Initial Attempt to Establish a Mathematical Model of the Effect of Global Warming on the Lithosphere

Cheng Q. and Bonham-Carter G., 2005, GIS and Spatial Analysis, Proc. of IAMG 2005, Publ. by Intern. Assoc. for Mathematical Geology pp. 1216-1221
ISBN: 0-97034220-1-7

This is an updated, supplemental version of the article that was submitted for consideration in Oslo in 2008 (33IGC; E1L01105L).

Effect of Plio-Quaternary glaciation on inertial characteristics of isostatically / eustatically affected parts of the lithosphere: the effect of global warming on the lithosphere

N. Parubets (1), Y. A. Tsekhanov (2), P. Parubets (3)

(1) Granton Institute of Technology, 263 Adelaide Street West, Toronto, M5H1Y3 Canada,
(2) Voronezh State University, 84 October Street, Voronezh, 394006, Russia,
(3) Dept.of Mech. Engineering, University of Waterloo, Waterloo, N2L3G1 Canada,


The mid-ocean ridge displacements (MORD) are perhaps the most remarkable formations on the ocean floor. Currently, the only explanation for the their existence is Euler’s theorem – a mathematical abstract, which does not satisfactorily reveal the underlying physical forces that could cause these displacements. The presented model links the origin of the MORD with the enormous mass redistribution of water among the tectonic plates during the Plio-Quaternary glaciations (P-QG). In the least, this paper presents possible examples of powerful physical forces, which are required to create these geophysical phenomena.


The analysis of the effect of P-QG on the lithosphere is derived from the critique of the origin of the MORD. The ridge displacements are unique geological formations that cover the entire ocean floor and can be measured in hundreds, or in some cases, thousands of kilometres; the centres of the ridges originated in recent geological time. The crucial question is: were the ridges’ centers (see Figure 1 and Figure 3) created in that strange, displaced manner (Figure 1A) with such great distances between them, or were they broken and displaced after their formation (Figure 1B to 1A)?

The Euler theorem’s geometry is the only cause cited for the origin of these displacements as they appear at the present time. Yet, as a matter of fact, this theorem only points to some abstract mathematical pole, when some point or points of a sphere (in the case of the applicability of the theorem to geophysics, a spherical shell) are moving on the surface of the sphere (Euler, 1755; Smirnov, 1964). What is most important, however, is that the theorem does not and could not reveal the forces that are moving these shells. Moreover, if Euler’s theorem is applied to all the actual ridge displacements on the entire ocean floor, the theorem reveals an unprecedented large number of the different “poles of rotation,” which should coexist somehow in the same geological time (See Figure 3).

Hence, if Euler’s theorem does not sufficiently explain the origin of these displacements, there are currently no other interpretation as to how these displacements occurred after their formation.

Fig. 1 – Comparison of the same area of the oceanic floor in the supposed Pliocene and Recent.

Since the age of the center of the ridges is Plio-Quaternary, the only way to discover the cause of these catastrophic displacements, is to examine the stratigraphical records of that period of time. Only two significant events in the Plio-Quaternary – both well documented – can be identified from the available stratigraphical data: the major glaciation that enveloped the North and South hemispheres and the orogenic processes on the continents at this time (see Figure 2). It is reasonable to assume that at least one of them and perhaps both are very causes of the ridge displacements.

Despite the fact that the first trace of Cenozoic glaciation appeared in the middle Eocene, only in the Pliocene did significant ice accumulation begin, with the reaching of the Glacial Maximum (GM) and then the Last Glacial Maximum (LGM) at the Pleistocene with a total ice sheet volume of up to 110 x 106 km3 (see Table 1). This means that, since the beginning of the glaciation, the Pacific tectonic plate, for example, was relieved of approximately 16 x 1015 kg of water pressure. The African plate was relieved of 4.8 x 1015 kg, and a total of up to 90 x 1015 kg of water had been relocated from the ocean onto the Eurasian, North American, and Antarctic tectonic plates during GM in the form of ice sheets. Such an enormous load redistribution among the tectonic plates had to bring correspondingly enormous isostatical/eustatical readjustments of different parts of the planet’s crust. Presented below are the mechanisms of the horizontal dislocation of some parts of the lithosphere due to glacio-isostasy/eustasy at the Plio-Quaternary and also projections of it in recent times.

