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Self-focusing instability in ionospheric plasma with thermal conduction Sodha, M. S., Sharma, A., Verma, M. P., & Faisal, M. (2007). Self-focusing instability in ionospheric plasma with thermal conduction. Physics of Plasmas, 14(5), 052901-052906. [052901]. DOI: 10.1063/1.2727449 Published in: Physics of Plasmas Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal General rights Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact [email protected]. Download date:09. Aug. 2017 PHYSICS OF PLASMAS 14, 052901 共2007兲 Self-focusing instability in ionospheric plasma with thermal conduction Mahendra Singh Sodhaa兲 Disha Academy of Research and Education, Disha Crown, Katchna Road, Shankar Nagar, Raipur 492 007, Chattisgarh, India Ashutosh Sharma, M. P. Verma, and Mohammad Faisal Ramanna Fellowship and DST Project Programme, Department of Education Building, Lucknow University, Lucknow 226 007, India 共Received 27 November 2006; accepted 19 March 2007; published online 8 May 2007兲 In this communication, an expression for the growth rate of self-focusing instability in the ionospheric plasma has been derived after taking finite thermal conduction into account. The instability arises on account of the depletion of electrons from regions where the irradiance of the perturbation is large. In contrast to earlier work, an appropriate energy balance equation for electrons and ions and the proper dependence of thermal conductivity on electron temperature have been used. The dependence of the growth rate of the filamentation instability on the background irradiation, thermal conductivity, and the wave number of transverse perturbation has been investigated. The mid-latitude daytime ionospheric model of Gurevich has been used for numerical computations, corresponding to a height of 200 km. The gradient of irradiance perturbations is assumed to be along the magnetic field of the Earth. The numerical results have been illustrated graphically and discussed. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2727449兴 I. INTRODUCTION There has been considerable interest in the growth of an instability 共associated with the propagation 共along the z axis兲 of a high power electromagnetic beam in a nonlinear medium兲, which is characterized by beam irradiance and electron density fluctuations in a direction 共x axis兲 transverse to that of the beam propagation. The fact that higher beam irradiance in a region causes lower electron density and consequently higher refractive index leads to the increase in irradiance in this region; this is synonymous with selffocusing. Hence, such instabilities are known as selffocusing instabilities. The saturated state of such an instability results in light filamentation. Several publications1–12 in this field have appeared in the scientific literature since 1962. Apart from the scientific point of view, the results in this field have a bearing on ionospheric modification experiments,13–19 beams from proposed satellite power stations passing through the ionosphere,20 and laser plasma interaction phenomena. It is of interest to consider the mechanism of selffocusing instability in a plasma. A small perturbation E1共r兲, superposed on a uniform beam, with electric vector E0, causes the electrons to concentrate towards the region, where E1 is maximum 共E1max兲. This redistribution of electrons causes a refractive index gradient, which if large enough to overcome diffraction, leads to a