1.3 Electrical environment of the nighttime upper atmosphere

In the mesosphere and the $D$ region of the ionosphere the properties of the upper atmosphere begin their transition from those of a neutral gas to a those of a weakly ionized plasma, as the plasma frequency becomes significant compared with the effective electron-neutral collision frequency. This region has also sometimes been called the ``ignorosphere'' because of its inaccessibility to in situ measurements by either high-altitude aircraft or orbiting spacecraft, or to remote sensing by ground-based radar or top-side sounding. Use of sounding rockets, optical remote sensing through lidar or the photometric and imaging techniques described in this work, and VLF radio studies lend themselves to investigations of these awkward altitudes characterized by their relatively low ($\leqslant$ ${\ensuremath{5\!\times\!10^{3}}}$ cm$^{-3}$) electron densities.

Figure 1.1 compares frequencies and rates of some physical processes in the lower ionosphere. The vertical green line shows $\omega_c$, the electron cyclotron frequency in the Earth's magnetic field. At 75 to 80 km, the nighttime electron density is similar to that of the solar wind at 1 AU. In great contrast to solar wind plasma, however, for altitudes up to $\sim$70 km under ambient nighttime conditions, and up to 90 km under an applied electric field near the breakdown threshold (discussed below in Section 2.1), the effective collision frequency $\nu_{{\rm eff}}$ for electrons is large compared to the cyclotron frequency. At night the electron number density is only 10$^{-16}$ times the neutral density at 70 km and $<$$10^{-10}$ times at 90 km, and on timescales greater than 10 $\mu$s, the electrons are in thermal equilibrium with the neutrals, which are typically at $<$300 K. This region may thus be described as a cold, collisional, weakly ionized electron plasma.

Figure 1.1: Important frequencies in the lower ionosphere. (a) Several characteristic frequencies between 50 and 100 km altitude. The red and gray curves show values for a ``normal'' nighttime ionosphere. In (b), the same values are shown with smaller axis ranges. $\omega_c$ is the electron cyclotron frequency, $\omega_{p}$ is the electron plasma frequency, $\nu_{\rm eff}$ is the effective collision rate, $\nu_a$ is the attachment rate, and $\ensuremath{E_{k}}$ is the conventional breakdown electric field (page ).

Representative values of the nighttime electron density^1.1 ( $\ensuremath{n_{e}}$) have been used in Figure 1.1. Shown in gray is the electron plasma frequency,

$$\ensuremath{\omega_{p}}=\sqrt{\frac{q_e^2\ensuremath{n_{e}}}{\ensuremath{\epsilon_{}}\ensuremath{m_{e}}}}$$ (1.7)

where $-q_e$ and $\ensuremath{m_{e}}$ are the charge and mass of the electron, respectively. In the $D$ region, $\omega_{p}$ changes abruptly as a function of altitude over one wavelength for an electromagnetic wave of frequency similar to $\omega_{p}$. For example, a radio wave of frequency 10 kHz has wavelength 30 km and thus sees an abrupt (i.e., over a spatial range $\ll$30 km) transition from essentially free space to the highly reflecting ionosphere (i.e., relatively high refractive index). Equivalently, the index of refraction $\mu$ for plane electromagnetic waves (ignoring collisions and the ambient magnetic field) is

$$\mu=\sqrt{\ensuremath{\epsilon_{}}\left(1 - \frac{\ensuremath{\omega_{p}}^2}{\omega^2}\right)}$$

and can be seen in this expression to change over less than one wavelength from $\sim$1 to 0 and then to an imaginary number. This sharpness accounts for the low-loss long distance propagation of VLF radio in the Earth-ionosphere cavity (Section 3.1). However, the effect of collisions greatly modifies the effective reflection height and causes some $D$ region radio absorption (Section 2.1.5).

