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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[1] A specially-designed sounding rocket experiment with a bare, conducting tether was recently suggested to test an idea that tether-produced energetic-electron beams can be used for remote sensing the neutral density in the E region. This letter contains the theoretical analysis of collisionless beam-plasma interactions (BPI) that complement direct impact of the energetic electrons with neutral particles. Collisionless effects are shown to play a significant and even major role in tether-induced aurora (TA). In the F region, BPI lead to appreciable green-line (557.7 nm) emissions at ∼200 km. Farther downward, BPI develop inside a local minimum in the plasma density between the E and F regions, the so-called valley. Here a thin layer of a strongly-elevated electron temperature and airglow is predicted. Neutralizing electric currents carried by ionospheric electrons can become unstable in the low-density valley. As a result, developing plasma waves inhibit the currents. In the extreme case, the beam might be locked (a ‘virtual cathode’). In addition to optical observations, these effects can also be observed by radio frequency methods.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[2] Martinez-Sanchez and Sanmartin [1997] (hereafter MSS97) showed that ion bombardment of a bare, conducting tether in the ionosphere and acceleration in the tether potential would produce energetic electron beams. In the mid-latitude F region (magnetic dip angles χ ∼ 40°–50° and the plasma density nc ≃ 106 cm−3), the tether of a length 20 km would yield electron fluxes qb ≃ 1010 cm−2s−1 and energies ɛb ≃ 3 keV. As in natural aurorae, collisions of the energetic electrons with neutral particles should produce airglow in the E region, i.e., tether-induced aurora (TA). Since the neutral density can be deduced from the auroral volume emission rates [e.g., Rees, 1989], MSS97 suggested that tether-produced electron beams can be used for remote sensing the neutral density in the E region.

[3] A specially-designed sounding rocket experiment with a bare, conducting tether of ∼1 km length and −3 kV potential was recently suggested to test this idea [Fujii et al., 2005]. To ensure successful observations, all possible interactions of the energetic electrons with the ionosphere should be accounted for. The TA-based method implies that unlike intense electron beams injected from spacecrafts [e.g., Grandal, 1982], tether-produced beams are not subject to collisionless beam-plasma interactions (BPI). However, it is established that BPI can play an important role in natural aurorae for similar beam parameters. In particular, this regards to thin layers of enhanced ionization/airglow and elevated electron temperatures in the auroral ionosphere [e.g., Stenbaek-Nielsen and Hallinan, 1979; Wahlund et al., 1989; Mishin and Telegin, 1989; Voronkov and Mishin, 1993; Schlesier et al., 1997].

[4] This letter contains the theoretical analysis of collisionless effects that complement direct impact of the tether-produced energetic electrons with neutral particles. These effects are shown to play a significant and even major role in the tether-induced aurora, leading to the beam's scattering and green-line emissions in the F region. Farther downward, BPI develop in the transition region between the E and F regions, the so-called valley [e.g., Titheridge, 2003]. As a result, a thin layer of a strongly elevated electron temperature (Te) and airglow develops well above the collisional TA, similar to those observed in the auroral ionosphere [Voronkov and Mishin, 1993; Schlesier et al., 1997].

[5] The beam current jb = e · qb ≃ 15 μ A/m2 must be compensated by ionospheric electrons. In the F region, the ionospheric current velocity uc ≃ 1010/nc is well below the current-driven instability threshold [e.g., Mikhailovskii, 1974]. However, the plasma density in the valley drops to ≤103 cm−3 [e.g., Titheridge, 2003], yielding uc of the order of the thermal electron velocity vT0 = equation image. As a result, plasma waves can develop and inhibit the currents. In the extreme case, the beam may be locked near the valley ‘bottom’ analogous to a virtual cathode. A schematic of TA that accounts for collisionless effects is depicted in Figure 1, where rL is the beam gyroradius and B is the geomagnetic field (cf. MSS97). In addition to optical observations, electrostatic plasma waves excited in the course of BPI and elevated electron temperatures can be detected by incoherent scatter radars [e.g., Schlesier et al., 1997].

image

Figure 1. Cartoon depicting the tether-induced airglow with collisionless interactions included. Inset shows some trajectories of electrons emitted from adjacent sites of the tether.

