The Terminator-Time Method and 3D Problem of Subionospheric

THE TERMINATOR-TIME METHOD AND 3D PROBLEM OF SUBIONOSPHERIC

RADIO WAVE PROPAGATION ACROSS THE SOLAR TERMINATOR

O.V. Soloviev(1), M. Hayakawa(2), O.A. Molchanov(3)

(1) Institute of Radio Physics, St.Petersburg State University, Ulyanovskaya 1/1, Petrodvorets, St.Petersburg,

198504, Russia; E-mail: [email protected]

(2) Department of Electronic Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu shi,

Tokyo 182-8585, Japan; E-mail: [email protected]

(3) United Institute of Physics of the Earth RAS, Bol. Gruzinskaya 10, Moscow,

123810, Russia; E-mail: [email protected]

ABSTRACT

This paper presents a mathematical model, an asymptotic theory and an appropriate numerical algorithm to study, in the scalar approximation, VLF point source field propagation problem within the non-uniform Earth-ionosphere waveguide allowing for 3D local ionosphere inhomogeneity upon a ground of the solar terminator transition. The local ionosphere inhomogeneity, which centre is situated above the model earthquake, is simulated by a bell-shaped perturbation of the ionospheric waveguide wall. Numerical results show that the emergence of the local ionosphere inhomogeneity on the radio wave propagation path deforms the curves of field amplitude and phase diurnal variations in accord with experimental data.

INTRODUCTION

The possibility of existence of ionospheric earthquake precursory anomalies is more and more widely accepted. Various ionospheric plasma phenomena have been detected by satellites and ground-based facilities over the epicentre of future shocks. The applications of the subionospheric VLF radio signals in a search for the seismic hazard precursors are widely discussed in the literature. Nevertheless up to now we still do not clearly understand the nature of seismic influence on the ionosphere.

The solar terminator is a part of terrestrial atmosphere situated between the space illuminated by the full ball of the sun and the complete shadow space. It is a layer between two conical surfaces; the layer boundaries are very diffuse. The strongest variations of temperature, pressure, electron density height profile, in the atmosphere during all natural day, occur at solar terminator passage. It conveys the suggestion that the ionospheric plasma is instable in period of terminator transition, and hopefully it is possible to receive an appreciable response to an extraneous action (started from seismic activity) into this region and this response may be powerful enough to be recorded.

Monitoring of the "OMEGA" signals at the Inubo observatory (35º42' N; 140º52' E) from the Tsushima transmitter (34º37' N; 129º27' E) at both 10.2 and 11.3 kHz frequencies revealed a time shift of the regular minima in the diurnal variations of phase and amplitude on the propagation path 1043 km long, which can be considered as short-distance propagation [1]. The minima in the diurnal phase and amplitude patterns generally occur when the terminator moves along the VLF propagation path. One minimum takes place around sunrise, and can be named morning "terminator time" (TT); another takes place around sunset, and can be named as evening TT. Their difference must be changed from day to day as the day length at the given geographic latitude, if propagation path is latitudinal. In the case of earthquake influence, it was occurred the abnormal behaviour of this morning and evening TT difference, that began a few days before the earthquake such that it had the lengthening of daytime conditions. The idea of the TT method consists in high sensitivity of VLF signals to changes in the lower ionosphere due to multi-mode interaction when the evening or morning terminator moves along a propagation path [2]. In the context of VLF radio wave propagation the most important ionospheric parameters are electron density and effective collision frequency height profiles. In regular conditions these parameters are determined by solar zenith angle in the essential to the VLF subionospheric propagation lower ionosphere domain. In perturbed conditions the spatially localized variations of electron density (due to various causes) are possible. MODEL, THEORY, NUMERICAL RESULTS AND DISCUSSION

