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GEODYNAMICS & TECTONOPHYSICS
PUBLISHED BY THE INSTITUTE OF THE EARTH’S CRUST SIBERIAN BRANCH OF RUSSIAN ACADEMY OF SCIENCES
2015 VOLUME 6 ISSUE 3 PAGES 387-408
ISSN 2078-502X
http://dx.doi.org/10.5800/GT-2015-6-3-0187
Genetic sources and tectonophysical regularities of
DIVISIBILITY OF THE LITHOSPHERE INTO BLOCKS OF VARIOUS RANKS AT DIFFERENT STAGES OF ITS FORMATION: TECTONOPHYSICAL ANALYSIS
S. I. Sherman
Institute of the Earth's Crust, Siberian Branch of RAS, Irkutsk, Russia
Abstract: The paper presents the first tectonophysical reconstruction of initial divisibility of the protolithosphere as a result of convection in the cooling primitive mantle. Initial division of the protolithosphere into separate masses, i.e. prototypes of the blocks, and their size are predetermined by the emerging Rayleigh-Benard convection cells. In studies of geology and geodynamics, the Rayleigh-Benard convection cells were first referred to as a factor to explain the formation of initial continental cores. Considering the Rayleigh-Benard cells and their structural relics can help clarify initial divisibility of the protolithosphere and the origin of the major lithospheric plates, i.e. prototypes of continents. In our opinion, the initial mega-scale block structure of the protolithosphere and the emerging lithosphere were predetermined by the Rayleigh-Benard cells as they were preserved in the emerging lithosphere and their lower boundaries corresponded to the core-mantle boundary, i.e. one of the major discontinuities of the planet. Our theoretical estimations are in good agreement with the number and sizes of the Earth's theorized first supercontinents, Vaalbara and Ur.
In our tectonophysical discussion of the formation of the lithospheric block structure, we analyze in detail the map of modern lithospheric plates [Bird, 2003] in combination with the materials from [Sherman et al., 2000]. In the hierarchy of the blocks comprising the contemporary lithosphere, which sizes are widely variable, two groups of blocks are clearly distinguished. The first group includes megablocks with the average geometric size above 6500 km. Their formation is related to convection in the Earth mantle at the present stage of the geodynamic evolution of the Earth, as well as at all the previous stages, including the earliest one, when the protolithosphere emerged. The second group includes medium-sized blocks with the average geometric size of less than 4500 km and those with minimum sizes, such as rock lumps. They reflect primarily the degradation of megablocks as a result of their destruction due to high stresses in excess of the tensile strength of the medium. This group may also include blocks which formation is related to convection in the upper mantle layer, asthenosphere. There are grounds to assume that through the vast intermediate interval of geologic time, including supercycles of Kenorlend, Rodin, and and partically Pangea, the formation of the large lithospheric blocks was controlled by convection, and later on, they were 'fragmented' under the physical laws of destruction of solid bodies. However, it is difficult to clearly distinguish between the processes that predetermine the hierarchy of formation of the block structures of various origins - sizes of ancient lithospheric blocks cannot be estimated unambiguously.
Thus, mantle convection is a genetic endogenous source of initial divisibility of the cooling upper cover of the Earth and megablock divisibility of the lithosphere in the subsequent and recent geodynamic development stages. At the present stage, regular patterns of the lithospheric block divisibility of various scales are observed at all the hierarchic levels. The areas of the lithospheric megaplates result from regular changes of convective processes in the mantle, which influenced the formation of plates and plate kinematics. Fragmentation of the megaplates into smaller ones is a result of destruction of the solid lithosphere under the physical laws of destruction of solid bodies under the impact of high stresses.
Key words: lithosphere, tectonic plates, blocks, convection, destruction, tectonophysics, divisibility of the lithosphere, Rayleigh-Benard cells, continents
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For citation: Sherman S.I. 2015. Genetic sources and tectonophysical regularities of divisibility of the lithosphere into blocks of various ranks at different stages of its formation: tectonophysical analysis. Geodynamics & Tectonophysics 6 (3), 387-408. doi:10.5800/GT-2015-6-3-0187.
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S.I. Sherman: Genetic sources and tectonophysical regularities of divisibility of the lithosphere...
Генетические источники и тектонофизические
ЗАКОНОМЕРНОСТИ РАЗНОРАНГОВОЙ БЛОКОВОЙ ДЕЛИМОСТИ ЛИТОСФЕРЫ НА РАЗЛИЧНЫХ ЭТАПАХ ЕЕ ФОРМИРОВАНИЯ: ТЕКТОНОФИЗИЧЕСКИЙ АНАЛИЗ
С. И. Шерман
Институт земной коры СО РАН, Иркутск, Россия
Аннотация: Впервые проводится тектонофизическая реконструкция формирования первичной делимости протолитосферы в результате конвекции остывающей примитивной мантии. Формирующиеся в ней конвективные ячеи Рэлея-Бенара предопределяют размеры первичного разделения протолитосферы на отдельные массы - прообразы блоков. Ячеи Рэлея-Бенара не впервые используются в геологии и геодинамике. Первоначально на них ссылались для объяснения формирования первичных континентальных ядер. Обращение к ячеям Рэлея-Бенара и их структурным реликтам способствует пониманию того, как зарождается первичная делимость протолитосферы, которая трансформируется в крупные литосферные плиты - прообразы континентов. Именно консервирующиеся в формирующейся литосфере ячеи Рэлея-Бенара, нижняя граница которых корреспондировала с одним из главных разделов планеты - границей ядра, - предопределили первоначальную мегамасштабную блоковую структуру протолитосферы и формирующейся литосферы. Проведенные теоретические оценки сопоставлены и хорошо согласуются с количеством и размерами площадей первых гипотетических континентальных структур - суперконтинентов Ваальбара и Ура.
Продолжение тектонофизического разбора формирования блоковой структуры литосферы реализовано на детальном анализе карты современных литосферных плит [Bird, 2003] с привлечением фактических материалов [Sherman et at., 2000]. В широкой по размерам площадей иерархии блоков в современной литосфере Земли отчетливо выделяются две группы. Первая - мегаблоки, среднегеометрический размер которых превышает 6500 км. Их формирование на современном этапе геодинамического развития Земли, а также на всех предшествующих, в том числе и на самом раннем, при зарождении протолитосферы связано с конвекционными процессами в мантии Земли. Вторая группа - блоки со среднегеометрическим размером менее 4500 км, вплоть до минимального, соответствующего кусковатости горных пород, отражают, прежде всего, деструкцию мегаблоков в результате их разрушения под действием высоких внутренних напряжений, превышающих предел прочности среды. В этой же группе могут быть блоки, формирование которых также связано с конвекцией, охватывающей верхний мантийный уровень - астеносферу. Можно предполагать, что в громадном промежуточном интервале геологического времени, охватывающем суперциклы Кенорленд, Родинию и, частично, Пангею, формирование крупных литосферных блоков контролировалось конвекцией, а их дальнейшее «дробление» регулировалось физическими законами разрушения твердых тел. Однако четкую границу между процессами, определяющими иерархию формирования блоковых структур разного генезиса в прошедшие времена, провести трудно из-за неопределенности размеров литосферных блоков далекого прошлого.
Таким образом, конвекция в мантии является генетическим эндогенным источником первичной делимости остывающей верхней оболочки Земли, а также мегаблоковой делимости собственно литосферы в последующие этапы ее геодинамического развития. На современном этапе закономерности разномасштабной блоковой делимости литосферы прослеживаются на всех иерархических уровнях. Площади мегаплит литосферы - результат закономерных изменений конвективных процессов в мантии и их воздействия на формирование и кинематику плит; деструкция мегаплит на меньшие по площади блоки - результат закономерного дробления твердых тел литосферы при высоких напряжениях.
Ключевые слова: литосфера, тектонические плиты, блоки, конвекция, деструкция, тектонофизика, делимость литосферы, ячеи Релея-Бенара, континенты
It is now evident that without understanding the Earth's evolution since the earliest stages when the covers of our planet and its continental crust were formed, it is difficult to determine locations where the major natural resources are accumulated and to reveal how various structural elements and a wide variety of igneous rocks were generated and continue their development.
