Appendix C: Population of N₂ Levels

• Computing the Radiative Intensity
• Atmospheric Attenuation

The electron energization is computed with the help of the Fokker-Planck code (described in Appendix B). For definiteness we assume that the emission region is located at z=80 km [Lyons , 1994; Sentman et al., 1995; Bossipio et al., 1995; Winckler et al., 1996].

  

Figure C.1: Energy levels diagram for the nitrogen electronic levels considered in the discussed model. The relevant radiative transitions are shown by arrows

Computing the Radiative Intensity

In the current model we have retained only the electronic levels of N₂ shown in Fig. C.1. The computation of the intensity of a radiative transition connecting the v-th and v$^{\prime }$-th vibrational levels of electronic states $\alpha$ and $\beta$ is accomplished as following. We first compute the excitation rate $\nu _{ex}^\alpha$ of the $\alpha$electronic level of N₂ by electron impact

$$\nu _{ex}^\alpha =4\pi N\int f(\mathrm{v})\mathrm{v}^3\sigma _{ex}^\alpha ( \mathrm{v})d\mathrm{v}$$

using the excitation cross section of the B, B$^{\prime }$, W, C and E electronic levels by electron impact from Cartwright et al. [1977]. While the excitation cross section of the N₂(D) electronic level is taken from Freund [1971] and normalized by using the peak value from Cartwright [1970]. The excitation cross section of the N₂⁺(B) electronic level by the electron impact from the ground state was taken from Van Zyl and Pendleton [1995].

  

Figure C.2: The excitation rates for the different leves: B, B', W, C, E. for the h = 80 km.

Figure C.2 shows the excitation rates for the relevant electronic levels of N₂ as a function of the electric field E_o at this height z=80 km. We have neglected effects caused by the W$\rightarrow$B transition compared with that due to the B$\rightarrow B^{\prime }$transition, both having similar excitation threshold but different excitation rates as revealed by Fig (C.2). We then obtain the population of vibrational levels inside each of the electronic states by solving the following set of stationary equations [Cartwright, 1978]  

$$\frac{dn_v^\alpha }{dt}=q_{ov}^{xa}\nu _{ex}^\alpha n_e+\sum... ...m_{\beta j}A_{vj}^{\alpha \beta }-k_{q,v}^\alpha Nn_v^\alpha =0$$ (42)

where:

• n_e is the electron density,
• $n_j^\alpha$ is the number density of the v-th vibrational level of electronic state $\alpha$,
• q_ov^xa is the Franck-Condon factor which shows the transition probability to the v vibrational level of the $\alpha$ electronic state from the 0 vibrational level of the ground state X (in a cold ambient gas only the lowest vibrational level is populated),
• $A_{vv^{^{\prime }}}^{\alpha \beta }$ is the Einstein spontaneous transition probability,
• $k_{q,v}^\alpha$ is the rate constant of collisional quenching of the v vibrational level of the $\alpha$ electronic state
• N is the air density.

Therefore, the first term in the right side of Eq. (C.1) shows the direct pumping of the v vibrational level of the $\alpha$ electronic state by the electron impact. While the second term shows cascade excitation, the third term describes the radiation losses. The last term reveals losses due to the collisional quenching. Note that the usage of the stationary equations for the population of vibrational level is justified by the fact that radiative lifetime of the relevant electronic states have to be shorter than the duration T of electromagnetic pulse from lightning in order to be effectively pumped. Therefore, a stationary distribution of $n_v^\alpha$ is established during the pulse.

From Eq. (C.1) we obtain now the population of the $\alpha$electronic level as

$$n_v^\alpha =\nu _{ex}^\alpha n_eF_{1,v}^\alpha +\sum_\beta \nu _{ex}^\beta n_eF_{2,v}^{\alpha \beta }$$

$$F_{1,v}^\alpha =\frac{q_{ov}^{xa}\tau _v^\alpha }{1+\tau _v^\alpha k_{q,v}^\alpha N}$$

 

$$F_{2,v}^{\alpha \beta }=\frac{\tau _v^\alpha }{1+\tau _v^\alpha k_{q,v}^\alpha N}\sum_jF_{1,j}^\alpha A_{jv}^{\alpha \beta }$$ (43)

here $\tau _v^\alpha =1/\sum_{j\beta }A_{vj}^{\alpha \beta }$ is lifetime of v-th vibrational level of the $\alpha$ electronic state. The coefficients $F_{1,v}^\alpha$ and $F_{2,v}^{\alpha \beta }$ reveal relative importance of the direct and cascade excitation of v-th vibrational level of state $\alpha$, and are calculated using the data from Gilmore et al., [1992]. Moreover, the quenching factor ($1+\tau _v^\alpha k_{q,v}^\alpha N)^{-1}$ was calculated for the relevant electronic levels using the quenching rate coefficient recently revised by Morill and Benesh [1996], see Fig ( C.3). The calculation revealed that the collisional quenching can be neglected at heights above 70 km, since the photon of all three bands of interests N₂(1P), N₂(2P), N₂⁺(1N) is not affected by the collisions.

