Geoscience Reference
In-Depth Information
Here it is traditional to define a
three-dimensional energy spectrum E(κ)
as the
integral of
φ
ii
/
2 over a sphere of radius
κ
,
φ
ii
(κ
1
,κ
2
,κ
3
)
2
E(κ)
=
dσ.
(2.62)
κ
2
κ
i
κ
i
=
The factor of 2 is used so that
E(κ)
integrates to the kinetic energy per unit mass:
∞
u
i
u
i
2
=
E(κ) dκ.
(2.63)
0
2.5.4 The eddy velocity scale u(r)
We can now use
E(κ)
to estimate
u(r)
,the
velocity scale
or
characteristic velocity
(the typical root-mean-square velocity, say) of eddies of spatial scale
r
, or, equiva-
lently, spatial wavenumber magnitude
κ
1
/r
. Following
Tennekes and Lumley
(
1972
), we define “an eddy of scale
r
” as one of spatial scale between roughly
r/
2
and 3
r/
2, so that it lies in a band of width
r
∼
r
about scale
r
. In wavenumber
terms, we take “an eddy of wavenumber magnitude
κ
∼
∼
1
/r
” to lie in awavenumber
band
κ
∼
κ
about wavenumber magnitude
κ
.Thenwehave
2
[
u(r)
]
∼
κE(κ),
κ
∼
1
/r.
(2.64)
To proceed further we need to know
E(κ)
. As we shall discuss in
Chapter 7
,
Kolmogorov
(
1941
) argued that for wavenumbers in the
inertial subrange
,1
/
κ
1
/η
,
E
depends only on
and
κ
and so on dimensional grounds has the form
2
/
3
κ
−
5
/
3
.
E(κ)
∼
(2.65)
Through the interpretation
r
∼
1
/κ
, the inertial subrange corresponds to scales
r
η
.Using
(2.65)
in
(2.64)
gives
2
/
3
r
5
/
3
r
1
/
2
E(
1
/r)
r
1
/
2
(r)
1
/
3
.
u(r)
∼
∼
∼
(2.66)
Equation (2.66)
holds for
υ
(Problem 2.11)
. This is a much wider range of applicability than we might have
expected.
The
turnover time
of an eddy of size
r
is defined as
r/u(r)
;itisoftentakenasa
rough estimate of the lifetime of an eddy of size
r
.
≥
r
≥
η
; that is, it yields
u()
=
u, u(η)
=
