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Appendix B: Absorption and Emissions

Once we have the fractal discharge structure, we must consider the propagation of the lightning related electromagnetic fields in the lower ionosphere. As the field propagate into lower ionosphere self-absorption must be considered. The fields energize the electrons generating highly non-Gaussian electron distribution functions. The electron energization is computed with the help of a Fokker-Planck code. This Fokker-Planck code include inelastic loses due to collisions with the neutrals. This is how the fields give energy to the medium. Such inelastic losses will produce the emissions. The present model consists of a plasma formation simulation and a field propagation simulation. Together they calculate self-consistently the propagation through the ionosphere by electromagnetic fields due to the lightning discharge, which for our purposes is fractal.

Self-Absorption

As the lightning related fields propagate in the upper atmosphere and lower ionosphere, the field changes the properties of the medium by heating the electrons and therefore inducing self-absorption. The electromagnetic field propagation is described by Maxwell's equations. We closely follow [ Gurevich 1978] and write Maxwell's equations as

\begin{displaymath} \mathbf{\nabla }\times \mathbf{B}=\frac{4\pi }c\mathbf{J}+\frac 1c\frac{\partial \mathbf{E}}{\partial t}\end{displaymath}

\begin{displaymath} \mathbf{\nabla }\times \mathbf{E}=-\frac 1c\frac{\partial \mathbf{B}}{\partial t}\end{displaymath}

\begin{displaymath} \nabla \cdot \mathbf{D}=4\pi \rho \end{displaymath}

\begin{displaymath} \nabla \cdot \mathbf{B}=0\end{displaymath}

this form will be specially relevant for our further studies. The medium is incorporated as  
 \begin{displaymath} \mathbf{D}=\widehat{\varepsilon }\mathbf{E}\qquad \mathbf{J}... ...{\partial t}(\frac{\widehat{\varepsilon }-1}{4\pi }\mathbf{E})\end{displaymath} (29)
where $\widehat{\sigma }$ and $\widehat{\varepsilon }$ are the conductivity and dielectric operators of the plasma, and in general depend on the field $\mathbf{E}$. The nonlinear wave equation describing E is then given by  
 \begin{displaymath} \nabla ^2\mathbf{E}-\nabla (\mathbf{\nabla }\cdot \mathbf{E}... ...{\partial ^2}{\partial t^2}(\widehat{\varepsilon }\mathbf{E})=0\end{displaymath} (30)
Suppose we can assume that $\widehat{\sigma }$ and $\widehat{\varepsilon }$reach a steady state faster than the relevant time evolution of Eq. (B.2), as it will occur in our case, then a wave incident at the boundary preserves the frequency $\omega $ [Gurevich 1978]. In this approximation the wave equations simplifies as

\begin{displaymath} \nabla ^2\mathbf{E}-\nabla (\nabla \cdot \mathbf{E})+\frac{\... ...varepsilon }+\frac{4\pi i}\omega \widehat{\sigma })\mathbf{E}=0\end{displaymath}

To get an idea of how the field propagates through the plasma we make the following assumptions: (1) we take $\widehat{\varepsilon }=\widehat{\varepsilon }(z,E)$ and $\widehat{\sigma }=\widehat{\sigma }(z,E)$, (2) $\vert\widehat{\varepsilon }\vert\gg \vert\frac{4\pi }\omega \widehat{\sigma }\vert$.Furthermore, if we assume a normal polarization of the wave, we then obtain  
 \begin{displaymath} \nabla ^2\mathbf{E}+\frac{\omega ^2}{c^2}(\widehat{\varepsilon }+\frac{4\pi i}\omega \widehat{\sigma })\mathbf{E}\simeq 0\end{displaymath} (31)
The solution in this ray approximation is then written as  
 \begin{displaymath} \mathbf{E}(s,t)\simeq \frac{\mathbf{E}(0,t-\frac sc)}{s^2}\t... ..._{\mathbf{s}}[i\widehat{n}-\widehat{\Gamma }]\cdot d\mathbf{s}}\end{displaymath} (32)
where $\mathbf{s}$ is the path of the ray. By applying this solution to Eq. ( B.3) we obtain