Academician M.I. Kuz'min [2014, p. 626]
1. Introduction
Initial divisibility of the Earth protolithosphere, i.e. the cooling outer hard cover of the planet, and its transformation with time into lithospheric blocks have
not been properly studied yet in terms of the geodynamics of faulting, and tectonic regularities in divisibility of the lithospheric blocks of various ranks still need to be clarified. In the outer cover of the Earth, initial divisibility of the protolithosphere was due to cooling
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of the primitive mantle as a result of heat-gravitational convection manifested by Rayleigh-Benard cells. In this study, we assess tectonophysical conditions for the generation of such cells and estimate potential sizes of the cells and amounts of primary proto-lithospheric cooling masses as prototypes of the blocks. Our estimations are consistent with the reconstructed first supercontinental cycles of the geodynamic evolution of the Earth. The block divisibility of the recent continental lithosphere is analysed in detail with reference to the map of present tectonic plates and blocks of the lithosphere and the cumulative plate count according to [Bird, 2003]. In addition to the data from [Bird, 2003] which mainly cover megaplates and blocks of medium sizes, we analyse the parameters specified in [Sherman et al., 2000] for medium- and small-size blocks resulting from destruction of megaplates and blocks of the continental lithosphere. We propose regression equations describing divisibility of the continental lithosphere into blocks in a wide scale range, from medium-to small-sized blocks and rock fragments in outcrops. Such fragments result from destruction of medium-and small-sized blocks of the 'solid' lithosphere which takes place when internal stresses exceed the rock breakdown point, as described by exponential functions.
The occurrence of megablocks is related to mantle convection at the early stage of the evolution of the protolithosphere and subsequent stages of its transformation into the lithosphere through the global super-cycles of the geodynamic evolution of the Earth. The scale and organization of mantle convection are factors that predetermine divisibility of the lithosphere into megablocks through all the recent stages of its development, including the present stage.
2. The primary hot cover of the Earth, its composition and thickness
One of the most recent theoretical reviews of the early stages in the evolution of the Solar System and the geological history of the Earth was published by M.I. Kuz'min [2014] who rightly notes that the global academic geological community is challenged to estimate the time when the first continental crust was formed on the Earth. He develops the concepts co-authored with V.V. Yarmolyuk [Yarmolyuk, Kuz'min, 2012] on the formation of the outer and deep covers of the Earth, mantle processes and their impacts on the occurrence of surface structures, igneous rocks and ores.
The review [Kuz'min, 2014] is based on the latest data on the origin of the Solar System and formation of the first continental rocks on the Earth, which contain zircon, the oldest mineral so far dated on the Earth. It is assumed that the Solar System formed from a gas-and-
dust nebula 4.568 Ga ago. The continental crust was gradually growing from its recorded peak size (4.25 Ga) till 4.1 Ga, i.e. completion of the first eon in Earth's history, the Hadean. It seems to be a critical milestone in the early geological history of the Earth, followed by the Archaen history [Kuz'min, 2014] - intensive cooling of the outer cover of the Earth commenced in this period. The heat flow was supported by the inner supply of heat generated due to gravitational compression of the planet while its solid body was formed [Schubert et al., 2001].
The estimated average temperatures of the mantle range from 1250-1350 °C to 1400 °С, and a roughly estimated temperature of the cooling Earth is 0 °С. In the present stage, the maximum temperature of the as-thenosphere top is about 1350-1400 °С, and this temperature level is supported by various endogenous heat sources of the Earth and compensates heat losses caused by cooling. At the early stage of the Earth evolution, temperatures range from 0°С (or slightly above 0°С) at the Earth's surface to ~1350-1400 °C at the depth levels whereat temperature changes in the period of cooling are less significant due to heat influx. Under this assumption, the cover can be viewed as a gradually cooling low-viscous fluid body comprising the lower and upper layers which temperatures are significantly different. At the first stage when the outer cover of the Earth was formed, convection was the major mechanism of heat energy dissipation. It can be assumed that convection commenced in the pre-Kat-archean eon and is underway until now, while the volumes and forms of convective flows have significantly changed with time. This time period agrees with the maximum age of about 4.1 Ga determined in [Kuz'min, 2014] for the start of the development of the protolithosphere that converted with time into the lithosphere which development is continued.
By its initial composition, the cooling upper cover of the Earth corresponds to the so-called primitive mantle, as evidenced by the composition of chondrites, i.e. stony (non-metallic) meteorites. The bulk composition of the primitive mantle is similar to the silicate cover of the Earth which was formed of the protoplanet material after the core had separated [Hofmann, 1997]. It is noteworthy that variaitons in the composition of the primitive mantle do not influence estimations of temperatures at the lower boundaries of the primary mantle masses at the cooling surface of the Earth.
Physical parameters and the composition of the primitive mantle changed in the post-Archean period [Vrevsky et al., 2010] including several large cycles of the geodynamic development of the Earth, which were also related to mantle convection. By the Archean period, the Earth's crust was completely formed, and the upper boundary of the mantle convection processes went down into the Earth interior by dozens of kilome-
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tres [Artemieva, 2011]. Responses of the upper solid part of the lithosphere as a rheological body to external impacts, especially external loading at different velocities, became different. In the former convective medium, convection in the cooling upper layer of the quasifluid was replaced by heat conduction/diffusion in the solid body. Studies of destruction of solid bodies and rank-variable fracturing and faulting are reported in many papers, including [Peive, 1990; Sherman et al., 1991, 1992, 1994; Seminsky, 2003; Sherman, 2002, 2012, 2014a, 2014b; and others], and it is established that physical laws of deformation and destruction of solid bodies are applicable.
A major challenge is to analyse, in a retrospective, the origin and initial formation of structures in the cooling protolithosphere which physical properties are assumed similar to those of the cooling low-viscous quasi-liquid mass. In such a medium, the maximum dissipation of energy was ensured by convection of various types, from structurally organized (Rayleigh-Benard cells) to chaotic. This long-term process in the upper cover of the Earth had its regular features due to convective flows of the mega mass that was cooling it The total thickness of the cooling mass of the primitive mantle can be assumed at 2900 km as the outer core boundary is located at this depth. Below we review hydrodynamic regularities in convection of cooling low-viscous materials and relic structures which are important for reconstructing the paleogeodynamic settings of the distant past and estimating probable dimensions of the primary blocks.
3. Key regularities in the formation of convective
CELLS IN COOLING LOW-VISCOUS MATERIALS AND RELIC
STRUCTURES
It is most reasonable to believe that energy in the cooling low-viscous medium is dissipated by mantle convection that is most common manifested by Ray-leigh-Benard cells. The cooling surface of the Earth is assumed to behave as the cooling low-viscous medium much time before the Katarchean. Convection is generally reviewed below as a prerequisite for an assessment of conditions for initial divisibility of the outer cover of the protolithosphere.
General convection equation. The onset of convection occurs when the Rayleigh number reaches some critical value. The Rayleigh number, Ra is a dimensionless number predetermining the behaviour of gas, fluid or mass of a very low viscosity at a specified temperature gradient When the Rayleigh number exceeds its critical value, the equilibrium of the cooling fluid is disturbed, which leads to the occurrence of convection flows and bifurcation. The bifurcation point is the critical value of the Rayleigh number:
n P^APL3
Ra =-------,
vx
(1)
where g is the gravity acceleration; L is the size of the fluid area; AT is the difference of temperatures at the surface and the lower layer of the fluid; v is the kinematic viscosity of the fluid; x is the heat conductivity of the fluid; is the coefficient of thermal expansion of the fluid.
If the Ra value is small, convection does not start. If values of Ra are average, conditions are favourable for heat convection. Chaos occurs at high values of Ra. Values of Ra depend on combinations of all other parameters in equation 1. However, considering cooling of the primitive mantle in the model discussed here, the difference of temperatures and thickness of the cooling layer are the main parameters. It should be noted that patterns of convective cells are significantly dependent on dimensions of the cooling area. In such cases, an additional parameter needs to be introduced - aspect ratio, G [Getling, 1998]:
G=L/h,
(2)
where L is horizontal size of an area (for a circular section, it corresponds to a radius); h is vertical size of the area.
The Grashof (Gr) and Prandtl (Pr) numbers are also widely used in studies of Rayleigh-Benard cells, but not in this review.
In the below discussion of the natural conditions, we refer to cases with large values of G. Horizontal projections of cells are called planforms. The planforms may significantly vary depending on parameters of the medium. Planforms of the cells which are typical observed in the experimental and natural settings are reviewed below.
Planforms of convective cells, and physical conditions for their formation and stability. Three types of cell planforms are typically observed in the experiments [Getling, 1998]: two-dimensional bars, hexagonal cells and square/rectangular cells (Fig. 1).
As shown in Fig. 1a, 2D bars are oriented along axis Х and parallel to axis У (x-bars) or vice versa, and a cell is formed by a pair of neighbouring bars that occupy the entire spatial interval. In the bars, fluid circulates in vertical plane X, Z as well as in the opposite directions.
Hexagonal cells (Fig. 1, b) are composed by the superposition of three systems of bars which are located at angles 2n/3 to each other. Such cells are characterised by periodicity in directions of axes Х and У and invariant in case of rotation by an angle of 60°. A hexagonal cell is classified in l-type in liquid convection cases (Fig. 1, b-l) or g-type in gas convection cases (Fig. 1, b-g) with regard to a velocity vector, i.e. depending on whether the liquid is ascending in the central part of
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I Fig-1- Schematics of convective cells: a - 2D bars; b - hexagonal cells of l- and g-types (according to [Getling, 1998]).
