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The reflection of the electromagnetic waves from the ionosphere with the impedance modified by internal gravity waves
Advances in Space Research 40 (2007) 404–408 www.elsevier.com/locate/asr
The reflection of the electromagnetic waves from the ionosphere with the impedance modified by internal gravity waves L.P. Kogan *, G.I. Grigor’ev, T.M. Zaboronkova Radiophysical Research Institute, 25 B. Pecherskaya St., Nizhny Novgorod 603950, Russia Received 28 October 2006; received in revised form 18 April 2007; accepted 25 April 2007
Abstract We have studied the influence of a smoothly inhomogeneous disturbance of the impedance of the ionosphere caused by internal gravity waves present in the reflection zone of propagating VLF electromagnetic waves. The case has been considered when the amplitude of impedance disturbance is commensurable with its regular value. The problem is solved with an auxiliary function satisfying the Dirichlet zero condition at the boundary. The Hertz potential formula has been obtained for the grazing propagation of electromagnetic waves. Ó 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Internal gravity waves; Impedance; Very low frequency electromagnetic waves; Hertz potential
1. Introduction It is known that IGW have a pronounced effect on the propagation of electromagnetic waves in a wide range of frequencies (Grigor’ev, 1999) resulting in Doppler frequency shift, arrival angle variations, etc. The influence of IGW on the radio-wave absorption has been estimated in Grigor’ev (2006). Here we investigate the reflection of VLF waves from the boundary disturbed by the action of IGW. The propagation of VLF waves is studied now using different models of the lower ionosphere. In some models the analysis of VLF wave propagation is conducted with an account of height distribution of different medium parameters (densities of charged and neutral particles, collision frequencies, temperature, etc.) (Lundborg and Thide, 1986) The alternative approach uses models where the ionosphere is substituted by an impedance reflecting surface. Such models have been well developed (Bezrodny et al., 1984) and they are used by many authors to analyse VLF wave propagation. For example, based on such a model *
Corresponding author. Tel.: +8312 683740; fax: +8312 369902. E-mail address: [email protected] (L.P. Kogan).
Remenets (2004) gave an interpretation of experimental data confirming the influence of ultra-relativistic particles on VLF wave band electromagnetic fields. In our article, we use the model of the impedance reflecting surface. Using the model with a sharp impedance boundary the authors were the first to estimate the influence of internal gravity waves (IGW) on the reflection of VLF waves from the lower ionosphere. The influence of IGW on the impedance was accounted for by the setting the impedance disturbance with parameters typical for IGW. In this approximation the waves reflected from the lower ionosphere can give information on the IGW existing at the reflection area. This information can be retained over multihop paths if there are no medium disturbances near the reflection points on the ground or sea surfaces. Here we consider the affect of IGW on VLF wave propagation without taking into account the influence of the direct wave. Direct wave formula (without taking into account IGW) is well known, so it is not given in the article. For example, VLF wave direct ray propagation over the path with the absence of IGW-type disturbances was analysed in detail in Al’pert et al. (1953). The following assumptions have been made in the article. The VLF wavelength is assumed to be much less than
0273-1177/$30 Ó 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.084
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
the horizontal scales of inhomogeneities. We assume the amplitude of the impedance disturbance to be less than or of the order of its regular value. The wave grazing angle along the inhomogeneous plane boundary is considered to be small. The field radiated by a vertical electric dipole located over a plane impedance boundary was determined by Maluzhints’s method (Al’pert et al., 1953) which was modified in Zaboronkova et al. (2003) for the case of an inhomogeneous boundary. Using the latter approach we estimate here the effect of IGW on the ionosphere boundary impedance disturbances affecting the propagation of a cylindrical electromagnetic wave. The approach applied in this work allows one first to pass from the problem on the wave reflection from the boundary with the nonuniform impedance given by a complicated expression (Neuman problems) to the solution of an easier problem of the auxiliary function satisfying zero boundary conditions (Dirichlet problem) (Kravtsov and Orlov, 1990). Second, the method applied allows one to avoid difficulties associated with the commensurability of the reflection zone sizes and the horizontal scale of the impedance inhomogeneity. This last fact explains the impossibility of solving the initial problem using well known methods such as the tangent plane or ray optics methods.
