Geoscience Reference
In-Depth Information
We represent the fluctuating horizontal velocity vector
u
=
(u, v)
at a point
x
=
(x, y)
in the plane and at time
t
by a sum of Fourier components of horizontal vector
wavenumber
κ
n
=
(κ
1
n
,κ
2
n
)
with random vector coefficients
a
n
=
(a
1
n
,a
2
n
)
,
N
a
n
(
κ
n
,z,t,α)e
i
κ
n
·
x
,
u
(
x
,z,t
;
α)
=
(10.54)
n
=−
N
with
α
the realization index. It follows that
∂u/∂x
, for example, is
N
∂u
∂x
=
iκ
1
n
a
1
n
e
i
κ
n
·
x
.
(10.55)
n
=−
N
With
(10.53)
-
(10.55)
we can now write
w
as
N
e
i
κ
n
·
x
z
κ
2
n
a
2
n
) dz
w(
x
,z,t
;
α)
=−
i (κ
1
n
a
1
n
+
0
n
=−
N
(10.56)
κ
1
n
z
0
a
2
n
dz
e
i
κ
n
·
x
.
κ
2
n
z
0
N
a
1
n
dz
+
=−
i
n
=−
N
At horizontal scales much larger than the height
z
(i.e., for
κ
n
z
1, with
κ
n
=|
κ
n
|
) we expect the Fourier coefficients
a
n
to vary only weakly with
z
because
turbulent eddies with large horizontal scales (small
κ
) are unlikely to have small
vertical scales. Thus, for
κ
n
z
1 we can approximate the integrals in
(10.56)
as
z
z
a
1
n
dz
a
2
n
dz
a
1
n
z,
a
2
n
z,
(10.57)
0
0
so that
Eq. (10.56)
becomes
N
(
κ
1
n
a
1
n
+
κ
2
n
a
2
n
)
e
i
κ
n
·
x
w(
x
,z,t
;
α)
−
iz
n
=−
N
(10.58)
N
iz
a
n
·
κ
n
e
i
κ
n
·
x
,κ
n
z
=−
1
.
n
=−
N
Thus, the Fourier coefficients of
w
are those of
(u, v)
dotted with
iz
κ
n
.Since
the spectrum at any wavenumber is proportional to the mean-square value of the
Fourier coefficients at that wavenumber, it follows that
−
spectrum of
w
(κ
n
z)
2
×
spectrum of
u, v,
κ
n
z
1
.
(10.59)
