Chapter 5: Runaway Discharge in a Magnetic Field

• Runaway Breakdown
  • Fully Ionized Case
  • Weakly Ionized Plasma
  • First Neglect Friction: Go to a Parallel Frame
• Boltzmann Equation
  • Mean Free Path
  • Scattering
  • Ionization
• Runaway Discharge in a B Field
  • The Electron Runaway Basin Boundary
  • Electron Runaway in Perpendicular Electric and Magnetic Fields
  • Spreading of the Runaway Discharge in the Presence of a Magnetic Field
  • Electron Runaway Under an Arbitrary Angle Between Electric and Magnetic Fields
• Importance of B Field

Relativistic Beams as a Possible Source of HAL Phenomena

Runaway Breakdown

A new type of electrical air breakdown, called runaway breakdown or runaway discharge, was discussed recently by Gurevich et al. [1992] and applied to the preliminary breakdown phase of a lightning discharge. This phase occurs in the cloud vicinity and marks the initiation of the discharge [Uman, 1987]. The important property of the runaway breakdown is that it requires a threshold field by an order of magnitude smaller than the conventional breakdown discharge under the same pressure conditions. However, its initiation depends on the presence of seed electrons with energy in excess of tens keV in the high electric field region. Such energetic electrons are often present in the atmosphere as secondaries generated by cosmic rays [Daniel and Stephens, 1974].

The possibility for influence of cosmic ray secondaries on the lightning discharges was first discussed in a speculative manner by Wilson [1924]. Recently McCarthy and Parks [1992] attributed X-rays observed by aircrafts in association with the effect of thundercloud electric field on runaway electrons. Gurevich et al. [1992] presented the first consistent analytic and numerical model of the runaway discharge and later on Roussel-Dupre et al. [1994] presented its detailed quantitative application to the X-ray observations.

The physics of the runaway discharge is based on the concept of electron runaway acceleration in the presence of a laminar electric field [Dreicer, 1960; Gurevich, 1960; Lebedev, 1965]. The runaway phenomenon is a consequence of the long range, small angle scattering among charged particles undergoing Coulomb interactions. The scattering cross section decreases with velocity as $\sigma \sim \mathrm{v}^{-4}$ [Jackson 1975]. As a result for a given electric field value a threshold energy can be found beyond which the dynamic friction, as shown in Fig (5.1), cannot balance the acceleration force due to the electric field resulting in continuous electron acceleration.

  

Figure 5.1: Schematic of the dynamical friction force as a function of the electron energy. Trace 1 corresponds to a cold fully ionized plasma (v<v_T), Trace 2 corresponds to high energy electrons. It is valid for any plasmas fully or low ionized. Here E_D is the Dreicer field, while E_cn and E_c0 are the critical and minimum runaway fields correspondingly.

Fully Ionized Case

Here we review the basic physics of the electron runaway in unmagnetized plasmas, starting with the electron acceleration in a fully ionized plasma. The cold electrons having mean directed velocity v less than the electron thermal speed $\mathrm{v}_T=\sqrt{T/m}$ undergo the dynamical friction force

$$F=m\nu _o\mathrm{v}$$

which is proportional to the electron velocity v as shown by trace 1 in 5.1, since at $\mathrm{v}<\mathrm{v}_T$ the electron collision frequency $\nu =\nu _o$ is constant, defined by the electron thermal speed. However, for fast electrons having the velocity larger than $\mathrm{v}_T$, the dynamical friction force given by [Jackson 1975]  

$$F=m\nu (\mathrm{v})\mathrm{v}=\frac{4\pi e^4Zn_e}{m\mathrm{v}^2}\ln \Lambda$$ (11)

reduces when the velocity increases as shown by the trace 2 on Fig. 5.1, where n_e is the electron density, and $\Lambda$ is the Coulomb logarithm. As a result the friction force has a maximum at $\mathrm{v}_T$.The electric field which balances the dynamical friction force at $\mathrm{v}=\mathrm{v}_T$ is known as Dreicer or critical field:

$$E_D=E_{ci}=\frac{4\pi e^3Zn_e}T\ln \Lambda$$

As illustrated by Fig. 5.1 the dynamical friction cannot confine the plasma electrons, which become runaway, if the electric field E applied to the plasma is higher than E_D. On the contrary, if the applied field is less than E_D, electrons are confined by the dynamical friction. At the same time the electrons acquire relatively small velocity $\mathrm{v}_E$,directed along E. Thus the plasma is heated resistively. However, even in this regime the friction force cannot confine fast electrons having energy $\epsilon \gt\epsilon _c\simeq T\frac{E_D}E$ (see Fig. 5.1). Such electrons are continuously accelerated by the electric field and run away. For instance, tokamaks usually operate in the regime of resistive heating, but under some conditions the runaway regime can also takes place in tokamaks.

Weakly Ionized Plasma

A similar situation occurs in a weakly ionized plasma. But unlike the fully ionized plasma, the collision frequency of the low velocities electrons in the weakly ionized gas is determined by the cross-section of the electron-neutral collision, rather than by the thermal electrons. However, for electrons with energies in excess of the ionization potential ($\epsilon \gt\epsilon _i$) the interactions with the nuclei and atomic electrons obey the Coulomb law, hence the dynamical friction force decreases with the electron energy, [Bethe and Ashkin, 1953] as given by Eq. (5.1 ). In this case the value of the critical electric field is given by Gurevich, [1960]

$$E_{cn}=\frac{4\pi e^3ZN}{\epsilon _i}k_n$$

Notice that N is the density of the neutral molecules and Z is the mean molecular charge, which for air is 14.5, and k_n is the numerical factor, determined by the type of the neutral gas. In fact, for hydrogen $k_n\simeq 0.33$, for helium $k_n\simeq 0.30$. If the electric field is larger than E_cn the whole bulk electrons are accelerated. If the field is less than E_cn, only a few electrons having energy higher than $\epsilon _c$ are accelerated

$$\epsilon \gt\epsilon _c=\frac{2\pi e^3ZN\ln \Lambda _n}E$$

where $\Lambda _n\simeq \epsilon _c/Z\epsilon _i$. These are the runaway electrons in the neutral gas.