2. Problem statement The initial two-dimensional problem for the Hertz potential was solved in the following way. In the beginning the vacuum half-space plane boundary with nonuniform impedance is transformed (in new coordinates) onto the curved impedance boundary of half-space with a nonuniform refraction coefficient using conformal mapping. The conformal mapping is chosen in such a way that the new boundary impedance should be constant. In the new curved coordinates we introduce the auxiliary function (G) taking zero value at the interface (Al’pert et al., 1953). After that using the inverse conform mapping we return back into the initial coordinates. In these coordinates function G satisfies the wave equation with vertically inhomogeneous refraction coefficient but a zero boundary condition (Dirichlet). The expression for the function G can be derived by the geometric optics method. The auxiliary function G is used to calculate the Hertz potential. The introduction of the auxiliary function G satisfying the zero boundary conditions removes one more difficulty in the initial problem setting for the Hertz potential. The difficulty is that at small grazing angles the reflecting area size is comparable with the horizontal scale of the disturbed impedance. This peculiarity is absent in the problem formulation for the auxiliary function G. Let us introduce the two-dimensional Green function (corresponding to Hertz potential P with unit amplitude) which allows one to determine the fields generated by real
405
sources. The wave equation for the Green function can be written in the following form: Dx;z P þ k 20 P ¼ J
ð1Þ
with the boundary condition oP ¼ ik 0 gðxÞP ; oz
ð2Þ
z¼h
as well as the radiation condition. Here Dx,z is the twopffiffiffiffiffiffiffiffiffi dimensional Laplace operator; k 0 ¼ 2p ¼ x e0 l0 ; k J = d(x)d(z), d is the Dirac delta-function, (x, y, z) is the Cartesian coordinate system (Fig. 1). We will omit the time dependence exp(ixt). The source J is located in the vacuum halfspace ffi qffiffiffilower z 6 h with the normalized on le00 inhomogeneity impedance g(x) = g0 + g1(x) simulating the presence of an internal gravity wave at the lower boundary of the Ionosphere. Here Re(g(x)) P 0, g0 = const, Max(|g1|) < |g0|. We consider the horizontal scale lg of the disturbance to be essentially larger than the electromagnetic wave length k. In this case we can neglect the depolarization effect (Al’pert et al., 1953). Components of the electromagnetic field can be expressed through the Hertz potential P. 3. Determination of the Hertz potential To solve the problem formulated we use the conformal map n(u) which transfers the area z 6 h of the complex plane u = x + iz to some one-to-one related area z1 6 f(x1) of the plane n = x1 + iz1 (Fig. 2). This mapping transfers the plane z = h to the curved surface z1 = f(x1). The boundary-value problem (1)–(2) is rewritten then in the form Dx1 ;z1 P þ k 20 n2 ðx1 ; z1 ÞP ¼ n2 ðx1 ; z1 ÞJ ðx1 ; z1 Þ; oP gðx1 ; z1 Þ ¼ ik 0 P : om bðx1 ; z1 Þ z1 ¼f ðx1 Þ
ð3Þ ð4Þ
Here omo is the normal derivative to the line of the disturbed boundary, and the values bðx1 ; z1 Þ ¼ bðxðx1 ; z1 Þ; zðx1 ; z1 ÞÞ ¼ jdn j and nðx1 ; z1 Þ ¼ jdu j are, respecdn du tively, the modules of the derivative of the map n(u) and its inverse transformation u(n) (obviously b ¼ 1n).
Fig. 1. Geometry of the problem in (X, Z) system.
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L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
we come to the following equation in the initial variables (x, z): 2 on 1 on2 oG ¼ g^J : ð8Þ G k 20 2 Dx;z G þ k 20 1 þ c oz c oz oz Further we will be interested in the case of grazing propagation when the horizontal distance x from the source to the observation point is much bigger than the distance along the vertical h from this point to the disturbed boundary. Here the grazing angle of the incident wave is hs 1. In such a case, to within the terms h2s , we have the following relation
Fig. 2. Geometry of the problem in curve system.
Since the function b(x1, z1) is not determined, let us choose it in the form such that the equality is true: oP ¼ cP ; ð5Þ om z1 ¼f ðx1 Þ (here c = ik0g0 = const). Thus, the function bðx1 ; z1 Þjz1 ¼f ðx1 Þ can be written bðx1 ; f ðx1 ; z1 ÞÞ ¼ 1 þ g1 ðx1g;f0 ðx1 ÞÞ. Thus, we have come to the problem of electromagnetic wave propagation in the halfspace with a known inhomogeneous refraction coefficient and one-dimensional disturbed boundary having a permanent impedance. To solve the boundary-value problem (3) and (5) let us act on the right and left parts of the Eq. (3) by the operator g^ ¼
o þ c: os
ð6Þ
