Аннотация: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 ≈ 150km. 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.
Keywords: Navier-Stokes equations, thermodynamically inhomogeneous viscous liquid, Boussinesq
approximation, dispersion equation, stable (unstable) motion, convection, magnetic field