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The Purpose of the Work: The modern mantle and crust have a complex structure and, in addition, contain both thermal and material heterogeneities, as evidenced by the results of seismic and electromagnetic studies. Changes are also reflected by the change in the mineralogical and chemical composition of the matter. This structure was formed for the long geological history of the planet’s development and the process continues at the present time. The system remains unsteady. To understand the evolution of such dynamic structures, information is needed about the initial state of the system, in our case, about the state of the Earth at the final stage of its formation. It can be obtained only by the results of numerical modeling based on the results of the investigation of the evolution of isotope systems. Therefore, the purpose of the work is to identify the features of the formation of mineral deposits in the early crust and mantle. For this, it is necessary to obtain variants of the numerical solution of the problem of the formation of the planet. Solution Methods: An algorithm for solving a non-linear system of differential equations for solving a 3D boundary dynamic problem in the sphere of an increasing radius is developed. The numerical method of “through account” is used in the work. Results: Based on methods for solving boundary value problems for a system of differential equations with the use of new results of mineralogical and isotope studies of the oldest material samples, quantitative variants of the thermal evolution of the Earth, directly determining the formation of early metallogeny, are constructed. It is shown that the random distribution of particles and bodies of a protoplanetary cloud during the accumulation of the planet causes the formation of a random material and temperature composition of the growing crust and mantle, which ensured a special metallogeny of the cratons and their framing, which no longer repeated in the geological history of the planet. A special role in it was played by changes in the gravitational field during the growth of the planet and the angular velocity of the Earth’s rotation. Further Research: It is proposed to extend the results obtained to the conditions for taking into account the dynamics of the double Earth-Moon system.

The study of regularities in the formation of mineral deposits has always been the main task of geology. By the end of the last century it was found out that the distribution of deposits obeys to certain regularities both in localization in the Earth’s crust and in age. It was realized that changes in the material composition of deposits are due to both the nature of tectonic movements in the evolution of the Earth, and the time of appearance of each shell of the Earth. The greatest success was achieved in the study of tectonic processes in the lithosphere. However, it became clear that it is necessary to clarify the mechanisms and energy of interaction between internal and surface structures. Initially, a one-dimensional model was used to solve this problem. Adiabatic compression and three-dimensional effects due to the deposition of bodies on the surface of a growing planet have not been taken into account. The main source of internal energy of the protoplanet at this stage of its formation was the release of heat accompanying the natural radioactive decay of long-living uranium and thorium. The contribution due to the release of heat from the impact of small particles and bodies of the protoplanetary cloud on the surface of the growing planet turned out to be very small, since most of it, according to Stefan’s law, was radiated into space. As a result, sufficiently low estimates of the temperature of the inner regions of the protoplanet were obtained by the end of the process of their accumulation (^{26}Al in the matter of the protoplanetary cloud [

“feeding” zone of the Earth were obtained. It was shown in [^{26}Al in the pre-planetary matter of the Earth.

The interest of geologists in the problems of geodynamics, the development of the structure of the interior regions of the planet has increased significantly after the publication of the paper [

The solution of the problem should describe the formation of a protoplanet from the matter of bodies and particles of a protoplanetary solar cloud. The statement of the problem is described in detail in [^{26}Al in the matter of the protoplanetary cloud [

∂ m ( t ) ∂ t = 2 ⋅ ( 1 + 2 ⋅ θ ) ⋅ R 2 ⋅ ω ⋅ ( 1 − m ( t ) M ) ⋅ σ , t ≥ 0 , m ( 0 ) = m 0 , R ( 0 ) = R 0 , (1)

where: ω is the angular velocity of the orbital motion, σ is the surface density of matter in the “feeding” zone of the planet, m(t) is the pre planetary mass, M is the modern mass of the planet, R is the radius of the growing body, θ is the statistical parameter taking into account the particle distribution by masses and velocities in the “feeding” zone, m_{0}, R_{0} is the mass and radius of the arising planet at the initial moment of time. For each achieved mass value, the distribution along the radius of density, hydrostatic pressure, and melting point in the inner regions is calculated. The density distribution in the mantle is calculated from the Williamson-Adams equation, which can be written as:

