Model of the Propagation of Very Low-Frequency Beams in the Earth

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Ann. Geophys., 38, 207–230, 2020 https://doi.org/10.5194/angeo-38-207-2020 © Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License.

Model of the propagation of very low-frequency beams in the Earth–ionosphere waveguide: principles of the tensor impedance method in multi-layered gyrotropic waveguides

Yuriy Rapoport1,6, Vladimir Grimalsky2, Viktor Fedun3, Oleksiy Agapitov1,4, John Bonnell4, Asen Grytsai1, Gennadi Milinevsky1,5, Alex Liashchuk6, Alexander Rozhnoi7, Maria Solovieva7, and Andrey Gulin1 1Faculty of Physics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 2IICBA, CIICAp, Autonomous University of the State of Morelos (UAEM), Cuernavaca, Morelos, Mexico 3Department of Automatic Control and Systems Engineering, The University of Shefﬁeld, Shefﬁeld, UK 4Space Science Laboratory, University of California, Berkeley, Berkeley, California, USA 5International Center of Future Science, College of Physics, Jilin University, Changchun, China 6National Space Facilities Control and Test Center, State Space Agency of Ukraine, Kyiv, Ukraine 7Shmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia

Correspondence: Yuriy Rapoport ([email protected])

Received: 20 March 2019 – Discussion started: 7 May 2019 Revised: 25 November 2019 – Accepted: 17 December 2019 – Published: 10 February 2020

Abstract. The modeling of very low-frequency (VLF) elec- 1 Introduction tromagnetic (EM) beam propagation in the Earth–ionosphere waveguide (WGEI) is considered. A new tensor impedance The results of the analytical and numerical study of very method for modeling the propagation of electromagnetic low-frequency (VLF) electromagnetic (EM) wave/beam beams in a multi-layered and inhomogeneous waveguide is propagation in the lithosphere–atmosphere–ionosphere– presented. The waveguide is assumed to possess the gy- magnetosphere system (LAIM), in particular in the Earth– rotropy and inhomogeneity with a thick cover layer placed ionosphere waveguide (WGEI), are presented. The ampli- above the waveguide. The influence of geomagnetic field in- tude and phase of the VLF wave propagates in the Earth– clination and carrier beam frequency on the characteristics ionosphere waveguide can change, and these changes may of the polarization transformation in the Earth–ionosphere be observable using ground-based and/or satellite detectors. waveguide is determined. The new method for modeling the This reflects the variations in ionospheric electrodynamic propagation of electromagnetic beams allows us to study characteristics (complex dielectric permittivity) and the in- the (i) propagation of the very low-frequency modes in the fluences on the ionosphere, for example, “from above” by Earth–ionosphere waveguide and, in perspective, their exci- the Sun–solar-wind–magnetosphere–ionosphere chain (Pa- tation by the typical Earth–ionosphere waveguide sources, tra et al., 2011; Koskinen, 2011; Boudjada et al., 2012; such as radio wave transmitters and lightning discharges, Wu et al., 2016; Yigit˘ et al., 2016). Then the influence on and (ii) leakage of Earth–ionosphere waveguide waves into the ionosphere “from below” comes from the most pow- the upper ionosphere and magnetosphere. The proposed ap- erful meteorological, seismogenic and other sources in the proach can be applied to the variety of problems related to lower atmosphere and lithosphere and Earth, such as cy- the analysis of the propagation of electromagnetic waves in clones and hurricanes (Nina et al., 2017; Rozhnoi et al., 2014; layered gyrotropic and anisotropic active media in a wide fre- Chou et al., 2017) as well as from earthquakes (Hayakawa, quency range, e.g., from the Earth–ionosphere waveguide to 2015; Surkov and Hayakawa, 2014; Sanchez-Dulcet et al., the optical waveband, for artificial signal propagation such 2015) and tsunamis. From inside the ionosphere, strong thun- as metamaterial microwave or optical waveguides. derstorms, lightning discharges, and terrestrial gamma-ray flashes or sprite streamers (Cummer et al., 1998; Qin et al.,

Published by Copernicus Publications on behalf of the European Geosciences Union. 208 Y. Rapoport et al.: Tensor impedance method

2012; Dwyer, 2012; Dwyer and Uman, 2014; Cummer et al., ered in Kuzichev and Shklyar(2010), Kuzichev et al.(2018), 2014; Mezentsev et al., 2018) influence the ionospheric elec- Lehtinen and Inan(2009, 2008) based on the mode presen- trodynamic characteristics as well. Note that the VLF signals tations and made in the frequency domain, we use the com- are very important for the merging of atmospheric physics bined approach. This approach includes the radiation con- and space plasma physics with astrophysics and high-energy dition at the altitudes of the F region, equivalent impedance physics. The corresponding “intersection area” for these four conditions in the lower E region and at the lower boundary of disciplines includes cosmic rays and currently very popular the WGEI, the mode approach, and finally, the beam method. objects of investigation – high-altitude discharges (sprites), This combined approach, finally, creates the possibility to ad- anomalous X-ray bursts and powerful gamma-ray bursts. The equately interpret data of both ground-based and satellite de- key phenomena for the occurrence of all of these objects tection of the VLF EM wave/beam propagating in the WGEI is the appearance of the runaway avalanche in the presence and those, which experienced a leakage from the WGEI into of high-energy seed electrons. In the atmosphere, there are the upper ionosphere and magnetosphere. Some other details cosmic-ray secondary electrons (Gurevich and Zybin, 2001). on the distinctions from the previously published models are Consequently, these phenomena are intensified during the air given below in Sect. 3. shower generating by cosmic particles (Gurevich and Zy- The methods of effective boundary conditions such as bin, 2001; Gurevich et al., 2009). The runaway breakdown effective impedance conditions (Tretyakov, 2003; Senior and lightning discharges including high-altitude ones can and Volakis, 1995; Kurushin and Nefedov, 1983) are well cause radio emission both in the high-frequency (HF) range, known and can be used, in particular, for the layered metal- which could be observed using the Low-Frequency Array dielectric, metamaterial and gyrotropic active layered and (LOFAR) radio telescope network facility and other radio waveguiding media of different types (Tretyakov, 2003; Se- telescopes (Buitink et al., 2014; Scholten et al., 2017; Hare nior and Volakis, 1995; Kurushin and Nefedov, 1983; Collin, et al., 2018), and in the VLF range (Gurevich and Zybin, 2001; Wait, 1996) including plasma-like solid state (Ruibys 2001). The corresponding experimental research includes the and Tolutis, 1983) and space plasma (Wait, 1996) media. measurements of the VLF characteristics by the international The plasma wave processes in the metal–semiconductor– measurement system of the “transmitted-receiver” pairs sep- dielectric waveguide structures, placed into the external mag- arated by a distance of a couple thousand kilometers (Biagi netic field, were widely investigated (Ruibys and Tolutis, et al., 2011, 2015). The World Wide Lightning Location Net- 1983; Maier, 2007; Tarkhanyan and Uzunoglu, 2006) from work is one of the international facilities for VLF measure- radio to optical-frequency ranges. Corresponding waves are ments during thunderstorms with lightning discharges (Lu et applied in modern plasmonics and in the non-destructive al., 2019). Intensification of magnetospheric research, wave testing of semiconductor interfaces. It is interesting to re- processes, particle distribution and wave–particle interaction alize the resonant interactions of volume and surface elec- in the magnetosphere including radiation belts leads to the tromagnetic waves in these structures, so the simulations of great interest in VLF plasma waves, in particular whistlers the wave spectrum are important. To describe such com- (Artemyev et al., 2013, 2015; Agapitov et al., 2014, 2018). plex layered structures, it is very convenient and effective The differences of the proposed model for the simulation to use the impedance approach (Tretyakov, 2003; Senior of VLF waves in the WGEI from others can be summa- and Volakis, 1995; Kurushin and Nefedov, 1983). As a rule, rized in three main points. (i) In distinction to the impedance impedance boundary conditions are used when the layer cov- invariant imbedding model (Shalashov and Gospodchikov, ering waveguide is thin (Senior and Volakis, 1995; Kurushin 2011; Kim and Kim, 2016), our model provides an optimal and Nefedov, 1983). One of the known exclusions is the balance between the analytical and numerical approaches. impedance invariant imbedding model. The difference be- It combines analytical and numerical approaches based on tween our new method and that model is already mentioned the matrix sweep method (Samarskii, 2001). As a result, this above and is explained in more detail in Sect. 3.3. Our new model allows for analytically obtaining the tensor impedance approach, i.e., a new tensor impedance method for model- and, at the same time, provides high effectiveness and stabil- ing the propagation of electromagnetic beams (TIMEB), in- ity for modeling. (ii) In distinction to the full-wave finite- cludes a set of very attractive features for practical purposes. difference time domain models (Chevalier and Inan, 2006; These features are (i) that the surface impedance character- Marshall and Wallace, 2017; Yu et al., 2012; Azadifar et izes cover layer of a finite thickness, and this impedance al., 2017), our method provides the physically clear lower is expressed analytically; (ii) that the method allows for and upper boundary conditions, in particular physically jus- an effective modeling of 3-D beam propagating in the gy- tified upper boundary conditions corresponding to the radi- rotropic waveguiding structure; (iii) finally that if the con- ation of the waves propagation in the WGEI to the upper sidered waveguide can be modified by any external influence ionosphere and magnetosphere. This allows for the determi- such as bias magnetic or electric fields, or by any extra wave nation of the leakage modes and the interpretation not only or energy beams (such as acoustic or quasistatic fields, etc.), of ground-based but also satellite measurements of the VLF the corresponding modification of the characteristics (phase beam characteristics. (iii) In distinction to the models consid-

