The emergence of Earth’s glaciation epochs is an oscillatory process as the periods of significant cooling-down are followed by appreciable warm-ups (interglacial stadials) which, in turn, are followed by cooling-down periods. For this reason it is practically impossible to explain this oscillatory process by smooth climatic changes (such as declinein the atmospheric pressure due to bacterial activity). In this case, it is important to consider to the behavior of the revolving Earth precession, which emerges in association with deflections of the Earth’s mass distribution from the spherical symmetry. Such deflections from symmetry are caused first of all by the Earth’s crust nonuniformity in its continental and oceanic segments (i.e., the position of continents and oceans on the Earth’s surface); the other reason is the potential density nonuniformity of mantle [1].

The average period of the Earth’s axis to go over the total precession circle is presently τ ≈ 25.7 - 26 thousand years [2]. The precession motions are superposed by smaller short-period nutation (“nodding”) fluctuations. They are perceived as the pole motions apparently caused by the tidal disturbances, movements of the internal Earth’s core and by the Earth-Moon system revolution around the common barycenter. A result is rather complex pattern of the Earth’s axis rotation [3-5].

The shape of Earth is very close to that of the revolution ellipsoid of a liquid body with the equatorial inertial swelling. The Earth’s equatorial radius (R_e = 6378.2 km) is greater than the polar radius R_p = 6356.8 km by 21.4 km, which corresponds to the compression

. That is the reason why a greater excess mass is concentrated at the equator and can have the gravity interaction with the other celestial bodies. These interactions tend to turn Earth so that her equatorial plane would coincide with the rotation plane of the disturbing body. Still, only the gravity pull by Moon and Sun play the main role in decreasing the precession angle (see Figure 1).

The gravitational pull of the Moon and Sun acts simultaneously on both sides of the Earth’s equatorial swelling tending to turn its revolution axis in the opposite directions. But the gravity action on the swelling side facing the Moon or Sun is slightly greater than on the opposite side (see Figure 1).

To determine the mass of the Earth’s equatorial swelling, it is necessary to determine its volume. The volume of the equatorial swelling is equal to the difference between the Earth’s revolution ellipsoid volume and the volume of a sphere inscribed in it:

Then the mass of the equatorial swelling is equal to:

where V_swell (≈2.8 g/cm³) is the average density of the swelling accounting for the oceanic water layer about 3 - 4 km deep and the underlying layer of the oceanic crust and the upper mantle (about 17 km). One half of this equatorial swelling is facing Moon, whereas the other half is on the opposite side of Earth (see Figure 1). Thus, the effective mass of each half is about 2 times smaller, just m_swell/2 ≈ 1 × 10²⁴ g. Then, the difference of the Lunar gravity forces acting on these halves is:

where γ (=6.67 × 10⁻⁸ сm³/g∙s²) is the gravitational constant; m_L (=7.35 × 10²⁵ g) is the Lunar mass; L_L (=3.844 × 10¹⁰ сm) is the distance between the Earth’s and Moon’s centers of mass; R (=6.371 × 10⁸ сm) is the average radius of Earth; Ψ is the precession angle (present value of Ψ = 23.44˚); λ (≈5˚) is the angle between Moon-aroundEarth rotation plane and the plane of the ecliptics.

Thus, ΔP_L ≈ 1.3725 × 10²¹ сm∙g/s².

Similarly, the difference in the Sun gravitational forces acting on the equatorial swellings is:

where (=1.99 × 10³³ g) is the mass of Sun; (≈ 1.496 × 10¹³ сm) is the distance between the Sun and Earth mass centers. Then, ≈ 0.6088 × 10²¹ cm∙g/s².

The lunar and solar gravitational forces must be applied to the center of mass of each half of the equatorial swelling. Each half is similar to a convex dome with the center of gravity from the center of the Earth at a distance of about 2/3 Earth’s radius (Figure 2). Therefore,

The difference of the momentums of force applied from the lunar side to the Earth’s equatorial swelling (see Figure 2) is equal to M_L = P_L∙h = 3.71 × 10²⁹ g∙сm²/s². From the solar side, ΔM_S = ΔP_S∙h = 1.72 × 10²⁹ g∙сm²/s². The combined effect gravitational pull of the Moon and Sun is Δ(M_L + M_S) = 5.43 × 10²⁹ g∙сm²/s².