We emphasize that the amplitude of the electric field leading to the electron runaway is limited, since only for nonrelativistic electrons the dynamical friction force drops when the electron energy increases [Bethe and Ashkin, 1953]. For the electrons having energy greater than $\epsilon \geq 10$ keV the dynamical friction force due to collisions with the neutral gas is given by [Bethe and Ashkin, 1953]

$$F=\frac{4\pi Ze^4N}{mc^2}a\;\Phi (\gamma )$$

 

$$a\Phi =\frac{\gamma ^2}{\gamma ^2-1}\{\ln \frac{mc^2\sqrt{\g... ...\ln 2}2+\frac 1{2\gamma ^2}+\frac{(\gamma -1)^2}{16\gamma ^2}\}$$ (12)

where $\gamma =1/\sqrt{1-\frac{\mathrm{v}^2}{c^2}}$ is the Lorenz factor, I=80.5 eV, $a\simeq 10.87$ in air, and $\Phi$($\gamma$) is F($\gamma$)normalized to unity. For nonrelativistic electrons the dynamical friction force rapidly decreases with the increase of the electron momentum

$$F=\frac{4\pi Ze^4Nm}{p^2}\ln (\frac{p^2}{2mI})$$

The dynamical friction force, Eq. (5.2), reaches its minimum value

$$F_{\min }=\frac{4\pi Ze^4N}{mc^2}a$$

at $\gamma \min =3.42$, $\epsilon _{\min }=1.2$ MeV, $p_{\min }=3.27$ mc. F then slowly (logarithmically) increases with $\gamma$ as it is shown in Fig. 5.1. Therefore, the minimum of the friction force F$_{\min }$is related to the minimum value of the electric field E_c0, which still generates the runaway

$$E_{c0}=\frac{4\pi Ze^3N}{mc^2}a$$

Therefore in the air the runaway electrons could appear in a wide range of electric field E_c0<E<E_cn which spans almost three orders of magnitude. Similar limitation on the electron runaway takes place for the electrons in a fully ionized plasma [Connor and Hastie, 1975].

The detailed discussion of the electron runaway in the air caused by the electric fields due to thunderstorm is presented by McCarthy and Parks [1992]. A new step in the theory of runaway electrons was made by Gurevich et al. [1992], who discussed the possibility of producing an avalanche of runaway electrons. The basic idea is that the fast electrons ionize the gas molecules producing a number of free electrons. Some of secondary electrons have energy higher than the critical energy of runaway. Those electrons are accelerated by the electric field and in turn are able to generate a new generation of fast electrons. This avalanche-like reproduction of fast electrons is accompanied by the exponential increase of the number of thermal secondary electrons, i.e. the electrical breakdown of gas occurs. Such kind of the runaway breakdown is often called runaway discharge. It has the following main properties:

• 1. The critical field of the runaway breakdown is an order of magnitude below the threshold of the conventional air breakdown.
• 2. The runaway discharge has to be triggered by the high energy electrons of $\epsilon \gt\epsilon _c$.
• 3. The runaway discharge develops inside the streamers directed along the electric field [Gurevich et al., 1994].
• 4. The runaway discharge is followed by the generation of x- and $\gamma$ -ray emissions [Roussel-Dupre et al., 1994].

These properties allow us to consider the runaway discharge as the possible mechanism which initializes the lightning discharge during thunderstorms.

To help with future calculations, we note that the value of the critical electric field as a function of altitude is given by $E_{c0}=\frac{4\pi Ze^3}{mc^2}Na\simeq 2\times 10^5\times e^{-h/H_o}$ V/m, where $H_o\simeq 6.5$ km is the atmospheric scale height.

First Neglect Friction: Go to a Parallel Frame

As the friction force becomes smaller with height, the magnetic field must be included in the analysis. This is especially true for the equatorial regions where the laminar electric field due to lightning is predominantly perpendicular to the magnetic field [Papadopoulos et al., 1996]. Note that in the case of E$\bot$B, a geometry expected in the equatorial region, the electrons will be accelerated only when E>B, if we neglect the dynamical friction. Suppose we first neglect the dynamical friction and quantify what is the E field required to produce infinite acceleration for a given E, B configuration where $\theta _o$ is the angle between the electric and geomagnetic fields.

We follow Papadopoulos et al., [1996] and study the runaway acceleration of a test electron in crossed static electric and magnetic fields by transforming the equations of motion to a reference frame moving with the velocity $\beta c$ relative to the ionospheric frame in which the transformed fields E$^{\prime }$, B$^{\prime }$ are parallel. In this frame the electrons can be treated as unmagnetized. Following Jackson [1975] the electric and magnetic field in a moving frame are  

$$\mathbf{E}^{\prime }=\gamma (\mathbf{E}+\mathbf{\beta }\time... ...2}{\gamma ^2+1}\mathbf{\beta }(\mathbf{\beta }\cdot \mathbf{E})$$ (13)

 

$$\mathbf{B}^{\prime }=\gamma (\mathbf{B}-\mathbf{\beta }\time... ...2}{\gamma ^2+1}\mathbf{\beta }(\mathbf{\beta }\cdot \mathbf{B})$$ (14)

$$\gamma =1/\sqrt{1-\beta ^3}$$

with $\mathbf{E}^{\prime 2}-\mathbf{B}^{\prime 2}=\mathbf{E}^2-\mathbf{B}^2$and $\mathbf{E} \cdot \mathbf{B=E}^{\prime }\cdot \mathbf{B}^{\prime }$. We must find the $\mathbf{\beta }$ required to have $\mathbf{E}^{\prime }\Vert \mathbf{B}^{\prime }$, however there is an infinite number of solutions since any frame parallel to the $\mathbf{E}^{\prime }\Vert \mathbf{B}^{\prime }$ direction will also preserve such relationship. We constrain the solution by requiring that $\mathbf{\beta \cdot E=\beta \cdot B=0}$ and $\mathbf{E}^{\prime }\mathbf{\times B}^{\prime }\mathbf{=0}$. Using Eq. (5.3) and Eq. (5.4) we obtain $\mathbf{(E+\beta \times B)\times (B-\beta \times E)=0}$ and with the help of the above constraints we get

$$\mathbf{\beta =E\times B}\frac{E^2+B^2-\sqrt{(E^2-B^2)^2+4(\mathbf{E\cdot B})^2}}{ 2(\mathbf{E\times B})^2}$$

but more relevant is the equation of the transformed fields

$$E^{\prime 2}=\frac 12[E^2-B^2+\sqrt{(E^2-B^2)^2+4(\mathbf{E\cdot B})}]$$

$$B^{\prime 2}=\frac 12[B^2-E^2+\sqrt{(E^2-B^2)^2+4(\mathbf{E\cdot B})}]$$

Notice that in the case $\mathbf{E\cdot B=0}$ we have the two limits:

• If E>B then $B^{\prime }=0$ and $E^{\prime }=\sqrt{E^2-B^2}.$
• If B>E then $E^{\prime }=0$ and $B^{\prime }=\sqrt{B^2-E^2}$

As a result there is not acceleration in the case of crossed electric and magnetic field with B>E. Therefore, the characteristic electric field in SI units is $E(\frac{kV}m)=30\;B(G)$ where B is the local magnetic induction. Since at the equator the magnetic field is B=0.25 G, the required field accelerate an electron corresponds to $E=7.5\;\frac{kV}m$.This threshold applies to the condition that $\mathbf{E\cdot B=0}$ which is the situation for electrons above a thunderstorm close to the equator.

This threshold field is also independent of height at long as the gyroradius is smaller than the mean free path of runaways which occurs at altitudes as low as 25 km for sensible electric fields [Longmire, 1978; Papadopoulos et al., 1994]. The above results can be extended to any angle between the electric and magnetic fields. In the parallel frame, the $E^{\prime }$ field still needs to beat the coulomb friction force, i.e. $E^{\prime }\gt E_{c0}^{\prime }$. Figure (5.2) shows the condition in the $(E/B_o,\theta _o)$ plane where $E^{\prime }=E_{c0}^{\prime }$. Note the constraint at $\theta _o=\pi /2$.

This is a qualitative analysis, that constraints the field to a threshold value $E\sim 7.5\;\frac{kV}m,$ which seems to be a characteristic threshold in the presence of the Earth's magnetic field. Of course the Coulomb friction term is not covariant, hence, its frame transformation is far from trivial. The detailed quantitative approach will be presented in the following sections.

  

Figure 5.2: The curve in the (E,$\theta$) plane where the transformed field is equal to zero.

Boltzmann Equation

The Boltzmann equation for the high energy ($\varepsilon \gt 10$ keV) electron distribution function where the interactions are primary Coulomb in nature can be written as [Roussel-Dupre et al., 1994]  

$$\frac{\partial f}{\partial t}-e(\mathbf{E+}\frac{\mathbf{v}}... ...partial f}{\partial \mathbf{u}}=\frac{\partial _ef}{\partial t}$$ (15)

$$\frac{\partial _ef}{\partial t}=\frac 1{\mathrm{u}^2}\frac{\... ...ial {\partial \mu }[(1-\mu ^2)\frac{\partial f}{\partial \mu }]$$

 

$$+\frac \beta {2a}\int d\Omega \int_{\gamma _i}^\infty d\gamm... ...{\prime })\widetilde{\sigma }(\gamma ^{\prime },\gamma ,\zeta )$$ (16)

where $\mathbf{u}=\gamma \mathbf{\beta }$, $\zeta$ is the angle between the incident and the scattered electron, $\widetilde{\sigma }(\gamma ^{\prime },\gamma ,\zeta )$ is the normalized dimensionless double differential ionization cross-section [Roussel-Dupre et al., 1994], $\gamma _i$corresponds to the ionization energy $\varepsilon _i$, and the time has been normalized to $t\rightarrow t/\tau _c$ with

$$\tau _c=8.5\,e^{h/H_o}\;(n\sec )$$

The three terms of the right part of Eq. (5.6) corresponds to:

• The change of energy due to the friction force for electrons moving through the neutral gas.
• The angular scattering effect.
• The production of secondary electrons creating the runaway avalanche if the secondary has energy greater than threshold energy.

The exact solution of this complicated equation is out of the scope of this work, but it is instructive to understand the time scales and relative importance of the different terms. The main questions is: what are the constraints imposed by the magnetic field?

Mean Free Path

In attempting to apply the concept of runaway breakdown driven by a laminar lightning induced vertical electric field at altitudes exceeding 30 km one is faced with a main difficulty. For such altitudes the effective mean free path $\lambda _R(\gamma )$ for runaway electrons given by

$$\lambda _R(\gamma )=\frac{\mathrm{v}}{\nu (\mathrm{v})}=\bet... ...a \frac{\sqrt{\gamma ^2-1}}{\gamma \Phi (\gamma )}e^{h/H_o}\;m$$

can exceed the electron gyroradius

$$\lambda _B(\gamma )=\beta \gamma \frac c{\Omega _B}\simeq 55\;\beta \gamma \;m$$

in the geomagnetic field.

  

Figure 5.3: (a) The height h at which the electron gyroradius becomes greater than the runaway mean free path. (b) The normalized field required to produce the runaway breakdown at a given energy.

The height h at which the electron gyroradius becomes greater than the runaway mean free path is shown in Fig. 5.3a as a function of the electron energy. So even at relatively low heights $h\sim 25$ km, the magnetic field becomes relevant. Conversely, we could insist in a low energy runaway at the cost of a high field. Figure 5.3b shows the electric field required to produce a runaway

$$\frac{E(\gamma )}{E_{c0}}=\Phi (\gamma )$$

as a function of $\gamma$, normalized by the minimum field E_c0 required to produce a runaway for $\gamma =3.4$. Notice that in this graph we are assuming that magnetic effects can be neglected. The large electric field required to produce a low energy runaway may become a relevant constraint, since lightning induced electric fields of that magnitude may be hard to produce.