Here oso is the tangent derivative to the curve s(x1, z1) (Fig. 2). The set of curves s should meet the following requirements: (i) There comes only one line s through every point of the plane (x1, z1). (ii) For the boundary points z1 = f(x1) the direction of the tangent to the corresponding curve of the set s coincides with the direction of the normal m along which the differentiation has been made in (5) (see Fig. 2 where all the curves s(x1, z1) are plotted by dotted lines). Thus, the equality oso ¼ omo holds true at the points of the curved boundary. For certainty let us choose such a set s all lines of which are mapped to the straight lines x = const by the inverse conformal mapping u(n) transferring the half-space z1 6 f(x1) of the complex area n = x1 + iz1onto the area z 6 h of the plane u = x + iz. In this case the line z1 = f(x1) is transformed into the straight line z = h. We have just applied the conformal map in the variables (x1, z1) and then return back to the initial coordinates (x, z). Now for the function G ¼ g^P ¼ ðnozo þ cÞP satisfying the zero Dirichlet condition at the upper boundary Gjz¼h ¼ 0
ð7Þ
oG ¼ ik 0 sin hs G oz and we can rewrite the Eq. (8) in the form: Dx;z G þ k 20 eðx; zÞG ¼ g^J ;
ð9Þ 1 on2
here eðx; zÞ ¼ 1 þ 2con þ ik 0 sin hs c2 oz . Function e(x, z) is oz interpreted as the dielectric permeability of some effective medium. Such a medium is stipulated by the problem solution procedure. Each type of impedance inhomogeneity corresponds to its own medium. To solve the auxiliary problem (9) and (7) we now may apply the geometrical optics method since k lg. Using the approach developed in Zaboronkova et al. (2003), we assume that in the area z P h there is an effective medium with the dielectric permeability e+ = e(x, z h) symmetric to the permeability e = e(x, z) of the medium for z 6 h. pffiffiffiffiffi Corresponding refractive indices we denote as n ¼ e . We then consider an imaginary source J 0 at the point M00 (0, 2h), symmetric with respect to the plane z = h to the source located at M0(0, 0). For this imaginary source the auxiliary function G 0 can be written as: G 0 = gˆ(z)P (Al’pert et al., 1953). It is evident that at the level z = h the sum of the two fields radiated by sources corresponding J and J 0 [see (9)] is equal to zero, and the boundary condition (7) is fulfilled automatically. Let us denote the auxiliary functions for the boundary z = h incident and reflected waves as G(inc) and G(ref) GðincÞ ¼ A ~ R1 ; 0 exp ik 0
Z
M 1 ðx=2;hÞ
! ðe ðx; zÞÞ0:5 dl ;
M 0 ð0;0Þ ðrefÞ
G
e ~ ¼A ~ R1 ; 0 q ~ R1 ; 0 A R1 ; ~ R "Z Z M 1 ðx=2;hÞ ðe ðx; zÞÞ0:5 dl þ exp ik 0 M 0 ð0;0Þ
#!
Mðx;0Þ 0:5
ðe ðx; zÞÞ dl
:
M 1 ðx=2;hÞ
ð10Þ
Here M1 = M1(x/2,h) is the central point of the area essential for the reflection from the ‘‘weak’’ boundary z = h (Ginzburg, 1970), ~ R1 ¼ ~ R1 ðx=2; hÞ, ~ R¼~ Rðx; 0Þ. The ðincÞ ~ value AðR1 ; 0Þ is the amplitude of the function G1 , the factors q and T are the reflection and transmission coeffie determines the convergence of the cients, the cofactor A ray reflected from the boundary (Kravtsov and Orlov, 1990).
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
By analogy we introduce the auxiliary function G 0 (tr) for the transient wave radiated by the imaginary source J 0 = d(x)d (z 2h) G
0ðtrÞ
R1 ; ~ ¼ A ~ R; ~ R00 T ~ R00 exp ik 0 þ
Z
"Z
M 1 ðx=2;hÞ
M 00 ð0;2hÞ
0:5
ðeþ ðx; zÞÞ dl #!
Mðx;0Þ 0:5
ðe ðx; zÞÞ dl
:
M 1 ðx=2;hÞ
ð11Þ
Here ~ R00 ¼ ~ R00 ð0; 2hÞ is the radius-vector of the point M00 0 determiningthe coordinates of the source J . Thus we can 0 0 e ~ ~ ~ ~ ~ ~ write that A R; R0 ¼ A R1 ; R0 A R; R1 . From the symmetry of the R problem it follows that R M1 M1 0:5 0:5 ðe ðx; zÞÞ dl ¼ ðe ðx; zÞÞ dl. That’s why the 0 þ M0 M0 (ref) 0 (tr) phases of the functions G and G are equaled and A ~ R1 ; ~ R00 ¼ A ~ R0 . R1 ; ~ As a result, we have Z Gðx; zÞ ¼ GðrefÞ þ G0ðtrÞ ¼ A ~ R; ~ R00 exp ik 0 þ
Z
M1
ðe ðx; zÞÞ0:5 dl
M0
M
0:5 ðe ðx; zÞÞ dl :
M1
ð12Þ
Here it was taken into account that 1 + q = T. Hence the our problem is reduced to the calculation of the amplitude and phase of a wave propagating in a smoothly inhomogeneous media with the dielectric permeability e±. In the approximation of the grazing propagation it is not difficult to obtain from the expression G = gˆP the formula for the Hertz potential P: Pðx; zÞ ¼
Gðx; zÞ þ Oðh2s Þ: ðik 0 Þn sin hs þ c
ð13Þ
On the basis of the obtained formulas (12) and (13), we investigate the influence of various types of IGW on the propagation of electromagnetic waves. In particular, we consider the influence of periodic and soliton-type IGW. 4. The analysis of the influence of periodic and soliton-type IGW on the radio wave propagation
lnðbðx; zÞÞ ¼
1 pg0
Z
þ1 1
407
gðx0 Þz dx0 2
ðx0 xÞ þ z2
:
ð15Þ
