Geoscience Reference
In-Depth Information
reducing squared uncertainty
e
2
. In large-eddy simulation of atmospheric boundary
layers, for example, we can average the calculated fields over homogeneous hori-
zontal planes. In this way one “snapshot” of a simulated field on a sufficiently large
plane can yield a good estimate of the ensemble average.
2.5 The turbulence spectrum and the eddy velocity scale
In
Chapter 1
we introduced the velocity scales
u, υ
and length scales
, η
of the
energy-containing and dissipative eddies, respectively, and showed that
/η
∼
R
3
/
t
, with
R
t
the large-eddy Reynolds number
u/ν
.
R
t
varies from less than 10
3
in some engineering flows to about 10
8
in the convective atmospheric boundary
layer and perhaps 10
10
in a supercell thunderstorm; over that range
/η
varies from
about 10
2
to 10
7
.
The
power spectral density
of the turbulent velocity field (loosely called “the
turbulence spectrum”) allows the velocity scale
u
of the energy-containing eddies
to be generalized to
u(r)
, the velocity scale of an eddy of size
r
, with
η
.We
shall present here an informal derivation of the turbulence spectrum, beginning for
simplicity with a homogeneous scalar function of a single variable.
Part III
contains
a more formal presentation.
≥
r
≥
2.5.1 The spectrum of a one-dimensional, real, random,
homogeneous scalar function
Let
f(x)
be a real, homogeneous function, the sumof an ensemble-mean part
F
and
a fluctuation
f(x)
, defined over record of length
L
. It could be a spatial record of
temperature or a velocity component in a turbulent flow, for example. We can
approximate
f(x)
through a Fourier series, a sum of sines and cosines of
wavelengths
L/n, n
=
0
,...,N
:
a
n
cos
2
πnx
L
b
n
sin
2
πnx
L
.
N
N
a
0
2
+
f(x)
+
(2.39)
n
=
1
n
=
1
The coefficients
a
n
and
b
n
are real numbers called
Fourier coefficients
.
Since each of the sine and cosine terms in
Eq. (2.39)
integrates to zero over the
record length, it follows that the average of
f(x)
over the record length is
L
1
L
a
0
f(x)dx
=
2
.
(2.40)
0
f(x)
with
x
.
The other Fourier components together represent the variation of