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2. BPI Effects in the F Region

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[6] In the proximity of the tether, the pitch-angle distribution of downward-propagating electrons emitted from unit length is fθ = equation image at χ ≤ θ ≤ equation image (MSS97), where nb = qb/vb ∼ 3 cm−3, vb = equation image. Note that for upward-propagating electrons θ [RIGHTWARDS ARROW] π - θ. Accelerated electrons follow their gyro-trajectories and quickly (Δt < 1 ms) form a curtain-like beam: Δy ∼ 2rL = 2vb/Ω ≪ ΔzLt and ΔxLt/tanχ (the inset in Figure 1). Here vb = vbsin θ, Ω ≃ 8.2 106 s−1 is the electron gyrofrequency, the axis equation image and equation image coincide with the local vertical and the horizontal geomagnetic component, respectively. Notice that the minimum of the beam dwell time τ ∼ 2rL/VR is of order 10 ms, provided the payload's speed along the equation image-axis is VR ∼ 1 km/s. As a result of the overlap of trajectories originating from adjacent (vertical) sites of the tether, the distribution function (DF) in the beam's core can be approximated as

  • equation image

[7] Here U = 〈vb〉 = equation imagevb ≈ 0.45 vb and U = 〈vb〉 = equation imageequation image(cos χ) · vb ≈ 0.86 vb are the average parallel and transverse to B velocities, ΔU = equation image ≈ 0.3 cos χ vb ≈ 0.22 vb and ΔU = equation image ≈ 0.1 vb are their velocity spreads, 〈〉 stands for averaging over pitch angles, and E(a) is the second order complete elliptic integral; χ = 45° is assumed for numerical estimates. The exact form of f and f in equation (1) is not important, only the conditions f∥,⊥ (∣x[RIGHTWARDS ARROW] ∞) [RIGHTWARDS ARROW] 0 and ∫ f · fd3v = 1 are assumed.

[8] Since U⊥,∥ > ΔU⊥,∥vT0, the distribution (1) is close to the so-called “warm” beam in v and to the beam of “oscillators” in v, respectively. The latter excites upper hybrid (uh) ωuhr = equation image and electron Bernstein (eb) waves, ωs(k)/Ω = s + δs(k). Here ωp ≈ 6 104equation image s−1, s = 1, 2,…, and ∣δs(k)∣ < 1/2. The growth rate of short-scale equation imagekrL ≫ 1 oscillations with k < kskequation image maximizes near the so-called double plasma resonance ωuhr(zs) = equation image [RIGHTWARDS ARROW] s · Ω [e.g., Mikhailovskii, 1974]

  • equation image

[9] For nb/nc = 3 10−6 cm−3, one gets γ ≃ 2 104 ≫ νe ∼ 103 s−1 (the elastic collision frequency of thermal electrons). The plasma density varies with altitude, and the frequency mismatch ∣ωuhr(z) − s · Ω∣ ≃ ∣Δz · equation imageωuhr∣ ≃ 2π∣Δz∣ · 40 equation image destroys the resonance ([RIGHTWARDS ARROW]0.1 Ω) at ∣Δzs∣ ≃ 3 km, provided the density gradient scale-length Ln = nc/∣∇nc∣ ≃ 50 km below the F-maximum. In the proximity of the F-maximum one has ∣Δzs∣ ≫ 10 km. Since γ · τ ≫ Λ ∼ 10 (the Coulomb logarithm), the unstable beam-wave system has time to reach saturation [e.g., Sagdeev and Galeev, 1969]. In the saturated state, the wave energy density Ws is of the order of the initial beam transverse energy [e.g., Aburdgania et al., 1978], i.e., Ws ≃ 0.3nbmU2 ≃ 103 eV/cm3. Note that the excitation of double-plasma resonances in the F region was observed during injections of weak electron beams from the shuttle [Gough et al., 1995].

[10] The r.m.s. amplitude, Es = equation image ≃ 2 V/m, is well beyond the threshold Epd ≃ 10−2equation image for parametric decay into secondary uh/eb waves [Zhou et al., 1994]. The waves' rise time, ∼2Λ(Epd/Es) νe−1, is much less than τ and hence tertiary waves will be generated, etc. The resulting wave energy spectrum consists of a few spectral peaks, each of order (Λνe)1/2Es (well-known weak turbulence “cascading” [e.g., Sagdeev and Galeev, 1969]). The r.m.s. amplitude of the total uh/eb field is EΣ ≃ 2Es.

[11] Dimant et al. [1992] and Grach [1999] showed that acceleration by eb/uh waves via cyclotron resonance v = (ω(k) − sΩ)/k is a key source of suprathermal electrons in ionospheric HF heating experiments. Photoelectrons present in the F region during twilight provide many more “seed” electrons for acceleration than are available with a Maxwellian distribution [e.g., Grach, 1999]. At k < ks and equation imagemv2 ∼ 10 eV, for acceleration to be efficient the frequency mismatch must be less than 0.1 Ω. This is valid along the whole turbulence layer. The accelerated electron population depends directly on the resonance waves' energy and the acceleration time ∼Δzs/v. Both the wave energy Ws and the layer's length Δzs essentially exceed those characteristic of the HF heating. Thus, the values of the density na ≃ (10−4 − 10−3) nc of accelerated electrons with energies up to ɛa ≃ 50 eV, obtained for the HF heating, can be considered as the lower limit.