Scattering

The magnetic field gyration can be considered as a form of scattering and should be compared with the scattering term of Eq. (5.6). Their ratio can be written as

$$\tau _s(\gamma )\frac{\Omega _B}\gamma \sim \frac 13\frac{\sqrt{\gamma ^2-1}}{(\frac Z2+1)\Phi (\gamma )}e^{h/H_o}$$

as long as the distribution function is not extremely structured in the angle $\mu$, i.e. no filaments exist. First if $\tau _s(\gamma )\frac{\Omega _B}\gamma <1$ the collision dominates the runaway process and the electrons can be considered as unmagnetized. If $\tau _s(\gamma )\frac{\Omega _B}\gamma \gt 1$ then the magnetization becomes a considerable factor. The height at which the gyrofrequency becomes more important than the scattering frequency, $\tau _s(\gamma )\frac{\Omega _B}\gamma =1$, is shown in Fig. 5.4 as a function of the electron energy.

  

Figure 5.4: The height at which the gyrofrequency becomes a relevant factor.

Again we reach the same conclusion that the magnetic field becomes very relevant at heights $h\geq 25$ km, and must be included in the analysis.

Ionization

The time scale for ionization can be found from the last term in the right of Eq. (5.6) and is given by $\tau _i\sim 10\times \tau _c$. Hence the time scale for the avalanche is slower than the time scale for the changes in energy or scattering. Therefore, we can study the runaway process and the threshold requirements for the runaway process using single particle trajectories, as we will do next. As a result, the electric and magnetic fields must be included in a theory of the runaway acceleration for heights above $h\sim 30$ km where the high altitude phenomena seems to occur. Furthermore, we can learn relevant properties of the runaway process by observing single particle trajectories.

gamma rays radio bursts blue jets Does not account very well for red sprites

Runaway Discharge in a B Field

In the presence of a magnetic field the conditions for electron runaway are different from those described for a pure static electric field. In order to discuss the effects caused by magnetic field we will study the motion of fast electrons in the air under the influence of both electric $\mathbf{E}$and magnetic field $\mathbf{B}$. The equation of motion can be found from the Boltzmann equation, Eq. (5.6), and is given by  

$$\frac{d\mathbf{p}}{dt}=e\mathbf{E}+\frac e{mc\gamma }(\mathb... ...es \mathbf{B})-\nu \mathbf{p}\qquad \nu (\gamma )=F(\gamma )/p$$ (17)

where $\mathbf{p}$ is the electron momentum, $\mathbf{\nu }$ is the electron collision frequency, F($\gamma$) is the dynamical friction force which is a function of electron energy and is given by Eq. (5.2). We consider now the stationary solution of Eq. (5.7)  

$$\mathbf{p}_{st}=\frac{eEp}{F_D(1+\omega _c^2/\nu ^2)}\{\wide... ...frac{\Omega _B}\nu \widehat{\mathbf{e}}_{\bot }\sin \theta _o\}$$ (18)

where $\widehat{\mathbf{e}}$, $\widehat{\mathbf{h}}$ and $\widehat{\mathbf{e}}_{\bot }$ are the unity vectors directed along $\mathbf{E}$, $\mathbf{B}$,and $\mathbf{E}\times \mathbf{B}$ correspondingly, $\theta _o$ is the angle between $\mathbf{E}$ and $\mathbf{B}$. The electron cyclotron frequency is $\Omega _B=eB/mc\gamma$, and taking into account Eq. (5.7) the ratio $\Omega _B/\nu$ can be presented as

$$\frac{\Omega _B}\nu =\frac{eB}F\frac p{mc\gamma }=\frac{eB}{F_D}\frac{\sqrt{\gamma ^2-1}}\gamma$$

The function $\Omega _B/\nu$ determines the effect caused by the magnetic field on the electron motion. Note that this ratio changes rapidly with the height and with the electron energy. We also have to mention that the momentum $\mathbf{p}_{st}$ is given by the solution of Eq. (5.8) which is an implicit function, since both the dynamical friction force F_D and collision frequency $\nu$ depend on the absolute value of momentum p. Actually, Eq. (5.8) represents a set of algebraic equations, which allows us to obtain $\mathbf{p}_{st}$. To solve this equation set, we consider first the equation for the absolute value of the momentum p  

$$1=\frac{eE}F\frac{\sqrt{1+(\frac{\omega _c^4}{\nu ^4}+\frac{... ...a _o+\frac{\omega _c^2}{\nu ^2}}}{1+\frac{\omega _c^2}{\nu ^2}}$$ (19)

where F, $\Omega _B$ and $\nu$ all depend on p. Note that in order to obtain the above equation we took into account the following relation

$$\vert\widehat{\mathbf{e}}+q\widehat{\mathbf{h}}-q_1\widehat{... ...bot }\vert=\sqrt{1+q^2+2q\cos \theta _o+q_1^2\sin ^2\theta _o}$$

where q and q₁ are certain functions. We solve Eq. (5.9) to obtain the absolute value of momentum p. Substitute it then into the right side of Eq. (5.8) to obtain the desired stationary solution in the form $\mathbf{p}_{st}=\mathbf{p}_{st}(\mathbf{E},\mathbf{B},N_m)$. Note that if the electric field is significantly higher than the critical field $E\geq 2E_{c0}$ the minimum electron kinetic energy required for runaway is $\epsilon _{st}<mc^2$. The minimum field requirement is increased in the presence of the magnetic field, hence $E\geq E_{c0}$ must at least be always satisfied.