continuous increase in E1max as the beam propagates; this phenomenon is referred to as self-focusing instability. This communication presents an investigation of the self-focusing instability in the ionosphere. The energy balance of the electrons includes contributions from the solar radiation, Ohmic heating, energy loss in collisions with neua兲 Electronic mail: [email protected] 1070-664X/2007/14共5兲/052901/6/$23.00 tral atoms/molecules/ions, and energy loss by thermal conduction. The energy balance for the ions is obtained by equating the energy gained from the electrons to the energy lost to neutral species. Since the heat capacity 共proportional to the number density兲 of the neutral species is several 共102 to 103兲 times the heat capacity of the electrons and ions, the neutral species act as a constant temperature sink. The midlatitude ionospheric model of Gurevich19 has been used to provide the basic data for the computations. Perkins and Valeo21 were the first to highlight the role of electronic thermal conduction in filamentation; however their analysis ignores the change in the temperature of the electrons on account of the main beam. As may be seen from the present analysis this is a serious omission. Tewari et al.22 have studied filamentation, in the case, when the electron energy loss by collisions has been neglected; further, the temperature of the electrons in the region where the irradiance of the filament is highest has been implicitly assumed as the temperature in the absence of the beam. Using the available theory of filamentation on account of ponderomotive nonlinearity and taking thermal conduction into account Schmidt23 has analytically and numerically studied the effect of induced spatial incoherence and random phase screen on controlling filamentation. Cornolti and Lucchesi24 have considered the energy loss by collisions as well as by electronic thermal conduction in the analysis of filamentation; however, like Perkins and Valeo,21 they have also ignored the change in the electron temperature on account of the main beam. Epperlein25 has, in his elegant analysis of filamentation, assumed the electron and ion temperature to be the same, which is not a good approximation for the ionospheric plasma subjected to the field of a high power electromagnetic wave. Ghanshyam and Tripathi26 have analyzed the phenom- 14, 052901-1 © 2007 American Institute of Physics Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 052901-2 Phys. Plasmas 14, 052901 共2007兲 Sodha et al. enon of filamentation in a collisional plasma, taking into account thermal conduction as well as collisions with heavier particles as the mechanisms of power loss by the electrons, heated by the beam; however, their energy balance equation 关Eq. 共42兲 of their paper兴 and the expression for the electron temperature in the absence of perturbation 关their Eq. 共45兲兴 are not correct. This is on account of the use of the modified parameter ␦⬘, which has justification when the space dependence of the amplitude of the field is the same as that of the perturbation E1 共this is not true for E0, which is uniform兲 and also when the dependence of thermal conductivity on the electron temperature is ignored. Further, this analysis does not take into account the change in the ion temperature on account of Ohmic heating of electrons by the electric field of the beam and subsequent collisions of these hot electrons with ions. These omissions necessitate an improved version of the analysis. This has