This is a commentary on DOI:10.1029/2008JD010468
Aerosol and Clouds
Comment on “‘Seed’ electrons from muon decay for runaway mechanism in the terrestrial gamma ray flash production,” by Gerson S. Paiva, Antonio C. Pavão, and Cristiano C. Bastos
Article first published online: 22 MAY 2010
Copyright 2010 by the American Geophysical Union.
Journal of Geophysical Research: Atmospheres (1984–2012)
Volume 115, Issue D10, 27 May 2010
How to Cite
2010), Comment on “‘Seed’ electrons from muon decay for runaway mechanism in the terrestrial gamma ray flash production,” by Gerson S. Paiva, Antonio C. Pavão, and Cristiano C. Bastos, J. Geophys. Res., 115, D10207, doi:10.1029/2009JD012255., and (
- Issue published online: 22 MAY 2010
- Article first published online: 22 MAY 2010
- Manuscript Accepted: 15 APR 2010
- Manuscript Received: 15 APR 2009
- gamma rays;
 In this comment, we show that the mechanism described by Paiva et al.  is not viable for explaining terrestrial gamma-ray flashes (TGFs). It includes errors in the properties of atmospheric cosmic ray secondary particles and runaway electrons, ignores other more important effects that render their model unnecessary, and makes unreasonable assumptions about the lightning change moment changes and the resulting electric fields, which in turn contradict other aspects of their model. Finally, the paper misrepresents its predictions and how they compare with TGF observations.
 Terrestrial gamma ray flashes (TGFs), discovered in 1994 using the CGRO/BATSE instrument [Fishman et al., 1994], are intense bursts of gamma rays, observed from space, that emanate from the Earth's atmosphere and are associated with positive intracloud lightning flashes [Stanley et al., 2006]. Recent studies of the atmospheric absorption of TGF gamma rays and their geographical distribution suggest that the gamma rays originate at or near the tops of thunderclouds [Dwyer and Smith, 2005; Carlson et al., 2007; Østgaard et al., 2008; Williams et al., 2006].
 To date, the only viable mechanism for generating TGFs is through relativistic runaway-electron avalanches. Runaway electrons are produced when the rate of energy gain from strong electric fields exceeds the rate of energy loss, predominantly from ionization losses [Wilson, 1925]. As the runaway electrons propagate through air, they occasionally experience hard elastic scattering with atomic electrons in air, resulting in energetic secondary electrons, called “knock-on” electrons, that may also run away. The result is an avalanche of high-energy electrons that grows exponentially with time and distance [Gurevich et al., 1992; Gurevich and Zybin, 2001]. This mechanism is called relativistic runaway electron avalanche (RREA) multiplication, sometimes referred to as “runaway breakdown.” As the runaway electrons propagate through air they experience bremsstrahlung interaction with air atoms, emitting X-ray/gamma rays up to several tens of MeV. The average energy of the runaway electrons is about 7 MeV, independent of the electric field, the density of air and the spectrum of the energetic seed particles [Dwyer, 2004; Lehtinen et al., 1999]. The RREAs successfully fit the TGF energy spectrum, lending strength to the basic argument that RREAs are involved [Dwyer and Smith, 2005; Dwyer et al., 2008; Grefenstette et al., 2008; Carlson et al., 2007; Hazelton et al., 2009; Dwyer, 2008].
 An important part of relativistic runaway electron avalanche production is the source of the energetic electrons that “seed” the avalanches. Some possible sources of the seed electrons are the ambient atmospheric cosmic ray flux, cosmic ray extensive air showers, runaway electron production in the high field regions associated with lightning leaders, and relativistic feedback, the latter of which involves backward propagating positrons and X-rays [Dwyer, 2003, 2007]. The roles of ambient atmospheric cosmic ray flux and cosmic ray air showers have been recently called into question by Dwyer , but these issues are still being debated.
