Parametric instabilities in ionospheric heating experiments

Transcription

1 Parametric instabilities in ionospheric heating experiments Spencer P. Kuo Department of Electrical and Computer Engineering, Polytechnic University 6 MetroTech Center, Brooklyn, NY, skuo@duke.poly.edu Abstract. Parametric instabilities excited in ionospheric heating experiments are studied. The initial processes excited directly by HF heating (pump) waves include parametric decay instabilities, which decay the HF heating wave to a frequency-downshifted Langmuir/upper hybrid sideband together with an ion acoustic/lower hybrid wave as the decay mode, and oscillating two stream instabilities, which decay the HF heating wave to two oppositely propagating Langmuir/upper hybrid sidebands and a purely growing mode/field-aligned density irregularities. These instabilities provide effective channels to convert electromagnetic heating waves to electrostatic plasma waves in the F region of the ionosphere. The instability thresholds, growth rates, angular distribution, and regions of excitation are determined. The high frequency sidebands (Langmuir waves and upper hybrid waves) of the initial parametric instabilities can be driven to large amplitudes. These waves then become new pump waves to excite follow-up parametric instabilities, which provide cascade channels to broaden the spectra of plasma waves and generate short-scale density irregularities. The follow-up parametric instabilities include cascades of Langmuir pump waves into Langmuir sidebands and ion acoustic waves, and the filamentation of those high frequency electrostatic waves to generate field-aligned density irregularities. Again, the thresholds and growth rates of these instability processes are determined. Contents I. Introduction II. Wave Propagation in a Non-uniform Plasma-Swelling Effect 5 A. Solution of the Wave Equation near a Turning Point B. A Specific Example on Wave Propagation in the Bottom-side of the Ionosphere 6 III. Parametric Coupling 7 A. Parametric Decay Instability (PDI) B. Oscillating Two Stream Instability (OTSI) C. Langmuir Cascade D. HFPLs IV. Coupled Mode Equations for Parametric Instabilities in Magneto Plasma 11 V. Initial Parametric Instability Processes Excited by HF Heating Waves Parametric Decay Instabilities (PDI) 14

2 A. Decay to Langmuir and Ion Acoustic Waves B. Decay to Upper Hybrid and Lower Hybrid Waves in High-latitude Upper Hybrid Resonance Region Oscillating two-stream instability (OTSI) 17 A. Excitation of Langmuir Waves together with Purely Growing Density Striations 17 B. Upper Hybrid Waves together with Field-aligned DensityIrregularities VI. Follow-up Parametric Processes excited by Electrostatic Pump Waves 0 1. Cascade of PDI and OTSI-excited Langmuir Waves via Ion Acoustic Waves 0 A. Non-resonant Decay B. Resonant Decay Filamentation instabilities 5 A. Filamentation of Langmuir Waves B. Filamentation of Upper Hybrid Waves VII. Discussion 8 Acknowledgements 9 References 9 I. Introduction Ionospheric heating and modification by powerful HF waves transmitted from the ground have been a very active research area over the past three decades. Early experiments immediately led to the observations of many unexpected phenomena, such as wideband attenuation of the ionosonde signals and artificially induced spread-f that were then realized to be associated with the generation of large and small scale field-aligned density irregularities. These experimental observations together with many supporting theories were reported in the special issues of the Journal of Geophysical Research (1970) and Radio Science (1974), as well as several separate theoretical (Fejer & Kuo 1973; Perkins et al. 1974) and experimental (Carlson et al. 197; Gordon & Carlson 1974; Utlant 1975) articles. Since then ionospheric heating has been recognized to be an ideal approach for experimental and theoretical investigation of the linear and nonlinear properties of the ionospheric plasma and considerable progress toward the understanding of the nonlinear plasma processes has been reported, for instance, in articles by Fejer (1979) on the theoretical side and by Carson and Duncan (1977) on the experimental side, and in a special issue of JATP (198). As powers of HF heating waves continuously increased and diagnostic systems were improved, new physical phenomena were constantly observed. Presently, all the heating facilities are located in the northern atmosphere, where the geomagnetic field is inclined downward. Thus, in most of the ionospheric heating experiments, right-hand circularly polarized high frequency (HF) heating waves with frequencies less than the maximum cutoff frequency of the ionosphere were used to maximize the modification effect on the F region of the ionosphere. Heating waves then converted to the O-mode in the region near the HF reflection height or converted to the Z-mode in the high latitude ionosphere (when the incident angle of the HF heater is close to the Spitze angle: θ s = sin -1 {[Ω e /(ω 0 + Ω e )] 1/ cosθ m },

3 here θ m is the magnetic dip angle) in the region above the HF reflection height. The HF reflection height is above the height of the upper hybrid resonance layer and overlaps with that of the electron plasma resonance layer. Therefore, the heating wave was accessible to the spatial regions, where linear mode conversion processes could occur and many parametric coupling conditions could be matched. Moreover, the heating wave electric field was enhanced by the swelling effect near its reflection height. Collision frequencies of charged particles in the F region of the ionosphere are too low to achieve effective collisional absorption of ground-transmitted HF heating waves. On the other hand, many phenomena signifying significant ionospheric modification by anomalous heating effects were observed. Processes leading to anomalous heating effects had to quickly convert heating waves into electrostatic plasma waves before heating waves were reflected back to the ground. Among those, parametric instabilities (Drake et al. 1974; Kaw et al. 1976; Schmidt 1979) provide the most effective channels to convert heating waves into electrostatic plasma waves of high and low frequencies. In other words, excitation of parametric instabilities is an essential step toward significant F-region ionospheric modification by HF heating waves. Many radar observations on plasma lines and ion lines (Carlson et al. 197; Showen & Kim 1978; Hagfors et al. 1983; Stubbe et al. 199; Westman et al. 1995; Lee et al. 1998, 1999) and ground measurements on stimulated electromagnetic emissions (SEEs) evidence the excitation of parametric instabilities in ionosphere heating experiments. In general, HF heating waves produce an abundance of wave phenomena through parametric instabilities. However, remote sensing by radars cannot explore all of them. For example, backscatter radars cannot detect upper hybrid waves, whose excitation by the HF heating waves in high-latitude region is inferred by the ground measurements on SEEs. They effectively detect those Langmuir waves having wave numbers twice the wave numbers of the probing radar signals and propagating parallel or antiparallel to the pointing directions of radars. Those recorded spectral lines were termed HF enhanced plasma lines (HFPLs). In the mid-latitude region such as at Arecibo, Puerto Rico, parametric instabilities were excited in the region slightly below the reflection height of the O-mode HF heating wave (Carlson et al. 197), where the heating wave became linearly polarized in the geomagnetic field direction. The sidebands excited by both parametric decay instability (PDI) and oscillating two stream instability (OTSI) were Langmuir waves, which were excited most strongly with propagation directions parallel to the geomagnetic field (Fejer & Kuo 1973; Perkins et al. 1974; Kuo et al. 1993; Kuo & Lee 1999). However, HFPLs observed in Arecibo heating experiments represent 40 0 plasma lines, which are the weak components in the spectrum of excited Langmuir waves. In the high latitude heating experiments, PDI and OTSI by the O-mode HF heating wave had to compete with thermal parametric instabilities excited in the upper hybrid resonance region located at a lower height, where the O-mode heating wave was still dominated by the field component of right-hand circular polarization. Sidebands of instabilities were upper hybrid waves propagating nearly perpendicular to the geomagnetic field (Lee & Kuo 1983; Stenflo 1985; Stenflo & Shukla 1988; Stenflo 1991; Huang & Kuo 1994; Zhou et al. 1994; Kuo & Huang 1996; Kuo et al. 1997). Upper hybrid waves were found to play a key role in the generation of stimulated electromagnetic emissions (SEEs), observed in Tromso heating experiments (Thide et al. 198; Stubbe et al. 198, 1984, 1994; Stenflo 1990; Stubbe & Kopka 1990; Kuo 1997). Due to the field-aligned nature of upper hybrid waves, these waves could not be detected directly by EISCAT s UHF and VHF radar and did not contribute to Tromso s 3