d ρ = ρ ( r ) g ( r ) Φ d r , 0 ≤ r ≤ R , ρ ( R ) = ρ 0 , (2)

where ρ ( r ) ―density, d ρ ―density increment, d r ―radius increment, ρ 0 ―

density on the Earth’s surface, g ( r ) = G m r 2 ―gravitational acceleration. Here G

is the gravitational constant, m is the mass enclosed inside the sphere of radius r. Φ ―a seismic parameter that can be defined as:

Φ = K ρ ( r ) , (3)

where K is the modulus of compression. The distribution of the hydrostatic pressure along the radius is found from the Euler equation describing the mechanical equilibrium of a liquid [

d p d r = ρ ( r ) g ( r ) , 0 ≤ r ≤ R , p ( R ) = 0 , (4)

where p(r) is the pressure. In the core, mainly of iron composition, the dependence of the melting point on pressure is calculated from [

k ρ G m R Δ r Δ t = ε σ [ T 4 − T 1 4 ] + ρ c P [ T − T 1 ] Δ r Δ t , (5)

where: ρ is the density of matter, G is the gravitational constant, m is the mass of the growing planet, R is its radius, T is the temperature at the outer boundary, T_{1} is the temperature at the sunflower point, ε is the transparency coefficient of the medium, c_{p} is the specific heat, k―the fraction of the potential energy converted into heat. The temperature distribution in the body of the increasing radius is found from the solution of the heat conduction equation with allowance for the possibility of a melt appearance without explicitly isolating the position of the phase transition boundaries, convective heat transfer in the melt [

c эф ρ ( ∂ T ∂ t + V ∇ T ) = ∇ ( λ эф ∇ T ) + Q , 0 ≤ r ≤ R , 0 < θ < 2 π , 0 < φ ≤ 2 π , T ( R , θ , φ , t ) = T R , T ( r , θ , φ , 0 ) = T 0 . (6)

Here T = T ( r , θ , φ , t ) is the temperature at time t, V ―here is the mass velocity of the substance, Q = Q ( r , θ , φ , t ) ―the capacity of the internal heat sources, c эф , λ эф the effective heat capacity and thermal conductivity, which take into account the heat of fusion in the Stefan problem [_{R} is the temperature on the surface of the growing planet, obtained from Equation (5), T_{0} is the initial temperature distribution, ∇ is the Nabla differential operator. The value of the effective heat capacity was calculated in accordance with the idea of the through-account method [

с эф = с P + L 2 Δ T , (7)

where L is the heat of the phase transition. Convective heat transfer in the melt layer is accounted for by the effective coefficient of thermal conductivity [

λ э ф = { λ ⋅ ( 1 + 0.2 R a − R a k 3 ) , R a > R a k λ , R a ≤ R a k , (8)

R a = α ⋅ g ⋅ Δ T ⋅ h 3 χ ⋅ ν , (9)

where Ra is the Rayleigh number, Ra_{k} is the critical value of the Rayleigh number; α is the coefficient of thermal expansion, h is the thickness of the convective layer; ΔT is the temperature difference at its boundaries; g is the magnitude of the gravitational acceleration; v is the effective viscosity in the layer. The regions of the melt were compared with the melting point and temperature, for a given moment of time. The power of internal heat sources is the sum of the heat release during radioactive decay of ^{26}Al and adiabatic compression:

Q = Q A 26 l + Q 1 , (10)

Q A 26 l = ρ ⋅ K Al 2 O 3 ⋅ 54 102 ⋅ N A μ Al 2 O 3 ⋅ A 26 l A 27 l ⋅ e − λ ⋅ t ⋅ q ⋅ λ , (11)

where K Al 2 O 3 is the concentration at the initial time; λ is the decay constant; q is the energy of decay; ^{26}Al/^{27}Al is the isotope ratio [_{1}―capacity of heat release by adiabatic compression [_{A}―number of Avogadro, μ Al 2 O 3 ―concentration of Al_{2}O_{3}, ρ―density.