We have accepted the impedance g(x) variation to be determined by horizontally propagating IGW having two horizontal scales lg and ls. The impedance corresponding to such a disturbance is set in the form ! a2 gðxÞ ¼ g0 1 þ a1 sinðk g x /0 Þ þ : 1 þ ðx x0 Þ2 =l2s Here a1 and a2 are the normalized to g0 amplitudes of periodic and soliton-type disturbances, k g ¼ 2p , lg is the lg spatial period of the given harmonic, and ls is the soliton horizontal scale, x0 is the soliton center; the phase /0 and x0 do not depend on x. The IGW frequency is extremely small in comparison with the electromagnetic wave frequencies, hence we can completely neglect the impedance g time variations in determination of the reflected radio wave amplitude and phase. The numerical calculations of a module of the normalized disturbed Hertz potential jPP0 j (here P0 is the undisturbed Hertz potential) have been performed on the basis of the obtained results [see (12) and (13)]. It should be remembered that the ionosphere boundary model was taken from (Bezrodny et al., 1984). Fig. 3 shows the dependence of jPP0 j on the distance x from the source to the observation point. Here the quantities lg = ls have the values 300 km (curve 1), 400 km (curve 2) and 500 km (curve 3). The regular value of the ionosphere impedance g0 = 0.45, the relative amplitudes a1 = a2 = 0.2, the ionosphere boundary height h = 100 km. The soliton geometric center coordinate x0 = 1200 km. In Fig. 4 lg = 300 km, ls = 300 km (curve 1), lg = 300 km, ls = 400 km (curve 2), lg = 300 km, ls = 500 km (curve 3). The analysis of the calculations performed leads us to the following conclusions. The total effect of both types of IGW on the field at the observation point does not equal to the sum of individual effects of the periodic and
We have obtained the Hertz potential P in the case of the electromagnetic wave reflection from a horizontally inhomogeneous ionosphere. We have to note here that for the conformal map n(u) derivative module b(x, z) we can formulate the following boundary-value problem Dx;z lnðbðx; zÞÞ ¼ 0; lnðbðx; 0ÞÞ ¼
gðxÞ : g0
ð14Þ
In this case it has been taken into account that the function ln(b(x, z)) is a harmonic one and its Laplace operator is equal to zero. The solution of this boundary problem is given by the Poisson integral:
Fig. 3. Hertz potential normalized to |P0| as a function of the distance x at lg = ls = 300 km (curve 1), lg = ls = 400 km (curve 2), lg = ls = 500 km (curve 3).
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L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
IGW affect on the VLF waves is quite possible for a direct wave propagating over the earth surface. Such a problem has not been solved yet and it requires a separate treatment. It should be noted that our calculations are applicable in the absence of intensive noise and artificial disturbances which may make difficult the registration of an effect of the IGW on a reflection of VLF waves. The research was performed with the financial support of the RFFI (Grants N 04-02-16344, 04-05-64140). References Fig. 4. Hertz potential normalized to |P0| as a function of distance x at lg = ls = 300 km (curve 1), lg = 300 km, ls = 400 km (curve 2), lg = 300 km, ls = 500 km (curve 3).
soliton-type IGW taken separately. The applied method of calculation may be generalized for the case of waveguide propagation of radio waves. It should be stressed that our calculation allows us to estimate the influence of stronger disturbance when the disturbance amplitude is commensurable with the impedance regular value. 5. Conclusion We estimated the influence of the IGW on the reflection of VLF waves from the lower boundary of the ionosphere. By analogy with the calculations presented in the article for the sharp ionospheric boundary with the given impedance (accounting IGW effect) one can perform calculations taking into account the presence of the earth surface as well. In the presence of gravity waves of orographic origin or generated by other sources (in particular, by lightning discharges in the vacuum gap between the Earth and the ionosphere) (Grigor’ev, 1999) one can assume that the
Al’pert, Yu.A., Ginzburg, V.L., Feinberg, E.L. Propagation of Electromagnetic Waves over the Earth surface. AN SSSR Press, Moscow, 1953. Bezrodny, V.G., Bliokh, P.V., Shubova, R.S., Yampolsky, Yu.M. Fluctuations on Very Long Radiowaves in Earth–Ionosphere Waveguide. Nauka Press, Moscow, 1984. Ginzburg, V.L. Propagation of Electromagnetic Waves in Plasmas. Pergamon Press, Oxford, 1970. Grigor’ev, G.I. Acoustic gravity waves in the earth’s atmosphere (review). Radiophys. Quantum Electron. 42, 1–21, 1999. Grigor’ev, G.I. Variations in the radiowave absorption in the ionosphere due to internal gravity waves. Radiophys. Quantum Electron. 49, 194– 198, 2006. Kravtsov, Yu.L., Orlov, Yu.I. Geometric Optics of Inhomogeneous Media. Springer-Verlang, Berlin, 1990. Lundborg, B., Thide, B. Standing wave pattern of HF waves in the ionospheric reflection region, II. Applications. Radio Sci. 21, 486–500, 1986. Remenets, G.F. Originality of sLW wave-method of electric features researching of average atmosphere with scattering of ultra-relativistic electrons. Part. 1. In: The proceeding of the YII International Baikal Youth Scientific School on the Fundamental Physics, the Siberian branch of RAS, Irkutsk, pp. 55–61, 2004. Zaboronkova, T.M., Kogan, L.P., Tamoikin, V.V. The field of an electric dipole in a half-space with a smoothly nonuniform perturbation of the boundary impedance. J. Commun. Technol. Electron. 48, 645–654, 2003.