4 HFPLs. On the other hand, HFPLs have been detected by EISCAT s 933 MHz (UHF) and 44 MHz (VHF) radar during Tromso heating experiments (Hagfors et al. 1983; Stubbe et al. 1985; Stubbe et al. 199; Westman et al. 1995; Stubbe 1996; Rietveld et al. 000), except that the zero off-set frequency plasma line was strong as monitored with the EISCAT UHF radar, but very weak or absent with the EISCAT VHF radar (Stubbe et al. 199). It suggested that OTSI, which had a higher threshold than that of PDI, were suppressed by instabilities draining heating wave energy in the upper hybrid resonance region. Usually, HFPLs observed in heating experiments do not contain many cascade lines. These radar-detected HFPLs in early heating experiments conducted in Tromso were obtained under the progressive increase from.5 % to 100 % of the maximum available HF heating powers (~ 60 MW ERP in the case of using UHF radar and 10 MW in the case of VHF radar), showing that the number of cascade lines was limited to two, independent of the input power (Stubbe et al. 199; Stubbe 1996). This was also the case as the heating power was increased slightly to 70 MW ERP (Rietveld et al. 000). As the heating power was further increased to 1. GW ERP, up to five cascade lines in HFPLs were then observed at Tromso (Westman et al. 1995). The most intense cascade lines in Tromso s measurements of HFPLs were found to originate from the matching height of the PDI line in HFPLs (Rietveld et al. 000), which is located at an altitude lower for the UHF radar-detected lines than for the VHF radar-detected lines (Stubbe 1996). A major facility for conducting ionospheric heating experiments is also available in Gakona, Alaska, as part of the High Frequency Active Auroral Research Program (HAARP) (Kossey et al., 1999). HAARP HF transmitting system is a phased-array antenna of 180 elements. Each element is a cross dipole, which radiates circularly polarized wave up to 0 kw in the frequency band from 3 MHz to 10 MHz. The antenna gain, which increases with the radiating frequency, varies from 15 db to 30 db. Thus an effective radiated power (ERP) of 90 dbw will be available in heating experiments. A backscatter radar (450 MHz), which is able to explore the spectral features of the HFPLs, has been installed near the heating site. Therefore, it is essential to identify and analyze the parametric instabilities, which are likely to be excited in HAARP heating experiments. In the present work, PDI and OTSI and the subsequent cascades via follow-up parametric instability processes are analyzed systematically to explore the underlying mechanism that impedes the cascade process in Tromso heating experiments, the mechanism that is responsible for the generation of a broad spectrum of frequency-downshifted HFPLs originating from a narrow altitude region in Arecibo heating experiments, as well as other relevant phenomena observed/to be likely observed in heating experiments. Those excited plasma waves will introduce significant perturbations to the background plasma, which, in turn, can further modify the spectral distribution in time. Such a self-consistent analysis (Kuo et al. 1987; 1990) leading to, such as, the plasma line overshoot phenomena observed in HF heating experiments (Showen & Behnke 1978; Showen & Kim 1978) is, however, not included in the present work. Analyses will be based on fluid formulation, thus instabilities involving electron Bernstein waves that require kinetic formulation are not included in the present work. The decay channels considered by Kaysmov et al. (1985) for the cascade of Langmuir waves are also not included in the present work. The other one that is not included in the present work is the electromagnetic filamentation instability (Kuo & Schmidt 1983), which requires full Maxwell s equations used in the formulation. The swelling effect on the electric field of the HF heating wave near its reflection height 4

5 is discussed in section II. In section III the frequency and wavevector matching conditions required in parametric coupling are illustrated graphically. The coupled mode equations for parametric instabilities are formulated in section IV. PDI and OTSI, the initial parametric processes that are excited directly by HF heating waves, are first discussed in section V. In section VI, various follow-up parametric processes that are excited by electrostatic pump waves are studied. Those include 1. Cascade of Langmuir waves through ion acoustic waves,. Cascade of Langmuir waves through lower hybrid waves, 3. Decay of upper hybrid waves to Langmuir sidebands and ion decay modes, and 4. Filamentation of high frequency electrostatic waves to generate field-aligned density irregularities (FAI). The results are discussed in section VII. II. Wave Propagation in a Non-uniform Plasma-Swelling Effect Wave propagation is governed by a wave equation given by d E/dz + k (z) E = 0 (1) In the case of wave propagation from ground upward to the ionosphere, k (z) = [ω 0 - ω p (z)]/c, where ω 0 is the wave frequency, ω p (z) = [4πn e (z)e /m e ] 1/ is the electron plasma frequency, and the electron density n e (z) increases with the altitude; geomagnetic field is not included in the formulation to simplify the analysis and presentation. Therefore, k(z) decreases as wave propagates upward. Considering the situation that ω is less than the maximum plasma frequency of the ionosphere (where is called F-peak), thus, wave can reach a layer at z = z 0, where ω p (z 0 ) = ω 0 and k(z 0 ) = 0. This point z 0 is called turning point, where wave reflection occurs. First, when the wave is not near a turning point (i.e., reflection point) z 0, it can be expressed approximately by the WKB solution A. Solution of the Wave Equation near a Turning Point E = A 0 (k) 1/ exp[±i z k(z) dz] () In the vicinity of k 0, the electron density can be assumed to have a linear increasing profile, i.e., n e (z) = n 0 [1 + (z z 0 )/L], where n 0 = n e (z 0 ) and L is the linear scale length. Hence, ω p (z) = ω 0 [1 + (z z 0 )/L], and k (z) = [ω 0 - ω p (z)]/c = β(z 0 z), where β = ω 0 /Lc. Eq. (1) is then reduced to d E/dz + β(z 0 z) E = 0 (3) Introduce new coordinate g = z 0 z, then (3) becomes d E/dg + βg E = 0 (4) A coordinate transformation, y = (/3)(βg 3 ) 1/, leads to d/dg β 1/ (3/β 1/ ) 1/3 y 1/3 d/dy and d /dg β(3/β 1/ ) /3 y /3 d /dy + (β 1/ /)(β 1/ /3) 1/3 y 1/3 d/dy; and let E = g 1/ F, i.e., E = (3/β 1/ ) 1/3 y 1/3 F, (4) is transformed to a Bessel equation d F/dy + y 1 df/dy + (1 1/9y )F = 0 (5) The solution of (5) is a Bessel function of order 1/3, i.e., 5

