It has been suggested recently that electron associative detachment from negative atomic oxygen ions provides an additional source of free electrons in electric discharges of the mesosphere, the sprites, and gigantic jets. Here we study attachment under some simplifying assumptions and show that the threshold field decreases with time and can reach values well below the conventional threshold field. The concept of a fixed threshold field therefore itself breaks down. We find that the growth rate decreases with decreasing electric field and that long exposure time of electric fields therefore is needed for electron avalanches to grow. Detachment is likely to affect the conductivity of streamer filaments and other long-lasting space charge structures like gigantic jets or the ionization state of the mesosphere when illuminated by thunderstorm fields. Detachment by itself does not directly affect small-scale streamer formation or explain the time delays of sprites as proposed by others.
 Sprites are electric discharges in the mesosphere powered by the quasi-electrostatic field following a positive cloud-to-ground lightning discharge in a cloud below (e.g., the reviews of Neubert et al. ; Pasko and Yair ). The threshold electric field for gas breakdown, Ek, is traditionally thought of as the field value where the creation of new electrons from electron impact ionization of neutrals balances the loss of free electrons from attachment to neutral species (e.g., Raizer ). The field scales with the neutral density, nn, such that Ek/nn is approximately constant with altitude. This convenient scaling law, coupled with the classical view of the threshold field, has been applied to the mesosphere with much success. In the dilute mesosphere, however, it has recently been suggested that the physics also includes the detachment process of electrons from negative ions [Gordillo-Vázquez, 2008; Luque and Gordillo-Vázquez, 2011]. Detachment is now included in models of lightning electric fields propagating into the mesosphere [Hiraki, 2009; Marshall, 2012; Liu, 2012] and of giant jets [Neubert and Chanrion, 2011], and is proposed to explain delayed sprites that are triggered at fields below the classical threshold field [Luque and Gordillo-Vázquez, 2011].
 In the paper we present here, we complement the study of Luque and Gordillo-Vázquez  and Liu  by deriving some simple analytical expressions describing the ionization state of the mesosphere. We find the relevant time constants and discuss the implications for electron avalanche-to-streamer transition and the generation of sprites.
2 Equations and Concepts
 We first look at the general problem of electron multiplication in a gas under the influence of a background electric field, E. The gas resembles the mesosphere with 80% N2 and 20% O2, and we assume that the densities and the electric field are constant in space and time. Following Neubert and Chanrion  and Luque and Gordillo-Vázquez , we assume that the dominant reactions are electron impact ionization of neutrals, dissociative attachment of electrons to molecular oxygen, and associative detachment from negative atomic oxygen:
where M is either oxygen or nitrogen. Attachment removes free electrons and builds up a reservoir of negative atomic oxygen ions, and detachment brings the electrons back in play again. The electron and negative ion densities, ne, ni, are then described by
with γa, γd, and γi as the attachment, detachment, and ionization rates, respectively. The time constants associated with the reaction rates (τ=1/γ) given in Neubert and Chanrion  are shown on Figure 1 as functions of the electric field for nn=1.4×1021m−3, corresponding to ≃70 km altitude (green curves). The field is normalized to the conventional value of the breakdown field, Ek, defined as the field magnitude where the attachment and ionization rates are equal. For fields above Ek, τi is the smallest and therefore ionization is the dominant process. Attachment dominates at intermediate fields and detachment at fields below ∼0.4 Ek.
 The condition ∂ne(t)/∂t=0 defines the threshold electric field. When γd=0, we have the conventional threshold, Ek, where γi(E)≃γa(E). The reduced threshold field, Ek/nn, is ∼100–120 Td, corresponding to ∼2.6–3.2 MV m−1 at sea level (e.g., Raizer ) and ∼119–146 V m−1 at 70 km altitude.
 In the mesosphere, where γd>0, the breakdown field is where
 This condition is time dependent. As the negative ion density builds up in time via the attachment process, breakdown may occur for lower values of γi and therefore for lower electric fields. With detachment, then, the concept of a fixed breakdown field itself breaks down.
 The equations (1) and (2) form a linear system that is solved by finding the eigenvalues and eigenvectors of its corresponding matrix:
where subscripts 1,2 correspond to the +,− signs, respectively, ne(0)=noe, ni(0)=noi, and λ1, λ2 are the eigenvalues. Our formulation is similar to that of Liu , except the terms of the equations are combined differently.
 For γd=0, we capture the conventional scenario and Γ simplifies to Γ =|γa−γi|. If E<Ek, we have λ1=0,λ2=−Γ, and equation (4) describes a decaying electron density. If E>Ek, then λ1=+Γ,λ2=0, which describes an exponentially increasing electron density. Figure 1 shows the growth and decay time constant, τ∗=1/Γ, as a function of the normalized electric field for neutral densities corresponding to ≃70 km altitude (dotted curve).
