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Oblique sounding of the ionosphere by powerful wave beams
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Advances in Space Research 47 (2011) 1181–1186 www.elsevier.com/locate/asr
Oblique sounding of the ionosphere by powerful wave beams I.A. Molotkov a, B. Atamaniuk b,⇑ a
Inst. of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, RAS, Troitsk, Moscow Region 142190, Russia b Space Research Center, PAS, 18A Bartycka, 00-716 Warsaw, Poland Received 31 July 2010; received in revised form 25 November 2010; accepted 26 November 2010 Available online 5 December 2010
Abstract The article is devoted to modeling the impact on the ionosphere powerful obliquely incident wave beam. The basis of this analysis will be orbital variational principle for the intense wave beams-generalization of Fermat’s principle to the case of a nonlinear medium (Molotkov and Vakulenko, 1988a,b; Molotkov, 2003, 2005). Under the influence of a powerful wave beam appears manageable the additional stratification of the ionospheric layer F2. Explicit expressions show how the properties of the test beam, with a shifted frequency, released in the same direction as the beam depend on the intensity of a powerful beam and the frequency shift. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Oblique wave beam; Ionosphere; Trajectory variational principle
1. Introduction The aim is to model the influence on the ionosphere of the powerful oblique wave beam with the working frequencies higher than the critical frequency of the F2 layer. We wish to study the variation of the parameters of both ionosphere layer itself and of the weak test beam which serves for the diagnostics and which has the same direction as the powerful beam. The investigation of the behavior of the beam in such conditions was performed in Bochkarev et al. (1982) in the framework of geometrical optics. The perturbation of the electron temperature in such a process was calculated in Field et al. (1990). Short summary of this theory can be found in Molotkov and Cherkashin (1994). The problem is very difficult and therefore below we shall limit ourselves to the use of a rather simple model which does not take into account all the details of waves propagation in ionosphere.
⇑ Corresponding author.
E-mail addresses: [email protected] (I.A. Molotkov), batama@ cbk.waw.pl (B. Atamaniuk).
Let us assume that the propagation of the wave beams in nonlinear smoothly inhomogeneous medium is described by the cubic-nonlinear Helmholtz equation Du þ x2 að! r Þju2 j þ bð! r Þ u ¼ 0; x 1; a > 0; ! r ¼ ðx; yÞ ð1:1Þ Here D- is the two-dimensional Laplacian, að! r Þ-strictly positive. Square brackets in (1.1) is the square of the refractive index of the nonlinear medium. We will look for the complex-valued solution of Eq. (1.1), concentrated in the vicinity of a curve l on the plane (x, y) and having a sense of a wave traveling along the l aðx; yÞ ¼ aðs; nÞ;
bðx; yÞ ¼ bðs; nÞ
The large dimensionless parameter, proportional to the frequency-x, has the meaning of the ratio of the scale of the inhomogeneity of the medium to the wavelength. For simplicity we consider the flat problem and stationary processes. The approximate problem is a classical one (see Akhmanov et al., 1966; Erokhin and Sagdeev, 1982). Having in mind the application to the upper ionosphere, with estimates as in Bochkarev et al. (1982), often we assume.
0273-1177/$36.00 Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.11.036
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að! r Þ ¼ a ¼ const; " !# 2 ðy m y Þ ! 2 bð r Þ ¼ bðyÞ ¼ B 1 q 1 ; H2
B ¼ const ð1:2Þ
Here b does not depend on the horizontal coordinate x and monotonously decreases with y for y < ym, q is the ratio of the critical frequency of the F2 layer to the working frequency, ym is the height corresponding to the maximum of the electron density, H is the half-width of the layer. Of course, Eq. (1.2) for b(y) is used for the heights satisfying the inequality ym H < y,
It is clear that the main difficulty of the problem is the account of self-action, i.e., the necessity to find simultaneously the axial line l and the nonlinear wave field concentrated around l. In this connection, in Molotkov and Vakulenko (1988a,b) for a number of nonlinearities, including nonlinear model (1.1), indicate the possibility of the possibility to separate the problem of the axial line from the problem of the wave field. This separation of problems is just by the construction of the appropriate (TVP). The construction of the (TVP) may be performed in two ways: either starting from some appropriate ansatz, or with the help of Whitham’s principle (Molotkov and Vakulenko, 1988a,b; Molotkov, 2003, 2005). Both paths lead to full matches results. The initial construction phase of (TVP), may be to find explicit expressions for the curvature at each point of the curve. The result for the model (1.1) shows that the axial line l is an extremal one-parameter variational principle Z d hðx; y; AÞdS ¼ 0 where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b þ b2 þ 4A2 a2 þ 43 A2 a2 H¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 b þ b2 þ 4A2 a2 The parameter A in the variational principle has the meaning of total intensity of the beam. In the ionospheric conditions, the coefficient A defines the nonlinearity is small, so that we have the inequality Aa B
ð2:1Þ
Under the condition (2.1) the variational principle discussed above takes the form Z pffiffiffi A2 a 2 d b 1 þ 2 dS ¼ 0 ð2:2Þ 6b where dS is the differential of the arc length. The variational principle (2.2) differs from the usual Fermat’s principle (the beam trajectory without nonlinearity) only by correction of integrand expression. 3. Axial line of powerful beam The application of the variational principle (2.2) leads to the following results. Suppose that in the absence of nonlinearity the rising branch of the beam, axial line is described by the equation y = y0(x). The first integral of the function y0 is pffiffiffiffiffiffiffiffiffiffiffi signum y 0 pffiffiffiffiffiffiffiffiffi 0 ffi ¼ bð0Þ cos c0 ¼ c0 bðy 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:1Þ 1 þ y 02 0 where c0 – exit angle of the beam. Eq. (3.1) can be integrated but the expression for y0 is very complicated and not very important for further analysis.The function y0(x)
I.A. Molotkov, B. Atamaniuk / Advances in Space Research 47 (2011) 1181–1186
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and depends on the field of the powerful beam. One can get a formula for this field in the coordinates (n, s)associated with the axial line l. Where n – perpendicular to the l. of the normal forms a sharp angle with the positive direction of the y-axis, s is the length of the arch along /, normal vector forms an acute angle with the positive direction of axis y, s is the length of the arch along. When the inequality (2.1) is satisfied the following asymptotic equation is valid (Molotkov, 2003, 2005) sffiffiffiffiffiffiffiffi 2Aa 1 u¼
b 2 2 Aaxn A a cosh pffiffi 1 2b2
Fig. 1.