6 which lead to Therefore, the solutions of (4) are found to be F = J ±1/3 (y) (6) E ± = A ± g 1/ J ±1/3 ((/3) βg 3 ), E = A + g 1/ J 1/3 ((/3) βg 3 ) + A g 1/ J 1/3 ((/3) βg 3 ) (7) B. A Specific Example on Wave Propagation in the Bottom-side of the Ionosphere Let the turning point at z = z 0 = 0 to simplify the expression, thus g = -z. Applying the continuity condition at z = 0, the solution of (4) in the region around z = 0 is obtained to be E 0 g 1/ [J 1/3 ((/3) βg 3 ) + J 1/3 ((/3) βg 3 )] for z 0 E = (8) E + x 1/ K 1/3 ((/3) βx 3 ) for z > 0 where E + = E 0 [J 1/3 (0) + J -1/3 (0)]/K 1/3 (0). The wave field function E(z) given by (8) is plotted in Fig. 1. As shown, the wave field amplitude near the turning point (i.e., the reflection height) is enhanced considerably by the cutoff effect. This is called swelling effect, which has significant positive effect on exciting instabilities in this region. Fig.1 Airy function distribution of the wave field near its turning point. 6

7 III. Parametric Coupling Plasma can support high frequency EM waves, as well as electrostatic (ES) plasma waves of high and low frequencies as plasma modes that oscillate in plasma as thermal fluctuations in the absence of external sources. In essence, plasma is a nonlinear medium. Therefore, parametric coupling among three modes can occur. When a large amplitude high frequency wave E p (ω 0, k p ) (either EM or ES) appears in plasma, this wave can act as a pump wave to excite plasma modes through parametric couplings. For example, this pump wave electric field can drive a nonlinear current in the electron density perturbation n s (ω s, k s ) of a low frequency plasma mode to produce beat waves E 1 (ω 1, k 1 ) and E 1 (ω 1, k 1 ). Since these are propagating waves, their wavevectors, in addition to the frequencies, also have to be matched in the couplings. In other words, both frequency and wavevector matching conditions: ω 0 = ω 1 + ω s * = ω 1 ω s and k p = k 1 + k s = k 1 k s are imposed in parametric couplings in plasmas. The strength of the coupling depends on the involved nonlinearities and the nature of the induced beat wave. The coupling is strong when the beat wave is resonant with plasma (i.e., a plasma mode). Beat waves, in turn, also couple with the pump wave to introduce a low frequency nonlinear force on electrons, which produces plasma density perturbation having the same frequency and wavevector as n s (ω s, k s ). Hence, this coupling produces a feedback to the original density perturbation n s (ω s, k s ). If the feedback is positive and large enough to overcome linear losses of coupled waves, the coupling becomes unstable and coupled waves grow exponentially in the expense of pump wave energy. This is called parametric instability, by which the pump wave E p (ω 0, k p ) decays to two sidebands E 1 (ω 1, k 1 ) and E 1 (ω 1, k 1 ) through a low frequency decay mode n s (ω s, k s ). This instability process involves the nonlinearity of the plasma and thus is a nonlinear instability. The parametric coupling is imposed by the frequency and wavevector matching conditions as well as a threshold condition on the pump wave field intensity. This process can be reduced to a three-wave coupling process when the decay mode n s (ω s, k s ) has a finite oscillating frequency. In this situation, two sidebands cannot satisfy the same dispersion relation simultaneously. Thus the frequency-upshifted sideband E 1 (ω 1, k 1 ) is off resonant with plasma and can be neglected in the coupling. The most effective parametric instabilities excited directly by the electromagnetic heating wave are 1) Parametric decay instability (PDI) and ) Oscillating two-stream instability (OTSI), in both mid-latitude and high-latitude regions. The sideband(s) in mid-latitude region is Langmuir wave. In high-latitude region the sideband(s) can be upper hybrid wave or Langmuir wave, however, the instabilities involving Langmuir wave as a sideband have to compete with those excited in the lower altitudes and having upper hybrid wave as a sideband. Dipole pump, i.e., its wavevector k 0 = 0, can be assumed. This is because these instabilities are excited in the region near the O-mode reflection height and the sideband(s) and decay mode are electrostatic waves, which have much larger wavenumbers. A. Parametric Decay Instability (PDI) This is a three-wave coupling process represented by EM Pump (ω 0, k 0 =0) Langmuir sideband (ω 1, k 1 ) + Ion acoustic wave (ω s *, ks ) where ks = -k 1. 7

