Geoscience Reference
In-Depth Information
After some calculations, it can be shown that (Treuhaft and Lanyi
1987
)
T
0
(
1
T
2
2
σ
L
(
T
)
=
T
−
t
)
D
L
(
t
)
d
t
.
(191)
In order to estimate
C
n
using observations of
2
L
we need to know the shape of the
C
n
profile. When we want to estimate a
C
n
that can be used for calculating
D
L
it
is however not critical to know the exact shape of the profile, most important is
then that the integrated value of
C
n
is correct. Thus we can for example make the
approximation that
C
n
has an exponential profile or, even more simple, assume that
C
n
is constant up to a height
H
and zero above as done by Treuhaft and Lanyi (
1987
).
Using the latter approximation,
C
n
at heights lower than
H
can be calculated from
σ
L
(
σ
T
)
C
n
=
(
3
d
z
d
z
d
t
.
(192)
0
)
0
0
2
1
/
3
z
)
2
z
|
2
/
(
T
−
t
z
−
+
(
vt
)
−|
z
−
The height
H
should be chosen such that the largest fluctuations in the refractive
index occur below
H
. For microwaves where the fluctuations in the wet delay is
dominating an appropriate choice is the scale height of water vapor (approximately
2km).
Another possible way to obtain
C
n
is to use vertical profiles of pressure, tempera-
ture, and humidity obtained from e.g. radiosonde measurements. As discussed above,
when turbulence is present, large-scale variations (gradients) in the atmosphere are
mixed and create small-scale fluctuations. Thus there will be variations caused by
both the large-scale gradients as well as turbulence. At large scales the variations due
to gradients will dominate, and at small scales turbulence. At some scale in between
the magnitudes of the large-scale gradients and by turbulence will be equal. This
scale is proportional to the so-called
outer scale of turbulence L
0
. Typically values
of
L
0
range between a few meters to several hundreds of meters. The value can vary
with time, but for the calculations of
C
n
typically a mean value is used (d'Auria et al.
1993
). Hence
C
n
L
2
/
3
2
L
0
.
∝
∇
n
(193)
0
Thus
C
n
could be calculated from the gradient in
n
. Since the gradient of the refractive
index is typically more than one order of magnitude larger in the vertical direction
than in the horizontal direction, we can approximate the refractive index gradient by
its vertical component. However, one problem that needs to be considered is that the
refractive index is not conserved in adiabatic motion in the atmosphere. When an air
parcel is moved up or down in the atmosphere (e.g. due to turbulence) its size will
change due to the change of atmospheric pressure with height. This in turn will cause
the temperature and partial pressure of water vapor—and thus also the refractivity—
to change. This needs to be corrected for when calculating
C
n
.Thewaytodothisis
to consider the vertical gradient in the refractive index caused by vertical gradients
in quantities conserved under adiabatic motion in the atmosphere, e.g. the potential