Ann. Geophys., 38, 207–230, 2020 www.ann-geophys.net/38/207/2020/ Y. Rapoport et al.: Tensor impedance method 209 and amplitude) of the VLF beam propagating in the waveg- since the relevant details will be presented in the following uide structure can be modeled. articles. In Sect. 4 the results of the numerical modeling are Our approach was properly employed and is suitable for presented. In Sect. 5 the discussion is presented, including the further development which will allow to solve also the an example of the qualitative comparison between the re- following problems: (i) the problem of the excitation of the sults of our theory and an experiment including the future waveguide by the waves incident on the considered structure rocket experiment on the measurements of the characteris- from above could be solved as well with the slight modifica- tics of VLF signal radiated from the artificial VLF transmit- tion of the presented model, with the inclusion also ingoing ter, which is propagating in the WGEI and penetrating into waves; (ii) the consideration of a plasma-like system placed the upper ionosphere. into the external magnetic field, such as the LAIM system (Grimalsky et al., 1999a,b) or dielectric-magnetized semiconductor structure. The electromagnetic waves radiated out- 2 Formulation of the problem side the waveguiding structure, such as helicons (Ruibys and = Tolutis, 1983) or whistlers (Wait, 1996), and the waveguide The VLF electromagnetic waves with frequencies of f modes could be considered altogether. An adequate bound- (10–100) kHz can propagate along the Earth’s surface for ary radiation condition on the upper boundary of the cover- long distances > 1000 km. The Earth’s surface of a high con- = ing layer is derived. Based on this and an absence of ingo- ductivity of z 0 (z is vertical coordinate) and the iono- = ing waves, the leakage modes above the upper boundary of sphere F layer of z 300 km form the VLF waveguide (see the structure (in other words, the upper boundary of cover- Fig.1). The propagation of the VLF electromagnetic ra- ing layer) will be searched with the further development of diation excited by a near-Earth antenna within the WGEI the model delivered in the present paper. Namely, it will be should be described by the full set of Maxwell’s equations possible to investigate the process of the leakage of electro- in the isotropic atmosphere at 0 < z < 60 km, the approxi- magnetic waves from the open waveguide. Then their trans- mate altitude of the nearly isotropic ionosphere D layer at formation into magnetized plasma waves and propagating 60km < z < 75km, and the anisotropic E and F layers of the along magnetic field lines, and the following excitation of ionosphere, due to the geomagnetic field H 0, added by the the waveguiding modes by the waves incident on the sys- boundary conditions at the Earth’s surface and at the F layer. tem from external space (Walker, 1976) can be modeled as In Fig.1, θ is the angle between the directions of the verti- a whole. Combining with the proper measurements of the cal axis z and geomagnetic field H 0. Note that the angle of phases and amplitudes of the electromagnetic waves, propa- θ is complementary to the angle of inclination of the geomagnetic field. The geomagnetic field H 0 is directed along gating in the waveguiding structures and leakage waves, the 0 0 0 model can be used for searching and even monitoring the ex- z axis and lies in the plane xz, while the planes x z and xz ternal influences on the layered gyrotropic active artificial or coincide with each other. natural media, for example, microwave or optical waveguides or the system LAIM and the Earth–ionosphere waveguide, 3 Algorithm respectively. An important effect of the gyrotropy and anisotropy is the The boundary conditions and calculations of impedance and corresponding transformation of the field polarization during beam propagation in the WGEI are considered in this sec- the propagation in the WGEI, which is absent in the ideal tion. The other parts of the algorithm, e.g., the reflection of metal planar waveguide without gyrotropy and anisotropy. the EM waves from the WGEI effective upper boundary and We will determine how such an effect depends on the car- the leakage of EM waves from the WGEI into the upper iono- rier frequency of the beam, propagating in the WGEI, and sphere and magnetosphere, will be presented very briefly as the inclination of the geomagnetic field and perturbations they are the subjects of the next papers. in the electron concentration, which could vary under the influences of the sources powerful enough placed “below”, 3.1 Direct and inverse tensors characterizing the “above” and/or “inside” the ionosphere. ionosphere In Sect. 2 the formulation of the problem is presented. In Sect. 3 the algorithm is discussed including the determi- In the next subsections we derived the formulas describing nation of the VLF wave/beam radiation conditions into the the transfer of the boundary conditions at the upper boundary upper ionosphere and magnetosphere at the upper boundary, (z = Lmax) (Fig.1), resulting in the tensor impedance condi- placed in the F region at 250–400 km altitude. The effective tions at the upper boundary of the effective WGEI (z = Li). tensor impedance boundary conditions at the upper boundary Firstly let us describe the tensors characterizing the iono- (∼ 85 km) of the effective Earth–ionosphere waveguide and sphere. the 3-D model TIMEB of the propagation of the VLF beam The algorithm’s main goal is to transfer the EM boundary in the WGEI are discussed as well. The issues regarding the conditions from the upper ionosphere at the height of Lz at ∼ VLF beam leakage regimes are considered only very briefly, 250–400 km to the lower ionosphere at Lz ∼ 70–90 km. All www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 210 Y. Rapoport et al.: Tensor impedance method

  ε1 εh 0 0 εˆ = −εh ε1 0 , 0 0 ε3 2 ωpe · (ω − iνe) ε = 1 − 1 − 2 − 2 · ((ω iνe) ωHe) ω 2 ω · (ω − iνi) − pi , − 2 − 2 · ((ω iνi) ωHi) ω εh ≡ ig; 2 ω · ωHe g = − pe − 2 − 2 · Figure 1. The geometry of the anisotropic and gyrotropic waveg- ((ω iνe) ωH e) ω uide. EM waves propagate in the OX direction. H is the exter- 2 · 0 ωpi ωHi nal magnetic field. The effective WGEI for EM waves occupies + , ((ω − iν )2 − ω2 ) · ω the region 0 < z < Lz. The isotropic medium occupies the region i Hi 0 < z < L , L < Lz. The anisotropic and gyrotropic medium 2 2 ISO ISO ωpe ωpi occupies the region L < z < Lmax. The covering layer occu- ε = 1 − − ; ISO 3 (ω − iν ) · ω (ω − iν ) · ω pies the region Lz < z < Lmax. WGEI includes the isotropic region e i 0 < z < L and a part of the anisotropic region L < z < L . 4πe2n ISO z max 2 = It is supposed that the anisotropic region is relatively small part ωpe , me of the WG, (L − L )/L ∼ (0.1–0.2). At the upper boundary of z ISO z 4πe2n the covering layer (z = L ) EM radiation to the external region 2 max ωpi = , (z > Lmax) is accounted for with the proper boundary conditions. mi The integration of the equations describing the EM field propaga- eH = 0 tion allows for obtaining effective impedance boundary conditions ωHe , mec at the upper boundary of the effective WG (z = Lz). These bound- eH0 ary conditions effectively include all the effects on the wave prop- ωHi = , (1) mic agation of the covering layer and the radiation (at z = Lmax) to the external region (z > Lmax). where ωpe, ωpi, ωHe and ωHi are the plasma and cyclotron frequencies for electrons and ions, respectively, me, mi, νe components of the monochromatic EM field are considered and νi are the masses and collision frequencies for elec- to be proportional to exp(iωt). The anisotropic medium is trons and ions, respectively, and n is the electron concen- inhomogeneous along OZ axis only and characterized by the tration. The approximations of the three-component plasma- permittivity tensor ε(ω,z)ˆ or by the inverse tensor β(ω,z)ˆ = like ionosphere (including one electron component, one ef- εˆ−1(ω,z): E = β(ω,z)ˆ ·D, where D is the electric induction. fective ion and one effective neutral component) and quasi- Below the absolute units are used. The expressions for the neutrality are accepted. The expressions for the components components of the effective permittivity of the ionosphere of the permittivity tensor ε(ωˆ , z) are obtained from Eq. (1) are in the coordinate frame X0YZ0, where the OZ0 axis is by means of multiplication with the standard rotation matri- aligned along the geomagnetic field H 0. ces (Spiegel, 1959) dependent on angle θ. For the medium with a scalar conductivity σ, e.g., lower ionosphere or atmosphere, the effective permittivity in Eq. (1) reduces to ε = 1 − 4πiσ/ω.

3.2 The equations for the EM ﬁeld and upper boundary conditions

In the case of the VLF waveguide modes, the longitudinal wave number kx is slightly complex and should be calculated accounting for boundary conditions at the Earth’s surface and the upper surface of the effective WGEI. The EM ﬁeld depends on the horizontal coordinate x as exp(−ikxx). Taking into account kx ≈ k0 (k0 = ω/c), in simulations of the VLF beam propagation, it is possible to put kx = k0. Therefore,

Ann. Geophys., 38, 207–230, 2020 www.ann-geophys.net/38/207/2020/ Y. Rapoport et al.: Tensor impedance method 211

Maxwell’s equations are ∂H ∂H    − y = x + = ik0Dx, ikxHz ik0Dy, i k2 β · β ∂H β ∂H ∂z ∂z E = β + x 12 21  y − 12 y  x 11 2 2 2 − = k0  k kx  ∂z kx ∂z  ikxHy ik0Dz 0 1 − β22 2 1 − β22 2 k0 k0 ∂Ey ∂Ex   − = −ik0Hx, + ikxEz = −ik0Hy, ∂z ∂z k k2 β · β − x β + x 12 23 H − = − 13 2 2 y ikxEy ik0Hz, (2) k0  k kx  0 1 − β22 2 k0 where Ex = β11Dx + β12Dy + β13Dz, etc. All components   of the EM field can be represented through the horizontal i β ∂H β ∂H = − 22 x + 21 y  Ey 2 2 components of the magnetic field Hx and Hy. The equations k0  kx ∂z kx ∂z  1 − β22 2 1 − β22 2 for these components are given below. k0 k0 kx β23     − Hy k2 ∂ β ∂H ∂ β ∂H k0 1 − β x  22 x  −  21 y  22 k2 2 2 0 ∂z  kx ∂z  ∂z  kx ∂z  1 − β22 2 1 − β22 2 (4) k0 k0   In the region z ≥ L the upper ionosphere is assumed to ∂ β max −  23  + 2 = be weakly inhomogeneous, and the geometric optics approx- ikx 2 Hy k0Hx 0 (3a) ∂z  kx  1 − β22 2 imation is valid in the VLF range there. We should note that k0      due to high inhomogeneity of the ionosphere in the vertical ∂ k2 β · β ∂H ∂ β ∂H direction within the E layer (i.e., at the upper boundary of β + x 12 21  y  −  12 x  11 2 2 2 the effective VLF WGEI) such an approximation is not ap- ∂z  k kx  ∂z  ∂z  kx ∂z  0 1 − β22 2 1 − β22 2 plicable. These conditions determine the choice of the upper k0 k0    boundary of z = Lmax at ∼ 250–400 km, where the condi- 2 tions of the radiation are formulated. The dispersion equa- ∂ kx β12 · β23 + ikx β + Hy  13 2 k2   tion which connected the wave numbers and the frequency ∂z k − x 0 1 β22 2 of the outgoing waves is obtained from Eq. (3a, 3b), where k0   Hx,y ∼ exp(−ikzz), while the derivatives like ∂β11/∂z and 2 kx β32 · β21 ∂Hy the inhomogeneity of the media are neglected. + ikx β +  −  31 2 k2  k0 1 − β x ∂z !! ! 22 k2 k2 k2 0 β k2 − k2 1 − β x · β 1 − β x β ∂H 22 z 0 22 2 11 22 2 − 32 x k0 k0 ikx 2 − kx ∂z ! 1 β22 2 k2 k0 x 2 + β12 · β21 k   k2 z 2 4 · 0 2 kx kx β23 β32 ! + k 1 − β − Hy = 0 2 0  33 2 4 k2  kx k0 k0 1 − β x + (β13 + β31) 1 − β22 + 22 k2 2 0 k0 (3b) k2 x (β · β + β · β )k k 2 12 23 32 21 x z The expressions for the horizontal components of the electric k0 field E , E are given below. ! ! !! x y k2 k2 k4 −k2 − β x − β x − x β · β − 0 1 33 2 1 22 2 4 23 32 k0 k0 k0 2 2 − (β21kz + β23kxkz) · (β12kz − β32kxkz) = 0 (5)

