23
The loops formed by the evolution of the QBO are represented in the
lower left panel of Fig. 6 by a series of dots which represent sequential
1-month means of zonal winds at two different levels in the equatorial
stratosphere. The dots tend to be clustered in preferred regions of this
particular phase space; i.e., near A and B. As the QBO executes a loop,
the trajectory tends to move relatively quickly between clusters. These
transitions take place when the layers of rather intense vertical shear in
Fig. 7 propagate downward through one or the other of the levels represented in Fig. 6. Hence the characteristic "square wave" shape of the time
series in the top panel (i.e., the presence of embedded higher harmonics)
is responsible for the irregularity of the loops in the bottom panels. Were
it not for these higher harmonics, the loops would be circular or elliptical
and the density of points would be uniform along the loops.
A simpler and, in some sense, more informative phase space representation of the QBO can be constructed by expanding the time series of zonal
wind at the seven levels represented in Fig. 7 into EOFs and using the time
series of the expansion coefficients (often referred to as "principal components" or PCs) of the two leading EOFs as the axes in the phase space
plot. In this particular example, the two leading EOFs account for 90% of
the variance of the monthly mean time series and over 95% of the 5-month
running mean time series of zonal wind at the seven levels. Hence, their
PC time series contain the information required to reconstruct the more
robust features of Fig. 7 (Wallace et al., 1993). The EOFs themselves,
shown in Fig. 12, resemble sinusoidal functions of height, in quadrature
with one another, with a vertical half-wavelength "-' 10km.
When represented in the phase space defined by the two leading PCs
(Fig. 13), the QBO appears simpler and more coherent than in Fig. 6. The
tendency for clustering is gone and the trajectories are much more circular 1 .
The inherent predictability of the QBO is evident in the regularity of the
circular orbit.
In perfectly periodic phenomena such as the annual march, the orbits
in phase space are perfectly reproducible from one cycle to the next. In
the absence of embedded higher harmonics, they can be fully represented
in a two-dimensional phase space in which the x and y axes can be iden1 Upon careful inspection it is evident that the circular loops in Fig. 13 are centered, not on the origin,
but on a point in the lower right quadrant and the density of points tends to be highest in that quadrant.
This asymmetry reflects the tendency for westerly regimes to propagate downward more rapidly than
easterly regimes in Fig. 7. The lower right quadrant in Fig. 13 corresponds to the phase of the QBO in
which easterlies overlie westerlies, and the phase progression is relatively slow.
The loops formed by the evolution of the QBO are represented in the
lower left panel of Fig. 6 by a series of dots which represent sequential
1-month means of zonal winds at two different levels in the equatorial
stratosphere. The dots tend to be clustered in preferred regions of this
particular phase space; i.e., near A and B. As the QBO executes a loop,
the trajectory tends to move relatively quickly between clusters. These
transitions take place when the layers of rather intense vertical shear in
Fig. 7 propagate downward through one or the other of the levels represented in Fig. 6. Hence the characteristic "square wave" shape of the time
series in the top panel (i.e., the presence of embedded higher harmonics)
is responsible for the irregularity of the loops in the bottom panels. Were
it not for these higher harmonics, the loops would be circular or elliptical
and the density of points would be uniform along the loops.
A simpler and, in some sense, more informative phase space representation of the QBO can be constructed by expanding the time series of zonal
wind at the seven levels represented in Fig. 7 into EOFs and using the time
series of the expansion coefficients (often referred to as "principal components" or PCs) of the two leading EOFs as the axes in the phase space
plot. In this particular example, the two leading EOFs account for 90% of
the variance of the monthly mean time series and over 95% of the 5-month
running mean time series of zonal wind at the seven levels. Hence, their
PC time series contain the information required to reconstruct the more
robust features of Fig. 7 (Wallace et al., 1993). The EOFs themselves,
shown in Fig. 12, resemble sinusoidal functions of height, in quadrature
with one another, with a vertical half-wavelength "-' 10km.
When represented in the phase space defined by the two leading PCs
(Fig. 13), the QBO appears simpler and more coherent than in Fig. 6. The
tendency for clustering is gone and the trajectories are much more circular 1 .
The inherent predictability of the QBO is evident in the regularity of the
circular orbit.
In perfectly periodic phenomena such as the annual march, the orbits
in phase space are perfectly reproducible from one cycle to the next. In
the absence of embedded higher harmonics, they can be fully represented
in a two-dimensional phase space in which the x and y axes can be iden1 Upon careful inspection it is evident that the circular loops in Fig. 13 are centered, not on the origin,
but on a point in the lower right quadrant and the density of points tends to be highest in that quadrant.
This asymmetry reflects the tendency for westerly regimes to propagate downward more rapidly than
easterly regimes in Fig. 7. The lower right quadrant in Fig. 13 corresponds to the phase of the QBO in
which easterlies overlie westerlies, and the phase progression is relatively slow.