\begin{displaymath} \widehat{n}^2=\widehat{\varepsilon }\end{displaymath}

\begin{displaymath} \{\widehat{n},\widehat{\Gamma }\}_{+}=\frac{4\pi }\omega \widehat{\sigma }\end{displaymath}

where {}+ is the anticommutator. We have assumed that

\begin{displaymath} \vert\frac{\partial (i\widehat{n}-\widehat{\Gamma })}{\parti... ...frac{\omega /c}{\vert i\widehat{n}-\widehat{\Gamma }\vert}\ll 1\end{displaymath}

Equation (B.4) is the solution to Maxwell's equations and will be used to describe the propagation and absorption of the fields generated by the fractal antenna. We must still estimate the dependence of $\widehat{\sigma }$ and $\widehat{\varepsilon }$ on the field E and the validity of the independence of $\widehat{\sigma }$ and $\widehat{\varepsilon }$ in time. Again following [Gurevich 1978] we will estimate the dependence of the $\widehat{\sigma }$ and $\widehat{\varepsilon }$ on the field strength by looking at the plasma average velocity and energy. When the field effects become important we will see later that the electron distribution will have a strong directional component with the average plasma velocity $\mathbf{v}$ given by

\begin{displaymath} m\frac{d\mathbf{v}}{dt}=-e\mathbf{E}-\frac ec\mathbf{v}\times \mathbf{B} _o-m\nu _e\mathbf{v} \end{displaymath}

where $\mathbf{E}$ is the oscillating field, and $\nu _e$ the averaged collisional frequency with the neutrals. Only collisions with neutrals are relevant at these heights. Similarly, we don't consider transport effects in this region $h\leq 100$ km. Note that without fields, v(t)= v$(0)e^{-\nu _et}$, therefore, the plasma velocity reaches a steady state velocity in $\tau _o\sim \frac 1{\nu _e}$. For the region of interest, $70\leq h\leq 90$ km, the background electron-neutral collisional frequency is $\nu _e\geq 10^5$ Hz and increases in the presence of an electric field. For the case of an oscillating field $\mathbf{E}\sim \mathbf{E}_oe^{i\omega t}$ and a constant magnetic field $\mathbf{B}_o$ the velocity of the plasma is given by  
 \begin{displaymath} \mathbf{v}_e=\frac{eE_o}{m[\Omega _B^2+(i\omega -\nu _e)^2]}... ...}}+\Omega _B(\widehat{\mathbf{e}}\times \widehat{\mathbf{b}})\}\end{displaymath} (33)
where $\Omega _B=\frac{eB_o}{mc}$, $\widehat{\mathbf{e}}$ is the unit vector in the direction of $\mathbf{E}_o$ and $\widehat{\mathbf{b}}$ is the unit vector in the direction of $\mathbf{B}_o$. As it can be seen in Eq. (B.1), we can obtain $\widehat{\sigma }$ and $\widehat{\varepsilon }$from the plasma current

\begin{displaymath} \mathbf{J}=-n_ee\mathbf{v}=\widehat{\sigma }\mathbf{E}-\frac{i\omega }{4\pi } (\widehat{\varepsilon }-1)\mathbf{E} \end{displaymath}

The explicit formulas of $\widehat{\sigma }$ and $\widehat{\varepsilon }$are not very illuminating, but we are interested in the case in which $\omega \ll \Omega _B,\nu _e$ under the influence of the field. Let's define $\phi $ to be the angle between the magnetic field and the horizontal. In this approximation the conductivity tensor is

\begin{displaymath} \widehat{\sigma }\simeq \frac{\omega _o^2\nu _e}{4\pi (\Omeg... ...2 & -\zeta \cos \phi & 1+\zeta ^2\sin ^2\phi\end{array}\right) \end{displaymath}

and the dielectric tensor is

\begin{displaymath} \widehat{\varepsilon }-1\simeq \frac{\omega _o^2\nu _e^2}{(\... ...& 1-\zeta ^2+\zeta ^2(3+\zeta ^2)\sin ^2\phi\end{array}\right) \end{displaymath}