been presented in this communication with reference to the ionosphere. Expressions for the growth rate of the instability and the conditions for the instability to occur, maximum value of the growth rate and the corresponding value of q⬜ have been derived and a discussion of numerical results 共corresponding to an ionospheric height of 200 km and the magnetic field of the earth along the x axis兲 has been presented. II. THE ANALYSIS Let the electric field of a beam of uniform illumination and that of a small perturbation 共filament兲 be represented by ĵE0 and ĵE1, respectively, so that the resultant field E propagating in the z direction through a plasma can be expressed as E = ĵ共E0 + E1兲 = ĵ共A0 + A1兲exp i共t − kz兲, 共1兲 where A0, without loss of generality, is a real constant and A1共兩A1 兩 A0兲 is complex, ĵ is a unit vector and k is the wave number defined later. Neglecting the small contribution A1A*1, one can write E · E* = A20 + A0共A1 + A*1兲. 共2兲 The dielectric function of the plasma depends on E · E* and hence can be expressed as ⑀共z,E · E*兲 = ⑀0共z兲 + ⑀2共z兲A0共A1 + A*1兲, where ⑀2 = 冋 ⑀ 共EE*兲 册 共3兲 . EE*=A20 The wave equation for A1, on neglecting 2A1 / z2 共assuming A1 to be slowly varying兲 and A1A*1, reduces to − 2ik A1 2 2 + ⵜ⬜ A1 + 2 ⑀2A20共A1 + A*1兲 = 0, z c 共4兲 2 = ⵜ 2 − 2 / z 2. where ⵜ⬜ To solve Eq. 共4兲, one can express the complex amplitude A1 of the perturbation as A1 = A1r + iA1i , where A1r and A1i are real. Assuming A1r and A1i to be independent of y and proportional to exp i共q储z + q⬜x兲; i.e., A1r ⬀ exp i共q储z + q⬜x兲, A1i ⬀ exp i共q储z + q⬜x兲, 共5兲 and following Ghanshyam and Tripathi,26 one obtains two homogeneous equations in A1r and A1i, which on elimination of these amplitudes lead to the dispersion relation q储 = 冉 − iq⬜ 2 2 2 2 ⑀2A20 − q⬜ 2k c 冊 1/2 共6兲 . For the sake of convenience, the magnetic field of the Earth is assumed to be along the x axis and the electron cyclotron frequency to be much less than . As discussed later, in the ionosphere thermal conduction is important only when the magnetic field lies nearly along the x axis 共otherwise it is negligible兲. The parameter q储 will be imaginary, when 2 ⬍ 2共2/c2兲⑀2A20 . q⬜ 共7兲 In this case the field of the filament, A1 exp iq储z will grow exponentially with z at a space rate ⌫ = iq储 = 冉 q⬜ 2 2 2 ⑀2A20 − q⬜ 2k c2 冊 1/2 . 共8兲 As pointed out earlier, Ghanshyam and Tripathi’s26 analysis of the dielectric function of the plasma determining ⑀2 and ⑀0 is not correct because the axial electron temperature Te0 关their Eq. 共45兲兴 in the absence of the perturbing field E1 involves the coefficient of thermal conductivity and the wave number q of the perturbing field through the modified electron energy transfer parameter ␦⬘ 关their Eq. 共43兲兴; this is obviously not correct because A0 and hence Te0 in the field of the main beam only, do not depend on 共x , y , z兲; hence, thermal conduction should not enter into the expression for Te0. Further, the use of ␦⬘ makes sense only when the strong dependence of the thermal conductivity on the electron temperature is neglected. This analysis also does not take into account the heating of ions by electron-ion collisions. The wave equation for the total field can be separated for A0 and A1. On choosing k= 冑 ⑀0 , c the wave equation for A0 yields a solution A0 = constant; without loss of generality, A0 can be assumed to be real. III. EVALUATION OF ⑀0 AND ⑀2 A. Energy balance Following Gurevich,19 the energy balance for the electrons in the ionosphere in the presence