I Рис. 1- Схематическое изображение конвективных ячеек: a - двумерные валы; b - шестиугольные ячейки l- и g-типа (по [Getling, 1998]).
the cell or gas is descending, which, in its turn, is related to the temperature dependence from the viscosity of the medium. As known, with higher temperatures, viscosity of fluids decreases, while viscosity of gases increases. A velocity vector depends on a sign of derivative dy/dT which is negative for liquids and positive for gases. In the ascending convective flow, the material is always warmer than in the descending flow. Respectively, the liquid viscosity in the central parts of l-cells is lower, while the gas viscosity in the central parts of g-cells is higher. Circulation tends to follow the direction where the viscosity is lower in the centre of the cell. The stability of circulation trends, as well as the unchangeability of stationary flows of bars in relation to variations of defining parameters are estimated in a wide range of Rayleigh and Prandtl numbers, taking into account k=2n/A, i.e. the number of waves of length A per 2n radians, or the number of spatial intervals of waves per on 2n radians.
Square cells form systems which directions are rotated by a n/4 angle with respect to the coordinate system (X, Y).
Cell planforms are significantly influenced by even minor changes in the physical conditions of the medium and variations of its parameters included in the general equation of convection (see Equation 1). Twodimensional bars are the main form of stationary convective structures produced by thermal-gravitational or thermocapillary convection mechanisms. In case of
thermal-gravitational convection, the scale of flow can increase depending on AT and G (see Equations 1 and 2).
For the geological interpretation of the significance of cellular structures developing in cooling masses of the primitive mantle material, two facts are of importance: (1) relic structures, such as boundaries of cellular structures in the cooling protolithosphere, and subsequently, in the upper part of the lithosphere, and (2) relic masses of the deep mantle material, which were delivered into the Earth's upper horizons and cooled in zones of inter-cell boundaries, i.e. relic structures at the boundaries of the primary cellular formations. Specialists in the laws of Rayleigh-Benard convection believe that two groups of boundaries in the Earth's upper horizons are of importance: boundaries between convective cells and border lines between orderly fragment-textures with different orientations of the bars, which comprise a more complex pattern
(Fig. 2).
The most significant structural boundaries are lines bordering fragments-textures with different orientations of the bars [Getling, 1998]. The duration of their existence predetermines the stability of stationary convection flows and the geological significance of convection. According to [Clever, Busse, 1996], the hexagonal cells can be stable at Pr>1.2 and Ra>3000. If Pr<10, the stability area of the hexagon cells looks like a band stretching from smaller to larger Ra values. If the
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a
Ъ
c
I Fig. 2. Defects of bar structures (lines show boundaries of bars): a - dislocation; b - disclination (singularities of the focus type are below); c - structural boundary (according to [Getling, 1998]).
I Рис. 2. Дефекты валиковых структур (линии соответствуют границам валов): a - дислокация; b - дисклинации (внизу сингулярности типа фокуса); c - структурная граница (по [Getling, 1998]).
Prandtl numbers are larger than 10, the area of stable hexagon cells is disturbed. In the experiments with Ra <3000, no stable hexagons are recorded.
The area of stable square cells is wider and covers the range of Ra from 4000 to 50000 with Pr varying from 2.5 to 16. When approaching the specified minimum value, the stability area narrows and becomes undetectable when Pr=2.5 [Busse, Clever, 1998].
The material conveyed by the convective flows is hardening in places where the flows begin to move downward due to lower temperatures. Solidification takes place at the walls of thermal convection cells. Thus, when the vertical walls of the cells (i.e. surfaces of basalt prisms) are solidified, thermal convection continues inside the prisms until complete solidification of all the lava components. A photo in Fig. 3 shows the tops of basalt pillars with sagging centres of the columns which were the last to cool and solidify.
Based on the brief review of convective planforms and conditions of their formation, it is possible to reveal a common pattern of convection processes taking place during cooling of the homogeneous medium. In space and time, convection is manifested by a highly ordered flow of cooling quasi-liquid masses. The stability area is wide. As the area occupied by the flow is narrowing, stability is reduced as the characteristic scale of inhomogeneity of the structure is reducing [Getling, 1998].
In standard experimental conditions, Rayleigh-Benard convection cells occupy the entire thickness of the cooling layer, and their typical horizontal size is
comparable to the vertical size or slightly exceeds it. In the majority of problems solved by geodynamics, it is assumed that convection cells occupy the entire mantle or partially occupy the layer or occur between the layers [Kirdyashkin, Dobretsov, 1991; Dobretsov et al., 2001; Trubitsyn V.P., Trubitsyn A.P., 2014]. In such conditions, the main factor predetermining convection is viscosity of the medium, which is included in equations of interrelated Rayleigh, Grashof and Prandtl numbers. It can significantly increase or decrease their values and change the stability of convection accordingly. When viscosity varies by two or three orders (which may take place in the cooling upper part of the protolithosphere), the main viscosity gradient is in the topmost layer, wherein viscosity is increasing relatively faster than in the lower layers, and a hard cover is thus formed. Convection goes downward. Moreover, in the discussed cases of ascending convection, the cooling masses drift towards borders of the cells and subside due to gravity, which leads to simultaneous thickening of the emerging vertical border zone (i.e. a plane), which substance is more viscous. As the process develops further, such planes create favourable conditions for initial faulting of the lithosphere, and the lithospheric plates and large blocks are thus bordered by faults that are stable in time.
As a result of gradual cooling a protective cap is formed over the cooling mass. As the process develops, the solid cap becomes thicker. The merger of the two descending cooling flows leads to further thickening of the emerging cap, and the partition between emerging blocks of the lithosphere is thus fixed.
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I Fig. 3. Solidified convection flows of basalt lava with 'sagging' surfaces in the centres of cells [Shumilov, 2009].
I Рис. 3. Застывшие конвекционные потоки базальтовой лавы с «проседанием» поверхностей в центрах ячей [Shumilov, 2009].
Over time, the cap thickens and evolves into the brittle part of the lithosphere, while convection flows continue to function in the mantle and gradually drift to the lower hypsometric levels. The above-mentioned processes take place as the equilibrium of the convection system under the cap is disturbed due to changes of temperature and viscosity gradients. Conditions that are favourable for convection are now found at larger depths, and heat energy dissipation is facilitated. In this time period, the dominating convection occupying the entire mantle may be either replaced by convection in two layers or occur in a more complicated pattern.
Regardless of their planforms, the Rayleigh-Benard cells give evidence that convection non-equilibrium can be a source of order. In comparison with the homogeneous hot mass, convective cells (regardless of their forms) can be regarded as highly organized structures facilitating the dissipation of energy and the formation of other, more stable forms in the cooling mantle. In this regard, the system remains open and continues to give the entropy.
Based on the above, cooling of the pre-Katarchean Earth's surface can be analysed with an assumption that the energy of the hot low-viscous body dissipated by thermal-gravitational convection. In this case, the Rayleigh-Benard cells can be viewed as initial structures in the cooling mantle of the Earth, and boundaries between the cells were the first to get solidified and thus predetermined the contours of the future huge masses/blocks of the protolithosphere, which were the basis of Vaalbara and, may be, other theorized first supercontinents of the Earth.
4. The origin of the first largest local structures in the protolithosphere as viewed under the concept of convection in the cooling primitive mantle: a tectonophysical approach to
PALEOGEODYNAMIC RECONSTRUCTIONS
In this study, a tectonophysical approach is proposed to analyse initial divisibility of the protolithosphere. The origin of the primary local structures / blocks is related to cooling of the Earth's surface layer composed by hot low-viscous masses of the primitive mantle. In such a medium, a relatively efficient way of heat dissipation is convection that is structurally arranged in Rayleigh-Benard cells. Their relics, i.e. protocontinents, can be regarded as residual structures giving evidence of intitial atectonic divisibility of the protolithosphere. In this assumption, laws of their formation are determined by the laws of convection.
The cooling upper layer covered the entire primary surface of our planet. Its thickness was limited by the outer boundary of the almost completely formed core which was located at a depth of about 2900 km. It is known that the evolution of convection and its patterns are significantly influenced by dimensions (diameter and depth) of the area involved in convection (see Equation 2). For circular cross-sections in the experiments, the horizontal size of convection cells corresponds to the depth of the layer wherein convection takes place. Original convection experiments are described in the book by N.L. Dobretsov et al. [2001] who estimated the minimum horizontal size of convection cells in the lower mantle: “As follows from results of the experiments and the classical laws of convection, a cell can be stable if its transverse size is only by a factor of 1.8 times (or less) larger than its thickness” (p. 161).
