Model of Propagation of VLF Beams in the Waveguide Earth-Ionosphere. Principles of Tensor Impedance Method in Multilayered Gyrotropic Waveguides

Abstract

Abstract. Modeling propagation of VLF electromagnetic beams in the waveguide earth-ionosphere (WGEI) is of a great importance because variation in the characteristics of these waves is an effective instrument for diagnostics the influences on the ionosphere from above (Sun-Solar Wind-Magnetosphere-Ionosphere), from below (the most powerful meteorological, seismogenic and other sources in the lower atmosphere and lithosphere/Earth, such as hurricanes, earthquakes, tsunamis etc.), from inside the ionosphere (strong thunderstorms and lightning discharges) and even from the far space (such as gamma-flashes, cosmic rays etc.). Thus, VLF became one of the most universal instrument for monitoringthe Space Weather in the direct sense of this term, i.e. the state of the Sun-Earth space and the ionosphere as it is particularly determined by all possible relatively powerful sources, wherever they are placed. This paper is devoted mostly to modelling VLF electromagnetic beam propagation in the WGEI. We present a new tensor impedance method for modelling propagation of electromagnetic beams (TIMEB) in a multi-layered/inhomogeneous waveguide. Suppose that such a waveguide, i.e. WGEI, possesses the gyrotropy and inhomogeneity with a thick cover layer placed above the waveguide. Note a very useful and attractive feature of the proposed TIMEB method: in spite of a large thickness of the waveguide cover layer, the proposed effective impedance approach reflects an impact of such a cover on the electromagnetic (EM) waves, which propagate in the waveguide. This impedance approach can be applied for EM waves/beams in layered gyrotropic/anisotropic active media in very wide frequency range, from VLF to optics. Moreover, this approach can be applied to calculations of EM waves/beams propagation in the media of an artificial origin such as metamaterial microwave or optical waveguides. The results of the modelling the propagation of VLF beams in the WGEI are included. The qualitative comparison between the theory and experimental observation of increasing losses of VLF waves in the WGEI is discussed. The new proposed method and its further development allows the comparison with the results of the future rocket experiment. This method allows to model (i) excitation of the VLF modes in the WGEI and their excitation by the typical VLF sources, such as radio wave transmitters and lightning discharges and (ii) leakage of VLF waves/beams into the upper ionosphere/magnetosphere.

Spectral Analysis and Information Entropy Approaches to Data of VLF Disturbances in the Waveguide Earth-Ionosphere

“…It will also be interesting to investigate in future works the possibility of the influence of the doubly resonant evanescent (reactive) Brunt–Väisälä mode on the spectrum of ULF modulation of the VLF EMW [ 59 ]. In the future, the approach developed in this paper leads to a very important problem of constructing a combined synergetic EMW-AGW model of wave disturbances as communication agents in the LEAIM system [ 13 , 18 , 34 , 39 , 42 , 61 , 70 , 77 ].…”

“…For the convenience of readers, here we very briefly outline the essence of what is presented in Appendix A , namely: (1) interaction between AGW existing in the atmosphere and VLF EMW propagating in the WGEI can be connected with the generation in the ionosphere of the currents on the combinations of the frequencies of VLF EMWs and of AGWs. Such the current can be caused by two factors (see relations (A5), (A6) in Appendix A ): (a) dragging of charged particles by means of AGWs against the background of ionospheric plasma with disturbances of charged particle concentration caused by VLF EMW; (b) motion of plasma particles with frequency of VLF EMW on the background of slowly varying plasma concentration, caused by AGW in the atmosphere–ionosphere; (2) AGW as global oscillations in the atmosphere–ionosphere are excited resonantly, and therefore relatively very efficiently, as the so-called reactive, or evanescent, modes [ 20 , 55 , 56 , 57 , 58 , 59 ]; an influence of the AGW packets containing such modes on the stable and unstable ionosphere has been investigated in [ 19 , 60 ] and [ 20 , 61 ], respectively. The estimation presented below would not cover comprehensively the all complex processes of the VLF modulation by AGW, outlined above.…”

“…Therefore, the relations presented below are rather a qualitative illustration of this fact than the very detailed description of the process of the generation of the combined frequencies. The equations for the main two types of the charged particles (electrons and “averaged” ions) have the form [ 61 , 62 ]. In the system (A1)–(A3), index (indices “ i ” and “ e ” correspond to “averaged” ionospheric ions and electrons, respectively); ; and are the charge, mass, concentration, velocity, and effective collision frequency characterizing the charge particles characterizing by the index “ l ”, respectively; and are geomagnetic field, electric field in the ionosphere, effective electric field which influences on the charged particles characterizing by index “j”, accounting for the effect of dragging of charged particles by AGWs in the ionosphere and the velocity of neutral particles in the atmosphere–ionosphere system under the propagation of AGWs, respectively.…”

“…Note also that the system of the equations for the motion of charges particles (A1)–(A3) should be, generally speaking, supplemented by the system of equations for the neutral particles in the ionospheric plasma. At this stage, we suppose that the solution of the hydrodynamic equations for neutral atmosphere–ionosphere component [ 56 , 61 , 62 , 63 ] is presented in (A1) and (A2) by the AGW velocity , as a part of effective electric field, responsible for the dragging of the charged particles. Later in the present Appendix, we will stay on the spectrum characteristics of the resonant modes of AGW, the most interesting in the context of the interpretation of the VLF spectrum data presented in this paper.…”

“…Accounting for the relations (A1)–(A4) and the equation of motion for neutral component (these equations are not presented here explicitly and can be found in [ 62 , 63 ], with inclusion of the effect of dragging of the neutral component by the plasma moving under the influence of VLF wave field, yields the presentation of the velocity and concentration for the plasma components in the form Hereafter, indices “AGW” and “VLF” in (A5) are used to indicate that the corresponding values are connected with ULF AGWs and VLF EMWs, respectively; indices “00” hereafter mean the complex amplitudes of the corresponding values. The amplitudes of the perturbations of the electron concentration and velocities of charged particles with the frequencies and wavenumbers of the AGW and VLF EMW in (A5), respectively, namely , can be obtained using the standard linearization procedure [ 56 , 61 , 62 , 63 ] for the relations (A1)–(A3) and are not presented here. Accounting for (A1)–(A5), one can obtain for the ionospheric current on the mixed VLF EMW-ULF AGW frequency (to which upper index “VLF+AGW” corresponds), by the order of value, the following expressions In (A6) and (A7) are the electric field of VLF EMW on the frequency without any perturbations (in particular, one caused by AGW), corresponding complex vector amplitude, corresponding polarization function with the proper altitude dependence and corresponding scalar complex amplitude, respectively.…”