 When γd>0, one can show that λ1≥0 and λ2≤0 and that they have finite values both below and above Ek. Defining the time constants τ1=1/λ1 and τ2=−1/λ2, this implies that component 2 of the solution disappears for t>>τ2 and that the electron and negative ion densities converge toward the same time constant of growth, τ1. The time constants are shown in Figure 1 (red curves) for the atmosphere corresponding to 70 km altitude. The time constants of reactions and growth/decay are consistent with those of Liu , Figures 1 and 9. Note that the reaction rates adopted here are positive even for E-fields approaching 0, such that the electron density will grow even for very low field values, although with a very large time constant. This can be understood in terms of statistics and probabilities of single electrons achieving an energy sufficient for ionization and attachment, or the ion for detachment.
 The electron and ion densities given by equations (4) and (5), and their ratio η(t)=ni(t)/ne(t), are shown on Figure 2 (top), as functions of time, with E/Ek=0.8,nio=0, and neo at some finite value. The electron density first decreases because of attachment, then, after a few τ2, the electron and ion densities grow with a time constant τ1, and η reaches an asymptotic value, ηa, given by
 One can also reform equation (3) and express the threshold for growth of the electron density in terms of ηt:
where the electron density will grow if η>ηt. In the following, we refer to this as the discharge regime.
 Figure 2 (bottom), shows ηa and ηt as functions of the electric field. With the adopted formalism for the reaction rates, they are independent on nn. One can show that ηa>ηt for all electric fields, and that the gas in principle will reach the discharge regime for any field value as long as the exposure time is long enough.
 We note that τ2 is the time scale to reach exponential growth in the electron and negative ion densities. When the electric field is below Ek, we have τ2<τ1, and the gas reaches the discharge regime faster than the time scale of growth. Sprites that are delayed relative to the parent lightning stroke and appear to be triggered at fields below Ek [Gamerota and Cummer, 2011] are therefore not delayed because it takes time to build up ni, as proposed by [Luque and Gordillo -Vázquez, 2011], since this time is of the order of τ2, but must be associated with the growth time τ1 and the time to build up the field to a sufficient magnitude which, on the other hand, is lower when detachment is included.
 While the gas is in the discharge regime for t>τ2, the growth rate is small and probably poorly characterized for low electric fields, and in a real mesosphere, the situation is more complex, as will be discussed in the following.
3 Application to the Mesosphere
 The ion chemistry of the mesosphere is quite complex and includes a multitude of processes not considered here (e.g., recombination) which will place a lower limit on the electric field required for the discharge regime. But more important is the limit imposed by the short exposure to the electric fields from thunderstorm processes.
 When an electric field is imposed on the mesosphere from a lightning discharge, electric currents are set up that will cancel the field. The dielectric relaxation time is defined as τσ=εo/σ, where εo is the vacuum permittivity and σ is the electric conductivity. The conductivity of the mesosphere generally increases with altitude such that the relaxation time decreases with altitude. The actual response time of the mesosphere to an imposed electric field may be longer and is of the order of a few τσ [Pasko et al., 1997, Figures 26 and 27]. It is therefore expected that processes driven by an electric field in general must have time constants of the order of, or smaller than, τσ, or they will not develop before the field is shorted out.
 To discuss the implications of the detachment process on the mesosphere, we adopt the formalism and approach of Neubert and Chanrion  to describe the dielectric relaxation time. The neutral atmosphere is the same as in Neubert and Chanrion , which has a total number density nn = 1.4 ×1021m−3at 70 km altitude. An ion density is assumed below 70 km altitude that gives conductivities similar to Pasko and Inan , profile “a”, with no negative O− ions. For the electron density, we use the formulation of Wait and Spies  that expresses the density (m−3) as a function of altitude, z (in km), in terms of the reflection height, h′ (in km), and the sharpness factor, β (in km−1): noe(z) = 1.43 ×1013exp (−0.15h′)exp(β−0.15)(z−h′). Values of h′ and β for the nighttime D-region have been estimated from ELF-VLF wave propagation characteristics at mid- and low latitudes in a number of studies [Thomson and McRae, 2009; Han and Cummer, 2010; Maurya et al., 2012]. We have chosen two profiles that represent nighttime conditions. The first profile has h′ = 85 km and β=0.38 as in Neubert and Chanrion  and the second profile has h′ = 85 km and β=0.65, as suggested by Thomson and McRae . The electron density mainly affects regions above ∼70 km altitude. To get the electric conductivity, we adopt the mobility of the charged species as given in Neubert and Chanrion .
