### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] Sprites are mesospheric discharges that carry significant electrical currents and produce electromagnetic radiation observed typically in the extremely low (ELF) to ultra low (ULF) frequency bands. In this letter, we present the first theoretical estimates of the electromagnetic radiation produced by individual sprite streamers using simulation results from a plasma fluid model. It is demonstrated that the spectral content of the radiation produced by sprite streamers is a function of the air density *N*and the lightning-induced quasi-static ambient electric field*E* in the regions of space where the sprite streamers are propagating. We demonstrate that the exponential growth of the current in sprite streamers at 75 km would be preferentially associated with electromagnetic radiation in the frequency range from 0 and up to ∼3 kHz, whereas the growth of the streamer current at 40 km could produce radiation with frequencies up to ∼300 kHz, consistently with the scaling of atmospheric air density. We further conjecture that the periodic branching of streamers may lead to a radiation spectrum enhancement in the very low (VLF) to low frequency (LF) range.

### 2. Model Formulation

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[3] In order to calculate the electric currents flowing in the body of sprite streamers, a two-dimensional cylindrically symmetric (*r*, *z* dependent) plasma fluid model developed by J. Qin et al. (Dependence of positive and negative sprite morphology on lightning characteristics and upper atmospheric ambient conditions, submitted to *Journal of Geophysical Research*, 2012) is used to simulate the dynamics of sprite streamers. In this model, the chemical reactions accounted for include the electron impact ionization of N_{2} and O_{2}, the electron dissociative attachment to O_{2}, and the electron detachment process O^{−} + N_{2} e + N_{2}O [*Qin et al.*, 2012, equations (1)–(4)]. Photoionization processes are included using the three-group SP_{3} model developed by *Bourdon et al.* [2007]. The motion of charged species is simulated by solving the drift-diffusion equations for electrons and ions coupled with the Poisson's equation [*Qin et al.*, 2012, equations (5)–(9)]. The transport equations for charged species are solved using a flux-corrected transport technique that combines an eighth-order scheme for the high-order fluxes and a donor cell scheme for the low-order fluxes.

[4] Sprite streamers are primarily vertical filamentary plasma discharges that have characteristic radii ∼100 m at ∼75 km [*Stenbaek-Nielsen and McHarg*, 2008], and become smaller at lower altitudes (e.g., ∼1 m at 40 km) according to the similarity laws [*Pasko et al.*, 1998; *Liu and Pasko*, 2004]. In order to calculate their far field radiation, sprite streamers can be considered as vertical antennas with current variation at any locations *I*(*z*, *t*) calculated by the plasma fluid model. This assumption is justified by our results showing that the radius of the streamer cross-section is at least three orders of magnitude smaller than the wavelength of any radiation under consideration. The analytical time-domain solution for the azimuthal magnetic component of the EM field from a finite antenna of length*H* (see Figure 1) is given by *Uman et al.* [1975]:

where *i* is the current flowing in the antenna, *θ* is the polar angle of the receiver with respect to the source location, *R* is the distance from the antenna to the receiver (see Figure 1), *μ*_{0} and *c* are, respectively, the permeability and speed of light in vacuum. Note that although it is included in our calculations, the contribution from the first term in equation (1) is negligible for the far field radiation. Fourier transform is then applied to calculate the spectrum of the sprite radiation using *B*_{ϕ}(*t*), which is obtained numerically by applying equation (1) with the current derived from the streamer modeling.

