Geoscience Reference
In-Depth Information
where
C
v
is the velocity structure constant.
It can be shown that the structure functions for other atmospheric variables, like
the temperature and the refractive index, are similar (i.e. also proportional to
2
/
3
).
R
Thus the fluctuations in refractive index between two locations,
r
and
r
+
R
, can be
described by the structure function
D
n
(
R
)
[
n
]
2
C
n
2
/
3
D
n
(
R
)
=
(
r
)
−
n
(
r
+
R
)
=
R
.
(183)
The constant
C
n
is called the refractive index structure constant. This equation is how-
ever not valid for large scales since it becomes infinite when the distance approaches
infinity, which is unrealistic. In order to fix this problem, Treuhaft and Lanyi (
1987
)
modified the expression by introducing a saturation length scale
L
[
n
]
2
2
/
3
R
C
n
D
n
(
R
)
=
(
r
)
−
n
(
r
+
R
)
=
2
/
3
.
(184)
R
1
+
L
will converge to
C
n
L
2
/
3
as
With this expression
D
n
(
goes to infinity.
Turbulence does not only cause spatial variations in the refractive index, but also
temporal variations. Oneway to describe the temporal variations is to assume Taylor's
frozen flow hypothesis (Taylor
1938
). In this hypothesis the turbulent variations in
the refractive index in the atmosphere are frozen and move with the wind velocity
v
,
i.e. it is assumed that
n
R
)
R
(
,
)
=
(
−
(
−
t
0
),
t
0
)
. This is an approximation which
works well over shorter time periods but may not be valid over longer time periods
(hours, days). By using Taylor's frozen flow hypothesis the temporal variations in
the refractive index over a time period
T
can be d escribed by the structure function
D
n
(
r
t
n
r
v
t
T
)
[
n
]
2
T
]
2
/
3
[
v
C
n
D
n
(
T
)
=
(
t
)
−
n
(
t
+
T
)
=
2
/
3
.
(185)
v
T
1
+
L
By combining Eqs. (
184
) and (
185
) we get a general expression for the structure
function for the fluctuations in the refractive index between
r
1
at time
t
1
and
r
2
at
time
t
2
[
n
]
2
]
2
/
3
r
1
−
r
2
−
(
t
1
−
t
2
)
[
v
C
n
D
n
(
r
1
,
t
1
;
r
2
,
t
2
)
=
(
r
1
,
t
1
)
−
(
r
2
,
t
2
)
=
2
/
3
.
n
r
1
−
r
2
−
v
(
t
1
−
t
2
)
1
+
L
(186)