This paper deals with VLF radio wave propagation problem within the non-uniform Earth-ionosphere waveguide. The propagation path lines up with the motion direction of the solar terminator. Upon a ground of the solar terminator a three-dimensional local ionosphere inhomogeneity belongs above the propagation line. This last one is a bell-shaped perturbation of the ionospheric waveguide wall, which centre is situated above the given point of the Earth's surface and the lateral dimensions are less than solar terminator transition width. The source is assumed to be a harmonic ()− ω exp i t electric dipole of moment P0 . The Earth's surface is considered as uniform all through the propagation δ path and characterized by homogeneous surface impedance e . The ionospheric surface impedance and waveguide height for terminator transition model are obtained from the given electron density and collision frequency height profiles [3], which depend on solar zenith angle that varies from 80º to 100º with step equal to 2º. It means we explore about 2500 km terminator transition, approximately 110 minutes in latitude near 35º N. The Earth's curvature is ignored as well as the geomagnetic field. Thus, we consider the point source field in the three-dimensional vacuum domain D δ bounded by Earth's plane and ionospheric surface Sit characterized by non-uniform impedance it . In the cylindrical coordinate system ()r,ϕ,z the dipole is chosen to be directed along the axis OZ , the plane z = 0 specifies the Earth's = ()ϕ δ = δ ()ϕ surface, the functions: zit zit r, , which specifies Sit , and it it r, ,zit do not vary in the perpendicular to the terminator motion direction. The local ionosphere perturbation deforms the surface Sit into the surface Si , which = ()ϕ δ ()ϕ is described by function zi zi r, and its electrical properties are specified by impedance i r, , zi . Thus, the ∈ surfaces Sit and Si coincide everywhere with the exception of the area S p Si , which defines the surface of the local inhomogeneity.

In the scalar approximation we may describe the electromagnetic field by the function Π()x, y,z , the vertical component of Hertz's vector, which satisfies inhomogeneous Helmholtz' equation in D ∈ R3 and the impedance = () boundary conditions on the surfaces z 0 and Si . The origin of Cartesian coordinate system x, y,z and the origin of the cylindrical coordinate system ()r,ϕ,z are the same. Using the second Green's formula, the unknown function Π()x, y,z at any point interior to the domain D may be represented as follows: r r ikε  ∂Π ()R,R′  Π()r = Π ()r + 0 Π()r′ δ ()r′ Π (r r′ )+ 0  ′ R 0 R ∫∫ R  i R 0 R,R i dS , (1) P k∂n′ 0 Si  

= ω ε µ r r ′ ′ where k 0 0 is the free-space wave number, R is the observation point, R is an integration point, n is a Π ()r normal directed outside the waveguide cavity D , 0 R is the field of considered point source in the regular flat ≥ δ δ = Π (r r ′) waveguide characterized by the constant height h zi and surface impedances e and i0 const, 0 R,R is the Π (r ) Π ( r r ′)= Π ( r ) r = r − r′ Green's function, which may be obtained from 0 R by formula 0 R, R 0 R1 , R1 R R . Along with the main problem and equation (1) let us write the equation for point source field propagation problem in the non- uniform Earth-ionosphere waveguide with solar terminator transition only, i.e. without local inhomogeneity: r r ikε  ∂Π ()R,R′  Π ()r = Π ()r + 0 Π ()r′ δ ()r′ Π (r r′ )+ 0  ′ t R 0 R ∫∫ t R  it R 0 R,R i dS . (2) P k∂n′ 0 Sit   The right-hand side surface integrals from (1) and (2) (so-called surface integrals of the first type) may be transformed to the ordinary two-dimensional integrals over area ()− ∞ < x′, y′ < +∞ at the plane z = const. Let us split the = Ω Ω Ω integration plane z const into two areas p and ∞ , where the first p is a projection of S p (local perturbation area) on this plane, the second Ω∞ is the all-remaining part of the plane. For x, y ∈Ω∞ the following equations are ()= () δ ()= δ () fulfilled: zi x, y zit x, y and i x, y it x, y . Subtracting equation (1) with transformed surface integral from similarly transformed equation (2) we finally arrive at r r ikε  ∂Π ()R,R′  Π()r = Π ()r + 0 Π()r′ δ ()r′ Π (r r′ )+ 0  + 2 + 2 ′ ′ + R 0 R ∫∫ R  it R 0 R,R i  1 pt qt dx dy P k∂n′ 0 Ω p +Ω∞   r r ikε  Π ()R,R′  + 0 Π()r′ δ ()r′ Π (r r′ )+ 0  + 2 + 2 − ∫∫ R  i R 0 R,R i  1 p p q p (3) P k∂n′ 0 Ω p   r r  ∂Π ()R,R′   − δ ()r′ Π (r r′ )+ 0  + 2 + 2  ′ ′  it R 0 R,R i  1 pt qt dx dy ,  k∂n′  

∂z′ ()x′, y′ ∂z′ ()x′, y′ r r r where p = it ,i , q = it ,i , R ∉ S . In a limiting case R → S an additional item Π()R 2 t ,p ∂x′ t ,p ∂y′ i i ∂Π ()r′ r ∂ ′ arises on the right hand side of (3) due to the jump of the normal derivative of the Green's function 0 R ,R n .