### 3. EM Radiation From the Exponentially Growing Sprite Streamers

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[5] Stable propagation of streamers requires an applied electric field that is greater than the critical fields *E*_{cr}^{±}. The minimum electric field for stable propagation of positive streamers is *E*_{cr}^{+} ≃ 4.4 *N*/*N*_{0} kV/cm and that for negative streamers is *E*_{cr}^{−} ≃ 12.5 *N*/*N*_{0} kV/cm, where *N* is the air density at the altitude of interest and *N*_{0} ≃ 2.688 × 10^{25} m^{−3} is its reference value at ground level [e.g., *Pasko et al.*, 2000]. Streamers propagating in an electric field higher than the critical fields *E*_{cr}^{±} experience an exponential growth of transverse physical dimension of their heads and propagation speed in time (∼exp(*ν*_{0}*t*), where *ν*_{0} is the exponential growth rate of streamers) [e.g., *Liu et al.*, 2009; *Celestin and Pasko*, 2011; *Kosar et al.*, 2012; *Qin et al.*, 2012]. *Liu et al.* [2009] and *Kosar et al.* [2012] have calculated that for realistic applied field magnitudes the growth rate *ν*_{0} of streamers at 75 km altitude is on the order of 10^{3} s^{−1}, which corresponds to frequencies of ∼1 kHz. *Kosar et al.* [2012] have also shown that the exponential growth of streamers propagating under given supercritical fields scales according to the similarity laws for streamer breakdown proposed by *Pasko et al.* [1998] (∼1/*N* at typical sprite altitudes of ∼40–90 km where quenching effects of photoionization are negligible [*Liu and Pasko*, 2004]). Note that the air density is *N*_{40} ≃ 8.31 × 10^{22} m^{−3} at 40 km and *N*_{75} ≃ 8.94 × 10^{20} m^{−3} at 75 km, that leads to a scaling factor of *N*_{40}/*N*_{75} ≃ 93. Therefore, one can expect that the growth rate of streamers at 40 km altitude is on the order of 10^{3} × *N*_{40}/*N*_{75} ≃ 10^{5} s^{−1} corresponding to frequencies of ∼100 kHz. It is the main purpose of this section to demonstrate that the exponential growth of the electric current in sprite streamers produces radiation up to ∼3 kHz (VLF) at 75 km and up to ∼300 kHz (LF) at ∼40 km.

[6] Figure 2 shows the vertical current densities *J*_{z}(*r*, *z*) of two sprite streamers developing at 75 km and 40 km, respectively, with the parameters of the simulation described in the caption. In each case, the current density *J*_{z}(*r*, *z*) is highly enhanced and rapidly varying in the streamer head that serves as the main radiator, followed by a streamer body with almost constant and relatively weak current density. It is shown that the size of the streamer head *R*_{s} (scales as 1/*N*) at 75 km is ∼100 times larger than that of the streamer head at 40 km, and the current density *J*_{z}(*r*, *z*) (scales as *N*^{2}) at 75 km is ∼10^{4} times weaker than that of the streamer at 40 km, which is consistent with the similarity laws (i.e., *N*_{40}/*N*_{75} ≃ 93).

[7] The vertical current *I*_{z}(*z*) flowing in the streamer body is calculated by integrating the current density *J*_{z}(*r*, *z*) along the radial direction and shown in Figure 3. Note that the current *I*_{z}(*z*) in the body of the streamer does not depend on air density, since *I*_{z}(*z*) = *J*_{z}(*r*, *z*) × *A*_{s}, where *J*_{z}(*r*, *z*) scales as *N*^{2} and the area of the streamer cross section *A*_{s} ≃ *π R*_{s}^{2} scales as *N*^{−2}. The streamer currents at 75 km and at 40 km are therefore comparable. On the other hand, the timescale of the streamer development, which scales as 1/*N*, at 40 km is ∼93 times shorter than that of the streamer at 75 km (see Figure 3). Therefore, the growth rate *ν*_{0} of the streamer current at 40 km is ∼93 times greater than that of the streamer current at 75 km [see also *Kosar et al.*, 2012, Figure 5].

[8] The top curve in Figure 4 shows the growth rate of the magnetic field *B*_{ϕ} corresponding to the current variation of a streamer developing at 75 km in an electric field of 1.0*E*_{k} (see Figure 3a). Note that *B*_{ϕ} is obtained by applying equation (1) using current derived from the streamer modeling. Similarly to the growth of radius [*Kosar et al.*, 2012], the growth rate of streamer current is not only a function of altitude (see Figure 3) but also a function of the applied electric field (see Figure 4).

[9] For a magnetic field *B*_{ϕ}(*t*) = exp(*ν*_{0}*t*) in arbitrary units that exponentially grows from *t* = 0 to *t*_{0}, its spectrum can be calculated analytically from its Fourier transform as follows:

The magnitude of *B*(*ω*) is:

where *ω* is the angular frequency, and for ≫ 1, one gets:

where *ω* = 2*πf* and *f* is the frequency. Note that the condition ≫ 1 can be fulfilled, for example at 75 km, if *t*_{0} ⪆ 0.3 ms since *ν*_{0} ≃ 10^{4} s^{−1} (see Figure 4). This spectrum is a square-root Lorentzian density function peaked at*f*= 0 Hz and decreasing rapidly at higher frequency with a half-width of*f*_{half} = *ν*_{0}(*h*, *E*)/2*π.* Table 1 lists the *f*_{half} values for altitudes *h* = 75, 40 km and for field from *E* = 0.5 to 1.0*E*_{k}. Note that the *f*_{half} values at 75 km in Table 1 are calculated using *ν*_{0} obtained directly from plasma fluid simulations of sprite streamers, whereas these at 40 km are estimated using the values at 75 km and the air density scaling factor *N*_{40}/*N*_{75} ≃ 93. The *f*_{half} values at other altitudes *h* between 40 and 75 km can be estimated using a scaling factor of ∼exp , where 7.72 km is the scale height of the atmospheric density. We test the accuracy of the estimates in Table 1 by calculating *f*_{half}(40 km, 1.0*E*_{k}) using the streamer current shown in Figure 3b. The result is 271 kHz, and it has a small 1.5% difference from 267 kHz listed in Table 1, which is due to the different ambient electron densities at 75 km and 40 km taken into account in the simulations (see the caption of Figure 2). An important conclusion based on the above results is that sprite streamers propagating at ∼40 km altitude would naturally produce LF radiation.