8 B. Oscillating Two Stream Instability (OTSI) This is a four-wave coupling process represented by Langmuir sideband (ω 1, k 1 ) + Purely growing mode (-iγ, k s ) EM Pump (ω 0, 0) Langmuir sideband (ω 1, -k 1 ) Purely growing mode (iγ, k s ) where again ks = -k 1. PDI and OTSI having the upper hybrid wave as a sideband have the similar representations. It is noted that the instability processes usually prefer the excited waves to be plasma modes. The instability threshold can increase considerably if the ω-k relationships of the excited waves are too much off from the dispersion relations of the plasma modes. For example, the threshold of OTSI is higher than that of PDI because purely growing mode is not a plasma mode and the sidebands are also slightly off from the Langmuir mode. How can the frequency and wavevector matching conditions of the two processes be satisfied through the coupled plasma waves is exemplified in Fig., which focuses on the special situation that k 1 (and k S ) is parallel to the geomagnetic field to simplify the plot. In the figure, the dispersion curves of the Langmuir wave (represented by the parabola, which asymptotically approaches to the line labeled by ω = kv te and its mirror image) and ion acoustic wave (represented by the straight line labeled by ω = kc S and its mirror image) are plotted on the ω-k plane, the dipole pump is represented by a point (ω 0, 0) on the vertical axis, and purely growing modes are located on the horizontal axis. Thus each wave (or mode) labeled by its frequency and wavenumber (ω, k) can be represented by a position vector of the corresponding point on the plane. In this representation, the frequency and wavenumber matching conditions are combined into a single vector matching condition. Consider PDI matching first. Starting at the point (ω 0, 0) a downward inclined line parallel to the line ω = kc S is drawn to intersect with the parabola. The intersecting point (ω 1, k 1 ) determines the sideband, which is an electron plasma (Langmuir) mode. The difference of the two vectors (ω 0, 0) and (ω 1, k 1 ) determines a position vector (ω S, k S ). Because this difference vector is parallel to the line ω = kc S, this point (ω S, k S ) is on the line ω = kc S. In other words, (ω S, k S ) is an ion acoustic mode. The vector relationships shown in the figure illustrate how to determine plasma modes to meet the frequency and wavenumber matching conditions in the PDI process. Next consider OTSI matching. Because the purely growing mode is on the horizontal axis, two symmetric vector matching relations involving two different sidebands as shown in the figure can be arranged. Thus OTSI is a four wave coupling process. Because the real frequency of the purely growing mode is zero that is less than the ion acoustic mode frequency, the sidebands real frequency also has to be slightly less than the corresponding Langmuir mode frequency. Using a different point (ω 0, 0) on the vertical axis to represent the pump wave, a horizontal line passing this point is drawn. Thus the sidebands locate at two mirror image points on this line and slightly outside the parabola. The exact locations of the sidebands require a detail analysis of the coupled mode equations. 8

9 Fig. Vector relations showing the frequency and wavevector matching arrangements to identify the plasma modes in the parametric coupling. Because the bottom-side ionosphere has an increasing plasma density profile, the parabola in Fig. can move up and down, with respect to a fix pump wave point (ω 0, 0), to represent different interaction layers. In both processes, the central point (ω p, 0) of the parabola has to be below the point (ω 0, 0). However, the point (ω p, 0) in OTSI is closer to the point (ω 0, 0) than that in PDI. The wavenumbers of the points determined by the vector matching condition vary with ω p. In heating experiments, the wavenumbers of HF enhanced plasma lines and ion lines (HFPLs and HFILs) are fixed by the wavenumber of backscatter radar. Thus the HFPLs and HFILs contributed by PDI and OTSI are originating from narrow regions below the O-mode reflection height. Langmuir waves excited by OTSI and PDI in the region below the HF reflection height can become pump waves to excite new parametric instabilities, which generate frequencydownshifted Langmuir waves to be their sidebands. This is called Langmuir cascade. Continuous cascade of Langmuir waves through new parametric instabilities broadens the downshifted frequency spectrum of Langmuir waves. Similar description is also applicable to the upper hybrid cascade. The permissible number of cascade and the required pump threshold field vary with each cascade process distinct by the low frequency decay mode. In the following, a Langmuir cascade process that involves an ion acoustic wave as the decay mode is discussed. C. Langmuir Cascade This three-wave coupling process is represented by Langmuir Pump (ω 1,k 1 ) Langmuir sideband (ω,k ) + Ion acoustic wave (ω s *,ks ) where ks = k 1 k k 1. 9

10 We now use Fig. 3 to explain how to determine the plasma modes in the cascade process. Starting at a point (ω 1, k 1 ) on the RHS of the parabola, which represents a Langmuir wave excited by PDI, a downward inclined line parallel to the line ω = kc S is drawn, the same procedure as that used in describing PDI. The intersecting point (ω, k ) on the LHS of the parabola determines the sideband. The vector between the two points (ω 1, k 1 ) and (ω, k ) can then be mapped on the ω = kc S line to identify the decay mode. The matching conditions are satisfied as indicated by the vector relations shown in the figure. Similar procedure starting at the point (ω, k ) can be applied to determine the plasma modes in the subsequent cascade. However, one can easily find that k 3 will be smaller than k (likewise, k < k 1 ). On the other hand, HFPLs have a fixed k value. Therefore, the parabola has to move down slightly in each subsequent cascade to keep the sideband to have a fixed k. In other words, the cascade lines in the HFPLs contributed by this cascade process are originating from a relatively thick layer. In the frequency spectrum of HFPLs, the first spectral peak having the highest frequency at ω = ω 1 is an OTSI line if ω 1 = ω 0, the heating wave frequency; and it is a PDI line if ω 1 is downshifted from ω 0 by ω = ω 0 ω 1 = ω S0 = k R C s, where k R is the wavenumber of the probing backscatter radar signal; the subsequent spectral peaks at ω, ω 3, then correspond to the first, second, cascade lines. The cascade lines are recognized by doubling their frequency downshift from the preceding line to ω S0. For example, if a spectrogram of HFPLs contains 7 spectral peaks starting at ω = ω 0, then the first two spectral peaks at ω = ω 0 and ω 0 - ω S0 are called OTSI and PDI line, respectively. The remaining 5 spectral peaks at ω 0 (n +1)ω S0, n = 1,, 5 are called cascade lines and are attributed to the PDI process. Therefore, the spectral width of HFPLs is about 11 ω S0. Fig. 3 Vector matching arrangement on the dispersion plane. 10