of the electric field E of a high irradiance electromagnetic wave may be expressed as Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 052901-3 Phys. Plasmas 14, 052901 共2007兲 Self-focusing instability in ionospheric plasma… Q+J·E=Q+ e 2N e e * EE 2m2 = Ne 兺 em␦em 冉 冉 Q+J·E=Q+ 3 3 k BT e − k BT 2 2 冊 冊 = Ne 3 3 + Neei¯␦ei kBTe − kBTi − · 共Ke Te兲, 2 2 It may be remembered 共Ref. 19, pp. 212–213兲 that in the ionosphere, the main contribution of the energy flux is made by thermal conduction Ke 共Te兲 and the contribution of thermal force is negligible. In particular, for heat propagation along the magnetic field of the Earth 共x axis兲, the thermal conduction is the dominant process for heights up to 500 km; the expression for the relevant coefficient of heat conduction is also the same as that for an isotropic plasma. The energy balance equation for the ions is 冉 冊 3 3 3 3 Neei¯␦ei kBTe − kBTi = Ne¯im␦im kBTi − kBT . 2 2 2 2 Hence, Ti = ei¯␦eiTe + ¯im␦imT . ei¯␦ei + ¯im␦im ei¯␦ei · im ¯␦ + ei ei im 冎冉 3 3 k BT e − k BT 2 2 冊 共11a兲 In the absence of the wave 共i.e., in undisturbed ionosphere兲 J · E = 0; hence, Eq. 共11a兲 reduces to • The subscript “00” corresponds to the regions away from the beam, • J · E is the time averaged Ohmic loss per unit volume, • vem is the collision frequency of electrons with the mth neutral species, • ␦em is the fraction of excess energy of an electron 共above that of neutral species兲 lost in a collision with the mth neutral species, • veiis the electron collision frequency with ions, • ␦ei is the mean fraction of excess energy of an electron 共above that of an ion兲 lost in a collision with an ion, • the collision and cyclotron frequency of the electrons is much less than the wave frequency , • Ke is the electronic thermal conductivity, • Ne is the number density of electrons, • Te is the electron temperature, • Ti is the ion temperature, and • T is the temperature of neutral species. 冊 兺 em␦em + − · 共Ke Te兲, 共9兲 where Q is the net power per unit volume gained by the electrons from solar radiation, on account of excess energy of electrons, produced by photo-ionization, above the thermal energy 3NekBTe00 / 2. 冉 再 e 2N e e * EE 2m2 Q = Ne00 冉 3 3 kBTe00 − kBT 2 2 冊再 兺 em0␦em0 + 冎 ei0¯␦ei0¯im0 . ei0¯␦ei0 + ¯im0 共11b兲 From Eqs. 共11a兲 and 共11b兲, one obtains ␣EE* = 2000−1 e − + 再冉 再冉 冊冉兺 Te −1 T Te00 −1 T em␦em + 冊冉 冊冉兺 冊冎 Ne00 Ne ei0␦ei0 · im0 ei0␦ei0 + im0 − ei␦ei · im ei␦ei + im em0␦em0 冎 2 共Ke Te兲 , 3kBNeT 冊 共12兲 where ␣ = 2000e2 / 3m2kBT and the factor 2000共⬇M H / m兲 has been introduced for numerical convenience. M H is the mass of a hydrogen atom. B. Distribution of electron density On account of the nonuniform spatial distribution of the irradiance of the instability, there is a corresponding nonuniform distribution of electron and ion temperatures, which creates pressure gradients of electron and ion gas. Initially these gradients push the electrons and ions from regions of higher irradiance and thus temperature 共and hence pressure兲 to regions of lower irradiance; this transport of electrons and ions continues until a steady state is reached; i.e., when these gradients are balanced by the space charge field. With this reasoning in mind and following earlier workers,27 one can write Te00 + Ti00 Ne = . Ne00 Te + Ti 共13兲 This equation is valid 共Ref. 19, p. 234兲 even when the electron-ion force is taken into account. However, the above equation is not quantitatively consistent with the condition of hydrodynamic equilibrium or kinetic theory 共for the simple dependence of e on electron density兲 based on Boltzmann’s transfer equation. 