b
(its graph – the curve 1 in Fig. 1) is defined on some interval [0, 2X], is symmetric and has a single extremum – the maximum at x = X. Considering the nonlinearity, we obtain an approximate equation for the l in the form Aa y ¼ y 0 ðxÞ þ ð3:2Þ y ðxÞ B 1 For the function y1(x)the variational principle (2.2) gives the linear inhomogeneous equation Lðy 0 Þy 1 ¼
y 00 b0 ðy 0 Þ B2 0 y y ¼ 1 1 2bðy 0 Þ 1 þ y 02 6b2 ðy 0 Þ 0
ð3:3Þ
Which should be supplemented by the condition y0(x) = 0. Here L(y0(x)) – linear operator with coefficients depending on y0(x). It is easy to verify that the function y1(x) is also defined on [0, 2X], nonnegative, symmetric and has a maximum at x = X. Axis x has direction along earth surface, axis y is height over earth surface. Beam axial lines: 1 – in linear theory, 2 – for powerful beam with taking into account the nonlinearity, 3 – for test beam when the frequency shift is absent. Thus the trajectory (curve 2 in Fig. 1) is placed higher than it follows from the linear theory. The shift of the turning point of the axial line of the powerful beam caused by nonlinearity is easy to investigate.Suppose that in the linear case turn pffiffiffiffiffiffiffiffiffithe ffi p ffiffiffiffiffiffiffiffiffi occurs at the point (X, Y), y0(X) = Y, bðY Þ ¼ bð0Þ cos y 0 . With the account of nonlinearity, the turn is at the point (X, g) and 2
g¼Y þ
ðAaÞ 1 >Y 3 bðY Þjb0 ðY Þj
ð3:4Þ
Due to nonlinearity the turning point is shifted upwards remaining at the same place in the horizontal direction.The displacement of the beam axial line and its turning point increases with the power of the beam increasing. 4. Controlled, additional stratification of the ionospheric F2 layer For the selected model of the medium, the square of the effective refraction index, is proportional to the quantity N 2 ¼ ajuj2 þ bðyÞ
ð4:1Þ
Z s pffiffiffi A2 a 2 b 1þ exp ix ds 2b2 0
ð4:2Þ
in which b = b(s) = b(y) b(y)jl. To obtain the function b(s) for a given function b(y)jl, one need to use the relation between the coordinates s and x to l , given by Z x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s¼ 1 þ y 0 ðxÞdx 0
where y0 (x)has to be taken from (3.2). From (4.1) and (4.2) we obtain
2Aa2 Aaxn A 2 a2 N2 ¼ b þ cosh2 pffiffiffi 1 2 ð4:3Þ b 2b b The second term in the right-hand side of (4.3) plays the role of a correction and is absent in the linear theory. However the derivative with respect to y (or n) of this term contains x in the denominator and for certain set of parameters may compete with the derivative of the first term. This testifies that under the action of the powerful wave an additional guided stratification of the ionosphere takes place. In other words, the redistribution of the concentration of the electron plasma in ionosphere occurs. If the intensity of the powerful radiation is so great that B q 2=3 Aa > ð4:4Þ 1=3 xH ð2aÞ then an additional wave guide and anti-wave guide are formed in the F2 layer. For rather high frequencies, the condition (4.4) is compatible with the condition (2.1). The stratification of F2 layer will be described in more detail.It can be shown that the maximum perturbed term ajuj2 is located at a height y = f (Fig. 2), where f ’ ym þ
H2 qy m
ð4:5Þ
The resulting expression shows that the stratification occurs namely at the top of the layer, where y > ym. Differentiating expression (4.1) with respect to y , we obtain a complex transcendental equation, determines the vertical coordinates of the axes y = Y1 and y = Y2 waveguide and anti-waveguide. Under the condition (4.4) we find the approximate formulas
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at x = X. For d =0 the curve l1 (curve 3 in Fig. 1) is placed below the curve l. For d < 0 the correction terms in (5.2) have opposite signs and may locally compensate each other. In the general case, the vertical coordinate g1 of the turning point l1 given by the asymptotic equation ðAaÞ2 1 þ 6A1 2d B bðY Þ ð5:3Þ g1 ¼ Y þ 0 x jb0 ðY Þj 6bðY Þjb ðY Þj The coincidence of the heights of the maximums of the trajectories l and l1 in the leading order takes place at d Aa2 ¼ x bðY ÞðB bðY ÞÞ
Fig. 2.
