Paleoclimate data have yielded variations with periods of ~23, ~40, and ~100 ky. Thermodynamic changes resulting from orbital eccentricity, obliquity, and precession have been ascribed as the cause of the variations although processes within the oceans and atmosphere may have too short memory to explain such variations. In this work, the dynamics of Sun-Moon gravitation (SMG) were explored for a rotating Earth and were determined to have a long memory in magma, a mostly ignored geophysical fluid with a mass ~3,400 times that of the atmosphere plus the oceans. Using the basic motion and gravitation (including obliquity) of the Sun and the Moon, we determined that SMG-induced magma motion could produce paleoclimatic variations with multiple periods (e.g., ~23, ~40, ~80, and ~100 ky), with considerable power for Earth’s heat. Such “reproducible” power could possibly maintain an energetic Earth against collapse, radioactivity, and cooling.
Orbital eccentricity, obliquity, and precession, with periodicities of ~100, ~41, and ~23 ky, respectively, are three important drivers of paleoclimatic variations [
A time series with the mean deducted (left column, data interval is 0.1 ky) and its squared amplitude spectrum (right column) for temperature (Row a), CO2 (Row b), insolation (i.e., Milankovitch cycle, Row c), and dust (Row d) from the EPICA Ice Core during July at 65°N for the past 800,000 years [
Past studies on slow processes within the atmosphere and oceans have indicated climatic variations of shorter periods from ~1 to ~100 ky [
However, the slow processes within the atmosphere and oceans may only explain shorter variation probably because the atmosphere and oceans have memories shorter than those of paleoclimatic variations from the viewpoint of dynamics [
Using a theoretical model established for SMG-induced magma motion (see Section
The following questions were further answered in this study. (1) How can orbital drivers (note that orbital obliquity was included while orbital eccentricity and precession were excluded) with limited periodicity produce period-abundant paleoclimatic variations within the PSMGIM? (2) Can the PSMGIM be significant for Earth’s heat budget? (3) How will PSMGIM variations contribute to paleoclimatic variations? The last question was partially explained and needed further studies.
This section contains three major parts, as follows: (1) the dynamic model for SMG-induced magma motion, (2) the periodicity calculations included for SMG-induced magma motion and the probability for a period to occur in SMG-induced magma motion, and (3) a spectrum analysis for data and modeling results.
As far as methodology is concerned, omitting small-magnitude SMG in climate is an extension of the scale-analysis method that has been effectively applied for linear processes and short-term weather systems. However, for climate and paleoclimate studies with a large space and a long time duration, small driving factors such as SMG that accumulatively and actively acts on climate and paleoclimate systems may not be omitted. The size of SMG is even comparable to Coriolis in the “SMG dynamical zone (SMDZ)” [
Obtaining an accurate nonlinear solution for magma motion under SMG in Eulerian system is currently impossible. SMG changes with a relative location between a float and the Sun or the Moon. For a numerical model to determine the changing location of a moving float, grid spacing in Eulerian system must be smaller than the distance the float moves within one time step. The longer the temporal scale is required to study or predict, the smaller the speed dynamically contributes to the corresponding temporal variation [
Basic (1st, 2nd, and 3rd) memorial periods versus the velocity of SMG-induced magma motion.
