Efficiency of the Earth--ionosphere waveguide excitation by...

3. Relations of Field Components Excited in the Ionosphere by Dipoles Located in the Waveguide

[8] The solutions of waveguide problems for the anisotropic, irregular along the q coordinate waveguide can be carried out approximating the irregular part by a finite number of regular parts [Rybachek et al., 1997a]. The waveguide problem for the regular part [Rybachek, 1995] is reduced to a solution of the Maxwell's equations in the waveguide cavity for electric dipoles

(3)

or for magnetic dipoles

(4)

In (3) and (4), k is the wave number of the free space and p_e and p_m are the volume densities of the dipole moments of the electric and magnetic dipoles, respectively. The boundary conditions at both boundaries of the waveguide have the following form:

(5)

Besides, the conditions of finiteness of the fields E and H under q=0, p should be fulfilled. In expressions (5), a is a matrix of the ionospheric admittance. The matrix d have the following form:

(6)

where d is the relative surface impedance of the ground which is taken the same for both polarizations.

[9] According to the principle of polarization duality, the general solution of the Maxwell's equations is given by a superposition of two fundamental solutions. The transverse magnetic and transverse electric fields are described by the electric P_e=P_e e_r and magnetic P_m=P_m e_r Hertz vectors, respectively, directed along the separation coordinate r. In the waveguide cavity outside the sources the electric and magnetic field vectors can be derived as

(7)

The solution of the problem is obtained by the normal waves method. The dependence of the potentials on the coordinates r and q with allowance for asymptotic presentations of the Legendre functions P_n (- cosq) applicable at |n|q

1, |n|(p-q)

1 [Erdelyi, 1953] for each normal wave is described by the following functions [Makarov et al., 1994; Rybachek et al., 1997a]:

(8)

Here n is the eigenvalue and R_n^(e)(kr) and R_n^(m)(kr) are the eigenfunctions of the radial operator of the problem satisfying the differential equations

and the boundary conditions at the surface of the ground at r = a following from (5) and (6)

(9)

where the prime denotes a derivative with respect to the argument and the impedance d is taken independent on the spectral parameter. The boundary conditions at the upper ionospheric boundary determined by the elements of the admittance a(n) make it possible to obtain the characteristic equation for the eigenvalues.

[10] According to (7), one can present the tangential components of the fields in the waveguide cavity outside the sources in the single-mode approximation, omitting the e and m indices at field components, in the following way:

(10)

The radial components can be expressed via potentials using (7) and the differential equations determining the transverse (radial) and longitudinal operators of the problem:

(11)

Using expressions (8), (10), and (11) and taking |n|

1, we find the ratios of the field components in the waveguide cavity

(12)

If the point of observation is located on the surface of the ground, then, because of the boundary conditions (9), the ratios (12) take the following form:

(13)

Ratios (12) and (13) are valid for any height of the emitter located in the waveguide cavity.

[11] The relations between the components of the fields excited in the ionosphere in point 2 with coordinates (r,q) by various emitters located in the waveguide cavity in point 1 with coordinates (b,0) (b leq d) can be obtained using expressions (1):

for the electric dipoles (oriented along the unit vector e_q and radial):

(14)

for the magnetic dipoles (oriented along the unit vector e_q and radial):

(15)

for the horizontal dipoles (electric oriented along the unit vector e_q and magnetic oriented along the unit vector e_j ):

(16)

for the horizontal dipoles (magnetic oriented along the unit vector e_q and electric oriented along the unit vector e_j ):

(17)

Here E_z^qx(1,2, H₀) and H_z^qx(1,2, H₀) are the z components of the electric and magnetic fields, respectively, excited in the ionosphere at point 2 under the given geomagnetic field H₀ by electric (q=e) or magnetic (q=m) dipoles oriented along the unit vector e_x and located in the waveguide cavity at point 1; E_x^iqz(2,1,- H₀) and H_x^iqz(2,1,- H₀) are the x components of the fields excited at point 1 under the geomagnetic field - H₀ by the electric or magnetic dipoles directed along e_z and located in the ionosphere at point 2 ( x and z take the values r, q, and j ). Using expressions (12), we rewrite the ratios of the ionospheric field components (14)-(17) in the following form:

(18)

where B^qx ( q=e, m; x=r, q, j) are six-element column matrices of electric and magnetic field components.

