16
is to be expected, since the power spectrum is the Fourier transform of the
autocorrelation function, and for each value of N, the sum of the mean
variance of samples of length N days plus the average variance of means
over N successive days (i.e., the sum of the variances plotted in the upper
right and lower left panels) is equal to the total variance of the daily data,
based on the full length of the record. Regardless of which way the data
are presented, the transition between deterministic, aperiodic behaviour
at the higher frequencies and white noise at the lower frequencies is quite
gradual: it is impossible to say at precisely what frequency or on what time
scale it occurs. The 'transition zone' in the frequency domain, in which
the behaviour is neither purely deterministic, nor completely random is
particularly broad in this example because the GeM mimics several types
of aperiodic variability in the real atmosphere, ranging from baroclinic
instability, with a characteristic time scale of a day or two, to a zonally
symmetric mode that varies on a time scale of a few weeks. The transition
zone in Fig. 10 may be viewed as representing a superposition of a number
of somewhat narrower zones, each bracketing the 'low-frequency cutoff' of
a particular mode of deterministic aperiodic variability. In the coupled
climate system, deterministic aperiodic variability extends out to much
longer time scales where it coexists with random sampling fluctuations
associated with the atmosphere's own internal variability.
Even in the case of this extremely long time series, the "white" segment
of the spectrum in Fig. 10 is barely resolved. It would be possible, in principle, to extend the spectrum to lower frequencies, but only at the expense
of reducing the number of degrees of freedom of the spectral estimates,
rendering them even more noisy than those in Fig. 10. If the time series
had been, say, 10,000 days long instead of 100,000 days, and/or if lower
frequency dynamical processes such as those associated with the ENSO cycle had been present in the model, as in the real world, the transition zone
would have extended to the lowest resolvable frequencies, as it typically
does in the analysis of observational data.
Even though the power spectrum of a "red noise" time series is featureless, such a time series can yield what could mistakenly be interpreted as
intriguing looking results when subjected to certain analysis techniques.
When a large ensemble of 'red noise' time series of length N is subjected
to EOF analysis in the time domain, one obtains a set of modes that
resemble the results of harmonic analysis except that the frequencies of
the family of waves are not constrained to be integral multiples of 27r/N.
is to be expected, since the power spectrum is the Fourier transform of the
autocorrelation function, and for each value of N, the sum of the mean
variance of samples of length N days plus the average variance of means
over N successive days (i.e., the sum of the variances plotted in the upper
right and lower left panels) is equal to the total variance of the daily data,
based on the full length of the record. Regardless of which way the data
are presented, the transition between deterministic, aperiodic behaviour
at the higher frequencies and white noise at the lower frequencies is quite
gradual: it is impossible to say at precisely what frequency or on what time
scale it occurs. The 'transition zone' in the frequency domain, in which
the behaviour is neither purely deterministic, nor completely random is
particularly broad in this example because the GeM mimics several types
of aperiodic variability in the real atmosphere, ranging from baroclinic
instability, with a characteristic time scale of a day or two, to a zonally
symmetric mode that varies on a time scale of a few weeks. The transition
zone in Fig. 10 may be viewed as representing a superposition of a number
of somewhat narrower zones, each bracketing the 'low-frequency cutoff' of
a particular mode of deterministic aperiodic variability. In the coupled
climate system, deterministic aperiodic variability extends out to much
longer time scales where it coexists with random sampling fluctuations
associated with the atmosphere's own internal variability.
Even in the case of this extremely long time series, the "white" segment
of the spectrum in Fig. 10 is barely resolved. It would be possible, in principle, to extend the spectrum to lower frequencies, but only at the expense
of reducing the number of degrees of freedom of the spectral estimates,
rendering them even more noisy than those in Fig. 10. If the time series
had been, say, 10,000 days long instead of 100,000 days, and/or if lower
frequency dynamical processes such as those associated with the ENSO cycle had been present in the model, as in the real world, the transition zone
would have extended to the lowest resolvable frequencies, as it typically
does in the analysis of observational data.
Even though the power spectrum of a "red noise" time series is featureless, such a time series can yield what could mistakenly be interpreted as
intriguing looking results when subjected to certain analysis techniques.
When a large ensemble of 'red noise' time series of length N is subjected
to EOF analysis in the time domain, one obtains a set of modes that
resemble the results of harmonic analysis except that the frequencies of
the family of waves are not constrained to be integral multiples of 27r/N.