11 D. HFPLs In general, HF heating waves produce an abundance of wave phenomena through parametric instabilities. However, remote sensing by radars cannot explore all of them. For example, backscatter radars cannot detect upper hybrid waves, whose excitation by the HF heating waves in high-latitude region is inferred by the ground measurements on stimulated electromagnetic emissions (SEEs). Radar detection is based on incoherent and coherent scatterings from the target. In probing background plasma, return signals come from incoherent scatterings. On the other hand, plasma waves can scatter radar signal coherently with much larger scattering cross sections, which make the return signals to have large signal to noise ratios. However, the coherent scattering has to satisfy the Bragg scattering condition, i.e., the frequencies and wavevectors of the scattered waves (ω B, k B ), plasma waves (ω L, k L ), and incident radar wave (ω R, k R ) are related by the relations ω B = ω R ω L and k B = k R k L. In general, ω R >> ω L, thus k B k R = ω R /c. Therefore, in the backscattering case, k B k R and k L! k R. In other words, radar can only effectively detect those Langmuir waves having wavenumbers twice the wavenumber of the probing radar signal and propagating parallel (upgoing) or anti-parallel (down-going) to the pointing direction of radar. These recorded spectral lines are termed HF enhanced plasma lines (HFPLs). The up-going plasma wave scatters the radar signal to produce a frequency-downshifted radar return (ω B, k B ), and the frequencyupshifted radar return (ω B+, k B+ ) is produced by scattering the radar wave over the down-going plasma wave. In general, the spectrogram of HFPLs records the spectrum of the radar return signals at frequencies offset by ω R ω 0. Therefore, the spectral lines of up-going plasma waves will appear in the spectrogram as cascading to the right hand side (positive frequency side) with increasing frequencies. Occasionally, spectral lines appearing in the negative frequency side are also recorded. These lines correspond to up-going frequency up-shifted plasma waves. Their generation mechanisms can be quite different from those of frequency downshifted plasma waves discussed in the preceding paragraphs. On the other hand, the spectral lines of downgoing plasma waves will appear in the spectrogram as cascading to the left hand side (negative frequency side) with decreasing frequencies. Again, if spectral lines appear in the positive frequency side, these lines correspond to down-going frequency up-shifted plasma waves. IV. Coupled Mode Equations for Parametric Instabilities in Magneto Plasma Parametric excitation of Langmuir/upper hybrid waves φ(ω, k) and low-frequency plasma waves n s (ω s, k s ) by electromagnetic or Langmuir/upper hybrid pump waves E p (ω 0, k p ) are considered in the following, where E p, φ, and n s denote electric field of a pump wave, electrostatic potential of a Langmuir/upper hybrid sideband, and density perturbation of a low frequency decay mode, respectively. Langmuir waves can have large oblique propagation angles (with respect to the background magnetic field B 0 = ^z B 0 ), upper hybrid waves have near 90 0 propagation angles, and low-frequency plasma waves include ion acoustic waves, purely growing modes, and lower hybrid waves. The coupled mode equation for the Langmuir/upper hybrid sideband is derived from the electron continuity and momentum equations, and Poisson s equation t n e + n e v e = 0, (9) 11

12 ( t + ν e ) n e v e + Ω e n e v e ^z = n e v e v e 3v te δn e (e/m e )n e E, (10) φ = 4πeδn e, (11) where n e = n 0 + δn e + n s ; n 0 and δn e are the unperturbed plasma density and electron density perturbation associated with Langmuir/upper hybrid waves, respectively; Ω e = eb 0 /m e c the electron cyclotron frequency; v te = (T e /m) 1/ the electron thermal speed; E = E P + E L and E L = φ; and the adiabatic relationship P e = 3T e δn e is used; ν e = [(ν en + ν ei ) + ν el ] 1/ is the effective electron collision frequency including electron-neutral elastic collision frequency (ν en ), electron-ion Coulomb collision frequency [ν ei =.63 (n 0 /T e 3/ ) lnλ 39.5(n 0 /T e 3/ ) (f p /T e 3/ ), here lnλ 15 is assumed; n 0 is in cm -3, T e is in K, and f p is the electron plasma frequency) and a phenomenological term of ν el = (π/) 1/ (ω 0 ω p /k z k v te 3 )exp( ω 0 /k z v te ) to account for the electron Landau damping effect. With the aid of (11) and the following two orthogonal components of (10), ( t + ν e ) n e v e ^z = Ω e n e v e n e v e v e ^z [3v te δn e + (e/m e )n e E] ^z and (1) ( t + ν e ) n e v ez = n e v e v ez [3v te z δn e + (e/m e )n e E z ], the three orthogonal components of (10) are combined into a single scalar equation ( t + ν e )[( t + ν e ) + Ω e ] n e v e = [( t + ν e ) + Ω e ][3v te δn e ω p δn e ] + Ω e [3v te δn e (e/m e )n 0 φ] [Ω e z n e v e v ez + Ω e ( t + ν e ) ^z ( n e v e v e ) + ( t + ν e ) ( n e v e v e )] (e/m e ){[( t + ν e ) + Ω e z ] (n s E P ) Ω e ( t + ν e ) ^z ( n s E P )}, (13) where ω p = (4πn 0 e /m e ) 1/ is the electron plasma frequency. The terms on the right hand side (RHS) of (13) are assembled into four groups of terms. The first two groups contain linear response terms and the last two contain coupling terms. The contribution to the parametric coupling from the third group of terms is much smaller than that from the fourth group and hence, the coupling terms in the third group will be neglected. Using (9), (11), and (13), the coupled mode equation for the Langmuir (or upper hybrid) sideband is then derived to be (Kuo et al. 1983) {[( t + ν e ) + Ω e ]( t + ν e t + ω p 3v te ) Ω e (ω p 3v te ) }φ = ω p {[( t + ν e ) + Ω e z ] E p n s */n 0 Ω e ( t + ν e )^z (n s */n 0 ) E P }, (14) where stands for a filter, which keeps only terms having the same phase function as that of the function φ on the left hand side. (14) is derived from the fluid equations, which does not include the kinetic effects. Hence, one of the lowest order kinetic effects (Dysthe et al. 1984) is included in (14) by adding a phenomenological term in ν e to account for the electron Landau damping. Both electrons and ions can effectively respond to low frequency wave fields. Hence, the formulation of the coupled mode equation needs to include both electron and ion fluid equations. Since electrons and ions tend to move together, the formulation can be simplified by introducing the quasi-neutral condition: n si n se = n s. The ion fluid equations are similar to (9) and (10), except that the subscript e is changed to i, and the charge e changed to e. Moreover, the collision terms ν e v e and ν i v i are replaced by ν ei (v e v i ) and ν ie (v i v e ) + ν in v i in the electron and 1