The reflection of the electromagnetic waves from the ionosphere with the impedance modified by internal gravity waves L.P. Kogan *, G.I. Grigor’ev, T.M. Zaboronkova Radiophysical Research Institute, 25 B. Pecherskaya St., Nizhny Novgorod 603950, Russia Received 28 October 2006; received in revised form 18 April 2007; accepted 25 April 2007
Abstract We have studied the influence of a smoothly inhomogeneous disturbance of the impedance of the ionosphere caused by internal gravity waves present in the reflection zone of propagating VLF electromagnetic waves. The case has been considered when the amplitude of impedance disturbance is commensurable with its regular value. The problem is solved with an auxiliary function satisfying the Dirichlet zero condition at the boundary. The Hertz potential formula has been obtained for the grazing propagation of electromagnetic waves. Ó 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Internal gravity waves; Impedance; Very low frequency electromagnetic waves; Hertz potential
1. Introduction It is known that IGW have a pronounced effect on the propagation of electromagnetic waves in a wide range of frequencies (Grigor’ev, 1999) resulting in Doppler frequency shift, arrival angle variations, etc. The influence of IGW on the radio-wave absorption has been estimated in Grigor’ev (2006). Here we investigate the reflection of VLF waves from the boundary disturbed by the action of IGW. The propagation of VLF waves is studied now using different models of the lower ionosphere. In some models the analysis of VLF wave propagation is conducted with an account of height distribution of different medium parameters (densities of charged and neutral particles, collision frequencies, temperature, etc.) (Lundborg and Thide, 1986) The alternative approach uses models where the ionosphere is substituted by an impedance reflecting surface. Such models have been well developed (Bezrodny et al., 1984) and they are used by many authors to analyse VLF wave propagation. For example, based on such a model *
Corresponding author. Tel.: +8312 683740; fax: +8312 369902. E-mail address: [email protected] (L.P. Kogan).
Remenets (2004) gave an interpretation of experimental data confirming the influence of ultra-relativistic particles on VLF wave band electromagnetic fields. In our article, we use the model of the impedance reflecting surface. Using the model with a sharp impedance boundary the authors were the first to estimate the influence of internal gravity waves (IGW) on the reflection of VLF waves from the lower ionosphere. The influence of IGW on the impedance was accounted for by the setting the impedance disturbance with parameters typical for IGW. In this approximation the waves reflected from the lower ionosphere can give information on the IGW existing at the reflection area. This information can be retained over multihop paths if there are no medium disturbances near the reflection points on the ground or sea surfaces. Here we consider the affect of IGW on VLF wave propagation without taking into account the influence of the direct wave. Direct wave formula (without taking into account IGW) is well known, so it is not given in the article. For example, VLF wave direct ray propagation over the path with the absence of IGW-type disturbances was analysed in detail in Al’pert et al. (1953). The following assumptions have been made in the article. The VLF wavelength is assumed to be much less than
0273-1177/$30 Ó 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.084
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
the horizontal scales of inhomogeneities. We assume the amplitude of the impedance disturbance to be less than or of the order of its regular value. The wave grazing angle along the inhomogeneous plane boundary is considered to be small. The field radiated by a vertical electric dipole located over a plane impedance boundary was determined by Maluzhints’s method (Al’pert et al., 1953) which was modified in Zaboronkova et al. (2003) for the case of an inhomogeneous boundary. Using the latter approach we estimate here the effect of IGW on the ionosphere boundary impedance disturbances affecting the propagation of a cylindrical electromagnetic wave. The approach applied in this work allows one first to pass from the problem on the wave reflection from the boundary with the nonuniform impedance given by a complicated expression (Neuman problems) to the solution of an easier problem of the auxiliary function satisfying zero boundary conditions (Dirichlet problem) (Kravtsov and Orlov, 1990). Second, the method applied allows one to avoid difficulties associated with the commensurability of the reflection zone sizes and the horizontal scale of the impedance inhomogeneity. This last fact explains the impossibility of solving the initial problem using well known methods such as the tangent plane or ray optics methods.