  

Figure C.3: The effective quenching factor for the transitions of interest: B, B', C, D

Note that collisional transfer between the N₂(B) and other excited N₂ states could affect the N₂ optical spectrum [Morill and Benesh, 1996] and is an important issue for quasistationary auroras. However, in order to affect sprite spectra, the collisions has to occur faster than the duration of a sprite, T$\sim 10$ m$\sec$, which requires that the density of the excited nitrogen molecules to be larger than $\frac 1{k_{tr}T}\simeq 2.5\times 10^{12}$ cm^-3. (Here the collision rate coefficient is assumed to be $k_{tr}\simeq 4\times 10^{-11}$ cm³s^-1 [Morill and Benesh 1996]). To produce such significant abundance of the electronically excited molecules about 10⁹ J km^-3 has to be released. Such value limits the application of the above effect to some local spots, since the total energy released when a charge of hundreds of Coulombs is transferred from cloud to ground is about 10¹¹ J [ Uman , 1987], and only a small fraction of this energy is absorbed in the ionosphere producing the red sprite.

The intensity of the radiative transition in Rayleighs connecting the v-th and v$^{\prime }$-th vibrational levels of electronic states $\alpha$ and $\beta$ by  

$$I_{\nu \nu ^{\prime }}^{\alpha \beta }(\lambda )=\frac{10^{-6}}{4\pi }\int n_v^\alpha A_{vv^{^{\prime }}}^{\alpha \beta }dl$$ (44)

A similar scheme is applied in order to obtain intensities of the N₂(2P) and the N₂⁺(1N) bands from which we obtain the spectrum $I_s(E_o,\lambda ).$

Atmospheric Attenuation

The observed spectrum depends on the location of the detector. If observed from space, the spectrum is the same as the source spectrum $I_s(E_o,\lambda )$, while if observed from either ground or airplane it will be distorted by atmospheric attenuation. Atmospheric attenuation depends on the zenith angle $\chi$ of the optical source, the altitude h_o of the detector, and on the properties of the atmosphere, such as relative humidity and aerosol density. We consider the following contributions to the attenuation: absorption by ozone, oxygen and water vapor, the Rayleigh scattering by air molecule, and Mie scattering by aerosols. The total attenuation of the optical emission is the result of the above contributions and is given by  

$$I(h,\chi ,\varepsilon ,\lambda )=I_s(\varepsilon ,\lambda )e^{-\tau (h,\chi ,\varepsilon ,\lambda )}$$ (45)

$$\tau (h,\chi ,\varepsilon ,\lambda )=\sec \chi \sum_s\sigma _{abs}^s(\lambda )\int_h^{h_o}N_s(z)dz$$

where h_o is the altitude of the sprite, I_s is the sprite source spectrum shown in Fig (4.1), $\sigma _{abs}^s(\lambda )$ is the corresponding effective wavelength dependent attenuation cross section, and N_s (z) is the density of particles that absorb or scatter the photons:

• The absorption by molecular oxygen is computed by taking into consideration the fact that the absorption spectrum of O₂ has four narrow peaks centered at $\lambda =$ 6872, 6893, 7608 and 7638 A [ Greenblatt et al. , 1990].
• The absorption by ozone is computed using the absorption cross-section from Lenoble [1993], and applying the mid-latitude ozone model [Bresseur and Solomon, 1984].
• The absorption caused by the water vapor is calculated using the cross section from Lenoble [1993]. We assume 80% relative humidity, take the dependence of the water vapor pressure on the temperature from Handbook of Chemistry and Physics [1983, Fig. 18-13], and assume also that the temperature in the troposphere follows the profile observed at Wallops Island (38 N^o) during summer time [Handbook of Geophysics, 1985, Fig. 15-13].
• The Rayleigh scattering was calculated using the wavelength dependence of the cross section given by Nicolet et al. [1982].
• Finally the Mie scattering was calculated by assuming the vertical distribution of the aerosol attenuation (at 0.55 $\mu$m) from the background spring-summer model [Handbook of Geophysics, 1985, pg. 18-13], and then we extend the attenuation to any wavelength by using the tropospheric aerosol model [Handbook of Geophysics, 1985, Fig. 18-21]. For the sake of definiteness we assumed 5,000 particles/cm³ number density of aerosols.