Y1 ffi f þ
4B y m Aax q2
ð4:6Þ
4B y m p Aax q2
ð4:7Þ
and Y2 ffi f þ
It is evident that with increasing intensity of high-power beam deflection (Y1 f) and (Y2 f) reduced. The described phenomena may be considered as another mechanism different from the is thermodiffusion leading to the formation of the dissipative structure in the ionosphere (Polyakov and Yakhno, 1980). 5. The axial line of the test wave beam. Taking into account the frequency shift Let us investigate the axial line of the test (diagnostic) wave beam. We assume that the test beam is released at the same angle as the powerful one but has a different frequency x1 = x + d. The additive contribution dmay be either positive, or negative, but has to be small by absolute value as compared with x: jdj x
ð5:1Þ
The test beam propagates in the medium having the refraction index squared equal to N 21 . The quantity N 21 differs from N2 (see (4.3)) by by replacing u to xand q and b to d 2d q1 ¼ q 1 ; b1 ¼ b þ ðB bÞ x x The axial line l1 of the test beam is close to l. Its equation may be written down making use of the function y0(x)and the two correction terms which are linear in small parameters (AaB1)2 and dx1: y ¼ y 0 ðxÞ þ
Aa d y ðxÞ þ y 3 ðxÞ B 2 x
ð5:4Þ
6. The axial line of the test wave beam. Taking into account the shift of the initial angle We continue to study the axial line of the test wave beam.Now consider that the test beam is produced with the same frequency x, that powerful, but with the exit angle c0 þ c1 ;
jc1 j c0
ð6:1Þ
shifted compared with the exit angle c0 powerful beam.Test beam propagates in a medium with the square of the effective refractive index (4.3). We looking for the axial line in the form 2 Aa c yðxÞ ¼ y 0 ðxÞ þ y 4 ðxÞ þ 1 y 5 ðxÞ ð6:2Þ B c0 To determine the function y4(x) and y5(x) we again use the first integral of Euler’s equation for functional (2.2). We obtain an equation similar to (3.3): c00 b0 ðy 0 Þ B2 0 y y ¼ ¼ f4 4 0 4 b ðy 0 Þ 1 þ c02 6b2 ðy 0 Þ 0
ð6:3Þ
c00 b0 ðy 0 Þ 0 y ¼ c0 tan c0 ¼ f5 y 5 b0 ðy 0 Þ 5 1 þ c02 0
ð6:4Þ
In the layer F2 in the left parts of Eqs. (6.3) and (6.4) the dominant terms are of the second.Therefore, the function y4(x) and y5(x) almost proportional to the right sides of f4 and f5. terms. For example, if y = ym we obtain y5 ffi 6ð1 2q2 Þc0 tan c0 ð6:5Þ y4 For real values of the quantities c0 and q, this fraction is very small. This means that changes the initial angle is relatively little effect on the amplitude of the probe beam.
ð5:2Þ
The variational principle (2.2) for the functions y2(x), y3(x) leads to linear equations similar to (3.3) but different from (3.3) by the form of the right-hand sides. The functions y2(x) and y3(x) are similar to the function y1(x) in (3.2). They are non-negative and have a maximum
7. Phase of test wave Let us analyze the phase of the test wave. The field of the test wave satisfies the linear Helmholtz equation with the refraction index N1 introduced above and the frequency x1. We take into account that the trajectory of l1 is close
I.A. Molotkov, B. Atamaniuk / Advances in Space Research 47 (2011) 1181–1186
to l, although it differs from it. For phase of the probe beam we have the formula rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2d ð6:6Þ W1 ¼ ðx þ dÞ b þ ðB bÞds x We shall calculate the phase W1 with the accuracy up to the terms AaB1 and dx1, neglecting higher powers of small parameters. Next we single out in (6.6) correction terms containing small parameters. We may consider that the integration in correction terms proceeds along the curve l, but in the principal term we have to take into account the difference between l1 and l which was pointed out in the preceding item and is given by Eq. (5.2). The equation for W1, obtained in this way contains not only the phase W of the powerful beam but also the complicated correction. These terms are due to the corrections to the effective refraction index N1, trajectory corrections in (5.2) and the corrections due to frequency shift. As in the previous item, the signs of the corrections are such that they may cancel each other. Of course, the difference between the phases, probing and powerful waves strongly depends on what the horizontal distance of this value is observed. Let us present the equation for W1 written down with the above accuracy for the finite cross section x = 2X: 4x tan c0 d 2 pffiffiffiffiffiffiffiffiffiffi b ðY Þ 2Aa2 W1 ð2X Þ ¼ Wð2X Þ þ 0 jb ðY Þj bðY Þ x ð6:7Þ As it was in the preceding item, one can see in (6.7) the possibility of the compensation of the parameters Aa and d. For d = 0 W1 ð2X Þ < Wð2X Þ
ð6:8Þ
This inequality means that the optical path length of the test beam is less than the optical path length of the powerful beam, though, the geometrical path of the test beam is larger. The inequality (6.1) is due to the behavior of the effective refraction index which decreases with the departure from the axis l.
