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In-Depth Information
Ω
increasing
(a)
(b)
(c)
(d)
1
10
-1
10
-2
10
-3
10
-4
10
-5
1
1
1
5
11
6
10
-1
10
-2
10
-3
10
-4
10
-5
10
-1
10
-2
10
-3
10
-4
10
-5
10
-1
10
-2
10
-3
10
-4
10
-5
4
10
15
12
18
-4.2
8
1
10
100
1
10
100
1
10
100
1
10
100
Zonal wavenumber
m
Figure 1.15.
Azimuthal wave number power spectra obtained from measurements of temperature in a rotating annulus as
is
increased through the regular wave regime toward fully developed “geostrophic turbulence”. Experiments were carried out at the
following points in
(
,
10
8
, (b) 0.344, 5.40
10
8
, 0.230, 8.07
10
8
, and (d) 0.086, 2.16
10
9
.
T
)
parameter space: (a) 1.44, 1.30
×
×
×
×
Adapted from
Buzyna et al.
[1984] with permission.
self-interactions, leading to a
K
−
5
/
3
continuum with maximum power over a range of rela-
tively low wave numbers dominated by the main baro-
clinic wave activity with a characteristic quasi-power law
decay toward the highest wave numbers that commonly
approached
k
−
3
3
inertial range (where
K
3
is the three-dimensional total wave number) if a suffi-
cient scale separation exists between energy injection and
large-scale dissipation and forcing. The existence or other-
wise of an inverse energy cascade in Earth's atmosphere or
oceans continues to be an area of active controversy, with
recent work suggesting that energy cascades upscale from
around the first internal baroclinic deformation radius
toward larger scales in the oceans [
Scott and Wang
, 2005;
Scott and Arbic
, 2007], though without a
k
−
5
/
3
inertial
range apparent (at least in the sea surface height signature
in satellite altimetric measurements). The development
of coherent structures at large scales in the atmosphere,
however, has been suggested to lead to a suppression of
spectrally local nonlinear interactions and consequently
the suppression of any significant inverse energy cascade
at scales larger than the main energy-containing baroclinic
scales in the atmosphere [e.g.
Schneider and Walker
, 2006;
O'Gorman and Schneider
, 2007].
Such issues were not considered in earlier experimen-
tal studies [
Hide et al.
, 1977;
Pfeffer et al.
, 1980;
Buzyna
et al.
, 1984;
Bastin and Read
, 1997, 1998], which only pre-
sented and discussed basic temperature variance spectra
and synoptic structures from their measurements. As is
evident from Figure 1.15, such spectra show little obvi-
ous evidence of the
K
−
5
/
3
inertial range that might indi-
cate an energy-dominated cascade at low wave numbers.
Wordsworth et al.
[2008] did explore aspects of spectral
energy transfers in a set of rotating annulus experiments,
both with and without sloping horizontal boundaries,
even though such experiments also showed no obvious
sign of the classical
K
−
5
/
3
k
−
4
[
Hide et al.
, 1977;
Pfeffer et al.
,
1980;
Buzyna et al.
, 1984] and even steeper in some cases
[
Pfeffer et al.
, 1980;
Buzyna et al.
, 1984]. Simple heuristic
arguments based on the Kolmogorov-Kraichnan theory
[
Kraichnan
, 1967, 1971] predicts energies to decay as
k
−
3
at high wave numbers in an enstrophy-dominated iner-
tial range, but this is not the only possible explanation.
The (possibly transient) formation of sharp fronts and
vorticity filaments within geostrophically turbulent flows
may lead to a kinetic energy spectrum with a slope as
steep as
k
−
4
[e.g.,
Saffman
, 1971;
Brachetetal.
, 1988]. The
formation of persistent, stable vortex structures within
geostrophically turbulent flows can also perturb the sim-
ple Kolmogorov-Kraichnan scaling arguments by intro-
ducing some spatiotemporal intermittency to the flow.
This effect can also apparently lead to spectral slopes of
k
−
3
−
k
−
4
or even steeper [e.g.,
Basdevant et al.
, 1981;
McWilliams
, 1984], although the presence of imposed
vorticity gradients (e.g., due to a
β
effect or topography)
may weaken such eddies and the flow reverts toward a
k
−
3
energy spectrum.
A further question regarding these highly turbulent
flows with strong background rotation is whether they
also exhibit any evidence for an inverse energy cas-
cading inertial range. Early theoretical work and mod-
els [e.g.,
Salmon
, 1978;
Rhines
, 1979;
Salmon
, 1980]
suggested that such inverse cascades would be rela-
tively common in stratified, quasi-geostrophic flows, with
energy being injected into the barotropic mode at around
the internal Rossby deformation radius via baroclinic
−
3
inertial range. Nevertheless,
these experiments did show evidence for a weak, spec-
trally local (eddy-eddy) upscale kinetic energy cascade in