The boundary value problem was solved numerically. When solving Equation (5), using a random function that is a standard random number generator, we get the distribution of thermal heterogeneities in terms of kinetic energy and over the surface of the growing planet. The boundary value problem for Equation (6) written in spherical coordinates was solved by the method of finite differences using the splitting scheme [

It is thermal and material heterogeneities that are the objects, which are registered by the methods of modern geophysics and geochemistry in the mantle. As can be seen from the presented results, they are formed during the accumulation of the Earth and form heterogeneities of various sizes and shapes. The resulting inhomogeneous inclusions change with time. In the presented model variants of their evolution are considered. which are obtained as a result of a numerical solution of the boundary value problem for a system of differential equations describing the change in the mass of a growing planet using the Safronov equation, the flow of matter using the Navier-Stokes equation, the motion of phase boundaries using the Stefan problem; On the surface of a growing planet, conditions are formulated that take into account the random distribution of bodies and particles as a function of the energy and the point of incidence released. Since this distribution of falling bodies is given randomly, and it randomly changes during the accumulation of the model of a growing planet, it was received a set of different solutions to the problem. As it shown by the different results of the solution, to the end of the planet’s accumulation a relatively thin heterogeneous crust forms, and in the upper mantle an asthenosphere layer. In the mantle, heterogeneous inclusions are observed, ranging in size from the first to several hundred kilometers.

The insets show the results on an enlarged scale. Radial chains of heterogeneities of the elevated temperature in the mantle are seen for the instantaneous time.

As it is shown in the upper inset to

inclusions under the action of the Stokes force begin to float, leading to the formation of jets or so-called mantle plumes. Floating local thermal inclusions and formed jets reduce the convective stability of the mantle and promote the development of regular convective currents. The very dynamics of heterogeneous inclusions, their random distribution in temperature and material composition, provided a special metallogeny of the cratons, which no longer repeated in the geological history of the planet. In particular, it caused the fact that contrary to the expected, the magmatism belonging to the effect of plumes of the same age and depth of embedding is not sufficient to form a close material composition arriving to the earth’s surface and this can be a consequence only of contamination processes.

On

The numerical solution of the formulated problem for a three-dimensional model with allowance for the random distribution of bodies and particles to the surface of a growing Earth for the first time made it possible to trace the further fate of thermal heterogeneities in the nucleus, mantle and crust in formation. The thermal conductivity of a large number of primary, due to the composition of the trapped bodies of the protoplanetary cloud, increased in comparison with

the host material, leads to the formation of jets, plumes, and the differentiation of accumulated matter in them in the gravitational field. Unlike the further formation of jets in the modern Earth structure, when they are formed mainly on the core-mantle section and the 680 km phase boundary in the mantle, these plumes could be formed at arbitrary depths depending on the size and position of the primary heterogeneities. Thus, at this stage of the development of the Earth P-T, the conditions for separation of the magmatic melt and its composition could be extremely diverse; this is the reason for its difference in composition from the modern one. This requires a careful study of the dynamics of the emerging binary system Earth-Moon. Since in the process of its formation, the distribution of pressure and temperature in the inner regions of the emerging Earth, the velocity of the planet’s orbital rotation may substantially differ from these thermodynamic parameters in the Earth as alone body.

These results had been achieved by support of the grant RFBR № 16-05-00540.

Khachay, Y. and Antipin, A. (2018) Features of the Formation of Mineral Deposits at the Initial Stages of Formation of the Earth’s Mantle and Crust. Open Journal of Geology, 8, 222-231. https://doi.org/10.4236/ojg.2018.83014