In the absence of magnetic field (B=0) Eq. (5.9) determines two stationary points at E>E_c0. The first of these points is reached for p_st<p_min given by Eq. (5.2). This is an unstable point. It means that the electrons having p<p_st are decelerated, while the electrons with p>p_st are accelerated and run away. The mentioned above limit is correct for the momentum parallel to the electric field. If the initial electron momentum possesses a component orthogonal to $\mathbf{E}$,a separatrix appears which separates the runaway electrons from those losing their energy [Gurevich et al., 1992; Roussel-Dupre et al., 1994]. The same picture is correct if the constant magnetic field $\mathbf{B}$ exists which is parallel to $\mathbf{E}$. However, if a component of $\mathbf{E}$ orthogonal to $\mathbf{B}$ appears, it can significantly change the above picture. Let us consider a case when $\mathbf{E}\bot$ $\mathbf{B}$. We first introduce the dimensionless

$$\delta _o=E/E_{c0}\qquad \eta _o=B/E_{c0}$$

parameters which allow us to rewrite Eq. (5.9) as  

$$\delta _o^2=\Phi ^2(\gamma )+\eta _o^2(1-1/\gamma ^2)$$ (20)

$$\Phi (\gamma _{\min })=1,\;\gamma _{\min }=3.42$$

where $\Phi (\gamma )=F_D(\gamma )/eE_{c0}$. This equation defines the value of $\gamma _{st}$ and correspondingly $\mathbf{p}_{st}$ for different parameters $\delta _o$ and $\eta _o$ as solution to Eqs. (5.8) and ( 5.9) respectively. We find next the dimensionless critical field $\delta _{c0}$ as the minimum value of $\delta _o(\gamma )$ which still allows solution of Eq. (5.10). Equating the derivative d$\delta _o^2$/d$\gamma$ to zero we find

$$\eta _o^2=-\gamma ^3\Phi (\gamma )\frac{d\Phi (\gamma )}{d\gamma }$$

and

$$\delta _{c0}=-\gamma (\gamma ^2-1)\Phi (\gamma )\frac{d\Phi (\gamma )}{d\gamma }+\Phi ^2(\gamma )$$

which determine an implicit form the dimensionless critical field $\delta _{c0}$ and minimum value $\gamma _c$, depending on the dimensionless magnetic field $\eta _0$. In fact, in the absence of a magnetic field ($\eta _o=0$) we obtain that $\delta _{c0}=1$, $\gamma _c=\gamma _{min}=3.42$.Figure 5.5 reveals that the critical electric field $\delta _{c0}$gradually increases, as the magnetic field $\eta _0$ rises.

  

Figure 5.5: Threshold electric field E_c0 versus magnetic field $\eta _o$obtained for $\theta _o$ = 90^o, 70^o, 45^o and 10^o (for curves from top to bottom respectively). A dashed trace shows analytical approximation valid at $\theta _o$ = 90^o for the nonrelativistic case.

We find now the asymptotic form for $\delta _{c0}$ at high values of $\eta _0$. In order to do it we take into account that at high $\eta _0$ the value (v/c)² $\ll$ 1, so the nonrelativistic dynamical friction force can be applied. Therefore the function $\Phi (\gamma )$ is rewritten as

$$\Phi (\gamma )=\Phi _o\frac{\gamma ^2}{\gamma ^2-1}\qquad \Phi _o\simeq 0.913$$

from which we obtain that  

$$\delta _{c0}=\frac{\sqrt{3}\Phi _o^{1/3}}{2^{1/3}}\eta _o^{2... ...amma _c^2-1}{\gamma _c^2}=\frac{\sqrt{3}\Phi _o}{ \delta _{c0}}$$ (21)

    

The asymptote given by Eq. (5.11) is shown by a dashed trace in Fig. 5.5. The above discussion was focused on an instructive case when $\mathbf{E}\bot$ $\mathbf{B}$. However, Eq. (5.9) allows us to obtain the critical electric field $\delta _{c0}$ as a function of the magnetic field $\eta _0$ for an arbitrary angle $\theta _o$ between the directions of the electric and magnetic field. This is shown in Fig. 5.5. In fact, for $\theta _o<45^o$ the critical electric field practically does not depend on the value of the magnetic field, which resembles the runaway as it occurs in the absence of the magnetic field and is driven by E_||. Note that the runaway electron moves at an angle $\alpha$ to the direction of the electric field, where the angle $\alpha$is obtained from Eq. (5.8). In fact, for $\mathbf{E}\bot$ $\mathbf{B}$, i.e. $\theta _o=90^o$ it acquires the following form

$$\mu =\cos \alpha =(1+\frac{\eta _o^2(\gamma ^2-1)}{\gamma ^2\Phi ^2(\gamma )})^{-1/2}$$

The conclusion is that electrons having low energy $(\gamma \simeq 1)$ move almost parallel to the direction of the electric field. This is due to the fact that for low electron energy the electron collision frequency is much higher than the cyclotron frequency $\omega _c$, thus the effect caused by the magnetic field on the electron motion is not significant. When the electron energy increases, the electron collision rate reduces rapidly. It leads to a deflection of the electron velocity from the direction of the electric field. As the magnetic field increases the angle $\alpha$gradually tends to $\pi /2$, i.e., in a strong magnetic field relativistic electrons start drifting in the $\mathbf{E}\times \mathbf{B}$ direction.

The Electron Runaway Basin Boundary

We study next the equation of the electron motion in order to obtain the separatrix between the two regimes: those electrons which possess trajectories that take them to higher energies, and the other electrons which possess trajectories leading to zero energy. Using the dimensionless variables $\delta _0$ and $\eta _0$ Eq. (5.7) is presented as

$$\frac{du_x}{d\tau }=\delta _o+\frac{\eta _o}\gamma u_y\sin \theta _o-\frac{\Phi (\gamma )}{\sqrt{\gamma ^2-1}}u_x$$

$$\frac{du_y}{d\tau }=-\frac{\eta _o}\gamma u_x\sin \theta _o+... ... u_z\cos \theta _o-\frac{\Phi (\gamma )}{\sqrt{\gamma ^2-1}}u_y$$

$$\frac{du_z}{d\tau }=-\frac{\eta _o}\gamma u_y\cos \theta _o-\frac{\Phi (\gamma )}{\sqrt{\gamma ^2-1}}u_z$$

 

$$\gamma =\sqrt{1+u_x^2+u_y^2+u_z^2}$$ (22)

where we use the dimensionless momentum by normalizing the conventional momentum over mc, i.e. $\mathbf{u=p}/mc$; $\tau$ is the dimensionless time

$$\tau =t/\tau _c$$

where $\tau _c=\frac{mc}{eE_{c0}}=8.5\,e^{h/H_o}\;(n\sec )$. Here the electric field goes along the x axis, while the magnetic field is located in the x-z plane. Equations (5.12) were integrated numerically. Results of the computation are discussed starting with two limit cases: $\mathbf{E}\bot \mathbf{B}$, i.e. $\theta _o$=90^o, and $\mathbf{E}\,\vert\vert$ $\mathbf{B}$, i.e. $\theta _o$=0.