There are at least three different mechanisms by which thundercloud charge configurations may impose electric fields on the upper atmosphere. Thundercloud charging as a result of convective charge separation occurs on time scales of $\sim$100 s. (Research into the mechanisms of charge separation and into the nature of charge configurations in thunderstorms has been ongoing for decades.) Secondly, sudden and large changes in electrical currents may radiate electromagnetic fields in all directions. A primary example of such strongly radiating processes, at least in the frequency range of interest here, is the return stroke of cloud-to-ground lightning, whose radiation spectrum peaks with period 50 to 100 $\mu$s. Third, continuing currents flowing to ground through return stroke channels may redistribute large quantities of charge on time scales of $\sim$0.5 ms to $>$100 ms.

The low-frequency conductivity of the atmosphere determines whether the electric field due to these charge configuration changes in thunderstorms can penetrate to high altitudes. In the absence of significant magnetic fields ( $\nabla\times {\bf B}= 0$), equation (1.3) along with the constitutive relation ${\bf J}=\sigma{\bf E}$ becomes

$$\ensuremath{\frac{\partial \bf E}{\partial t}}=-\frac{\sigma}{\ensuremath{\epsilon_{}}}{\bf E}$$ (1.8)

indicating that an applied electric field locally relaxes exponentially with a time constant $\index{electric field!relaxation time ($\tau_{\rm E}$)}\ensuremath{\tau_{\rm E}}=\ensuremath{\epsilon_{}}/\sigma$, regardless of the complexities of conductivity gradients. Figure 1.1 shows this electric relaxation rate $\index{electric field!relaxation time ($\tau_{\rm E}$)}\ensuremath{\tau_{\rm E}}^{-1}=\sigma/\ensuremath{\epsilon_{}}$ due to the sum of the electron and ion conductivities, both under ambient conditions and with ambient electron density in the presence of an imposed electric field equal in magnitude to the conventional breakdown field $E_{k}$ (see Section 2.1.4). These profiles are discussed further in Section 2.4.

From these simple considerations, some important phenomenological classifications of upper-atmospheric discharges can be presaged. Electric fields due to growing thundercloud charge configurations, which may involve charge centers of hundreds of coulombs [Marshall et al., 1996] but which accumulate over time scales of many tens of seconds, do not affect altitudes much above the troposphere. Space charge developed by currents flowing in accordance with equation (1.8) screen these fields from the thin upper atmosphere. In close vicinity to the tops of thunderclouds, however, such fields are thought to initiate upward streamer-like discharges which can propagate into the stratosphere; these have been denoted blue jets and blue starters [Pasko et al., 1996a; Wescott et al., 1995].

When these same thundercloud charge accumulations are partially neutralized or redistributed by lightning return strokes and their continuing currents on much faster timescales than those on which the charges are built up, the effects penetrate to much higher regions of the atmosphere. Because of the space charge that builds up in conjunction with thundercloud charge separation, the upper atmosphere sees an increase in electric field as a result of any sudden redistribution of charge, even if the new configuration causes reduced electric fields in the troposphere. For instance, a large positive cloud-to-ground return stroke may drain an extensive positive charge region of $>$100 C to the conducting Earth over 1 ms. On short time scales in the mesosphere, this is entirely equivalent to placing a negative charge of identical magnitude in the thundercloud. Considering the electric relaxation rates in Figure 1.1, a new charge configuration in the troposphere is effective in allowing the penetration of electric fields up to 85 km if the change occurs faster than $\sim$1 ms, but only to 70 km if the change occurs on timescales on the order of $\leqslant$10 ms. This principle has been used in many theoretical studies of sprites (Section 2.3).

On even faster timescales, radiated electromagnetic fields at VLF frequencies may penetrate to a height^1.2 roughly determined by a comparison of their frequency with the time scale $\sigma/\ensuremath{\epsilon_{}}$. As discussed above, the electromagnetic pulse from lightning is largely reflected in the $D$ region, but the penetration of these fields above the reflection height results, due to the finite conductivity, in heating of the electron population (Section 2.1.5). For strong radiated fields, this energy deposition can produce the phenomenon known as elves.

