2.4 Electromagnetic model

The effect of vertical tropospheric lightning currents on the electron population at altitudes up to 100 km is modeled in this dissertation with a finite-difference time domain calculation in cylindrical coordinates, adapted from that used by Veronis et al. [1999]. The model solves Maxwell's equations around a vertical symmetry axis, solving for the vertical and radial electric field, azimuthal magnetic field, electron density, and ion and electron conduction currents. The Earth's magnetic field is neglected, as justified in Figure 1.1 for altitudes up to $\sim$95 km. Optical emissions are calculated from the electron density and the net electric field, and instrumental response to the emissions is calculated for a given geometry and field-of-view.

These calculations are carried out in three steps. The electron density $\ensuremath{n_{e}}$ and electric field $E$ are calculated as a function of time and space in the cylindrical geometry. Next, the volume optical emission rates are calculated from $\ensuremath{n_{e}}$ and $E$ in the same geometry. Finally, the optical signal seen from a chosen vantage point is calculated using three dimensional geometry. The curvature of the Earth is taken into account only in this last step; however, the resulting inaccuracy is small for radial distances $<$$\sim$300 km in the cylindrical geometry.

A cloud-to-ground lightning return stroke (CG) is modeled by imposing a current between the ground and a spherical gaussian charge distribution at 10 km altitude. For lightning currents of $\sim$30 $\mu$s duration, mesospheric electric fields are dominated by those of the lightning electromagnetic pulse (EMP), while for $\sim$500 $\mu$s currents, radiation effects are negligible and the quasielectrostatic (QE) field dominates. Both EMP and QE fields are inherently accounted for in this fully electromagnetic model [Veronis et al., 1999].

The initial conditions of $\rho$$=$0, $\ensuremath{{\bf E}}$$=$0 allow $\rho(t)$ not to be recorded explicitly in the calculation. Instead, all changes to the charge density are accounted for implicitly by the displacement current, and the resulting fields are fully in accord with Maxwell's equations, including contributions to the current density ${\bf J}$ from both electrons and ambient (not modified) ion conductivity.

However, this model is not faithful to the Boltzmann equation (1.6). The $[{\bf v}\cdot\nabla_{\bf r}]f$ term, which accounts for diffusion, is ignored, as is the vector velocity dependence of the distribution function $f$. The effect of the total electric field $\vert\ensuremath{{\bf E}}\vert$ along with all inelastic collisions is accounted for only through the swarm parameters for mobility, attachment, and ionization rates. As a result, changes to the explicitly calculated $n_{e}$ result from ionization and attachment, but not from the relatively small contribution of $\nabla \cdot \ensuremath{{\bf J}}$. Nevertheless, ``space charge'' effects on the electric field are accounted for via equation (1.3) and are evident in the results shown below.

Even more than this absence of fluid (electron transport) properties, the primary feature which renders the model used here incapable of reproducing streamer behavior is the impracticality of numerical solutions with extremely high spatial resolution. The huge conductivity gradients which intensify the electric field at the head of a streamer must be resolved (and managed in a numerically stable way) in order to produce a ``self-propagating'' discharge [Pasko et al., 1998a; Dhali and Williams, 1985]. While this can be done for modeling streamer development at a given altitude and for a given externally applied initial electric field [Pasko et al., 1998a], it is impractical to model simultaneously the full temporal and spatial development of the QE field over the full mesospheric altitude range. Our model also does not account for a wide variety of inelastic processes, such as photoionization, which can play a role in creating free electrons ahead of a streamer and which become important for the distribution function at the high values of $E/\ensuremath{E_{k}}$ in a streamer head. While ambipolar diffusion is not accounted for in the current densities used here, it has been shown to be a negligible consideration even in streamers [Dhali and Williams, 1985], as compared with the electrostatic effects of unbalanced charge.

Sections 2.4.1, 2.4.2, and 2.4.3 give the numeric values of various coefficients and cross sections used to account for elastic and inelastic collision processes and to calculate optical emissions in the model. The ionization and attachment coefficients and the rates of molecular excitation responsible for optical emissions have been updated from those used by Veronis et al. [1999], and are based on the compilations and calculations of Pasko et al. [1999a].

