Geoscience Reference
In-Depth Information
Since our wave-cutoff filter makes r and s quantities uncorrelated, for the term
in question in
Eq. (6.30)
we can write
−
u
i
(u
i
u
j
)
,j
=−
u
i
[
(u
i
u
j
)
,j
−
(u
i
u
j
)
,j
]=−
u
i
(u
i
u
j
)
,j
u
i
[
(u
i
+
u
i
)(u
j
+
u
j
)
u
i
[
u
i
u
j
+
u
i
u
j
+
u
i
u
j
+
u
i
u
j
]
,j
=−
]
,j
=−
=−
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
.
(6.86)
The corresponding result for the subfilter-scale TKE budget
(6.36)
is
u
i
(u
i
u
j
)
,j
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
−
u
i
u
j
u
i,j
.
−
=−
(6.87)
We expect interscale-transfer terms, like the shear production and viscous dissi-
pation terms in the TKE budget
(5.42)
, to have the form of a stress tensor contracted
with a strain-rate tensor.
Equations (6.86)
and
(6.87)
contain velocity gradients, not
strain rates, but we can express a velocity gradient as a sum of strain-rate and
rotation-rate tensors
s
ij
and
r
ij
:
u
i,j
+
u
j,i
u
i,j
−
u
j,i
u
i,j
=
+
=
s
ij
+
r
ij
.
(6.88)
2
2
s
ij
and
r
ij
are symmetric and antisymmetric, respectively. Since a stress ten-
sor is symmetric, its contraction with
r
ij
vani
shes.
Thus,
on the right side of
Eq. (6.87)
, for example, we can replace
u
i
u
j
u
i,j
with
u
i
u
j
s
ij
.
We now rewrite
(6.86)
and
(6.87)
as the sum of interscale-transfer and turbulent-
transport terms, using some guidelines. The interscale-transfer terms must contain
both r and s elements, because they occur in both equations, and they must sum to
zero. Keeping the terms in the same order and indicating with brackets terms that
we have broken into two parts, we rewrite
Eqs. (6.86)
and
(6.87)
as
1
1
u
i
(u
i
u
j
)
,j
2
(u
i
u
i
u
j
)
,j
−
u
i
u
j
s
ij
−
2
(u
i
u
i
u
j
)
,j
−
=−
(u
i
u
i
u
j
)
,j
−
u
i
u
j
s
ij
,
−
(6.89)
(u
i
u
i
u
j
)
,j
−
u
i
u
j
s
ij
1
u
i
(u
i
u
j
)
,j
2
(u
i
u
i
u
j
)
,j
−
=−
−
1
−[
u
i
u
j
s
ij
]−
2
(u
i
u
i
u
j
)
,j
.
(6.90)
Grouping like parts gives
1
u
i
(u
i
u
j
)
,j
u
i
u
j
s
ij
+
u
i
u
j
s
ij
−
2
(u
i
u
i
u
j
)
,j
−
(u
i
u
i
u
j
)
,j
,
−
=−
(6.91)
1
u
i
(u
i
u
j
)
,j
u
i
u
j
s
ij
−
u
i
u
j
s
ij
−
2
(u
i
u
i
u
j
)
,j
−
(u
i
u
i
u
j
)
,j
.
−
=
(6.92)
