Relativistic Runaway Electrons Above Thunderstorms

Home , Electron avalanche, Runaway breakdown

Relativistic Runaway Electrons above Thunderstorms

Nikolai G. Lehtinen

Physics Department STAR Laboratory Stanford University

Advisers: Umran S. Inan, EE Department, Timothy F. Bell, EE Department, Roger W. Romani, Physics Department Plan

1. Introduction

2. Monte Carlo model of runaway electron avalanche

3. Fluid model of runaway electrons above thunderstorms

4. Effects of runaway electrons in the conjugate hemisphere Lightning-mesosphere interaction phenomena

~ 2000 cm-3 Electron density Β 100 km THERMOSPHERE Elves

80 km Sprites MESOSPHERE γ-rays 60 km Runaway ~100 MV 40 km E ~ 103 V/m electrons STRATOSPHERE at 40 km Blue Jet 20 km Cameras + + + + + TROPOSPHERE +CG 0 km - --- -

γ-ray flash Red Sprites (BATSE observation)

40 Elves 30

Sprites 10

Rate (counts/0.1ms) 0 015205 10 1996.204.07.17.38.792 Time (ms) Red Sprites

Red Sprites: - altitude range ~50-90 km - lateral extent ~5-10 km - occur ~1-5 ms after +CG discharge - last up to several 10 ms

Aircraft view Ground View

90-95 km

horizon

Space Shuttle View

sprite

thunderstorm Examples of Terrestrial Gamma Ray Flashes (BATSE Data)

Terrestrial Gamma Rays: - time duration ~1 ms - energies 20 keV—2 MeV - hard spectrum (bremsstrahlung) Rate (counts/0.1ms) Rate (counts/0.1ms) Rate (counts/0.1ms)

Time (ms) Time (ms) Time (ms) Time (ms) C. T. R. Wilson, 1925

While the electric force due to the thundercloud falls off rapidly, ... the electric force required to cause sparking ... falls off still more rapidly. Thus, ... there will be a height above which the electric force due to the cloud exceeds the sparking limit ...

The rate of loss of energy ... varying approximately as the inverse square of the velocity ...... Thus, §-particles ... [will] have already acquired energies exceeding those of the fastest known §-particles from radioactive substances ... Runaway and conventional breakdown fields

Fd ~ Nm

Et=Fd,min /e

Ec 40 Et Altitude, km 20

0 0 2 4 6 10 10 10 10 Electric Fields, V/m Production of accelerating E field BEFORE DISCHARGE AFTER DISCHARGE E (small) Large E causes runaway breakdown h h

negative negative screening charge screening charge

10 km + Q 10 km 5 km - Q 1 ms 5 km - Q

Runaway electron avalanche

Electron thermalizes due to collisions

Ionization

Cosmic ray shower E incident primary particle e-

e- -5 3 Cosmic rays produce ~10 /cm -sec - e - >1 MeV electrons at ~10 km altitude e Runaway electron avalanche studies

Previous runaway avalanche models: ¥ Analytical [Gurevich et al., 1996; Sizykh et al., 1993; Bulanov et al., 1997] ¥ Kinetic [Symbalisty et al., 1998] ¥ Monte Carlo [Shveigert, 1988]

Goals of runaway avalanche studies: ¥ Avalanche rates ¥ Distribution functions

Applications: ¥ Atmosphere ¥ High energy plasmas (astrophysics, fusion)

Motivation for this work: ¥ The previous kinetic and Monte Carlo models do not include magnetic field ¥ Discrepancies between different models Monte Carlo model

z 1. Relativistic equations of motion:

dp e = −eE − p × B + Γ(t) dt mγ B E θ dr p = dt mγ x 2. Production of new electrons in the ionization process. y

Forces due to scattering:

Γ= F d,excl + Γion + Γel dynamic friction ionization elastic scattering

F = F − F 1 d,excl d d,ion 10 -1 kg 2

0 10

runaways eE

Energy Loss, MeV m F d eE F t -1 d,excl 10 -2 0 2 10 10 10 Energy, MeV

Et ~ 0.1Ec; Ec is the conventional breakdown (sparking) field Growth of the number of particles

γ Rt NR(t) ~ e

5 10

E/Et=2 E/Et=5 E/Et=8 4 10

3 10 Number of particles

2 10 0 1 2 3 4 5 t/τ

τ 172 ns at sea level γ Runaway avalanche growth rate R (B⊥E)