Thus, generally Eq. (5) determined the wave numbers for the outgoing waves to be of the fourth order (Wait, 1996). The boundary conditions at the upper boundary z = Lmax within the ionosphere F layer are the absence of the ingoing waves, i.e., the outgoing radiated (leakage) waves are present only. Two roots should be selected that possess the negative imaginary parts Im(kz1,z2) < 0; i.e., the outgoing waves dissipate www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 212 Y. Rapoport et al.: Tensor impedance method upwards. However, in the case of VLF waves, some simpli- ary conditions are rewritten in terms of Hx and Hy only. fication can be used. Namely, the expressions for the wave ∂H numbers k , are obtained from Eq. (3a, 3b), where the de- x 1 2 = −i(kz1A1 + kz2α2A2) pendence on x is neglected: |k1,2| k0. This approximation ∂z is valid within the F layer where the first outgoing wave cor- i = − ((k − α α k )H + α (k − k )H ) responds to the whistlers with a small dissipation; the sec- 1 z1 1 2 z2 x 2 z2 z1 y ond one is the highly dissipating slow wave. To formulate ∂Hy = −i(kz1α1A1 + kz2A2) the boundary conditions for Eq. (3a, 3b) at z ≥ Lmax, the EM ∂z field components can be presented as i = − ((kz2 − α1α2kz1)Hy + α1(kz1 − kz2)Hx) (12) −ikz1z˜ −ikz2z˜ 1 Hx = A1e + α2A2e ,

−ikz1z˜ −ikz2z˜ The relations in Eq. (12) are the upper boundary conditions Hy = α1A1e + A2e . (6) of the radiation for the boundary z = Lmax at ∼ 250–400 km. These conditions will be transformed and recalculated using In Eq. (6), z˜ = z−Lz . Equation (3a, 3b) are simplified in the approximation described above. the analytical numerical recurrent procedure into equivalent impedance boundary conditions at z = Lz at ∼ 70–90 km. 2 2 ∂ H ∂ Hy Note that in the “whistler–VLF approximation” is valid β x − β + k2H = 0, 22 ∂z2 21 ∂z2 0 x at frequencies ∼ 10 kHz for the F region of the ionosphere. 2 2 In this approximation and kx ≈ 0, we receive the dispersion ∂ Hy ∂ H β − β x + k2H = 0 (7) equation using Eqs. (5), (8), (9), in the form 11 ∂z2 12 ∂z2 0 y k0 2k2 = k2g2, (13) The solution of Eq. (7) is searched for as Hx,y ∼ exp(−ikzz)˜ . z 0 The following equation has been obtained to get the wave where k2 = k2 + k2 = k0 2 + k0 2 and k0 and k0 are the trans- numbers kz1,z2 from Eq. (7): x z x z x z verse and longitudinal components of the wave number rel- 4 2 κ − (β22 + β11)κ + β11β22 − β12β21 = 0, ative to geomagnetic field. For the F region of the iono- 2 sphere, where νe ω ωH e, Eq. (13) reduces to the stan- 2 k0 0 κ = . (8) dard form of the whistler dispersion equation |kz|k = k0|g|, k2 2 2 0 2 z g ≈ −ωpe/(ωωHe) and ω = c k|kz|(ωH e/ωpe). In a special case of the waves propagating along geomagnetic field, k0 = Therefore, from Eq. (8) follows, x 0, for the propagating whistler waves, the well-known disper- 2 0 2 2 !1/2 sion dependence is ω = c k (ωHe/ω ) (Artsimovich and β + β β + β 2 z pe κ2 = 11 22 ± 11 22 + β β ; Sagdeev, 1979). For the formulated problem we can rea- 1,2 2 2 12 21 sonably assume kx ≈ 0. Therefore Eq. (13)√ is reduced to 4 2 4 2 2 k cos θ = k g . As a result, we get kz1 = g/cosθk0 and β22 − κ β z √ 0 α = 1 = 12 ; k = −i g/cosθk , and then, similar to the relations in 1 β − 2 z2 0 21 β11 κ1 Eq. (12), the boundary conditions can be presented in terms 2 β11 − κ β of the tangential components of the electric field as α = 2 = 21 ; 2 2 β12 β22 − κ 2 ∂U ˆ 2 + BU = 0; 2 k0 ∂z kz1,z2 = . (9) κ2 E 1,2 U = x ; Ey The signs of kz1,z2 have been chosen from the condition 1r g 1 + i −1 − i Im(kz1,z2) < 0. From Eq. (5) at the upper boundary of z = Bˆ = k . (14) 0 1 + i 1 + i Lmax, the following relations are valid: 2 cosθ Conditions in Eq. (12) or Eq. (14) are the conditions of ra- Hx = A1 + α2A2,Hy = α1A1 + A2. (10) diation (absence of ingoing waves) formulated at the upper From Eq. (10) one can get boundary at z = Lmax and suitable for the determination of the energy of the wave leaking from the WGEI into the upper −1 −1 A1 = 1 (Hx − α2Hy); A2 = 1 (Hy − α1Hx); ionosphere and magnetosphere. Note that Eqs. (12) and (14)

1 = 1 − α1α2. (11) expressed the boundary conditions of the radiation (more ac- curately speaking, an absence of incoming waves, which is Thus, it is possible to exclude the amplitudes of the outgoing the consequence to the causality principle) are obtained as a waves A1,2 from Eqs. (9). As a result, at z = Lmax the bound- result of the passage to limit |kx/kz| → 0 in Eq. (5). In spite