An important complication to the conclusions above results from the fact that the conductivity itself may change under an applied electric field. The isotropic conductivity due to electrons alone is $\sigma={\vert}q_e{\vert}\ensuremath{n_{e}}\ensuremath{\mu_{e}}$, and the electron mobility $\ensuremath{\mu_{e}}=q_e\ensuremath{\nu_{\rm eff}}^{-1}\ensuremath{m_{e}}^{-1}$ in turn is a decreasing function of the electric field. The application of an electric field heats the electron population (Section 2.1) and increases the collision frequency $\nu_{\rm eff}$, as shown in Figure 1.1 for a representative electric field value, $\ensuremath{E_{k}}$. Enhanced values of $\nu_{\rm eff}$ in turn lead to the decrease of the mobility and thus the conductivity (Figure 1.1), thus allowing better penetration of transient electric fields to higher altitudes.

On the other hand, the electric field can also lead to the modification of the electron density $\ensuremath{n_{e}}$ through impact ionization of the neutrals by accelerated electrons and through the enhancement of electron attachment to neutrals. For example, if $E$$>$ $\ensuremath{E_{k}}$ then $\ensuremath{n_{e}}$ increases, leading to enhanced conductivity and reversing the effect of heating described above. Ionization and heating effects both turn out to be of key importance in sprites and elves, hence the need arises for detailed modeling to account self-consistently for the nonlinear effect of an intense and varying electric field. Such modeling has now been carried out by several groups, as mentioned in Sections 2.2 and 2.3, and an electromagnetic model which accounts for these processes is described in Section 2.4.

Two more curves in Figure 1.1 require discussion. Under an appreciable electric field, the two-body reaction

$${\rm O}_2 + e^- +3.6 {\rm eV} \longrightarrow {\rm O}^- + {\rm O}$$ (1.9)

is the dominant mechanism for the loss of free electrons. The rate $\nu_{\rm a}$ of this ``dissociative attachment'' process exhibits a peak at an electric field value slightly below $E_{k}$, and the rate shown for $\nu_{\rm a}$ in Figure 1.1 corresponds to its maximum. Under ambient electric field conditions, three-body attachment processes involving either O$_2$ and N$_2$ or two O$_2$ molecules dominate instead [Glukhov et al., 1992], and also dominate over diffusion. As a result, it is the ``three-body'' rate shown in Figure 1.1 which defines the relaxation rate for ionization changes at high altitude following a discharge such as sprites or elves. For comparison, it may be noted that a typical time between lightning flashes in a given location in a storm is on the order of tens to $>$100 seconds, and a typical time between return strokes in a multiple-stroke flash is $\sim$40 ms [Uman, 1987, p. 14]. Therefore only above $\sim$80 km are ionization changes likely to accumulate from upper atmospheric discharges associated with successive lightning flashes. This possibility is discussed in Sections 2.5.3 and 4.4.

The intersection of the curves showing electric relaxation rate $\sigma/\ensuremath{\epsilon_{}}$ (a function of electron density) and maximum dissociative attachment rate $\nu_a$ (a function of neutral density) in Figure 1.1 also defines an important boundary for breakdown phenomena [Pasko et al., 1998a]. Above this altitude (76 to 82 km) an applied electric field relaxes in accordance with equation (1.8) before much electron attachment can occur. Below this altitude the electric field relaxes slowly compared with the attachment rate; thus one can expect free electrons to be largely depleted (immobilized as negative ions) immediately after any transient electric field. As a result, any electron density enhancements are highly transient below $\sim$75 km. In addition, an electric field increasing in intensity on timescales comparable to the relaxation rate in this region causes a depletion of the free electron population before the electric field reaches its peak, so any resultant discharge process occurs in a gas nearly devoid of free electrons. This results in a diffuse region of sprites above 75 km and a streamer region below [Pasko et al., 1998a], as observed in Section 5.1.

It may be concluded that the mesophere and lower ionosphere is a region where both the average electron energy and the electron density may vary strongly in response to transient electric fields. The physics of discharges in a weakly ionized gas is treated more quantitatively in Section 2.1, and Section 3.1 further discusses low frequency radio propagation below the ionosphere.