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values we obtain f = 308 km, Y2 = 309 km, Y1 = 317 km. The latter height, practically coincides with the height of the trajectory of a powerful beam.The trajectory of the probe beam at d = 0 is always above the trajectory of a powerful beam, although very close to the latter. Now let d – 0. If jSj = 50 kHz (Bochkarev et al., 1982), then the change of g1 due to the term with d/xis very small. As far as the influence of such frequency shift on the phase difference W1(2X) W(2X) is concerned, it is for AaB1 = 0.2 about 100 times larger than the contribution of 2Aa2.
9. Conclusion Let us formulate some consequences of the above calculations. 1. Accounting for nonlinear self-powerful wave beam leads to the upward displacement of the beam axial line and the turning point compared to the linear theory results (see (3.2) and (3.4)). 2. The propagation of the powerful beam in ionosphere layer leads to additional stratification of the medium. If the intensity of the beam is large, enough (see condition (4.4)) the guided waveguide and anti-waveguide are formed in ionosphere. 3. In absence of the frequency shift, the trajectory of the test beam lies a bit higher than the trajectory of the powerful beam. 4. In presence of the frequency shift, the rise of the test beam trajectory, depends both on the intensity of the powerful beam and on the frequency shift. The relation between these factors (see (5.4)), when they cancel each other can be determined. 5. The phase difference of the test and powerful waves depends on the same factors (see (6.7)). In absence of frequency shift, this phase difference is negative, since the optical path length of the probe beam, is less than the optical length of the powerful beam. 6. The influence of the shift of the initial angle of the trajectory of the probe beam was considered. The small influence of this factor on the trajectory was shown.
8. Numerical estimates Let us turn to the numerical evaluation of the effects. If we take in the linear theory in Eq. (1.2) ym = 255 km, H = 70 km, q = 0.36 and c0 = 17, b(0) = B, we obtain for the height of the trajectory of the powerful beam the value Y 216 km. The true altitude g of this trajectory depends on the intensity of the powerful beam. For AaB1 = 0.2 (this corresponds to the power of radiation equal to 6 MW) in accordance with (3.4) g Y 8 km. For AaB1 = 0.4 the nonlinear rise of the trajectory is g Y 32 km. For these parameters, the condition (4.4), formation of additional waveguide inside the layer F2, always satisfied. Position of the axes of the waveguide and anti-waveguide is defined by (4.6) and (4.7). For selected above numerical
References Akhmanov, S.A., Sukhorukov, A.P., Khokhlov, R.V. Self-focusing and trapping of intense light beams. JETP 50, 6, 1966. Bochkarev, G.S., Eremenko, V.A., Lobachevsky, L.A., Ljannoy, B.E., Migulin, V.V., Cherkashin, Y.u.N. Nonlinear interaction of decametre radio waves at close frequencies in oblique propagation. J. Atmos. Terr. 44, 12, 1982. Erokhin, N.S., Sagdeev, R.Z. Features self-focusing and energy absorption of high-power wave beams in an inhomogeneous plasma. JETP T. 83, 1, 1982. Field Jr., E.C., Bloom, R.M., Kossey, P.A. Ionospheric heating with oblique high-frequency waves. J. Geophys. Res. 96, 12, 1990. Molotkov, I.A., Vakulenko, S.A. Sosredotochennyje nelinejnyje volny. Izdatelstvo Leningrad. University Press, 1988a.
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Molotkov, I.A., Vakulenko, S.A. Wave beams in an inhomogeneous medium with saturated nonlinearity. Wave Motion 17, 3, 1988b. Molotkov, I.A., Cherkashin, Yu.N. Some nonlinear effects of wave beam propagation in the ionosphere. J. Atmos. Terr. 56, 11, 1994. Molotkov, I.A. Analitycheskije metody teorii nelinejnyh voln. M. Izd, Fizmatlit, 2003.
Molotkov, I.A. Analitical Methods in the Nonlinear Waves Theory. Publisher House Pensoft, Sofia, 2005. Polyakov, S.V., Yakhno, V.G. On the thermal diffusion mechanism of generation of inhomogeneities of electron density in F-layer of the ionosphere. Plasma Phys. T. 6, 2, 1980.
