Geodynamics

Fys, M.; Yurkiv, Mariana; Brydun, A. M.; Lozynskyi, V.

doi:10.23939/jgd2016.02.036

Cited by 4 publications

(4 citation statements)

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“…The described method is more general in comparison with those described and implemented in (Fys et al, 2016). The generalizations relate primarily to the construction of matrix relations i A with (17) and obtaining an approximate solution of (17) regardless of the conditions of its existence.…”

Section: Resultsmentioning

confidence: 99%

“…For systems (17) at different values i t and i r , the following cases are possible: the system has a single solution, no solutions (overridden), multiple solutions (undefined). Numerical experiments in Fys et al 2016) show that the only solution exists when 3 n = and 4 n = (except for I in ( 16)).…”

Section: Theoretical Basis Of the Approximate Construction Methods Of The Gradient And The Mass Distribution Function Of The Ellipsoidal mentioning

confidence: 99%

“…For the basic function in the construction we take the function of mass distribution in the form (Meshcheryakov, 1991): (Dzewonski & Anderson, 1981) In the article (Fys et al, 2016), a method for determining these elements by a phased solution of systems of equations with one free unknown and its further determination from condition (14) for 2,0 tpqs ==== has been proposed To do this, we form a matrix of equations according to formulas (1) and free terms according to (2), whence we obtain the necessary surface moments ,4, pqs pqs s ++= ,, pqspairwise, according to which with (24) we calculate the power moments (6), then using the power moments and formula (2) -coefficients…”

Section: The Implementation Of the Approximate Construction Methods Algorithmmentioning

confidence: 99%

“…As a result, we obtain (Fys et al, 2016) We construct the next approximation by the Stokes constants of the third and fourth orders.…”

Section: Mnk Amnk ++£mentioning

confidence: 99%

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Geodynamics

Fys¹,

Brydun²,

Yurkiv³

et al. 2020

JGD

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Purpose. To create an algorithm for constructing a three-dimensional masses distribution function of the planet and its derivatives taking into account the Stokes constants of arbitrary orders. Being based on this method, the task is to perform the research on the internal structure of the Earth. Methodology. The derivatives of the inhomogeneous mass distribution are presented by linear combinations of biorthogonal polynomials which coefficients are obtained from the system of equations. These equations follow from integral transformations of the Stokes constants, the calculation process is carried out by a sequential approximation and for the initial approximation we take a one-dimensional density model that is consistent with Stokes constants up to the second inclusive order. Next, the coefficients of expansion of the potential of higher orders are determined up to a predetermined order. In this case, the information on the power moments of the density of surface integrals makes it possible to analyze and control the iterative process. Results. The results of calculations using the software according to the described algorithm are obtained. A fairly high degree of approximation (sixth order) of three-dimensional mass distributions function is achieved. Carto diagrams were created being based on the values of deviations of the three-dimensional average distributions (“isodens”), which give a fairly detailed picture of the Earth’s internal structure. The presented maps of “inhomogeneity’s” at characteristic depths (2891 km core – mantle, 5150 km internal – external core) allow us to draw preliminary conclusions about global mass movements. At the same time, the information on derivatives is significant for interpretation. First of all, it should be noted that the gradient of “inhomogeneity’s” is directed toward the center of mass. The presented projections of this gradient on a plane perpendicular to the rotation axis (horizontal plane) show the tendency of spatial displacements. Scientific novelty. Vector diagrams of the gradient, in combination with carto diagrams, give a broad picture of the dynamics and possible mechanisms of mass movement within the planet. To a certain extent, these studies confirm the phenomenon of gravitational convection of masses. Practical significance. The proposed algorithm can be used in order to build regional models of the planet, and numerical results can be used to interpret global and local geodynamic processes inside and on the Earth’s surface.

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Section: Resultsmentioning

confidence: 99%

Section: Theoretical Basis Of the Approximate Construction Methods Of The Gradient And The Mass Distribution Function Of The Ellipsoidal mentioning

confidence: 99%

Section: The Implementation Of the Approximate Construction Methods Algorithmmentioning

confidence: 99%

“…As a result, we obtain (Fys et al, 2016) We construct the next approximation by the Stokes constants of the third and fourth orders.…”

