Geoscience Reference
In-Depth Information
The one-dimensional streamwise wavenumber spectrum
F
c
is related to
φ
(
2
)
by
∞
φ
(
2
)
(
κ
h
)dκ
2
.
F
c
(κ
1
)
=
(15.96)
−∞
We'll assume isotropy in the horizontal plane (i.e., axisymmetry in
z
), so that
φ
(
2
)
(
κ
h
)
φ
(
2
)
(κ
h
)
.We'lldefinea
two-dimensional spectrum E
(
2
)
=
by integrating
c
over circular rings in the horizontal plane:
2
π
E
(
2
c
(
κ
h
)
φ
(
2
)
(κ
h
)κ
h
dθ
2
πκ
h
φ
(
2
)
(κ
h
).
=
=
(15.97)
0
It also integrates to the variance,
∞
E
(
2
c
(κ
h
)dκ
h
=
c
2
.
(15.98)
0
From
Eqs. (15.96
),
(15.97)
, and axisymmetry it follows that
∞
E
(
2
c
(κ
h
)
2
πκ
h
F
c
(κ
1
)
=
dκ
2
.
(15.99)
−∞
Kelly and Wyngaard
(
2006
) showed that this can be inverted to yield
∞
d
dκ
h
2
κ
1
F
c
(κ
1
)
(κ
1
−
E
(
2
c
(κ
h
)
=−
κ
h
)
1
/
2
dκ
1
.
(15.100)
κ
h
Equation (15.100)
provides a way to determine the two-dimensional spectrum
from measurements of the one-dimensional spectrum, under the assumption of
axisymmetry.
15.6.2 The inertial range
At energy-containing wavenumbers spectra depend strongly on the stability state of
the ABL, but we've seen that in the inertial subrange and beyond they can approach
universality. We'll assume the turbulence at these smaller scales is isotropic in the
plane. It is particularly convenient to find variables whose spectra in the plane
have axisymmetric forms in an isotropic field, for such spectra depend on a single
wavenumber.
Let us assume that at wavenumbers
κ
|
and
h
is the boundary-layer depth, the vertical inhomogeneity of the boundary layer
is unimportant and the spectral density in the horizontal plane is the integral of the
=|
κ
|=|
1
/h
,where
κ
(κ
1
,κ
2
,κ
3
)
