The analytical form of absorption index appropriate to a certain range of height of the upper atmosphere plasma medium during ordinary wave propagation under quasi-transverse approximations has been utilized to develop an integral equation through Wentzel-Kramers-Brillouin method of solution. The integral equation is useful for the determination of effective electron collision frequency. Numerical results are presented graphically along with the results of some earlier work.Primjenjujemo analitički izraz za apsorpcijski koeficijent prikladan za širenje običnih elektromagnetskih valova u razmatranom rasponu visina gornje atmosferske plazme. Primijenili smo kvazi-poprečno približenje za izvod integralne jednadžbe pomoću Wentzel-Kramers-Brillouinove metode. Integralna jednadžba je pogodna za određivanje efektivne frekvencije elektronskih sudara. Načinili smo numeričke analize i ishodi se uspoređuju grafički s nekim ranijim rezultatima

Bandyopadhya, R.

De, M.

De, S. S.

Dubey, S. K.

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ISSN1330–0008CODENFIZAE4SOME STUDIES ON THE VARIATION OF ELECTRON COLLISIONFREQUENCY AND ABSORPTION COEFFICIENT WITHIN THEANISOTROPIC ATMOSPHERES. S. DEa, M. DEa, S. K. DUBEYa, and R. BANDYOPADHYAYbaCentre of Advanced Study in Radio Physics and Electronics,University of Calcutta, Calcutta 700009, IndiabDepartment of Mathematics, University of Jadavpur, Calcutta 700032, IndiaReceived 23 March 1999; revised manuscript received 11 December 2000Accepted 12 February 2001The analytical form of absorption index appropriate to a certain range of heightof the upper atmosphere plasma medium during ordinary wave propagation underquasi-transverse approximations has been utilized to develop an integral equationthrough Wentzel-Kramers-Brillouin method of solution. The integral equation isuseful for the determination of effective electron collision frequency. Numerical re-sults are presented graphically along with the results of some earlier work.PACS numbers: 87.15.Rn, 87.50.-a UDC 535.217, 539.21Keywords: upper atmosphere plasma, wave propagation, absorption index, electron colli-sion frequency1. IntroductionThe presence of the geomagnetic field in the upper atmosphere makes themedium magneto-active. Electromagnetic waves split into ordinary and extraor-dinary waves with different refractive indices and absorption coefficients.The formula for the refractive indices of a cold magnetoplasma, as deducedby Appleton and Hartree, is applicable to the case where only electrons are effec-tive and the electron-neutral-particle collision frequency is assumed to be indepen-dent of the electron velocity. Application of the Appleton and Hartree formula tothe problems of propagation through the ionised medium is unsuitable. Therefore,quasi-longitudinal (QL) and quasi-transverse (QT) approximations are made.The refractive indices depend on the angle between the normal to the wavefront and the geomagnetic field. They also depend on electron density, electron-neutral particle collision frequency, electro gyro-frequency and some other param-eters. Moreover, the effects of ions and the dependence of collision frequencies onFIZIKA A (Zagreb) 9 (2000) 4, 187–192 187de et al.: some studies on the variation of electron collision frequency . . .their respective velocities are important. Based on these considerations, differentformulae of refractive indices have been developed [1–4], which are valid underappropriate circumstances.Some workers [5–8] generalized the Appleton-Hartree magnetoionic formulathrough the solution of the Boltzmann equation following the Chapman-Enskogmethod. In these articles, the effects of the velocity dependence of the collision fre-quency are derived in closed analytical forms. The results also yield the Appleton-Hartree form on elimination of the velocity dependence of collision frequency.Budden [9] critically examined the Appleton-Hartree results of QL and QT ap-proximations and pointed out certain limitations in their use. He presented somealternative angular approximations in place of traditional approximations. To jus-tify different approaches, a more general method has been given for waves in acold magnetoplasma [10]. In the work oh Heading [11], the propagation equationsare generalized to n-dimensions from which any particular algebraic form can beobtained. In the