Geoscience Reference
In-Depth Information
1.3.4. Wave Number Selection
parameters, often exhibit hysteresis [e.g.,
HideandMason
,
1975;
Sitte and Egbers
, 2000;
von Larcher and Egbers
,
2005] in that the location of a transition in parameter
space depends upon the direction from which that tran-
sition is approached (e.g., transitions from
m
=3
Within the regular baroclinic wave regime, the flow
tends to equilibrate typically (in the absence of a strong
β
-effect, e.g., associated with topographically sloping
boundaries) to a state dominated by a single azimuthal
wave number and its harmonics, which may be steady,
quasi-periodic, or chaotic. The mechanisms by which
rotating annulus waves select which wave number to favor
at fully nonlinear equilibration are still not fully under-
stood but seem likely to share some aspects in common
with mechanisms identified in simple, weakly nonlinear,
spectrallytruncated models of baroclinic instability. Such
models [e.g., see above and
Pedlosky
, 1970;
Drazin
, 1970;
Pedlosky
, 1971] represent only the leading order nonlinear
interactions between a single mixed baroclinic-barotropic
traveling wave and the background (
m
= 0) zonal flow
(i.e., suppressing quadratic and higher order wave-wave
interactions). The nonlinear self-interaction of a growing,
linearly unstable wave generates a correction to the
m
=0
zonal flow (at second order in wave amplitude) that feeds
back on the growth rate, eventually reducing it to zero
(a steady wave state, for which the modal amplitude
equations may asymptotically reduce to a set of Landau
equations in the presence of some frictional damping), or
with a more complicated, quasi-periodic or chaotic time
dependence [e.g., see
Lovegrove et al.
, 2001, 2002], for
which the modal amplitude equations may reduce asymp-
totically to the classical real or complex Lorenz equations.
When more than one distinct wave number mode is
able to grow from infinitesimal amplitude on a given
zonal flow, weakly nonlinear models do not provide a
unique answer as to what mechanism will act to select
the dominant mode. However, one commonly found fac-
tor is for the flow to preferentially select the mode that
is capable of releasing the most available potential energy
(APE) from the initial flow [
Hart
, 1981]. In practice, this
may correspond to the mode that can reach the largest
barotropic amplitude [
Hart
, 1981;
Appleby
, 1988] pro-
vided nonlinear wave-wave interactions are absent. Where
wave-wave interactions are permitted, the mode selection
may become hysteretic such that a nonoptimal wave mode
(i.e., one that does not release the maximum possible
APE) may persist as the dominant mode if it was pre-
viously dominant under more favorable conditions at an
earlier time. This is found to manifest itself within the
regular flow regime as intransitivity (i.e., multiple equi-
librium states), in which two or more alternative flows
with differing azimuthal wave number
m
can occur for
a given set of parameters [e.g.,
Hide
, 1970;
Hide and
Mason
, 1975]. The state obtained in any particular exper-
iment implicitly depends upon the initial conditions. In
addition, transitions between different states in the reg-
ular regime, achieved by slowly changing the external
→
4do
not occur at the same point as
m
=4
3). In a situation
where the forcing that maintains the background zonal
state is varied cyclically with time over a range that crosses
the boundary between two or more optimal modes, this
can lead to complex and chaotic behavior as the flow pat-
tern flips erratically from one dominant mode to another
[e.g., see
Buzyna et al.
, 1978].
Another issue is how an initially dominant wave flow
may retain its dominance and remain indefinitely stable?
The presence of wave-wave and higher order nonlinear
interactions might be expected to permit the possibility
of secondary instabilities of the primary dominant wave
mode, at least in principle, thereby preventing the sus-
tained dominance of a single baroclinic wave mode.
Hide
[1958] and
Hide and Mason
[1975] showed empirical evi-
dence from a range of early experiments that, depending
upon the radius ratio between inner and outer cylinders,
there was a maximum azimuthal wave number of a sta-
ble and persistent dominant wave such that its azimuthal
wavelength always seemed to exceed roughly 1.5 times the
radial extent of the wave, i.e.,
→
2
π(b
+
a)
3
(b
m
max
a)
.
(1.13)
−
The prototypical idealized model for such a situation
considers the stability of the basic Rossby-Haurwitz (RH)
mode on the sphere to wavelike barotropic perturbations
[
Lorenz
, 1972;
Hoskins
, 1973;
Baines
, 1976], although this
has also been generalized to investigate baroclinic pertur-
bations and instabilities of the basic RH wave [e.g., see
Kim
, 1978;
Grotjahn
, 1984a, 1984b]. The principal crite-
rion for barotropic stability of the RH wave can be inter-
preted in relation to Fjørtoft's theorem for energy transfer
in a quasi-geostrophic flow [
Fjørtoft
, 1953], for which both
energy and squared vorticity must be conserved in non
dissipative nonlinear interactions. This essentially requires
that a given wave mode must lose energy simultaneously
to both a higher
and a lower
wave number mode. Thus,
the longest wave number modes capable of fitting into the
domain tend to be relatively stable because of the unavail-
ability of longer wavelength modes to which they can lose
energy in an instability. Such an interpretation appears to
be consistent with the criterion in equation (1.13) found
by
Hide
[1958] and
Hide and Mason
[1975] for the maxi-
mum azimuthal wave number that can sustain a persistent
dominant wave flow.
Recent work by
Young and Read
[2013] suggests that
barotropic instability may not be the only possible mech-
anism for breakdown of regular baroclinic wave flows.
