In this way we may relate macroscopic observables to the fundamental calculable parameter and the electron distribution function . In addition, we derive herein a fundamental scaling law and the time scale over which the electron distribution function thermalizes. 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. Immediately apparent is the timescale over which the electron distribution function adjusts to a new electric field.
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. While this value is several orders of magnitude larger than the value for elastic collisions , it is small enough to justify the assumptions made above for most of the electric field intensities expected to result in the mesosphere from lightning electromagnetic pulses. 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 .