Electron Conductivity

The electron conductivity, $\sigma_{e}$, is related to the electron density, $n_e$, by:

$$\sigma_{e} = q_{e}\mu_{e}n_e$$ (2.18)

where $q_{e}$ is the fundamental unit of charge (1.6$\times$10$^{-19}$ C) and $\mu_{e}$ is the electron mobility. $\mu_{e}$ is a function of the air density (Section 2.2.2) and $E/E_k$ (see below).

The electron mobility is inversely related to the effective collision frequency of electrons with neutrals, $\nu_m$ (Equation 2.2). $\nu_m$ is proportional to the thermal velocity of electrons, $\bar{v}$ (Equation 2.3). $\bar{v}$ can be expressed as a function of $E/E_k$ according to the relationship $\varepsilon\!=\!\frac{1}{2}m{v}^2$ and Equation 2.6. As $E/E_k$ is increased, $\mu_{e}$ will decrease as $\bar{v}$ and hence $\nu_m$ increases. This will decrease the electron conductivity, $\sigma_{e}$, according to Equation 2.18.

For altitudes of 64 km and below, an $n_e$ profile identical to that of Pasko et al. (1997b) is used:

$$n_e = 8.0\times10^{-3}e^{(\frac{z-64}{10.1})} \quad [{\scriptstyle cm^{-3}}] \qquad \qquad z < \mbox{64~km}$$ (2.19)

At altitudes of 80 km and higher, the IRI-95 model of electron density is used (Rawer et al., 1978). An IRI-95 profile was obtained from http://nssdc.gsfc.nasa.gov/
space/model/models/iri.html. Input variables consist of latitude, longitude, date, and time. The location of Langmuir Laboratory (34.0$^{\circ}$ N, ) was input into the model as well as a date and time of 1997, October 7, 4 UT. This particular time was chosen since it represents a reasonable average value for the high-speed video observations of sprites (Chapter 5). The IRI-95 model predicts that the electron density would be 3 cm$^{-3}$ at 80 km altitude for the above input parameters. Changing the location and time inputs to correspond with particular high-speed video sprite events in Chapter 5 would result in an electron density at 80 km altitude which would be as high as 9 cm$^{-3}$ at 3:03:59 UT (Section 5.2.3) or as low as 1 cm$^{-3}$ for sprites which occur after 4:00 UT.

The value $n_e\!=\!8.0\times10^{-3}$ cm$^{-3}$ in Equation 2.19 at 64 km altitude and the IRI-95 model value of $n_e\!\simeq\!3$ cm$^{-3}$ at 80 km altitude were joined by assuming a constant exponential scale height increase of electron density between these altitudes. The exponential scale height for $n_e$ between 64 km and 80 km altitude was only 2.65 km.

The dotted blue line in Figure 2.5 shows the electron conductivity component calculated with $\mu_e$ corresponding to essentially ``cold'' electrons with a temperature $T_e$ the same as that of the air molecules, $T_{air}$. Once the breakdown field ratio, $E/E_k$, exceeds $\sim5\times10^{-4}$ Pasko et al. (1997b), $\mu_e$ will decrease as $\nu_m$ increases (see previous discussion). The decrease in $\mu_e$ and hence $\sigma_e$ (Equation 2.18) can be substantial at $E\!=\!E_k$. This is shown by the dotted red line in Figure 2.5, which corresponds to the electron conductivity if somehow $E\!=\!E_k$ at all altitudes (corresponding to $T_e\!=\!T_e(E_k)\!\gg\!T_{air}$).

Neither the ``cold'' electrons, $T_e\!=\!T_{air}$, or ``hot'' electrons, $T_e\!=\!T_e(E_k)\!\gg\!T_{air}$, will correspond to the actual state of electrons at most altitudes. Rather, the actual $T_e$ will likely lie somewhere in between for $E\!<\!E_k$. The dot-dash line in Figure 2.5 corresponds to conditions expected immediately after the onset of a sprite-producing CG with a maximum $E/E_k$ of 0.8 at $\simeq\,79$ km altitude. Above 79 km altitude, the electric field is significantly attenuated by the larger conductivities (see Section 2.4.4). This causes the electron conductivity to rapidly cross from a ``hot'' profile to a ``cold'' one, resulting in a sharper conductivity ledge at the base of the ionosphere.