Advances in Space Research 47 (2011) 1181–1186 www.elsevier.com/locate/asr
Oblique sounding of the ionosphere by powerful wave beams I.A. Molotkov a, B. Atamaniuk b,⇑ a
Inst. of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, RAS, Troitsk, Moscow Region 142190, Russia b Space Research Center, PAS, 18A Bartycka, 00-716 Warsaw, Poland Received 31 July 2010; received in revised form 25 November 2010; accepted 26 November 2010 Available online 5 December 2010
Abstract The article is devoted to modeling the impact on the ionosphere powerful obliquely incident wave beam. The basis of this analysis will be orbital variational principle for the intense wave beams-generalization of Fermat’s principle to the case of a nonlinear medium (Molotkov and Vakulenko, 1988a,b; Molotkov, 2003, 2005). Under the influence of a powerful wave beam appears manageable the additional stratification of the ionospheric layer F2. Explicit expressions show how the properties of the test beam, with a shifted frequency, released in the same direction as the beam depend on the intensity of a powerful beam and the frequency shift. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Oblique wave beam; Ionosphere; Trajectory variational principle
1. Introduction The aim is to model the influence on the ionosphere of the powerful oblique wave beam with the working frequencies higher than the critical frequency of the F2 layer. We wish to study the variation of the parameters of both ionosphere layer itself and of the weak test beam which serves for the diagnostics and which has the same direction as the powerful beam. The investigation of the behavior of the beam in such conditions was performed in Bochkarev et al. (1982) in the framework of geometrical optics. The perturbation of the electron temperature in such a process was calculated in Field et al. (1990). Short summary of this theory can be found in Molotkov and Cherkashin (1994). The problem is very difficult and therefore below we shall limit ourselves to the use of a rather simple model which does not take into account all the details of waves propagation in ionosphere.
⇑ Corresponding author.
E-mail addresses: [email protected] (I.A. Molotkov), batama@ cbk.waw.pl (B. Atamaniuk).
Let us assume that the propagation of the wave beams in nonlinear smoothly inhomogeneous medium is described by the cubic-nonlinear Helmholtz equation Du þ x2 að! r Þju2 j þ bð! r Þ u ¼ 0; x 1; a > 0; ! r ¼ ðx; yÞ ð1:1Þ Here D- is the two-dimensional Laplacian, að! r Þ-strictly positive. Square brackets in (1.1) is the square of the refractive index of the nonlinear medium. We will look for the complex-valued solution of Eq. (1.1), concentrated in the vicinity of a curve l on the plane (x, y) and having a sense of a wave traveling along the l aðx; yÞ ¼ aðs; nÞ;
bðx; yÞ ¼ bðs; nÞ
The large dimensionless parameter, proportional to the frequency-x, has the meaning of the ratio of the scale of the inhomogeneity of the medium to the wavelength. For simplicity we consider the flat problem and stationary processes. The approximate problem is a classical one (see Akhmanov et al., 1966; Erokhin and Sagdeev, 1982). Having in mind the application to the upper ionosphere, with estimates as in Bochkarev et al. (1982), often we assume.
0273-1177/$36.00 Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.11.036
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að! r Þ ¼ a ¼ const; " !# 2 ðy m y Þ ! 2 bð r Þ ¼ bðyÞ ¼ B 1 q 1 ; H2
B ¼ const ð1:2Þ
Here b does not depend on the horizontal coordinate x and monotonously decreases with y for y < ym, q is the ratio of the critical frequency of the F2 layer to the working frequency, ym is the height corresponding to the maximum of the electron density, H is the half-width of the layer. Of course, Eq. (1.2) for b(y) is used for the heights satisfying the inequality ym H < y,
It is clear that the main difficulty of the problem is the account of self-action, i.e., the necessity to find simultaneously the axial line l and the nonlinear wave field concentrated around l. In this connection, in Molotkov and Vakulenko (1988a,b) for a number of nonlinearities, including nonlinear model (1.1), indicate the possibility of the possibility to separate the problem of the axial line from the problem of the wave field. This separation of problems is just by the construction of the appropriate (TVP). The construction of the (TVP) may be performed in two ways: either starting from some appropriate ansatz, or with the help of Whitham’s principle (Molotkov and Vakulenko, 1988a,b; Molotkov, 2003, 2005). Both paths lead to full matches results. The initial construction phase of (TVP), may be to find explicit expressions for the curvature at each point of the curve. The result for the model (1.1) shows that the axial line l is an extremal one-parameter variational principle Z d hðx; y; AÞdS ¼ 0 where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b þ b2 þ 4A2 a2 þ 43 A2 a2 H¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 b þ b2 þ 4A2 a2 The parameter A in the variational principle has the meaning of total intensity of the beam. In the ionospheric conditions, the coefficient A defines the nonlinearity is small, so that we have the inequality Aa B
ð2:1Þ
