624
von Dobeneck and Schmieder
Schwarzacher 1993) and known solar activity cycles (e.g. sunspot, Hale and Gleissberg cycles) frequently detected in varved sediments have periods
<200 years (Glenn and Kelts 1991), whichperiodicities may actually be expected in the spectral
range from 1 to 18 kyr? Besides free, quasi-periodic climate fluctuations, the most likely candidates
are multiples and combination tones of Milankovitch
frequencies originating from nonlinear transformations in the response chain of insolation, climate,
proxy physics and sedimentation.
'Harmonics' result from nonlinear response to
a single orbital cycle, while 'intermodulation frequencies' arise from nonlinear coupling of two or
more basic orbital cycles. The frequencies of these
additional modes are simply integer (k, I, m, n, 00')
combinations ofthe fundamental forcing frequenciesJ;,J;.,h,,/4' 00. (Le Treut and GhiI1983):
f(k,l,m,n, 00') = kfJ +Ijz +mf3 +nf4 +00.;
k,l,m,n,oo. E {00.,-1, 0,1, 2,00.};
order = Ikl + III + Iml + Inl+oo
The number of potential combination tones within
an investigated frequency band can be boosted ad
lib by admitting high order combinations involving
negative coefficients. Concerning orbital forcing,
only the lowest order combinations within a given
frequency range are bound to represent substantial spectral power (e.g., Yiouetal. 1991). Allloworder (:<;; 3) combination tones calculated from obliquity and precession and their realization and significance in the spectra of various astronomical time
series are shown in Fig. 13. The five most important eccentricity frequencies can be obtained as linear combinations ofthe basic precession frequencies (e.g., 1/404 kyrl = 1122.4 kyrl- 1123.7 kyrl;
Berger and Loutre 1992). It is therefore unnecessary to include them separately in the calculation
of potential orbital combination tones. However, it
is essential to distinguish not just two, but all three
major precession frequencies.
Even pure obliquity and precession signals
(Fig. 13c, d) and their transformation into latitude
dependent insolation (Berger and Loutre 1994) involve (slightly) nonlinear process (Berger and
Pestiaux 1984) and display sub-Milankovitch peaks
above the noise floor of numerical precision
(Fig. 13b). From the viewpoint of physical climate
models, non-linearresponse is rather the rule than
the exception. Not just ice-sheet dynamics (Le
Treut and GhiI1983), but also factors controlling
monsoon intensity (Short et al. 1991; Crowleyet
al. 1992) favor the generation of orbital harmonics. Periods of 10-12 kyr, corresponding to a doubled precession cycle (simulated here by squaring
the precession index; Fig. 13e), appear in insolation (Fig. 13b; Berger and Loutre 1997) and proxy
spectra (Pestiaux et al. 1988; Park et al. 1993;
Hagelberg et al. 1994), particularly at low latitudes.
The coupling of obliquity and precession (Fig. 13f)
generates characteristic peaks in the 13-15 kyr
band, which are also found in sediments (e.g.,
Berger et al. 1991) and ice core records (Yiou et
al. 1991). A simple non-linear mathematical transform of an astronomical insolation signal (Berger
and Loutre 1991) yields a nearly complete spectrum of all theoretically derived orbital combination
tones (Fig. 13a) and may be used as unbiased reference to identify significant sub-Milankovitch
peaks of proxy spectra. (Fig. 14a). In order to provide equivalent frequency resolution with respect
to the CEARIS spectra, the underlying time series
was reduced to the same time range (377 - 6 ka;
Fig. 14b,e).
The spectral amplitudes of insolation (Fig. 13b)
fade steeply in the sub-Milankovitch range, while
those of its nonlinear transform (Fig. 13a/14a) and
the CEARIS record (Fig. 14c) decrease much less.
This discrepancy implies spectral power channeling
into the sub-Milankovitch band. By visual comparison ofthe synthetic spectra (Fig. 14b,e) with those
of the CEARIS stack (Fig. 14c,f) and the stack of
the eight individual CEARIS spectra (Fig. 14d)
many analogous peaks can be detected. The most
prominent sub-Milankovitch peaks appear near 9,
10.5, 11.5, 13 and 15 kyr. They coincide with the
earlier mentioned low-order combination tones and
harmonics of obliquity and precession. A combined
evaluation of model and proxy spectra and their coherence spectrum (Fig. 14g) indicates, that the coherence at these four sub-Milankovitch frequencies is significant at an 80% level and thus nearly
as high as at the basic Milankovitch frequencies.