2. Problem statement The initial two-dimensional problem for the Hertz potential was solved in the following way. In the beginning the vacuum half-space plane boundary with nonuniform impedance is transformed (in new coordinates) onto the curved impedance boundary of half-space with a nonuniform refraction coefficient using conformal mapping. The conformal mapping is chosen in such a way that the new boundary impedance should be constant. In the new curved coordinates we introduce the auxiliary function (G) taking zero value at the interface (Al’pert et al., 1953). After that using the inverse conform mapping we return back into the initial coordinates. In these coordinates function G satisfies the wave equation with vertically inhomogeneous refraction coefficient but a zero boundary condition (Dirichlet). The expression for the function G can be derived by the geometric optics method. The auxiliary function G is used to calculate the Hertz potential. The introduction of the auxiliary function G satisfying the zero boundary conditions removes one more difficulty in the initial problem setting for the Hertz potential. The difficulty is that at small grazing angles the reflecting area size is comparable with the horizontal scale of the disturbed impedance. This peculiarity is absent in the problem formulation for the auxiliary function G. Let us introduce the two-dimensional Green function (corresponding to Hertz potential P with unit amplitude) which allows one to determine the fields generated by real
405
sources. The wave equation for the Green function can be written in the following form: Dx;z P þ k 20 P ¼ J
ð1Þ
with the boundary condition oP ¼ ik 0 gðxÞP ; oz
ð2Þ
z¼h
as well as the radiation condition. Here Dx,z is the twopffiffiffiffiffiffiffiffiffi dimensional Laplace operator; k 0 ¼ 2p ¼ x e0 l0 ; k J = d(x)d(z), d is the Dirac delta-function, (x, y, z) is the Cartesian coordinate system (Fig. 1). We will omit the time dependence exp(ixt). The source J is located in the vacuum halfspace ffi qffiffiffilower z 6 h with the normalized on le00 inhomogeneity impedance g(x) = g0 + g1(x) simulating the presence of an internal gravity wave at the lower boundary of the Ionosphere. Here Re(g(x)) P 0, g0 = const, Max(|g1|) < |g0|. We consider the horizontal scale lg of the disturbance to be essentially larger than the electromagnetic wave length k. In this case we can neglect the depolarization effect (Al’pert et al., 1953). Components of the electromagnetic field can be expressed through the Hertz potential P. 3. Determination of the Hertz potential To solve the problem formulated we use the conformal map n(u) which transfers the area z 6 h of the complex plane u = x + iz to some one-to-one related area z1 6 f(x1) of the plane n = x1 + iz1 (Fig. 2). This mapping transfers the plane z = h to the curved surface z1 = f(x1). The boundary-value problem (1)–(2) is rewritten then in the form Dx1 ;z1 P þ k 20 n2 ðx1 ; z1 ÞP ¼ n2 ðx1 ; z1 ÞJ ðx1 ; z1 Þ; oP gðx1 ; z1 Þ ¼ ik 0 P : om bðx1 ; z1 Þ z1 ¼f ðx1 Þ
ð3Þ ð4Þ
Here omo is the normal derivative to the line of the disturbed boundary, and the values bðx1 ; z1 Þ ¼ bðxðx1 ; z1 Þ; zðx1 ; z1 ÞÞ ¼ jdn j and nðx1 ; z1 Þ ¼ jdu j are, respecdn du tively, the modules of the derivative of the map n(u) and its inverse transformation u(n) (obviously b ¼ 1n).
Fig. 1. Geometry of the problem in (X, Z) system.
406
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
we come to the following equation in the initial variables (x, z): 2 on 1 on2 oG ¼ g^J : ð8Þ G k 20 2 Dx;z G þ k 20 1 þ c oz c oz oz Further we will be interested in the case of grazing propagation when the horizontal distance x from the source to the observation point is much bigger than the distance along the vertical h from this point to the disturbed boundary. Here the grazing angle of the incident wave is hs 1. In such a case, to within the terms h2s , we have the following relation
Fig. 2. Geometry of the problem in curve system.
Since the function b(x1, z1) is not determined, let us choose it in the form such that the equality is true: oP ¼ cP ; ð5Þ om z1 ¼f ðx1 Þ (here c = ik0g0 = const). Thus, the function bðx1 ; z1 Þjz1 ¼f ðx1 Þ can be written bðx1 ; f ðx1 ; z1 ÞÞ ¼ 1 þ g1 ðx1g;f0 ðx1 ÞÞ. Thus, we have come to the problem of electromagnetic wave propagation in the halfspace with a known inhomogeneous refraction coefficient and one-dimensional disturbed boundary having a permanent impedance. To solve the boundary-value problem (3) and (5) let us act on the right and left parts of the Eq. (3) by the operator g^ ¼
o þ c: os
ð6Þ