2. Paper by Paiva et al. 
 In their paper, Paiva et al.  suggested an alternative mechanism for generating the energetic seed electrons for RREA. They proposed that cosmic ray muons are decelerated by large electric fields generated by lightning discharges. According to these authors, the muons are brought to rest, at which point they decay and produce the energetic seed electrons. According to their calculations for a 1 km2 area at 15 km altitude, 107 such seed electrons are produced in 1 ms. They then postulate that with an additional 1010 avalanche multiplication factor for runaway electrons, a TGF can be explained by their mechanism.
 In this comment, we shall argue that this proposed mechanism is not viable and that the authors' calculations involve errors in cosmic ray and runaway electron physics and unreasonable assumptions about lightning discharges.
 The authors estimate that the muon flux at a 15 km altitude is 1.5 × 105 m−2 s−1, using 104 m−2 s−1 for the sea level muon flux and an order-of-magnitude increase to 15 km. More appropriate numbers would be ∼130 m−2 s−1 (i.e., approximately 1 cm−2 min−1) and a factor of 5 increase for hard muons [see, e.g., Hillas, 1972, p. 50]. As a result, the correct flux of muons at 15 km is about 103 m−2 s−1, more than 2 orders of magnitude smaller than the number calculated by Paiva et al. .
 In fact, the electron and positron cosmic ray secondary component dominates at 15 km and has a flux at least several times larger than the muon component [Hillas, 1972, p. 50]. As a result, when estimating the flux of energetic seed particles, the muons are usually neglected due to their small contribution at thundercloud altitudes. Furthermore, minimum-ionizing muons will produce energetic knock-on electrons (also called delta rays) through hard elastic scattering with air atoms. Even if one were inclined to consider just the contribution from the muons, these secondary electrons would be comparable in number to those produced by muon decays, even if 100% of the muons were brought to a complete halt and decayed at the right spot.
 The interaction length (g/cm2) for a relativistic muon to produce an energetic knock-on electron is given by
where A is the average molar mass of air, Z is the average atomic number of air, NA is Avogadro's number, m is the mass of the electron, E is the kinetic energy of the knock-on electron, and re is the classical electron radius [Hayakawa, 1969]. If we take E = 1 MeV, commonly used in the literature as the kinetic energy of seed electrons for runaway electron avalanches [e.g., Dwyer, 2003, 2007], then equation (1) gives 13 g/cm2, which corresponds to about 100 m at sea level and 700 m at an altitude of 15 km. These numbers are the same as the runaway electron avalanche (e-folding) length for the sea-level equivalent electric field of 350 kV/m [Dwyer, 2003]. In other words, each muon that passes through the runaway electron avalanche region will produce, on average, one seed electron within the first avalanche length through hard elastic scattering. A similar calculation applies to the much more numerous electron and positron cosmic ray secondary component.
 On the other hand, Paiva et al.  estimate that for every incident muon, 15% will have sufficiently low energy to be stopped by the electric field and half of those will be of the correct charge sign. As a result, optimistically, at most 7.5% will be stopped. Considering this factor and the greater number of secondary electrons created directly by cosmic rays at this altitude, the mechanism proposed by Paiva et al. is almost 2 orders of magnitude smaller than the seed electron contribution from other cosmic ray sources.
 The electrical characteristics of the lightning and associated fields are even more unrealistic. The assumed charge and charge moment are too high; the resulting field is unrealistic; and the assumption of this field is even internally inconsistent, since it would produce unreasonably large amounts of runaway electron avalanche multiplication.
 In order to slow down muons, the authors assume a potential difference of 4.0 GV above the thundercloud, resulting from a lightning current that moves 450 C over 7 km vertically within the cloud, producing a charge moment change of 3200 C-km (using equation (2) of Paiva et al. ). This charge moment change is about a factor of 25 larger than the values reported in association with TGFs [Cummer et al., 2005]. Paiva et al. considered positive polarity energetic intracloud lightning (+EIC) as responsible for TGFs and hence their charge moment changes (see paragraph 14 of Paiva et al.). However, typical energetic intracloud flashes (also known as narrow bipolar events, or NBE) have been calculated to move, instead, approximately 0.3 C of charge over about half that distance [Eack, 2004]. Furthermore, the value of 450 C was derived by Paiva et al. by taking the peak current of 450 kA from Inan and Lehtinen  and multiplying it by the duration of 1 ms. It should be pointed out that this peak current was not an observation associated with any TGF-related lightning; rather, it was a value required by Inan and Lehtinen's electromagnetic pulse model of TGFs.