[12] Atomic oxygen O is the main neutral constituent in the F region above ∼200 km. Transitions from the O(1S) state to O(1D) yield 557.7-nm (green-line) photons. Collisional quenching of the O(1S) state in the F region is negligible. Its volume excitation rate is Q1S = NO · 4π equation image σ1S(ɛ) Φa(ɛ)dɛ ≃ 10−9naNO cm−3s−1. Here NO is the oxygen density, Φa(ɛ) ≃ naequation image/(4πɛa) is the differential number flux, σ1S ≃ 310−18 cm2 at 7–20 eV, and ɛ1S ≈ 4.2 eV [e.g., Bernhardt et al., 1989]. Figure 2 shows the energy loss of accelerated electrons in eV/km at different altitudes, calculated with Majeed and Strickland's [1997] cross-sections for winter conditions at mid latitudes. Given the acceleration region near 300 km, accelerated electrons propagate downward to za ∼ 200–220 km. For ɛa in the range 10–50 eV and na ∼ 103 cm−3, the column emission rate at magnetic zenith is estimated as 10−6equation imageQ1Sdz · τ ≃ 103.5 τ ≃ 30 Rayleighs (R) at τ = 10 ms. Although small, it is well above the sensitivity threshold of contemporary optical imagers of order a few R [e.g., Pedersen and Carlson, 2001].

image

Figure 2. Energy loss of suprathermal electrons at different altitudes (the left axis). The dashed line shows the ratio δ between inelastic and elastic collision frequencies at 250 km.

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[13] Note that the distribution of upward-propagating electrons is also described by equation (1), where U is replaced by −U. Therefore, their collisionless effects in the F-region are roughly the same as those of downward-propagating electrons. Second, the beam density nb is proportional to the plasma density near the tether nc = n0 [MSS97]. Thus, the growth rate (2)equation image does not change with n0, while Ws and, most significantly, na do. As a result, the intensity of green-line emissions in the weakening F region will decrease, reaching the observational threshold at n0n* = 105 cm−3.

3. Collisionless Effects in the E Region

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[14] Since relaxation over transverse velocities does not affect the beam distribution over v, Langmuir waves with phase velocities ωl/k < U − ΔU can grow with the rate γ ≃ ωpequation image2 ≫ νe. For the waves to develop within the dwell time, γ · τ must exceed Λ. This yields nc < nc* ≃ 2 104nb2 cm−3. For the unstable beam-wave system to reach saturation in inhomogeneous plasmas, it is necessary that the condition Ln > Lb ≃ Λequation image is fulfilled [Vedenov and Ryutov, 1975]. Since Lb(nc*) ∼ 300 km, the warm-beam instability in the F region is inhibited by the plasma inhomogeneity, except for the proximity of the F-maximum if nmax < nc*.

[15] In the so-called valley below ∼200 km the mean plasma density is nv ≃ (1–1.5) 103 cm−3 [Titheridge, 2003], resulting in Lb(nv) ≃ 15 km. The density altitude-profile has at least one minimum near the entry into the valley. Here the beam instability is particularly favored as ∇nc [RIGHTWARDS ARROW] 0 [cf. Voronkov and Mishin, 1993]. From the quasilinear theory [e.g., Vedenov and Ryutov, 1975] it follows that the wave energy density W in the saturation is Wqlequation imagenbɛbequation image ≃ 5105equation image (cf. nvTe0 ≃ 75 equation image at Te0 = 0.05 eV). In fact, at W ≥ (M/m)1/3nvTe0 the aperiodic parametric instability develops, and electrons are heated to Te = TwW/(2nv) within the time τp ≃ 10(M/m)1/3ωp−1 due to wave breaking [De Groot and Katz, 1973; Bauer et al., 1992]. Here M is the mean ion mass. Since γτp > 1, the waves continue rising during the nonlinear heating, resulting in Twequation image(M/m)1/3Te0 ≃ 5 eV within the layer Δz = Lh ≃ Λ equation image ≃ 0.5 km.

