Scattering of VLF field at a three-dimensional ionospheric...

2. Formulation of Electrodynamical Problem of Scattering

[12] The problem of the field of a harmonic ( exp(-i wt )) vertical electric dipole in the Earth-ionosphere waveguide with a three-dimensional local irregularity is considered in the impedance formulation for the parallel-plain model of the waveguide channel. The published estimates of the vertical and horizontal dimensions of sprites modeled by our local irregularity show that the dimensions do not exceed 50-100 km. This means that the electromagnetic field scattered at such irregularity may be considerable (and thus may be observed and registered) only at short paths of ~1000-1500 km for which one can neglect the curvature of the Earth's surface. The neglecting by the Earth's curvature makes the problem visual not breaking the general character of the proposed solution algorithm. A transition to the spherical model may be done using the formulae obtained by Soloviev [1990] and Soloviev and Agapov [1997]. In the considered model of the waveguide, the lower wall described by a plain surface S_g is assumed to be homogeneous, its properties being determined by the surface impedance d_g. The waveguide is limited from the top by the surface S_i, its properties being determined (in its regular part) by the impedance d_i. The three-dimensional local irregularity directly adjoining the upper wall of the waveguide is chosen in the form of a finite by a height cylinder, the shape of the cross section of it being arbitrary. The surface of the upper wall of the cylinder coincides with the surface S_i, whereas the surface of the lower wall S_p is located in the plane parallel to the surface S_i. We will denote the side surface of the cylinder as S_l. The properties of the waveguide space D R³ limited by the waveguide walls and surfaces of the model irregularity coincide with the properties of the vacuum, its dielectric and magnetic permeability and wave number being e₀, m₀, and k, respectively.

Figure 1

[13] In the cylindrical coordinate system (r, j,z) with the z axis going through the source, the surfaces S_g and S_i are described by the equations z=0 and z=h, respectively. The source of the field (a vertical electrical dipole with the dipole moment P₀ ) is located in the point (0,0,z_t ). The surface S_p lies in the plane z=z_p, and the surface S_l is parallel to the z axis. The problem geometry is shown in Figure 1. In a scalar approximation (neglecting field depolarization at the scatter at the three-dimensional irregularity) the electromagnetic field excited by such source is described by the vertical component of the Hertz vector which in the D region satisfies the inhomogeneous Helmholtz equation and the following boundary conditions:

(1)

where n is the external normal to the boundary surfaces of the waveguide volume, d(M)=d_g for M

S_g, d(M)=d_i for M

S_i, d(M)=d_l for M

S_l, and d(M)=d_p(r, j) for M

S_p. Conditions at the infinity require attenuation of P(r, j, z) at r

. In this case there is no need to attract additionally a boundary condition at the edge (the line of crossing of S_p and S_i, S_p, and S_l ) because the available conditions guarantee unambiguousness of the solution. It should be noted that actually in the VLF range, there are the following estimates of the impedance values for the TM polarization field generated by the considered source: |d_g,i,p|<1 and |d_l|>1. If the vertical component of the Hertz vector is known, the vertical component of the electric field may be calculated by

[14] For determination of the values of the parameters of the inhomogeneous impedance model of the waveguide channel Earth-ionosphere, we attracted known from publications [Orlov et al., 2000] vertical profiles of the concentration N_e(z) and effective collision frequency n_e(z) of electrons. For the given frequency and angle of incidence of the electromagnetic wave at an inhomogeneous layer Y, the equation for the impedance matrix

is integrated numerically from a height of ~100 km downward to ~40 km, covering with a guaranty the region important for formation of the reflected from the ionosphere VLF field [Galiuk et al., 1989; Kirillov, 1979, 1981]. We take the determination of the impedance in the form E_tg= Z₀d_i[ H_tg times

n], where Z₀ =(m₀/e₀)^1/2, and E_tg and H_tg are the tangential to the considered surface components of the electric and magnetic fields, respectively. The values d_i= d_d of the impedance matrix components obtained at the lower boundary of integration z=z_d are recalculated to the height z=h in the vacuum. The formulae for such recalculation we obtain using the following matrix of the reflection index of the plane wave

on the boundary of the plasma semispace in vacuum on the vertical coordinate

Substituting the relation of the reflection index and surface impedance matrix at the plasma semispace-vacuum boundary

where

into the expressions for the components of the impedance matrix, we obtain formulae for recalculation of the components of the impedance matrix to the height z=h in vacuum

where

[15] The height z=h corresponds to the altitude where the combination of the impedance matrix d= d^(e) - d₁₂ d₂₁/ d^(m) in a minimum way depends on the angle Y. For realization of such algorithm together with the equation for the matrix d_i, the equation for its derivative partial d_i/ partial Y, is integrated. Also the formulae of recalculation of the derivative in vacuum were derived and used. The value of the Earth's surface impedance d_g is determined by the given values of the relative dielectric permeability and conductivity for the two-layer model of the underlying surface.

[16] In this paper we are not interested in what degree the initial TM field at the scatter at the irregularity will be reexcited into the field of the TE polarization. We will assume this effect to be negligible. So the anisotropy related to the presence of the geomagnetic field will be approximately taken into account. It is known that in a general case the TM and TE components of the electric field are described by the electric P^(e) and magnetic P^(m) Hertz vectors and are related. For the field in the vacuum cavity this relation is expressed by the following boundary conditions at the ionospheric wall of the waveguide:

So the impedance we use in the scalar problem is taken in the form