Besides the effect of Moon and Sun, the Earth’s precession is influenced by the asymmetry of continental positions on the globe’s surface. The combined mass of the continents m_cont is equal to approximately 2.25 × 10²⁵ g [6], the average thickness of the continental crust is H_cont = 40 km = 4 × 10⁶ сm and their average elevation (stand) above sea level is Δh = 875 m = 8.75 × 10⁴ cm. The effect of continents on the Earth’s asymmetry is due to their centers of mass being positioned slightly above the center of mass of the mantle matter displaced by the continents. Under the continents’ isostatic equilibrium the mass of the displaced mantle m_mant is equal to the mass of the continents m_cont = 2.25 × 10²⁵ g. Judging from the elevation of continent, the continents’ center of mass is above the center of mass of the displaced mantle by h ≈ 500 - 600 m. Thus, the continental crust excess mass is Δm_cont ≈ m_cont∙h/H_cont ≈ 3.38 × 10²³ g. With the centrifugal acceleration g_ctrf = Ω²∙R∙cosφ, where Ω = 7.27 × 10⁻⁵ rad/s is the Earth’s revolution angular velocity and is the latitude of the continents’ center of mass. At φ = 30˚, g_ctrf = 2.9 сm/s² and ΔP_cont = Δm_cont∙g_ctrf ≈ 9.8 × 10²³ g∙сm/s². Depending on the distance between the center of mass of the entire continental ensemble and the Earth’s center, the value ΔM_cont is between 10²⁹ and 10³⁰ - 10³¹ g∙сm²/s². The mantle is non-uniform, especially with respect to the positions of the lighter ascending mantle flows and the heavier descending ones. Besides, the nonuniformity is added by the core surface topography. The result is that the combined effect of all these factors is not clear. An indirect estimate, based on a comparison of the theoretical temperature climate fluctuations in Pleistocene with the isotopic temperatures of the Antarctic ice cover (see Figure 3), shows that the present-day ΔM_cont+m is equal approximately to (0.2 tо 0.6) × 10²⁹ сm²∙g/s².

Using the theory of free gyroscopes the authors determined the average rate of the revolution of Earth’s axis rotation around the intersection line of the equatorial plane with the lunar orbit plane around Earth and the Earth orbit plane around Sun:

where I (=8.04 × 10⁴⁴ g∙сm²) is the Earth’s moment of inertia; Ω = 7.27 × 10⁻⁵ rad/s) is the angular velocity of the Earth’s revolution. The present-day Earth’s rotation rate, with the precession angle of 23.44˚, is ω ≈ 4.83 × 10⁻¹² rad/s or 1.53 × 10⁻⁴ rad/year (rotation by 1˚ takes approximately 716.5 years). The Earth rotation from the present precession angle of 23.44˚ to the lunar orbit inclination angle to the ecliptics occurs asymptotically and gradually approaches 5.1˚. This takes millions of years. Thus, the function of Earth’s rotation rate vs. the precession angle is quite nonlinear. The Earth rotation time may be found from the following equation:

Inasmuch as the ω = dψ/dt, the precession angle correlation vs. time is (Figure 4)

At the equilibrium of Lunar-Solar gravity pull on the Earth equatorial swelling with the effect of the continents and mantle,

ω = 0, and the precession angle acquires its equilibrium value at t → ∞. For the present-day positions of the continents, ψ_∞ ≈ 2.5˚ (Figure 4). Currently, the precession angle declines at a rate of about 6 × 10⁻⁴ deg/year.

On assuming that the Earth’s albedo А = 0.3, the value of Earth’s effective temperature Т_е = 263.6 К. Then, the tropospheric temperature (including the surface temperature) can be expressed as planet’s effective temperature:

According to the empirical calculations, the presentday average near-surface Earth temperature at р = р₀ = 1 atm and ψ = 23.44˚ is approximately equal to: T_s ≈ 288 Кor +15˚ Сassuming T_s = 288.2 К. The proportionality coefficient at ψ = 23.44˚, Т_е = 263.6 К and T_s = 288.2 K, and at ψ = 0˚, Т_е = 255 К and T_s = 278.6 К. Then, within a wide range of the precession angles for the Earth, b^α = 1.093.

Using Equation (9), one can establish the correlation of the near-surface temperature vs. time (Figure 5).

As the plane of the Earth’s equator approaches the plane of the lunar orbit around Earth and the ecliptics, the Moon’s and Sun’s external influence on the Earth’s equatorial swelling substantially decreases. According to Equation (9), a noticeable cooling of climate occurs as the result. As soon as the average near-surface temperature reaches some critical level, glaciations begin on the highlatitude continents (for the northern region, the critical value of the Earth average temperature will to be close to 9˚C - 10˚C).

Indeed, the emergence of ice sheets and growth in the polar areas unavoidably disrupts the Earth’s revolution equilibrium and leads to a renewed rapid increase of the

precession angle. To determine the direction of the action of ice sheets on the Earth’s revolution regime, the glaciations in both Northern and Southern hemispheres must be accounted for (Figure 6).

However, the Antarctic ice mass increment was limited by the finite size of the underlying continent and mostly occurred in the nearshore areas and in the West Antarctic where humid cyclone penetration was common. The Eastern Antarctic ice dome has a high stand (up to 4 km). For this reason, it is dominated by anticyclones and the snow mass increment in its central areas is mostly due to the hoar-frost precipitation out of a relatively dry air. This increment is almost totally compensated by the plastic flow of ice from the central areas to the shores. Besides, Antarctic (the main ice accumulator in the southern hemisphere) is almost symmetrical relative to the South Pole and does not cause substantial disruptions in the Earth’s axial symmetry.

The main ice mass in the Northern Hemisphere was accumulated over the Canadian Shield. Thus, two force momentums, directed against each other must have been acting on the body of Earth in Late Pleistocene. One was

caused by the Canadian glaciation and the other, by the West Antarctic glaciation as shown in Figure 7. The momentum of force from the Northern Hemisphere clearly prevailed.