13 ion fluid equations, respectively, where ν in is the ion-neutral collision frequency. The ion Landau damping rate ν il / (π/8) 1/ (ω s /k z v s )[(m/m) 1/ + (T e /T i ) 3/ exp( ω s /k z v ti )] of the ion acoustic wave will be included phenomenologically in the coupled mode equation only for the ion acoustic wave by replacing ν in = (ν in + ν il ) 1/ = ν i, where v s = (T e /M) 1/. Also included in the formulation is the electron thermal energy equation (Braginskii, 1965) t T e + (T e0 /3) v e = (/3n e ) (κ z z + κ )T e ν e (m/m)(t e T e0 ) + ν e m v e /3, (15) where κ z = 3n 0 T e0 /mν e, κ = (ν e /Ω e ) κ z, and T e0 is the unperturbed electron temperature. Using the same procedure as that outlined in the early work (Kuo 1996), the coupled mode equation for the low frequency mode in the collisional case is derived to be t 3 {( t + ν e )[ t ( t + ν i ) C s ] + Ω e Ω i t } + Ω e {( t + Ω i )[ t ( t + ν i ) C s ] + Ω i C s } z (n s /n 0 ) = (m/m)[( t + Ω i ) z + t ][ t ( t + ν e ) (a p + δt e /m) + Ω e ( z a pz t J B /n 0 ) Ω e t a p ^z ], (16) where Ω i the ion cyclotron frequency, C s = [(T e + 3T i )/M] 1/ the ion acoustic speed, v ti = (T i /M) 1/ is the electron thermal speed, and M the ion (O + ) mass; the coupling terms a p = v e v e and J B = n e v e arise from plasma nonlinearities and can be expressed explicitly by using only the linear part of electron velocity and density responses to high frequency wave fields; and δt e = T e T e0 is the result of the differential Ohmic heating, which is significant only for the field-aligned purely-growing modes and can be evaluated from (15). Both (14) and (16) will be simplified for each parametric coupling process to be studied in the following sections. Moreover, in the initial processes, k p 0 will be assumed for dipole pump fields. For follow-up processes, pumps are electrostatic waves and E p = φ p will be employed. In the analyses, the spatial and temporal variation of physical functions in the form of p = p exp[i(κ r ϖt)] will be assumed to reduce the coupled differential equations (14) and (16) to coupled algebraic equations leading to the dispersion relation for each parametric instability, where κ and ϖ are the appropriate wavevector and frequency of each physical quantity. The results of analyses will be applied to understand observations on HFPLs measured by Arecibo, EISCAT and HAARP backscatter radars. The relevant parametric values for heating experiments conducted at Arecibo, Puerto Rico, Tromso, Norway, and Gakona, Alaska, respectively, will be used. In Arecibo heating experiments, the parameters are: ω 0 /π = 5.1 MHz, Ω e /π = 1.06 MHz, T e = T i = 1000 K, v te m/s, v ti = m/s, C s m/s, and k π (i.e., λ 0 = m = λ R /sinθ m, where λ R = 0.7 m is the wavelength of the 430MHz radar signal and θ m = 50 0 is the magnetic dip angle); ν in = 0.5 s 1 and ν il ~ (k s /k R ) s 1, where k s and k R are the wavenumbers of ion acoustic wave and radar signal; ν el << ν en < ν ei, and ν e ν en + ν ei 500 s 1. In Tromso heating experiments, the parameters are: ω 0 /π = 5.43 and 6.77 MHz, Ω e /π = 1.35 MHz, T e = 1500 K, T i = 1000 K, ν in = 0.8/0.5 s 1, v te m/s, v ti = m/s, C s m/s. In the case of 933 MHz radar, k 1 = 1.17π (i.e., λ 1 = m), ν il1 ~ (k s1 /k R1 ) s 1 ; in the case of 4 MHz radar, k =.9π (i.e., λ = m), ν il ~ (k s /k R ) s 1 ; electron Landau damping rate can be neglected in both cases of heater 13

14 frequencies, thus ν e = ν en + ν ei ~ 600 s -1. In HAARP heating experiments, the parameters are: ω 0 /π = 5 MHz, Ω e /π = 1.4 MHz, T e = 1500 K, T i = 1000 K, v te m/s, v ti = m/s, C s m/s, and k π (i.e., λ 0 = m = λ R /sinθ d, where λ R = 0.67 m is the wavelength of the 450MHz radar signal and θ m = is the magnetic dip angle); ν in = 0.5 s 1 and ν il ~ (k s /k R ) s 1. Again, ν el << ν en < ν ei, and ν e ν en + ν ei 600 s 1. V. Initial Parametric Instability Processes Excited by HF Heating Waves 1. Parametric decay instabilities (PDI): An HF heating wave of right-hand circular polarization propagates toward its reflection height, its wavevector and group velocity decrease gradually to zero. The accumulation of its energy flux leads to significant enhancement in the wave electric field. The wave polarization is also changed to the O-mode polarization in that region. Thus the HF heating wave electric field exceeds the threshold fields of different parametric instabilities. In the high-latitude region, right-hand circularly polarized heating waves can also excite parametric instabilities in the upper hybrid resonance region before heating waves reach reflection heights and convert to the O- mode polarization. Among them PDI is a favorable one excited by the heating wave, as will be shown in the following and as also evidenced by experimental results. It is a process effectively converting the transmitted electromagnetic wave energy into plasma wave energy. Consider the decay of a dipole pump E p (ω 0, k p =0) into a Langmuir/upper hybrid sideband φ(ω, k) and an ion acoustic/lower hybrid decay mode n s (ω s, k s ), where E p (= ^z E p near the reflection height and = (^x i^y )E p in high-latitude upper hybrid resonance region), φ, and n s denote pump wave field, sideband s electrostatic potentials, and ion acoustic/lower hybrid mode s density perturbation, respectively; k = ^z k z + ^x k ; frequency and wavevector matching conditions lead to ω = ω 0 ω s * and k s = k. A. Decay to Langmuir and ion acoustic waves: The coupled mode equation (14) is then reduced to {[( t + ν e ) + Ω e ]( t + ν e t + ω p 3v te ) Ω e (ω p 3v te ) }φ = ω p [( t + ν e ) + Ω e ] z E p n s */n 0, (17) for the Langmuir sideband. As the oblique propagation angle θ (with respect to the magnetic field) of the Langmuir sideband is not close to 90 0, the ion acoustic decay mode is mainly driven by the parallel (to the magnetic field) component of the ponderomotive force induced by high frequency wave fields. Moreover, t << Ω e z, thus the coupled mode equation (16) for the ion acoustic mode is reduced to {( t + Ω i )[ t ( t + ν i ) C s ] + Ω i C s } z (n s /n 0 ) = (m/m) [( t + Ω i ) z + t ] z a pz, (18) where a pz = z v ez / for the present case that E p = ^z E p. 14