2.4.1 Heating

Aside from those involving sources and losses of free electrons, all changes to the electron distribution function are accounted for in the model by changes to the electron mobility. We use the form of $\ensuremath{\mu_{e}}(E/N)$ provided by Pasko et al. [1997b], which is a polynomial fit to experimental data. Pasko et al. [1997b] provide references to the experimental data, as well as a comparison with kinetic calculations.

The electron conductivity follows from the mobility and electron density as $\sigma=q_e\ensuremath{n_{e}}\ensuremath{\mu_{e}}$. Conduction current in the model is calculated using both electron and ambient ion conductivity [Pasko et al., 1997b].

2.4.2 Ionization

Changes in electron density due to electron impact ionization and dissociative attachment are calculated using

$$\ensuremath{\frac{{d}\ensuremath{n_{e}}}{{d}t}}=(\ensuremath{\nu_{\rm i}}-\ensuremath{\nu_{\rm a}})\ensuremath{n_{e}},$$ (2.15)

where

$$\ensuremath{\nu_{\rm i}}= \begin{cases} 0&{\rm for \;} \frac... ...isplaystyle{\frac{N}{N_}} 10^{p} & {\rm otherwise,} \end{cases}$$

and where

$$p=\sum_{i=0}^{3}a_i\left[ \log_{10}\left( \frac{E}{({\rm 1 \ensuremath{{\rm V\hbox{-}m}^{-1}}\xspace })}\frac{1}{N/N_} \right) \right]^i$$

with $N/N_$ being the neutral density normalized to sea level, and $a_i$ given as follows:

$$\begin{array}{\vert r\vert c\vert c\vert c\vert c\vert c\ver... ...32.878& %%008064484 1.4546 %%3344779757 \ \hline \end{array}$$

Similarly,^2.6

$$\ensuremath{\nu_{\rm a}}= \begin{cases} 0&{\rm for \;} \frac... ...isplaystyle{\frac{N}{N_}} 10^{p} & {\rm otherwise,} \end{cases}$$

where

$$p=\sum_{i=0}^{4}b_i\left[ \log_{10}\left( \frac{E}{({\rm 1 \ensuremath{{\rm V\hbox{-}m}^{-1}}\xspace })}\frac{1}{N/N_} \right) \right]^i$$

and $b_i$ is as follows:

$$\begin{array}{\vert r\vert c\vert c\vert c\vert c\vert c\ver... ...& %% 898829656 -1.35113 %% 633977548}, \ \hline \end{array}$$

Values of $\nu_{\rm i}$ and $\nu_{\rm a}$ are shown in Figure 2.4. The point at which $\ensuremath{\nu_{\rm i}}=\ensuremath{\nu_{\rm a}}$ corresponds to $E=\ensuremath{E_{k}}$. The fits given here are appropriate for $E\lesssim5\ensuremath{E_{k}}$ [Pasko et al., 1997b].

Figure 2.4: Model ionization and dissociative attachment rates. Also shown are the electron impact excitation coefficients for the states producing optical emissions in the ${\rm N}_2(1{\rm P})$ and ${\rm N}_2(2{\rm P})$ bands.

For small electric fields, the three-body attachment process dominates attachment (Section 1.3), and

$$\nu_{\rm a(3\hbox{-}body)}\simeq\Bigl(\frac{N}{N_}\Bigr)^2\;{\ensuremath{4\!\times\!10^{7}}}\; {\rm s}^{-1}$$ (2.16)

[Glukhov et al., 1992, and Victor Pasko, private communication]. However, neither this relatively slow effect nor steady-state ionization processes which determine the ambient ionization level are included in the model.

2.4.3 Optical emissions

Optical emissions are calculated only from two molecular bands of neutral N$_2$ which are expected to dominate the instrument responses of our photometers (Taranenko et al. [1992] and Section 3.3). These optical bands are known as the first positive (1P) and second positive (2P) bands and they result from transitions between electronic states of N$_2$ which have the following designations and threshold energies:

$$\begin{array}{rlrcll} {\ensuremath{{\rm N}_2(2{\rm P})}}:& \e... ...\mu{\rm s}\quad}& A^3\Sigma_{\rm u}^+ &(6 {\rm eV}) \end{array}$$ (2.17)

The spectra from these transitions are complex as a result of vibrational sub-states with additional energies of up to $\sim$2 eV [Green et al., 1996; Stanton and St John, 1969].