Ann. Geophys., 38, 207–230, 2020 www.ann-geophys.net/38/207/2020/ Y. Rapoport et al.: Tensor impedance method 213 of the disappearance of the dependence of these boundary ten as conditions explicitly on kx, the dependence of the character- ˆ istics of the wave propagation process on kx, as a whole, is H j = bj · H j+1, j = N,...1 (17a) accounted for, and all results are still valid for the descrip- Hxj+1 = b11j+1H1 + b12j+1H2; tion of the wave beam propagation in the WGEI along the H + = b + H + b + H ; horizontal axis x with a finite kx ∼ k0. yj 1 21j 1 1 22j 1 2 H1 ≡ Hxj ; 3.3 Equivalent tensor impedance boundary conditions H2 ≡ Hyj . (17b) The tensor impedance at the upper boundary of the effective This method is a variant of the Gauss elimination method for WGEI of z = L (see Fig.1) is obtained by the conditions of z the matrix three-diagonal set of Eq. (16). The value of bˆ is radiation in Eqs. (12) or (14), recalculated from the level of N obtained from the boundary conditions (12) as z = Lmax ∼ 250–400 km, placed in the F region of the ionosphere, to the level of z = Lz ∼ 80–90 km, placed in the E re- − αˆ ( ) · H +α ˆ (0) · H = . gion. N N−1 N N 0 (18) The main idea of the effective tensor impedance method ˆ = − ˆ (0) −1 · ˆ (−) ˆ is the unification of analytical and numerical approaches Therefore bN (αN ) αN . Then the matrices bj have and the derivation of the proper impedance boundary con- been computed sequentially down to the desired value of z = ditions without “thin-cover-layer” approximation. This ap- Lz = h · Nz, where the impedance boundary conditions are ˆ proximation is usually used in the effective impedance ap- assumed to be applied. At each step the expression for bj proaches, applied either for artificial or natural layered gy- follows from Eqs. (16), (17a) and (17b) as rotropic structures (see e.g., Tretyakov, 2003; Senior and ˆ (0) + ˆ (+) · ˆ · = − ˆ (−) · = Volakis, 1995; Kurushin and Nefedov, 1983; Alperovich (αj αj bj+1) H j αj H j−1 0. (19) and Fedorov, 2007). There is one known exception, namely the invariant imbedding impedance method (Shalashov and ˆ = − ˆ (0) + ˆ (+) · Therefore, for Eq. (17a, 17b), we obtain bj (αj αj Gospodchikov, 2011; Kim and Kim, 2016). The compari- − bˆ )−1 ·α ˆ ( ). The derivatives in Eq. (4) have been approxi- son of our method with the invariant imbedding impedance j+1 j mated as method will be presented at the end of this subsection. Equa- tion (3a, 3b) jointly with the boundary conditions of Eq. (12) ∂H (H ) + − (H ) have been solved by finite differences. The derivatives in x ≈ x Nz 1 x Nz ∂z h Eq. (3a, 3b) are approximated as Nz (b − 1) · (H ) + b · (H ) Nz+1 11 x Nz Nz+1 12 y Nz ∂ ∂Hx 1 (Hx)j+1 − (Hx)j = , C(z) ≈ C(zj+1/2) h ∂z ∂z h h (20) (Hx)j − (Hx)j−1 −C(zj− / ) 1 2 h ∂Hy and a similar equation can be obtained for ∂z . Note Nz ∂ 1 that as a result of this discretization, only the values at the (F (z)Hx) ≈ (F (zj+1)(Hx)j+1 ∂z 2h grid level Nz are included in the numerical approximation − F (zj−1)(Hx)j−1). (15) of the derivatives ∂Hx,y/∂z at z = Lz. We determine tensor impedance Zˆ at z = L at ∼ 85 km. The tensor values are = · + z In Eq. (15) zj+1/2 h (j 0.5). In Eq. (10) the approxima- included in the following relations, all of which are corre- ≈ [ − ] tion is ∂Hx/∂z (Hx)N (Hx)N−1 /h. Here h is the dis- sponded to altitude (in other words, to the grid with the num- cretization step along the OZ axis, and N is the total number ber Nz and corresponding to this altitude). of nodes. At each step j the difference approximations of Eq. (3a, 3b) take the form ˆ n × E = Z0 · H, n = (0,0,1) or ˆ (−) · + ˆ (0) · + ˆ (+) · = = + ; = − − αj H j−1 αj H j αj H j+1 0, (16) Ex Z021Hx Z022Hy Ey Z011Hx Z012Hy (21) Hx The equivalent tensor impedance is obtained using a two-step where H j = , j = N − 1,N − 2,...,1, zj = h · j and ˆ hy procedure. (1) We obtain the matrix bj using Eq. (3a, 3b) Lz = h·N. Due to the complexity of expressions for the ma- with the boundary conditions of Eq. (12) and the procedure trix coefficients in Eq. (16), we have shown them in Ap- of Eqs. (17)–(19) described above. (2) Placing the expres- pendix A. The set of the matrix Eq. (16) has been solved by sions of Eq. (21) with tensor impedance into the left parts the factorization method also known as an elimination and and the derivatives ∂Hx,y/∂z in Eq. (20) into the right parts matrix sweep method (see Samarskii, 2001). It can be writ- of Eq. (4), the analytical expressions for the components of www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 214 Y. Rapoport et al.: Tensor impedance method the tensor impedance are and Hy at the upper boundary of the VLF waveguide at 80– 90 km. The naturally chosen direction of the recalculation   of the upper boundary conditions from z = Lmax to z = Lz, i β β = −  21 · − 22 · −  i.e., from the upper layer with a large impedance value to the Z011 2 b21 2 (b11 1) , k0h  kx kx  lower-altitude layer with a relatively small impedance value, 1 − β22 2 1 − β22 2 k0 k0 provides, at the same time, the stability of the simulation pro-  cedure. The obtained components of the tensor impedance i β21 ∂Hy | | ≤ Z = −  · (b − 1) are small, Z0αβ 0.1. This determines the choice of the 012  k2 22 k0h 1 − β x ∂z upper boundary at z = Lz for the effective WGEI. Due to 22 k2 0 the small impedance, EM waves incident from below on  this boundary are reflected effectively back. Therefore, the β22 β23 ≤ ≤ − · b − kxh · , region 0 z Lz indeed can be presented as an effective k2 12 k2  1 − β x 1 − β x WGEI. This waveguide includes not only lower boundary at 22 k2 22 k2 0 0 L at ∼ 65–75 km with a rather small losses, but it also in-  ISO cludes thin dissipative and anisotropic and gyrotropic layers i k2 β · β Z = β + x 12 21 · b between 75 and 85–90 km. 021 11 2 2 21 k0h  k kx 0 1 − β22 2 Finally, the main differences and advantages of the pro- k0  posed tensor impedance method from other methods for β impedance recalculating, and in particular invariant imbed- − 12 · (b − 1), k2 11  ding methods (Shalashov and Gospodchikov, 2011; Kim and − x 1 β22 2 Kim, 2016), can be summarized as follows: k0   1. In contrast to the invariant imbedding method, the cur- i k2 β · β Z = β + x 12 21  · (b − 1) rently proposed method can be used for the direct recal- 022  11 2 k2  22 k0h k0 1 − β x culation of tensor impedance, as is determined analyti- 22 k2 0 cally (see Eq. 22).    2 k β12 · β23 β12 2. For the media without non-locality, the proposed −k h · β + x  − · b . x 13 2 2 2 12  k kx  kx  method does not require the solving of one or more in- 0 1 − β22 2 1 − β22 2 k0 k0 tegral equations. (22) 3. The proposed method does not require forward and re- The proposed method of the transfer of the boundary con- flected waves. The conditions for the radiation at the upper boundary of z = L (see Eq. 12) are determined ditions from the ionosphere F layer at Lmax = 250–400 km max through the total field components of H , which sim- into the lower part of the E layer at Lz = 80–90 km is sta- x,y ble and easily realizable in comparison with some alternative plify the overall calculations. approaches based on the invariant imbedding methods (Sha- 4. The overall calculation procedure is very effective and lashov and Gospodchikov, 2011; Kim and Kim, 2016). The computationally stable. Note that even for the very stability of our method is due to the stability of the Gauss low-loss systems, the required level of stability can be elimination method when the coefficients at the matrix cen- achieved with modification based on the choice of the tral diagonal are dominating. The last is valid for the iono- maximal element for matrix inversion. sphere with electromagnetic losses where the absolute values of the permittivity tensor are large. The application of the 3.4 Propagation of electromagnetic waves in the proposed matrix sweep method in the media without losses gyrotropic waveguide and the TIMEB method may require the use of the Gauss method with the choice of the maximum element to ensure stability. However, as our Let us use the transverse components of the electric Ey and simulations (not presented here) demonstrated, for the elec- magnetic Hy fields to derive equations for the slow vary- tromagnetic problems in the frequency domain, the simple ing amplitudes A(x,y,z) and B(x,y,z) of the VLF beams. Gauss elimination and the choice of the maximal element These components can be represented as give the same results. The accumulation of errors may oc- 1 iωt−ik x cur in evolutionary problems in the time domain when the Ey = A(x,y,z) · e 0 + c.c., 2 Gauss method should be applied sequentially many times. 1 iωt−ik x The use of the independent functions Hx and Hy in Eq. (3a, Hy = B(x,y,z) · e 0 + c.c. (23) 3b) seems natural, as well as the transfer of Eq. (17a), be- 2 cause the impedance conditions are the expressions of the Here we assumed kx = k0 to reflect beam propagation in the electric Ex and Ey through the magnetic components Hx WGEI with the main part in the atmosphere and lower iono-

Ann. Geophys., 38, 207–230, 2020 www.ann-geophys.net/38/207/2020/ Y. Rapoport et al.: Tensor impedance method 215 sphere (D region), which are similar to free space by its electromagnetic parameters. The presence of a thin anisotropic (" ! !# and dissipative layer belonging to the E region (Guglielmi 1 k k k2 E = ε ε − ε + x y · ε − y E and Pokhotelov, 1996) of the ionosphere causes, altogether x 13 32 12 2 33 2 y 1y k k with the impedance boundary condition, the proper z depen- 0 0 ! ) dence of B(x,y,z). Using Eqs. (21) and (22), the bound- i k2 ∂H k k ∂E + ε − y y + x ε · H + i y ε y ary conditions are determined at the height of z = L for 33 2 13 y 2 13 z k0 k ∂z k0 k ∂z the slowly varying amplitudes A(x,y,z) and B(x,y,z) of 0 0 (" ! 2 !# the transverse components Ey and Hy. As it follows from 1 kxky ky Ez = ε31 ε12 + − ε32 ε11 − Ey Maxwell’s equations, the components Ex and Hx through Ey 2 2 1y k0 k0 and Hy in the method of beams have the form ! i ∂H k k2 − ε y − x ε − y H i ∂E β˜ ∂H 31 11 2 y y 33 y ˜ k0 ∂z k0 k0 Hx ≈ − ,Ex ≈ γ12Ey + i + β13Hy, (24) k0 ∂z k0 ∂z 2 ! ) ky ky ∂Ey −i ε11 − . = −1 − ˜ = −1 ˜ = k2 k2 ∂z where γ12 10 (ε13ε32 ε12ε33), β13 10 ε13 and β33 0 0 −1 = − 10 ε33; 10 ε11ε33 ε13ε31. From Eqs. (21) and (24), the (27) boundary conditions for A and B can be defined as k2 k2 In Eq. (27), 1 ≡ ε − y · ε − y − ε · ε . The i ∂A y 11 k2 33 k2 31 13 A − Z011 · + Z012 · B ≈ 0, 0 0 k0 ∂z equations for Ey and Hy obtained from the Maxwell equa- i ∂A tions are γ · A + Z · + (β˜ − Z ) 12 k 021 ∂z 13 022 0 2 i ∂B ∂ 2 2 ∂Ez ˜ −k − k Ey + iky − ikxEx − ikyEy · B + β33 · ≈ 0. (25) ∂z2 x y ∂z k0 ∂z ∂E + k2D = 0; −ik x + k k E + k2H = 0. (28) The evolution equations for the slowly varying amplitudes 0 y 0 ∂z x 0 z 0 y A(x,y,z) and B(x,y,z) of the VLF beams are derived. The monochromatic beams are considered when the frequency ω After the substitution of Eq. (27) for Ex and Ez into is fixed and the amplitudes do not depend on time t. Looking Eqs. (28), the coupled equations for Ey and Hy can be de- for the solutions for the EM field as E,H ∼ exp(iωt−ikxx− rived. The following expansion should be used: kx = k0 + ikyy), Maxwell’s equations are δkx, |δkx| k0; also |ky| k0. Then, according to Weiland and Wilhelmsson(1977), ∂H − ik H − y = ik D , y z ∂z 0 x ∂ ∂ −i · δkx → ,−i · ky → . (29) ∂Hx ∂x ∂y + ik H = ik D , ∂z x z 0 y The expansions should be used until the quadratic terms in ky − ikxHy + ikyHx = ik0Dz and the linear terms in δkx. As a result, parabolic equations ∂Ey − ikyEz − = −ik0Hx, (Levy, 2000) for the slowly varying amplitudes A and B are ∂z derived. In the lower ionosphere and atmosphere, where the ∂Ex ε(ω,z) + ikxEz = −ik0Hy, effective permittivity reduces to a scalar , they are in- ∂z dependent. − ikxEy + ikyEx = −ik0Hz. (26) 2 2 ∂A i ∂ A ∂ A ik0 Here Dx = ε11Ex +ε12Ey +ε13Ez. From Eq. (21), the equa- + + + · (ε − 1)A = 0 ∂x 2k ∂y2 ∂z2 2 tions for Ex and Ez through EyandHy are 0 ∂B i 1 ∂ ∂B ∂2B ik + β + + 0 · (ε − )B = , 2 1 0 ∂x 2k0 β ∂z ∂z ∂y 2 (30a)

