#### 3.1. Fluid Model of RRE Avalanche

[6] We use a fluid model, neglecting the RRE velocity dispersion, to estimate TGF production by lightning EMP. The necessary condition for avalanche production of RRE is for EMP electric field to exceed the runaway threshold *E*_{t} with the reduced value *E*_{t}/*N*_{m} ≈ 8 Td, where *N*_{m} is the air molecule number density and 1 Td = 1 townsend = 10^{−21} V m^{2}. Without magnetic field, the RRE avalanche growth rate can be approximated by a polynomial based on the Monte Carlo calculations of *Lehtinen et al.* [1999]:

where *Z*_{m} ≈ 14.5 is the average molecular nuclear charge, *r*_{0} ≈ 2.82 × 10^{−15} m is the classical electron radius and *c* is the speed of light.

[7] The RRE number density *N* satisfies the equation

where *S* is the external source.

[8] A constant uniform source of seed MeV electrons was used in earlier works [*Lehtinen et al.*, 1999]. However, *Gurevich and Zybin* [2001] recognized that RRE avalanche may be seeded not necessarily by the continuous flux of bulk cosmic rays, but also by individual particles of energy _{CR} ∼ 10^{8}–10^{10} MeV. The flux of particles ≳10^{10} MeV is ∼0.1 km^{−2} s^{−1} [*Eidelman et al.*, 2004, chapter 24]. For a thundercloud area of ∼100 km^{2} [*Lehtinen et al.*, 1997], an extensive air shower may occur every ∼100 ms, and may coincide with a high QES and EMP electric field.

[9] The initial seed for an RRE avalanche is the maximum number of electrons in a cosmic ray shower [*Hillas*, 1982]:

where _{c} ≈ 80 MeV is the critical energy in air, which gives *N*_{0} ∼ 10^{5}–10^{7} for _{CR} ∼ 10^{8}–10^{10} MeV.

[10] If the source *S* consists only of the *N*_{0} electrons initially created by the cosmic ray air shower at position **r**_{0}, then the total number of RRE is

where the position of avalanche **r**(*t*) is found by integrating (*t*) = **v**(**E**) with **r**(0) = **r**_{0} and **v**(**E**) being the RRE drift velocity of ≈(0.5–0.9)*c*. The electron motion at >30 km altitude is constrained along **B**_{E}, so that the effective *E* in (1) is the projection of **E** onto **B**_{E} [*Gurevich et al.*, 1996].

[11] The number of γ-photons emitted by the RRE beam per unit time is

where *v* = ∣**v**∣, and χ ≈ 10^{−28} m^{2} is the bremsstrahlung cross section [*Heitler*, 1954, p. 245] integrated over the RHESSI photon detection range of 0.003–17 MeV [*Lin et al.*, 2002], assuming ∼35 MeV electrons [*Smith et al.*, 2005].

[12] The total number of photons in an average RHESSI TGF is ≳3 × 10^{15}, assuming isotropicity [*Smith et al.*, 2005]. However, bremsstrahlung emission is likely to be anisotropic due to collimated motion of electrons and since the bremsstrahlung cross section is forward-directed within ∼γ^{−1} ≈ 1/70 radians, where γ is the relativistic factor of ∼35 MeV electrons. Thus, the requirement on total number of photons is reduced to ∼3 × 10^{15}(1/70)^{2} ≈ 10^{12}, leading to

for initial electron number *N*_{0} = 10^{5}–10^{7}.

[13] Higher altitudes are favored in our model due to a slow 1/*R* decrease of *E* for an EMP. Since photons are produced at altitudes >35 km, propagation through the atmosphere does not significantly reduce their observed flux.

#### 3.2. Production of TGF by an EMP From a Rapidly Moving Return Stroke

[14] The electric field of the EMP radiated by a lightning return stroke propagating upward with velocity *v*_{rs} is [*Krider*, 1992]:

where *R* is the distance from source, θ is the zenith angle and β = *v*_{rs}/*c*, assuming that the EMP is emitted close to the surface of the Earth. The current *I*(*t*) is assumed to have a non-zero constant value *I*_{p} during a time interval of ∼50 μs. This corresponds to the propagation time to the altitude *h* = 15 km [*Rakov and Tuni*, 2003] for β ≈ 1, which approximately corresponds to the upper cloud boundary (which can be as high as 20 km). We note that the typical lightning channel length is usually shorter, ∼7.5 km.

[15] The electric field given by (4) is higher in comparison to a nonrelativistic case by a factor (1 − β^{2} cos^{2}θ)^{−1} > 1, relaxing the peak current requirements to exceed RRE threshold *E*_{t}, at the altitudes indicated in Figure 1. The validity of equation (4) is discussed in detail in Section 4.1.

[16] The ground-observed average values of β are ∼0.55 for −CG and ∼0.3 for +CG [*Rakov and Uman*, 2003, p. 232]. VHF emissions from return strokes [*Shao et al.*, 2003] suggest an average value of β = 0.75, as determined from the beam pattern observed by the FORTE satellite. Under certain conditions, values as high as 0.99 have been considered [*Rakov and Tuni*, 2003]. We note, however, that the high measured values of β ≈ 1 might result from the initial bidirectional development of the return-stroke channel [*Rakov and Uman*, 2003, p. 414]. The EMP electric field for *I*_{p} = 200 kA and β = 0.99 in Figure 2 demonstrates that the RRE avalanche threshold *E*_{t} is exceeded at a wide range of altitudes. Discharge currents of 200 kA are fairly common for +CG discharges [*Rakov and Uman*, 2003, p. 215]. There currently exist no direct current measurements >300–350 kA. Currents up to 500 kA were reported by the National Lightning Detection Network (NLDN), with an expected maximum of ∼800 kA (K. Cummins, personal communication, 2005). However, these estimates can involve an up to an order of magnitude uncertainty, depending on *v*_{rs}. Furthermore, the NLDN field-to-current conversion equation is calibrated only for smaller negative subsequent strokes, as opposed to first negative or positive strokes, the latter type being considered herein.

[17] Figure 3 shows the calculated number of γ-photons, maximized over all possible RRE avalanche starting points **r**_{0}. The optimal **B**_{E} dip angle (at which RRE motion along **B**_{E} partially synchronizes the electrons with the wave) is 45° for −CG and 0° for +CG. The case of −CG is illustrated in Figure 2, showing that electrons accelerated by the *E* field also acquire a velocity component along the direction of the EMP. For +CG, the direction of electron motion is reversed, resulting in a different optimal **B**_{E} dip angle. There is a dramatic increase in TGF production for β = 0.995. Electrons moving along **B**_{E} at 45° do not escape the atmosphere since, as schematically shown in Figure 2 for a −CG, their upward motion is interrupted at the axis of the discharge. This important feature of our mechanism is consistent with the fact that previously predicted upward escaping energetic electron beams or “curtains” [*Lehtinen et al.*, 2000] are not observed on satellites such as SAMPEX.