where $\omega _o^2=\frac{4\pi e^2n_e}m$ is the electron plasma frequency, and $\zeta =\frac{\Omega _B}{\nu _e}$. If we take the angle $\phi \simeq 90^o $ and since the electric fields are mostly horizontal for a horizontal fractal discharge, we can finally obtain that

\begin{displaymath} \widehat{\sigma }\simeq \frac{\omega _o^2\nu _e}{4\pi (\Omeg... ...\zeta & 0 \ -\zeta & 1 & 0 \ 0 & 0 & 1\end{array}\right) \end{displaymath}

If we do not consider the diffraction effects so that $\widehat{\varepsilon } \simeq 1$ (for $\omega _o^2<\Omega _B^2,\nu _e^2$), then

\begin{displaymath} \frac \omega c\widehat{\Gamma }\simeq \frac{\omega _o^2\nu _... ...\zeta & 0 \ -\zeta & 1 & 0 \ 0 & 0 & 1\end{array}\right) \end{displaymath}

Therefore, the field power density behaves as  
 \begin{displaymath} E^2(\widehat{\mathbf{r}}s,t)=\frac{E^2(0,t-\frac sc)}{s^2}\theta (t-\frac sc)e^{-\csc (\chi )\int_0^zK(z,E^2)dz}\end{displaymath} (34)
where K(z,E2)= $\frac{\omega _o^2\nu _e}{c(\Omega _B^2+\nu _e^2)}$, $\sin (\chi )=\frac z{\sqrt{x^2+y^2+z^2}}$ is the elevation angle of the point $\mathbf{r}=\widehat{\mathbf{r}}s=\{x,y,z\}$ and $\theta (t)$ is the step function. Equation (B.6) is a nonlinear equation for the field power density since the non-Maxwellian nature of the distribution function under an intense electric field (see latter the Fokker-Planck formalism) is incorporated through $\nu _e=\nu _e(z,\vert E\vert)$. The propagation of the fields through the lower ionosphere is computed including the loss due to self-absorption. This is how the field changes the properties of the medium. We will assume that the field power density is below the ionization threshold so that we don't have to estimate the time spatio-temporal dependence of the electron density.

Electron Distribution and the Fokker-Planck Approach

The plasma distribution function in the presence of an electric field is strongly non-Maxwellian, therefore, a kinetic treatment to compute the electron distribution function must be used. The kinetic treatment will be necessary to find the relevant parameters for the field propagation, such as the averaged total electron-neutral collisional frequency $\nu _e=\nu _e(z,\vert E\vert)$, or for the optical emissions, such as the excitation rates of the different electronic levels. We use an existing Fokker-Planck code, which has been developed for the description of ionospheric RF breakdown [Short et al. , 1990; Tsang et al., 1991; Papadopoulos et al., 1993a], and later used for studying such phenomena as a triggered atmospheric breakdown [ Papadopoulos et al., 1993b], and remote photometry of the atmosphere [ Papadopoulos et al., 1994]. The code includes electron energization by collisional absorption of the EM power in the presence of inelastic losses due to molecular N2 and O 2. It has been an invaluable tool in producing a fully kinetic description of ionospheric EM breakdown and has been successfully benchmarked against the experimental data. The Fokker-Planck formalism [Gurevich 1978] starts by taking the spatially homogeneous Boltzmann equation  
 \begin{displaymath} \frac{\partial f}{\partial t}-e(\mathbf{E}+\frac 1{mc}\mathb... ...}_o)\cdot \frac{\partial f}{\partial \mathbf{v}}=S\vert _{coll}\end{displaymath} (35)
For a weekly ionized plasma the distribution of electrons is mainly governed by the interaction with neutrals, therefore, the electron-neutral collisional term  
 \begin{displaymath} S\vert _{coll}=\int \int d\mathbf{v}_1d\Omega \frac{d\sigma ... ...athbf{v}_1)-f(\mathbf{v} ^{\prime })F(\mathbf{v}_1^{\prime })\}\end{displaymath} (36)
include all the elastic and inelastic losses due to collisions with the neutrals. In this collision integral, $F(\mathbf{v})$ is the distribution function of the neutrals in the atmosphere, v$^{\prime }$ and v are the velocity of the electron before and after the collision, v$_1^{\prime }$ and v1 are the velocity of the neutral before and after the collision. The Scattering cross-section $\frac{d\sigma (\theta ,\phi )}{d\Omega }$ relates the angles between the initial v$^{\prime }-$v$_1^{\prime }$ and final v-v1 relative velocities. In the presence of an electric field, we expand the Boltzmann equation in terms of directional terms [Gurevich 1978]