This result is statistically valid, but is based upon a
synthetic and somewhat arbitrary reference curve.
von Dobeneck and Schmieder
Schwarzacher 1993) and known solar activity cycles (e.g. sunspot, Hale and Gleissberg cycles) frequently detected in varved sediments have periods
<200 years (Glenn and Kelts 1991), whichperiodicities may actually be expected in the spectral
range from 1 to 18 kyr? Besides free, quasi-periodic climate fluctuations, the most likely candidates
are multiples and combination tones of Milankovitch
frequencies originating from nonlinear transformations in the response chain of insolation, climate,
proxy physics and sedimentation.
'Harmonics' result from nonlinear response to
a single orbital cycle, while 'intermodulation frequencies' arise from nonlinear coupling of two or
more basic orbital cycles. The frequencies of these
additional modes are simply integer (k, I, m, n, 00')
combinations ofthe fundamental forcing frequenciesJ;,J;.,h,,/4' 00. (Le Treut and GhiI1983):
f(k,l,m,n, 00') = kfJ +Ijz +mf3 +nf4 +00.;
k,l,m,n,oo. E {00.,-1, 0,1, 2,00.};
order = Ikl + III + Iml + Inl+oo
The number of potential combination tones within
an investigated frequency band can be boosted ad
lib by admitting high order combinations involving
negative coefficients. Concerning orbital forcing,
only the lowest order combinations within a given
frequency range are bound to represent substantial spectral power (e.g., Yiouetal. 1991). Allloworder (:<;; 3) combination tones calculated from obliquity and precession and their realization and significance in the spectra of various astronomical time
series are shown in Fig. 13. The five most important eccentricity frequencies can be obtained as linear combinations ofthe basic precession frequencies (e.g., 1/404 kyrl = 1122.4 kyrl- 1123.7 kyrl;
Berger and Loutre 1992). It is therefore unnecessary to include them separately in the calculation
of potential orbital combination tones. However, it
is essential to distinguish not just two, but all three
major precession frequencies.
Even pure obliquity and precession signals
(Fig. 13c, d) and their transformation into latitude
dependent insolation (Berger and Loutre 1994) involve (slightly) nonlinear process (Berger and
Pestiaux 1984) and display sub-Milankovitch peaks
above the noise floor of numerical precision
(Fig. 13b). From the viewpoint of physical climate
models, non-linearresponse is rather the rule than
the exception. Not just ice-sheet dynamics (Le
Treut and GhiI1983), but also factors controlling
monsoon intensity (Short et al. 1991; Crowleyet
al. 1992) favor the generation of orbital harmonics. Periods of 10-12 kyr, corresponding to a doubled precession cycle (simulated here by squaring
the precession index; Fig. 13e), appear in insolation (Fig. 13b; Berger and Loutre 1997) and proxy
spectra (Pestiaux et al. 1988; Park et al. 1993;
Hagelberg et al. 1994), particularly at low latitudes.
The coupling of obliquity and precession (Fig. 13f)
generates characteristic peaks in the 13-15 kyr
band, which are also found in sediments (e.g.,
Berger et al. 1991) and ice core records (Yiou et
al. 1991). A simple non-linear mathematical transform of an astronomical insolation signal (Berger
and Loutre 1991) yields a nearly complete spectrum of all theoretically derived orbital combination
tones (Fig. 13a) and may be used as unbiased reference to identify significant sub-Milankovitch
peaks of proxy spectra. (Fig. 14a). In order to provide equivalent frequency resolution with respect
to the CEARIS spectra, the underlying time series
was reduced to the same time range (377 - 6 ka;
Fig. 14b,e).
The spectral amplitudes of insolation (Fig. 13b)
fade steeply in the sub-Milankovitch range, while
those of its nonlinear transform (Fig. 13a/14a) and
the CEARIS record (Fig. 14c) decrease much less.
This discrepancy implies spectral power channeling
into the sub-Milankovitch band. By visual comparison ofthe synthetic spectra (Fig. 14b,e) with those
of the CEARIS stack (Fig. 14c,f) and the stack of
the eight individual CEARIS spectra (Fig. 14d)
many analogous peaks can be detected. The most
prominent sub-Milankovitch peaks appear near 9,
10.5, 11.5, 13 and 15 kyr. They coincide with the
earlier mentioned low-order combination tones and
harmonics of obliquity and precession. A combined
evaluation of model and proxy spectra and their coherence spectrum (Fig. 14g) indicates, that the coherence at these four sub-Milankovitch frequencies is significant at an 80% level and thus nearly
as high as at the basic Milankovitch frequencies.
This result is statistically valid, but is based upon a
synthetic and somewhat arbitrary reference curve.