Fig. 2 - Probable global model of Cenozoic Glaciation

Fig. 2 illustrates the most probable global model of Cenozoic glaciation, based on summarized and mean data selected from well-known and accepted ice sheets models and works in the field: B.G. Anderson and H. W. Borns Jr. (1994), P.J. Barrett (1989), J.C. Behrendt and A. Cooper (1991), D.I. Benn and D.J.A. Evans (1998), C.R. Bentley (1997), M.J. Bentley (1999), J.E. Brotchie and R. Silwester (1969), P.E. Calkin and G. Young (2002), C.H. Denton and T.J. Hughes (1981), R.L. Edwards (1995), J. Ehlers (1996), R.H. Fillon (1972), P-L Forsström et al. (2003), L.A. Frakes et al. (1992), S. Funder (1989), B.U. Haq and F.W.B. Van Eysinga (1998), Y. Herman et al. (1989), T.J. Hughes (1985 and 1998), P. Huybrechts (1990), P. Huybrechts et al. (2003), W.O. Kupsch (1967), E. Le Meur and P. Huybrechts (2003), C.H. Lear et al. (2000), J. Menzies and T.J. Hughes (2002), K.G. Miller et al. (1991), K. Moran et al. (2006), M. Nakada and K. Lambeck (1988), A. Nesje and S.O. Dahl (2000), W.R. Peltier (1988), L. Polyak et al. (2001), M.C. Serreze (2007), N.J. Shackleton (1987), M.J Siegert (2001), A.M Tushingham and W.R Peltier (1991), P. Wu and W.R. Peltier (1983 and 1984), J.C., Zachos et al. (1993 and 1996), C. Zweck and P. Huybrechts (2003).

Table 1: Volume of ice in the ice sheets and water in the ocean during Plio-Quarternary glaciation .

Table 1 was created using the same data as in Figure 2.

Click here for larger version
Fig. 3 – The Modification of the Geological Map of the Ocean Floor indicating Euler Poles

Fig 3. illustrates the Modification of the Geological Map of the Ocean Floor (2008). The Green squares with arrows indicate Euler poles for different sections of the MORD. The existence of a great number of “poles of rotation” hampers efforts to apply the Euler theorem concept to reveal the physical forces leading to the formation of ridge displacements.

Proposed model of oceanic floor displacements resulting from the P-QG.

A) During the development of the presented model, we proceed that the thickness ratio of the combined lithosphere and crust, hL, to the radius of the planet R, is approximately 150km/6380km or 0.024. In mechanics of thin-walled membranes, this shell is modeled such that forces can only act on it in tension or compression. Bending loads are ignored. We also applied the results of well known geophysical models as well as postulates of isostasy and rheology, which directly or/and indirectly demonstrates three categories of features of the lithosphere and asthenosphere:

■ The deformation and/or depression of some parts of the crust from 400 to 800m resulting from imposed loading during the LGM (Mörner, 1980; Svendsen and Mangerud, 1987; Kupsch, 1967; Daly, 1934). Comparing the extent of glaciation/loading in the LGM and GM, we can extrapolate that the crust could have depressed by up to one km during the GM. This depth could be considered to be even greater if the rebounding processes of adjacent areas are taken into account. This assumption is indirectly supported by data indicating that present-day glacial isostatic adjustments reach 7mm/year in some areas (Khan et al., 2007) or even more, up to 10mm/year (Sella et al., 2007). It appears that since GM, some areas of the planet have depressed/rebounded up to 1300m magnitude.

■ The balance between the loaded parts of the crust and their surroundings: this balance occurs, not in the uniform, thin, and elastic lithosphere, but in the viscously fluid asthenosphere (Ehlers, 1996; Officer et al., 1988). This means that, not only the crust under the superimposed load sank, but also parts of the lithosphere under the crust sank into the asthenosphere, probably to same degree.

■ A slow response to the lithospheric movements on the isostasy/eustacy, loading/reloading of the crust and lithosphere (Mörner, 1980; Walcott, 1970; Brotchie and Silvester, 1969; Houghton et al., 2001).