Therefore, when convection takes place in a rather thick layer, a pattern of convection cells is compatible with the thickness of the layer. In our study of the mega-scale case, the horizontal size of the layer subject to convection is much larger than its vertical size, and the layer can thus be considered as a cooling flat body. The radius of the first round-shaped convective cells can amount to 2900-3000 km, and the distance between the emerging cooling boundaries of areas with descending masses (i.e. cell diameter) can be about 6000 km. In this case, the cell diameter can be numerically similar to the radius of the cooling Earth, and the cell area can be determined as the area of a spherical segment (Ss=2nRh, where R is the Earth radius, and h is the thickness of the cooling layer, i.e. R/2). It amounts to nR2 or about 3 steradians1. The total area of the Earth surface is 4nR2 (or 4n Q). Under ideal conditions,
1 Steradian, sr (fi) a solid angle at the centre of a sphere subtending a section on the surface equal in area to the square of the radius of the sphere.
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I Fig. 4. Principal scheme of Rayleigh-Benard cells on the spherical surface.
I Рис. 4. Принципиальная схема конвективных ячей Рэ-лея-Бенара на сфере.
as a maximum, four mega-large convective cells with an average area of about three steradians can form in the cooling upper cover of the Earth. It should be noted that the boundary of the bottom surfaces of the cells is the outer core of the Earth, which square area is four times smaller than that of the Earth surface. As the area of the bottom surfaces of the cells is restricted, the cells cannot achieve their potential maximum area on the surface. If a minimum surface area of a cell is one steradian, 12 cells as a maximum can form at the Earth surface. Therefore, the total number of the primary cells predetermining the initial divisibility of the emerging protolithosphere can range between 3-4 (minimum) to 12 (maximum) (Fig. 4).
Paleogeodynamic reconstructions shows that the first supercontinent Vaalbara (2.8-3.6 Ga) consisted of two major structures, i.e. protocontinents/cratons Kaapval and Pilbara [Hazen, 2012], which reflect the divisibility of the already solid protolithosphere and, may be, the divisibility of the emerging lithosphere. The concept that convection took place in the primitive mantle at the very early stages of the protolithosphere is not denied, although not widely discussed in the press [Nebel et a/., 2014]. An acceptable argument is proposed by I. Artemieva and B. Mooney [2001] who consider the 'thermal' age of the Archean formations of the Earth.
There are grounds to state that convective processes in the Earth's mantle played the major role and were the basic criterion for the formation of the first largest block structures of the protolithosphere and also for subsequent transformations of such blocks into the lithospheric structures. According to the reconstructions, after Vaalbara supercontinent reconstruct-ted Ura (about 3 Ga) and younger Kenorland (21002700 Ma), Columbia (1500-1800 Ma), Rodinia (7501050 Ma) and Pangea (200-300 Ma) supercontinents [Li et al, 2008; Lubnina, 2011; Hazen, 2012]. About 200 min years ago, Pangea broke apart to form Laurasia and Gondwana, i.e. groups of the southern and northern continents, respectively. The lithospheric megablocks have not been completely formed and destruct-ed yet, and these processes are still ongoing at the present stage that is transitional to the formation of a new supercontinent. In the long-term geological history of the Earth, the number and dimensions of the lithospheric plates, which were actively involved in super cycles, were variable, but the blocks and plates per se have never disappeared completely (!). Since the Ar-chean, these integral large bodies have been subject to many geodynamic catastrophes and reconstructions [Li et al., 2008], and their initial masses were partially 'lost' in some of the geodynamic cycles, regained in the others, converted into the six major lithospheric plates and survived! Strongly metamorphosed rocks of the Archaean age are observed in huge areas of the six continental lithospheric plates (Africa, Antarctica, North America, Eurasia, Australia and South America).
It is challenging to conduct a proper mathematical analysis of changes in the number of the lithospheric plates and their areas from one super cycle to another. The major cycles in the geodynamic evolution of the Earth and corresponding lithospheric plates of different shapes and kinematics are revealed by paleo-geodynamic reconstructions. The lithospheric plates can provide for stable mantle convection at Ra<106, but it becomes unsteady at Ra>107 [Getling, 1998]. In response to changes in the convection pattern, the system of interacting lithospheric plates has to readjust itself. Numerical solutions of equations of energy, mass and momentum transfer suggest that mantle convection takes place while a set of plates is self-generated. According to [Trubitsyn V.P., Trubitsyn A.P., 2014], “the set of plates is inevitably generated, without requiring any additional boundary and initial conditions” (p. 146). The foregoing explains why the Earth has been subject to numerous transformations, including the catastrophic ones, in the natural course of its geodynamic evolution. Is this not a proof of the law of self-organized criticality which is discussed in [Bak, 1996]?
Obviously, mantle convection has been the major long-term genetic source providing for the cyclic geo-
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dynamic development of the entire lithosphere and its mega-block pattern.
In this regard, results of studies by P. Bird and coauthors [Bird, 1988, 1998, 2003; Bird, Rosenstock, 1984; Bird et al., 2002] are noteworthy - the hierarchy of the lithospheric plates and blocks is mathematically analyzed and the cumulative-number/area distribution is established. In our study, their results are complemented by experimental data published in [Sherman et al., 2000] (Table) and compared with other quantified information on the formation of plates and the fault-block structure of the lithosphere which vary in ranks at the present stage of development. We apply tectono-physical methods to analyse the hierarchy of the lithospheric plates and blocks with regard to their square areas at the present stage of the geodynamic evolution of the lithosphere. The analysis is aimed at identification of genetic sources and patterns of the lithosphere block divisibility at various hierarchical levels in different stages of the lithosphere evolution.
5. Recent hierarchic divisibility of the fault-block
STRUCTURE OF THE LITHOSPHERE: TECTONOPHYSICAL
ANALYSIS
The subject of our analysis and the basis of further reconstructions is the map of lithospheric blocks of the Earth which is published in [Bird, 2003] (Fig. 5). Its main original features are (1) the pattern of the plates on the present surface of the Earth, and (2) calculations of plate areas in steradians (Table). To analyse ratios between areas and boundaries of the lithospheric plates, P. Bird showed them on the map close to each other in an arbitrary reconstruction (no more! - S. Sh.) of the 'intact' surface of the Earth. His map shows 52 plates of various ranks, including ‘Manus microplate’ which area is the smallest (0.0002 sr, see Table). Another specific feature of the map is the use of steradian, a dimensionless unit to estimate areas of plates and blocks, thus avoiding some skewing in quantitative comparisons of plates and blocks located at different latitudes of the sphere. Using this method, P. Bird presented in digital form a global set of boundaries of the present lithospheric plates and blocks of various sizes and ranks and estimated plate sizes. He established a mathematical regularity in the abrupt, non-uniform decrease of plate areas on the present Earth's sphere (columns 2 and 3 in Table). In Figure 6, the cumulative plate count as a function of plate areas in steradians is shown in the bilogarithmetic scale.
Based on Table (Nos. 1-52, column 3) supplemented by data from [Bird, 1988, 1998, 2003; Bird, Rosenstock, 1984; Bird et al., 2002], regression equations show that areas of the plates and blocks are decreasing with increasing numbers in the hierarchy, i.e. with transition
from mega structures to regional and local ones (Fig. 6). The detailed interpretation of the plot is given in [Bird, 2003], and a brief is given in the figure caption (Fig. 6). The main regression line based on the data from Table has two bends.
According to P. Bird [2003], when plotted with logarithmic scales, the plates of areas between 0.002 and 1 steradian (from the relatively small Jian Fernandes plate, JZ to the large South America plate, SA) occur in numbers that roughly obey a power law:
(cumulative count) N~7 (steradians)-1/3
or N*7fi-V3. (3)
This equation clearly reflects the scale relationship between the increase in the number of plates and a proportional reduction in their areas. This is characteristic of plates which areas are smaller than one steradi-an, i.e. plates with an average lateral size of about 4000 km (see Table).
For a more detailed tectonophysical analysis of the relationships between sizes of the lithospheric plates and blocks, the data published by P. Bird are supplemented by results of similar studies focused on intracontinental areas [Sherman et al., 2000]. The consolidated database provides for analyses of tectonophysi-cal regularities in the block divisibility of the present 'solid' lithosphere. For now, seven major lithospheric plates (Nos. 1 to 7 in Table) are outside the scope of analysis and considered below in the closing statements. These major plates are viewed as original indicators of the initial and subsequent stages of lithosphere divisibility, and it seems more reasonable to consider them after reviewing and 'excluding' quantitative data on plates dominating in number in other hierarchic levels.
