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Scientific requirements for a rapid polar shift

Motivation

During the Last Glacial Maximum, about 20'000 years ago, arctic East Siberia was not ice covered, while in the west the ice sheet reached the region of New York. This asymmetry suggests that the North Pole was situated somewhere in Greenland. If this had been the case, the North Pole must have moved geographically to its present position in the Arctic Sea at the end of the Pleistocene, about 13'000 years ago. This problem motivated theoretical investigations since the end of the 19^{th} century with negative result. However, a recent study using rough approximations claims that a scenario for a rapid geographic polar shift compatible with laws of nature and observations is possible . An example of a proposed polar shift is shown in Fig. 1.

Fig. 1: The asymmetric ice cover is shown and the path of the North Pole, which starts in Greenland as a spiral, which after several diminishing turns ends in the Arctic Sea. A turn lasts about 400 days. |Caption = Path of the North Pole after a stretching deformation of the globe in an oblique direction. Arbitrarily, a relaxation time of 1000 days for a global deformation has been assumed.

If the Earth were a perfect sphere, an arbitrarily small deformation, e.g. a slight stretching in an oblique direction to the rotation axis would lead to a geographic wandering of the poles around that direction . In space the polar axis would stay fixed. The geographic motion of the poles is produced by a rotation of the globe. This follows from the law of free motion of a rigid body.

Earth's shape

Owing to its rotation Earth's radius at the equator is larger by 21 km than at the poles. The excess mass at the equator stabilizes the rotation around the axis through the poles. Actually, due to perturbations, the poles show a minute, but measurable motion, Chandler's wobble. This is an irregular circular motion with a period of about 400 days. For a significant wandering of the poles, a deformation of the Earth is required, which can compete with the stabilizing exess mass around the equator. For Fig. 1, a stretching of the globe in a direction 30° away from the polar direction was assumed, such that the radius in that direction was increased by 6 km. The following two sections resume the mathematics which leads to Fig.1.

Inertial tensor and geographic movement of the poles

The mass distribution of a solid body enters into the law of motion through the inertial tensor $I$. Its components are determined from the mass density $\rho ({\vec {r}})$ by the integral

$I_{jk}=\int d^{3}r(r_{j}^{2}\delta _{jk}-r_{j}r_{k})$

There exists a cartesian coordinate system fixed to the solid body in which $I$ is diagonal. In this coordinate ystem $I$ is represented by lengths along the three axes. In the case of the unperturbed Earth, two axes have equal length, $a=b$, and they lie in the plane of the equator, while the third, $c$, in the polar direction is longer.

In a coordinate system fixed to the solid, the motion of the angular rotation vector ${\vec {\omega }}(t)$ is determined by Euler's equation , a vector equation, in which the square braket is a cross product:

${\frac {d{\vec {I\omega }}}{dt}}=[{\vec {I\omega }},{\vec {\omega }}]$

When the rotation axis coincides with this third axis, the motion is stable. However, when the instantaneous rotation axis makes a small angle with the third axis, it will precess around this third axis forming a cone. The period of this precession is larger than the rotation period (= 1 day) by the factor $c/(c-a)$. For the Earth this precession period is about 400 days. During the precession the rotation axis moves geographically. A deformation of the Earth modifies its inertial tensor. Since the Earth has its equatorial bulge, a sizable deformation in a oblique direction is required to obtain a notable change of the direction of the large axis. As an example, it has been estimated that a one per mil stretching in a direction 30 degrees from the previous large axis would modify the inertial tensor in such a way that the direction of the new largest axis made an angle of about 17 degrees with the previous large axis.

Deformation and relaxation

A rapid deformation of the globe, compatible with the continuation of life on Earth, could be due to a close passage of a planet sized object. The tidal forces would stretch the Earth in the direction of the object. This situation was considered in , where the authors estimate that a Mars sized mass passing at 20'000 km from Earth's center with a relative speed of about 40 km/s would produce the deformation mentioned above. This estimate is not detailed, but since the action lasts for about 10 minutes only, simple inertial considerations may cover dominant parts.