20 E=15E t

τ 15 R =γ R,d γ 10

5 E=5E t

0 0 5 10 15 20 25 30 cB/E t

¥ the avalanche grows faster at higher electric fields ¥ high perpendicular magnetic field quenches the runaway electron avalanche Electron distribution function (B=0)

-eE θ p

E=2Et E=15Et 2 10 θ=0 o runaways θ=45 1 θ=60o 10

0 10

), arbitrary units -1 θ

θ =0 10 θ=45o θ o f(E, =60 runaways -2 10 0 1 2 3 4 5 10 10 10 10 10 10 10 0 10 1 10 2 10 3 10 4 10 5 Energy, keV Energy, keV

¥ the runaway electron beam is more bunched forward at higher electric fields Electrons in momentum space for E⊥B E=5Et

E E 10 10

/mc 0 /mc 0 z z p p -10 -10 0 -20 -40 p /mc -10 0 10 x p /mc y E -10

/mc 0 y p 10 B=0 0 -20 -40 p /mc x

E E 5 5 0 B 0 B /mc /mc z z p -5 p -5 -10 -10 0 -5 -10 -15 -20 -10 0 p /mc p /mc x y -20 E B /mc y -10 |B|=|E|/c p

0 0 -5 -10 -15 p /mc x

|B|=2|E|/c: No avalanche Probability for an electron to run away and the runaway region boundary,

E/Et=5

3 0.8 2 without scattering with scattering [Gurevich et al, 1992] 0.7 1 0.6 0.9 0.7 0.60.40.2 0.8 0.5 qE 0.8 0.5

/ mc 0 0.9 || 0.9

p 0.4

0.6

0.1 0.7 0.3 0.2 -1 0.4 0.3 0.7

0.8 0.5 0.2 -2 0.8 0.9 0.6 0.9 0.1 -3 0.7 0 -3 -2 -1 0 1 2 3 p⊥ / mc The boundary of the runaway region

p electric force qE ||

dynamic friction Fd

diffusion due to elastic collisions, <θ> described by D(p) ps

p⊥

The Fokker-Planck equation ∂f 1 ∂ 2 − ∂ − 2 −qE ∂f = 2 p (Fd qEµ)f + (1 µ ) f+D ∂t p ∂p ∂µ p ∂µ motion along p angularmotion

2 dσion 3 1 d Θ + NmZmv 3 f(p )d p ;(µ = cos θ, D = ) d p 4 dt p →p ionization,f(p )=Nδ(p − p0)

Angular equilibrium:

qE ∂f − f + D =0 → average µ = cos θ as a function of p. p ∂µ

Equilibrium along p:

Fd (p)=qEµ→runawayregion boundary ps γ Avalanche rate R found from the Fokker-Planck equation and comparisons to other papers

Fokker-Planck equation integrated over momentum space: 2 ∂NR 1 mc = γRNR, where γR ≈ , ∂t τ Es

Es — energy corresponding to the boundary of the runaway region.

3 10

2 10 τ R 1

=γ 10 R,d γ

0 10 FP equation [Lehtinen et al., 1999] FP without elastic scattering [Lehtinen et al., 1999] Monte Carlo [Lehtinen et al., 1999] Gurevich et al. [1994], Roussel-Dupre et al. [1994] Symbalisty et al. [1998] -1 10 0 2 4 6 8 10 δ =E/E 0 t

τ 172 ns at sea level λ = cτ 51 m Formation of the runaway electron beam

Schematics of the beam formation

100 km B Ionosphere 0

80 km

60 km

40 km Avalanching runaway

20 km electrons h+=10 or 20 km +Q -Q h =5 km - +CG

Two-dimensional modelling

¥ cartesian ¥ cylindrically symmetric translationally symmetric along y axis

— arbitrary direction of the — vertical magnetic field magnetic field — localized charge — horizontally extended charge configuration

>100 km + +++ + +++++++ + +++ + + + ------+CG Estimate of runaway electron flux into the ionosphere

5 10 -3 point charge h+=20 km point charge h+=10 km 0 disk charge 10 h+=10 km disk charge h+=20 km at 80 km altitude, m R N -5 10 200 400 600 800 1000 Q, C

dN v R = γ (E,N )N + S (z) R dz R m R 0

Electric field is calculated on the basis of point/disk charge fields in a horizontally stratified atmosphere with conductivity

