Nonlinear vortex solution for perturbations in the Earth's ionosphere

There is much observational evidence of different fine structures in the ionosphere and magnetosphere of the Earth. Such structures are created and evolve as a perturbation of the ionosphere's parameters. Instead of dealing with a number of linear waves, we propose to investigate and follow up the perturbations in the ionosphere by dynamics of soliton structure. Apart from the fact that this is a more accurate solution, the advantage of soliton solution is its localization in space and time as a consequence of the balance between nonlinearity and dispersion. The existence of such a structure is driven by the properties of the medium. We derive the necessary condition for having a nonlinear soliton wave, taking the vortex shape as a description of the ionosphere parameter perturbation. We employ a magnetohydrodynamical description for the ionosphere in plane geometry, including rotational effects, magnetic field effects via ponderomotive force, and pressure and gravitational potential effects, treating the problem self-consistently and nonlinearly. In addition, we consider compressible perturbation. As a result, we have found that Coriolis force and magnetic force on the one hand and pressure and gravity on the other hand determine dispersive properties. Dispersion at higher latitudes is mainly driven by rotation, while near the Equator, within the E and F layers of the ionosphere, the magnetic field modifies the soliton solution. Also, a very general description of the ionosphere results in the conclusion that the unperturbed thickness of the ionosphere layer cannot be taken as an ad hoc assumption: it is rather a consequence of equilibrium property, which is shown in this calculation.

Nonlinear vortex solution for perturbations in the Earth’s Ionosphere

There are many observational evidences of different fine structures in the ionosphere and magnetosphere of the Earth. Such structures are created and evolve as a perturbation of the ionosphere’s parameters. Instead of dealing with number of linear waves, we propose to investigate and follow up the perturbations in the ionosphere by dynamics of soliton structure. Apart of the fact that it is more accurate solution, the advantage of soliton solution is its localization in space and time as consequence of balance between nonlinearity and dispersion. The existence of such structure is driven by the properties of the medium. We derive necessary condition for having nonlinear soliton wave, taking the vortex shape, as description of ionosphere parameters perturbation. We employ magnetohydrodynamical description for the ionosphere in plane geometry, including rotational effects, magnetic field effects via ponderomotive force, pressure and gravitational potential effects, treating the problem self-consistently and nonlinearly. In addition, we consider compressible perturbation. As a result, we have obtained that Coriolis force and magnetic force at one side, and pressure and gravity on the other side, determine dispersive properties. Dispersion at higher latitudes is mainly driven by rotation, while near the equator, within the E and F-layer of ionosphere, magnetic field modifies the soliton solution. Also, very general description of the ionosphere results in the conclusion that the unperturbed thickness of the ionosphere layer cannot be taken as ad hoc assumption, it is rather consequence of equilibrium property, which is shown in this calculation.

Vortex Dynamics on the f and beta Plane and Wave Radiation by Vortices

Quasi-geostrophic dynamics being essentially the vortex dynamics, the main notions of vortex dynamics in the plane are introduced in this chapter. Dynamics of vorticity is treated both in Eulerian and Lagrangian descriptions. Dynamics of point vortices and vortex patches (contour dynamics) are recalled, as well as discretisations of the vorticity equation preserving Casimir invariants, which reflect Lagrangian conservation of vorticity. The influence of the beta effect upon vortices is illustrated, and exact modon solutions of the QG equations on the f and beta planes are constructed. Basic notions of turbulence and specific features of two dimensional turbulence are reviewed for future use. Lighthill radiation of gravity waves by vortices is illustrated on the example of a pair of point vortices, and back-reaction of the radiation upon the vortex system is demonstrated and analysed. Influence of rotation upon the Lighthill radiation is explained. Construction of the Kirchhoff vortex solution is proposed as a problem.

Magnetic monopole versus vortex as gauge-invariant topological objects for quark confinement

First, we give a gauge-independent definition of chromomagnetic monopoles in [Formula: see text] Yang–Mills theory which is derived through a non-Abelian Stokes theorem for the Wilson loop operator. Then we discuss how such magnetic monopoles can give a nontrivial contribution to the Wilson loop operator for understanding the area law of the Wilson loop average. Next, we discuss how the magnetic monopole condensation picture are compatible with the vortex condensation picture as another promising scenario for quark confinement. We analyze the profile function of the magnetic flux tube as the non-Abelian vortex solution of [Formula: see text] gauge-Higgs model, which is to be compared with numerical simulations of the [Formula: see text] Yang–Mills theory on a lattice. This analysis gives an estimate of the string tension based on the vortex condensation picture, and possible interactions between two non-Abelian vortices.