Derivation of equations of the model of the dynamic behaviour of the three-dimensional atmospheric cloud of electrically charged ice crystals under the influence of electrostatic forces

Abstract

The purpose of this research is to create a theoretical model describing the dynamics of a cloud of electrically charged ice crystals in a three-dimensional space. The linear motion of the crystals was declined and considerations were limited to rotation of these crystals under the influence of electrostatic forces. The crystals were modeled as thin circular discs to simulate hexagonal plate ice crystals. This has reduced problem to the problem of two degrees of freedom (that allow for rotation around its center of gravity) per crystal. The basic assumption of the model is the crystals in the cloud are synchronized, which means that the differences in rotation of adjacent crystals are small. With this assumption, it was possible to derive continuous differential equations describing the dynamic behavior of such a medium on a macro scale from equilibrium of moments (from electrostatic forces) acting on the single crystal and discrete equations of motion of the cloud. Finally, the system of differential equations is second order with respect to time and space with trigonometric variable coefficients is derived. Examples of solutions of the system of equations are shown and compared with both solutions of similar theoretical problems available in the literature and optical effects in the clouds described by observers. It can be used in the future to better understand the dynamic optical phenomena occurring in the atmosphere.