To solve (3), we apply the asymptotic method explicitly described in [4,5]. First of all, we select and exclude from Π( r ) Π (r ) Π (r r ′) R , 0 R , 0 R,R the rapidly oscillating factors that are corresponding to the point source field over the perfectly conducting plane for r >> z . Thus, we arrive at the equation for slowly varying attenuation function:

−1 r  P exp()ikr  r V ()R =  0  Π()R πε 2 0 r 

[]()′ + − = 2 + ′2 − ′ ()ϕ − ϕ′ that integrand has exponential factor exp ik r r1 r , where r1 r r 2rr cos . Then we introduce an elliptic coordinate system ()u,v on the surface z = const, and substitute the variables of integration: r r r′ = ()ch u + cos v , r = ()chu − cosv , dS′ = r r′du dv , − ∞ < u < +∞ , 0 ≤ v ≤ π . 2 1 2 1 In this elliptic coordinate system the noted exponential factor takes the form exp[]ikr()chu − 1 , and if kr >> 1 this factor may be assumed as being rapidly oscillating with respect to u (across the wave propagation path) as compared with the remaining cofactor determined by the attenuation function of the unknown Hertz's vector component. This allows us to use the stationary-phase method for evaluating the integral over u . To avoid losing the information on the transverse (in the horizontal direction) structure of the irregularity, we must perform the calculations with accuracy up −1 −3 2 to the terms of order O(()kr ), neglecting higher-order terms O(()kr ) only. As a result, the two-dimensional integral equation (3) is transformed into the equation with contour integrals along two lines: the projection of = ∂ Ω propagation path onto z const plane and the linear boundary S p of the local inhomogeneity area p . Such one- dimensional integral equation takes into account the geometry and the electrical properties of both the localized ionosphere perturbation and the solar terminator transition. A smooth shape and sizes of S p may be arbitrary. To solve the last equation, we use the numerical-analytical method of semi-inversion [4], which combines the direct inversion of the dominant part of the integral operator of the problem, that is a Volterra operator, with the iterative process by which the remaining part of the integral operator is inverted through successive approximations.

As an example, we present the results of numerical simulation of radio wave propagation process on the Tsushima- Inubo path. The source frequency is f = 10.2 kHz. Earth's wall impedance is determined by Earth's conductance σ = 4S/m, the ionospheric waveguide wall properties - as has been stated above. The figure below illustrates the temporal behaviour (during sunrise) of the amplitude of the attenuation function on the path 1043 km long, without and with three-dimensional local ionosphere disturbance. This local ionosphere inhomogeneity, which centre is situated above the model earthquake, is simulated by a bell-shaped perturbation of the ionospheric waveguide wall:  ()()− 2 + − 2   x x p y y p  ∆h = A exp − , p  2   rp  = = where Ap -3 km, rp 200 km, x p and y p correspond to the horizontal coordinates of the great Hyogo-ken Nambu (Kobe) earthquake (January 17, 1995) epicenter in the chosen coordinate system.

Considering the results of numerical simulation it is arguable that the emergence of the local ionosphere inhomogeneity on the radio wave propagation path deforms the curves of field amplitude and phase diurnal variations in accord with experimental data [1,2].

REFERENCES

[1] M. Hayakawa, O. A. Molchanov, T. Ondoh, and E. Kawai, "Precursory signature of the Kobe earthquake on VLF subionospheric signal," J. Atmos. Electr., vol. 16, No. 3, pp. 247-257, 1996. [2] O.A. Molchanov, and M. Hayakawa, "Subionospheric VLF signal perturbations possibly related to earthquakes," J. Geophys. Res., vol. 103 (A8), pp. 17489-17504, August 1, 1998. [3] A.B. Orlov, A.E. Pronin, and A.N. Uvarov, "The electron density of lower ionosphere profile modelling according to the VLF propagation data," in Problemy Difraktsii i Rasprostraneniya Voln, vol. 28. St.Petersburg: St.Petersburg Univ. Press, pp. 83-114, 2000. [4] O.V. Soloviev, and V.V. Agapov, "An asymptotic three-dimensional technique to study radio wave propagation in the presence of a localized perturbation of environment," Radio Sci., vol. 32, pp. 515-524, March-April 1997. [5] O.V. Soloviev, "Low frequency radio wave propagation in the Earth-ionosphere waveguide perturbed by a large- scale three-dimensional inhomogeneity," Izv. Vyssh. Uchebn. Zaved. Radiofiz. (Radiophys. and Quantum Electronics), vol. 41, pp. 588-604, May 1998.