σ = σ(0)ez/H,H=10km

Point charge ﬁeld in conducting atmosphere:

√ q − 2 2 Φ(r, z)= √ e ( r +z +z)/2H 2 2 4π0 r + z

Point charge ﬁeld in vacuum:

q Φ(r, z)= √ 2 2 4π0 r + z

Take into account the mirror charges and the fact that removal of the charge is equivalent to placing an opposite charge into the same location. FLUID MODEL OF RUNAWAY AVALANCHE

• Quasi-electrostatic ﬁeld: E = −∇φ, φ = φ(r,t)

∂ρ ρextσ0 + ∇·J + ∇·JR = , JR = −evRNR ∂t 0 ρ + ρ ∇·E = ext 0 JR, vR, NR — current, velocity and density of runaway electrons

ρext — source thundercloud charge, ∇·Jext = −ρextσ0/0

• Runaway electrons: ∂N R + ∇·(v N )=γ N + S (z) ∂t R R R R o γR — avalanche rate [Monte Carlo, Lehtinen et al., 1999]

vR — runaway velocity [Monte Carlo, Lehtinen et al., 1999]

So(z) — source due to cosmic rays Runaway electron avalanche in the middle atmosphere: Cartesian model results

Q=12 C/km, h+=10 km, t=3 ms

Runaway velocity Electric field

70 (a) B 70 60 60 50 50 40 40 z, km z, km 30 30 20 20 10 + 10 (b) + 0 — 0 — 0 20 40 60 80 1000 20 40 60 80 100 x, km x, km

Runaway density: Runaway density: 2D structure time evolution log(N ), m-3 ; N =767.03 m-3 R Rmax 5 10 2 70 (d) 60 0 100 50 -3 -2 40 , m R z, km N 30 -5 t=.5 ms 10 -4 t=1 ms 20 t=2 ms 10 t=3 ms (c) + -6 — -10 10 20 40 60 80 0 20 40 60 80 100 x, km x, km OPTICAL EMISSIONS

• Optical emissions are due to — thermal electrons, driven by the electric ﬁeld, — suprathermal electrons (∼> 10 eV), created by the runaways, and are calculated on the basis of known molecular excitation cross- sections and emission rates.

• Steady-state emissions:

— First Positive N2 group (1P)

— Second Positive N2 group (2P) + — First Negative N2 group (1N) + —N2 Meinel group (M) + — First Negative O2 group (1N) Optical emissions in the First Positive N2 Band: Cartesian model results

Q=12 C/km, h+=10 km, t=3 ms

Runaway and Conventional Thresholds

70 E > E (conventional) 60 c 50 40 E < E < E (runaway breakdown) t c z, km 30

20 E < E t 10 (a) + 0 — 20 40 60 80 x, km

Total optical emissions Runaway optical emissions log (I ) for 1P N band, Rayleighs log (I ) for 1P N band, Rayleighs 10 run 2 10 run 2 6 70 70 2 4 60 60 50 2 50 0 40 40 z, km 0 z, km -2 30 30 -2 20 20 -4 10 10 (b)+ -4 (c) + — — -6 20 40 60 80 20 40 60 80 x, km x, km Terrestrial Gamma Ray Flashes

γ-ray flash Satellite configuration (BATSE observation) CGRO 40

RCGRO 30 γ-rays (>300 keV) 20 Runaway ~500 km electrons 10

Rate (counts/0.1ms) 0 015205 10 Time (ms)

Simulated BATSE data at ~45¡ magnetic north latitude in energy interval 100-300 keV

γ-ray emissions are calculated using Heitler [1954] bremsstrahlung cross-sections.