共10兲 C. Electron temperature im is the collision frequency of an ion with a neutral molecules/atoms and ␦im 共⬇1兲 is the fraction of excess energy transferred per ion-neutral species collision. Substituting for Ti from Eq. 共10兲 in Eq. 共9兲 , one obtains The electron-ion collision frequency and electronic thermal conductivity28 共for heat propagation along the direction of the magnetic field兲 may be expressed as ei = ei00共Ne/Ne00兲关Te/Te00兴−3/2 , 共14兲 Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 052901-4 Ke = Phys. Plasmas 14, 052901 共2007兲 Sodha et al. 5NekB2Te , 2me 共15兲 and e = ei + 兺 em . 共16兲 When the magnetic field B makes an angle with the direction of heat propagation 共say, the x axis兲, 1 / e in Eq. 共15兲 is replaced by e / 共2e + B2 sin2兲, where B = eB / mc. Since e B for ionosphere, thermal conduction is important only when B lies almost along the x axis. The temperature dependence of em, ␦em, and im has been tabulated by Gurevich.19 In the presence of the electric field with the perturbation given by Eqs. 共1兲 and 共5兲, the electron temperature can be expressed as 共17兲 Te = Te0 + Te1 exp共iq储z + iq⬜x兲, where Te1 Te0. To proceed further, it is useful to express 共empirically兲 e 关兺em␦em + ei␦ei · im / 共ei␦ei + im兲兴 and Te + Ti as a polynomial in Te / Te00, using Eqs. 共10兲 and 共12兲–共17兲 and the database of Gurevich19 共Tables 1, 2, 6, 8, and 11兲 for an ionospheric height of 200 km. Thus, 冉 D = d1 , where the parameters a0, a1, a2, a3, b0, b1, b2, d0, and d1 can be evaluated, as indicated above. Substituting for Ne / Ne00, Ke, Te, 关兺em␦em + ei␦ei · im / 共ei␦ei + im兲兴, e, and Te + Ti from Eqs. 共13兲, 共15兲, 共17兲, and 共18a兲–共18c兲 in Eq. 共12兲 and equating the terms with and without exp共iq储z + iq⬜x兲 on both sides of the resulting equation, one obtains Te0 = T Te + Ti = 冋 册 Te Te00 T 共2ei␦ei + im兲 + im Te00 Te00 共ei␦ei + im兲 =d+ where a = a0 + a1 B = b1 + 2b2 d = d0 + d1 and 共18c兲 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 A = a1 + 2a2 b = b0 + b1 Te1 D exp共iq储z + iq⬜x兲, Te00 Te0 Te0 + a2 Te00 Te00 2 + a3 Te0 Te0 + 3a3 Te00 Te00 Te0 Te0 + b2 Te00 Te00 Te0 , Te00 Te0 , Te00 2 , 2 , Te0 Te00 Te00 −1 T 冊 冊 Ti00 b + 1 Te00 Te00 冧 冥 b +1 , a 共19兲 共20兲 where 冤 F共q⬜兲 = 2000 − 共18a兲 共18b兲 冉 冉 Te1 ␣A0共A1 + A*1兲 = , T F共q⬜兲 冊 Te1 B exp共iq储z + iq⬜x兲, Te00 ⌳d and ␦ · Te1 A exp共iq储z + iq⬜x兲, 兺 em␦em + ei␦ ei + im = a + Te00 ei ei im e = b + 冤冦 ␣A20 + 2000 ⌳= 冉兺 冦 冉 冊再 冉 冊冉 a T b Te00 ⌳ T b Te00 冉 em0␦em0 + Te00 A B + − T a b 冉 冊冉 Bd Te00 −1 b T Ti00 + 1 Te00 Te00 D− 冊 ei0␦ei0 · im0 ei0␦ei0 + im0 冊 冧 冊 Te0 −1 T + 冊冎 冥 5kBT 2 Te0 q , 3mb2 ⬜ T 共21a兲 共21b兲 and q⬜ q储. D. Dielectric function The dielectric function ⑀ can now be obtained from the equation ⑀ = 1 − 4Nee2/m2 3 , by expressing Ne in terms of the electron temperature Te 关Eq. 共13兲兴 and using Eqs. 共17兲 and 共20兲; Thus ⑀=1− 2p 共Te00 + Ti00兲 2p 共Te00 + Ti00兲D ␣A0共A1 + A*1兲 + 2 . 2 d d2 F共q⬜兲 共22兲 Comparing Eqs. 共3兲 and 共22兲, one obtains ⑀0 = 1 − 2p 共Te00 + Ti00兲 2 d 共23a兲 and ⑀2 = 2p 共Te00 + Ti00兲D ␣ , 2 d2 F共q⬜兲 共23b兲 where p is the plasma frequency in absence of the electromagnetic field. Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 052901-5 Phys. Plasmas 14, 052901 共2007兲 Self-focusing instability in ionospheric plasma… TABLE II. Dependence of b0, b1, and b2 on height 关refer to Eq. 共18b兲兴; mid-latitude daytime ionosphere. IV. GROWTH OF INSTABILITY The condition for the growth is given by Eq. 