Table 1. Half-Width of the Spectrum of*B*_{ϕ} (kHz) as a Function of Altitude and Applied Electric Field | 1.0*E*_{k} | 0.9*E*_{k} | 0.8*E*_{k} | 0.7*E*_{k} | 0.6*E*_{k} | 0.5*E*_{k} |
---|

75 km | 2.87 | 2.09 | 1.56 | 1.17 | 0.84 | 0.59 |

40 km | 267 | 194 | 145 | 109 | 78 | 55 |

### 4. EM Radiation From the Periodic Branching of Sprite Streamers

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[10] It should be emphasized that the spectrum distribution shown by equation (4) only accounts for the EM radiation from the exponential growth of the current in sprite streamers. Moreover, when the radius of the streamer head becomes several times larger than the photoionization range *L*_{ph}, the stable growth of sprite streamers is no longer possible, because the density of photoelectrons created ahead of the streamer is insufficient to support the streamer advancement [*Liu and Pasko*, 2004]. These streamers with large heads will split into several small-scale streamers with each head being able to propagate and grow exponentially again [*Liu and Pasko*, 2004]. Note that the photoionization range *L*_{ph}, that is determined by the absorption length of UV photons by oxygen molecules, is inversely proportional to the air density *N*and therefore decreases exponentially with decreasing altitude. This streamer branching phenomenon has been commonly observed in high-speed video observations of sprites [e.g.,*Cummer et al.*, 2006; *McHarg et al.*, 2010], and it is also clearly shown in these videos that sprite streamers branch more frequently when propagating toward lower altitudes where the photoionization range *L*_{ph} is shorter.

[11] It is reasonable to assume that the vertical current *I*_{z}(*z*), that increases exponentially before the sprite streamers branch (see Figure 3), may experience a sudden decrease during the extremely rapid process of branching, since right after branching most of the small-scale streamers do not propagate vertically (see the streamer branching process shown in*McHarg et al.* [2010]). After this sudden decrease, the vertical current *I*_{z}(*z*) starts another round of exponential growth before the sprite streamers branch again. When propagating downwards, the branching of sprite streamers occurs more frequently and introduce periodicity in the variation of vertical streamer current that may lead to narrowband enhancements in the spectrum of sprite radiation.

[12] The time interval between two successive streamer branching (i.e., the period *T*_{b} of the vertical current variation that determines the peak frequencies in the spectrum of sprite radiation) is a function of air density *N*, and its order of magnitude can be estimated using the following two different approaches leading to similar results.

[13] The first approach is based on the theoretical work of *Liu and Pasko* [2004] in which the photoionization range *L*_{ph} is stated to be (*χ*_{min}*p*_{O2})^{−1} and *L*_{ph} ≃ 0.2 cm at ground pressure in air, where *χ*_{min} and *p*_{O2} are the absorption coefficient and the partial pressure of molecular oxygen, respectively. *Celestin and Pasko* [2011] estimated that the value of the branching radius *R*_{b} is ∼0.5 cm at ground pressure and then found the corresponding time of streamer branching *T*_{b0} ≃ 11.2 ns and the corresponding streamer length *L*(*t*_{b}) = 7 cm. Using the air density scaling factors, while neglecting the non-similarity of photoionization, we find that at 40 km*T*_{b40} ≃ 3.62 *μ*s corresponding to frequencies of ∼276 kHz, and at 75 km *T*_{b75} ≃ 0.34 ms corresponding to frequencies of ∼2.97 kHz.

[14] The second approach is based on the experimental work of *McHarg et al.* [2010]in which successive branching of sprite streamer propagating at ∼75 km has been clearly documented with sub-millisecond time resolution. The example used in this estimate is the one presented in*McHarg et al.* [2010, Figure 2], which shows that a sprite streamer split at ∼76 km into five small streamers that propagated down to ∼70 km altitude where they split again into smaller streamers. Assuming that the streamer velocity was 1.8 × 10^{7} m/s [*McHarg et al.*, 2010], the time interval between successive branching at ∼75 km is 0.33 ms corresponding to frequencies of ∼3.00 kHz. Using the air density scaling factor of ∼93, at 40 km the time interval is ∼3.58 *μ*s corresponding to frequencies of ∼279 kHz.