where β ≡ ε−1. Accounting for the presence of the gyrotropic layer and the tensor impedance boundary conditions at the upper boundary of z = Lz of the VLF waveguide, the equations for the slowly varying amplitudes in the general www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 216 Y. Rapoport et al.: Tensor impedance method case are z = Lz, and with the boundary conditions at the Earth’s surface (Eq. 31b), are used to simulate the VLF wave propaga- ∂A i ∂2A ∂2A ik + + + 0 · (ε˜ − ) · A tion. The surface impedance of the Earth has been calculated 2 2 22 1 ∂x 2k0 ∂y ∂z 2 from the Earth’s conductivity (see Eq. 31a). The initial con- γ21 ∂B ik0 ditions to the solution of Eqs. (30a, 30b), (25) and (31b) are + + · γ B = 0 2 ∂z 2 23 chosen in the form ∂B i 1 ∂ ∂B ∂2B + β˜ + A(x = 0,y,z) = 0, ˜ 33 2 ∂x 2k0 β11 ∂z ∂z ∂y 2n B(x = 0,y,z) = B0 exp(−((y − 0.5Ly)/y0) ) i ∂ 1 ∂ ˜ ik0 + (γ12A) + (β13B) + γ32A+ 2n ˜ ˜ ˜ · exp(−((z − z1)/z0) ), 2β11 ∂z 2β11 ∂z 2β11 = β˜ ∂B ik 1 n 2. (32) + 31 + 0 · − 1 · B = 0. (30b) ˜ ˜ 2β11 ∂z 2 β11 In the relations of Eq. (32), z1, z0, y0 and B0 are the vertical position of maximum value, the vertical and transverse In Eq. (30b), characteristic dimensions of the spatial distribution and the ε13 · ε32 − ε12 · ε33 maximum value of H , respectively, at the input of the sys- γ ≡ , y 12 1 tem, x = 0. The size of the computing region along the OY ε · ε − ε · ε axis is, by the order of value, L ∼ 2000km. Because the γ ≡ 23 31 21 33 , y 21 1 gyrotropic layer is relatively thin and is placed at the upper ε · ε − ε · ε part of the VLF waveguide, the beams are excited near the γ ≡ 21 13 23 11 , 23 1 Earth’s surface. The wave diffraction in this gyrotropic layer ε · ε − ε · ε along the OY axis is quite small; i.e., the terms ∂2A/∂y2 and γ ≡ 31 12 32 11 , 32 1 ∂2B/∂y2 are small there as well. Contrary to this, the wave ε β˜ ≡ 11 , diffraction is very important in the atmosphere in the lower 11 1 part of the VLF waveguide near the Earth’s surface. To solve ˜ ε13 the problem of the beam propagation, the method of splitting β13 ≡ , 1 with respect to physical factors has been applied (Samarskii, ˜ ε31 β31 ≡ , 2001). Namely, the problem has been approximated by the 1 finite differences ˜ ε33 β33 ≡ , 1 A ∂C ˆ ˆ C ≡ , + LyC + LzC = 0. (33) 1 ≡ ε11 · ε33 − ε13 · ε31; B ∂x − + − ε21(ε13ε32 ε12ε33) ε23(ε31ε12 ε32ε11) ˆ ε˜22 ≡ ε22 + . In the terms of LyC, the derivatives with respect to y are 1 ˆ included, whereas all other terms are included in LzC. Then Equation (30b) is reduced to Eq. (30a) when the effec- the following fractional steps have been applied; the first one tive permittivity is scalar. At the Earth’s surface of z = 0, is along y, and the second one is along z. the impedance conditions are reduced, accounting for the p+1/2 − p medium being isotropic and the conductivity of the Earth be- C C ˆ p+1/2 + LyC = 0, ing finite, to the form hx + + 1/2 p 1 − p 1/2 iω C C ˆ p+1 + LzC = 0 (34) Ey = Z0EHx,Ex = −Z0EHy,Z0E ≡ , (31a) h 4πσE x The region of simulation is 0 < x < L = 1000–2000 km, where σ ∼ 108 s−1 is the Earth’s conductivity. The bound- x E 0 < y < L = 2000–3000 km and 0 < z < L = 80–90 km. ary conditions (31a) at the Earth’s surface, where Z = y z 022 The numerical scheme of Eq. (34) is absolutely stable. Here Z ≡ Z , Z = Z = 0, β = ε−1(z = 0), γ = 0 021 0E 012 021 33 12 h is the step along the OX axis, and x = ph and p = and β˜ = 0 can be rewritten as x p x 13 0,1,2,.... This step has been chosen from the conditions of i ∂Ey the simulation result independence of the diminishing hx. Ey + Z0E = 0, k0 ∂z On the simulation at each step along the OX axis, the correction on the Earth’s curvature has been inserted in an i ∂Hy + Z0EHy = 0. (31b) adiabatic manner applying the rotation of the local coordi- ε(z = 0)k0 ∂z nate frame XOZ. Because the step along x is small hx and Equation (30a, 30b), combined with the boundary conditions ∼ 1km Lz, this correction of the C results in the multi- of Eq. (25) at the upper boundary of the VLF waveguide of plier exp(−ik0 · δx), where δx = z · (hx/RE) and RE Lz

Ann. Geophys., 38, 207–230, 2020 www.ann-geophys.net/38/207/2020/ Y. Rapoport et al.: Tensor impedance method 217

atively accurate match between the regions, described by the full-wave electromagnetic approach with Maxwell’s equations and the complex geometrical optics (FWEM-CGO approach). All of these materials are not included in this paper, but they will be delivered in the two future papers. The first paper will be dedicated to the modeling of VLF waves propagating in the WGEI based on the field expansion as a set of eigenmodes of the waveguide (the mode presentation approach). The second paper will deal with the leakage of the VLF beam from the WGEI into the upper ionosphere and magnetosphere and the VLF beam propagation in these media. Here we describe only one result, which concerns the mode excitation in the WGEI, because this result is princi- pally important for the justification of the TIMEB method. It was shown that more than the five lowest modes of the WGEI are strongly localized in the atmosphere or lower ionosphere. Their longitudinal wave numbers are close to the corresponding wave numbers of EM waves in the atmosphere. This fact Figure 2. The rotation of the local Cartesian coordinate frame demonstrates that the TIMEB method can be applied to the at each step along the Earth’s surface hx on a small angle δϕ ≈ propagation of the VLF electromagnetic waves in the WGEI. 1x/RE in radians, while 1x = hx. The following strong inequali- ties are valid for hx Lz RE. At the Earth’s surface z = 0. 4 Modeling results is the Earth’s radius (see Fig.2 and the caption to this fig- The dependencies of the permittivity components ε1, ε3 and ure). At the distances of x ≤ 1000km, the simulation results εh in the coordinate frame associated with the geomagnetic do not depend on the insertion of this correction, whereas field H 0 are given in Fig.3. The parameters of the iono- at higher distances a quantitative difference occurs: the VLF sphere used for modeling are taken from Al’pert(1972), beam propagates more closely to the upper boundary of the Alperovich and Fedorov(2007), Kelley(2009), Schunk and waveguide. Nagy(2010), and Jursa(1985). The typical results of simulations are presented in Fig.4. The parameters of the iono- 3.5 VLF waveguide modes and reflection from the VLF sphere correspond to Fig.3. The angle θ (Fig.1) is 45 ◦. The waveguide upper effective boundary VLF frequency is ω = 105 s−1 and f = ω/2π ≈ 15.9 kHz. The Earth’s surface is assumed to as ideally conductive at the In general, our model needs the consideration of the waveg- level of z = 0. The values of the EM field are given in abso- uide mode excitations by a current source such as a dipole- lute units. The magnetic field is measured in oersted (Oe) or like VLF radio source and lightning discharge. Then, the re- gauss (Gs) (1Gs = 10−4 T), whereas the electric field is also −1 flection of the waves incident on the upper boundary (z = Lz) in Gs, 1Gs = 300 Vcm . Note that in the absolute (Gaus- of the effective WGEI can be considered. There it will be sian) units of the magnitudes of the magnetic field compo- possible to demonstrate that this structure indeed has waveg- nent |Hy| are the same as ones of the electric field component uiding properties that are good enough. Then, in the model |Ez| in the atmosphere region where the permittivity is ε ≈ 1. described in the present paper, the VLF beam is postulated The correspondence between the absolute units and practical already on the input of the system. To understand how such SI units is given in the Fig.4 caption. a beam is excited by the, say, dipole antenna near the lower It is seen that the absolute values of the permittivity com- boundary of z = 0 of the WGEI, the formation of the beam ponents increase sharply above z = 75 km. The behavior structure based on the mode presentation should be searched. of the permittivity components is step-like, as seen from Then the conditions of the radiation (absence of ingoing Fig.3a. Therefore, the results of simulations are tolerant to waves; Eq. 12) can be used as the boundary conditions for the choice of the upper wall position of the Earth’s surface– the VLF beam radiated to the upper ionosphere and magne- ionosphere waveguide. The computed components of the tosphere. Due to a relatively large scale of the inhomogeneity tensor impedance at z = 85 km are Z011 = 0.087 + i0.097, in this region, the complex geometrical optics (Rapoport et Z021 = 0.085+i0.063, Z012 = −0.083−i0.094 and Z022 = al., 2014) would be quite suitable for modeling beam prop- 0.093 + i0.098. So, a condition of |Z0αβ | ≤ 0.15 is satis- agation, even accounting for the wave dispersion in magne- fied there, which is necessary for the applicability of the tized plasma. The proper effective boundary condition, sim- boundary conditions in Eq. (25). The maximum value of −5 ilar to Rapoport et al.(2014), would allow for making a rel- the Hy component is 0.1Oe = 10 T in Fig.4a for the ini- www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 218 Y. Rapoport et al.: Tensor impedance method tial VLF beam at x = 0. This corresponds to the value of fects the excitation of the Ey component within the waveg- −1 the Ez component of 0.1Gs = 30Vcm . At the distance of uide rather weakly. x = 1000 km the magnitudes of the magnetic field Hy are of −5 about 3 × 10 Oe = 3nT, whereas the electric field Ey is − 3 × 10 6 Gs ≈ 1mVcm−1. 5 Influence of the parameters of the WGEI on the The wave beams are localized within the WGEI at 0 < z < polarization transformation and losses of the 75km, mainly in the regions with the isotropic permittivity propagating VLF waves (see Fig.4b–e). The mutual transformations of the beams of different polarizations occur near the waveguide upper An important effect of the gyrotropy and anisotropy is the boundary due to the anisotropy of the ionosphere within the corresponding transformation of the field polarization during thin layer at 75km < z < 85km (Fig.4b, d). These trans- the propagation in the WGEI, which is absent in the ideal formations depend on the permittivity component values of metal planar waveguide without gyrotropy and anisotropy. the ionosphere at the altitude of z > 80km and on the com- We will show that this effect is quite sensitive to the carrier ponents of the tensor impedance. Therefore, the measure- frequency of the beam, propagating in the WGEI, and the ments of the phase and amplitude modulations of different inclination of the geomagnetic field and perturbations in the EM components near the Earth’s surface can provide infor- electron concentration, which can vary under the influences mation on the properties of the lower and middle ionosphere. of the powerful sources placed below, above and inside the In accordance with boundary condition in Eq. (32), we ionosphere. In the real WGEI, the anisotropy and gyrotropy suppose that when entering the system at X = 0, only one of are connected with the volume effect and effective surface the two polarization modes is excited, namely, the transverse tensor impedances at the lower and upper surfaces of the ef- magnetic (TM) mode, i.e., at x = 0, Hy 6= 0 and Ey = 0 fective WGEI, where z = 0 and z = Lz (Fig.1). For the cor- (Fig.4a). Upon further propagation of the beam with such responding transformation of the field polarization determi- boundary conditions at the entrance to the system in a ho- nation, we introduce the characteristic polarization relation mogeneous isotropic waveguide, the property of the electro- |Ey/Hy|(z;y = Ly/2;x = x0), taken at the central plane of magnetic field described by the relation Hy 6= 0 and Ey = 0 the beam (y = Ly) at the characteristic distance (x = x0) will remain valid. The qualitative effect due to the presence from the beam input and VLF transmitter. The choice of the of gyrotropy (a) in a thin bulk layer near the upper bound- characteristic polarization parameter (|Ey/Hy|) and its de- ary of WGEI and (b) in the upper boundary condition with pendence on the vertical coordinate (z) is justified by con- complex gyrotropic and anisotropic impedance is as follows: ditions (1)–(3). (1) The WGEI is similar to the ideal pla- during beam propagation in the WGEI, the transverse electric nar metalized waveguide because, first, the tensor ˆ is dif- (TE) polarization mode with the corresponding field compo- ferent from the isotropic Iˆ only in the relatively small up- nents, including Ey, is also excited. This effect is illustrated per part of the WGEI in the altitude range from 75–80 to in Fig.4b and d. 85 km (see Fig.1). Second, both the Earth and ionosphere The magnitude of the Ey component depends on the val- conductivity are quite high, and corresponding impedances ues of the electron concentration at the altitudes z = 75– are quite low. In particular, the elements of the effective 100 km. In Fig.5a and b the different dependencies of the tensor impedance at the upper boundary of the WGEI are electron concentration n(z) are shown (see the solid, dashed small, |Z0αβ | ≤ 0.1 (see, for example, Table1). (2) Respec- and dotted lines 1, 2 and 3, respectively). The corresponding tively, the carrier modes of the VLF beam are close to the dependencies of the component absolute values of the per- modes of the ideal metalized planar waveguide. These un- mittivity are shown in Fig.4c and d. coupled modes are subdivided into sets of (Ex,Hy,Ez) and The distributions of |Ey| and |Hy| at x = 1000km are (Hx,Ey,Hz). The detailed search of the propagation of the given in Fig.6. Results in Fig.6a and b correspond to the separate eigenmodes of the WGEI is not a goal of this paper solid (1) curve n(z) in Fig.5; Fig.6c and d correspond to and will be the subject of the separate paper. (3) Because we the dashed (2) curve; Fig.6e and f correspond to the dot- have adopted the input boundary conditions in Eq. (32) (with ted (3) curve in Fig.5. The initial Hy beam is the same and Hy 6= 0, Ey = 0) for the initial beam(s), the above-mentioned is given in Fig.4a. The values of the tensor impedance for value |Ey/Hy|(z;y = Ly/2;x = x0) characterizes the mode these three cases are presented in Table1. coupling and corresponding transformation of the polariza- The distributions of |Ey| and |Hy| on z at x = 1000 km in tion at the distance of x0 from the beam input due to the the center of the waveguide, y = 1500 km, are given in Fig.7. presence of the volume and surface gyrotropy and anisotropy These simulations show that the change in the complex ten- in the real WGEI. The results presented below are obtained sors of both volume dielectric permittivity and impedance at for x0 = 1000 km, that is, by the order of value, a typical the lower and upper boundaries of the effective WGEI in- distance, for example, between the VLF transmitter and re- fluence the VLF losses remarkably. The modulation of the ceiver of the European VLF/LF radio network (Biagi et al., electron concentration at the altitudes above z = 120 km af- 2015). Another parameter characterizing the propagation of the beam in the WGEI is the effective total loss parameter