Under the condition (2.1) the variational principle discussed above takes the form Z pffiffiffi A2 a 2 d b 1 þ 2 dS ¼ 0 ð2:2Þ 6b where dS is the differential of the arc length. The variational principle (2.2) differs from the usual Fermat’s principle (the beam trajectory without nonlinearity) only by correction of integrand expression. 3. Axial line of powerful beam The application of the variational principle (2.2) leads to the following results. Suppose that in the absence of nonlinearity the rising branch of the beam, axial line is described by the equation y = y0(x). The first integral of the function y0 is pffiffiffiffiffiffiffiffiffiffiffi signum y 0 pffiffiffiffiffiffiffiffiffi 0 ffi ¼ bð0Þ cos c0 ¼ c0 bðy 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:1Þ 1 þ y 02 0 where c0 – exit angle of the beam. Eq. (3.1) can be integrated but the expression for y0 is very complicated and not very important for further analysis.The function y0(x)
I.A. Molotkov, B. Atamaniuk / Advances in Space Research 47 (2011) 1181–1186
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and depends on the field of the powerful beam. One can get a formula for this field in the coordinates (n, s)associated with the axial line l. Where n – perpendicular to the l. of the normal forms a sharp angle with the positive direction of the y-axis, s is the length of the arch along /, normal vector forms an acute angle with the positive direction of axis y, s is the length of the arch along. When the inequality (2.1) is satisfied the following asymptotic equation is valid (Molotkov, 2003, 2005) sffiffiffiffiffiffiffiffi 2Aa 1 u¼
b 2 2 Aaxn A a cosh pffiffi 1 2b2
Fig. 1.
b
(its graph – the curve 1 in Fig. 1) is defined on some interval [0, 2X], is symmetric and has a single extremum – the maximum at x = X. Considering the nonlinearity, we obtain an approximate equation for the l in the form Aa y ¼ y 0 ðxÞ þ ð3:2Þ y ðxÞ B 1 For the function y1(x)the variational principle (2.2) gives the linear inhomogeneous equation Lðy 0 Þy 1 ¼
y 00 b0 ðy 0 Þ B2 0 y y ¼ 1 1 2bðy 0 Þ 1 þ y 02 6b2 ðy 0 Þ 0
ð3:3Þ
Which should be supplemented by the condition y0(x) = 0. Here L(y0(x)) – linear operator with coefficients depending on y0(x). It is easy to verify that the function y1(x) is also defined on [0, 2X], nonnegative, symmetric and has a maximum at x = X. Axis x has direction along earth surface, axis y is height over earth surface. Beam axial lines: 1 – in linear theory, 2 – for powerful beam with taking into account the nonlinearity, 3 – for test beam when the frequency shift is absent. Thus the trajectory (curve 2 in Fig. 1) is placed higher than it follows from the linear theory. The shift of the turning point of the axial line of the powerful beam caused by nonlinearity is easy to investigate.Suppose that in the linear case turn pffiffiffiffiffiffiffiffiffithe ffi p ffiffiffiffiffiffiffiffiffi occurs at the point (X, Y), y0(X) = Y, bðY Þ ¼ bð0Þ cos y 0 . With the account of nonlinearity, the turn is at the point (X, g) and 2
g¼Y þ
ðAaÞ 1 >Y 3 bðY Þjb0 ðY Þj
ð3:4Þ
Due to nonlinearity the turning point is shifted upwards remaining at the same place in the horizontal direction.The displacement of the beam axial line and its turning point increases with the power of the beam increasing. 4. Controlled, additional stratification of the ionospheric F2 layer For the selected model of the medium, the square of the effective refraction index, is proportional to the quantity N 2 ¼ ajuj2 þ bðyÞ
ð4:1Þ
Z s pffiffiffi A2 a 2 b 1þ exp ix ds 2b2 0
ð4:2Þ
in which b = b(s) = b(y) b(y)jl. To obtain the function b(s) for a given function b(y)jl, one need to use the relation between the coordinates s and x to l , given by Z x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s¼ 1 þ y 0 ðxÞdx 0
where y0 (x)has to be taken from (3.2). From (4.1) and (4.2) we obtain
2Aa2 Aaxn A 2 a2 N2 ¼ b þ cosh2 pffiffiffi 1 2 ð4:3Þ b 2b b The second term in the right-hand side of (4.3) plays the role of a correction and is absent in the linear theory. However the derivative with respect to y (or n) of this term contains x in the denominator and for certain set of parameters may compete with the derivative of the first term. This testifies that under the action of the powerful wave an additional guided stratification of the ionosphere takes place. In other words, the redistribution of the concentration of the electron plasma in ionosphere occurs. If the intensity of the powerful radiation is so great that B q 2=3 Aa > ð4:4Þ 1=3 xH ð2aÞ then an additional wave guide and anti-wave guide are formed in the F2 layer. For rather high frequencies, the condition (4.4) is compatible with the condition (2.1). The stratification of F2 layer will be described in more detail.It can be shown that the maximum perturbed term ajuj2 is located at a height y = f (Fig. 2), where f ’ ym þ
H2 qy m
ð4:5Þ
The resulting expression shows that the stratification occurs namely at the top of the layer, where y > ym. Differentiating expression (4.1) with respect to y , we obtain a complex transcendental equation, determines the vertical coordinates of the axes y = Y1 and y = Y2 waveguide and anti-waveguide. Under the condition (4.4) we find the approximate formulas
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at x = X. For d =0 the curve l1 (curve 3 in Fig. 1) is placed below the curve l. For d < 0 the correction terms in (5.2) have opposite signs and may locally compensate each other. In the general case, the vertical coordinate g1 of the turning point l1 given by the asymptotic equation ðAaÞ2 1 þ 6A1 2d B bðY Þ ð5:3Þ g1 ¼ Y þ 0 x jb0 ðY Þj 6bðY Þjb ðY Þj The coincidence of the heights of the maximums of the trajectories l and l1 in the leading order takes place at d Aa2 ¼ x bðY ÞðB bðY ÞÞ
Fig. 2.