共7兲, with ⑀2 expressed by Eq. 共23b兲. Substituting for ⑀2 from Eq. 共23b兲 in Eq. 共8兲, one obtains an expression for the growth rate ⌫ of the perturbation in terms of q. Thus, q c ⌫= 冑 2 ⑀0 冉 2 2 2p 共Te00 + D␣A2 Ti00兲 2 0 d F共q兲 − q2 冊 1/2 共24兲 , where q = 共c / 兲q⬜, 冤 F共q兲 = 2000 − 冦 冉 冊再 冉 冊冉 a T b Te00 ⌳ T b Te00 冉 Te00 A B + − T a b 冉 冊冉 Bd Te00 −1 b T Ti00 + 1 Te00 Te00 D− 冊 冊 Te0 −1 T 冊冎 冧 冥 + q 2 Te0 T , d⌫ = 0, dq which yields 2p D␣A20 共T + T 兲 F1 , e00 i00 2 d2 b0 80 141 666.66 977 954.54 −13 257.57 90 100 25 983.33 9 116.66 144 990.15 41 827.42 −2 200.75 −611.36 共25兲 where TABLE I. Dependence of a0, a1, a2, and a3 on height 关refer to Eq. 共18a兲兴; mid-latitude daytime ionosphere. b1 b2 110 3 465.00 9 762.04 −144.31 120 1 949 4 782.40 −72.5 130 150 1 070.83 761.16 1 951.70 606.23 −29.35 −6.40 200 516.73 −16.68 6.31 冤 F1 = 2000 and  = 共10kBT2 / 3mc2兲共1 / b2兲. For the instability to occur, ⌫ ⬎ 0; thus, the critical value of q = 共c / 兲q⬜, below which the instability occurs 共viz., qcritical兲 is given by putting the right-hand side of Eq. 共24兲 equal to zero. For ⌫ to be maximum, the optimum value of q; i.e., qopt is given by 2 关F共qopt兲兴2 = qopt Height 共km兲 − 冦 冉 冊再 冉 冊冉 a T b Te00 ⌳ T b Te00 冉 Te00 A B + − T a b 冉 冊冉 Bd Te00 −1 b T Ti00 + 1 Te00 Te00 D− 冊 冊 Te0 −1 T 冧冥 冊冎 . For qopt, given by Eq. 共25兲, the maximum growth rate, namely, ⌫max is given by 冉 冊再 c qopt ⌫max = 2 冑⑀ 0 1+ 2 2qopt F1 冎 1/2 共26兲 . V. NUMERICAL RESULTS AND DISCUSSIONS All the numerical results obtained by us correspond to a height of 200 km for the mid-latitude daytime ionospheric model of Gurevich;19 the collision frequency and the cyclotron frequency of the electrons are also assumed to be much less than the wave frequency. Computations for other cases can be likewise made by using Tables I–III for the characterization of the electron temperature dependence on the parameters a , b , and d occurring in Eqs. 共18a兲–共18c兲 at different heights. Figure 1 illustrates the dependence of the critical value of the wave number of the disturbance qcritical on the dimensionless irradiance ␣A20 of the main beam. This may be TABLE III. Dependence of d0, and d1 on height 关refer to Eq. 共18c兲兴; midlatitude daytime ionosphere. Height 共km兲 a0 a1 a2 a3 Height 共km兲 d0 d1 80 90 100 110 120 130 150 200 5.8⫻ 103 8.3⫻ 102 2.15⫻ 102 1.1⫻ 102 30.8 9.9 −2.9 0.05 −1156.9 −226.4 −60.6 −40.0 −10.1 −3.2 4.46 0.2 218.9 47.8 13.7 7.87 2.634 1 −1.16 −0.05 −7.6 −2.1 −0.6 −0.35 −0.1 0 0.1 0.067 80 90 100 110 120 130 150 200 180 190 210 270 360 460 670 1100 180 200 240 320 400 500 800 1300 Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 052901-6 Phys. Plasmas 14, 052901 共2007兲 Sodha et al. FIG. 1. Dependence of critical wave number qcritical of a perturbation on dimensionless background irradiance ␣A20 in daytime mid-latitude ionosphere at a height 200 km. readily understood as follows. As q increases, the width of the disturbance 共⬀ the wavelength兲 decreases, diffraction becomes more important, requiring larger magnitudes of nonlinearity, corresponding to larger values of the irradiance ␣A20 of the main beam. Figure 2 illustrates the dependence of the growth rate ⌫ of the self-focusing instability with the wave number q = 共c / 兲q⬜ in the transverse direction for different values of the dimensionless background irradiance ␣A20 = 1, 5, 10, 20, 30, 40, 50, and 100 for an ionospheric height of 200 km in the daytime. It is seen that corresponding to a certain value of the beam irradiance 共␣A20兲, there is an optimum value of q for which the growth rate is maximum; this is evident from a cursory look at Eq. 