In a weakly ionized gas, electron-neutral and electron-ion collisions greatly dominate over electron-electron collisions. As mentioned in Section 1.3, the electron densities encountered in this work are generally smaller than the neutral/ion density by a factor 10. The electric fields routinely found in models of sprites and elves would be adequate to completely ionize the gas if they were sustained for long enough, but they are necessarily transient. Indeed, the higher the electron density, the shorter can an electric field persist, as a result of enhanced conductivity and decreased relaxation time .
Investigations pertaining to discharges in weakly ionized gases have historically focused on ``glow discharges'' in which the properties of the cathode and anode play an important role. These studies have been motivated largely by interest in high voltage insulators and, more recently, plasma processing. In the 1940's a theoretical understanding of a qualitatively different process, ``streamer breakdown,'' emerged [Bazelyan and Raizer, 1998, p. 12]. In the following, generalizations applicable to the heating of an ionized gas under a moderate electric field are developed and related to high-altitude discharges. Section 2.1.5 applies some of these results to the absorption of a radio wave in a collisional and unmagnetized ionosphere.
The meanings of ``breakdown'' and ``discharge'' are somewhat variable, and possibly unclear in a high-altitude context. Bazelyan and Raizer [1998] use ``breakdown'' to refer to the short-circuiting of some external voltage source. As such a consideration is not applicable to the case of lightning effects on the upper atmosphere, breakdown can alternately be defined as the ``fast formation of a strongly ionized state under the action of applied electric or electromagnetic field'' [Bazelyan and Raizer, 1998]. ``Discharge'' is a more general term describing the release of electric (or electromagnetic) energy into some medium.
In this section we discuss some semi-analytical considerations relating mostly to glow discharges. The complexities of the ``spark'' -- comprising corona, streamer, leader, and arc processes -- are still under considerable study. Bazelyan and Raizer [1998] provide a modern overview.
Below, we base our definitions of some important parameters on the most measureable macroscopic2.1 quantities, namely the current density (inferable from a total current), the electric field , the electron density , and the neutral density .
Since , where is the mean electron (drift) velocity, we define the electron mobility,
In addition, we define an effective electron-neutral collision frequency . Assuming no electron-electron or electron-ion (``Coulomb'') collisions,2.2 the force balance resulting in the net drift velocity is the requirement that
(2.4) |
In this way we may relate macroscopic observables to the fundamental calculable parameter and the electron distribution function . The latter, while conceptually fundamental, is not easily calculable from fundamental principles nor is it easily measureable; however, many macroscopic measurements may shed light on it.
Lastly, as a redundant parameter, the mean free path may be defined by
It remains to justify the assumption of a small drift velocity in equation (2.5). In addition, we derive herein a fundamental scaling law and the time scale over which the electron distribution function thermalizes.2.3 Both in the following and preceding discussions, we ignore the weak velocity dependence of the scattering cross section, in order easily to deduce some approximate behaviors.
For an electron having an energy , a fraction of its energy is lost per effective collision. The rate of energy gain by the electron is thus the difference between the collision term and that due to the electric field, , as long as . This condition remains to be justified below in the context of the following discussion for 3 eV; however, if elastic collisions dominate the effective collision term, for air [e.g., Goldstein, 1980, p. 118]. Using (2.1) and (2.3) for we have
Immediately apparent is the timescale
Equation (2.10) exhibits a key feature of electric discharges in a weakly ionized gas. Many discharge behaviors scale linearly as , or for stronger fields as some other function of . As a result, processes occurring in the relatively dilute upper atmosphere may be studied experimentally on smaller spatial scales by using stronger electric fields at atmospheric pressure.
For a Maxwellian distribution,
To check whether , we use (2.1) and (2.3) to find . With (2.11), (2.10) gives
(2.13) |
Referencing Figure 1.1 again and using , we see that with the thermalization time is 10 s at 90 km and much faster at lower altitudes. These simple considerations suggest that during heating driven at 90 km altitude by a 20 kHz electromagnetic pulse, the electron energy distribution is maintained essentially in equilibrium. However, at lower values of and , increases and may become slow compared with the electric field variation. This issue has been explored in detail and is discussed in Section 2.2.1. Using a detailed model for the electron distribution function, Taranenko et al. [1993a] found that an equilibrium mean energy of 4.3 eV was reached in 10 s for an electric field of 10 at 90 km altitude.
Lastly, we note that a pleasing form of (2.10) is obtained using (2.6):
As a result, on short timescales, ions and neutrals are in a separate thermal equilibrium from that of the electrons. The value remains very small as long as only elastic collisions are accessible ( 1.8 eV), and the direct proportionality between and from equation (2.10) remains strictly true. However, for eV inelastic processes with N become available, and for average energy 0.5 eV, electrons lose 90% of the energy gained from heating by an electric field to the excitation of molecular vibrations [Bazelyan and Raizer, 1998, p. 22]. For in the range of 10 to 15 eV, electronic levels, which are largely responsible for optical emissions, are excited, and above 12.2 eV for O and 15.6 eV for N, molecular ionization is possible. At these energies inelastic energy losses dominate over elastic ones and tends to 1, making modeling based on the simple considerations used in Section 2.1.2 essentially invalid.
