Geoscience Reference
In-Depth Information
With
(15.60)
this implies that
f
and
g
are related through
r
2
∂f
∂r
1
2
r
∂
∂r
(r
2
f),
g(r)
=
f
+
=
(15.65)
so that in an isotropic field
R
ij
is determined by a single scalar function, either
f
or
g
.
G. I.
Taylor
(
1935
) defined the length scale
λ
from the curvature of
g
at the origin:
0
2
∂
2
g
∂r
2
1
/
2
λ
=
−
.
(15.66)
Today we call
λ
the Taylor microscale. He misinterpreted
λ
, stating that it “may
roughly be regarded as a measure of the diameters of the
smallest
eddies which
are responsible for the dissipation of energy.” In fact from the definition of
g
,
Figure 15.3
,
we can express
Eq. (15.66)
as
(Problem 15.20)
2
u
2
∂u
n
∂x
p
u
2
/ν
.
∂u
n
∂x
p
λ
2
=
∼
(15.67)
We can rewrite this as
R
−
1
/
2
ηR
1
/
4
λ
=
=
,
(15.68)
t
t
which shows that the length scale
λ
lies between
and
η
, much closer to the latter
(Problem 15.14)
. It is considered to be more significant in a time characteristic of
the viscous dissipation,
τ
d
∼
λ
2
/ν
, than as a length scale.
We can also use isotropy to relate the one-dimensional spectra, which we can
now write as
u
2
/
∼
+∞
+∞
u
2
f(r)
e
iκr
F
11
(κ) dκ,
u
2
g(r)
e
iκr
F
11
(κ) dκ.
=
=
(15.69)
−∞
−∞
From
(15.69)
we can write
+∞
u
2
∂f
e
iκr
iκF
11
(κ) dκ.
∂r
=
(15.70)
−∞
Thus, the inverse relation is
+∞
1
2
π
e
−
iκr
u
2
∂f
iκF
11
(κ)
=
∂r
dr.
(15.71)
−∞