Length of period (ky) | ≤5 | 8 | 15 | 30 | 60 | 100 | 200 | 1000 | ≥2000 | |||
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Under the gravitation of the Moon | 1st | Half a month for any |
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2nd | 0° | The size of |
≥183 | 114 | 60.8 | 30.4 | 15.2 | 9.13 | 4.56 | 0.91 | ≤0.46 | |
±15° | ≥176 | 110 | 58.8 | 29.4 | 14.7 | 8.81 | 4.41 | 0.88 | ≤0.44 | |||
±30° | ≥158 | 99.0 | 52.7 | 26.3 | 13.2 | 7.90 | 3.95 | 0.79 | ≤0.40 | |||
±45° | ≥129 | 80.7 | 43.0 | 21.5 | 10.8 | 6.45 | 3.23 | 0.65 | ≤0.32 | |||
±60° | ≥91.3 | 57.0 | 30.4 | 15.2 | 7.60 | 4.56 | 2.28 | 0.46 | ≤0.23 | |||
±75° | ≥47.2 | 29.5 | 15.7 | 7.87 | 3.94 | 2.36 | 1.18 | 0.24 | ≤0.12 | |||
3rd | The size of |
≥183 | 114 | 60.8 | 30.4 | 15.2 | 9.13 | 4.56 | 0.91 | ≤0.46 | ||
|
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Under the gravitation of the Sun | 1st | Half a year for any |
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2nd | 0° | The size of |
≥130 | 81.2 | 43.3 | 21.7 | 10.8 | 6.50 | 3.25 | 0.65 | ≤0.33 | |
±15° | ≥125 | 78.4 | 41.8 | 20.9 | 10.5 | 6.27 | 3.14 | 0.63 | ≤0.31 | |||
±30° | ≥113 | 70.3 | 37.5 | 18.8 | 9.37 | 5.62 | 2.81 | 0.56 | ≤0.28 | |||
±45° | ≥91.9 | 57.4 | 30.6 | 15.3 | 7.65 | 4.59 | 2.30 | 0.46 | ≤0.23 | |||
±60° | ≥65.0 | 40.6 | 21.7 | 10.8 | 5.41 | 3.25 | 1.62 | 0.33 | ≤0.16 | |||
±75° | ≥33.6 | 21.0 | 11.2 | 5.60 | 2.80 | 1.68 | 0.84 | 0.17 | ≤0.08 | |||
3rd | The size of |
≥130 | 81.2 | 43.3 | 21.7 | 10.8 | 6.50 | 3.25 | 0.65 | ≤0.33 |
Note: the unit of speed is
To avoid difficulties associated with the Eulerian approach and to try to shed some light on SMG dynamic effects for paleoclimate, the dynamic equations of the SMG were established within the Lagrangian system via local coordinates on a rotating Earth. With a small Coriolis force, the widely used geostrophic balance approximation [
For magma with a lower speed and a large friction,
Here,
The PSMGIM for horizontal motion can be written as follows:
The basic (named 1st, 2nd, and 3rd, resp.) period components included in (
For a period range
Spectrum methods [
The discrete periods and the corresponding square amplitudes were, respectively, as follows:
Theoretically, the integer
(Some publications have not applied DFT correctly. As a result, the amplitude of time series reconstruction is much larger than that of the time series. Here, using a sample size of 8,000, time series reconstructions were almost identical to the time series and provided accurate spectra with a correlation coefficient of 100% and a confidence of 99.99% for all of the presented figures).
Magma is located within the lower mantle (at ~900 to ~2,900 km in depth and has a mass of ~2.94 × 1024 kg and a mass-weighed-mean radii of ~4,580 km) and the outer core (at ~2,900 to ~5,100 km in depth, with a mass of ~1.84 × 1024 kg and a mass-weighed-mean radii of ~3,260 km). Magma became the major fluid left within Earth as Earth cooled and collapsed with time and contributes to Earth’s heat budget via formation and disintegration [
The “buffer” and “quick release” characteristics determined for the PSMGIM (with a smaller amplitude and a longer phase for negative anomaly phases as compared to positive anomaly phases) were derived from the experimental dissipation coefficient (no data available) set from 10−4 to 10−3 s−1 for magma within the Outer Core and enlarged by 100 to 200% for magma within the lower mantle (Figure
SMG drives magma to move nonlinearly because the location of magma relative to the Sun or Moon changes with time. Momentum can be accumulated in moving magma under a rotating Earth on multiple periods that may be much more abundant than the original periods of the SMG, depending on orbital properties, including the following: (1) the revolution angular velocity of the Sun or Moon, (2) the relative velocity and the dissipation of magma motion, and (3) Earth’s radius and rotational velocity (as defined in (
Period ranges (PR,
Squared-amplitude spectra for the total PSMGIM time series (TW2). The experimental dissipation coefficient for magma within the outer core was set at 10−4 (left column) and 10−3 s−1 (right column) and enlarged by 100% (Row 1), 150% (Row 2), and 200% (Row 3) for magma within the lower mantle. The total PSMGIM time series (not plotted) was 100% and was correlated with its reconstruction via the Discrete Fourier Transform (DFT) with a confidence of 99.99% and was evaluated using the following: (1) the maximum size of the positive/negative amplitude of its anomaly (“
What is the connection between the PSMGIM and paleoclimatic variations? Does the connection occurs through slow heat-transport or sporadic heat-eruptions associated with magma convection [
The authors declare that there is no conflict of interests regarding the publication of this paper.
Natural Science Foundation of China (41222037) provided the publication fee. Uncountable supports were provided by Zhiren’s wife (Hong) and their families. Dr. Kimberly Mace provided English language advice.