In estimating the effect of ice sheet on the Earth’s precession angle, the glacier masses positioned symmetrically with respect to the poles may be disregarded as they mutually balance each other and do not have any significant effect on the Earth’s revolution regime.

According to our estimate, the excess mass of the ice sheets (taking the southern glaciers into account) in Late Pleistocene was located at about 70˚N over Canada and was approximately 2 × 10²² g. The centrifugal acceleration at 70˚ latitude is equal to about g_strf ≈ 1.15 сm/s². In this case, ΔP_glacier ≈ 2.3 × 10²² g·сm/s², and the distance between center of gravity of the excess mass and the North Pole was close to h_glacier ≈ 2.2 × 10⁸ cm or 2200 km. Then the additional momentum of force attached to Earth is equal to ΔМ = ΔР_glacier·h ≈ 5.1 × 10³⁰ g∙сm²/s². Then, using Equation (7), ω ≈ 7.9 × 10⁻¹¹ rad/s and the characteristic warming time τ ≈ 2500 years. This, of course, is a very approximate estimate, but it provides the order of a characteristic warming and glaciers degrading time of about a few thousand years, which was actually observed. Quoting Kotlyakov [7], “the disintegration of a giant Pleistocene glaciation in the Northern Hemisphere occurred very rapidly, just over a few thousand years”.

To calculate temperature climate change, it is necessary to estimate how it is affected by the Earth’s orbit precession during its rotation around the Sun (Milankovitch cycles). Main harmonics of the Milankovitch cycles have periods of about 41,000 and 23,000 years [8]. Their effect causes temperature changes on the order of ±2˚C to ±3˚C. The average Late Pleistocene temperature considering such fluctuations and in comparison with the isotopic temperature of the Antarctic ice at the Vostok Station is shown in Figure 6.

It should be noted here that the isotope temperatures within the Antarctic ice describe not the ice cover formation temperature but the average World ocean surface water temperature; their evaporation preserved the deuterium/hydrogen isotope ratios typical of these waters. After having been precipitated in the Antarctic these ratios were remembered by the corresponding layers of the ice cover.

As shown in Figure 6, correlation of the theoretical temperature curves with the experimental data is quite good although the theoretical curves are somewhat smoothed.

Cycle and the independent geologic data on the ice sheet distribution in Canada and the US as presented in 1988 by Imbry and Imbry (Figure 3).

Both curves are similar, with slight deviations in ages and scales. Our curve is tied with the age scale of the Antarctic temperature fluctuations. It is also possible that some glacier traces (end moraines) of somewhat earlier glaciation phases (for instance, the phases about 70,000 years ago) were obliterated by the last-phase glaciation, which substantially overlaid the area of earlier glacier phases.

The shape of the theoretical curve depends substantially on 1) the phase relationships between the main cycles of the Lunar-Earth and “glacier” temperature fluctuitions; 2) precession cycles of the Earth-around-Sun revolution orbit; and 3) precession of the Earth own revolution. Based on these, the writers selected the best fit between the theory and experimental isotope temperature determinations in the Antarctic ice sheet from the phase shift of the component climatic fluctuations. Thus, the future climate changes can be forecast (as an example of such forecast see Figure 8).

The Pleistocene/Holocene boundary (we are living in Holocene) is usually drawn at the boundary between the latest and most significant phase of the Wurm (Valday) glaciation and the present-day interglacial stadial. Based on the data in Figure 8 (tied-in with the age determinations of temperature fluctuations in the Antarctic ice cover at the Vostok station) the age of this boundary is

approximately 12,000 - 11,000 years. The later values are close to the value of 10,000 years determined by the INKVA Holocene commission [9].

On the other hand, the warm climate during the second half of Mesozoic was due to the formation at that time of the supercontinent Pangaea and to the accelerated oxygen generation caused by the explosion of flowering plants, which temporarily compensated the decline in the nitrogen partial pressure. After Pangaea began to break down and oxygen pressure stabilized, a new cooling phase started in Cenozoic despite a slight increase in the solar luminosity. The authors estimate that the average surface temperature at the end Mesozoic reached +18˚C to +19˚C (the present-day value is +15˚C).

As a result of the Moon-Earth interaction, slow but regular climate cooling episodes periodically occurred in Pleistocene. Every one reached 8˚C to 10˚C and lasted 100 to 120 thousand years. After the emergence of thick ice sheets there was a rapid, within just a few thousand years, climate warming by the same 8˚C - 10˚C, with equally rapid degradation of glaciers. Therefore, the Moon-Earth connections in combination with the Earth’s glaciations initiated substantially nonlinear self-oscillatory climatic processes so typical of the entire Late Pleistocene. The future climatic forecast is cooling, probably, the most severe of all preceding cooling episodes. According to Sorokhtin et al., 2010, cooling occurs due to life activity of the nitrogen-consuming bacteria, which continuously lowers the partial pressure of nitrogen and subsequently the general atmospheric pressure. The atmospheric pressure decline leads to the Earth’s climate cooling (see Equation (9)).