共24兲. Figure 3 expresses the dependence of the maximum growth rate ⌫max on the background irradiance ␣A20 for the daytime mid-latitude ionospheric height of 200 km. The monotonic dependence is also evident from Eq. 共26兲. It may be reiterated that in the upper ionosphere 共height, greater than 150 km兲, the electron collision frequency is much less than the electron cyclotron frequency due to the FIG. 2. Dependence of the dimensionless growth rate 共c / 兲⌫ of selffocusing instability on q, the dimensionless wave number of the perturbation in the transverse direction corresponding to the dimensionless background irradiance ␣A20 = 1, 5, 10, 20, 30, 40, 50, and 100 in daytime mid-latitude ionosphere at a height 200 km. FIG. 3. Dependence of the dimensionless maximum growth rate 共c / 兲⌫max dimensionless background irradiance ␣A20 in daytime mid-latitude ionosphere at a height 200 km. Earth’s magnetic field. Hence, if the magnetic field is not along the x axis, the relevant thermal conductivity, for conduction along the x axis will be much lower than that in the case considered here and the role of thermal conduction will be at best marginal. ACKNOWLEDGMENT The authors are grateful to Department of Science and Technology, Government of India for financial support. G. A. Askaryan, Sov. Phys. JETP 15, 1088 共1962兲. V. I. Talanov, Sov. Radiophys. 9, 260 共1966兲. 3 A. J. Palmer, Phys. Fluids 14, 2714 共1971兲. 4 P. Kaw, G. Schmidt, and T. Wilcox, Phys. Fluids 16, 1522 共1973兲. 5 C. E. Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 209 共1974兲. 6 M. S. Sodha and V. K. Tripathi, J. Appl. Phys. 48, 1078 共1977兲. 7 M. S. Sodha, R. P. Sharma, K. P. Maheshwari, and S. C. Kaushik, Plasma Phys. 20, 585 共1978兲. 8 W. L. Kruer, U. Ruffna, and E. Westerhof, Comments Plasma Phys. Controlled Fusion 9, 63 共1985兲. 9 R. L. Berger, B. F. Lasinski, T. B. Kaiser, E. A. Williams, A. B. Langdon, and B. I. Cohen, Phys. Fluids B 5, 2243 共1993兲. 10 S. Wilks, P. E. Young, J. Hammer, M. Tabak, and W. L. Kruer, Phys. Rev. Lett. 73, 2994 共1994兲. 11 F. W. Vidal and T. W. Johnston, Phys. Rev. Lett. 77, 1282 共1996兲. 12 A. Bendib, F. Buzid, K. Bendib, and G. Matthieussent, Phys. Plasmas 6, 4008 共1999兲. 13 T. M. George, J. Geophys. Res. 75, 6436 共1970兲. 14 W. F. Utlaut and R. Cohen, Science 174, 245 共1971兲. 15 P. N. Guzdar, P. K. Chaturvedi, K. Papadopoulos, and S. L. Ossakow, J. Geophys. Res. 103, 2231 共1998兲. 16 M. J. Keskinen and S. Basu, Radio Sci. 38, 2906 共2003兲. 17 N. A. Gondarenko, S. L. Ossakow, and G. M. Milikh, J. Geophys. Res. 110, A09304 共2005兲. 18 F. W. Perkins and M. V. Goldman, J. Geophys. Res. 86, 600 共1981兲. 19 A. V. Gurevich, Nonlinear Processes in Ionosphere 共Springer Verlag, Berlin, 1978兲. 20 W. C. Brown, IEEE Spectrum 10, 38 共1973兲. 21 F. W. Perkins and E. J. Valeo, Phys. Rev. Lett. 32, 1234 共1974兲. 22 D. P. Tewari, J. K. Sharma, S. C. Kaushik, and R. P. Sharma, Opt. Acta 24, 1131 共1977兲. 23 A. J. Schmidt, Phys. Fluids 31, 3079 共1988兲. 24 F. Cornolti and M. Lucchesi, Plasma Phys. Controlled Fusion 31, 213 共1989兲. 25 E. N. Epperlein, Phys. Rev. Lett. 65, 2145 共1990兲. 26 X. Ghanshyam and V. K. Tripathi, J. Plasma Phys. 49, 243 共1993兲. 27 M. S. Sodha and A. Sharma, Radio Sci. 41, 4008 共2006兲. 28 F. Seitz, The Modern Theory of Solids 共Tata Mc-Graw Hill, New York, 1940兲. 1 2 Downloaded 09 May 2007 to 202.141.41.67. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp