# Millennial cycles of mean sea level excited by earths orbital variations.

1. INTRODUCTIONNowadays, it is well known that the continental ice sheets reached latitudes between 40[degrees] and 50[degrees] in the North America and between 50[degrees] and 60[degrees] in EuroAsia during the last glacial maximum. Roughly, this area is almost circular with radius 4400 km (corresponding to 40[degrees] over the meridian). The center of this area is shifted by 10[degrees] from the North Pole to the Greenland direction. The South Pole ice sheet covers almost circular area with latitude above 50[degrees] (Paul and Schafer-Neth, 2003; Schafer-Neth and Paul, 2003), thus its radius is approximately the same as the North Pole ice sheet. These polar ice sheets are approximately 3 km thick and they lead to 110 m decrease of the mean sea level. In the Pleistocene, the Earth's climate was mostly in ice ages state, and the ice ages cycles have periods of 40 ka or 100 ka. The glaciations cause significant MSL variations, water redistribution between the ocean and polar ice cap, and change of the Earth's axial moment of inertia, which is connected with long periodical variations of the Earth's rotation, expressed as Length of Day (LOD) and Universal Time (UT1) oscillations. It is possible to determine the long-term glacial variations of LOD and UT1 by means of the reconstructed paleo MSL data.

Recently, the MSL variations have been reconstructed for the last 380 ka by Siddall et al. (2003). Highstands of sea level are reconstructed from dated fossil coral reef terraces, and these data are complemented by a compilation of global sea-level estimates based on ratios [[delta].sup.18]O of deep-sea oxygen isotopes [sup.18]O and [sup.16]O fixed in shells in sedimentary data. The value of [[delta].sup.18]O is used as a proxy of the global temperature using the fact that more [sup.18]O implies more ice in continents and more [sup.16]O implies less ice. Their reconstruction is accurate to within [+ or -]12 m, and gives a centennial-scale resolution.

Jouzel et al. (2007) estimate the mean temperature variations T by means of a high-resolution deuterium profile from Antarctica Dome C ice core, determined with a high relative accuracy of about 0.5 [per thousand]. They extend this deuterium climate record back to ~800 ka ago and provide temperature interpretation of the data by an atmospheric general circulation model including water isotopes. Their time scale is in excellent agreement with the Dome Fuji and Vostok ice core time scales within 1 ka for significant data parts (Parrenin et al., 2007). This difference is a systematic long term shift between different data sets and it is visible like a time lag in comparison with the millennial cycles of climate, MSL and insolation time series, while the time scale random errors stay at centennial level. The MSL and T variations are strongly correlated. It is possible to reconstruct MSL variations for the period 380-800 ka before present (BP) by a regression model MSL=7.4T-18.7, where MSL variations are in meters and T in [degrees]C (Fig. 1).

[FIGURE 1 OMITTED]

A century ago, Croll (1864) and Milankovich (1920) suggested that the ice ages have astronomical reasons. Nowadays, the science community accepts that the fundamental sources of ice ages are the long term variations of the Earth's orbit according to Milankovich theory (Milankovich, 1998). Significant oscillations in total and local Earth's insolation are caused by the variations of the following orbital elements:

* the eccentricity has several periods around 100 ka;

* the obliquity of the Earth's axis, that is the tilt with respect to the normal to the ecliptic plane, oscillates with a period of about 41 ka;

* the precession of the equinoxes with period of 25.8 ka;

* the 'climatic' precession (Loutre, 2009), defined by formula e sin (w), where e means eccentricity and w the longitude of perihelion from the moving equinox, presenting two main modes at 23 ka and 19 ka.

Smaller changes of the insolation are associated with other Earth's Keplerian elements, namely semimajor axis, inclination of the ecliptic in relation to J2000 reference frame and longitude of perihelion, whose variations can modulate the local Earth's insolation. The most accurate eccentricity, climatic precession, obliquity, and insolation for the past several millions years have been proposed by Laskar et al. (2011). The other elements: semi-major axis, inclination and longitude of perihelion are calculated by the program Mercury 6 (Chambers and Migliorini, 1997). The most important factors of ice age cycles are the local summer insolation at 65[degrees] N latitude, and the climatic precession.

Some authors consider that the orbital variations may explain only one half of reconstructed temperature variations derived from sea sediments and polar ice core drilling (Imbrie et al., 1992; 1993). They propose a function of the orbital components, which approximates the observed paleoclimatic variations better than the Milankovich theory (Imbrie, 1985; Imbrie et al., 1992, 1993). Another important step toward the improvement of the astro-paleo theory is made in the publications of Shopov et al. (1999, 2001, 2004), where the influence of some millennial cycles with periods up to 11500 years of the solar activity on the paleo climate is suggested.

2. DATA AND METHOD OF ANALYSIS

The latest high-precise series of the Earth's orbital and insolation changes mentioned above are shown in Figures 2 and 3. The semi-major axis (Fig. 2) remains very close to 1au, presenting relative variations up to [10.sup.-5] au. Its oscillations are not smooth, so that they are not readily describable by Fourier series. On the contrary, the variations of the other Keplerian elements (Fig. 2) and of the obliquity (Fig. 3) are rather smooth, yielding harmonics below 25 ka. As a function of the obliquity (Fig. 3) and of all orbital elements, the summer insolation at 65[degrees]N latitude (Fig. 3) has a lot of cycles below 25 ka.

The millennial astronomical and climate cycles are represented by partial Fourier series, the coefficients of which are derived by the Least-Squares Method (LS).

The partial sum of Fourier series F (t) of discrete data [f.sub.i] is given by

F (t) = [f.sub.o] + [f.sub.1] (t - [t.sub.0]) + [n.summation over (k=1)] [a.sub.k] sin [k 2[pi]/[t.sub.E]--[t.sub.B] (t - [t.sub.0])] + + [b.sub.k] cos, [k 2[pi]/[t.sub.E] - [t.sub.B] (t - [t.sub.0])] (1)

where [t.sub.0], [t.sub.B] and [t.sub.E] are the mean, first and last epochs of observations, respectively, [f.sub.0],[f.sub.1], [a.sub.k] and [b.sub.k] are unknown coefficients and n is the number of harmonics of the partial sum, which covers all oscillations with periods between ([t.sub.E]-[t.sub.B])/n and ([t.sub.E]-[t.sub.B]). The application of the LS estimation of Fourier coefficients needs at least 2n+2 observations, so the number of harmonics n is chosen significantly smaller than the number N of sampled data [f.sub.i]. The small number of harmonics n yields to LS estimation of the coefficient errors, too. The period of the first long periodical harmonic in (1) depends on the observational time span in case of classic Fourier approximation, but here it is possible to decrease this value so that the estimation may cover the whole set of desired frequencies. This method allows a flexible separation of the harmonic oscillations into different frequency bands by the formula

B (t) = [[m.sub.2].summation over (k=[m.sub.1]) [a.sub.k] sin [k 2[pi]/[t.sub.E] - [t.sub.B] (t--[t.sub.0])] + + [b.sub.k] cos [k 2[pi]/[t.sub.E] - [t.sub.B] (t - [t.sub.0])],

where the desired frequencies [[omega].sub.k] = k 2[pi]/[t.sub.E]-[t.sub.B] are limited by the bandwidth

2[pi][m.sub.1]/[t.sub.E]-[t.sub.B] [less than or equal to] [[omega].sub.k] [less than or equal to] 2[pi][m.sub.2]/[t.sub.E]-[t.sub.B] (3)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

After estimating the Fourier coefficients, it is possible to identify a narrow frequency zone presenting significant amplitude, and defining a given cycle. Then this cycle can be reconstructed in time domain as the partial sum limited to the corresponding frequency bandwidth. Doing this for climate, orbital and solar insolation time series, we shall identify their respective cycles, isolate and compare the common ones.

3. TIME SERIES SPECTRA

As already pointed out, the paleo variations of semi-major axis a (Fig. 2) are not sufficiently smooth and its spectrum is determined by the Fast Fourier Transform (FFT). The amplitude spectra of inclination i, longitude of perihelion w on one hand (Fig. 4), MSL, summer insolation at 65 N latitude and climatic precession on the other hand (Fig. 5) are determined by partial sums of Fourier series covering 800 ka data and composed of 200 harmonics. The amplitudes [A.sub.k] of these harmonics are calculated from the Fourier coefficients [a.sub.k] and [b.sub.k] by the expression

[A.sub.k] = [square root of [a.sup.2.sub.k] + [b.sup.2.sub.k] (4)

and their accuracy is given in Table 1.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Comparing the spectra of orbital elements and MSL, we put forward 6 common frequency bands with periods below 25 ka: 21-25 ka; 18.6-19.6 ka; 13-17 ka; 11.3-11.8 ka; 7.5-7.8 ka; 5.9-6.0 ka. Two of these bands include the periods of 6ka and 11.5 ka, the origin of which could be linked with the solar activity (Xapsos and Burke, 2009; Shopov et al. 1999, 2001, 2004). Their solar activity origin can be checked by detecting discrepancy between orbital insolation and MSL variations. Additional frequency band with periodicity 2.25-3.0 ka (not presented in spectra of Figs. 4 and 5) is also involved, because it contains the well-established millennial cycle of the solar activity (Hallstatt cycle of 2.3 ka).

4. COMMON MILLENNIAL CYCLES OF MSL, INSOLATION AND ORBITAL ELEMENTS

All millennial cycles above 5 ka of Earth's orbital elements, summer insolation at 65[degrees]N and MSL are determined by recontructing the partial Fourier series in time domain, whose periods belong to the chosen band.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

4.1. PERIOD BAND 21-25 KA

The millennial cycles of MSL, insolation, climatic precession and eccentricity in the band 21-25 ka (Fig. 6) are determined by 8 harmonics of Fourier approximations with numbers k between 31 and 38. The MSL variations express excellent agreement in phase and amplitude with the insolation and climatic precession over 200 ka time scales. The comparison of MSL and eccentricity shows phase inversion and amplitude disagreement, so we will compare MSL only with insolation and climatic precession.

4.2. PERIOD BAND 18.6-19.6 KA

The common orbital and climate cycles with periods from the band 18.6-19.6 ka (Fig. 7) are determined by 3 harmonics of Fourier approximations with numbers k between 41 and 43. The MSL cycles express good agreement with the insolation and climatic precession with negligible phase and amplitude differences over 800 ka time span, so we may suppose that these cycles have an orbital origin only.

4.3. PERIOD BANDS 13-15 KA AND 15-17KA

Whereas these bands are common in spectra of Earth's orbital elements and climatic indices, it was not possible to find any combination of harmonics with periods between 13 ka and 17 ka, whose superposition yields similar behavior between the orbital and climatic time series. This probably results from the existence of one or more disturbing cycles having no relation with orbital modulation of the insolation. The solar activity is a good candidate.

4.4. PERIOD BAND 11.3-11.8 KA

The common orbital and climate cycles with periods in the band 11.3-11.8 ka (Fig. 8) are determined by 4 harmonics of Fourier approximations with numbers k between 68 and 71. The MSL phase variations express excellent agreement with the insolation over 800 ka time span, so the presence of any oscillation originating in the solar activity is very unlikely, in contrary to the previous band. Obviously, the MSL oscillations with period of about 11.5 ka are of orbital origin, connected with the half period of climatic precession (23 ka).

4.5. PERIOD BAND 7.5-7.8 KA

The common orbital and climate cycles with periods in the band 7.5-7.8 ka (Fig. 9) are determined by 5 harmonics of Fourier approximations with numbers k between 103 and 107. The MSL exhibits phase reversal with the insolation, so that the solar activity can eventually perturb this cycle.

4.6. PERIOD BAND 5.9-6.0 KA

The MSL and insolation cycles in the band 5.96.0 ka (Fig. 10) are determined by 4 harmonics of Fourier approximations with numbers k between 133 and 136. This band should contain the 6 ka solar activity cycle mentioned in (Xapsos and Burke, 2009; Shopov et al., 1999, 2001, 2004). Actually, the phase variations of MSL and insolation over 800 ka time span are in good agreement, only a small amplitude deviation between 200 ka and 400 ka BP exists. So, we conclude that if the solar activity cycle exists with the exact period of 6 ka, then its amplitude is small compared to orbital influences. Most probably, such solar cycle have rather long period, between 6 ka and 7.8 ka.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

4.7. PERIOD BAND 2.25-3.0 KA

As this period band is rather narrow, the corresponding oscillations are determined by superposing only 3 harmonics with numbers k 100, 101 102 in the partial sum of Fourier series of MSL and insolation data for the period 0-230 ka BP (Fig. 11). As expected, the orbital-climatic influence is strongly disturbed by the Hallstatt solar cycle in this band, as evidenced by the strong phase and amplitude discrepancy between the summer 65 N insolation and MSL, for this band.

