Geoscience Reference
In-Depth Information
Thus under isotropy the spectral evolution
equation (16.13)
is
∂φ(κ)
∂t
2
κ
2
F(κ)
2
γκ
2
φ(κ).
=
2
Co
c,s
(κ)
+
−
(16.17)
In
Chapter 15
we integrated
φ
over spherical shells of radius
κ
, writing
4
πκ
2
φ(κ).
E
c
(κ)
=
κ
2
φ(κ)dσ
=
(
15
.
39
)
κ
i
κ
i
=
We'll extend that to the other terms in
Eq. (16.17)
, defining a “production” spectrum
P(κ)
and a “transfer” spectrum
T(κ)
:
κ
i
κ
i
=
κ
2
2
Co
c,s
(κ) dσ
8
πκ
2
Co
c,s
(κ),
P(κ)
=
=
κ
i
κ
i
=
κ
2
2
κ
2
F(κ)dσ
=
8
πκ
4
F(κ).
T(κ)
=
(16.18)
The spherically integrated version of
Eq. (16.17)
is then
∂E
c
(κ)
∂t
2
γκ
2
E
c
(κ).
=
P(κ)
+
T(κ)
−
(16.19)
This says that the equilibrium three-dimensional scalar spectrum
E
c
(κ)
results
from the balance of three terms:
P(κ)
, the rate of gain by production, centered
near
κ
e
∼
1
/
;
T(κ)
, the net rate of gain by transfer from other wavenumbers; and
the rate of loss by molecular destruction, centered near
κ
d
1
/η
oc
, with
η
oc
the
Obukhov-Corrsin scale
(Chapter 7)
. These are sketched in
Figure 16.1
.
∼
Figure 16.1 A schematic diagram of the terms on the right side of
Eq. (16.19)
,the
spectral budget of scalar variance in steady, isotropic turbulence of large turbulence
Reynolds and Corrsin numbers.