\begin{displaymath} f(\mathbf{v},t)=f_o(\mathrm{v},t)+\frac{\mathbf{v}}{\mathrm{v}}\cdot \mathbf{\ \ \ f}_1(\mathrm{v},t)+... \end{displaymath}

By taking the zeroth and first velocity moments, i.e. integrate the angular variables over the shell $\vert\mathbf{v}\vert=\mathrm{v,}$ of the resulting equation we obtain  
 \begin{displaymath} \frac{\partial f_o}{\partial t}-\frac e{3m\mathrm{v}^2}\frac... ...{v}}(\mathrm{v}^2\mathbf{E}\cdot \mathbf{f}_1)=S_0\vert _{coll}\end{displaymath} (37)
 
 \begin{displaymath} \frac{\partial \mathbf{f}_1}{\partial t}-e\mathbf{E}\frac{\p... ...e{mc}\mathbf{B}_o\times \mathbf{f}_1=\mathbf{S} _1\vert _{coll}\end{displaymath} (38)
where $S_0\vert _{coll}=\int d\Omega S\vert _{coll}$ and $\mathbf{S}_1\vert _{coll}=\int d\Omega \frac{\mathbf{v}}{\mathrm{v}}S\vert _{coll}$ are the collisional integrals respectively.

Since for weakly ionized plasmas the first moment of the collisional integral will converge faster than the zeroth moment [Huang, 1987], we expect that $S_1\vert _{coll}=-\nu (\mathrm{v})\mathbf{f}_1$, which is usually the closure scheme for expansion series of the above form. Such form for the first moment of the collisional integral is also suggested by the fact that the electron speed is larger than the neutral speed. The velocity dependent effective collisional frequency $\nu (\mathrm{v})=N\mathrm{v}\sigma _{tot}( \mathrm{v}) $ includes both elastic and inelastic processes, and $\sigma _{tot}$ is the effective transport cross section. The elastic contribution to the collisional integral gives [Gurevich 1978]  
 \begin{displaymath} S_0^e\vert _{coll}=\frac 1{2\mathrm{v}^2}\frac \partial {\pa... ...{kT}m\frac{\partial f_o}{\partial \mathrm{v}}+\mathrm{v}f_o]\}\end{displaymath} (39)
where T is the temperature of the neutrals, and $\delta _e$ is the fraction of the electron energy lost in an inelastic collision. The total inelastic collision term is written as a sum of the following processes: rotational, vibrational, optical, dissociation, attachment (including dissociative and three-particle process), and ionization

S0i|coll=Lrot+Lvib+Lopt+Ldis+Latt+Lion

We use Eq.(B.8) adapted for discrete energy transitions to obtain the vibrational, dissociation and optical loses written as

\begin{displaymath} L=L_{vib}+L_{opt}+L_{dis}+L_{rot}=-\frac 2{m\mathrm{v}}\sum_... ...epsilon _{kj})-\epsilon f_o(\epsilon )\sigma _{kj}(\epsilon )] \end{displaymath}

representing loss due to a transition (excitations and de-excitations) from the state $k\rightarrow k+j$ and a gain due to a transition that ends with an electron energy $\epsilon .$ The $\sigma _{ij}$ are the total cross sections for the respective processes.

