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were noisier but suggested that they were coinciding with
their respective potential energy terms or very slightly
delayed. An idealization of Figures 17-19 from
Pfeffer
et al.
[1973] in our Figure 3.5 illustrates the various energy
terms in (a) and the main energy conversion terms in (b).
A positive value in Figure 3.5b corresponds to an energy
flow in the direction given by the arrow in the annotation
and a negative value indicates a reverse energy flow. In par-
ticular, Figure 3.5b shows that the energy transfer between
the potential energy forms appears to be always from
the zonal to the eddy potential energy and the transfer
between the two eddy energies appears to be always from
the eddy potential energy to the eddy kinetic energy. In
contrast, the transfer between the two zonal energy forms
changes sign: The zonal kinetic energy receives energy
from the zonal potential energy during minimum energy
transfer from the zonal potential energy to eddy poten-
tial and then to eddy kinetic energy. This then changes
to a drain from the zonal kinetic energy to the zonal
kinetic energy at times when the energy transfer from
this zonal kinetic energy to the other forms of energy is
large. The transfer between the two kinetic energy terms
was always very small but appeared to be in the direc-
tion from eddy kinetic to zonal kinetic energy. The reversal
of the energy transfer between the kinetic energy terms is
illustrated in Figure 3.6.
Mechanical
forcing
Thermal
forcing
Friction
through
Ekman
layers
Zonal
kinetic
energy
Zonal
potential
energy
Friction
through
Ekman
layers
Eddy
kinetic
energy
Eddy
potential
energy
Figure 3.4.
Routes of energy transfer from the forcing of the
vertical shear in the two-layer experiment or from the imposed
baroclinicity in the thermally driven annulus to dissipation of
the kinetic energy and incorporating the thermal forcing rel-
evant for
Pfeffer et al.
[1973]. Adapted from Fig. 6 of
Hart
[1976]. Copyright © American Meteorological Society. Used
with permission.
cases, the baroclinic instability releases the zonal potential
energy stored in the sloped isotherms or isopycnals and
transfers this to the eddy potential energy and then the
kinetic energy of growing wave modes. These lose energy
through friction from the Ekman layers and through hor-
izontal viscous diffusion but also feed back into the zonal
flow. The feedback which can lead to an equilibration to
a steady wave arises from the fact that the energy transfer
from the eddies to the zonal flow reduces the baroclinicity
until a balance between the energy supply from the forc-
ing is balanced by energy loss through Ekman friction and
diffusion.
Amplitude vacillation can set in when this balancing
point of forcing and dissipation becomes unstable, and a
slight increase in wave amplitude does not lead to a suf-
ficient reduction in zonal potential energy and vice versa.
Pfeffer et al.
[1973] used experimental data to test a sug-
gestion by
Pfeffer and Chiang
[1967] that the main energy
conversion resulting in amplitude vacillation would be the
two routes between zonal and eddy potential energy and
between eddy potential and kinetic energy. The observa-
tions showed that the two potential energy terms were
shifted by a quarter vacillation period, or phaseshifted by
π/
2, with the zonal potential energy leading. This means
that the time of maximum zonal potential energy coin-
cided with increasing eddy potential while the maximum
eddy potential energy coincided with decreasing zonal
potential energy. The results for the kinetic energy terms
3.3.MODELLING APPROACHES
3.3.1. Computational Fluid Dynamics
Hignett et al.
[1985] succeeded in reproducing a
realistic amplitude vacillation in the finite-difference
Navier-Stokes model for the Oxford annulus filled with
a fluid of Prandtl number of around Pr
13 from
James
et al.
[1981] even at a relatively low resolution of 16 grid
cells in the radial and vertical, respectively, and 64 in the
azimuthal direction using a stretched grid to resolve the
boundary layers adequately. A later version of this model,
now known as MORALS, was used by
Young and Read
[2008] to construct a more detailed regime diagram for this
apparatus and found very good agreement in the structure
of the regime diagram, similar to that of Figure 3.3b.
Lu et al.
[1994] developed a numerical model using
finite-difference discretization in the radial and vertical
directions but a spectral representation in the azimuthal
direction to model the larger Florida annulus with a rela-
tively narrow gap and filled with a viscous fluid of Prandtl
number 73 (experiment B in
Pfeffer et al.
[1980b]) and
found good agreement, in particular the fact that vacillat-
ing flows were extremely common and steady waves very
rare.
Lu and Miller
[1997] then analyzed two particular
vacillating cases, one classified as amplitude vacillation
and the other as structural vacillation. In particular, they
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