The excitation, quenching, and cascading processes involved in emission in these bands are discussed by Pasko et al. [1997b]. We make use of the fact that the lifetimes of the states ${{\rm C}^3\Pi_{\rm u}}$ and ${{\rm B}^3\Pi_{\rm g}}$, given in (2.17), are fast compared with the variations in electric field and with the thermal relaxation time $\ensuremath{{\tau_{\rm th}}}$ of the distribution function. This fact justifies the assumption that instantaneously the population $n_k$ of the excited state $k$ is constant: $\partial n_k/\partial t\simeq0$, where $k$ corresponds to ${{\rm B}^3\Pi_{\rm g}}$ or ${{\rm C}^3\Pi_{\rm u}}$. The population and depopulation terms for excited state $k$ are [Sipler and Biondi, 1972]:

$$\ensuremath{\frac{\partial n_k}{\partial t}}=\nu_k\ensuremath{n_{e}}- n_k(A_k +\alpha_1 N_{{\rm N}_2} + \alpha_2 N_{{\rm O}_2}) + \sum_m{n_m A_m}$$

where $\nu_k\ensuremath{n_{e}}$ is the electron impact excitation rate, $A_k$ is the Einstein coefficient for radiative relaxation, and $\alpha_i$ are the coefficients for collisional deexcitation (commonly called ``quenching'') with nitrogen and oxygen. The sum is carried out over other states which radiatively relax (``cascade'') into state $k$; for instance, ${\rm N}_2(2{\rm P})$ emission populates the state ${{\rm B}^3\Pi_{\rm g}}$, which in turn may radiate in the ${\rm N}_2(1{\rm P})$ band.

The stationary condition mentioned above gives

$$n_k \simeq \frac{\ensuremath{n_{e}}\nu_k +\sum n_m A_m } {A_k + \alpha_1 N_{{\rm N}_2} + \alpha_2 N_{{\rm O}_2} }$$ (2.18)

and the volume photon emission rate is $A_k n_k$. The numerical values of the various coefficients used for the two bands studied are as follows [Pasko et al., 1997b]:

$$\begin{array}{r\vert c\vert c\vert c\vert c} & \alpha_1 & \... ...^{-1} & {\rm by} {\ensuremath{{\rm N}_2(2{\rm P})}} \end{array}$$

The values of the excitation coefficient $\nu_k$ were calculated according to the following polynomial fits [Pasko et al., 1999a]:

$$\nu_{_{\ensuremath{{{\rm C}^3\Pi_{\rm u}}}}}= \begin{cases} ... ...isplaystyle{\frac{N}{N_}} 10^{p} & {\rm otherwise,} \end{cases}$$

where

$$p=\sum_{i=0}^{4}c_i\left[ \log_{10}\left( \frac{E}{({\rm 1 \ensuremath{{\rm V\hbox{-}m}^{-1}}\xspace })}\frac{1}{N/N_} \right) \right]^i$$

and $b_i$ is as follows:

$$\begin{array}{\vert r\vert c\vert c\vert c\vert c\vert c\ver... ...2044.92& -426.910& 39.6648& -1.38351 \hline \end{array}$$

The coefficient $\nu_{_{\ensuremath{{{\rm B}^3\Pi_{\rm g}}}}}$ is calculated the same way but with the substitution of $d_i$ for $c_i$:

$$\begin{array}{\vert r\vert c\vert c\vert c\vert c\vert c\ver... ...1909.46& -390.447& 35.5826& -1.21911 \hline \end{array}$$

These calculations are done assuming the same air composition (i.e., 78% N$_2$, etc.) at high altitude as at sea level, a good assumption for the altitudes of interest. The quenching rate terms in the denominator of (2.18) become comparable to the Einstein coefficient $A_k$ only below altitudes of 32 km for ${\rm N}_2(2{\rm P})$ and 50 km for ${\rm N}_2(1{\rm P})$ so that the emissions produced in the lower nighttime ionosphere are not significantly quenched.