Here oso is the tangent derivative to the curve s(x1, z1) (Fig. 2). The set of curves s should meet the following requirements: (i) There comes only one line s through every point of the plane (x1, z1). (ii) For the boundary points z1 = f(x1) the direction of the tangent to the corresponding curve of the set s coincides with the direction of the normal m along which the differentiation has been made in (5) (see Fig. 2 where all the curves s(x1, z1) are plotted by dotted lines). Thus, the equality oso ¼ omo holds true at the points of the curved boundary. For certainty let us choose such a set s all lines of which are mapped to the straight lines x = const by the inverse conformal mapping u(n) transferring the half-space z1 6 f(x1) of the complex area n = x1 + iz1onto the area z 6 h of the plane u = x + iz. In this case the line z1 = f(x1) is transformed into the straight line z = h. We have just applied the conformal map in the variables (x1, z1) and then return back to the initial coordinates (x, z). Now for the function G ¼ g^P ¼ ðnozo þ cÞP satisfying the zero Dirichlet condition at the upper boundary Gjz¼h ¼ 0
ð7Þ
oG ¼ ik 0 sin hs G oz and we can rewrite the Eq. (8) in the form: Dx;z G þ k 20 eðx; zÞG ¼ g^J ;
ð9Þ 1 on2
here eðx; zÞ ¼ 1 þ 2con þ ik 0 sin hs c2 oz . Function e(x, z) is oz interpreted as the dielectric permeability of some effective medium. Such a medium is stipulated by the problem solution procedure. Each type of impedance inhomogeneity corresponds to its own medium. To solve the auxiliary problem (9) and (7) we now may apply the geometrical optics method since k lg. Using the approach developed in Zaboronkova et al. (2003), we assume that in the area z P h there is an effective medium with the dielectric permeability e+ = e(x, z h) symmetric to the permeability e = e(x, z) of the medium for z 6 h. pffiffiffiffiffi Corresponding refractive indices we denote as n ¼ e . We then consider an imaginary source J 0 at the point M00 (0, 2h), symmetric with respect to the plane z = h to the source located at M0(0, 0). For this imaginary source the auxiliary function G 0 can be written as: G 0 = gˆ(z)P (Al’pert et al., 1953). It is evident that at the level z = h the sum of the two fields radiated by sources corresponding J and J 0 [see (9)] is equal to zero, and the boundary condition (7) is fulfilled automatically. Let us denote the auxiliary functions for the boundary z = h incident and reflected waves as G(inc) and G(ref) GðincÞ ¼ A ~ R1 ; 0 exp ik 0
Z
M 1 ðx=2;hÞ
! ðe ðx; zÞÞ0:5 dl ;
M 0 ð0;0Þ ðrefÞ
G
e ~ ¼A ~ R1 ; 0 q ~ R1 ; 0 A R1 ; ~ R "Z Z M 1 ðx=2;hÞ ðe ðx; zÞÞ0:5 dl þ exp ik 0 M 0 ð0;0Þ
#!
Mðx;0Þ 0:5
ðe ðx; zÞÞ dl
:
M 1 ðx=2;hÞ
ð10Þ
Here M1 = M1(x/2,h) is the central point of the area essential for the reflection from the ‘‘weak’’ boundary z = h (Ginzburg, 1970), ~ R1 ¼ ~ R1 ðx=2; hÞ, ~ R¼~ Rðx; 0Þ. The ðincÞ ~ value AðR1 ; 0Þ is the amplitude of the function G1 , the factors q and T are the reflection and transmission coeffie determines the convergence of the cients, the cofactor A ray reflected from the boundary (Kravtsov and Orlov, 1990).
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
By analogy we introduce the auxiliary function G 0 (tr) for the transient wave radiated by the imaginary source J 0 = d(x)d (z 2h) G
0ðtrÞ
R1 ; ~ ¼ A ~ R; ~ R00 T ~ R00 exp ik 0 þ
Z
"Z
M 1 ðx=2;hÞ
M 00 ð0;2hÞ
0:5
ðeþ ðx; zÞÞ dl #!
Mðx;0Þ 0:5
ðe ðx; zÞÞ dl
:
M 1 ðx=2;hÞ
ð11Þ
Here ~ R00 ¼ ~ R00 ð0; 2hÞ is the radius-vector of the point M00 0 determiningthe coordinates of the source J . Thus we can 0 0 e ~ ~ ~ ~ ~ ~ write that A R; R0 ¼ A R1 ; R0 A R; R1 . From the symmetry of the R problem it follows that R M1 M1 0:5 0:5 ðe ðx; zÞÞ dl ¼ ðe ðx; zÞÞ dl. That’s why the 0 þ M0 M0 (ref) 0 (tr) phases of the functions G and G are equaled and A ~ R1 ; ~ R00 ¼ A ~ R0 . R1 ; ~ As a result, we have Z Gðx; zÞ ¼ GðrefÞ þ G0ðtrÞ ¼ A ~ R; ~ R00 exp ik 0 þ
Z
M1
ðe ðx; zÞÞ0:5 dl
M0
M
0:5 ðe ðx; zÞÞ dl :
M1
ð12Þ
Here it was taken into account that 1 + q = T. Hence the our problem is reduced to the calculation of the amplitude and phase of a wave propagating in a smoothly inhomogeneous media with the dielectric permeability e±. In the approximation of the grazing propagation it is not difficult to obtain from the expression G = gˆP the formula for the Hertz potential P: Pðx; zÞ ¼
Gðx; zÞ þ Oðh2s Þ: ðik 0 Þn sin hs þ c
ð13Þ
On the basis of the obtained formulas (12) and (13), we investigate the influence of various types of IGW on the propagation of electromagnetic waves. In particular, we consider the influence of periodic and soliton-type IGW. 4. The analysis of the influence of periodic and soliton-type IGW on the radio wave propagation
lnðbðx; zÞÞ ¼
1 pg0
Z
þ1 1
407
gðx0 Þz dx0 2
ðx0 xÞ þ z2
:
ð15Þ