 The dielectric time constants , and are shown on Figure 1 as functions of the electric field at 70 km altitude. At this altitude, the ion conductivity dominates and there is little difference between the two. The condition τ1<τσ requires E/Ek>0.73. The time constants are also shown on Figure 3 from 50 to 80 km altitude for three values of the electric field: E/Ek = 0.73, 1, and 2. One can see that the condition τ1<τσ is met at progressively lower altitudes as the field is lowered. However, this condition is not sufficient for reaching a significant and measurable effect. Instead, we adopt the approach used when discussing the formation of streamers from single electron avalanches. A streamer carries a significant space charge field in the tip, which means that once a streamer is formed, the self-generated field relaxes the requirement on the background electric field for sustained propagation. The background field for downward propagating, positive streamers of sprites is ∼Ek/6 [Raizer, 1997]. We use the criterion for streamer formation proposed by Raether and Meek in the approximation that the dynamics is determined by free electrons. It requires that the electron avalanche grows for a time ∼20 τ1 creating 108−109electrons. Further discussions on the avalanche-to-streamer transition are found in Montijn and Ebert  and Qin and Celestin . We emphasize that we at this point are not considering streamer formation, we only adopt the criterion of streamer formation to estimate conditions for significant effect of electron avalanches on the mesosphere.
 In Figure 4 is shown τ1 as a function of electric field and altitude (color coded). Also shown are the functions τ1=τσ and τ1=τσ/20 for the two conductivity profiles. The top panel is for the classical view that assumes γd=0 and the bottom panel for the finite values of γd adopted in the present paper. The Raether-Meek criterion is satisfied in the region below τ1=τσ/20. Comparing the two panels of Figure 4, we see that the region is not affected much by the detachment process when the field is well above Ek. Here, the region extends to 75–80 km altitude, where the electron conductivity plays a major role as seen in the difference between and . For fields around Ek and below, however, there is significant effect of detachment and the Raether-Meek criterion is satisfied for fields down to ∼0.7 Ek, reached at 50 km altitude, with τ1< 1 ms. Discharge initiation will still be favored at higher altitudes (as observed) because the normalized field from a thunderstorm lightning discharge, E/Ek, increases with altitude until ∼75–80 km altitude, above which the high electron conductivity shorts out the field.
 Electric fields in the mesosphere are primarily from two sources: one is the large-scale lightning field propagating to the mesosphere from the clouds below, and the other is the small-scale space charge field of local streamers.
 The lightning field may last for several tens of milliseconds for long-duration continuum currents. In addition, the temporal variation of the source field may grow in time to partly cancel the neutralizing dielectric currents in the mesosphere, thereby extending in time the field of the mesosphere to 100 ms or more [Gamerota and Cummer, 2011]. A long-duration, large-scale field from lightning may, therefore, stimulate electron avalanches down to ∼0.7Ek. The lowest value of the threshold is uncertain because it depends on the source field pulse duration and the conductivity in the lower mesosphere, which is not well characterized and may show large variability.
 The electric field of the ionization front of a streamer tip exposes a volume of the atmosphere for a very short time. We can estimate the exposure time from the velocity of the ionization front and its dimension in the direction of the velocity. The velocity measured in sprites is ∼107ms−1 and the sheath dimension is estimated to ∼10–100 m [Stenbaek-Nielsen and McHarg, 2008], which gives an exposure time of ∼10 μs. For electron growth to have significance, the exposure time must be of the order of τ1 or shorter. From Figure 3, it is clear that this requires the field to reach E/Ek∼2 or larger, consistent with estimates based on observations Adachi , Kanmae and Stenbaek-Nielsen , Kuo and Hsu , and Liu et al. . At these field magnitudes and temporal scales, detachment is likely to play a minor role. We further note that even if the mesosphere is preconditioned with negative oxygen ions from long-duration fields, this will not affect streamer formation and propagation because nio does not enter into the formula for the growth rate.
 However, the field inside the longer lasting streamer filaments is expected to be below Ek because of the enhanced conductivity and may be in the range where detachment dominates, 0.1−0.4Ek. The detachment process is therefore likely to increase the conductivity of filaments and other structures such as the gigantic jet, which will further decrease the electric field and may possibly have an indirect effect on the dynamics of the ionization fronts [Gordillo-Vázquez and Luque, 2010].
 From the above arguments we conclude that the observations of delayed streamer formation at E/Ek down to ∼0.15 reported in Gamerota and Cummer  are not explained by the detachment process as proposed by Luque and Gordillo-Vázquez . This means that neither the delay, as discussed earlier, nor the generation at low electric fields are influenced by detachment. We suggest instead, as recently also proposed by Liu and Kosar ; Kosar and Liu , that for low streamer initiation fields, patches of pre-ionization exist in the mesosphere with densities that allow the streamers to be formed directly rather than from a single electron avalanches. In this case, the lower threshold for propagation, E≃Ek/6, and the curves τ1=τσ apply. We propose that the delay of the streamer generation is an effect of the temporal variation of the source electric field as modeled by Luque and Gordillo-Vázquez . Since the fields are shorted out faster at higher altitudes, long duration fields will tend to favor lower altitudes as observed.
 The authors thank A. Schou-Jacobsen who did initial analytical work on this problem. The work was developed in the framework of the European Science Foundation (ESF) Research Network Project (RNP) Thunderstorm Effects on the Atmosphere-Ionosphere System (TEA-IS).
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.