[16] Along with the heating, appear short-scale (of order the Debye length) density oscillations with the r.m.s. amplitude δn ≃ (m/M)1/6nc [Volokitin and Mishin, 1978]. Conversion on the oscillations leads to the Langmuir waves' damping rate γcnv ≃ (m/M)1/3ωp. At γcnv > γ the beam instability is inhibited in the heated layer. Besides, the oscillations lead to the beam scattering over pitch-angles [e.g., Khazanov et al., 1993]. Thus, the BPI begins farther upstream, where the whole process repeats again and again. The propagation speed of this ‘heating’ wave appears to be uhLhp ≃ 103 km/s (cf. the ‘relaxation wave’ in the pure quasilinear BPI theory [Vedenov and Ryutov, 1975]). As a result, the heated layer ∼10 km near the valley's entry is formed within the dwell time, if the altitude extent of the density minimum Δzm exceeds 10 km. Otherwise, the layer's thickness is ∼ Δ zm. Note that this scenario holds as long as γ (nb, nv) ≃ 104 · nb > max (Λ/τ, ve), requiring nb > 0.4 cm−3 or n0/n* > 1.5. Voronkov and Mishin [1993] and Schlesier et al. [1997] suggested a similar scenario, except for the nonlinear heating and conversion, to explain thin layers of elevated Te observed in the valley during auroral precipitation events. Enhanced plasma waves in the valley were observed during a weak keV- electron precipitation [Kelley and Earle, 1988].

[17] Up to this point, it is inferred that the beam current is readily neutralized by ionospheric electrons or that their current velocity uc = qb/nc is below the threshold of current-driven instabilities. In the heated Tw-layer, it is the ion acoustic (IA) instability with uth(ia)equation image [e.g., Mikhailovskii, 1974]. In the background (Te0Ti0) plasma, it is the Buneman instability with uth(B)vT0 ≃ 107 cm/s. The latter develops if nv < qb/vT0 ≃ 103 cm−3, i.e., only near the density minimum, nm < 103 cm−3. Because of the Buneman instability, the effective electron collision frequency becomes νef ≃ ωpequation image [e.g., Galeev and Sagdeev, 1984], and the downward (∥ B) electric field E ≃ 4π equation imagejbp ≃ 0.01equation image V/m is established. The anomalous Joule heating Qef = j · E increases mainly Te, so that the IA instability develops shortly (Δt ≪ τ/Λ).

[18] The saturated spectrum of ion acoustic turbulence is subjected to a complex play of various conditions [e.g., Galeev and Sagdeev, 1984]. In the case where ucuth(ia), the ion-acoustic spectrum depends mainly on whether or not a non-Maxwellian tail of energetic ions that absorb the wave energy is formed. In the ionosphere, it is mainly defined by ion-neutral collisions [Mishin and Fiala, 1995]. Roughly, at altitudes below ∼150 km the tail cannot be created. In this case one has νefequation imageωp (Sagdeev's formula). The resulting electric field is E ≃ 0.05Te1/2(103/nm)3/2 V/m, provided TiTi0 due to ion-neutral collisions.

[19] In the Tw-layer, one has Te = Tw. If there are two density minima in the valley [e.g., Titheridge, 2003], in the lower one Te is defined by the condition Qef = nmνeδ(Te)Te. Here νe ≃ 210−7Nnequation image and δ(Te) is the coefficient of inelastic losses. At Nn ≃ 51010 cm−3 (at ∼150 km) one gets Te(m) ∼ 6 eV and E(m) ∼ 0.1 (103/nm)5/2V/m. At nm ≃ 600 cm−3 and Δzm ∼ 10 km, the potential drop E(m) · Δzm exceeds ɛb/e and thus the beam will be locked in this region.

[20] In the weakening ionosphere, the development of the IA instability in the Tw-layer is practically not affected until n0 > n*. However, the development of the Buneman instability is inhibited, unless nm < 10−3n0. Finally, electrostatic plasma waves excited in the course of BPI and elevated electron temperatures can be detected by the radar technique [e.g., Schlesier et al., 1997].

4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[21] It is shown that during a sounding rocket experiment with a conducting tether, collisionless effects can play a key role in the interaction of the tether-created, energetic electron beam with the ionosphere. In the F region, BPI will lead to losses of the beam transverse energy and green-line (557.7 nm) emissions. In the valley between the E and F regions, BPI will result in losses in the parallel beam energy, leading to strong electron heating and airglow. Finally, the development of current-driven instabilities in the low-density valley may inhibit neutralizing currents, thereby locking the beam near the valley ‘bottom’.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[22] We thank the referees for constructive comments. E. V. M. was supported in part by AFRL contract F19628-02-C-0012 with Boston College. G. V. K. was funded in part by the In-Space Propulsion Technology Program of NASA's Science Mission Directorate through the In-Space Propulsion Technology Office at Marshall Space Flight Center under the Technical Task Agreement M-ISP-04-37.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. BPI Effects in the F Region
  5. 3. Collisionless Effects in the E Region
  6. 4. Conclusion
  7. Acknowledgments
  8. References
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