We have accepted the impedance g(x) variation to be determined by horizontally propagating IGW having two horizontal scales lg and ls. The impedance corresponding to such a disturbance is set in the form ! a2 gðxÞ ¼ g0 1 þ a1 sinðk g x /0 Þ þ : 1 þ ðx x0 Þ2 =l2s Here a1 and a2 are the normalized to g0 amplitudes of periodic and soliton-type disturbances, k g ¼ 2p , lg is the lg spatial period of the given harmonic, and ls is the soliton horizontal scale, x0 is the soliton center; the phase /0 and x0 do not depend on x. The IGW frequency is extremely small in comparison with the electromagnetic wave frequencies, hence we can completely neglect the impedance g time variations in determination of the reflected radio wave amplitude and phase. The numerical calculations of a module of the normalized disturbed Hertz potential jPP0 j (here P0 is the undisturbed Hertz potential) have been performed on the basis of the obtained results [see (12) and (13)]. It should be remembered that the ionosphere boundary model was taken from (Bezrodny et al., 1984). Fig. 3 shows the dependence of jPP0 j on the distance x from the source to the observation point. Here the quantities lg = ls have the values 300 km (curve 1), 400 km (curve 2) and 500 km (curve 3). The regular value of the ionosphere impedance g0 = 0.45, the relative amplitudes a1 = a2 = 0.2, the ionosphere boundary height h = 100 km. The soliton geometric center coordinate x0 = 1200 km. In Fig. 4 lg = 300 km, ls = 300 km (curve 1), lg = 300 km, ls = 400 km (curve 2), lg = 300 km, ls = 500 km (curve 3). The analysis of the calculations performed leads us to the following conclusions. The total effect of both types of IGW on the field at the observation point does not equal to the sum of individual effects of the periodic and
We have obtained the Hertz potential P in the case of the electromagnetic wave reflection from a horizontally inhomogeneous ionosphere. We have to note here that for the conformal map n(u) derivative module b(x, z) we can formulate the following boundary-value problem Dx;z lnðbðx; zÞÞ ¼ 0; lnðbðx; 0ÞÞ ¼
gðxÞ : g0
ð14Þ
In this case it has been taken into account that the function ln(b(x, z)) is a harmonic one and its Laplace operator is equal to zero. The solution of this boundary problem is given by the Poisson integral:
Fig. 3. Hertz potential normalized to |P0| as a function of the distance x at lg = ls = 300 km (curve 1), lg = ls = 400 km (curve 2), lg = ls = 500 km (curve 3).
408
L.P. Kogan et al. / Advances in Space Research 40 (2007) 404–408
IGW affect on the VLF waves is quite possible for a direct wave propagating over the earth surface. Such a problem has not been solved yet and it requires a separate treatment. It should be noted that our calculations are applicable in the absence of intensive noise and artificial disturbances which may make difficult the registration of an effect of the IGW on a reflection of VLF waves. The research was performed with the financial support of the RFFI (Grants N 04-02-16344, 04-05-64140). References Fig. 4. Hertz potential normalized to |P0| as a function of distance x at lg = ls = 300 km (curve 1), lg = 300 km, ls = 400 km (curve 2), lg = 300 km, ls = 500 km (curve 3).
soliton-type IGW taken separately. The applied method of calculation may be generalized for the case of waveguide propagation of radio waves. It should be stressed that our calculation allows us to estimate the influence of stronger disturbance when the disturbance amplitude is commensurable with the impedance regular value. 5. Conclusion We estimated the influence of the IGW on the reflection of VLF waves from the lower boundary of the ionosphere. By analogy with the calculations presented in the article for the sharp ionospheric boundary with the given impedance (accounting IGW effect) one can perform calculations taking into account the presence of the earth surface as well. In the presence of gravity waves of orographic origin or generated by other sources (in particular, by lightning discharges in the vacuum gap between the Earth and the ionosphere) (Grigor’ev, 1999) one can assume that the
Al’pert, Yu.A., Ginzburg, V.L., Feinberg, E.L. Propagation of Electromagnetic Waves over the Earth surface. AN SSSR Press, Moscow, 1953. Bezrodny, V.G., Bliokh, P.V., Shubova, R.S., Yampolsky, Yu.M. Fluctuations on Very Long Radiowaves in Earth–Ionosphere Waveguide. Nauka Press, Moscow, 1984. Ginzburg, V.L. Propagation of Electromagnetic Waves in Plasmas. Pergamon Press, Oxford, 1970. Grigor’ev, G.I. Acoustic gravity waves in the earth’s atmosphere (review). Radiophys. Quantum Electron. 42, 1–21, 1999. Grigor’ev, G.I. Variations in the radiowave absorption in the ionosphere due to internal gravity waves. Radiophys. Quantum Electron. 49, 194– 198, 2006. Kravtsov, Yu.L., Orlov, Yu.I. Geometric Optics of Inhomogeneous Media. Springer-Verlang, Berlin, 1990. Lundborg, B., Thide, B. Standing wave pattern of HF waves in the ionospheric reflection region, II. Applications. Radio Sci. 21, 486–500, 1986. Remenets, G.F. Originality of sLW wave-method of electric features researching of average atmosphere with scattering of ultra-relativistic electrons. Part. 1. In: The proceeding of the YII International Baikal Youth Scientific School on the Fundamental Physics, the Siberian branch of RAS, Irkutsk, pp. 55–61, 2004. Zaboronkova, T.M., Kogan, L.P., Tamoikin, V.V. The field of an electric dipole in a half-space with a smoothly nonuniform perturbation of the boundary impedance. J. Commun. Technol. Electron. 48, 645–654, 2003.