Electron Runaway in Perpendicular Electric and Magnetic Fields

In this case the momentum is fading along the axes z, so essentially electrons are moving in the x-y plane. At low magnetic field $\eta _o<\delta _o$ two kind of trajectories occur depending on the initial conditions. An electron having low initial energy will lose its energy and eventually stops, while an electron having high enough initial energy runs away along an almost linear trajectory in the $\widehat{e}-\widehat{e}_{\bot }$ plane, and gains the energy. This regime resembles the runaway process as it happened in the absence of a magnetic field. The picture changes when the magnetic field increases so that $\eta _o\geq \delta _o$. In this case three different types of trajectories occur depending on the initial conditions, as it shown in Fig. 5.6 along with the corresponding temporal evolution of the electron kinetic energy. In some cases an energetic electron starts in the u_x-u_y plane and then rapidly losses its energy and eventually stops (top two panels of Fig. 5.6). In other cases the electron along a spiral trajectory while the electron kinetic energy rapidly increases (at $t\sim t_o$) and reaches then its steady state value after making several oscillations (Middle two panels of Fig. 5.6).

  

Figure 5.6: (Top two pannels) Electron trajectory in the u_x-u_y plane along with the temporal evolution of its kinetic energy obtained for $\mathbf{E}\bot \mathbf{B}$ at $\delta _o$ = 5, $\eta _o$ = 7, and for the initial values p_x^o=0.3 and u_y^o=-0.3. (Middle two pannels) trajectory obtained at u_x^o=0.3 and u_y^o=-0.6, (Bottom two panels) trajectory obtained for $\mathbf{E}\bot \mathbf{B}$ at $\delta _o$ = 5, $\eta _o$ = 7.5, for the initial values u_x^o=0.3 and u_y^o=-0.65

This regime is strongly different from what happened in the absence of a magnetic field where the runaway electrons reach very high energies, while in the $\mathbf{E}\bot \mathbf{B}$ field a steady state can be reached at a much smaller electron energy. We describe also the third kind of trajectories when the electron moves along the spiral trajectory losing its energy and eventually stops (Bottom two panels of Fig. 5.6).

This happens when the u_y momentum component reaches such negative value that the first and second terms in right part of the first of Eqs. (5.12) cancel each other ($u_y\sim -\gamma \delta _o/\eta _o$ ) leading to the exponential temporal decay of the u_x component. This is followed by the temporal decay of the u_y component as comes from the second of Eqs. ( 5.12). However, when relativistic electrons gain and lose energy they can generate Bremsstrahlung emission, and might produce secondary runaway electrons.

We proceed by defining the separatrix as a line in the $\mathrm{v}_x^o=\mathrm{v}_x(t=0)$ and $\mathrm{v}_y^o=\mathrm{v}_y(t=0)$ plane which separates the initial electron velocities leading to the runaway regime from those leading to the electron deceleration in a given electric and magnetic field. This is shown in Fig. 5.7 calculated for the normalized electric field $\delta _0=5$, and for few different values of normalized magnetic field ($\eta _0$ = 6.0, 6.5, 7.0, and 7.5). For each of these cases the runaway process occurs for $\mathrm{v}_x^o,\mathrm{v}_y^o$ located inside the domain bounded by the corresponding runaway separatrix. Note that when the applied magnetic field increases, the region of runaway shrinks. Finally, $\eta _0$ reaches the maximum value $\eta _{c0}$( $\delta _0$) when the runaway ceases. In fact, at $\delta _0=5$ the runaway ceases at $\eta _0=7.8$, which is in a considerable agreement with the critical value $\delta _{c0}(\eta _0=7.8)=4.9$ (see Fig. 5.5), found above by using some simplifications.

  

Figure 5.7: (a) Separatrix of runaway regime for $\mathbf{E}\bot \mathbf{B}$ in the (v_x^o/c, v_y^o/c) plane obtained for $\delta _0$ = 5 and $\eta _0$= 6.0, 6.5, 7.0, and 7.5. Separatrix of runaway breakdown. (b) Same as above except using the additional condition that the steady-state kinetic energy of the runaway electron is twice as large as its initial value.

Note that a primary runaway electron is able to produce a secondary electron which also runs away, if the kinetic energy of the primary electron is at least twice that required for runaway. This is the condition of the runaway breakdown [Gurevich et al., 1994]. The separatrix of runaway breakdown is obtained as it was done for the runaways, but using an additional condition that the steady state kinetic energy of the runaway electron is twice as large as its initial value. It is shown in Fig. 5.7b for the same values of electric and magnetic field as in Fig. 5.7a. Since the requirements for runaway breakdown are stronger than for just runaway, the corresponding domain is smaller than that for the runaways. In fact the runaway discharge developed at $\delta _0=5$ ceases if $\eta _0\gt 7$.

Spreading of the Runaway Discharge in the Presence of a Magnetic Field

We consider now the runaway discharge stimulated by a seed high energy electron. In the absence of the magnetic field the runaway discharge spreads inside a cone stretched along the direction of the electric field [Gurevich et al., 1994]. Below we discuss how the magnetic field affects the structure of the runaway discharge, and the dynamics of its spreading. We concentrate mainly on the case when the electric and magnetic field are parallel to each other. The motion of runaway electrons is studied in the spherical coordinate frame, in which both E and B vectors are directed along the x axis. The electron momentum evolves with an angle $\theta$ with the x axis, while its projection on the plane z-y evolves with an angle $\varphi$with the y axis. In this frame Eqs. (5.12) can be represented as

$$\frac{du}{d\zeta }=\gamma [\delta _o\mu -\Phi (\gamma )]$$

$$\frac{d\mu }{d\zeta }=\delta _o\gamma \frac{1-\mu ^2}u$$

 

$$\frac{d\varphi }{d\zeta }=\eta _o$$ (23)

where $\mu =cos\theta$, $u=\sqrt{u_x^2+u_y^2+u_z^2}$, and $\zeta$ is the proper time, and $\gamma =\sqrt{1+u^2}$. Since $\mu$ is a monotone function of $\zeta$, we can represent the trajectory in the ($u,\mu )$ plane, which is then described by the following equation

$$\frac{du(\mu )}{d\mu }=\frac 1{\delta _o(1-\mu ^2)}[\delta _o\mu -\Phi (\gamma (\mu ))]$$

