Geoscience Reference
In-Depth Information
If
E
c
is the spectral density of a conserved scalar (other than vorticity) in two-
dimensional turbulence, then the corresponding Kolmogorov hypothesis for its
inertial subrange is
E
c
=
E
c
(κ, χ
c
,χ
ω
).
Then on dimensional grounds it follows
that
χ
c
χ
−
1
/
3
κ
−
1
,
E
c
∼
(7.62)
ω
which is a generalization of
Eq. (7.60)
.
For the eddy velocity and scalar intensity scales
u(r)
and
c(r)
in the inertial
range of two-dimensional turbulence we write
χ
1
/
2
rχ
1
/
ω
,
χ
−
1
/
6
u(r)
=
u(r, χ
ω
)
∼
(r)
=
c(r, χ
ω
,χ
c
)
∼
.
(7.63)
c
ω
The
r
-dependencies of these two-dimensional scales and the three-dimensional
ones,
Eqs. (2.66)
and
(7.10)
, are strikingly different. In two dimensions the velocity
scale
u(r)
decreases much faster (
r
as opposed to
r
1
/
3
) with decreasing scale.
The difference is manifested in the
κ
−
5
/
3
energy spectrum in three-dimensional
turbulence but its much steeper
κ
−
3
falloff in two dimensions.
∼
7.5.4 Forward and inverse cascades
McWilliams
(
2006
)and
Vallis
(
2006
) discuss the properties of two-dimensional
turbulence driven by forcing over a narrow range of eddy scales. They show that the
energy- and enstrophy-conservation arguments applicable during the early, inviscid
stages imply there is a downscale, or
forward
(toward smaller scales) cascade of
enstrophy and an upscale, or
inverse
cascade of energy. Numerical experiments
confirm this picture, showing forward and inverse cascades emanating from the
wavenumber
κ
f
of the forcing. Thus the energy spectrum behaves as
2
/
3
κ
−
5
/
3
,
χ
2
/
3
κ
−
3
,
E(κ)
∼
κ<
f
;
E(κ)
∼
κ>
f
.
(7.64)
ω
From
Eqs. (6.93)
and
(6.95)
the energy and scalar variance cascade rates
and
χ
c
can be written
u
i
u
j
s
ij
−
u
i
u
j
s
ij
,
2
c
r
u
j
c
,j
−
2
c
s
u
j
c
,j
.
=
I
=
ω
=
I
c
=
(7.65)
Since it involves contractions of scalar flux and scalar gradient,
I
c
has the same
nature in two- and three-dimensional turbulence. But as a contraction of a stress
and a strain rate
I
is quite different in the two. In three dimensions
I
has two types
of terms, four involving interactions entirely in the horizontal plane and five that
involve vertical velocity; the latter group is presumably related to vortex stretching.
In two dimensions only the first four can act, and so presumably the mechanisms
involved in interscale transfer of kinetic energy are different in the two cases.