B) We also proceed with the following generalizations: the asthenosphere is initially considered to be a sealed spherical shell, which is filled by a viscous fluid. It is then possible to assume that the pressure, as result of 110 x 106 km3 of ice accumulation in the polar regions during P-QG, must have led to an equal increase in the hydrostatic pressure inside the fluid asthenosphere by ∆q, as defined below:

∆q = ρ1 g h ≈ 10MPa (1)

where: ρ1-density of ice = 103kg/m3
g – gravitational constant = 9.81m/s2
h – mean thickness of ice accumulation ≈ 1km

As a result of the polar loading/depression, unloaded areas (especially in equatorial regions) are subject to additional tensions inside the lithosphere. As the asthenosphere is not an absolutely sealed shell, part of the additional hydrostatical pressure build-up would be reduced as pressure is released through the plume’s veins in the lithosphere. However, even this smaller increased pressure inside the asthenosphere must result in additional tension stress in the crust and lithosphere:

∆σ = (∆q/2) R/ hL = (10MPa /2) 6380km/150km = 212MPa (2)

At much thinner parts of crust and lithosphere, say 15km thick, the stress could be as great as 2120MPa.. In the thinnest parts of the lithosphere near the ridges, where it is only 1.5km thick, the stress could be even greater, perhaps 2x104 MPa.

If only a small part, say 10 percent, of the increasing hydrostatical pressure inside the asthenosphere ∆q (1) does not escape through the lithosphere and crust, the stress build-up could be estimated to be 21.2MPa. However, the imposed stress still exceeds the tensile strength of basalt (≈14.5MPa) and greatly exceeds the basalt’s cohesive tensile strength by several orders of magnitude.

Thus, the max ice load at GM and LGM could have led to the rupture of the thinnest parts of the lithosphere. In mechanics, the failure mode for materials which accumulate tension and compressive loads propagate perpendicularly to the maximum tension loads. If this is so, it is clear why the fracture zones are mostly perpendicular to the MOR. Once they appear, the fracture zones remain the weakest parts of the lithosphere and crust and thus are more susceptible to horizontal displacements of the lithosphere due to its vertical movement as described in the following.

C) As previously mentioned, it is legitimate to assume that growing ice sheets during the P-QG depressed some parts of the crust and lithosphere underneath by a depth of up to 1,300m. The geophysical meaning of this is that some parts of the lithosphere were slowly moved toward the planet’s centre, by up to that distance. This decreasing distance to the planet’s centre led to the subsequent increase in the kinetic energy of those ice loaded parts of the lithosphere.

Moment of inertia of this mass: I = mr2 (6)
Moment of rotational kinetic energy L = Iω = mωr2 (7)
The mass radius of rotation r = Rcosφ(8)
where R is the planet’s radius
cosφ is the angle from the equator.

During time dt, mass m moved towards the planet’s center by dr: From equation 8, we have dr = dRcosφ and it follows that: dL =2mωcos2φRdr (10).

The application of an external impulse over time, Mdt, is required, so we have: Mdt = 2mωcos2φRdR/dt (12).

dR/dt is the vertical velocity of the ice-loaded mass. Vn and therefore, M =2mωcos2φRVn (14).

Mechanically, that moment is created by applying tangent force Ft, where M=Ftr (15). Substituting (15) into (14), it is evident that the tangential force, Ft, is dependant on the vertical velocity by: Ft = 2mωcosφVn (16).

For each lithosphere mass unit, dm = dxdyh ρ, acts force dFt = 2ρdxdyhωcosφVn.

Substituting the appropriate constants, we get the total dynamic tension on dM is 7x10-5 Pa only. Thus, the dynamic tension in the lithosphere and crust due to the lithosphere’s vertical velocity is insignificant. The vertical shift of square kilometer of lithosphere would exceed 1.19x1010 kJ in one year. Part of this energy had been transformed into heat, however, part of it into potential energy, which builds up over time. Even if only a fraction of that energy is transformed into elastic strain in the lithosphere and crust, then since GM, energy of up to 1.5x1011 MJ per km2 of lithosphere was directed towards horizontal displacements in some areas due to their vertical movements. This accumulation of energy since GM is critical and is perhaps enough to exceed the lithosphere’s elastic flexure limit and the shear resistance of the surrounding rocks. The escalating tearing process of the lithosphere should have occurred mainly along the already existing, weak oceanic fracture zones as indicated in Section B. The combination of the hydrostatic-induced stresses discussed in Section A and the inertial-induced stresses discussed in Section B constitute the base for the presented model.