Therefore, $\mu$ also serves as parametrization of the problem with

$$\frac{d\zeta (\mu )}{d\mu }=\frac u{\delta _o\gamma (1-\mu ^2)}$$

Correspondingly, the separatrix which separates in this plane the accelerating and decelerating electron trajectories is defined by the equation

$$\Phi (\gamma )=\delta _o\mu$$

Figure 5.8 shows the minimum initial electron energy required for runaway as a function of initial electron direction $\mu$. This is calculated for a few different values of the normalized electric field ( $\delta _o$= 2, 3, 4, 5 and 10). Shown by a dashed trace is the same separatrix obtained by Gurevich et al. [1994] for $\delta _o$= 2.

  

Figure 5.8: Minimum electron energy required for runaway at E||B, versus the direction of the initial electron $\mu =cos\theta$. Obtained at $\delta _0$ = 2, 3, 4, 5, and 10. Shown by a dashed line is the analytical approximation obtained at $\delta _0$ = 2 [Roussel-Dupre et al., 1994].

We consider next the diffusion of runaway electrons which occurs in the plane perpendicular to $\mathbf{E}$, and caused by the fact that secondary electrons appear at an arbitrarily angle. As a result of this diffusion the runaway discharge caused by a single seed electron acquires a conical shape as shown by Gurevich et al., [1994] in the absence of magnetic field. The runaway electron possesses two velocity components in the (y, z) plane

$$\mathrm{u}_z=\frac{d\widetilde{z}}{d\zeta }=u\sqrt{1-\mu ^2}\sin \varphi$$

$$\mathrm{u}_y=\frac{d\widetilde{y}}{d\zeta }=u\sqrt{1-\mu ^2}\cos \varphi$$

where y, z are given in the dimensionless units, $\widetilde{y}=y/c\tau _c$, $\widetilde{z}=z/c\tau _c$. In order to obtain the mean free path of the runaway electron we integrate this two equations using Eq. (5.13), and take into account that the secondary electron which is born at the spot $\mu _o$ close to separatrix propagates freely till the spot $\mu _1$, when its energy increases to twice its initial value. Therefore we have

$$r_\mu ^2=(\overline{\Delta \widetilde{y}})^2+(\overline{\Delta \widetilde{z}} )^2$$

$$\Delta \widetilde{y}=\int_{\mu _o}^{\mu _1}\frac{u(\mu )^2}{... ...c{\sin (\varphi _o+\eta _o\zeta (\mu ))}{\sqrt{1-\mu ^2}} d\mu$$

$$\Delta \widetilde{x}=\int_{\mu _o}^{\mu _1}\frac{u(\mu )^2}{... ...c{\cos (\varphi _o+\eta _o\zeta (\mu ))}{\sqrt{1-\mu ^2}} d\mu$$

where $\varphi _o=\varphi (t=0)$ and the bar shows averaging over $\varphi _o$. The characteristic time $\Delta t$ needed for electron to propagate from point $\mu _o$ to $\mu _1$ can be obtained from Eq. (5.13) as

$$\Delta t=\int_{\mu _o}^{\mu _1}\gamma \frac{d\zeta }{d\mu }d... ...\int_{\mu _o}^{\mu _1}\frac{u(\mu )d\mu }{\delta _o(1-\mu ^2)}$$

The diffusion coefficient D is then found to be

$$D=(c^2\tau _c)\frac{r_\mu ^2}{2\Delta t}$$

where the factor of 2 is due to the averaging over $\varphi _o$, and the result is shown in Fig. 5.9 for different values of electric and magnetic fields. Note that in the absence of magnetic field our results coincide with that obtained by Gurevich et al. [1994]. Figure 5.9 reveals that the magnetic field reduces the diffusion coefficient and confines the runaway discharge. The confinement is the only effect caused by the magnetic field parallel to the electric field, since the magnetic field cannot affect the electron kinetic energy. Note that if the magnetic field is directed at a certain angle to the electric field, the runaway discharge acquires the shape of the cone having an elliptical cross section in the plane perpendicular to $\mathbf{E}$. The semimajor axis is directed parallel to the projection of $\mathbf{B}$ on this plane, while the small semiaxis is perpendicular to this projection.

  

Figure 5.9: Dimensionless diffusion coefficient in the plane perpendicular to E||B obtained at $\eta _o$ = 0, 1, 2, 3, and 5 as a function of $\delta _0$.

Electron Runaway Under an Arbitrary Angle Between Electric and Magnetic Fields

In a general case when the angle between the vectors $\mathbf{E}$ and $\mathbf{B}$ is 0 $<\theta _o<$ 90, three different ranges of the angle $\theta _o$ were distinguished based on the physical properties of the runaway process. They are illustrated by the runaway trajectories shown in Figs. 5.10a, 5.10b, 5.10c obtained for different $\theta _o$.

  

Figure 5.10: Trajectories of runaway electron in (u_x,u_y,u_z) space obtained at $\delta _0$ = 5, 0 = 7.5, for the initial conditions u_x^o=0.3, u_y^o=-0.2, u_z^o=0.2 , and for different angle $\theta _o$ between the electric and magnetic fields: a) $\theta _o$ = 85, b) $\theta _o$ = 80, c) $\theta _o$ = 6 0.

If the angle $\theta _o$ ranges between 80^o and 90^o the runaway process differs significantly from that which occurs in the absence of the magnetic field. First, it develops only if the ratio E/B is greater than a certain threshold value, as was shown in Section 2. Second, contrary to the runaway electrons in the absence of the magnetic field where the energy gain is almost unlimited [ Roussel-Dupre et al., 1994], runaway electrons at 80 ^o $<\theta _o<$ 90^o reach a steady state, at which point they orbit across the magnetic field with a constant kinetic energy, as it is shown in Fig. 5.10a obtained for $\theta _o$= 85^o.