method, the refractive indices of the upper atmospheric plasmaand their different approximations can be justified for a specific dimension.The solution of the wave equation for the ionospheric plasma having slowlyvarying indices with height may be chosen appropriately by the Wentzel-Kramers-Brillouin (WKB) method. In this work, a suitable expression for refractive indicesof the ionospheric plasma in the QT mode of wave propagation is applied in thederivation of an integral equation through the use of the WKB solution. The ef-fective collision frequency at the different height ranges can be determined froman integral equation. The results of some numerical computations are presentedgraphically.2. Mathematical formulationFor an ordinary wave in the upper atmospheric plasma, assuming the QT ap-proximation, the expression for the refractive indices can be written as (Eq. (29)of Ref. [9])n2 =(1 − XU) (1 +XUcos2 θ +X2(2U2 + Y 2) − XU(U2X2)U2Y 2cos4 θ . . .), (1)wheren = (µ − iχ), X = Ne2mω2=ω2Nω2, U = 1 − iζ, ζ = νeffω, Y =eBmω.θ is the angle between the normal to the wave front and the direction antiparallelto the geomagnetic field B, n is the complex refractive index, µ the wave refractiveindex, χ the absorption index, ωN the electron-plasma circular frequency, ω thecircular signal frequency, N the number of electrons per unit volume, m the electronmass, νeff the effective collision frequency and e the electron charge. In Eq.(1), onlyelectrons are treated, hence the suffix e has been dropped.188 FIZIKA A (Zagreb) 9 (2000) 4, 187–192de et al.: some studies on the variation of electron collision frequency . . .Ionospheric plasma is a slowly varying medium, for which it is often appro-priate to use the WKB approximations for the solution of the problems of wavepropagation. The WKB solution for the y-component of the electric field of a wavepropagating through the ionospheric plasma, traversed by the geomagnetic field,can be written as (Ref. [12], Eq. 9.11)Ey(z) = A expiωt − kz∫z1n(z)dz , (2)Ey(z) = A exp−kz∫z1χ(z)dz exp−iωt − kz∫z1µ(z)dz , (3)where k = ω√µ0ε0 = ω/c, µ0 is the vacuum permeability, ε0 the vacuum permit-tivity, z the height of the considered atmospheric layer and z1 is the referentialheight, i.e. the bottom of the ionosphere.The power reflection coefficient in the process of total reflection from the iono-spheric layer at z = zmax, where X = 1, i.e., ωN = ω, is given by (Ref. [12], Eq.9.37)R(ω) = exp−4kzmax∫z1χ(z)dz . (4)After taking the logarithm, Eq. (4) may be expressed asR1(ω) = 4ω2ω∫χ(ωN , ω)dz(ωN )dωNdωN , (5)where R1(ω) = Cω lnR(ω).χ(ωN , ω) can be expressed from Eq. (1) asχ(ωN , ω) =1√2√M −√M2 + 2N , (6)whereM = 1 − X1 + ζ2sin2 θ − X2(1 − ζ2)(1 + ζ2)2cos2 θ +1(1 + ζ2)3Y 2×[{A(1 − ζ2) − 2Bζ}(1 + ζ2 − X) + 2Aζ2 + B(1 − ζ2)] cos4 θ ,N =ζX1 + ζ2(sin2 θ +2X(1 + ζ2)cos2 θ)− 1(1 + ζ2)3Y 2[{2Aζ + B(1 − ζ2)}(1 + ζ2 − X) − AXζ(1 − ζ2) + 2BXζ2] cos4 θ ,FIZIKA A (Zagreb) 9 (2000) 4, 187–192 189de et al.: some studies on the variation of electron collision frequency . . .A = X2{2(1 − ζ2) + Y 2} − X(1 − ζ2 − X2) − 2ζ2X ,B = 2ζX − 4ζX2 + Xζ(1 − ζ2 − X2) .The other symbols have been explained earlier.The expression in Eq. (6) can be simplified asχ =1√2[{b1 + a20(1 + b0)b2} cos2 θ + {a4(b3 − b4) − 2a40b22 −12b20−b5 − 2a11b0b6 + a12} cos4 θ − 12√2a0 + b0 + b20 , (7)whereP = 1 + ζ2, Q = 1 − ζ2,a0 =XP, a7 = 2Aζ2 + BQ, a8 = (A − B)Q − 2(A + B)ζ,a1 = ζ − 1, a9 = (A − 2BX)ζ2, a10 = (B + AXζ)Q,a2 =XQP, a11 =X4P 4Y 2, a12 =a2Y 22− QPY2X+P 2Y 22X2,a3 = Q − 2ζ, a13 = PX4, b0 = a20(1 +2a1a0),a4 =1P 3Y 2, b1 = a0(1 − a2), b2 = 1 + a3 + a1a0 − a2,a5 = AQ − 2Bζ, b3 = a5a6 + a7, b4 = a6a8 + 2a9 + a10,a6 = P − X, b5 = a11a12, b6 = a13a6a8 + 2a0 + a10.Substitution of (7) into (5) would result in the Volterra integral equation of thethird kind.3. ResultsIn the numerical analyses, the effective collision frequency has been taken asνeff = νen + νei, where νen and νei are the electron–neutral and electron–ion col-lision frequencies, respectively. Figure 1 shows the curves of the effective collisionfrequency νeff versus height in the range 50 km to 140 km of the upper atmosphere.The full curve (a) of Fig. 1 shows the collision frequency values determined fromthe IRI [13] and CIRA [14] data under the assumptions of the daytime geophysicalconditions for the stated range of height.190 FIZIKA A (Zagreb) 9 (2000) 4, 187–192de