Geoscience Reference
In-Depth Information
15.4.2.1 The spectral density tensor
As we discussed in
Chapter 14
, as a second-order tensor function of a vector
φ
ij
(
κ
)
has the isotropic form
φ
ij
=
J(κ)κ
i
κ
j
+
K(κ)δ
ij
,
(
14
.
9
)
with
J(κ)
and
K(κ)
functions to be determined. The requirement of zero velocity
divergence implies, from the definition
(15.46)
of
R
ij
,
that
∂R
ij
/∂r
j
=
0. This, in
turn, implies from
(15.46)
that
κ
j
φ
ij
=
0, which from (14.9) gives the constraint
κ
i
J(κ)κ
2
K(κ)
=
K(κ)
κ
2
+
0
,
so that
J(κ)
=−
.
(15.52)
Thus,
φ
ij
becomes
K(κ)
δ
ij
−
κ
2
.
κ
i
κ
j
=
φ
ij
(15.53)
Using the form
(15.53)
in
(15.51)
then gives
κ
i
κ
i
=
κ
2
K(κ)dσ
4
πκ
2
K(κ),
E(κ)
=
=
(15.54)
E/(
4
πκ
2
)
and we have finally
so
K(κ)
=
4
πκ
2
δ
ij
−
κ
2
E(κ)
κ
i
κ
j
φ
ij
(
κ
)
=
(15.55)
as the form of the spectral density tensor in an isotropic turbulent velocity field.
In an isotropic field we have
u
i
u
j
=
u
α
u
α
δ
ij
,
so that, for example,
u
1
u
3
=0.From
Eq. (15.46)
we also have
+∞
u
1
u
3
=
=
0
φ
13
(
κ
)d
κ
.
(15.56)
−∞
One might expect that
Eq. (15.56)
is enforced in an isotropic field by making
φ
13
=
0, but
Eq. (15.55)
shows this is not the case:
4
πκ
2
κ
1
κ
3
κ
2
.
E(κ)
φ
13
(
κ
)
=−
(15.57)
Instead, the integral
(15.56)
vanishes because
Eq. (15.57)
shows that
φ
13
is odd in
κ
1
and
κ
3
. Evidently the continuity constraint
u
i,i
=
0prevents
φ
13
from vanishing
(Problem 15.24)
.