Table consolidates three data sets - [Bird, 2003] (Nos. 1 to 52, columns 1, 2 and 3), [Cheremnykh, 1998] and [Sherman et al., 2000] (Nos. 53 to 225) and thus provides for a more detailed consideration of the ratios of areas of medium- and small-sized lithospheric plates from [Bird, 2003] and the ratios of areas of continental lithospheric blocks from other publications. For an adequate comparison of plate areas in different hierarchical levels, the areas calculated in steradians are converted to measurement system SI and given in square kilometres and corresponding characteristic linear dimensions (columns 4 and 5, Table). In the recalculations, it is assumed that the Earth's radius is R=6371 km, and the equation from [Sadovsky et al., 1987; Sadovsky, Pisarenko, 1991] is used to calculate linear dimensions Lb and plate/block areas, Sb:
Lb = V^. (4)
Regression equations are calculated for comparative analyses of the three above-mentioned data sets:
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Fig. 5. Lithospheric plates mapped by P. Bird [Bird, 2003].
The 52 plates of his model PB2002 are shown with contrasting colours. Two-letter plate identifiers are explained in Table. The 13 cross-hatched areas are ‘orogens’ in which an Eulerian plate model is not expected to be accurate. Labels of small plates and orogens are offset (with leader lines) for clarity. Mercator projection.
Рис. 5. Карта литосферных плит по П. Бёрду [Bird, 2003].
Цветом выделены 52 плиты по модели РВ2002. Наименование плит дано двойными буквами в соответствии с таблицей. Заштрихованные квадратной сеткой площади 13 районов соответствуют «орогенам», для которых модель вращения вокруг Эйлеровых полюсов не совсем точна. Двухбуквенное наименование мелких плит вынесено за их границы. Карта дана в проекции Меркатора.
(1) Regression by P. Bird, in its middle part showing areas from Jian Fernandes plate (JZ) to South America plate (SA), i.e. from mega- to medium-sized plates and blocks (Nos. 8 to 52, Table; equation 3 (in sr), and NC=2259.3L-067 (5) (symbol 1 in Fig. 7);
(2) Regression for the additional data on medium-and small-sized blocks (Nos. 53 to 225, Table; Nc=7049.1L-091 (6) (symbol 2 in Fig. 7). An important indicator is an inclination angle of the regression curve. Equation 6 differs from Equation 5 by an increase of the inclination angle. Taking into account that Equation 6 is based on numerous data from geological and structural maps of continents in various scales, it can be noted that 'small-sized' blocks are more numerous that 'large' ones. This is a logical consequence following the ratio of data obtained by direct field observations that always record more small blocks on sites than large ones. The same is valid for even rock outcrops and evidenced by the plot from [Bird, 2003] (see Fig. 6) at the second bend of the regression line;
(3) Regression for the consolidated data on plates and blocks of different characteristic sizes (No. 8 to 225, Table; NC=3080.8L-072 (7) (symbol 3 in Fig. 7). Regression (7) shows a significantly smoothed transition from small-sized lithospheric blocks to intra-continental blocks divisibility.
In general, equations 3, 5, 6 and 7 are similar, which suggests that the fragmentation of 'solid' rocks follows a physically uniform pattern, and, in more general terms, there is a tectonophysical law of the fault-block divisibility of the lithosphere which is valid for lithospheric blocks of a wide range of areas, from blocks which size is compatible with North and South America continents, i.e. nearly as big as lithospheric plates, to lump of rocks observed on small outcropped sites.
With account of our detailed studies of fault-block continental lithosphere structures of different ranks and in view of the identity of the physics of the process and the similarity of mathematical equations 5, 6 and 7, we construct a continuation of the regression curve
Geodynamics & Tectonophysics 2015 Volume 6 Issue 3 Pages 387-408
Areas and dimensions of lithospheric plates and blocks
Параметры площадей и размеров литосферных плит и блоков литосферы
# Names of plates and blocks Identifiers Area, steradian Area, km2 Average geometric size, km
1 Pacific PA 2.57685 104593416 10227.09
2 Africa AF 1.44065 58475466 7646.925
3 Antarctica AN 1.43268 58151967 7625.744
4 North America NA 1.36559 55428808 7445.053
5 Eurasia EU 1.1963 48557388 6968.313
6 Australia AU 1.13294 45985628 6781.27
7 South America SA 1.03045 41825596 6467.271
8 Somalia SO 0.47192 19155063 4376.65
9 Nazca NC 0.39669 16101505 4012.668
10 India IN 0.30637 12435448 3526.393
11 Sunda SU 0.21967 8916326 2986.022
12 Philippine Sea PS 0.13409 5442665 2332.952
13 Amur AM 0.13066 5303442 2302.92
14 Arabia AR 0.12082 4904040 2214.507
15 Okhotsk OK 0.07482 3036917 1742.675
16 Caribbean CA 0.07304 2964667 1721.821
17 Cocos CO 0.07223 2931790 1712.247
18 Yangtze YA 0.05425 2201988 1483.91
19 Scotia SC 0.0419 1700706 1304.111
20 Caroline CL 0.03765 1528200 1236.204
21 North Andes ND 0.02394 971716 985.7566
22 Altiplano AP 0.0205 832087.6 912.1884
23 Banda Sea BS 0.01715 696112.3 834.3335
24 New Hebrides NH 0.01585 643345.8 802.0884
25 Anatolia AT 0.01418 575561.1 758.6574
26 Birds Head BH 0.01295 525635.9 725.0075
27 Burma BU 0.0127 515488.4 717.9752
28 Kermadec KE 0.01245 505341 710.8734
29 Woodlark WL 0.01116 452980.4 673.0382
30 Mariana MA 0.01037 420914.6 648.7793
31 Molucca Sea MS 0.0103 418073.3 646.5859
32 North Bismarck NB 0.00956 388037 622.9261
33 Timor TI 0.0087 353129.9 594.2473
34 Okinawa ON 0.00802 325528.9 570.5514
35 Aegean Sea AS 0.00793 321875.9 567.341
36 South Bismarck SB 0.00762 309293.1 556.1412
37 Panama PM 0.00674 273574.2 523.0432
38 Juan de Fuca JF 0.00632 256526.5 506.4845
39 Tonga TO 0.00625 253685.3 503.6718
40 Balmoral Reef BR 0.00481 195236.2 441.8554
41 Sandwich SW 0.00454 184277 429.2749
42 Easter EA 0.00411 166823.4 408.4402
43 Conway Reef CR 0.00356 144499.1 380.1304
44 Solomon Sea SS 0.00317 128669.2 358.7048
45 Niuafo'ou NI 0.00306 124204.3 352.4263
46 Maoke MO 0.00284 115274.6 339.5211
47 Rivera RI 0.00249 101068.2 317.9123
48 Juan Fernandez JZ 0.00241 97821.03 312.7635
49 Shetland SL 0.00178 72249.56 268.7928
50 Futuna FT 0.00079 32065.82 179.0693
51 Galapagos GP 0.00036 14612.27 120.8812
52 Manus MN 0.0002 8117.928 90.09955
53 Angara-Ilim-9 IA9 47520 218
54 Angara-Ilim-13 IA13 34120 185
55 Angara-Ilim-3 IA3 33620 183
56 Angara-Ilim-12 IA12 31430 177
57 Prisayan-Enisei-2 IPE2 28100 168
58 Baikal-Patom-3 IIIBP3 27450 166
59 Angara-Ilim IA7 27380 165
60 Mirny IM1 25113 158
61 Stanovoy IV1 24000 155
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Continuation of Table
Продолжение таблицы
# Names of plates and blocks Identifiers Area, steradian Area, km2 Average geometric size, km
62 Angara-Ilim IA11 23790 154
63 Angara-Ilim IA5 21770 148
64 Prisayan-Enisei IPE1 20700 144
65 Aldan II1 20680 144
66 Angara-Ilim IA10 18940 138
67 Selenga-Yablonovy IIISYa18 17730 133
68 Mirny IM21 17480 132
69 Mirny IM25 17300 131
70 Barguzin IIIB3 17060 131
71 Selenga-Yablonovy IIISYa15 16780 129
72 Angara-Ilim IA15 16380 128
73 Aldan II2 16370 128
74 Baikal-Patom IIIBP1 15700 125
75 Mirny IM11 15340 124
76 East Sayan IIIES2 15500 124
77 Selenga-Yablonovy IIISYa5 15200 123
78 Mirny IM7 14820 122
