Chapter 3: Modeling Red Sprites

• The Model
• N₂(1P) Emissions from the Stochastic Model
• Importance of the Fractal Nature of Lightning

The Model

We now combine the ideas developed in the previous chapters and apply them to the modeling of red sprites. We have seen that the radiation pattern and intensity is critically dependent on the fractal model, e.g. the fractal dimension of the discharge. Therefore, we expect that the lightning parameters required to produce the red sprites are critically dependent on the discharge fractal dimension as well. We will study the current threshold required to produce the sprite for different fractal dimensions.

The modeling of the effects of the fractal lightning discharge on the lower ionosphere involves a series of steps, as it is shown in Fig. 3.1:

• Task 1: Modeling of the fractal lightning discharge and its spatio-temporal current profile as developed in Section 2.3.1. The phase coherence over the extent of the fractal will become extremely relevant. In fact, it will be the self-similarity over the fractal that will produce the spatially structured field profile.
• Task 2: Computation of the electromagnetic field in the lower ionosphere from the spatio-temporal current profile. The computation of the electromagnetic field must consider self-consistently the propagation of the lightning related EMP fields in the lower ionosphere as the field changes the properties of the medium by energizing the electrons and generating highly non-Gaussian distribution functions. The propagation and absorption of the electromagnetic fields is developed in Appendix B. The electron energization in the presence of inelastic losses due to collisions with neutrals is computed with the help of a Fokker-Planck code. The Fokker-Planck formalism is also developed in Appendix B. Once we have such a model we can estimate the radiation pattern from the fractal discharge structure that are projected into the upper atmosphere and lower ionosphere.
• Task 3: Calculation of the intensity of the stimulated high altitude optical emissions, using the spatio-temporal electric field profile. The energized electrons collide with the neutrals inducing excited states which are then followed by emissions. The computation of the optical emissions from N₂(1P) from the field pattern in the lower ionosphere is described in Appendix B. The computed optical emissions can be compared with the observations.

Therefore, the main parameters controlling the model are:

• The fractal structure of the model as represented by its fractal dimension. The fractal dimension should be one of the most relevant parameters controlling the spatio-temporal optical emissions since it controls the strength of the field's Fourier components (as seen before). For simplicity the size R of the discharge is taken as R=10 km as developed in Chapter 2.
• The current pulse peak and width, hence the total charge discharged. The altitude of the stroke taken as z_o=5 km with its image discharge at z_o=-5 km.
• The ambient electron density height profile, generally taken as the tenuous night time profile.

  

Figure 3.1: A diagram of the tasks involved in the treatment. From the fractal structure, we compute the fields generated and their interaction with the medium in the lower ionosphere

The spatial geometry is defined with z as the height from the ground and x-y as the constant height cross-section. The horizontal fractal lightning discharge is constructed in the x-y plane at a height of z_o=5 km from the ground (with the image discharge at z_o=-5 km). The point x=0 km and y=0 km corresponds to the center of the discharge.

N₂(1P) Emissions from the Stochastic Model

The stochastic fractal model is specially suitable for understanding the dependence on the dimension of the discharge, since $D(\eta )$ can be easily parametrized as is plotted in Fig. 2.16. The spatio-temporal emission pattern from the fractal discharge model can now be computed from the field pattern that includes self-absorption. As an example, we take a discharge current of $I_o=200\,$ kA and $\beta ={\rm v}/c=0.025$. Since the decay time is $\alpha =1000$ s^-1, the total amount of charge discharged is Q=200 C. The statistical relevance of such values for lightning and the generation of sprites will be discussed latter.

For this case the field is below the ionization threshold in the lower ionosphere. For intensities exceeding the ionization threshold, the time evolution of the electron density must be included.

  

Figure 3.2: (a) The instantaneous field profile with self-absorption included for $\eta =3$. (b) The instantaneous emissions of the N₂(1P). (c) The number of photons averaged over time as a function of height in units of kR/km.

The fractal dimension for the lightning discharge is $D\sim 1.25$. The instantaneous field amplitude at z=85 km and x=0 km, including self-absorption, is shown in Fig. 3.2a and the instantaneous optical emission of the N₂(1P) in Fig. 3.2b.

