Geoscience Reference
In-Depth Information
To verify that the filtering operation
(6.18)
meets the first requirement we take
the partial derivative of
Eq. (6.18)
with respect to time:
∂u
i
∂t
r
G(
x
∂u
i
x
)
∂u
i
(
x
,t)
∂t
d
x
=
∂t
=
−
.
(6.19)
In an unbounded domain filtering commutes with spatial differentiation as well
(Problem 6.2)
.
To meet the second requirement we choose the filter scale
to be much less than
the scale
of the large eddies so that the filter does not affect them significantly. We
also choose
to be much greater than the dissipative eddy scale
η
so it removes
the eddies that are impossible to resolve numerically at large
R
t
. The filter-scale
requirement is then
R
3
/
4
η
, which requires that
/η
∼
be sufficiently
t
large.
Satisfying the third requirement means, in particular, that filtering a constant
C
leaves the constant unchanged. This implies that
∞
x
)Cdx
1
dx
2
dx
3
C
r
=
C
=
G(
x
−
−∞
(6.20)
C
∞
−∞
x
)dx
1
dx
2
dx
3
,
=
G(
x
−
so that
∞
x
)dx
1
dx
2
dx
3
=
G(
x
−
1
,
(6.21)
−∞
which poses a requirement on
G
.
Using this filter on the equations of motion
(6.16)
yields
1
u
i,t
+
(u
i
u
j
)
,j
ρ
p
,i
+
νu
i,jj
+
β
i
,u
i,i
=
=−
0
.
(6.22)
We take the stochastic forcing
β
i
to be of a spatial scale large enough so that it
is resolved perfectly by the filter; i.e.,
β
i
β
i
. In the energy-containing range of
the filtered motion, where the eddy velocity and length scales are
u
and
, viscous
effects are negligible because
R
t
is large. At the smaller resolvable scales we can
use the result of
Chapter 2
,
Subsection 2.5.4
, that eddies of scale
r
have a velocity
scale
u(r)
=
u(r/)
1
/
3
. This implies that the ratio of the inertial and viscous terms
for these eddies is
u(r)r/ν
∼
(r/)
4
/
3
R
t
, which is smallest when
r
,thecutoff
scale. If
(/)
4
/
3
is 10
−
2
and
R
t
exceeds 10
4
, say, then even at the cutoff scale
the viscous term in
Eq. (6.22)
is negligible. We neglect it, taking as our large-eddy
equations
∼
∼
1
u
i,t
+
(u
i
u
j
)
,j
ρ
p
,i
+
=−
β
i
,
i,i
=
0
.
(6.23)
