Geoscience Reference
In-Depth Information
Filtering
Eq. (6.49)
in the horizontal plane and using the constraints
(6.50)
gives,
assuming that the filter passes the mean fields perfectly,
u
r
3
U
1
,
3
δ
i
1
+
u
i
u
j
r
1
u
i,t
+
U
1
u
i,
1
+
ρ
p
,i
,
,j
−
(u
i
u
3
)
,
3
=−
(6.51)
the filtered viscous term being negligible. Multiplying
Eq. (6.51)
by
u
i
, ensemble
averaging, and using
u
i,i
=
0 yields
U
1
u
i
u
i
2
u
i
u
i
,t
=−
p
r
u
r
3
,
3
−
1
2
1
ρ
U
1
,
3
u
r
1
u
r
3
−
(u
i
u
j
)
,j
u
i
.
,
1
−
(6.52)
The first term on the right side vanishes through horizontal homogeneity. From
Appendix 6.2
we can write the last term on the right as
u
i
u
i
u
3
,
3
−
u
i
u
i
u
3
,
3
.
1
2
(u
i
u
j
)
,j
u
i
=−
−
I
−
(6.53)
Combining
Eqs. (6.52)
and
(6.53)
then gives the resolvable-scale TKE budget:
1
2
u
i
u
i
u
3
,
3
.
(6.54)
The first four terms on the right represent the mean rates of shear production,
pressure transport, interscale transfer, and turbulent transport. The fifth is a variant
of turbulent transport resulting from the r-s decomposition. The interscale transfer
term here is that derived in
Appendix 6.2.
The corresponding equation for
u
i
u
i
u
i
,t
=−
U
1
,
3
u
r
1
u
r
3
−
p
r
u
r
3
,
3
−
I
−
u
i
u
i
u
3
,
3
−
1
ρ
1
2
is
u
3
U
1
,
3
δ
i
1
+
u
i
u
j
s
1
u
i,t
+
U
1
u
i,
1
+
ρ
p
,i
+
νu
i,jj
.
=−
(6.55)
,j
the subfilter-scale TKE budget:
1
2
u
i
u
i
,t
=−
p
s
u
3
,
3
+
2
u
i
u
i
u
3
,
3
−
u
i
u
i
u
r
3
,
3
−
1
ρ
1
U
1
,
3
u
1
u
3
−
.
(6.56)
The interpretation of the first five terms on the right side is the same as in the
resolvable-scale budget
(6.54)
; the last term is viscous dissipation.
When the resolvable and subfilter-scale TKE
equations (6.54)
and
(6.56)
are
added the interscale transfer terms disappear, the two pressure-transport terms sum
to one, the four turbulent-transport terms also sum to one, and the familiar TKE
budget emerges
(Problem 6.17)
:
1
2
(u
i
u
i
)
,t
=−
I
−
1
ρ
(pu
3
)
,
3
−
1
2
(u
i
u
i
u
3
)
,
3
−
U
1
,
3
u
1
u
3
−
.
(6.57)
