1. The single-mode vortex in a nonlinear medium
For two reasons we shall consider here a single-mode vortex. The first reason: this spatial configuration has shown the greatest stability in experiments with the equations of Maxwell-Hertz. The second reason: all the details of evolution of electrical and magnetic fields can be easily observed in this elementary vortex.
The single-mode vortical structure is the lowest configuration and corresponds to the attached Legander polynomial. The starting position of fields is presented schematically in Fig.20. Here the electric field E has the shape of the dipole, oriented across the spin axis. The magnetic field H has the shape of a closed ring, the axis of which is perpendicular to the spin axe and the electric-dipole axe. The symbol P designates Poiting vector in the corresponding places of a vortex.
All the calculations were done in conditions of a scalar approximation of dielectric and magnetic conductivity. Parameters of the induced-by-medium nonlinearity are chosen so that e and m in a vortex zone do not exceed 1 in the area of small amplitudes and 3,5 in the area of their maxim. The tensor representation of the medium parameters has been difficult mainly for technical reasons: it requires much greater on-line storage of a computer. But modern personal computers are essentially limited with regard to this characteristics. As result, all the described below numerical experiments are practically illustrative.
Stationary values are imposed on variable fields. The electric field has a radial direction (charge), and a magnetic field has an axial direction (magnetic moment). Both have 5 % of maxima of the variable-components amplitudes. Being set artificially, these fields remain unchanged further on, in the course of a numerical experiment. Possibly, the process of their relaxation up to any equilibrium natural state requires much more time, than half the turn of a vortex.
The evolution of a single-mode vortex in case it makes a turn within the limits of 0 - p are represented in Fig. 21 by means of isosurfaces of squared values of E (in a red colour) and H (dark blue). Numbers of cycles are designated by numerals by means of which the corresponding position is fixed. Here you can observe the way the general field configuration rotates.
Fig. 22 illustrates the situation in the middle of a vortex in a plane, which is in the middle of it and perpendicular to the axis z. The tic mark is the same as in Fig. 21. Here vectors indicate an electric field, and their color and intensity indicate the distribution of the amplitude z – the components of a magnetic field.
Let us note once again the fact that the approximate solution is presented here. When calculating, the exact equations of Maxwell-Hertz in the form of (Ï3.8) from Appendix 3 have been used. But initially tensor parameters of the medium were replaced by scalars e and m. Therefore, we could not see the picture at a turning angle, exceeding 180°.