Commonly some form of time-series or spectral analysis is used for extracting periodic signatures from a set of data pertaining to climate variations.  However, spectral analysis is, like statistical analysis, a black art rather than a fully respectable branch of mathematics or science.  What comes out depends very much on how the analysis is applied.
In trying to extract weather and climate periodicities from data extending back many centuries, researchers have made claims for such a wide variety of periods that the results should be viewed with great skepticism.  Few of the analyses agree with each other.  Some find the expected solar and lunar cycles; others do noteven using the same kinds of data.
In extracting periodic signatures from climate data, three important factors must be observed:
Temporal inhomogeneity has always plagued long runs of climate data.  The data run is seldom long enough to extract the really interesting long periods.  Temporal events such as the Little Ice Age and other slow variations may disturb the shorter periods, or even establish a new state with totally different cyclic behavior.  The circulation and weather systems may have changed qualitatively with the end of the Little Ice Age; so the short-term variationssuch as the ENSOmay be substantially different from what they were several hundred years ago.
Temporal inhomogeneity may distort historical climate data because the instruments and methods change over time.
Spatial inhomogeneity can be a serious problem whenever two sets of data from widely separated stations are mixed.  Sometimesas in the analysis of tree ring datainvestigators may combine data from a broad area in order to get a satisfactorily "clean" signal.  However, the weather patterns and climate cycles may be substantially different in, say, coastal areas, mountain areas, and continental interiors.  Results which are claimed to apply to a broad area, containing many kinds of topography and environment, are always suspect.
Methods for analysing temporal data have not been standardized, and investigators are often drawn to the newest, most "trendy" methods.  Keep in mind that spectral analysis methods were developed for extracting weak signals from very long runs of data.  They work very well when they are called on to extact the blips of a pulsar or a clandestine radio transmitter from days to months of radio transmissions; they don't work so well when the blip only occurs two or three times in the entire set of data.  Often a set of climate data does not span a long enough time to confidently extract any but the single strongest signal.  The results may suggest other signals, but they may be totally spurious.
As an example, consider several sets of tree ring data, which have been
widely published and referenced [from the work of Fritz and Shao].
The figure below shows inferred precipitation (in light blue) and
temperature (gray) for the Northern Great Plains.  At the top are
bars, denoting the historical
droughts since 1800, and inferred droughts for earlier times.
Three of the historical droughts were especially long-lived and
severe.  The drought from 1840 to 1864 resulted in displacement and
hardship for native peoples.  Several explorers came to the
Northern Plains, including Isaac Stevens and John Palliser, and saw
evidence of extreme aridity.  The years 1860 and 1861 were among
the driest ever recorded there.  The dustbowl years of the 1920's and
1930's lasted almost as long, and included some extremely dry years.
The most recent drought, which began about 1996, is almost as severe;
it could last as long as the others.  Over the past several centuries
great historical droughts in the interior of North America have occurred at
intervals of 75 years.
The data here are obviously very "ragged." The old Blackman and Tuckey
(BT) method, based on simple Fourier deconstruction of the signal, is
instructive, because it shows the need for very long series of
data.  The BT method involves a fourier transform of the
autocorrelation function, shown below for the data sets, plus a data
set for the Southwest Deserts.
The plotted curves show little order at short lag times.  Only at lag times greater than 200 years do the autocorrelations begin to show regular variations that might yield a genuine spectral feature.  But the value of the analysis is called into question because the total data run is only slightly more than 200 years.  The data here have been folded to overcome the limitations of the data sets.  So the regular oscillations beyond 200 years may simply be an artifact of the method of analysis.  Such data folding may be implicit in spectral analysis methods, and is a potential source of aliasing at long periods.  Researchers should always examine the autocorrelation function to see whether they indeed have a long enough data sample that real periodicities begin to show up.
The cross correlations of the three data sets show that there is a high
degree of synchronism at short periods and very long periods.  At
lag times between 120 and 250 years the correlations are poor.
Finally, a fourier analysis of the Northern Plains autocorrelation
yields a power spectrum, with peaks whose sharpness is dependent on the
lag windows used.  The spectra are unnormalized, so the relative
heights of peaks are not important.
For comparison, see the following spectrum, derived by one of the more
"modern" methods, the Auto Regressive (AR) method (which is similar to
Maximum Entropy methods).  The mathematical method here is quite
different, so the peaks show a dramatically different shape.  In
the AR method, choosing a large number of poles is equivalent to
choosing a long lag window.  The secondary peaks appear sharper,
but they are not necessarily more real. Presumably one could "refine"
the analysis to produce ever sharper peaks, but there is a great risk
of fooling ourselves.
Many papers have been published on drought and other climate cycles
showing a series of sharp peaks at the frequencies determined by the
mathematical analysis.  But the peaks shown above have a finite
width.  The mathematics of Fourier analysis says that the power
spectral frequency peaks cannot be narrower than
Beware of published analysis which show spectral peaks represented by narrow lines!
The data sets shown above are clearly not long enough to reveal a 75
year period; but the first pronounced peak is at half that period: 37
years (with an uncertainty of about 7 years).  Lesser peaks appear
at about 18 years.  Though the autogression analysis suggests very
sharp secondary peaks at the shorter periodsapproximately 18 and 10
years; they are probably not meaningful.
A useful test of the results of spectral analysis is to overlay the data with a periodic "comb," represented above by alternating green and brown bands.  (This is equivalent to the old "periodogram" method.)  Here some of the peaks and valleys (for the Northern Plains) agree moderately well with the comb.  With such a short run of data, the results of analysis can be trusted only if they can be demonstrated by such a simple "eyeball" procedure.
A physical cause for 18 and 37 year periods may be related to the lunar tidal cycle of 18.6 years.  There is no evidence here of an 11-year sunspot cycle.  See a discussion of droughts on the Great Plains.
The possibility of a 37 year resonance with the lunar tidal cycle is
intriguing; confirmation would require a homogeneous data set running
over many hundreds of years.  The tree ring data are probably not
sufficiently homogeneous.  There is an excellent theoretical reason
why such the lunar tidal resonance should be strongest at the double
period of 37 years: in 37 years the tidal cycle repeats
at the same time of the year (much like the 18 year Saros cycle of
eclipses).  A plausible physical mechanism has not yet been found,
though 35 to 40 year weather and precipitation cycles have been frequently
noted, and several meteorologists have suggested that the lunar tidal cycle
modifies ocean currents in the northern ocean basins.
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