Geoscience Reference
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the parameter space too sparsely to address the prob-
leminmuchdetail[e.g.,see
Showman et al.
, 2010;
Read
,
2011, for reviews]. The growing number of discoveries of
planets around other stars (e.g., see http://exoplanet.eu/
and
Schneider et al.
[2011]) offers the eventual prospect
of sampling parameter space much more densely, but the
available measurements are as yet much too crude to be
able to provide quantitative characterization of circula-
tion regimes. So at the present time (and for the foreseeable
future) the only way of addressing and characterizing the
diversity of planetary circulation regimes and bifurcations
is through the use of numerical model simulations.
To date, however, relatively little has been done to
define and sample an appropriate parameter space for
planetary circulations that comes anywhere near match-
ing what has been achieved in the laboratory, at least in
terms of breadth and detail with respect to the dominant
dimensionless parameters. The early work of
Geisler et al.
[1983] laid some of the foundations for this approach in
using a stripped-down version of an atmospheric global
circulation model (GCM) to investigate a range of proto-
typical circulations with an imposed equator-pole thermal
contrast and varying rotation speeds of an Earth-like
planetary atmosphere. By explicit analogy with laboratory
rotating annulus experiments, they presented their results
with respect to two dimensionless parameters:
(i) A thermal Rossby number defined as
S
=
gH
f
0
L
2
flow was dominated by near-monochromatic waves, peak-
ing in amplitude at midlatitudes and drifting in longitude
at a roughly steady rate. These waves were either steady in
amplitude or apparently undergoing periodic oscillations
reminiscent of the amplitude vacillation behavior seen in
the annulus.
A significant difference from the laboratory systems
was found, however, at the highest values of
S
, which
would lie above the corresponding upper symmetric tran-
sition in the laboratory and would therefore be expected to
exhibit axisymmetric flow. Instead, the flow in the spher-
ical shell was found to transition from a predominantly
baroclinic wave flow to a barotropically unstable flow, also
with zonally propagating waves of relatively low zonal
wave number
m
2 drifting around an intense polar
vortex. This kind of behavior has since been confirmed
in more recent work [e.g.,
Mitchell and Vallis
, 2010] in
which such barotropically unstable low at low plane-
tary rotation speeds may also be associated with strongly
superrotating zonal low at low latitudes. Such a low
appears be consistent with the strongly superrotating cir-
culations found on very slowly rotating planets such as
Venus and Titan.
Since Geisler et al.'s early study, there has been a steady
trickle of other work [e.g.,
Williams and Holloway
, 1982;
Del Genio and Suozzo
, 1987;
Williams
, 1988a, 1988b;
Jenkins
, 1996;
Navarra and Boccaletti
, 2002;
Barry et al.
,
2002;
Schneider and Walker
, 2006] exploring other areas
of parameter space, including cases corresponding to even
faster rotation speeds (lower values of
S
) than Earth.
Another early pioneer of this kind of modeling study was
Gareth Williams [
Williams and Holloway
, 1982;
Williams
,
1988a, 1988b], who presented results from an Earth-like
GCM for which the planetary rotation rate was varied
between
E
/
16 and
E
×
T
T
r
,
(1.60)
where
H
is a pressure scale height
(
=
RT
r
/g)
,
R
is the gas
constant,
f
0
=2
sin
φ
0
(
φ
0
was taken to be a latitude of
45
◦
),
L
is a horizontal length scale,
T
is the imposed
equator-pole temperature contrast at the surface, and
T
r
is a reference temperature.
(ii) A Taylor or inverse squared Ekman number,
defined as
8(where
E
is Earth's rota-
tion speed). At higher rotation speeds than that of Earth,
Williams' model simulations suggested that the dominant
scale of baroclinic instability would continue to decrease
with increasing
, but with a tendency (at
2
E
)
for the subtropical zonal jet stream to break up into a
set of two or more parallel jets associated with paral-
lel trains of baroclinically unstable eddies. At the highest
rotation speeds, up to seven or eight parallel jets were
obtained in each hemisphere, resulting in a circulation pat-
tern that bore a strong resemblance to that of Jupiter's
or Saturn's cloud bands. Williams did not attempt to
locate his simulations in a dimensionless parameter space,
but
Read
[2011] computed approximate values of
S
and dissipation parameters to locate these experiments
retrospectively. In common with some more recent work,
the results appear to suggest that the multiple jets organize
themselves on a scale comparable with the Rhines scale
and are largely generated and controlled by the nonlinear
interactions between eddies and the zonal flow.
T
E
−
2
=
2
H
4
K
v
,
(1.61)
where
K
v
is a vertical “eddy viscosity” coefficient.
For various practical reasons
Geisler et al.
[1983] only
studied cases equivalent to an Earth-like planet rotating at
the same speed as or slower than Earth itself. But this did
enable them to demonstrate the existence of a “lower sym-
metric” regime at relatively small
T
, where wavy flows
gave way to axisymmetric circulations, the boundary of
which was found close to the line defined by
S
2
.
This roughly emulates the lower symmetric regime bound-
ary found in rotating annulus experiments using relatively
high Prandtl number fluids [
Fein
, 1973]. They also found
evidence for a regular baroclinic wave regime at higher val-
ues of
S
than for Earth itself (
S
10
5
E
0.05), where the
