Geoscience Reference
In-Depth Information
The mean-square derivative is
∂u
∂x
2
κ
2
2
,
κ
2
cos
2
(κx
=
+
=
φ)
(2.19)
so it increases as the square of wavenumber. We shall see in
Chapter 5
that this
property of a derivative allows the small (large wavenumber), weak, Kolmogorov-
microscale eddies to keep turbulence in equilibrium by dissipating kinetic energy
and diffusing away scalar fluctuations at the same mean rate they are produced at
the large scales.
The mean of the product of
u
and
∂u/∂x
is
u
∂u
κ
2
sin 2
(κx
∂x
=
κ
sin
(κx
+
φ)
cos
(κx
+
φ)
=
+
φ)
=
0
.
(2.20)
Thus, here
u
and
∂u/∂x
are
uncorrelated
. This also follows directly from the
homogeneity of
u
:
∂u
2
∂x
u
∂u
1
2
∂x
=
=
0
.
(2.21)
2.3 Ergodicity
Let's assume we want the mean value of a variable
α)
that has stochastic
variations in both
x
and
t
and is also random - different in each realization
α
.We
assume further that its ensemble mean
U
, which in the most general case depends
on both
x
and
t
,
u(x, t
˜
;
α
=
N
1
N
U(x,t)
=
lim
N
→∞
u(x, t
˜
;
α),
(2.22)
α
=
1
is impractical to determine. But we can determine its time average at position
x
in
a single realization
n
, say,
T
1
T
U
T
(x,t,T
t
;
n) dt
,
;
n)
=
u(x, t
˜
+
(2.23)
0
and its spatial average at time
t
in a single realization
m
:
L
1
L
U
L
(x,t,L
x
,t
m) dx
.
;
=
˜
+
;
m)
u(x
(2.24)
0
U(x)
, we intuitively expect this time
average in a single realization to converge to the ensemble average as the averaging
time
T
increases:
If
u
is stationary in time, so that
U
˜
=
U
T
(x,t,T
lim
;
n)
=
U(x).
(2.25)
T
→∞
