Abstract

Problem- The mystery of origin Earth's magnetic field is connected with the solution of the problem of convection in the mantle and in the outer liquid core. However, the mathematical analysis of the stability conditions of convection in the core and the mantle of the Earth still insufficiently developed. Even in the classical formulation of the problem the main roots of the dispersion equation has not been studied. Purpose- It should be more fully and rigorously consider the problem of convection in rotational thermodynamically inhomogeneous viscous liquid. In this task, you must obtain a general dispersion equation and perform a mathematical analysis of the roots of this equation. You also need find the shape of convective cells in the bowels of the Earth. Approach- We start from the Navier-Stokes equations for thermodynamically inhomogeneous viscous fluid. In the Boussinesq approximation, neglecting the square of the velocity of fluid flow was obtained and carefully studied the cubic dispersion equation. The roots of the dispersion equation are investigated in order to detect the areas of stability and instability of convective currents. Findings- In the complex domain is made analysis of roots of the cubic dispersion equation. It is shown that on the chart "angular velocity – temperature gradient" there are areas of stable and unstable fluid motion. The Earth’s daily rotation does not affect the convection in the mantle, where convective cells are almost cubic with a characteristic size  But the convection in the Earth's outer liquid core is divided into a plurality of thin "pipes." These tubes are parallel to the Earth's axis, and currents up and down them alternate. The role of the daily rotation of the Earth in the creation of such mechanism of convection is very important. Implications- The method is applied to the study of convection in the mantle and liquid core of the Earth. We find some important properties of the convective cells in the mantle and the liquid core. The results are in satisfactory agreement with geological data. We draw attention to the fact that the structure of "tubes" in the liquid core is such that it stands out particularly equatorial toroidal ring. In this ring the convection is absent and, most likely, the liquid of the core there is in a highly turbulent state. We assume that this is the zone of turbulence ring may be responsible for the emergence of Earth's magnetic field. Originality- These results are new in mathematical physics and dynamics of the Earth's shells.

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