
b(x
;
α)
,and
c(x
;
α)
that we represent
through complex Fourier series. We write
a
as
N
α)e
iκ
k
x
,
a(x
;
α)
=
N
ˆ
a(κ
k
;
(6.58)
k
=−
and similarly for
b
and
c
.Then
abc
is
N
N
N
m
=−
N
ˆ
a(κ
k
) b(κ
l
)
c(κ
m
)e
i(κ
k
+
κ
l
+
κ
m
)x
.
abc
=
ˆ
(6.59)
k
=−
N
l
=−
N
Since by homogeneity
abc
does not depend on
x
, in the terms that contribute to
the sum in
Eq. (6.59)
the exponential must be 1, which happens if a
nd o
nly if
κ
k
+
κ
l
+
κ
m
0. It follows that the only nonzero contributions to
abc
come
from wavenumbers that satisfy
κ
m
=
κ
l
)
. Such groups of three Fourier
components whose wavenumbers sum to zero are called
triads
. Therefore we can
write
Eq. (6.59)
as
=−
(κ
k
+
N
N
a(κ
k
) b(κ
l
)
abc
=
N
ˆ
c(
ˆ
−
κ
k
−
κ
l
).
(6.60)
k
=−
N
l
=−
The result
(6.60)
extends to three dimensions, and thus to the expression
(6.42)
for
the rate of interscale transfer in three-dimensional homogeneous turbulence:
u
i
u
j
s
ij
−
u
i
u
j
s
ij
I
=
u
j
s
p
s
i,j
r
q
,
N
N
N
N
u
i
κ
k
ˆ
u
j
κ
l
ˆ
s
i,j
κ
s
m
−
u
i
κ
n
ˆ
r
r
=
N
ˆ
N
ˆ
ˆ
κ
κ
k
=−
N
l
=−
n
=−
N
p
=−
r
r
s
s
s
r
k
+
κ
l
+
κ
m
=
0
,
n
+
κ
p
+
κ
q
=
0
,
(6.61)
κ
κ
k
, for example, is a wavenumber vector that lies in the resolved domain.
where
κ