In the range of 0^o $<\theta _o<$ 60^o the runaway process resembles that which occurs in the absence of magnetic field, namely the electrons are moving along the direction of the magnetic field driven by a E_|| component of the electric field. This is illustrated by Fig. 5.10b, obtained at $\theta _o$= 60^o. The latter resembles a trajectory which is almost a straight line in the (p_x,p_y,p_z) space with a small effect of magnetic field at low momentum. However, in this case the magnetic field manifests itself by confining the runaway process, as discussed in Section 4.

In the transient range 60^o $<\theta _o<$ 80^o the runaway electron trajectories are twisted by the magnetic field when the electrons start the acceleration and have relatively low energy. The electron then gains energy along a straight trajectory, as shown by Fig. 5.10c obtained at $\theta _o=80$ .

The effect caused by the angle between the electric and magnetic fields on the runaway process is also illustrated by Fig. 5.11, which reveals the kinetic energy of a runaway electron as a function of the angle $\theta _o$. The kinetic energy was calculated for same initial conditions, and for the time equal to that required to reach a steady state at $\theta _o$=90^o. Note that $\epsilon _{kin}(\theta _o=90)/\epsilon _{kin}(\theta _o=0)$ has a small, but finite value, in fact at $\delta _o=5,\eta _o=7.5$ it is of the order of 10^-2.

  

Figure 5.11: Kinetic energy of the runaway electron as a function of the angle $cos\theta _o$, obtained at $\delta _o$ = 5, $\eta _o$ = 7 and at initial values p_x^o=0.3, p_y^o=-0.2, p_z^o=0.2.

Figure 5.11 shows that at $\cos \theta _o\gt.5$ (i.e. $\theta _o<60^o$) the runaway is driven mostly by the $E_{\vert\vert}=E\cos \theta _o$ component of the electric field, and it is not strongly different from that which occurred in the absence of a magnetic field; while at 0 < cos $\theta _o<$0.16 (i.e. at 80^o $<\theta _o<$ 90^o) the effect of the magnetic field becomes very important; and at 0.16 $<\cos \theta _o<$ 0.5 (i.e. at 60^o $<\theta _o<$ 80^o) a transient region between these two regimes exists.

The runaway boundary for an arbitrarily angle $\theta _o$ could also be investigated using the following approach. We consider an ensemble of N₀ electrons moving in air in the presence of electric and magnetic fields. The electrons which don't interact with each other, are uniformly distributed in space, as well as in the energy range, which we consider for definiteness as 1 $<\gamma <$ 3.2. The trajectories of the electrons were studied using Eqs. (5.12), and the trajectories which take electrons to higher energy were then distinguished from those which lead to zero energy. Figure 5.12 reveals the fraction of electrons, N/N₀, that runaway, as a function of $\delta _0$. The calculation was made for the angle $\theta _o$= 90^o, and from left to right the value of $\eta _0$ changes from 0 to 10 with the step 1. In the absence of magnetic field, shown by the very left trace, the separatrix resembles that obtained by Roussel-Dupre et al. [1994], while the increase of the magnetic field leads to the significant reduction in the fraction of runaway electrons.

  

Figure 5.12: Fraction of runaway electrons as a function of the electric field $\delta _o$ obtained for different values of the magnetic field $\eta _o$, at $\beta$=90. From left to right the value of $\eta _o$ changes from 0 to 10 with a step of 1.

Finally, knowing the electron runaway boundary, one can find the characteristic ionization time in the discharge caused by the runaway electrons by using the fluid approximation, assuming that the electron distribution function is a delta-function, i.e. consider a monoenergetic flux of electrons. Note that of particular interested is the production rate of secondary runaway electrons, since their production leads to the development of the runaway breakdown.

Importance of B Field

We state now the main features regarding the behavior of runaway electrons in the constant magnetic field.

• 1. When the magnetic field is less than the critical field $\eta _o=B/E_{c0}<1$ the affect of the magnetic field on the electron runaway is almost negligible.
• 2. The value of the threshold electric field E$_\eta$ required for the electron runaway in the presence of a magnetic field increases with the increase of $\eta _0$ (see Fig. 5.5). For high values of $\eta _0\gg$1 the threshold field always tends to the constant value

  $$E_\eta =E_{c0}/\cos \theta _o$$

  where $\theta _o$ is the angle between $\mathbf{E}$ and $\mathbf{B.}$ This equation shows that at $\eta _0\gg$1 the threshold electric field increases with $\theta _o$, which means that conditions of runaways are hindered.
• 3. In a case of perpendicular $\mathbf{E}$ and $\mathbf{B}$ fields, the value of threshold electric $E_\eta$ field increases smoothly with $\eta _o$ :

  $$E_\eta \sim \eta _o^{2/3}E_{c0}$$

Since the critical field E_c0 is approximately an order of magnitude less than the threshold of the conventional breakdown, it follows from the last equation that in a case of perpendicular electric and magnetic fields the runaway breakdown is hardly possible if $\eta _o\geq 30$.

In conclusion, the role played by the geomagnetic field in the runaway process discussed above for heights less than 20 km is negligible. Nevertheless the geomagnetic field plays a noticeable role at heights which ranges from 20 to 30 km. In fact, it significantly changes the threshold electric field E$_\eta$ for $\theta _o\geq 45^o$. At the height above 40 km the effect of geomagnetic field dominates at large angles $\theta _o$ and the conditions of runaway breakdown becomes even more hindered.

Therefore at high altitudes, (z > 40 km) for angles $\theta _o$ between $\mathbf{E}$ and $\mathbf{B}$ close to $\pi$/2, the runaway breakdown is hindered, while for $\theta _o\simeq 0$ between $\mathbf{E}$ and $\mathbf{B}$, the runaway process can proceed freely. Thus taking into consideration that the static electric field due to thunderclouds is directed almost vertically one can expect a significant difference in the parameters of high altitude discharges as they occur in the equatorial and midlatitudes.

Finally, we obtained the runaway separatrix which separates momentum space into two regimes: those electrons which possess trajectories that take them into higher energies, and other electrons which possess trajectories leading to zero energy. Using this separatrix, the characteristic ionization time required for the creation of a secondary runaway electron can be estimated.