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Figure 3. (a) The vertical dependencies of the modules of components of the permittivity in the frame associated with the geomagnetic ﬁeld |ε1|, |ε3| and |εh| with curves 1, 2 and 3, correspondingly. (b)–(g) The real (corresponding lines with the values denoted by one prime) and imaginary parts (corresponding lines with the values denoted by two primes) of the components ε1, ε3 and εh in general and detailed views.

Figure 4. Panel (a) is the initial distribution of |Hy| at x = 0. Panels (b) and (c) are |Ey| and |Hy| at x = 600km. Panels (d) and (e) are |Ey| −6 −1 and |Hy| at x = 1000km. For the electric ﬁeld, the maximum value (d) is 3 × 10 Gs ≈ 1mVcm ; for the magnetic ﬁeld, the maximum −5 5 −1 ◦ value (e) is 3 × 10 Gs ≈ 3nT. At the altitudes z < 75km,|Ez| ≈ |Hy|. ω = 1.0 × 10 s and θ = 45 .

of |Hymax(x = x0)/Hymax(x = 0)|. Note that this parameter and its position shifts to the lower altitudes with increasing characterizes both dissipative and diffraction losses. The last frequency. At the higher frequency (ω = 1.14 × 105 s−1), the are connected with beam spreading in the transverse (y) di- larger maximum of the polarization parameter corresponds rection during the propagation in the WGEI. to the intermediate value of the angle θ = 45◦ (Fig.8b); for In Fig.8 the polarization and loss characteristic dependen- the lower frequency (ω = 0.86×105 s−1), the largest value of cies on both the carrier beam frequency and the angle θ be- the first (higher) maximum corresponds to the almost vertical tween the geomagnetic field and the vertical directions (see direction of the geomagnetic field (θ = 5◦; Fig.8a). For the Fig.1) are shown. intermediate value of the angle (θ = 45◦), the largest value of In Fig.8a–c the altitude dependence of the polarization the main maximum corresponds to the higher frequency (ω = 5 −1 parameter |Ey/Hy| exhibits two main maxima in the WGEI. 1.14 × 10 s ) in the considered frequency range (Fig.8c). The first one lies in the gyrotropic region above 70 km, while The total losses increase monotonically with increasing fre- the second one in the isotropic region of the WGEI. As seen quency and depend weakly on the value of θ (Fig.8d). from Fig.8a and b, the value of the (larger) second maximum To model the effect of increasing and decreasing the elec- increases, while the value of the first maximum decreases, tron concentration ne in the lower ionosphere on the polar- www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 220 Y. Rapoport et al.: Tensor impedance method

Table 1. The values of tensor impedance components corresponding to the data shown in Fig.5.

Component of the tensor impedance Z011 Z021 Z012 Z022 Undisturbed concentration (curve 1 in Fig.5) 0 .088 + i0.098 0.085 + i0.063 −0.083 − i0.094 0.093 + i0.098 Decreased concentration (curve 2 in Fig.5) 0 .114 + i0.127 0.107 + i0.079 −0.105 − i0.127 0.125 + i0.125 Increased concentration (curve 3 in Fig.5) 0 .067 + i0.0715 0.061 + i0.051 −0.060 − i0.070 0.069 + i0.072

tion have been reported as a result of seismogenic phenomena, tsunamis, particle precipitation in the ionosphere due to wave–particle interaction in the radiation belts (Pulinets and Boyarchuk, 2005; Shinagawa et al., 2013; Arnoldy and Kintner, 1989; Glukhov et al., 1992; Tolstoy et al., 1986), etc. The changes in the |Ey/Hy| due to an increase or decrease in electron concentration vary by absolute values from dozens to thousands of a percent. This can be seen from the comparison between Fig.9b and c (lines 3) and Fig.8c (line 3), which corresponds to the unperturbed distribution of the ionospheric electron concentration (see also lines 1 in Figs.5b and9a). It is even more interesting that in the case of decreasing (Fig.9a, curve 2) electron concentration, the main maximum of |Ey/Hy| appears in the lower atmosphere (at the altitude around 20 km, Fig.9b, curve 3, which corresponds to ω = 1.14 × 105 s−1). In the case of increasing electron concentration (Fig.9a, curve 3), the main maximum of |Ey/Hy| appears near the E region of the ionosphere (at the altitude around 77 km, Fig.9c). The secondary maxi- Figure 5. Different profiles of the electron concentration n(z) mum, which is placed, in the absence of the perturbation of used in simulations: solid, dashed and dotted lines correspond to undisturbed, decreased and increased concentrations, respectively. the electron concentration, in the lower atmosphere (Fig.8c, Shown are the (a) detailed view and (b) general view. Panels (c) curves 2 and 3) or mesosphere and ionosphere D region and (d) show the permittivity |ε3| and εh modules. ((Fig.8c, curve 1) practically disappears or just is not seen in the present scale in the case under consideration (Fig.9c, curves 1–3). ization parameter, we have used the following parameteriza- In Fig. 10, the real (a) and imaginary (b) parts of the sur- tion for the ne change 1ne = ne(z) − n0e(z) of the electron face impedance at the upper boundary of the WGEI have a concentration, where n0e(z) is the unperturbed altitude dis- quasi-periodical character with the amplitude of oscillations tribution of the electron concentration. occurring around the effective average values (not shown explicitly in Fig. 10a and b), which decreases with an increas- 1ne(z) = n0e(z)8(z); ing angle of θ. The average Re(Z011) and Im(Z011) val- 2 2 (z − z2) (z − z1) ues in general decrease with an increasing angle of θ (see 8(z) = [F (z)] − [F (z1)] − [F (z2)]; = 1z2 1z2 Fig. 10a and b). The average values of Re(Z011) at θ 5, 12 12 30, 45 and 60◦ (lines 1–4 in Fig. 10a) and Im(Z ) at + 011 −2 z1 z2 = ◦ = ◦ F (z) = f00 · cosh z − /1z θ 45 and θ 60 (curves 3 and 4 in Fig. 10b) increase 2 with an increasing frequency in the frequency range (0.86– 5 −1 (35) 1.14) ×10 s . The average Im(Z011) value at θ = 5 and 30◦ changes in the frequency range (0.86–1.14) ×105 s−1 In Eq. (35), 1z ≡ z − z ; 1z is the effective width of 12 2 1 non-monotonically with the maximum at (1–1.1) ×105 s−1. the electron concentration perturbation altitude distribution. The value of finite impedance at the lower Earth–atmosphere The perturbation 1n is concentrated in the range of alti- e boundary of the WGEI influences the polarization transfor- tudes z ≤ z ≤ z and is equal to zero outside this region, 1 2 mation parameter minimum near the E region of the iono- 1n (z ) = 1n (z ) = 0, while 8(z ) = 8(z ) = 0. e 1 e 2 1 2 sphere (lines 1 and 2 in Fig. 10c). The decrease of sur- The change in the concentration in the lower ionosphere face impedance Z at the lower Earth–atmosphere bound- causes a rather nontrivial effect on the parameter of the polar- 0 ary of the WGEI by two orders of magnitude produces the ization transformation |Ey/Hy| (Fig.9a–c). Note that either an increase or a decrease in the ionosphere plasma concentra-

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Figure 6. Panels (a), (c) and (e) are dependencies of |Ey|. Panels (b), (d) and (f) are dependencies of |Hy| at x = 1000 km, with ω = 1.0 × 105 s−1 and θ = 45◦. The initial beams are the same as in Fig.4a. Panels (a) and (b) correspond to the solid (1) curves in Fig.5. Panels (c) and (d) are for the dashed (2) curves. Panels (e) and (f) correspond to the dotted (3) curves there.

