13
the time series to drop off by a factor of e) is given by Oil( -Ina), where
6t is the time increment between successive data points. For purposes of
significance testing, the effectively time between independent data samples
in the time series is 27 (Leith 1973). The expected value of the variance
of a such a "red noise" time series increases with sampling interval, but
the rate of increase slows as the sampling interval becomes much longer
than the decorrelation time. In the limiting case a = 1, the time series can
be modelled as a 'random walk' process whose expected variance increases
linearly with sampling interval.
The expected variance of means of N successive data points in a "red"
time series decreases with increasing N at a rate more gradual than liN,
but approaches liN as the averaging interval becomes much longer than
the decorrelation time. In other words, for averaging intervals much longer
than 7, non-overlapping means of a "red" time series are linearly independent: their variability must be regarded as random and inherently unpredictable sampling fluctuations associated with whatever physical processes
happen to be operative at higher frequencies. In a similar manner, the
power in the spectrum of a "red" time series increases with decreasing frequency, but the rate of increase slows as frequency becomes smaller than
the inverse of the decorrelation time and it eventually levels off. Hence, at
sufficiently low frequencies, even 'red' time series exhibit 'white' spectra.
Madden (1976) expressed concern that the month-to-month and winterto-winter variability inherent in the climate record might be nothing more
than sampling variability associated with the presence of higher frequency
phenomena such as baroclinic waves with characteristic time scales of days
and sporadic blocking episodes that might last as long as a week or two.
To illustrate the validity of his concern, we will make use of a synthetic
"climate", whose low frequency behaviour is known with much greater precision than that of the real atmosphere. The results presented in this subsection are based on a 100,000 day, 'perpetual January' simulation with a
low resolution (rhomboidal 15 truncation) GFDL general circulation model
(GCM) run with fixed climatological mean SST. If a single winter is regarded as being 100 days in length, this run provides a sample size equivalent to 1000 winters: roughly 20 times as many as in the observational
record.
The statistics presented in Fig. 10 are based on the simulated 500-hPa
height field poleward of 20oN. The calculations described below are roughly
equivalent to what would be obtained if they were performed on the time
the time series to drop off by a factor of e) is given by Oil( -Ina), where
6t is the time increment between successive data points. For purposes of
significance testing, the effectively time between independent data samples
in the time series is 27 (Leith 1973). The expected value of the variance
of a such a "red noise" time series increases with sampling interval, but
the rate of increase slows as the sampling interval becomes much longer
than the decorrelation time. In the limiting case a = 1, the time series can
be modelled as a 'random walk' process whose expected variance increases
linearly with sampling interval.
The expected variance of means of N successive data points in a "red"
time series decreases with increasing N at a rate more gradual than liN,
but approaches liN as the averaging interval becomes much longer than
the decorrelation time. In other words, for averaging intervals much longer
than 7, non-overlapping means of a "red" time series are linearly independent: their variability must be regarded as random and inherently unpredictable sampling fluctuations associated with whatever physical processes
happen to be operative at higher frequencies. In a similar manner, the
power in the spectrum of a "red" time series increases with decreasing frequency, but the rate of increase slows as frequency becomes smaller than
the inverse of the decorrelation time and it eventually levels off. Hence, at
sufficiently low frequencies, even 'red' time series exhibit 'white' spectra.
Madden (1976) expressed concern that the month-to-month and winterto-winter variability inherent in the climate record might be nothing more
than sampling variability associated with the presence of higher frequency
phenomena such as baroclinic waves with characteristic time scales of days
and sporadic blocking episodes that might last as long as a week or two.
To illustrate the validity of his concern, we will make use of a synthetic
"climate", whose low frequency behaviour is known with much greater precision than that of the real atmosphere. The results presented in this subsection are based on a 100,000 day, 'perpetual January' simulation with a
low resolution (rhomboidal 15 truncation) GFDL general circulation model
(GCM) run with fixed climatological mean SST. If a single winter is regarded as being 100 days in length, this run provides a sample size equivalent to 1000 winters: roughly 20 times as many as in the observational
record.
The statistics presented in Fig. 10 are based on the simulated 500-hPa
height field poleward of 20oN. The calculations described below are roughly
equivalent to what would be obtained if they were performed on the time