Geoscience Reference
In-Depth Information
16.2.1 Spatial averaging of conserved scalars
Flow sensors typically average the measured variable over a small region of space.
For example, a resistance-wire temperature sensor averages temperature over its
length; infrared-absorption sensors average the mixing ratio of water vapor and
carbon dioxide over the length of their transmission path. We can determine ana-
lytically how this line averaging impacts some statistics of the signal.
We'll consider first averaging the fluctuating scalar field
c
over a vector path
L
of length
L
. We'll associate the measurement with the midpoint
x
of the path.
Denoting
c
m
as the measured value of
c
,
s
as the vector along the path from that
midpoint, and
s
the distance along that path, we have
L/
2
1
L
c
m
(
x
,
L
,t)
=
c(
x
+
s
,t)ds.
(16.46)
−
L/
2
In this path averaging, scalar fluctuations having along-path length scales small
compared to
L
average to nearly zero and, hence, are greatly attenuated. Scalar
fluctuations in directions normal to the path are unaffected.
The Fourier-Stieltjes representation of the scalar field allows us to quantify
these averaging effects. We assume homogeneous turbulence so we can represent
the true and measured fluctuating scalar fields as (we shall hereafter suppress the
dependence on time):
e
i
κ
·
x
dZ(
κ
),
e
i
κ
·
x
dZ
m
(
κ
,
L
).
c
m
(
x
,
L
)
=
=
c(
x
)
(16.47)
Using
(16.46)
we can write the path-averaged scalar signal as
L/
2
e
i
κ
·
(
x
+
s
)
dZ(
κ
)
ds
1
L
c
m
(
x
,
L
)
=
−
L/
2
1
L
L/
2
e
i
κ
·
(
x
+
s
)
ds
dZ(
κ
).
=
(16.48)
−
L/
2
The averaging integral is
L/
2
1
L
sin
(
κ
·
L
/
2
)
e
i
κ
·
(
x
+
s
)
ds
e
i
κ
·
x
,
=
(16.49)
κ
·
L
/
2
−
L/
2
so
Eq. (16.48)
can be written
e
i
κ
·
x
sin
(
κ
·
L
/
2
)
c
m
(
x
,
L
)
=
dZ(
κ
).
(16.50)
κ
·
L
/
2