D) In the case of Antarctica (Parubets, et al., 2005) the moment of inertia (I) for a spherical semi-shell is

Where: m is a mass of the portion of the spherical shell; R is the planet’s radius; r is the radius of the “simplified Antarctica” (2,600 km); Θ1 is the angle between the equatorial plane and the edge of Antarctica; Θ2 is the angle between the equatorial plane and the axis of rotation. If the portion of the crust is 5,200 km in diameter, located at the site of Antarctica, and assumed to be the Antarctica tectonic plate, the moment of its inertia is

The volume and the mass of this part of the crust are 743 x 106 km3 and 0.208 x 1022 kg respectively; the mass of the portion of the lithosphere under it is: 1.0494 x 1022 kg.

Hence, the total mass of this portion of the lithosphere is 1.2574 x 1022 kg, and the moment of the inertia of the “simplified” Antarctica tectonic plate is I = 0.07 x 1.2574 x 1022 R2 kg.m2 = 88 x 1019 R2 kgm2. In a case of Antarctica’s depression during GM down to 400 m, the moment of inertia is: Id-400 = 88 x 1019 x (R – 400)2 kgm2 = 3.5839 x 1034 kgm2. In the case of Antarctica’s depression down to 1,300m the moment of inertia is: Id-1,300 = 88 x 1019 x (R – 1,300)2 kgm2 = 3.5826x1034 kgm2.

It is important to note that the greatest contribution to this decrease in rotation inertia is attributed to the isostatic inward bending and depression of the crust furthest from the axis of rotation. Isostatically depressed regions near the axis have no effect on this decrease.

As a result, Antarctica’s rotational velocity should increase if conservation of rotational kinetic energy is observed (LAnt = ½Iω2). However, the shear force imposed by the surrounding crust and lithosphere (approximately 16,300km in length, 35km thick crust, and 150km thick lithosphere) prevent this increase in velocity, which results in a substantial energy release. This energy release could be calculated from the change in rotational kinetic energy due to a change in rotational inertia without the corresponding change in rotational velocity. Accordingly, the differences in rotational kinetic energy before and after depression are: LAnt b/d – LAnt dif – 400 = ½ω2 (IA b/d – I dif – 400) – in a case of depression down to 400m, and LAnt b/d – LAnt dif-1,300 =½ ω2 (IA b/d – Idif – 1,300) – in a case of depression down to 1,300m, where ω is a planet’s angular velocity (ω = 7.2722 x 105 rad/sec). Hence, an energy of 2 x 1016 MJ had been released while Antarctica had depressed down to 400m, and released energy up to 5 x 1016 MJ while Antarctica had depressed down to 1,300m.

As a result, energy up to 16,500MJ per m2 had been extinguished, and the Antarctica tectonic plate should have been turned clockwise (from the perspective of the South Pole).

In the case of similar but eustatical movements of the lithosphere near the equator during Plio-Quaternary glaciation/deglaciation, a significant energy release of up to 351.7 x 108 MJ per square km of oceanic crust should have occurred.

During the interglacial period, including the recent one, the process of glacial isostasy still continues and Antarctica should have a tendency to move the opposite way, counter-clockwise (from the perspective of the South Pole). It seems that such “back and forth” movement could be the hypothetical main cause or at least a trigger factor for ridge displacements.

In reality, these simplified crust movements are greatly affected as well by the processes of denudation and sedimentation of the continents into the ocean; by volcanic activities and landslides; by still unknown movements of the fluid asthenosphere; and by changes in the salinity and temperature of the ocean water.