79 Selenga-Yablonovy IIISya21 14810 122
80 Stanovoy IV2 14580 121
81 Aldan IIe 14360 119
82 Tunguska IT1 13900 118
83 Barguzin (ШБ2) IIIB2 13875 118
84 Mirny IM5 13800 117
85 Mirny IM23 13460 116
86 Aldan II5 13400 116
87 Selenga-Yablonovy IIISYa19 13470 116
88 Angara-Ilim IA1 13280 115
89 Aldan II16 13150 115
90 Baikal-Patom IIIBPs 13320 115
91 Selenga-Yablonovy IIISYa13 13200 115
92 Khentei-Dauria VKhD2 13260 115
93 Mirny IM2 13053 114
94 Barguzin IIIB13 12700 113
95 Angara-Ilim IA4 12470 112
96 East Sayan IIIES10 12400 111
97 East Transbaikalie VET3 12100 110
98 Barguzin IIIB5 11700 108
99 Selenga-Yablonovy IIISYa4 11600 108
100 Stanovoy IV10 11450 107
101 Mirny IM9 11170 106
102 Selenga-Yablonovy IIISYa20 11220 106
103 Angara-Ilim IA16 11020 105
104 Angara-Ilim IA8 10570 103
105 Sayan-Altai IIISA3 10320 101
106 Selenga-Yablonovy IIISYa16 10260 101
107 Stanovoy IV7 10300 101
108 Mirny IM3 9800 99
109 Mirny IM14 9730 99
110 Baikal-Patom IIIBP4 9800 99
111 Selenga-Yablonovy IIISYa23 9420 97
112 East Sayan IIIES9 9190 96
113 Selenga-Yablonovy IIISYa14 9200 96
114 Baikal-Patom IIIBP11 8900 94
115 Sayan-Altai IIISA2 8530 92
116 Barguzin IIIB1 8490 92
117 Barguzin IIIB11 8550 92
118 Baikal-Patom IIIBP12 8500 92
119 Stanovoy IV8 8530 92
120 East Transbaikalie VET1 8600 92
121 Mirny IM6 8250 91
122 Barguzin IIIB19 8100 90
123 Baikal-Patom IIIBP2 8100 90
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Continuation of Table
Продолжение таблицы
# Names of plates and blocks Identifiers Area, steradian Area, km2 Average geometric size, km
124 Selenga-Yablonovy IIISYa24 8080 90
125 Selenga-Yablonovy IIISYay 8000 89
126 Barguzin IIIB7 7700 88
127 Baikal-Patom IIIBP5 7600 87
128 Aldan II9 7315 86
129 Aldan II18 7400 86
130 Angara-Ilim IA14 7180 85
131 Barguzin IIIB21 7200 85
132 Baikal-Patom IIIBP15 7200 85
133 Selenga-Yablonovy IIISYa9 7300 85
134 Barguzin IIIB23 6850 83
135 East Sayan IIIES12 6750 82
136 Barguzin IIIB15 6750 82
137 Baikal-Patom IIIBP7 6700 82
138 East Transbaikalie VET20 6800 82
139 Dzhida (IIID1) 6600 81
140 Khentei-Dauria (VKhD1) 6500 81
141 East Transbaikalie (VET16) 6500 81
142 East Transbaikalie (VET17) 6500 81
143 Angara-Ilim (IA2) 6400 80
144 Mirny (IM24) 6400 80
145 Aldan (II13) 6075 78
146 Barguzin (IIIB10) 6000 77
147 Mirny (IM4) 5800 76
148 Barguzin (IIIB14) 5800 76
149 Baikal-Patom (IIIBP8) 5850 76
150 Selenga-Yablonovy (IIISYan) 5800 76
151 Selenga-Yablonovy (IIISYa12) 5800 76
152 Baikal-Patom (IIIBP9) 5625 75
153 Dzhida (IIID2) 5600 75
154 Mirny (IM22) 5300 73
155 Stanovoy (IV13) 5380 73
156 Stanovoy (IV14) 5140 72
157 Barguzin (IIIB12) 5100 71
158 Barguzin (IIIB22) 5000 71
159 Stanovoy (IV 5) 5000 71
160 Angara-Ilim (IA18) 4900 70
161 Aldan (II8) 4855 69
162 Aldan (II17) 4700 69
163 Stanovoy (IV e) 4700 69
164 Mirny (IM16) 4490 67
165 Pribaikalsky fault zone 4500 67
166 Stanovoy 4500 67
167 East Transbaikalie VET8 4500 67
168 Angara-Ilim IAe 4350 66
169 East Sayan IIIES8 4400 66
170 Baikal-Patom IIIBP13 4275 65
171 Mirny IM13 4040 64
172 Aldan II12 4050 64
173 Barguzin IIIB20 4150 64
174 Mirny IM26 4040 63
175 Baikal-Patom IIIBP10 4000 63
176 Selenga-Yablonovy IIISYa8 4000 63
177 East Transbaikalie VET12 4000 63
178 East Transbaikalie VET15 4000 63
179 Barguzin IIIB8 3800 62
180 Stanovoy IV4 3800 62
181 East Transbaikalie VET13 3800 62
182 Aldan II10 3740 61
183 Barguzin IIIB24 3600 60
184 Stanovoy IV12 3590 60
185 East Transbaikalie VET7 3590 60
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End of Table Окончание таблицы
# Names of plates and blocks Identifiers Area, steradian Area, km2 Average geometric size, km
186 Aldan (II11) II11 3350 58
187 Baikal-Patom IIIBP14 3350 58
188 Aldan II15 3200 57
189 East Transbaikalie VET4 3300 57
190 East Transbaikalie VET21 3230 57
191 Aldan II14 3100 56
192 East Sayan IIIES11 3000 55
193 East Sayan IIIES6 2900 54
194 Barguzin IIIB16 2900 54
195 Stanovoy IV11 2900 54
196 Angara-Ilim IA17 2680 52
197 Mirny IM10 2690 52
198 Mirny IM15 2690 52
199 East Sayan IIIES7 2700 52
200 East Sayan IIIES13 2700 52
201 Barguzin IIIB4 2700 52
202 Selenga-Yablonovy IIISYa17 2700 52
203 Khentei-Dauria VKhD4 2690 52
204 East Transbaikalie VET9 2690 52
205 Mirny IM17 2470 50
206 East Sayan IIIES3 2475 50
207 Selenga-Yablonovy IIISYax 2400 49
208 Khentei-Dauria VKhD3 2400 49
209 East Sayan IIIES4 2250 47
210 Selenga-Yablonovy IIISYa22 2240 47
211 Selenga-Yablonovy IIISYa3 2100 46
212 Mirny IM18 2000 45
213 Barguzin IIIB9 1800 42
214 East Sayan IIIES5 1700 41
215 Barguzin IIIB17 1600 40
216 Mirny IM27 1570 39
217 Selenga-Yablonovy IIISYaxo 1500 39
218 Mirny IM8 1390 37
219 Barguzin IIIB6 1300 36
220 Barguzin IIIB18 1200 35
221 East Transbaikalie VET5 1100 33
222 East Transbaikalie VET11 1100 33
223 Mirny IM19 898 30
224 Mirny IM12 540 23
225 East Transbaikalie VET10 494 22
Notes. The table consolidates the following data: Nos. 1 to 52 - from [Bird, 2003] with recalculation for area and linear sizes; Nos. 53 to 225 - according to [Cheremnykh, 1998; Sherman et at., 2000]. Names of the largest plates are bold printed; the general laws of destruction of solid bodies do not apply to such plates. Names and two-letter identifiers of megablocks correspond to [Bird, 2003]. Names of regional blocks are from the catalogue by A.V. Cheremnykh.
Примечание. Данные с № 1 по 52 - по [Bird, 2003] с пересчетом на площадные и линейные меры; с № 53-225 - по [Cheremnykh, 1998; Sherman et al., 2000]. Жирным шрифтом выделены самые крупные литосферные плиты, не вписывающиеся в общие закономерности деструкции твердого тела. Названия мегаблоков соответствуют карте [Bird, 2003], названия региональных блоков даны по авторскому каталогу А.В. Черемных.
derived from equation 3 (Fig. 7) in the direction that has been substantiated by P. Bird (see Fig. 6). The two regression lines based on equations 3 and 7 have the same physical meaning in bilogarithmical scales, and their middle parts are identical in Figure 6 and generally similar in Figure 7. The regression lines and the equations reflect destruction patterns of the 'solid' lithosphere in a wide variety of scales.
The area estimations obtained by different methods provide for well-reasoned general conclusions concerning regular patterns of the block destruction of the Earth's lithosphere at specified hierarchical levels. The reviewed results complement each other and extend our knowledge of destruction of the lithosphere as the solid cover of the Earth. It is now reasonable to briefly review the known laws of destruction of solid bodies, i.e. rocks.
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Fig. 6. Number of plates plotted as a function of area [Bird, 2003, fig. 19].
Curve PB2002 (green) refers to the model in Fig. 7. A relatively steady slope of the curve for plate areas between 0.002 and 1 steradian suggests a power law relationship between the number of plates and their minimum size. Flattening in the left segment of the curve is due to the model incompleteness, i.e. there are many plates of smaller sizes which are not included in model PB2002. An abrupt variation of the slope in the right segment of the curve suggests that very large plates are limited in their area because of the finite area of the Earth, and perhaps also by mantle convection tractions.