Y1 ffi f þ
4B y m Aax q2
ð4:6Þ
4B y m p Aax q2
ð4:7Þ
and Y2 ffi f þ
It is evident that with increasing intensity of high-power beam deflection (Y1 f) and (Y2 f) reduced. The described phenomena may be considered as another mechanism different from the is thermodiffusion leading to the formation of the dissipative structure in the ionosphere (Polyakov and Yakhno, 1980). 5. The axial line of the test wave beam. Taking into account the frequency shift Let us investigate the axial line of the test (diagnostic) wave beam. We assume that the test beam is released at the same angle as the powerful one but has a different frequency x1 = x + d. The additive contribution dmay be either positive, or negative, but has to be small by absolute value as compared with x: jdj x
ð5:1Þ
The test beam propagates in the medium having the refraction index squared equal to N 21 . The quantity N 21 differs from N2 (see (4.3)) by by replacing u to xand q and b to d 2d q1 ¼ q 1 ; b1 ¼ b þ ðB bÞ x x The axial line l1 of the test beam is close to l. Its equation may be written down making use of the function y0(x)and the two correction terms which are linear in small parameters (AaB1)2 and dx1: y ¼ y 0 ðxÞ þ
Aa d y ðxÞ þ y 3 ðxÞ B 2 x
ð5:4Þ
6. The axial line of the test wave beam. Taking into account the shift of the initial angle We continue to study the axial line of the test wave beam.Now consider that the test beam is produced with the same frequency x, that powerful, but with the exit angle c0 þ c1 ;
jc1 j c0
ð6:1Þ
shifted compared with the exit angle c0 powerful beam.Test beam propagates in a medium with the square of the effective refractive index (4.3). We looking for the axial line in the form 2 Aa c yðxÞ ¼ y 0 ðxÞ þ y 4 ðxÞ þ 1 y 5 ðxÞ ð6:2Þ B c0 To determine the function y4(x) and y5(x) we again use the first integral of Euler’s equation for functional (2.2). We obtain an equation similar to (3.3): c00 b0 ðy 0 Þ B2 0 y y ¼ ¼ f4 4 0 4 b ðy 0 Þ 1 þ c02 6b2 ðy 0 Þ 0
ð6:3Þ
c00 b0 ðy 0 Þ 0 y ¼ c0 tan c0 ¼ f5 y 5 b0 ðy 0 Þ 5 1 þ c02 0
ð6:4Þ
In the layer F2 in the left parts of Eqs. (6.3) and (6.4) the dominant terms are of the second.Therefore, the function y4(x) and y5(x) almost proportional to the right sides of f4 and f5. terms. For example, if y = ym we obtain y5 ffi 6ð1 2q2 Þc0 tan c0 ð6:5Þ y4 For real values of the quantities c0 and q, this fraction is very small. This means that changes the initial angle is relatively little effect on the amplitude of the probe beam.
ð5:2Þ
The variational principle (2.2) for the functions y2(x), y3(x) leads to linear equations similar to (3.3) but different from (3.3) by the form of the right-hand sides. The functions y2(x) and y3(x) are similar to the function y1(x) in (3.2). They are non-negative and have a maximum
7. Phase of test wave Let us analyze the phase of the test wave. The field of the test wave satisfies the linear Helmholtz equation with the refraction index N1 introduced above and the frequency x1. We take into account that the trajectory of l1 is close
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to l, although it differs from it. For phase of the probe beam we have the formula rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2d ð6:6Þ W1 ¼ ðx þ dÞ b þ ðB bÞds x We shall calculate the phase W1 with the accuracy up to the terms AaB1 and dx1, neglecting higher powers of small parameters. Next we single out in (6.6) correction terms containing small parameters. We may consider that the integration in correction terms proceeds along the curve l, but in the principal term we have to take into account the difference between l1 and l which was pointed out in the preceding item and is given by Eq. (5.2). The equation for W1, obtained in this way contains not only the phase W of the powerful beam but also the complicated correction. These terms are due to the corrections to the effective refraction index N1, trajectory corrections in (5.2) and the corrections due to frequency shift. As in the previous item, the signs of the corrections are such that they may cancel each other. Of course, the difference between the phases, probing and powerful waves strongly depends on what the horizontal distance of this value is observed. Let us present the equation for W1 written down with the above accuracy for the finite cross section x = 2X: 4x tan c0 d 2 pffiffiffiffiffiffiffiffiffiffi b ðY Þ 2Aa2 W1 ð2X Þ ¼ Wð2X Þ þ 0 jb ðY Þj bðY Þ x ð6:7Þ As it was in the preceding item, one can see in (6.7) the possibility of the compensation of the parameters Aa and d. For d = 0 W1 ð2X Þ < Wð2X Þ
ð6:8Þ
This inequality means that the optical path length of the test beam is less than the optical path length of the powerful beam, though, the geometrical path of the test beam is larger. The inequality (6.1) is due to the behavior of the effective refraction index which decreases with the departure from the axis l.