Assuming that Plio-Quaternary glaciation is a main factor in ridge displacements that have been observed, and taking into account that 33 x 106 km3 of ice still accumulated in the remaining ice sheets, and that global warming continues to accelerate, it is appropriate to develop the following conclusions and even warnings.

According to the Global Warming Greenpeace Report (J. Leggett, 1990), in the recent interglacial period, the global average temperature changed at the rate of one degree Celsius every 500 years. However, the change in the registered temperature since it was established and the interpolation of the temperature through the use of the Global Circulation Model’s Predictions and the IPCC predictions for the 21st century (Houghton et al., 1990; Houghton et al., 2001) show a rate of global temperature change during the 21st century of one degree Celsius every 30 or even 20 years. This number reflects a rise in the average global temperature at the present time that is up to 25 times faster than it was 10,000 years ago, during the same, recent interglacial period.

What is most significant is that, in the near future, the rise of the global temperature could be even more rapid – up to 100 times faster. The recent models confirm such predictions (Stewart and Wiener, 2003; Drake, 2001; Houghton et al., 2001).

Some authors, however, have argued that the recent warming is simply a continuation of the last interglacial stage, and reflects the cyclical nature of the glacial-interglacial stages. Their assertion is that a new Ice Age will replace the present warming stage (Cracknell, 2001; Benn and Evans, 1998). It is very important to note that the suggestion to attach importance to the cyclical nature of glaciation/global climatic changes does not have any convincing confirmation in the geological records. Thus, the last known glaciation prior to the Cenozoic one ceased at Chaughsingian, and then, for 215 million years, there was an absence of permanent glaciation anywhere on the greenhouse planet during the Mesozoic and early Cenozoic periods (single asterisk on Fig. 4). Of course, some oscillational, Milankovich-like, effects did affect, to some degree the planet’s average temperature. But some other, more powerful processes led to the beginning and ending of the Palaeozoic as well Cenozoic glaciations indeed.

It does not seem wise to adhere to the idea that some weak process of oscillation in the temperature has predominated in the formation of the global climate throughout the history of the planet, including the present time. It is very likely that the planet’s last glaciation, which started some 45 million years ago and which had started to lose most of its ice sheets 20,000 years ago, is now at the stage that it is ready to lose the last 34 x 106 km3 of the total ice accumulation on the planet.

Many of the latest publications are pointing to the fact that the cause of the recent rise in global temperature is human activity during the past 20,000 years and especially during the recent age of industrial development. These assumptions supports the idea that weak oscillational processes do not form the planet’s climate at the present time. Instead, the climate is probably formed by constantly growing processes that could finally renew the early Cenozoic/Mesozoic-like greenhouse climate on the planet.

Fig. 4 – The most probable global model of Phanerozoic glaciation**, based on summarized and mean data taken from selected well-known and generally accepted ice sheet models and works in this field completed by: Bentley (1999), Calkin and Young (2002), Frakes et al. (1992), Hughes (1998), Moran et al. (2006), Polyak et al. (2001), Visser (1993), Zachos et al. (1993).
* Existing only in short periods in the Mesozoic, periglacial setting with seasonal winter ice but not one of permanent glaciers.
** The alleged late Ordovician–early Silurian, and late Devonian–early Carboniferous glaciations are not well corroborated and lack sufficient stratigraphic determination and thus are not included in the graph.

At the present time, the planet’s ice sheets and glaciers still contain up to 34 x 106 km3 of ice, which could add an additional 31 x 1012 tons of water to the ocean or 8 x 107 kg of additional pressure per km2 of ocean floor if it melted. This continuing load shift could lead to further isostatic readjustments of all tectonic plates, with the most significant effects along fracture zones and plate boundaries. If human civilization does not reduce its contribution to the process of climatic warming, and if that contribution is the main cause of the rise in temperature, the acceleration of the rise in temperature will sooner or later inevitably also accelerate the glacio-isostatic/eustatic processes, including perhaps the intensification of the oceanic ridge displacements. This movement in turn could amplify the orogeny processes to the level of the rapid formation of new thrusts and fault-block areas and perhaps the building of new mountains. The combination of all these devastating processes, including climatological processes, could push the planet to the very brink of the Holocene and to the onset of a new Postholocene epoch.


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