et al.: some studies on the variation of electron collision frequency . . .Equation (5) relates the reflection coefficient to νeff , and so in principle hasbeen used to determine νeff from signal strength measurements for the consideredheight range. This is presented by the full curve (b) in Fig. 1. These curves aresuperimposed on the results of an earlier work [15]., depicted by the dashed curve(c) in Fig. 1. The propagation frequencies for different heights are chosen in therange 0.01 to 2.5 MHz for numerical calculations.Because of the weak density-dependent characteristics of the collision frequencyat higher altitudes, some differences between results are observed, while at lowerheights, the present result agrees well with the previous work.Fig. 1. Variation of effective collision frequency with altitude in the range 50 km to140 km of the upper atmosphere. The curve (a) shows the variation based on IRI[13] and CIRA [14] data, while (b) shows the calculated values of signal strengthmeasurements. They are superimposed on the results of Thomas [15], representedby the curve (c).Fig. 2 (right). Plot of the imaginary part of the refractive index in QT approxi-mation for ordinary ray propagating in ionospheric plasma as a function of X forY = 0.5 and ζ = 0.01. The full curve is the result of the present analysis, while thedashed curve is due to an earlier work of Chandrasekhar et al.The values of the ratio of square of the plasma frequency to signal frequency(X) are chosen for different heights. The ordinary ray absorption coefficient (χ)was numerically computed for Y = 0.5 and ζ = 0.01 from Eq. (7) as a function ofX. The results are presented by the full curve of Fig. 2. These values are comparedto the work of Chandrasekhar et al. [16], shown by the dashed curve in Fig. 2. Xwas chosen to be larger than 0.035. The partial disagreement may be attributed tothe local changes in the electron density profiles.FIZIKA A (Zagreb) 9 (2000) 4, 187–192 191de et al.: some studies on the variation of electron collision frequency . . .4. ConclusionThe disagreement in the variation of the collision frequency with height repre-sented by the three curves in Fig. 1 is probably due to the anisotropic effects at thecorresponding range of height. Unlike other techniques, the present analysis canbe taken as an alternative approach to study the variation of certain ionosphericparameters.References[1] E. V. Appleton and F. W. Chapman, Proc. Phys. Soc. 44 (1932) 246.[2] A. V. Phelps and J. L. Patrick, Phys. Rev. Lett. 3 (1959) 340.[3] H. K. Sen and A. A. Wyller, Radio Sci. D 69 (1965) 95.[4] P. L. Dyson and S. M. Newbery, J. Atmos. Ter. Phys. 44 (1982) 363.[5] R. Jancel and T. Kahan, Nuovo Cim. 12 (1954) 573.[6] H. K. Sen and A. A. Wyller, J. Geophys Res. 65 (1960) 3931.[7] W. F. Allis and J. L. Delcroix, Proc. of Paris Summer School on Plasma Physics, Vol.1, p. 45[8] K. G. Budden, Radio Sci. D 69 (1965) 191.[9] K. G. Budden, J. Atmos. Ter. Phys. 45 (1983) 213.[10] H. G. Booker and H. O. Vats, Indian J. Radio and Space Phys. 19 (1990) 105.[11] J. Heading, J. Atmos. Ter. Phys. 51 (1989) 699.[12] K. G. Budden, Radio Waves in the Ionosphere, Cambridge Univ. Press, Cambridge(1961) p. 82.[13] International Reference Ionosphere (IRI), compilation K. Raver, S. Ramkrishnan andD. Bilitza, URSI, Brussels, Belgium (1978).[14] Cospar International Reference Atmosphere (CIRA), ed. A. C. Stickland, AkademieVerlag, Berlin (1978).[15] L. Thomas, J. Atmos. Ter. Phys. 30 (1968) 1211.[16] R. Chandrasekhar, Madhusudana, D. N. Rao and K. V. V. Ramana, J. Atmos. Ter.Phys. 43 (1981) 59.PROUČAVANJE PROMJENA FREKVENCIJE ELEKTRONSKIH SUDARA IAPSORPCIJSKOG KOEFICIJENTA U NEIZOTROPNOJ ATMOSFERIPrimjenjujemo analitički izraz za apsorpcijski koeficijent prikladan za širenjeobičnih elektromagnetskih valova u razmatranom rasponu visina gornje atmosferskeplazme. Primijenili smo kvazi-poprečno približenje za izvod integralne jednadžbepomoću Wentzel-Kramers-Brillouinove metode. Integralna jednadžba je pogodnaza odred–ivanje efektivne frekvencije elektronskih sudara. Načinili smo numeričkeanalize i ishodi se uspored–uju grafički s nekim ranijim rezultatima.192 FIZIKA A (Zagreb) 9 (2000) 4, 187–192

Proučavanje promjena frekvencije elektronskih sudara i apsorpcijskog koeficijenta u neizotropnoj atmosferi

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