Geoscience Reference
In-Depth Information
t
0
+
T
t
0
+
T
1
T
1
T
u
T
u(t
)
dt
−
u(t
)dt
,
−
U
=
[
U
+
]
U
=
(2.28)
t
0
t
0
which is a random variable with an ensemble mean of zero. One measure of
u
T
−
U
is its variance
σ
2
, which can be written as
T
2
t
0
+
T
t
0
+
T
1
σ
2
(u
T
u(t
)u(t
)dt
dt
.
U)
2
≡
−
=
(2.29)
t
0
t
0
u(t
)u(t
)
is called the
autocovariance
of
u(t)
. For a stationary process it is a
function only of the time difference
t
−
t
, so we write it as
u
2
ρ(t
−
t
),
u(t
)u(t
)
=
(2.30)
with
u
2
=
u(t)u(t)
the variance of
u(t)
and
ρ
its
autocorrelation function
.
Equation (2.30)
shows that
ρ
is an even function; i.e.,
ρ(t
−
t
)
ρ(t
−
t
)
.
=
Thus, we can write
Eq. (2.29)
as
T
2
t
0
+
T
t
0
+
T
u
2
σ
2
ρ(t
−
t
)dt
dt
.
=
(2.31)
t
0
t
0
t
−
t
One can reduce
Eq. (2.31)
to a single integral by changing variables to
η
=
t
and doing the integration over
ζ
, noting that the limits of that
integral then depend on
η
(
Problem 2.4
.) The result is
t
+
and
ζ
=
1
ρ(t)dt.
T
2
u
2
T
t
T
σ
2
=
−
(2.32)
0
The Eulerian
integral time scale τ
,
†
defined as
∞
ρ(t)dt
=
τ,
(2.33)
0
is a measure of the “memory time” of the Eulerian velocity fluctuation
u(t)
.For
averaging time
T
much longer than the integral scale
τ
we can approximate
Eq. (2.32)
as
T
2
u
2
T
2
u
2
τ
T
σ
2
ρ(t)dt
=
.
(2.34)
0
This is quite an important result, for with
Eq. (2.29)
it quantifies the statistical
difference between the ensemble average, which we can seldom use in applications,
and the time average that we usually use instead.
†
We'll introduce the Lagrangian integral time scale in
Chapter 4
.