  

Figure 3.3: The time averaged emission pattern. The temporal emission pattern has been time averaged for about a millisecond (duration of sprite). The column integrated emission intensity was about 30 kR for an optimal optical path.

To compare the results with the sprite observations, we must average the photon flux over the time scale of sprites. The averaged number of photons per sec per cm³ can now be calculated as  

$$<N(s^{-1}cm^{-3})\gt=\frac 1{\Delta t}\int_0^{\Delta t}\nu _{ex}^{1p}(E(t))n_edt$$ (9)

where n_e is independent of time since we stay below the ionization threshold. The number of photons averaged over time as a function of height is shown in Fig. 3.2c in units of kR/km. The units of kR/km is defined in terms of the detector optical path (column integration), so that, a sprite with a horizontal size of 20 km having an average of 10 kR/km would measure an intensity of $I(kR)\simeq 200$ kR for an optimal detector inclination.

  

Figure 3.4: The emission intensity for the optimal optical path as a function of the current strength I_o. The ionization theshold occurs at I_o=170 kA.

Of course, this value is computed locally at a single point, and we must repeat this procedure in space to deduce the spatial dependence of the emissions. The time averaged emission pattern from the lower ionosphere, for the horizontal cross-section along the line y=10 km, is shown in Fig. 3.3 for the discharge current of I_o=200 kA. The color coding represents the number of photons emitted per sec per cc in dBs normalized by its average over the image region. The maximum intensity is about 100 kR for column integration along the x axis. We did check that the field intensities at all times were below the ionization threshold for this current of I_o=200 kA. The fractal nature of the discharge can in fact produce a non-uniform emission pattern.

A plot of the number of emitted photons in units of kR/km as a function of the discharge current I_o is shown in Fig. 3.4 for the position at the core of the sprite, i.e. as given for Fig.3.2. If the field becomes larger than the ionization threshold, we must incorporate the time evolution of the electron density. The ionization threshold is reached when $\widetilde{\epsilon }\geq 0.1$ eV which for our fractal model with the propagation speed $\beta =0.025$ occurs at a critical current $I_o\sim 200$kA. Since the power density scales as $S(W/m^2)\sim I_o^2\beta ^2$, we can use Fig.3.4 as a reference for the requirements of the lightning parameters to generate the sprite. Hence, for a faster discharge a sprite can be produced with a lower current amplitude.

We proceed next to determine the spatial structure of the optical emissions as a function of the dimension D. We consider the 4 fractal discharges $\eta =1,2,3,\infty$ shown in Fig. 3.5 with dimensions $D=1.55,\;1.3,\;1.2,\;1.0$ respectively, where the thickness of the line corresponds to the strength of the current.

  

Figure 3.5: The fractal structures for dimensions D= 1.55, 1.3, 1.2, 1.0 ( $\eta =1,2,3,\infty$) respectively . The thickness of the lines corresponds to the current strength, and current conservation has been satisfied at each branching point.

The emission patterns along the cross-section x=10 km, averaged over the duration of the discharge using Eq. (3.1) is shown in the four panels of Fig 3.6. The velocity of the discharge was taken as $\beta =0.025$ and the amount current as I_o=200 kA. The emission rate, e.g. number of photons per cc per second, is computed in dB with respect to the averaged emission rate over the image area.

  

Figure 3.6: The emissions pattern as a function of dimension. $\beta =0.025$.

The maximum intensity in kR for a optimal column integration along the x axis, is shown in Fig. 3.7 as a function of the dimension of the discharge $D(\eta )$. Since the optical emission intensity is extremely sensitive to the power density, a factor of 2 on the electric field strength can have profound effects on the emission pattern of a given fractal discharge. Hence the sensitivity of the optical emission pattern on the dimension of the discharge.

  

Figure 3.7: The maximum intensity in kR as a function of the dimension. The graph has been interpolated (actual points are shown by asterix *).

  

Figure 3.8: The emissions patterns at the line x=10 km corresponding to the four fractal structures of Fig 3.5 which have dimensions D= 1.55, 1.3, 1.2, 1.0 respectively. The current is chosen so that the emission intensity is about I(kR)$\simeq$100 kR, and $\beta =0.025$.