Even for low enough electric fields such that the electron distribution function remains highly isotropic ( ), an applied electric field can cause the shape of to depart significantly from a Maxwellian. Because slower electrons participate only in elastic collisions (with inefficient energy transfer to neutrals) while energetic electrons may lose energy (or be attached) in inelastic processes, the high-speed end of the distribution can be greatly diminished as compared with a Maxwellian [Chapman, 1980, p. 124]. The resulting so-called Druyvesteyn distribution, in which rather than the Maxwellian form of , has a steeper ``tail'' and has been often used in glow discharge studies [Meek and Craggs, 1978, p. 110]. However, a detailed calculation of the distribution function from the Boltzmann equation which takes into account an appropriate set of inelastic collisions may result in a slightly more complex and structured distribution, for instance that calculated by Taranenko et al. [1993a].
The dominant inelastic processes for energized electrons in weakly ionized air are ionization and electronic excitation of neutrals, as already mentioned above, and electron attachment to neutrals. As a result of the third classical mechanics fact listed above, electrons cannot combine with electronegative species such as O or positive ions in a two-body collision. As a result, in order to recombine, cold electrons must undergo a three body collision such as
In accordance with equation (2.10), the rate coefficients and for dissociative attachment and molecular ionization in an electrically heated ionized gas scale as a function of . The electric field at which surpasses is known as the conventional breakdown electric field and denoted ; it follows that is proportional to . In a steady electric field above this threshold, and, since the ionization rate is proportional to the electron density, tends to increase exponentially. This electron avalanche process was first described by J. Townsend in 1910 [reprinted in Rees, 1973], and is applicable to all of the high altitude discharges modeled in this work. Wilson [1925] realized that at some altitude would be less than the electric field due to the charge configuration in a thundercloud, as shown in Figure 2.1. He thus predicted an electrical breakdown and ensuing optical emissions.
At much higher electric fields or over long distances and high neutral densities such that m, air breakdown may occur instead in the form of (corona) streamers [Bazelyan and Raizer, 1998, p. 11] or for distances and durations adequate to significantly heat the neutral gas, in the form of leaders [Bazelyan and Raizer, 1998]. Streamers are ionization waves which can propagate as narrow channels through regions where . This self-propagation is due to highly nonuniform electric fields which result from significant , or space charge. Streamer breakdown is not addressed in any detail in this work, but Section 2.3 provides references to recent overviews and to studies relating to sprites. Nevertheless, the issue of streamer initiation is addressed in the context of the observations presented in subsequent sections.
The requirements for streamer initiation have mostly been discussed in the context of spark-gap experiments. For instance, Raizer et al. [1998] and Bazelyan and Raizer [1998, p. 77] describe the critical number of avalanching electrons and a critical (minimum) radius of the avalanche region needed to transition from an avalanche to a streamer. Such considerations are appropriate for an avalanche starting from a narrow point and expanding in a gas of uniform density. In the case of sprites, streamers may sometimes form at the boundary of very large regions of enhanced ionization (Sections 2.5.1 and 5.1). Raizer et al. [1998] suggest that streamers in sprites are initiated by patches of electron temperature and density perturbations caused by the interference pattern from radiation due to complex horizontal intracloud lightning channels. An observed spatial association between a bulk discharge in the lower ionosphere and the formation of streamer channels is discussed in Section 5.1, and is not consistent with the proposal of Raizer et al. [1998].
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Figure 2.1 shows electric field thresholds required for air breakdown as a function of altitude. Conventional breakdown occurs at 32 at ground level and follows the neutral density to 2 at 70 km and 8 at 90 km altitude. Once a streamer is initiated, it may propagate in electric fields lower than . As shown, positively charged streamers, which propagate parallel to the electric field, have a lower propagation threshold than negatively charged (antiparallel to ) ones. The electric field threshold for runaway avalanche varies between the ``relativistic'' and ``thermal'' limits, and depends on the energy of available high-energy electrons. Above the relativistic runaway threshold, electrons with MeV do not thermalize because the electric force outweighs that due to collisions. At the thermal runaway threshold, this is true for electrons with eV, and above this threshold, it is true for all electrons. At tropospheric pressures streamers may develop into leaders, which can propagate in even lower electric fields than streamers can, due to thermal ionization of the neutral gas; lightning is an example. This leader development is seen to occur in electric fields greater than 1 kV/cm [Bazelyan and Raizer, 1998, p. 256]. This value does not scale simply with neutral density and leaders are not thought to occur at high altitudes [Pasko et al., 1998a].
Also shown in Figure 2.1 is the electric field magnitude that would be observed in free space due to a charge of 200 C placed at 10 km altitude above a conducting ground. The field drops off with altitude as due to the dipole resulting from the single image charge. When combined with the electric relaxation times shown in Figure 1.1 and discussed in Section 1.3, these considerations point to three likely scenarios for breakdown in the mesosphere and lower ionosphere. Transient electric fields following large charge redistributions (1000 ) in clouds may
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The value is plotted in Figure 2.2 using values of both under ambient electric field and at (see Figure 1.1 on page ). It can be seen that wave energy in the VLF frequency range, where the spectrum of lightning peaks, is largely absorbed over a very narrow altitude range. For low wave electric fields, this altitude is at 80 to 84 km, while for wave electric fields strong enough to cause a considerable ionization ( ), the altitude is near 87 to 90 km.
These conclusions take into account collisions not considered in the discussion on page (Section 1.3), but still ignore the Earth's magnetic field. Inan [1990] discusses reflection and absorption of the ordinary and extraordinary wave modes using the index of refraction given in a full magnetoionic treatment [e.g., Budden, 1985].