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values we obtain f = 308 km, Y2 = 309 km, Y1 = 317 km. The latter height, practically coincides with the height of the trajectory of a powerful beam.The trajectory of the probe beam at d = 0 is always above the trajectory of a powerful beam, although very close to the latter. Now let d – 0. If jSj = 50 kHz (Bochkarev et al., 1982), then the change of g1 due to the term with d/xis very small. As far as the influence of such frequency shift on the phase difference W1(2X) W(2X) is concerned, it is for AaB1 = 0.2 about 100 times larger than the contribution of 2Aa2.
9. Conclusion Let us formulate some consequences of the above calculations. 1. Accounting for nonlinear self-powerful wave beam leads to the upward displacement of the beam axial line and the turning point compared to the linear theory results (see (3.2) and (3.4)). 2. The propagation of the powerful beam in ionosphere layer leads to additional stratification of the medium. If the intensity of the beam is large, enough (see condition (4.4)) the guided waveguide and anti-waveguide are formed in ionosphere. 3. In absence of the frequency shift, the trajectory of the test beam lies a bit higher than the trajectory of the powerful beam. 4. In presence of the frequency shift, the rise of the test beam trajectory, depends both on the intensity of the powerful beam and on the frequency shift. The relation between these factors (see (5.4)), when they cancel each other can be determined. 5. The phase difference of the test and powerful waves depends on the same factors (see (6.7)). In absence of frequency shift, this phase difference is negative, since the optical path length of the probe beam, is less than the optical length of the powerful beam. 6. The influence of the shift of the initial angle of the trajectory of the probe beam was considered. The small influence of this factor on the trajectory was shown.
8. Numerical estimates Let us turn to the numerical evaluation of the effects. If we take in the linear theory in Eq. (1.2) ym = 255 km, H = 70 km, q = 0.36 and c0 = 17, b(0) = B, we obtain for the height of the trajectory of the powerful beam the value Y 216 km. The true altitude g of this trajectory depends on the intensity of the powerful beam. For AaB1 = 0.2 (this corresponds to the power of radiation equal to 6 MW) in accordance with (3.4) g Y 8 km. For AaB1 = 0.4 the nonlinear rise of the trajectory is g Y 32 km. For these parameters, the condition (4.4), formation of additional waveguide inside the layer F2, always satisfied. Position of the axes of the waveguide and anti-waveguide is defined by (4.6) and (4.7). For selected above numerical
References Akhmanov, S.A., Sukhorukov, A.P., Khokhlov, R.V. Self-focusing and trapping of intense light beams. JETP 50, 6, 1966. Bochkarev, G.S., Eremenko, V.A., Lobachevsky, L.A., Ljannoy, B.E., Migulin, V.V., Cherkashin, Y.u.N. Nonlinear interaction of decametre radio waves at close frequencies in oblique propagation. J. Atmos. Terr. 44, 12, 1982. Erokhin, N.S., Sagdeev, R.Z. Features self-focusing and energy absorption of high-power wave beams in an inhomogeneous plasma. JETP T. 83, 1, 1982. Field Jr., E.C., Bloom, R.M., Kossey, P.A. Ionospheric heating with oblique high-frequency waves. J. Geophys. Res. 96, 12, 1990. Molotkov, I.A., Vakulenko, S.A. Sosredotochennyje nelinejnyje volny. Izdatelstvo Leningrad. University Press, 1988a.
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Molotkov, I.A., Vakulenko, S.A. Wave beams in an inhomogeneous medium with saturated nonlinearity. Wave Motion 17, 3, 1988b. Molotkov, I.A., Cherkashin, Yu.N. Some nonlinear effects of wave beam propagation in the ionosphere. J. Atmos. Terr. 56, 11, 1994. Molotkov, I.A. Analitycheskije metody teorii nelinejnyh voln. M. Izd, Fizmatlit, 2003.
Molotkov, I.A. Analitical Methods in the Nonlinear Waves Theory. Publisher House Pensoft, Sofia, 2005. Polyakov, S.V., Yakhno, V.G. On the thermal diffusion mechanism of generation of inhomogeneities of electron density in F-layer of the ionosphere. Plasma Phys. T. 6, 2, 1980.