Geoscience Reference
In-Depth Information
so that a requirement for the time constant of the sensor is
r
2
πU
1
.
τ
w
(16.74)
This means that the smaller the separation
r
and the larger the mean wind speed
U
1
the smaller the required time constant. If we interpret “
”in
Eq. (16.74)
as
meaning a factor of 10 less, then for
r
=0.6mand
U
1
=10ms
−
1
this criterion
says that
τ
w
should be less than 10
−
3
s.
16.2.3 The two-point difference and the spatial derivative
Since cos 2
x
2sin
2
x
, we can write
Eq. (16.63)
for locally isotropic
=
1
−
turbulence as
r
2
∞
∞
sin
2
(κr/
2
)
(κr/
2
)
2
(c)
2
r
2
4
sin
2
(κr/
2
)F (κ) dκ
κ
2
F(κ)dκ.
(16.75)
=
=
−∞
−∞
In the limit as
r
approaches zero the difference variance divided by
r
2
is the spatial
derivative variance, which in an isotropic field is
∂c
∂x
2
∂c
∂y
2
∂c
∂z
2
(c)
2
r
2
lim
r
→
=
=
=
.
(16.76)
0
Since the spatial derivative variance is the integral of
κ
2
times the one-dimensional
spectrum,
∂c
∂x
2
∞
κ
2
F(κ)dκ,
=
(16.77)
−∞
we can write a derivative variance measured as in
(16.75)
as
∂c
∂x
2
m
∞
(c)
2
r
2
sin
2
(κr/
2
)
(κr/
2
)
2
κ
2
F(κ)dκ.
=
=
(16.78)
−∞
Thus sin
2
(κr/
2
)/(κr/
2
)
2
is the transfer function of the difference approximation
to the spatial derivative.
16.2.4 Application to velocity fields
The approach of the last three sections is applicable to turbulent velocity fields
as well. Again those applications that concern the smaller-scale structure are
traditionally handled with the assumption of local isotropy.