4.8. RELATIONS TO THE VARIATIONS OF EARTH ROTATION AND PRINCIPAL MOMENTS OF INERTIA

The Earth shape, gravity and rotation are highly affected by climatic variations associated with the glacial cycles in the late Pleistocene. The processes of glaciation, followed by ice melting and significant changes of the mean sea level redistribute great amount of water masses between oceans and ice sheets, which lead to changes of the principal moments of inertia. Chapanov and Gambis (2010, 2011) proposed models of Length of Day LOD, Universal Time UT1 and Earth principal moments of inertia A, B, C as polynomial functions of the MSL. As illustrated in Figure 12 over the period 0-800 ka BP, these models reproduced almost linearly the shape of the MSL. Thus, all MSL millennial harmonics with periodicities below 25 ka, should be valid for the millennial cycles of Earth rotation and principal moments of inertia. This may help in better reconstructing and understanding paleo Earth rotation.

5. CONCLUSIONS

The Mean Sea Level and climatic insolation are decomposed in partial Fourier series over the last 800 ka. This allows us to identify 7 common millennial cycles with periods below 25 ka, and given by the bands 21-25 ka, 18.6-19.6 ka, 13-17 ka, 11.3-11.8 ka, 7.5-7.8 ka, 5.9-6.0 ka and 2.25-3.0 ka. The MSL oscillations express good agreement with the corresponding insolation in the bands 21-25 ka, 18.6-19.6 ka, 11.3-11.8 ka and 5.9-6.0 ka, leading us to conclude that MSL millennial cycles are mostly due to orbital variations. However, the MSL and insolation oscillations have different amplitude and phase behavior for the period bands 13-17 ka, 7.5-7.8 ka and 2.25-3.0 ka. The last band contains the well-known Hallstatt solar cycle, which produces significant disturbances of orbital-climatic influences. The discrepancy between the orbital and MSL cycles in the band 13-17 ka and 7.5-7.8 ka evidences an additional excitation. As the 2.3 ka Hallstatt cycle of solar activity disturbs the agreement between MSL and orbital insolation index, we assume that this additional excitation could originate in hypothetic solar cycles with the periods of 7-8 ka and 13-17 ka.

[FIGURE 12 OMITTED]

DOI: 10.13168/AGG.2015.0028

ARTICLE INFO

Article history:

Received 14 January 2015

Accepted 9 June 2015

Available online 13 July 2015

ACKNOWLEDGEMENTS

This research was financially supported by the grant No. 13-15943S "Geophysical excitations in the motion of Earth's axis of rotation" awarded by the Grant Agency of the Czech Republic. The support given to authors by Bulgarian and Czech Academies of Sciences in frame of Joint Research Project "Periodical and impulse variations of geodetic time series" is appreciated. We express our thanks to Ch. Bizouard and an anonymous reviewer, whose constructive criticism and comments helped improve the text significantly.

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Yavor CHAPANOV [1] *, Cyril RON [2] and Jan VONDRAK [2]

[1] National Institute of Geophysics, Geodesy and Geography, BAS Acad. G. Bonchev, Str. Bl.3, Sofia 1U3, Bulgaria

[2] Astronomical Institute, Czech Academy of Sciences, Bocni II1401, 141 00 Prague 4, Czech Republic

Corresponding author's e-mail: yavor.chapanov@gmail.com

(1) ka (kilo-annum) thousand of years

Table 1 Accuracy of estimated amplitudes by the Least Squares Method. Element Harmonics 1-10 Harmonics 11-200 Longitude of 0.3 / 2.1 0.28 perihelion [degree] Inclination [degree] 5 x [10.sup.-6] / 4.2 x [10.sup.-6] 32 x [10.sup.-6] MSL [cm] 8 / 54 6 Insolation 65[degrees]N 0.4 / 2.8 0.3 [mW/[m.sup.2]] Climatic 2 x [10.sup.-4] / 1 x [10.sup.-4] precession x 100 1 x [10.sup.-3]

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Title Annotation: | Original paper |
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Author: | Chapanov, Yavor; Ron, Cyril; Vondrak, Jan |

Publication: | Acta Geodynamica et Geromaterialia |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2015 |

Words: | 3550 |

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