Geoscience Reference
In-Depth Information
15.5 Joint vector and scalar functions of space and time
15.5.1 The covariance, spectral density pair
With the Fourier-Stieltjes formalismfor stationary, homogeneous turbulent velocity
and scalar fields,
+∞
+∞
e
i
κ
·
x
dZ
i
(
κ
;
e
i
κ
·
x
dZ(
κ
;
u
i
(
x
;
t)
=
t),
θ(
x
;
t)
=
t),
(15.88)
−∞
−∞
we can define a joint spectrum:
0
,
κ
=
κ
,
dZ
i
(
κ
)dZ
∗
(
κ
)
=
κ
=
κ
;
φ
i
(
κ
)d
κ
,
+∞
e
i
κ
·
r
φ
i
(
κ
)d
κ
,
u
i
(
x
+
r
)θ(
x
)
=
R
i
(
r
)
=
−∞
(
2
π)
3
+∞
1
e
−
i
κ
·
r
R
i
(
r
)d
r
.
φ
i
(
κ
)
=
(15.89)
−∞
O
i
,
the sum of even and odd parts. The Fourier
transform of
R
i
is then, from
(15.89)
,
Again we can write
R
i
=
E
i
+
(
2
π)
3
+∞
1
e
−
i
κ
·
r
[
E
i
(
r
)
φ
i
(
κ
)
=
+
O
i
(
r
)
]
d
r
−∞
(
2
π)
3
+∞
(
2
π)
3
+∞
1
i
=
cos
(
κ
·
r
)E
i
(
r
)d
r
−
sin
(
κ
·
r
)O
i
(
r
)d
r
−∞
−∞
=
Co
i
(
κ
)
−
iQ
i
(
κ
),
(15.90)
the sum of real and imaginary parts. Since
Q
i
(
κ
)
is an odd function, we have from
Eq. (15.89)
+∞
+∞
u
i
(
x
)θ(
x
)
=
φ
i
(
κ
)d
κ
=
Co
i
(
κ
)d
κ
.
(15.91)
−∞
−∞
Thus
Co
i
(
κ
)
represents the spectral density of contributions to the scalar flux
u
i
θ
.
15.5.2 Isotropy
In an isotropic field
Co
i
(
κ
)
has the form
Aκ
i
(Chapter 14)
, with
A
a scalar. The
zero-divergence constr
aint
on
u
i
requires that
κ
i
Aκ
i
=
=
0, so that
A
0. Hence
Co
i
(
κ
)
=
0
,
and thus
u
i
θ
=
0
.
The last follows more directly in physical space:
as an isotropic vector,
u
i
θ
=
0
.