Figure 7. The dependencies of EM components at altitude z in the center of the waveguide at y = 1500 km for the different proﬁles of the electron concentration. The solid (1), dashed (2) and dotted (3) curves correspond to the different proﬁles of the electron concentration in Fig.5. Panels (a) and (b) are the same kinds of curves, with ω = 1.0 × 105 s−1 and θ = 45◦.

100 % increase of the corresponding |Ey/Hy| minimum at ponent and is positive, which means that it is directed ver- z ∼ 75 km (Fig. 10c). tically downward, as in the fair-weather current (curve 1, Fig. 12b). Then suppose that the surface seismogenic current is directed in the same way as the mesospheric current. We 6 Discussion first consider the case when the mesospheric current is zero and only the corresponding seismogenic current is present The observations presented in Rozhnoi et al.(2015) show a near the Earth. The corresponding mesospheric electric field possibility for seismogenic increasing losses of VLF waves under the condition of a given potential difference between in the WGEI (Fig. 11; see details in Rozhnoi et al., 2015). We the Earth and the ionosphere (curve 2, Fig. 12b) is directed discuss the qualitative correspondence of our results to these opposite to those excited by the corresponding mesospheric experimental data. current (curve 1, Fig. 12b). As a result, in the presence of The modification of the ionosphere due to electric field ex- both mesospheric and a seismogenic surface current, the total cited by the near-ground seismogenic current source has been mesospheric electric field (curve 3, Fig. 12b) is smaller in ab- taken into account. In the model (Rapoport et al., 2006), the solute value than in the presence of only a mesospheric cur- presence of the mesospheric current source, which followed rent (curve 1, Fig. 12b). It has been shown by Rapoport et al. from the observations (Martynenko et al., 2001; Meek et al., (2006) that the decrement of losses |k00| for VLF waves in the 00 00 2004; Bragin et al., 1974) is also taken into account. It is WGEI is proportional to |k | ∼ |ε | ∼ Ne/νe. Ne and νe de- assumed that the mesospheric current only has the z com- crease and increase, respectively, due to the appearance of a www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 222 Y. Rapoport et al.: Tensor impedance method

Figure 8. Characteristics of the polarization transformation parameter |Ey/Hy| (a–c) and the effective coefﬁcient of the total losses at the distance of x0 = 1000 km from the beam input (d). Corresponding altitude dependence of the electron concentration is shown in line 1 of Fig.5b. Panels (a), (b) and (d) show dependences of the polarization parameter (a, b) and total losses (d) on the vertical coordinate and frequency for different θ angles, respectively. Black (1), red (2), green (3) and blue (4) curves in panels (a), (b) and (d) correspond to 5, 30, 45 and 60◦, respectively. Panels (a) and (b) correspond to the frequencies ω = 0.86 × 105 s−1 and ω = 1.14 × 105 s−1, respectively. Panel (c) shows the dependence of the polarization parameter on the vertical coordinate for the different frequencies. Black (1), red (2) and green (3) lines correspond to the frequencies 0.86 × 105, 1 × 105 and 1.14 × 105 s−1, respectively, and θ = 45◦.

Figure 9. (a) Decreased and increased electron concentration (line 2, red) and (line 3, blue) correspond to f00 = −1.25 and f00 = 250, respectively, relative to the reference concentration (line 1, black) with parametrization conditions (see Eq. 35) z1 = 50 km, z2 = 90 km and 1 = 20 km. Panels (b) and (c) are the polarization parameter |Ey/Hy| altitude distribution for decreased and increased electron concentration, respectively. In (b) and (c) lines 1, 2 and 3 correspond to ω values 0.86×105 s−1, 1.0×105 s−1 and 1.14×105 s−1, respectively. Angle θ is equal to 45◦. seismogenic surface electric current, in addition to the meso- b), corresponds to an increase in losses with an increase in spheric current (curve 3, Fig. 12b). As a result, the losses the absolute value of the imaginary part of the dielectric con- increase compared with the case when the seismogenic cur- stant when a near-surface seismogenic current source appears rent is absent and the electric ﬁeld has a larger absolute value (curve 3 in Fig. 12b) in addition to the existing mesospheric (curve 1, Fig. 12). The increase in losses in the VLF beam, current source (curve 2 in Fig. 12b). This seismogenic in- shown in Fig. 13 (compare curves 2 and 1 in Fig. 13a and

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Figure 10. (a, b) Frequency dependencies of the real (a) and imaginary (b) parts of the effective tensor impedance Z011 component at the ◦ upper boundary (z = Lz, see Fig.1) of the WGEI. Lines 1 (black), 2 (red), 3 (blue) and 4 (green) correspond to θ = 5, 30, 45 and 60 , 5 −1 ◦ respectively. Panel (c) shows the polarization parameter |Ey/Hy| altitude dependency at the frequency 0.86×10 s and angle θ = 45 for −4 9 −1 the isotropic surface impedance Z0 at the lower surface of the WGEI equal to 10 . Earth conductivity σ is equal to 10 s for line 1 and −2 7 −1 Z0 = 10 (σ = 10 s ) for line 2.

Figure 11. Averaged residual VLF and LF signals from ground-based observations at the wave paths of JJY-Moshiri, JJI-Kamchatka, JJY- Kamchatka, Australia-Kamchatka and Hawaii-Kamchatka. Horizontal dotted lines show the 2σ level. The color-ﬁlled zones highlight values exceeding the −2σ level. In panel b Dst (disturbance storm time index) variations and earthquakes magnitude values are shown (from Rozhnoi et al., 2015, see their Fig. 1 but not including the Detection of Electro-Magnetic Emissions Transmitted from Earthquake Regions (DEMETER) data; the work of Rozhnoi et al., 2015, is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0)). See other details in Rozhnoi et al.(2015). crease in losses corresponds qualitatively to the results pre- side the WGEI (see Figs.4–7). Nevertheless, the TIMEB sented in Rozhnoi et al.(2015). method described by Eqs. (15)–(19), (22)–(24), (27), (30a) The TIMEB is a new method of modeling characteristics and (30b) and allows for the determination of all ﬁeld com- of the WGEI. The results of beam propagation in WGEI ponents in the range of altitudes of 0 ≤ z ≤ Lmax, where modeling presented above include the range of altitudes in- Lmax = 300 km. The structure and behavior of these eigen- www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 224 Y. Rapoport et al.: Tensor impedance method

Figure 12. Modification of the ionosphere by the electric field of seismogenic origin based on the theoretical model (Rapoport et al., 2006). (a) Geometry of the model (Rapoport et al., 2006) for the determination of the electric field excited by the seismogenic current source Jz(x,y) and penetrated into the ionosphere with isotropic (I) and anisotropic (II) regions of the “atmosphere–ionosphere” system. (b) Electric field in the mesosphere in the presence of the seismogenic current sources only in the mesosphere (1), in the lower atmosphere (2), and both in the mesosphere and in the lower atmosphere (3). (c) Relative perturbations caused by the seismogenic electric field, normalized on the corresponding steady-state values in the absence of the perturbing electric field, denoted by the index “0”, electron temperature (Te/Te0), electron concentration (Ne/Ne0) and electron collision frequency (νe/νe0).

Figure 13. Altitude distributions of the normalized tangential (y) electric (a) and magnetic (b) VLF beam field components in the central plane of the transverse beam distribution (y = 0) at the distance of x = 1000 km from the input of the system. Line 1 in (a, b) corresponds to the presence of the mesospheric electric current source only with a relatively small value of Ne and a large νe. Line 2 corresponds to the presence of both mesospheric and near-ground seismogenic electric current sources with a relatively large value of Ne and small νe. Lines 1 and 2 in (a) and (b) correspond qualitatively to the lines 1 and 3, respectively, in Fig. 12b, with ω = 1.5 × 105 s−1 and θ = 45◦. modes in the WGEI and leakage waves will be a subject of (Figs.4b, c), the few lowest modes of the WGEI along the separate papers. We present here only the final qualitative re- z and y coordinates still persist. At distance x = 1000 km sult of the simulations. In the range Lz ≤ z ≤ Lmax, where (Figs.4d, c;6e, f; and7a, b), only the main mode persists Lz = 85 km is the upper boundary of the effective WGEI, all in the z direction. Note that the described field structure cor- field components (1) are at least one order of altitude less respond to the real WGEI with losses. The gyrotropy and than the corresponding maximal field value in the WGEI and anisotropy causes the volume effects and surface impedance, (2) have the oscillating character along the z coordinate and in distinction to the ideal planar metalized waveguide with describe the modes leaking from the WGEI. isotropic filling (Collin, 2001). Let us make a note also on the dependences of the field The closest approach of the direct investigation of the components in the WGEI on the vertical coordinate (z). VLF electromagnetic field profile in the Earth–ionosphere The initial distribution of the electromagnetic field with al- waveguide was a series of sounding rocket campaigns at titude z (Fig.4a) is determined by the boundary conditions mid and high latitudes at Wallops Island, Virginia, USA, and of the beam (see Eq. 32). The field component includes Siple Station in Antarctica (Kintner et al., 1983; Brittain et higher eigenmodes of the WGEI. The higher-order modes al., 1983; Siefring and Kelley, 1991; Arnoldy and Kintner, experienced quite large losses and practically disappear after 1989). Here single-axis E field and three-axis B field anten- beam propagation on a 1000 km distance. This determines nas, supplemented in some cases with in situ plasma den- the change in altitude (z) and transverse (y) distributions of sity measurements, were used to detect the far-field fixed- the beam field during propagation along the WGEI. In par- frequency VLF signals radiated by US Navy and Stanford ticular, at the distance of x = 600 km from the beam input ground transmitters.

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The aim of this experiment will be the investigation of the VLF beams launched by the near-ground source and VLF transmitter with the known parameters and propagating both in the WGEI and leaking from WGEI into the upper ionosphere. Characteristics of these beams will be compared with the theory proposed in the present paper and the theory on leakage of the VLF beams from WGEI, which we will present in the next papers.