Section: Mnk Amnk ++£mentioning

confidence: 99%

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Geodynamics

Fys¹,

Brydun²,

Yurkiv³

et al. 2020

JGD

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Geodynamics

Fys¹,

Brydun²,

Yurkiv³

2019

JGD

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Purpose. To investigate the technique for constructing a three-dimensional distribution function for the masses of the interior of the Earth and its derivatives, coordinated with the parameters of the planet's gravitational field to fourth order inclusive. By using the mass distribution function constructed, to make an interpretation of the features of the internal structure of an ellipsoidal planet. Methodology. Based on the created initial approximation of the function, which includes a reference density model, further refinements are built. Using Stokes constants up to the second order inclusive, we give the following approximation, which we subsequently take as zero. In this case, the use of Stokes constants up to the fourth order inclusive leads to the solution of systems of equations. It is established that the addition of one identity leads to uniqueness of the solution. One system with Stokes constants 404244 ,,. ccc is an exception. It is necessary to note that the computation process is controllable, since the power moments of the density derivatives are reduced to quantities that take into account the value of the density on the surface of the ellipsoid. Results. In contrast to the second-order model describing gross global inhomogeneities, the obtained distribution function gives a detailed picture of the location of the density anomalies (the deviation of the three-dimensional function from the averaged over the sphere is "isodense") Analysis of maps at different depths 2891 km (core-mantle), 5150 km (internal-external core) allows us to draw preliminary conclusions about the global mass redistribution due to the rotating component of the force of gravity over the entire radius, as well as due to the horizontal components of the density gradient. On the contrary, the minimum of such a deviation is observed in the polar parts of the Earth, which also has its explanation: the magnitude of the rotational force decreases when approaching the pole. The mass distribution function is constructed using the proposed method to describe in more detail the picture of the mass distribution. Of particular interest are sketch maps of the components of the density anomaly function gradient, namely the component which coincides with the axis Oz-for the upper part of the shell which is negative, and for the lower part it is positive. This means that the gradient vector is directed toward the centre of mass. The nature of the values for other two components is different both in sign and in magnitude and depends on the placement point. The cumulative consideration and consideration of all the quantities makes possible a more complete interpretation of the processes inside the Earth. Originality. In contrast to the traditional approach, the changes for the density derivatives of one variable (depth), obtained from the Adams-Williams equation, in this paper made an attempt to obtain derivatives with Cartesian coordinates. Used in the described method, the parameters of the gravitational field up to the fourth order inclus...

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Geodynamics

Fys¹,

Brydun²,

Yurkiv³

et al. 2021

JGD

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Purpose. To investigate the features of the algorithm implementation for finding the derivatives of the spatial distribution function of the planet's masses with the use of high-order Stokes constants and, on the basis of this, to find its analytical expression. According to the given methodology, to carry out calculations with the help of which to carry on the study of dynamic phenomena occurring inside an ellipsoidal planet. The proposed method involves the determination of the derivatives of the mass distribution function by the sum, the coefficients of which are obtained from the system of equations, which is incorrect. In order to solve it, an error-resistant method for calculating unknowns was used. The implementation of the construction is carried out in an iterative way, while for the initial approximation we take the three-dimensional function of the density of the Earth's masses, built according to Stokes constants up to the second order inclusive, by dynamic compression by the one-dimensional density distribution, and we determine the expansion coefficients of the derivatives of the function in the variables to the third order inclusive. They are followed by the corresponding density function, which is then taken as the initial one. The process is repeated until the specified order of approximation is reached. To obtain a stable result, we use the Cesaro summation method (method of means).. The calculations performed with the help of programs that implement the given algorithm, while the achieved high (ninth) order of obtaining the terms of the sum of calculations. The studies of the convergence of the sum of the series have been carried out, and on this basis, a conclusion has been made about the advisability of using the generalized finding of the sums based on the Cesaro method. The optimal number of contents of the sum terms has been chosen, provides convergence both for the mass distribution function and for its derivatives. Calculations of the deviations of mass distribution from the mean value ("inhomogeneities") for extreme points of the earth's geoid, which basically show the total compensation along the radius of the Earth, have been performed. For such three-dimensional distributions, calculations were performed and schematic maps were constructed according to the taken into account values of deviations of three-dimensional distributions of the mean ("inhomogeneities") at different depths reflecting the general structure of the Earth's internal structure. The presented vector diagrams of the horizontal components of the density gradient at characteristic depths (2891 km - core-mantle, 700 km - middle of the mantle, also the upper mantle - 200, 100 km) allow us to draw preliminary conclusions about the global movement of masses. At the same time, a closed loop is observed on the “core-mantle” edge, which is an analogy of a closed electric circuit. For shallower depths, differentiation of vector motions is already taking place, which gives hope for attracting these vector-grams to the study of dynamic motions inside the Earth. In fact, the vertical component (derivative with respect to the z variable) is directed towards the center of mass and confirms the main property of mass distributions - growth when approaching the center of mass. The method of stable solution of incorrect linear systems is applied, by means of which the vector-gram of the gradient of the mass distribution function is constructed. The nature of such schemes provides a tool for possible causes of mass redistribution in the middle of the planet and to identify possible factors of tectonic processes in the middle of the Earth, i.e indirectly confirms the gravitational convection of masses. The proposed technique can be used to create detailed models of density functions and its characteristics (derivatives) of the planet's interior, and the results of numerical experiments - to solve tectonics problems.

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Geodynamics

Cited by 4 publications

References 1 publication

Geodynamics

Geodynamics

Geodynamics

Geodynamics

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