[15] The results obtained from the above two different approaches agree well. However, we emphasize that existing high-speed observations are not yet able to resolve the branching dynamics at low altitudes and the actual frequencies of the radiation from the periodic branching of sprite streamers at 40 km may be lower than the above estimates because the reduced electric field at 40 km produced by the lightning discharge is small, which leads to slow growth of sprite streamers. Nevertheless, the above estimates, along with the calculations inSection 3, predict that in the case of +CGs associated with large charge moment changes, sprite streamers that are able to propagate down to ∼40 km altitude can radiate electromagnetic field with frequencies in the LF range (30–300 kHz).

### 5. Observability of Sprite Radiation

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[16] We note that stable propagation of streamers requires the lightning-induced quasi-static electric field to be larger than the critical field*E*_{cr}^{±} as discussed in Section 3. Therefore, for streamers to propagate down to ∼40 km, a large charge moment change of ∼1000 C km is necessary. Besides, a large charge moment change that leads to a large streamer initiation region is also more likely to produce a large constellation of sprites (e.g., jellyfish sprites) [*Qin et al.*, 2011, 2012], which may be one necessary condition for sprite radiation to be detectable (see figures in *Cummer et al.* [1998]). Note that when measured several hundred kilometers away from the parent lightning discharge, the typical magnitude of the magnetic field radiated by sprite current is on the order of 10^{−9} T [e.g., *Cummer et al.*, 1998; *Cummer*, 2003]. The number of streamers required to produce this amount of radiation can be estimated using Figures 3a and 4. In Figure 3a, the length of the streamer grows exponentially in time, and using extrapolation (see *Liu et al.* [2009] and discussion therein) it is found that it takes ∼1 ms for this streamer to become 6 km long. In ∼1 ms, the magnetic field shown by the top curve in Figure 4 grows up to ∼1.2 × 10^{−12} T. Having assumed that this is the largest magnitude of the magnetic field radiated by an individual streamer, since branching occurs after streamer propagates ∼6 km at 75 km altitude [*McHarg et al.*, 2010], it requires ∼1000 individual streamers to produce the magnitude of the sprite radiation measured by *Cummer et al.* [1998] and *Cummer* [2003].

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. EM Radiation From the Exponentially Growing Sprite Streamers
- 4. EM Radiation From the Periodic Branching of Sprite Streamers
- 5. Observability of Sprite Radiation
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[17] In the present study, we demonstrate that the spectral content of the electromagnetic radiation from sprite streamers is highly dependent on the air density *N*, i.e., sprite streamers at lower altitudes with higher air density *N* produce a higher frequency radiation. We calculate that the exponential growth of the streamer current with a rate *ν*_{0} of ∼10^{3} s^{−1} at ∼75 km produces radiation with frequencies up to ∼3 kHz, whereas at ∼40 km with a rate *ν*_{0} of ∼10^{5} s^{−1}it produces a radiation with frequencies up to ∼300 kHz. The ideal spectrum due to this exponential growth is a “square-root” Lorentzian density function peaked at*f*= 0 Hz and decreasing rapidly at higher frequency with a half-width of*f*_{half} = *ν*_{0}/2*π*, where *ν*_{0} scales as the air density *N*and is a function of the lightning-induced electric field*E*. We also conjecture that the streamer branching process may introduce periodicity in the streamer current variation thus would lead to sprite radiation enhanced in the ELF (10 Hz–3 kHz) range for sources at 75 km and enhanced in the LF (30–300 kHz) range for sources at 40 km. The above-discussed frequency variation at different altitudes obeys the similarity laws for streamer breakdown proposed by*Pasko et al.* [1998], and therefore frequencies of streamer radiation at other altitudes can be estimated using linear scaling with air density *N.* Frequencies in the LF range associated with sprites have already been detected and interpreted as produced by relativistic electron beams [*Fullekrug et al.*, 2010, 2011]. The present study shows that sprite streamers could be responsible for at least part of this radiation. We further note that the charge moment change of the sprite-causative lightning discharge that determines the altitude range of the streamer propagation determines the spectrum of the sprite radiation. A lightning discharge associated with a large charge moment change not only enable streamer propagation down to low altitudes but also would be more likely to produce a large constellation of sprites (e.g., jellyfish sprites). In turn, this would likely produce electromagnetic radiation with a detectable magnitude.