The attachment loss term is

\begin{displaymath} L_{att}=-(N_{O_2}\sigma _{att2}(\mathrm{v})+N_{O_2}^2\sigma _{att3}(\mathrm{\ \ v }))\mathrm{v}f_o \end{displaymath}

which includes dissociative attachment (1st term) along with triple attachment (2nd term).

The losses due to ionization are given by

\begin{displaymath} L_{ion}=\sum_kN_k\int_{\mathrm{\nu }_{ion}}^{\infty \prime 3... ...rm{v}^{\prime })f_o(\mathrm{v}^{\prime })d\mathrm{v}^{\prime } \end{displaymath}

with v$_{ion}=\sqrt{2\epsilon _i/m}$ and $\epsilon _i$ the ionization energy. In this case the solution to Eq. (B.10) is $\mathbf{f}_1=- \mathbf{u}(\mathrm{v})\frac{\partial f_o}{\partial \mathrm{v}}$ with $\mathbf{u}(\mathrm{v})$, the directed electron velocity, satisfying

\begin{displaymath} \frac{d\mathbf{u}}{dt}=-\frac{e\mathbf{E}}m-\frac e{mc}\mathbf{u}\times \mathbf{B}_o-\nu (\mathrm{v})\mathbf{u} \end{displaymath}

and the solution is given by Eq.(B.5), from which we derive that

\begin{displaymath} \mathbf{u}\cdot \mathbf{E}=\frac{eE_o^2\nu }{m[\Omega _B^2+\nu ^2]}\{1+\frac{ \Omega _B^2}{\nu ^2}\cos ^2\theta _o\} \end{displaymath}

Therefore, the Fokker-Planck equation for the oscillating electric field is given by  
 \begin{displaymath} \frac{df_o}{dt}-\frac 1{3m\mathrm{v}^2}\frac \partial {\part... ...v}))\frac{ \partial f_o}{\partial \mathrm{v}})=S_0\vert _{coll}\end{displaymath} (40)
where $\widetilde{\epsilon }(E,\nu )=\frac{e^2E_o^2}{2m[\Omega _B^2+\nu ^2]} \{1+(\frac{\Omega _B}\nu )^2\cos ^2\theta _o\}$ is the quiver energy that depends nonlinearly on the collisional frequency $\nu $, the field Eo and the height h in the atmosphere. The averaged electron-neutral collisional frequency over the resulting distribution function is then denoted by $\nu _e$.

The Fokker-Planck equation, given by Eq.(B.12), is solved numerically. The major assumption involved in the derivation of Eq.(B.12) are:

We must now relate the results from Eq.(B.12) to the spatio-temporal field. The spatial variable does not affect the model since transport has been neglected, and the field is below the ionization threshold. Some relevant issues to notice:

All of the above assumptions are well satisfied for the electric field intensities expected. In addition, in the implementation of the numerical solution, the neutral atmosphere is assumed to consist of molecular oxygen and nitrogen only, excluding trace neutral constituents. Finally, using the Fokker-Planck code with the electric field distribution from the fractal antenna including the absorption, the intensities of the spatio-temporal optical emissions will be obtained.

Optical Emissions of N2(1P)

Consider the excitation of the 1st positive ($B^3\Pi _g$) level of molecular nitrogen N2 which has an excitation energy of 7.35 eV, and a lifetime of 8 $\mu \sec $. The emission from the 1st positive band is predominantly in the red. The other relevant line in the red part of the spectrum is the $b^{\prime }\Sigma _g^{+}$ of molecular oxygen O2 which has an excitation energy of 1.63 eV, but a very long lifetime of 12 $\sec $, and it is collisionally quenched at the relevant heights. We can compute the excitation rate of the 1st positive of N2 from the Fokker-Planck code for a given field power density

\begin{displaymath} \nu _{ex}^{1p}=4\pi N_{N_2}\int f(\mathrm{v})\mathrm{v}^3\sigma _{ex}^{1p}( \mathrm{v})d\mathrm{v} \end{displaymath}

where $\sigma _{ex}^{1p}$ is the excitation cross-section. Under a steady state situation, which is consistent with the lifetime $\tau =8\mu \sec $ of this N2(1P), the excitations are then followed by optical emissions where the number of photons emitted per sec, per cm3 is given by $\nu _{ex}^{1p}n_e$ for an given electron density ne. In order to compare with observations, we will need to average the number of photons in this band at a given position in space over a time $\Delta t$. Hence

\begin{displaymath} <\nu _{ex}^{1p}n_e\gt=\frac 1{\Delta t}\int_0^{\Delta t}\nu _{ex}^{1p}[E(t)]n_edt \end{displaymath}