15 Let E p = E p exp( iω 0 t) + c.c. and the spatial and temporal variation of perturbations have the form of p = p exp[i(κ r ϖt)], where κ and ϖ are the appropriate wavevector and frequency of each perturbation, reduces (17) and (18), respectively, to and [ω(ω + iν e ) ω kθ ]φ = i(k z /k )ω p E p (n s */n 0 ) (19) [ω s (ω s + iν i ) k C s ](n s /n 0 ) = i(k z k ω p /4πn 0 Mω 0 ω)e p φ*, (0) where ω kθ = ω p + 3k v te + Ω e sin θ and sin θ = k /k. Eqs. (19) and (0) are combined to obtain the dispersion relation [ω(ω + iν e ) ω kθ ][ω s *(ω s * iν i ) k C s ] = (k z ω p 4 /4πn 0 Mω 0 ω) E p, (1) We now set ω = ω r + iγ k and ω s = ω sr + iγ k in (1) to evaluate the threshold field ε pth = E pth (k, θ) and growth rate γ k (θ) of the instability excited at an arbitrary height h 1, where ω p (h 1 ) = ω r 3k 1 v te Ω e sin θ 1 is the matching height of the (k 1, θ 1 ) Langmuir wave. Thus in the general case that when the sideband and decay wave of the instability are driven waves, rather than eigen modes of plasma, the threshold field and growth rate of the instability are obtained to be () and E pth (k,θ; k 1,θ 1 ) = (1 + ω 1 4 /ω 0 ν e ) 1/ (mm/e ) 1/ (ν e ν i ω sr ω 0 3 ) 1/ /kcosθω p γ k [(ν e ν i /4)(E p /E pth ) + (ν e ν i ) /16] 1/ (ν e + ν i )/4, (3) where ω 1 = ω kθ ω r = 3(k k 1 )v te + Ω e (sin θ sin θ 1 ); ω sr = k C s ω sr ν i ω 1 /ω r ν e. It is shown by () that the threshold field varies with the propagation angle θ and wavelength λ 1 of the Langmuir sideband as well as the location of excitation. When the instability is excited at the matching height h of its Langmuir sideband (k, θ), i.e., ω 1 = 0, the threshold field is the minimum given by ε pthm = E pth (k,θ) m = (mm/e ) 1/ (ν e ν i ω sr ω 0 3 ) 1/ /kcosθω p. (4) The parametric values given in section IV for heating experiments are used to evaluate the threshold field of PDI, which generates HFPL at its matching height. However, ion acoustic wave is heavily damped by the ion Landau damping process, thus ν i is governed by the ion Landau damping rate, which is proportional to ω sr. Therefore, E pth (k,θ) m is independent of k, namely, independent of the radar frequency. In Arecibo heating experiments, ν i ~ Hz (k s = k = k R ), the minimum threshold field ε pthm = E pth (k 1 =5.71π,θ 1 =40 0 ) m of PDI for generating the HFPL is about V/m. In Tromso heating experiments, ν i Hz and ν i Hz correspond to the ion Landau damping rates of the ion acoustic waves having twice the wavenumbers of 933 MHz radar and 4 MHz radar, respectively. The minimum threshold fields evaluated from (4) are E pth (k 11 =1.44π,θ 1 =1 0 ) m = E pth (k 1 =.99π,θ 1 =1 0 ) m = 0.7/0.3 V/m for f 0 = 5.43 and 6.77 MHz, respectively. In HAARP heating experiments, ν i ~ Hz, the minimum threshold field E pth (k 1 =6π,θ 1 = ) m of PDI for generating the HFPL is about 0.4V/m. In the same region, the nonresonant (k, θ) sideband can also be excited. Likewise, HFPLs can also be excited in the region outside of their matching heights. However, the threshold field of the instability is increased by a mismatch factor (1 + ω 1 4 /ω 0 ν e ) 1/ ( k 1 cosθ 1 / kcosθ) and 15

16 consequently, the growth rate of the instability decreases as the mismatch frequency ω 1 of the sideband increases. HFPLs excited by the PDI process do not have the lowest threshold field and highest growth rate. They are most favorably excited in their matching height, which is below the reflection height of the O-mode heating wave by a distance d L(1k R v te + Ω e sin θ 0 )/ω 0, where L is the linear scale length of the background plasma, k R is the wavenumber of the backscatter radar signal, and θ 0 is the conjugate angle to the magnetic dip angle θ m. As the Langmuir waves excited by PDI grow to large amplitudes, they become pumps of follow-up parametric instabilities, which broaden the spectral and angular distribution as well as the frequency bandwidth of Langmuir waves. B. Decay to upper hybrid and lower hybrid waves in high-latitude upper hybrid resonance region: For the upper hybrid sideband, the coupled mode equation (14) is reduced to [( t + ν e )( t + ν e t + ω p 3v te ) + Ω e t ]φ = ω p [( t + ν e ) E p n s */n 0 Ω e (n s */n 0 )E P ^z ] = ω p ( t + ν e + iω e )( x i y ) E p n s */n 0. (5) For the lower hybrid decay mode, Ω i << ω s << Ω e, and t and Ω e z are in the same order of magnitude. Thus the coupled mode equation (16) is reduced to t { ( t + ν e )[ t ( t + ν i ) C s ] + Ω e Ω i ( + M z /m) t }(n s /n 0 ) = (m/m) [( t + ν e ) t a p + Ω e z a pz Ω e t J B /n 0 Ω e t a p ^z ], (6) where Ω e << ω p has been assumed. It is found that the third and fourth coupling terms on the RHS of (6) cancel each other. This leaves the first two terms on the RHS of (6), attributed to the transverse convective force and parallel ponderomotive force, to be the dominant coupling terms. Using the similar approach as that used in part A and the condition k z << k, (5) and (6) are reduced to and [ Γ + iν e ω( ω k /ω )]φ = i[ω p (ω + iν e Ω e )/ωk]e p (n s */n 0 ) (7) [ω s + i( ξ 1 )ν e ω s ω Lks ](n s /n 0 ) = i[k 3 ω p ω/4πn 0 Mω u (ω 0 + Ω e )]E p φ*, (8) where Γ = ω uk ω, ω uk = ω k + Ω e + ν e, ω k = ω p + 3k v te, and ω u = ω p + Ω e ; ω Lks = ω LH ξ + k C s, ξ = 1 + (M/m)(k z /k ), and ω LH = ω pi /(1 + ω p /Ω e ) Ω e Ω i. k 3x k is substituted. Equations (7) and (8) are combined to obtain the dispersion relation (Kuo 199) [ Γ + iν e ω(1 + Ω e /ω uk )][ω s i( ξ 1 )ν e ω s ω Lks ] [k ω p 4 (ω 0 Ω e )/4πn 0 Mω u (ω 0 + Ω e )] E p. (9) We now set ω = ω uk + iγ k and ω s = ω Lks + iγ k in (9) and evaluate the threshold field ε pth = E pth and the growth rate γ k of the instability excited in upper hybrid resonance region. Thus (9) reduces to [γ k +(1 1/ξ)ν e ][γ k +(1+Ω e /ω uk )ν e /] (e/m) [k (ω 0 Ω e )ω Lks /ξω 0 Ω e (ω 0 +Ω e )] E p (30) 16