7 Conclusions

1. We have developed the new and highly effective ro- bust method of tensor impedance for the VLF electromagnetic beam propagation in the inhomogeneous waveguiding media: the “tensor impedance method for modeling propagation of electromagnetic beams” in a multi-layered and inhomogeneous waveguide. The main differences and advantages of the proposed tensor impedance method in comparison with the known Figure 14. Proposed VIPER trajectory. method of impedance recalculating, in particular invariant imbedding methods (Shalashov and Gospodchikov, 2011; Kim and Kim, 2016), are the following: (i) our method is a direct method of the recalculation of ten- The most comprehensive study of the WGEI will be pro- sor impedance, and the corresponding tensor impedance vided by the ongoing NASA VIPER (VLF Trans-Ionospheric is determined analytically (see Eq. 22); (ii) our method Propagation Experiment Rocket) project (PI John Bon- applied for the media without non-locality does not nell, University of California, Berkeley; NASA grant no. need a solution for integral equation(s), as in the in- 80NSSC18K0782). The VIPER sounding rocket campaign variant imbedding method; and (iii) the proposed ten- consists of a summer nighttime launch during quiet magne- sor impedance method does not need to reveal the for- tosphere conditions from Wallops Flight Facility, Virginia, ward and reflected waves. Moreover, even the condi- USA, collecting data through the D, E and F regions of the tions of the radiation in Eq. (12) at the upper boundary ionosphere with a payload carrying the following instrumen- z = L are determined through the total field compo- tation: 2-D E and 3-D H field waveforms (DC-1 kHz); 3-D max nents H that makes the proposed procedure techni- waveforms ranging from an ELF (extremely low frequency) x,y cally less cumbersome and practically more convenient. to VLF (100 Hz to 50 kHz); 1-D wideband E field measurements of plasma and upper hybrid lines (100 kHz to 4 MHz); 2. The waveguide includes the region for the altitudes and Langmuir probe plasma density and ion gauge neutral- 0 < z < 80–90 km. The boundary conditions are the ra- density measurements at a sampling rate of at least tens of diation conditions at z = 300 km, which can be recal- Hz. The VIPER project will fly a fully 3-D EM field measure- culated to the lower altitudes as the tensor relations be- ment, direct current (DC) through VLF, and relevant plasma tween the tangential components of the EM field. In an- and neutral particle measurements at mid latitudes through other words, the tensor impedance conditions have been the radiation fields of (1) an existing VLF transmitter (the used at z = 80–90 km. very low-frequency shore radio station with the call sign 3. The application of this method jointly with the previous NAA at Cutler, Maine, USA, which transmits at a frequency results of the modification of the ionosphere by seismo- of 24 kHz and an input power of up to 1.8 MW; see Fig. 11) genic electric field gives results which qualitatively are and (2) naturally occurring lightning transients through and in agreement with the experimental data on the seismo- above the leaky upper boundary of the WGEI. This is sup- genic increasing losses of VLF wave/beam propagation ported by a vigorous theory and modeling effort in order to in the WGEI. explore the vertical and horizontal profile of the observed 3- D electric and magnetic radiated fields of the VLF transmit- 4. The observable qualitative effect is the mutual transfor- ter and the profile related to the observed plasma and neutral mation of different polarizations of the electromagnetic densities. The VLF wave’s reflection, absorption and trans- field occur during the propagation. This transformation mission processes as a function of altitude will be searched of the polarization depends on the electron concentra- making use of the data on the vertical VLF E and H field tion, i.e., the conductivity, of the D and E layers of the profile. ionosphere at altitudes of 75–120 km. www.ann-geophys.net/38/207/2020/ Ann. Geophys., 38, 207–230, 2020 226 Y. Rapoport et al.: Tensor impedance method

5. Changes in complex tensors of both volume dielec- respond to θ equal to 45 and 60◦, and these in- tric permittivity and impedances at the lower and up- crease with an increasing frequency in the consid- per boundaries of effective the WGEI influence the VLF ered frequency range of 0.86–1.14 ×105 s−1. The losses in the WGEI remarkably. average value of Im(Z11) corresponds to θ equal to 5 and 30◦ and changes in the frequency range of 6. An influence is demonstrated on the parameters charac- 0.86–1.14 ×105 s−1 non-monotonically while hav- terizing the propagation of the VLF beam in the WGEI, ing maximum values around the frequency of 1– in particular, the parameter of the transformation po- 1.1 ×105 s−1. larization |Ey/Hy| and tensor impedance at the upper boundary of the effective WGEI, the carrier beam fre- (iv) The value of finite impedance at the lower Earth– quency, and the inclination of the geomagnetic field and atmosphere boundary of the WGEI quite observ- the perturbations in the altitude distribution of the elec- ably influences the polarization transformation pa- tron concentration in the lower ionosphere. rameter minimum near the E region of the ionosphere. The decrease of surface impedance Z at the (i) The altitude dependence of the polarization param- lower Earth–atmosphere boundary of the WGEI in eter |Ey/Hy| has two main maxima in the WGEI: two orders causes the increase of the corresponding the higher maximum is in the gyrotropic region minimum value of |Ey/Hy| in ∼ 100 %. above 70 km, while the other is in the isotropic re- ≤ ≤ = gion of the WGEI. The value of the (larger) sec- 7. In the range Lz z Lmax, where Lz 85km is the ond maximum increases, while the value of the upper boundary of the effective WGEI, all field com- first maximum decreases, and its position shifts to ponents (a) are at least one order of altitude less than the lower altitudes with increasing frequency. In the corresponding maximal value in the WGEI and the frequency range of ω = (0.86–1.14) ×105 s−1, (b) have the oscillating character (along the z coor- at the higher frequency, the larger maximum polar- dinate), which describes the modes leaking from the ization parameter corresponds to the intermediate WGEI. The detail consideration of the electromagnetic value of the angle θ = 45◦; for the lower frequency, waves leaking from the WGEI will be presented in a the largest value of the first (higher) maximum cor- separate paper. The initial distribution of the electro- responds to the nearly vertical direction of the geomagnetic field with z (vertical direction) is determined magnetic field. The total losses increase monotoni- by the initial conditions on the beam. This field in- cally with increasing frequency and depend weakly cludes higher eigenmodes of the WGEI. The higher- on the value of θ (Fig.1). order modes, in distinction to the lower ones, have quite large losses and practically disappear after a beam prop- (ii) The change in the concentration in the lower agation for 1000 km distance. This circumstance deter- ionosphere causes a rather nontrivial effect on mines the change in altitude (z) distribution of the field the parameter of the polarization transformation of the beam during its propagation along the WGEI. In | | Ey/Hy . This effect does include the increase and particular, at the distance of x = 600 km from the beam decrease of the maximum value of the polariza- input, the few lowest modes of the WGEI along z co- | | tion transformation parameter Ey/Hy . The corre- ordinates still survived. Further, at x = 1000 km, prac- sponding change of this parameter has large values tically, only the main mode in the z direction remains. from dozens to thousands of a percent. In the case This fact is reflected in a minimum number of oscilla- of a decreasing electron concentration, the main tions of the beam field components along z at a given | | maximum of Ey/Hy appears in the lower atmo- value of x. sphere at an altitude of around 20 km. In the case of an increasing electron concentration, the main max- 8. The proposed propagation of VLF electromagnetic imum of |Ey/Hy| appears near the E region of the beams in the WGEI model and results will be use- ionosphere (at the altitude around 77 km), while the ful to explore the characteristics of these waves as secondary maximum practically disappears. an effective instrument for diagnostics of the influ- (iii) The real and imaginary parts of the surface ences on the ionosphere from above in the Sun–solar- impedance at the upper boundary of the WGEI wind–magnetosphere–ionosphere system; from below have a quasi-periodical character with the ampli- from the most powerful meteorological, seismogenic tude of “oscillations” occurring around some ef- and other sources in the lower atmosphere and litho- fective average value decreases with an increas- sphere and Earth, such as hurricanes, earthquakes and ing angle of θ. Corresponding average values of tsunamis; from inside the ionosphere by strong thunderstorms with lightning discharges; and even from far Re(Z11) and Im(Z11), in general, decrease with an space by gamma flashes and cosmic rays events. increasing angle of θ. Average values of Re(Z11) ◦ for θ equal to 5, 30, 45 and 60 and Im(Z11) cor-

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Appendix A: The matrix coefﬁcients included in Eq. (16)

Here the expressions of the matrix coefﬁcients are presented that are used in the matrix factorization to compute the tensor impedance (see Eq. 16).

ihz ihz ! 1 + (k − α α k ); α (k − k ) − −1; 0 αˆ (0) = 1 1 1 2 2 1 2 2 1 , αˆ ( ) = ; 1 ≡ 1 − α α ; N ihz − ; + ihz − N 0; −1 1 2 1 α1(k1 k2) 1 1 (k2 α1α2k1)

 β22 β21 ikx hz β23  ( 2 )j−1/2; −( 2 )j−1/2 + ( 2 )j−1 kx kx 2 kx 1−β22 1−β22 1−β22  k2 k2 k2  0 0 0  2   β12 ikx hz β32 kx β12·β21  (−) −( 2 )j−1/2 + ( 2 )j−1; (β11 + 2 2 )j−1/2−  αˆ = kx 2 kx k kx j  1−β22 1−β22 0 1−β22  k2 k2 k2  0 0 0   2 2   ikx hz kx β12·β23 ikx hz kx β32·β21  − (β13 + 2 2 )j−1 − (β31 + 2 2 )j  2 k kx 2 k kx  0 1−β22 0 1−β22 k2 k2 0 0  β22 β21 ikx hz β23  ( 2 )j+1/2; −( 2 )j+1/2 − ( 2 )j+1 kx kx 2 kx 1−β22 1−β22 1−β22  k2 k2 k2  0 0 0  2   β12 ikx hz β32 kx β12·β21  (+) −( 2 )j+1/2 − ( 2 )j+1; (β11 + 2 2 )j+1/2+  αˆ = kx 2 kx k kx j  1−β22 1−β22 0 1−β22  k2 k2 k2  0 0 0   2 2   ikx hz kx β12·β23 ikx hz kx β32·β21  + (β13 + 2 2 )j+1 + (β31 + 2 2 )j  2 k kx 2 k kx  0 1−β22 0 1−β22 k2 k2 0 0  β22 β22 2 2 β21 β21  −( 2 )j−1/2 − ( 2 )j+1/2 + k hz; ( 2 )j−1/2 + ( 2 )j+1/2 kx kx 0 kx kx 1−β22 1−β22 1−β22 1−β22  k2 k2 k2 k2  0 0 0 0  2 2   β12 β12 kx β12·β21 kx β12·β21  (0)  ( 2 )j−1/2 + ( 2 )j+1/2; −(β11 + 2 2 )j−1/2 − (β11 + 2 2 )j+1/2+ αˆ = kx kx k kx k kx j  1−β22 1−β22 0 1−β22 0 1−β22  k2 k2 k2 k2  0 0 0 0   2 4   2 2 kx kx β23·β32  +k hz · (1 − β33 2 − 4 2 )j  0 k k kx  0 0 1−β22 k2 0 (A1)

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