Geoscience Reference
In-Depth Information
To help interpret
Eq. (2.34)
we define the rms uncertainty
e
of the time mean
through
Eq. (2.29)
:
(u
T
U)
2
1
/
2
−
σ
U
.
e
≡
=
(2.35)
U
e
is a measure of the fractional error inc
u
rred in taking the finite-time mean as
the ensemble mean; a small
e
means that
u
T
is a good approximation to
U
.From
Eqs. (2.34)
and
(2.35)
the averaging time required for determining to an rms frac-
tional uncertainty
e
the mean of a signal
u(t)
whose ensemble mean is
U
and
˜
integral scale is
τ
is therefore
e
2
u
2
U
2
.
2
τ
T
=
(2.36)
Equation (2.36)
says that the required averaging time is
• proportional to
τ
,
the integral scale of the time series;
• proportional to
u
2
, the variance of the time series; and
• inversely proportional to
e
2
, the square of the rms fractional uncertainty in the time mean.
As it turns out, obtaining atmospheric turbulence statistics of low rms uncertainty
from stationary sensors can require unreasonably long averaging times; for this
reason poor estimates of statistics plague atmospheric turbulence research. This
makes it difficult to use atmospheric observations to develop and test models of
atmospheric turbulence
(Part II)
. Measurements made in a laboratory model of
the atmospheric boundary layer converge very much faster than the corresponding
atmospheric measurements
(Problem 2.8)
.
For an average of a homogeneous turbulence signal of integral scale
over a line
of length
L
, the one-dimensional result
(2.36)
can be written as
L
u
2
U
2
.
e
2
(
line average
)
(2.37)
The corresponding expressions for averaging homogeneous turbulence over a
square of area
L
2
and a cube of volume
L
3
are
L
2
L
3
u
2
U
2
,
u
2
U
2
,
e
2
(
plane average
)
2
(
volume average
)
(2.38)
where for simplicity we have taken
to be the same in each direction. Since in
general
/L
is small, this shows the advantages of area and volume averaging in
