Geoscience Reference
In-Depth Information
The scalar-variance budget in the inertial subrange is analogous:
rate of gain of scalar variance by transfer from larger scales
=
rate of loss of scalar variance by transfer to smaller scales
=
χ
c
.
(7.4)
The balances
(7.3)
and
(7.4)
are the foundation of Kolmogorov's scaling arguments
for the inertial subrange of the energy spectrum and their extension by
Obukhov
(
1949
)and
Corrsin
(
1951
) to the scalar spectrum.
7.1.2 The velocity spectrum
In
Chapter 2
we introduced the three-dimensional velocity spectrum
E(κ)
,which
has the property
∞
u
i
u
i
2
=
E(κ) dκ.
(2.63)
0
Kolmogorov
(
1941
) hypothesized that the only scaling parameters for its inertial
subrange are the energy cascade rate,
, and wavenumber,
κ
. If so, on dimensional
grounds it has the form
α
2
/
3
κ
−
5
/
3
,
E(κ)
=
(7.5)
with
α
now known as the Kolmogorov constant. This prediction was first confirmed
in the large-
R
t
flow in a tidal channel (
Grant
et al
.
,
1962
). Kolmogorov further
hypothesized that the scales of the smallest, dissipative eddies depend only on
and
ν
; if so, they must be
ν
3
1
/
4
velocity scale
υ
=
(ν)
1
/
4
,
length scale
η
=
.
(1.34)
In his book on G. I. Taylor,
Batchelor
(
1996
) gives some historical background
on these developments by Kolmogorov:
when I arrived in Cambridge in April 1945 to work for a PhD on turbulence under his
supervision …I cast about for new ideas, and in the course of searching the literature came
across two brief papers published in the literature in the USSR in 1941 in which Kolmogorov
put forward the idea of statistical equilibrium of the small-scale components of turbulent
motions and which had miraculously made their way safely to a Cambridge library. I was
excited by them, and told G.I. what I had found.
A little later, during the summer of 1945, …German atomic scientists W. Heisenberg and
C. F. von Weizsacker described to G.I. some new ideas on …the transfer of energy from
large to small-scale components, and …it was evident to G.I. and me that they had points
in common with Kolmogorov's theory. …it was later noticed that in 1945 the distinguished
physical chemist L. Onsager at YaleUniversity had published an abstract inwhich essentially