Рис. 6. График взаимосвязи количества плит как функции их площади (по [Bird, 2003, fig. 19]).
Кривая РВ2002 (зеленый цвет) отражает ситуацию по данным рисунка 7. Относительно постоянный наклон кривой между границами площадей 0.002 и 1.000 стерадиан отражает взаимоотношения между числом плит и их минимальным размером. Изменение угла наклона в левой части графика отражает недостаточное число наблюдений, то есть имеются еще более мелкие, не учтенные в авторской модели блоки. Резкое изменение угла наклона в правой части графика показывает, что крупные плиты ограничены в своих размерах конечной площадью Земли, а также, возможно, конвекционными мантийными потоками.
6. Genetic sources of the lithosphere
DIVISIBILITY OF VARIOUS RANKS
It can be stated that Equation 7 and its specific variants (see Fig. 7) are sufficient to fully describe relatively small lithospheric plates and intraplate blocks of the 'solid' lithosphere up to rock lumps. The experimental methods have yielded similar equations that mathematically reflect the physics of destruction of solid bodies in a wide range of scales (from a medium-size block which size amounts to a few thousand kilometres, to a rock fragment which diameter is a dozen centimetres), and there are ground to state that the block divisibility of the lithosphere follows laws of self-similarity. In the physics of destruction of solid and viscoelastic bodies, self-similarity patterns have been noted long ago at different scale levels. However, self-simila-
rity in a very wide spectrum of hierarchical levels, from centimetres to mega sizes, has not been considered yet, and this paper is the first attempt in this respect.
Based on results of independent studies, A.N. Kolmogorov [1941] and A.F. Filippov [1962] established that rocks are subject to fragmentation according to the law of destruction of solid bodies, and the following exponential expression is linear in coordinates lgN and lgL:
lgN=f(lgL), (8),
where L is the sample's arbitrary size, and N is the number of samples.
This conclusion includes the above-described empirical relationships that refer to large-size objects, such as lithosphere blocks of various ranks and, partly,
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Fig. 7. Curves of relationships between medium lateral dimensions of lithospheric plates and blocks: 1 - according to [Bird, 2003]; 2 - according to [Cheremnykh, 1998; Sherman et al., 2000]; 3 - integrated regression line based on data (1) and (2); 4 - extrapolated regression line according to equation 1; 5 - extrapolated regression line according to equation 3. 3. Nc -row of lithospheric plates and blocks by average characteristic sizes L, km (analogues to reconstructions by P. Bird in stera-dians).
Рис. 7. Графики взаимосвязи средних поперечных размеров плит и блоков литосферы по: 1 - по данным [Bird, 2003]; 2 - по данным [Cheremnykh, 1998; Sherman et al., 2000]; 3 - совмещенная линия регрессии по данным [1 и 2]; 4 - экстраполяция линии регрессии по данным уравнения (1); 5 - экстраполяция линии регрессии по данным уравнения (3). Nc - последовательность литосферных плит и блоков в порядке увеличения усредненных характерных размеров L, км (по аналогии с построениями П. Бёрда в стерадианах).
faults in the upper brittle layer of the lithosphere, which cross each other to form the corresponding hierarchic group of the fault-block structures of the solid cover of the Earth [Sherman, 1977; Sherman et al., 2000, 2004; Seminsky, 2001, 2003; and many others]. Obviously, this property mitigates issues related to 'buckling' of the regression lines at the boundaries of Juan Fernandez (JZ) and Shetland (SL) plates (0.00241 and 0.00178 sr, respectively). Buckling occurs due to the lack of quantitative data. In our study, this issue is completely eliminated after the additional data are supplemented (Table).
It can thus be stated that deformation and destruction of the Earth's solid cover under the influence of external loads take place according to a regular pattern. From certain hierarchical levels, residual destruction (in the form of blocks varying in ranks) is distributed in accordance with the characteristic sizes of the blocks and their number, as described by Equation 7.
The regression is buckling at the transition from medium-sized plates to large ones (see Fig. 6 and 7) due to other reasons.
In terms of physics, buckling of the regression line showing the hierarchy of plate sizes versus plate areas (see Fig. 6 and 7) is related to sources of rank-variable destruction of the lithosphere as a solid body. The buckle occurs abruptly at the transition from a few very large lithospheric plates to medium-sized plates and small blocks that are statistically abundant. The regression curve is buckled from a very steep angle to a more gentle one at the boundary between South America (area of 1.03 sr) and Somali plates (area of 0.47 sr). A jump occurs when the square area is doubled. For six large plates (Africa, Antarctica, North America, Eurasia, Australia and South America; data on Pacific plate are not taken into account), the regression curve is a steep, almost vertical line. Their areas are nearly similar and almost unchangeable. Naturally, they are governed by
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another law of formation of solid lithospheric masses, which is not consistent with the laws of destruction of solid bodies. In this respect, P. Bird [2003] also notes that the fact that large plates are clearly established is indicative of a very weak dependence of plate areas from any quantitative parameters, except for the Earth radius. When the Earth radius is used as a natural independent unit to measure solid angles in steradians, it is clearly revealed that very large areas of the above-mentioned six plates (Pacific plate is an exception) are almost similar. According to P. Bird [2003], typical areas of the large lithospheric plates correlate with mantle convection: “....this characteristic size seems more consistent with whole mantle convection than with layered convection”. Indeed, the formation of the largest blocks (i.e. primary lithospheric plates originating at the stage when the protolithosphere was formed and later periods) is more in line with mantle convection than destruction of the solid body. The author fully shares these assumptions by P. Bird as they are supported by all the above discussed data, arguments and tectonophysical calculations.
As shown by the analysis of areas of the rank-variable present lithospheric plates (see Table), the total area of the seven largest plates (less than 14 % of the total number of plates) is about 10.15 steradians, and they occupy almost 81 % of the entire surface of the Earth. The area of the six continental plates (Africa, Antarctica, North America, Eurasia, Australia and South America) is more than 60 % of the planet's surface area. They are well traced in the history of Pangea and less evidently in the more distant past. Through the history of the Earth, these huge integral bodies were subject to numerous geodynamic catastrophes and transformations. Some of them lost much of the initial mass, others completely 'disappeared' in the mantle, while some of the plates have grown in size. Therefore, from one super-continental cycle to another, the lithospheric blocks differ in number and kinematics. Based on the available data, it is impossible to state for sure which of them were formed before the Archaean and are still in place, and which of them were more or less transformed. It is challenging to restore the genesis of structural relics in the cooling medium of the protolithosphere and those in the lithosphere medium with reference to the very distant past. It is an inverse problem with unambiguous solutions. As a definite and indisputable conclusion cannot be drawn, one can only assume that convection is the most probable and better argumented physical process that took place when the low-viscous medium of the protolithosphere was cooling down to form large masses in the first hypothetical (?) supercycles of Vaalbara and Ur, and, finally, convection can be viewed as a mechanism predetermining the present shape of the Earth's continents. It is most probable that the megablocks were fragmented
in the similar pattern in Kenorlend and later supercycles. However, based on the available materials, it is possible only to justify the block structure of the present stage.
It is reasonable here to quote the book by N.L. Dob-retsov et al. [2001]: “In the history of the Earth, convection in two layers was replaced by convection in the entire mantle and vice versa, and this might have taken place several times. Therefore, it is most likely that in the Earth's history, convection in the entire mantle was replaced by two-layer convection. Another possible scenario is more complicated: about 2.5 Ga ago, two-layer convection was replaced by convection in the entire mantle [Maruyama, 1994]; after a period of one billion years, two-layer convection occurred again [Honda, 1995], and, finally, a transition back to convection in the entire mantle took place in the past 150-100 Ma [Tru-bitsyn, Rykov, 2000]” (p. 111). Convection predetermines the major cycles in the geodynamics of the Earth and fragmentation of the lithosphere into megablocks. The latter are destructed under the physical laws of destruction of solid rocks. Currently, destruction of the lithosphere is actively continued in seismic zones of the continental lithosphere and zones of subduction and spreading at the margins of the lithospheric blocks.
7. Discussion
Publications on convection in the Earth's mantle and its role in global geodynamic processes are quite numerous. In the majority of papers, recent geodynamic processes are considered with an assumption that the evolution of convection in the mantle and astheno-sphere took place in two- or three-layer or more complicated patterns [Lobkovsky, 1988; Lobkovsky, Kotelkin, 2000; Lobkovsky et al., 2004; Rykov, Trubitsyn, 1994a, 1994b; Trompert, Hansen, 1988; Trubitsyn, Rykov, 2002; Trubitsyn V.P., Trubitsyn A.P., 2014]. Mantle convection was discussed in a number of publications many years ago, the most famous of which are [Pekeris, 1935; Molnar et al., 1979].