[12] If the emitter is located on the surface of the ground (b=a), it follows from expressions (9) and (18) that

(19)

It is worth noting that in the case of a waveguide irregular on q, the value d in expressions (19) should be considered as the relative surface impedance of the ground in the regular part of the waveguide where the transmitter is located.

[13] Now we discuss the relationship between the components of the fields excited in the ionosphere by an emitter of any type located in the waveguide cavity. Methods and algorithms of field calculation in the waveguide cavity and in the ionosphere were developed in detail [Rybachek, 1995; Rybachek et al., 1997a, 1997b], so we do not discuss these methods. As it has been already mentioned, determination of the fields in the ionosphere is reduced to an integration of the system of differential equations for the tangential components of the fields coinciding by the form with the equations describing propagation of plane waves in plane-stratified media. The difference is that the ionospheric properties and the complex angle of wave incidence on the ionosphere a(n, r) depend on the radial coordinate. The a(n, r) angle is determined by the relation sin a (n, r)=n/(kr). The radial components of the fields are found from the Maxwell's equations for the ionospheric plasma. The asymptotic separation of variables in these equations leads to the field dependence on the coordinate q of the type (8) [see, e.g., Makarov et al., 1994], so the radial field components of each of normal waves are described by the following expressions:

(20)

(21)

Here e_xz ( x, z=r, q, j ) are the elements of the relative dielectric permittivity tensor of the ionosphere and r is the radial coordinate of the observation point.

[14] At low altitudes where the ionospheric properties are close to the properties of the free space, it follows from (20) that the components of the fields E_r and H_j are related by a simple formula krE_r simeq -n H_j. In the general case, the simple relation determined by formula (21) exists only between the components H_r and E_j. Between the other components, there exists a complicated relation due, in particular, to a sphericity of the waveguide, inhomogeneity of the ionosphere, and its anisotropy. However, one can expect that at large distances from any emitter located near the ground surface, the fields in the ionosphere would be close to the fields of plane waves propagating in a magnetoactive plane-stratified medium. In this case the relation between the field components are considerably simplified.

[15] Actually, in the region of the ionosphere where the quasi-longitudinal approximation

(22)

is valid the relations between the field components of a plane wave have the following form [Budden, 1961]:

(23)

(24)

(25)

The designations accepted in the magneto-ionic theory are used in formulae (22)-(25):

(26)

In (25) and (26): e, m_e are the charge and the mass of an electron, respectively, N is the electron concentration, n_e is the effective collision frequency of electron with neutral particles and ions, n is the refractive index of the ionosphere, and Y_L and Y_T are the longitudinal and transverse components of the Y vector. In the case of a regular waveguide N and n_e depend only on the radial coordinate. For an irregular waveguide these parameters depend in addition on the angular coordinate q. Expressions (23)-(25) are written in the spherical coordinates with the aim to use them for the problems being studied.

[16] Relations (24) and (25) complicated enough because of the complex parameter U, are simplified if besides (22) the following inequalities are fulfilled:

(27)

In this case we have

(28)

(29)

It is worth noting that as it follows from formulae (26) and (29) for n², the values of the refractive indices may be either real or imaginary. Because of this, out of two propagating upward characteristic waves only one wave with the real n is of a significantly propagating character.

[17] It follows from expression (28) with an allowance for (26) that

(30)

where H_{0 L} and H_{0 T} are the longitudinal (radial) and transverse components of the geomagnetic field, respectively.

[18] Taking into account the relation which follows from (21) and (23)

and expression (29), we find

(31)

Here |n| is computed from (26) and (29) as

(32)

where c is the free space velocity of electromagnetic waves.