The intensity of the radiative transition in Rayleighs is then given by  
 \begin{displaymath} I(R)=\frac{10^{-6}}{4\pi }\int <\nu _{ex}^{1p}n_e\gt dl\end{displaymath} (41)
where the integral is carried along the visual path of the detector (column integrated). Therefore, a Rayleigh (R) is the number of photons per sec per cm2 column integrated and normalized to 106 photons.


  
Figure B.1: (a) The two electrond density profiles as a function of heigh. (c) The absoption of the E field is followed by optical emissions of the N2 (1P). (d) The exitation rates. Plots correspond to the tenous night-time ne(1) electron density profile (dotted line) and dense night-time ne(2) electron density profile (dashed line).
\begin{figure} \center \includegraphics [width=6in,height=4in]{images/dipole.eps}\end{figure}

Consider first the field due to a point radiator, i.e. $E\sim \frac{E_o}he^{i\omega t}$, we can then compute the field propagation including self-absorption, as shown in Fig B.1b, for the two electron density n e profiles shown in Fig. B.1a, typical tenuous and dense electron density profiles respectively. Similarly, we can compute the optical emission intensity for the two profiles as shown in Fig. B.1 c. The excitations rates of N2(1P) for the different altitudes is given in Fig. B.1d as a function of the power density S(W/m2)=c$\varepsilon _o$E2(V/m).

From now on we take the tenuous electron density profile ne=ne(1). We can compute the height dependent field propagation and photon emissions using this simple monopole model for different electric power density profiles starting at the height h=60 km. Figure B.2 shows the field propagation and emissions for different field strengths propagating from h=60 km. It gives us an idea of the electric field power intensity required to produce observable emissions.


  
Figure B.2: (a) The field propagation for the point dipole model as a function of height for different power densitties hitting h=60 km. (b) The emissions produced by the fields shown in part (a). The three curves correspond to different field power densities at h=60 km as seen in Fig. B.2.
\begin{figure} \center \includegraphics [width=5in,height=4in]{images/fe.eps}\end{figure}

Ionization Threshold

As we commented above, the quiver energy $\widetilde{\epsilon }$ can parametrize the steady state distribution function. As we increase the power density S(W/m2), we will reach a threshold value of $\widetilde{\epsilon }$ where the electrons will gain enough energy to induce significant ionization. This threshold value occurs at $\widetilde{\epsilon } (eV)=1.7\,(2\pi \times 10^6)^2\frac{\mathrm{E}^2(V/m)}{[\Omega _B^2+\nu _e^2] }\{1+(\frac{\Omega _B}{\nu _e})^2\cos ^2\theta _o\}\sim 0.1$ eV [ Papadopoulos et al., 1993a] which can be computed at a given height and electric power density.

Taking $\theta _o=\pi /2$, we can estimate the power density required at h=60 km, from our point radiator model, to produce ionization at some height above 60 km. Of course ionization will not occur at h=60 km for these power densities, but assuming the propagation shown in Fig. B.2 we can estimate whether ionization will occur at a higher height for a given field strength at h=60 km. Figure B.3 shows the quiver energy height profile for the field profiles shown in Fig. B.2. Therefore, the ionization threshold will be reached at some height ($h\sim 90$ km) if $S\simeq 2$ W/m2 at a height of h=60 km.


  
Figure B.3: The quiver energy as a function of height for the field intensities shown in Fig. B.2a. The three curves correspond to different field power densities at h=60 km corresponding to Fig. B.2.
\begin{figure} \center \includegraphics [width=5in,height=4in]{images/quiv.eps}\end{figure}


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Previous: Fields from a Fractal