Geoscience Reference
In-Depth Information
The record average commutes with differentiation with respect to independent
variables except that in which it is recorded. Thus for
f(x,t)
(Problem 3.10)
∂f
∂t
∂f
∂x
f
rec
rec
f
rec
rec
∂
∂t
∂
∂x
=
=
,
but
.
(3.28)
The “local average” at any point is an average over a neighborhood of that point:
/
2
1
f
loc
(x,t,)
x
,t)dx
.
=
f(x
+
(3.29)
−
/
2
This does not satisfy rule
(2.6) (Problem3.11)
, but it commuteswith both derivatives
(Problem 3.12)
.
Reynolds
(
1895
) averaged the equation of motion “over a small region of the
flow.” This could be, for example, the local average
3
/
2
/
2
/
2
−
/
2
f(
x
1
f
loc
(
x
,t,)
x
,t)dx
1
dx
2
dx
3
.
=
+
(3.30)
−
/
2
−
/
2
This essentially removes eddies that are much smaller than the cube side
and
minimally affects those that are much larger than
. By contrast, ensemble aver-
aging removes all eddies. Thus the two become more alike when
- i.e.,
in coarse-resolution models. Perhaps this is why the type of averaging used in
coarse-resolution numerical models is not always explicitly discussed.
3.4.1 The generalization to spatial filtering
By the early 1970s increases in computer size and speed made it possible to resolve
the energy-containing eddies in an evolving turbulent flow. A conceptual turning
point was Leonard's (
1974
) generalization of Reynolds' (
1895
) notion of local
averaging in space, as in
Eq. (3.30)
, to spatial
filtering
:
∞
∞
∞
f
filt
(
x
,t)
f(
x
x
,t)G(
x
x
)dx
1
dx
2
dx
3
.
=
+
−
(3.31)
−∞
−∞
−∞
G
is called the
filter function
. This is the local average of
Eq. (3.30)
if
G
is taken
as 1
/
3
within a cube of side
centered at the origin and 0 outside it. As we shall
discuss in
Chapter 6
, other filter functions are now also used, and we now interpret
“local averaging” more broadly as “spatial filtering.” In the usual (
low-pass
)form
of this filtering Fourier components of small wavenumber
(κ
1
/)
are retained
and those of larger wavenumber are removed.