According to [Molnar et al., 1979], large sizes of continental plates are related to convection in the entire mantle, and horizontal dimensions of the plate depend on the depth of the convective process. Their analysis is based on the proportional change in the length of the subduction zone on the surface and the velocity of its sinking which depends on heat assimilation in the absence of a barrier at the border with the lower mantle.
The two-layer convection in the Earth mantle is described by L.P. Zonenshain and M.I. Kuz'min [1993] and reconstructed in a world-known scheme by S. Maruya-ma [1994].
To sum up this very brief information on the two-layer models of convection in the Earth mantle, it is
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worthy to refer again to the book by N.L. Dobretsov et al. [2001] who give much attention to convection as an important component of geodynamic processes. In their book, prior to the geodynamic analysis of processes in the mantle, asthenosphere and lithosphere, they provide a comprehensive description of the two main models of thermal-gravitational convection which provide the basis for a variety of geodynamic reconstructions. In the first model, convection take place in the entire mantle, from boundaries of the lithosphere base (30 to 100km) to the upper limit of the core (about 2900 km). The second model shows convection in the upper and lower mantle without any significant mass transfer between the two layers. In [Dobretsov et al., 2001], the authors focus on the complexity of geodynamic processes in the mantle and discuss their evolution up to the present stage of the Earth development. The concept of two-layer mantle convection is convincingly proved by results of many original experiments conducted by the authors with the use of specially designed installations [Kirdyashkin, Dobretsov, 1991; Dobretsov, Kirdyashkin, 1993]. Besides, they estimated parameters of convection. Specifically, based on the video records, they revealed flow lines in the two-layer fluid convection system and estimated horizontal and vertical velocities of the flows. Their very important observation is that convection flows above and below the interface go in different directions, and this is an evidence that vector directions of horizontal convection flows in the layers located one upon another are not interrelated. According to [Dobretsov et al., 2001], vectors of the vertical flows are similar, and this fact emphasises the major role of convection in the mantle which provides for dissipation of heat energy. Important are the digital parameters and vectors of flow velocities and cell sizes. In the bottom layer (which is thicker), cell sizes are proportional to the layer's thickness, and flow velocities are lower. In [Dobretsov et al., 2001], the concept of the two-layer convection in the Earth's mantle is well established by the experimental and actual observation data. Nonetheless, the authors note: “In the majority of cases, the geochemical data support the two-layer convection model, while much geophysical data may be interpreted in favour of convection in the entire mantle” (p. 108).
In [Trubitsyn V.P., Trubitsyn A.P., 2014], a digital model is described in detail to show that the present set of the lithospheric plates is a result of the evolution of convection. It provides an insight into the possible mode of flows in the entire mantle, movements of the masses between the upper and lower mantle, as well as between the central and lateral limits of the convection cells. Using equations, V.P. Trubitsyn and A.P. Trubitsyn calculated temperature, viscosity and velocity of mantle convection flows generated by effective diffusion-dislocation creeping in the absence of pseudo-
plastic deformation and in case of the very hard lithosphere. In particular, their temperature distribution pattern shows that small cold descending flows, i.e. small-scale convection, occur under the lithosphere. The estimated scheme of convection in the entire mantle in [Trubitsyn V.P., Trubitsyn A.P., 2014] is consistent with our ideas of the primary genetically emerging block divisibility of the protolithosphere. In the initial state, the protolithosphere remains uniform at the surface, has a roughly constant thickness and increased quasi-strength at the primary inter-cell boundaries whereat the viscosity of the medium is increasing due to cooling of the protolithosphere.
Regretfully, initial divisibility of the emerging upper solid cover of the Earth is taken into account by few researchers in their estimations, while such divisibility is one of the most likely results of convection in the cooling protolithosphere. This is obviously due to the absence of direct geological materials and the lack of appropriate paleo-reconstruction methods that can integrate geological, geophysical and geochemical databases. As shown by our study, computational methods are helpful, to a certain extent, in solving ill-conditioned inverse problems to reconstruct processes and structures of the distant past.
The concept of the single-layer convection, assuming that convection took place through the %-radius depth at the initial stage when the Earth lithosphere was formed, is acceptable and seems quite realistic, even though the literature is still insufficient on this subject. This justifies the author's efforts to clarify the origin of divisibility of the primary non-solid, almost continuous cover of the Earth which evolution history includes periods of the Phanerozoic and contemporary faulting in the lithosphere and its fragmentation under the laws of destruction of solid bodies. It can be noted in general that the upper cover of the Earth is subject to destruction in a regular pattern, from larger to smaller masses.
8. Conclusion
This study is pioneering in tectonophysical reconstruction of initial divisibility of the protolithosphere as a result of convection in the cooling primitive mantle. Initial division of the protolithosphere into separate masses, i.e. prototypes of the blocks, and their size was predetermined by the emerging Rayleigh-Benard convection cells. In studies of geology and geodynamics, the Rayleigh-Benard convection cells were first referred to as a factor to explain the formation of initial continental cores. Considering the Rayleigh-Benard cells and their structural relics can help clarify initial divisibility of the protolithosphere and the origin of the major lithospheric plates, i.e. prototypes of continents.
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In our opinion, the initial mega-scale block structure of the protolithosphere and the emerging lithosphere were predetermined by the Rayleigh-Benard cells as they were preserved in the emerging lithosphere and their lower boundaries corresponded to the coremantle boundary, i.e. one of the major discontinuities of the planet. Our theoretical estimations are in good agreement with the number and sizes of the Earth's theorized first supercontinents, Vaalbara and Ur.
In our tectonophysical discussion of the formation of the lithospheric block structure, we analyse in detail the map of modern lithospheric plates [Bird, 2003] in combination with the materials from [Sherman et al., 2000]. In the hierarchy of the blocks comprising the present lithosphere, which sizes are widely variable, two groups of blocks are clearly distinguished. The first group includes megablocks with the average geometric size above 6500 km. Their formation is related to convection in the Earth mantle at the present stage of the geodynamic evolution of the Earth, as well as at all the previous stages, including the earliest one, when the protolithosphere emerged. The second group includes medium-sized blocks with the average geometric size of less than 4500 km and those with minimum sizes, such as rock lumps. They reflect primarily the degradation of the megablocks as a result of their destruction due to high internal stresses in excess of the tensile strength of the medium. This group may also include blocks which formation is related to convection in the upper mantle layer, asthenosphere. There are grounds to assume that through the vast intermediate interval of geologic time, including supercycles of Kenorlend, Rodin, and Pangea, the formation of the large lithospheric blocks was controlled by convection, and later on, they were ‘fragmented’ under the physical laws of destruction of solid bodies. However, it is difficult to
clearly distinguish between the processes that predetermine the hierarchy of formation of the block structures of various origins - sizes of ancient lithospheric blocks cannot be estimated unambiguously.
Thus, mantle convection is a genetic endogenous source of initial divisibility of the cooling upper cover of the Earth and megablock divisibility of the lithosphere in the subsequent and recent geodynamic development stages. In the present stage, regular patterns of the lithospheric block divisibility of various scales are observed at all the hierarchic levels. The areas of the lithospheric megaplates result from regular changes of convection processes in the mantle, which influenced the formation of plates and plate kinematics. Fragmentation of the megaplates into smaller ones is primarily a result of destruction of the solid lithosphere under the physical laws of destruction of solid bodies under the impact of high stresses.
9. Acknowledgements
The author wishes to express his sincere gratitude to his colleagues: M.I. Kuz'min, Academician of RAS, E.V. Sklyarov, Corresponding Member of RAS, V.A. San'kov, Head of the Laboratory of Recent Geodynamics, IEC SB RAS, professor R.M. Lobatskaya and Ph.D. A.I. Kiselev for constructive scientific discussions that were helpful for improving the paper. The author appreciates the help provided by A.V. Cheremnykh in consolidating the input data and measurements of continental lithospheric blocks varying in hierarchic levels.
The study was conducted under the Research Plan of the Laboratory of Tectonophysics, IEC SB RAS with a partial financial support by the Russian Foundation for Basic Research (Grant No. 15-55-53023).
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Sherman, Semen I., Academician of the Russian Academy of Natural Sciences,
Doctor of Geology and Mineralogy, Professor, Chief Researcher Institute of the Earth's Crust, Siberian Branch of RAS 128 Lermontov street, Irkutsk 664033, Russia Tel.: (3952)428261; И e-mail: ssherman@crust.irk.ru
Шерман Семен Иойнович, академик Российской академии естественных наук, докт. геол.-мин. наук, профессор, г.н.с.
Институт земной коры СО РАН
664033, Иркутск, ул. Лермонтова, 128, Россия
Тел.: (3952)428261; И e-mail: ssherman@crust.irk.ru