17 and E pth and γ k of the instability are determined to be and E pth (m/e)(ξ 1) 1/ (1 + Ω e /ω 0 ) 1/ (ν e Ω e / k )[ω 0 (ω 0 + Ω e )/ω Lks (ω 0 Ω e )] 1/ (31) γ k = (ν e / )[( E p /E pth + 1/8) 1/ 3 /4]. (3) Using the parameters for Tromso/Haarp heating experiments, the threshold field (31), which turns out to be not very sensitive to ω 0, can be approximately expressed for all cases as ε pth = E pth.16ξ 1/4 /k V/m. (33) Thus the threshold field is very small in exciting short wavelength waves. For example, ε pth 0.3 V/m for ξ = 10 and k = 4π. Since, in general, E p /E pth >> 1, the growth time of the instability is less than 1 ms.. Oscillating two-stream instability (OTSI): HF heating waves can also excite OTSI. In the following we show that due to the geomagnetic field OTSI can be excited in a sizable region below the HF reflection height. This is in contrast to the unmagnetized case that OTSI can only be excited in a narrow region near the HF reflection height. The wavenumber spectrum of Langmuir sidebands excited in each height region, again, has an angular distribution centered at an oblique propagation angle (with respect to the magnetic field) of a Langmuir eigenmode at that height. Moreover, in the high latitude region OTSI can also occur in the upper hybrid resonance layer, which is well below the HF reflection height, to excite upper hybrid sidebands together with short-scale field-aligned density irregularities. A. Excitation of Langmuir waves together with purely growing density striations: OTSI process involves the decay of a dipole pump E p (ω 0, k p =0) into two Langmuir sidebands φ 1 (ω 1, k 1 ) and φ 1 (ω 1, k 1 ) and a purely growing mode n s (ω s =iγ s, k s ), where E p = ^z E p, φ 1 and φ 1, and n s are pump wave field, sidebands electrostatic potentials, and purely growing mode s density perturbation, respectively; γ s is the growth rate of the instability and k 1 = ^z k 0 + ^x k ; frequency and wavevector matching conditions lead to ω 1 = ω 1 = ω 0 + iγ s and k 1 = k s = k 1. From (14), the coupled mode equations for Langmuir sidebands are {[( t + ν e ) + Ω e ]( t + ν e t + ω p 3v te ) Ω e (ω p 3v te ) }φ 1± = ω p [( t + ν e ) + Ω e ] z E p n s± /n 0, (34) where the notations φ 1+ = φ 1, φ 1 = φ 1, and n s+ * = n s = n s are used. The parallel (to the magnetic field) component of the wavevector of the short scale purely growing mode is not negligibly small, thus the short scale purely growing mode, similar to the ion acoustic mode, is also mainly driven by the parallel component of the ponderomotive force induced by the high frequency wave fields. Therefore, the coupled mode equation for the purely growing mode has the same form as (19) Equations (34) and (19) are analyzed in the same way as that for (18) and (19). The dispersion relation of OTSI is then derived to be {(γ s + Ω i )[γ s (γ s + ν i ) + k 1 C s ] Ω i k C s } 17

18 = (e /mm)k 1 cos θ(γ s + Ω i cos θ){ ω /[ ω 4 + ω 0 (γ s + ν e ) ]} E p, (35) where ω = ω p + 3k 1 v te + Ω e sin θ ω 0, and θ = sin 1 (k /k 1 ). We first set γ s = 0 in (35) to determine the threshold condition of the instability. The threshold field is obtained to be ε pth = E p (θ) th = (mm/e ) 1/ C s [( ω 4 + ω 0 ν e )/ ω ] 1/ /cosθ. (36) Similar to () for PDI, (36) shows that the threshold field of OTSI also varies with the propagation angle θ and wavelength λ 1 of the Langmuir sidebands as well as the location of excitation. For each propagation angle θ and wavelength λ 1, the instability has the minimum threshold field ε pthm = E p (k 1, θ) m = (mm/e ) 1/ C s (ω 0 ν e ) 1/ /cosθ (37) when it is excited in a preferential height layer with ω (k 1,θ) = ω 0 ν e, i.e., ω p (h) = ω p (k 1,θ) = ω 0 (ω 0 + ν e ) 3k 1 v te Ω e sin θ, where h is altitude of the preferential layer. In other words, the spectral lines of the Langmuir sidebands excited by OTSI have an angular (θ) and a spectral (k 1 ) distribution, as well as a spatial (h) distribution in a finite altitude region. This minimum threshold field (37) increases with the oblique propagation angle θ of (k 1, θ) lines, but it is independent of k 1. The altitude h of the preferentially excited region for (k 1, θ) lines moves downward as the oblique propagation angle θ of these lines increases. The maximum growth rate γ sm (k 1,θ) of the instability and its excitation region ω (k 1,θ) are determined by taking partial derivative of (35) on ω and setting γ s / ω = 0 in the resultant. It leads to ω 3 (k 1,θ) = ω 0 (γ sm + ν e ), and γ sm + γ sm k 1 C s (ν e k 1 C s /)( E p / E p (k 1,θ) m ) 0, where γ sm >> Ω i, ν e are assumed (i.e. E p / E p (k 1,θ) m >> 1). This quadratic equation for γ sm has a real solution γ sm = (G + H) 1/3 (G H) 1/3, (38) where G = [(k 1 C s /3) 3 + H ] 1/ and H = (ν e k 1 C s /4)( E p / E p (k 1,θ) m ). Then (38) can be simplified in two pump power regimes. In the moderate heating power regime that E p / E p (k 1,θ) m < k 1 C s /ν e (i.e., γ sm < k 1 C s ), (38) reduces to γ sm ~ (ν e /)( E p / E p (k 1,θ) m ). On the other hand, it reduces to γ sm ~ (H) 1/3 in the strong power regime that E p / E p (k 1,θ) m > k 1 C s /ν e (i.e., γ sm > k 1 C s ). It is noted that the altitude h 1 of the maximum growth rate layer (i.e., ω p (h 1 ) = ω p (h 1 ) + ω 0 γ sm ) is slightly higher than the altitude h 1 of the minimum threshold layer (i.e., h 1 > h 1 ). The height difference is given by h = h h γ sm L/ω 0, where L is the linear scale length of the plasma density distribution. It shows that the height of the preferential (maximum growth rate) layer of the instability tends to shift upward from the matching height of the instability sidebands as the heating power (i.e., γ sm ) increases. However, in the moderate power regime this shift is negligibly small. Considering a family of spectral lines having k 0 = ^z k 0 as the common parallel component of their wave vectors, the oblique angle θ of (k 1, θ) lines increases with k (i.e., k 1 = k 0 + k ). The threshold field for the (k 1, θ) lines excited at the matching height h 10 of the (k 10, θ 0 ) lines (i.e., ω p (h 10 ) = ω p (k 10, θ 0 )) can be expressed in terms of the minimum threshold field E p (k 0, 0) m of the (k 0, 0) line as E p (k 1, θ) th = f 1 (θ,θ 0 ) E p (k 0, 0) m, where E p (k 0, 0) m = (mm/e ) 1/ C s (ω 0 ν e ) 1/, f 1 (θ,θ 0 ) = (cosθ) 1 [1 + ( ω m ω m0 ) /ω 0 ν e (ω 0 ν e + ω m ω m0 )] 1/, (39) 18