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they exhibit a continuum of phases. Standing oscillations can be represented just as well by individual time series or by a single linear combination of time series, whereas progressive oscillations can be conveniently
represented by a pair of time series which oscillate in quadrature with one
another, each of which may consist of a linear combination of the original
time series included in the analysis. A phase-space representation offers
'value added' only for progressive oscillations.
3.1 Progressive oscillations in two-dimensional phase-space
An individual time series x(t) can be viewed as a trajectory in a onedimensional phase-space and a pair of time series x(t) and y(t) define a
two-dimensional phase-space, for which the lower panels of Fig. 6 serve
as a convenient example. The x and y axes in these panels represent the
zonal wind component at two different levels that experience the equatorial stratospheric QBO. Referring to the time-height section in Fig. 7,
it is readily verified that an individual cycle of the QBO is represented
by a clockwise or counterclockwise loop in the diagram, depending upon
whether the upper or lower level zonal wind value is plotted on the x-axis.
In this particular example, the trajectory in phase-space executes a series
of counterclockwise loops.
The appearance of the loops in such a plot obviously depends upon the
vertical separation between the levels selected for the phase-space representation. If they are close together (e.g., zonal wind at the 30-mb plotted
versus zonal wind at the 40-mb level) the two time series will be positively
correlated and the loops will tend to be elongated along a line with a positive slope in the (x, y) plane. In the limit of no separation between levels,
the trajectories would collapse to that line. If the levels are chosen such
that one lies near the top ofthe domain of Fig. 7 (say, 10-mb) and the other
near the bottom (say, 50- or 70-mb) the negative correlation between the
time series will be reflected in an elongation of the loops along a straight
line with a negative slope in the (x, y) plane. The evolution of the QBO
is most clearly revealed in plots whose axes correspond to levels separated
by about 5 km, which corresponds to roughly one quarter of the effective
vertical wavelength of the QBO. The zonal wind fluctuations at such pairs
of levels tend to occur in quadrature; i.e., with one lagging the other by
1/4 period. They are neither positively nor negatively correlated with one
another: i.e., they are mutually orthogonal.
they exhibit a continuum of phases. Standing oscillations can be represented just as well by individual time series or by a single linear combination of time series, whereas progressive oscillations can be conveniently
represented by a pair of time series which oscillate in quadrature with one
another, each of which may consist of a linear combination of the original
time series included in the analysis. A phase-space representation offers
'value added' only for progressive oscillations.
3.1 Progressive oscillations in two-dimensional phase-space
An individual time series x(t) can be viewed as a trajectory in a onedimensional phase-space and a pair of time series x(t) and y(t) define a
two-dimensional phase-space, for which the lower panels of Fig. 6 serve
as a convenient example. The x and y axes in these panels represent the
zonal wind component at two different levels that experience the equatorial stratospheric QBO. Referring to the time-height section in Fig. 7,
it is readily verified that an individual cycle of the QBO is represented
by a clockwise or counterclockwise loop in the diagram, depending upon
whether the upper or lower level zonal wind value is plotted on the x-axis.
In this particular example, the trajectory in phase-space executes a series
of counterclockwise loops.
The appearance of the loops in such a plot obviously depends upon the
vertical separation between the levels selected for the phase-space representation. If they are close together (e.g., zonal wind at the 30-mb plotted
versus zonal wind at the 40-mb level) the two time series will be positively
correlated and the loops will tend to be elongated along a line with a positive slope in the (x, y) plane. In the limit of no separation between levels,
the trajectories would collapse to that line. If the levels are chosen such
that one lies near the top ofthe domain of Fig. 7 (say, 10-mb) and the other
near the bottom (say, 50- or 70-mb) the negative correlation between the
time series will be reflected in an elongation of the loops along a straight
line with a negative slope in the (x, y) plane. The evolution of the QBO
is most clearly revealed in plots whose axes correspond to levels separated
by about 5 km, which corresponds to roughly one quarter of the effective
vertical wavelength of the QBO. The zonal wind fluctuations at such pairs
of levels tend to occur in quadrature; i.e., with one lagging the other by
1/4 period. They are neither positively nor negatively correlated with one
another: i.e., they are mutually orthogonal.