 From electrostatics, the potential difference that must have existed within the thundercloud before the lightning discharge would have had to be at least twice that above the cloud after the lightning discharge, i.e., 8.0 GV. This potential difference is much larger than is thought to occur in thunderclouds and is much larger than values used for other TGF models [Dwyer, 2008]. Furthermore, this 8.0 GV potential difference between 8.0 and 15 km (as used in the paper), gives an average electric field of 1.1 × 106 V/m, which is about twice the local conventional breakdown field at those altitudes. Note that this very large field would extend over 7 km vertically. No such strong electric field has ever been observed in thunderclouds [Rakov and Uman, 2003; MacGorman and Rust, 1998].
 Based upon Monte Carlo calculations, the runaway electron avalanche multiplication factor is given by
where ΔV is the potential difference of the avalanche region in volts and I is the column depth of the avalanche region in g/cm2 [Dwyer, 2003, 2007]. Between 8.0 and 15 km, I = 245 g/cm2. Using ΔV = 8.0 × 109 V gives the unreasonable value of NRE = e1024 = 10445 within the thundercloud. Similarly, the avalanche multiplication factor in the TGF region above the cloud, postulated by the authors, would be about e525 = 10228, which is also not physical. Since large amounts of avalanche multiplication quickly discharge the field, very large multiplication factor would never occur in practice. The relativistic feedback mechanism also prevents such large avalanche multiplication factors from occurring [Dwyer, 2007, 2008]. Furthermore, the amount of avalanche multiplication required in order to stop the muons makes this proposed mechanism completely unnecessary, since even one seed electron would produce more than enough runaway electrons to make a TGF. In other words, in order for the muon mechanism to work, such a large field is required as to make the mechanism unnecessary.
 Finally, in order to use their model to explain the ground level TGF observed by Dwyer et al. , they postulate the existence of a positron avalanche but do not explain the details of the mechanism. Fast electrons (and positrons and muons) easily produce energetic knock-on electrons by ejecting them from atoms. But in order to produce another positron, a fast positron must create a very energetic bremsstrahlung gamma ray, which must, in turn, pair-produce on the nucleus of an atom of air. Neither process has a high probability. Thus there is no such thing as a positron avalanche per se, although there will be a few downward beamed gamma rays from the small positron population that is part of an electron avalanche. For instance, Monte Carlo simulations show that the ratio of positrons to electrons is roughly 1/1000 [Babich et al., 2005]. It is much more likely that the event seen from the ground was an electron avalanche produced by an upward pointing electric field, which accelerates electrons downward, than that it was produced by the tiny population of downward positrons that is part of an upward electron avalanche.
 In summary, the paper by Paiva et al.  assumes unrealistically large electric fields and cosmic ray fluxes to produce an electron source that, even at this exaggerated level, would be insignificant compared with the electrons produced more directly by the same cosmic rays. Even with this artificially high level of seed particles, an unreasonably large avalanche multiplication factor must still be assumed in order to explain TGFs. We note that the 1010 avalanche multiplication factor that the authors claim to be a prediction of their model (in their section 4) is actually a parameter that was adjusted to fit the TGF observations (calculated in their section 3). It is incorrect and a circular argument to claim that this parameter is in good agreement with observations and other work, as they do in their conclusion. Furthermore, the 35 MeV electron energy predicted by the authors (see their section 4) is not a characteristic energy of runaway electron seeds in any model; rather it is close to the highest energy reached in a fully evolved avalanche (well above the average energy of 7 MeV), and therefore irrelevant to a discussion of seeding.
 This work was supported by the NSF grant ATM 0607885 and DARPA grant HR0011-08-1-0088.
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