A deformed rotating globe will at each instant with rotation vector ${\vec {\omega }}(t)$ tend to relax in direction to a shape with the equilibrium bulge in the equator plane perpendicular to ${\vec {\omega }}(t)$. Let the corresponding inertial tensor be $I_{0}[{\vec {\omega }}(t)]$. In assume that the relaxation can be described by a single relaxation time $\tau$. The relaxation is applied directly to the inertial tensor by:

${\frac {dI}{dt}}=-{\frac {I(t)-I_{0}[{\vec {\omega }}(t)]}{\tau }}$

Initially, the bulge at the old equator is still the dominant deformation, while the stretching in an oblique direction moved the largest axis of $I$ away from the initial ${\vec {\omega }}(t)$. Therefore the motion of ${\vec {\omega }}(t)$ begins essentially as a geographic precession around the new major axis of $I$ with a period of about 400 days. If $\tau$ were much smaller than 400 d, the movement would not go far enough. This would be the case for a liquid globe, since the distances of the deformation would be covered by liquid flow in days at most. A sufficiently large $\tau$ may exist, if the streching broke solid structures, which thereafter plastically deformed into the new equilibrium form. The authors of solved the two differential equations numerically to obtain the geographic path of the North Pole ${\vec {\omega }}(t)$. For $\tau =1000$ days, the path is shown in Fig.1. Its result is a shift of the North pole from a position in Greenland, at the center of the observed largest glaciation, to the present position in the Arctic Sea.

Postulated scenario

In the scientific community a rapid polar shift, consistent with the laws of nature and the known facts, is considered impossible.The obvious difficulty with the quoted model is that the Mars sized object does no longer exist. The authors of , undertaken after the disintegration of the Levy-Schumaker comet passing near Jupiter, shows in Fig. 13 of that a Mars sized pile of stones (and therefore probably also a fluid sphere) passing near the Earth would not disintegrate into several fractions. Well, perhaps this might happen to the massive object, if the preheating of its interior brought it close to instability, so that a slight deformation from its spherical form resulted in an expansion of its interior. The assumed fractions have a reduced escape velocity, which limits the evaporation from a heavy object. Therefore the evaporation rate of the fractions is increased sufficiently to make a disappearance of the object within 10'000 years plausible.

Consequences of these requirements

Since the object in its eccentric orbit evaporated, a disk shaped cloud of ions, which move around the Sun, is formed. This reduces the solar irradiation on the Earth and the global temperature, when Earth's orbit is in the cloud. Muller et al have pointed out, that the angle of Earth's orbit with the ecliptic i.e. the inclination, varies with a period of about 100'000 years. This is the dominant period of the Ice Ages over the last 800'000 years. Since this is otherwise difficult to explain, Muller et al. postulated the existence of such a cloud. When Earth's orbit is outside the cloud, the global temperature can be enhanced due to scattered light from the cloud. Such increased temperatures between cold periods have been observed . The cloud itself has dynamics. The evaporation from the object leads to a steady buildup of the clouds density up to the point, when inelastic scatterings between the ions result in a collapse of the cloud. The resulting sequence of gradual reductions of the global temperature followed by rapid increases is identified with the Dansgaard-Oeschger temperature variations . The inclusions in the ice of Greenland and of Antarctis are vastly denser during cold periods, i.e. when Earth's orbit is in the cloud around the Sun. This refers to small grains with diameters less than 5 microns. They have a consistent size distribution and may have formed in Earth's atmosphere . The larger grains have been carefully examined and found to be terrestrial. In the Holocene, the fractured objects evaporate or even dispel by eruptions more material, however, in an orbital plane, which makes an angle of several degrees with the Ecliptic. Apart from sporadic effects on the Earth, the mean influence of the martial is therefore small.

Conclusion

The old request for a rapid geographic polar shift from the center of the Ice Age glaciations to the present position may not be impossible. However, the conditions that this imposes allow for a narrow possibility only. If the crude estimates turn out to be optimistic, this narrow window may close. Improved simulations are necessary to settle this question. In the scientific community the model has not been discussed. The model postulates an object, which no more exists, and of a type, which in the whole lifespan of the Earth was probably unique, considering that an actual collision with the Earth would have stopped life. This singular feature makes the theory unattractive. On the other hand, the necessary requirements lead to a situation before the pole shift, which produces the 100 kyr period and the violent Dansgaard-Oeschger temperature changes of the Ice Era of the Pleistocene.

References

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