Geoscience Reference
In-Depth Information
The autocorrelation function often emerges in statistics of derivatives and
integrals of
u(t).
For example
(Problem 15.3)
,
b
u(t) dt
2
b
b
u
2
b
a
b
u(t
)u(t
)dt
dt
=
ρ(t
−
t
)dt
dt
.
(15.3)
=
a
a
a
a
The variance of the derivative is
(Problem 15.4)
du(t)
dt
2
t
=
0
d
2
ρ(t)
dt
2
=−
u
2
.
(15.4)
The quantities in
(15.3)
and
(15.4)
define two time scales associated with
ρ(t)
:
∞
t
=
0
=−
d
2
ρ(t)
dt
2
2
λ
2
.
ρ(t)dt
=
τ,
(15.5)
0
τ
and
λ
are known as the integral scale and the microscale, respectively. In tur-
bulence the corresponding scale
λ
x
of a spatial record
u(x)
is called the Taylor
microscale.
Figure 15.1
shows a typical autocorrelation function and its scales
λ
and
τ
.As
indicated there,
λ
is the time of the zero crossing of the parabola fit to
ρ
at the
origin, and
τ
is defined through the area under the curve. In large-
R
t
flows
λ
is
small compared to
τ
(Problem 15.5)
. Trends in atmospheric data can prevent
ρ(t)
from going to zero and thus make
τ
difficult to determine.
15.2.2 Fourier representation of a real, stochastic scalar function
Following
Batchelor
(
1960
)and
Lumley and Panofsky
(
1964
), we'll use the
Fourier-Stieltjes representation for our random, stochastic fields. As the latter
Figure 15.1 A sketch of an autocorrelation function
ρ
, with its microscale
λ
and
integral scale
τ
shown. The area of the rectangle is the area under the
ρ
curve.
From
Lumley and Panofsky
(
1964
).