As seen in Fig. 3.6 different fractals require different current peaks (or propagation speed) to produce similar emissions intensities. For the 4 fractals of Fig. 3.6 we find the necessary current peak I_o needed to produce an emission intensity of about 100 kR. The corresponding emission patterns are shown in Fig. 3.8 with their peak current I_o. We note that the emission pattern corresponding to the fractal $\eta =3$has considerable spatial structure as compared with the other cases in the figure. We see that by having a spatially structured radiation pattern, the fractals can increase the power density locally in specific regions of the ionosphere and generate considerable optical emissions with relatively low (more realistic) lightning discharge parameters.

Importance of the Fractal Nature of Lightning

We have generated a novel model of red sprites that relies on the fractal structure of the lightning discharge. Such fractal structure is reflected in the fine structure of the subsequent optical emission pattern. The incorporation of the fractal structure of lightning provides a clear method for the generation of the fine structure of red sprites. For an optimal configuration, so that the fields get projected upwards, the lightning discharge must be horizontal, i.e. the so called intracloud lightning or ''spider lightning'' [Lyons, 1994]. It is important to notice that in our model with $\beta =0.025$ the ionization starts occurring for I_o>200 kA (i.e. for I_o>200 kA the equation for the evolution of the electron density n_e should be included).

We can compare these results with the simple tortuous random walk models (see Section 2.2.2). The maximum field strength at x=10 km y=10 km, z=60 km is shown in Fig. 2.20a and the emission strength in Fig. 2.20b as a function of the pathlength or dimension.

Therefore, we can obtain a significant increase in the strength of the radiated fields by treating the discharge as a fractal. Factors of 5 in the power density are not rare.

  

Figure 3.9: (a) The random walk model. (b) The time averaged intensity in kR/km for the random walk model as a function of the path length. The path length is increased after each succesive sudivision.

Certain fractals can radiate more effectively than others, but in general this problem is very complicated. The power density, and thus the emission pattern and intensity, scales as $S(W/m^2)\sim \beta ^2I_o^2\,f(D,L,\omega )\,g(\theta ,D,\beta ,\omega )$, where $\,f(D,L,\omega )$ is the efficiency function of the different fractals. $g(\theta ,D,\beta ,\omega )$ represent the spatial structure of the radiation pattern with $\theta$ as the angular position. There is some dependence in the radiation fields as a function of $\beta$ as can be observed from Fig. 2.22. Clearly, Q, $\beta$, I_o, D are the relevant parameters that control the optical emission pattern and intensity.

Statistics of intracloud lightning are in the best cases incomplete, but information about some independent measurements can be found in the book by Uman [1987]. It seems to suggest that in extreme cases the intracloud discharge can reach $Q\sim 100$ C or 100 kA, with a length of 10s of kms. Some rough estimates can be made of the relevance of the model discharge parameters by comparing with the statistics of cloud-to-ground discharges given in Uman [1987].

Such estimates can be made more precise if we assume the qualitative model suggested by Lyons [1996] where the sprite generating +CGs are associated with intracloud spider or dendritic lightning known to accompany many +CG events. He presents a qualitative model which is based on the fact that horizontal discharges of the order of 100 km have been observed in connection with +CG events. The model starts with the initial spider lightning followed by the positive leader toward ground, which in turn is followed by the positive return stroke. The latter generates an intracloud lightning discharge which propagates along the (fractal) spider channel. Therefore, in this model, the statistics of +CG current amplitudes could be related to the particular intracloud events responsible for the sprite generation. In cloud-to-ground discharges, a current peak of $I_o\sim 100$ kA, consistent with a charge transfer of Q=100 C, occurs between 1-5 % of the time. The speed of propagation of a cloud-to-ground is about $\beta \sim 0.5$, but intracloud discharges seem to propagate with speeds of an order of magnitude smaller [Uman, 1987]. Consequently, our model discharge parameters seem to agree with the sprite occurrence [Lyons, 1994] and the statistics of lightning discharge parameters [Uman, 1987].

We live in a world where dielectric discharges seem to have $D\sim 1.6$ [ Sanders 1986, Niemeyer et al., 1984]. On the other hand lightning discharges seem to show lower dimensions, a fact that might become relevant due to the sensitivity of the emission strength on the fractal dimension of the discharge. The optimal emissions intensity is obtained for dimensions $D\sim 1.3$ for the fractal models used above.