E Counts per 0.1 ms γ-rays 500

200 discharge N B 150 0 W E km Runaway 100 electrons S 50 Lightning discharge -500 -500 0 500 km Runaway electron avalanche in the middle atmosphere: cylindrically-symmetric model results

Q=420 C, h+=20 km, t=3 ms

Runaway velocity Electric field

60 60

40 40

z, km B z, km

20 20 (a) (b) 0 0 -40 -20 0 20 40 -40 -20 0 20 40 r, km r, km

Runaway density: Runaway density: 2D structure time evolution 105 log(N ), m-3 ; N =118.64 m-3 2 R Rmax t=.5 ms 1 t=1 ms t=2 ms 0 0 60 10 t=3 ms

-1 -3 , m

40 R

z, km -2 N 10-5 20 -3 (c) +— -4 -40 -20 0 20 40 (d) r, km -5 10-10 0 10 20 30 40 50 60 r, km Optical emissions in the First Positive N2 Band: cylindrically-symmetric model results

Q=420 C, h+=20 km, t=3 ms

Runaway and Conventional Thresholds

70 E > E (conventional) 60 c 50 E < E < E 40 t c z, km (runaway breakdown) 30 20 + E < E — t 10 (a) -40 -20 0 20 40 r, km

Total optical emissions Runaway optical emissions log (I ) for 1P N band, Rayleighs log (I ) for 1P N band, Rayleighs 10 run 2 10 run 2 6 2

60 4 60 0 40 2 40 z, km z, km 0 -2 20 + 20 (b) — (c) +— -2 -4 -40 -20 0 20 40 -40 -20 0 20 40 r, km -4 r, km Runaway electron propagation in the magnetosphere and conjugate effects

cloud-to-ground magnetic field discharge line energetic electrons electrons interact with plasma waves

conjugate effects

Conjugate effects include: ¥ optical emissions ¥ gamma ray emissions ¥ ionization Interaction of runaway electrons with magnetosphere

• Dispersion relation: (ω, k) = 0, where 2 − 2 3/2 − ω0 − 2 NR (1 β ) f(p)dp (ω, k)=1 2 ω0 2 ω N0 (ω − kcβ)

9 −3 N0 10 m — magnetospheric plasma density, 2 ω0 = e N0/(0me) — magnetospheric plasma frequency.

• Log-normal ﬁt to Monte Carlo distribution:

0.08 Monte Carlo analytical fit 0.07

0.06

0.05

0.04

0.03 f(p), dimensionless 0.02

0.01

0 -1 0 1 2 10 10 10 10 p/mc, dimensionless

NR −1 −3 ⇒ Im ω 25 ω0 0.05NR sec ,NR in m N0

−3 • Interaction with plasma becomes important for NR ∼> 100 m . Curtain formation

5 −3 NR =10 m Nonlinear interaction⇒isotropization and thermalization ∼90% stay out of the loss cone and form a curtain ∼10% precipitate at the conjugate point

(a) 1 s (b) 300 s discharge

0.2 MeV ~10 km ~ 2 MeV

Flux of electrons at energy ∼1 MeV of

−2 −2 −1 −1 ΦE 7×10 el-cm -s -keV

2 with an energy dependence for Emec

2 −E/(2 MeV) ΦE ∝E e Optical emissions by precipitating electrons

Emission rate, m-3 s-1 (N 1P for N =104 m-3 ) 2 precip 11 x 10 4

0 100 Altitude, km 50 8 4 6 Time, ms 0 0 2

N =104 m-3 precip 100 + N2 M 90 N 2P 80 2 + O2 1N 70

60 N 2P + 2 Altitude, km N2 M 50 + O2 1N + 40 N2 1N

N2 1P 30 -2 0 2 4 10 10 10 10 Integrated emission, R s Gamma ray emissions from precipitating electrons Satellite configuration γ-ray energy spectrum 2 10

1 10

0 CGRO 10

-1 10 Precipitating RCGRO ), arbitrary units

electrons ph -2 10 f(E γ-rays (>300 keV) ~500 km -3 10

-4 10 0 1 2 3 4 10 10 10 10 10 E , keV ph

Time and radius dependence

6 6 x 10 x 10 16

14 14

12 12 /s /s 2 2 10 10

8 8

6 6 Photon flux, ph/m Photon flux, ph/m 4 4

2 2

0 0 0 500 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 R , km Time, ms CGRO Ionization and conductivity enhancement

N = 10 4 m-3 ; τ=1 ms precip

Ionization enhancement

100

Altitude, km 40 ∆ N e 20 Nighttime Daytime

0 -2 0 2 4 6 10 10 10 10 10 N , ∆ N , cm-3 e e Conductivity enhancement 120 Modified nighttime conductivity Nighttime 100 Daytime

Altitude, km 40

0 -15 -10 -5 0 10 10 10 10 Conductivity, mho/m Time evolution of ionization

Calculated taking into account the time dynamics of ¥ electrons ¥ positive ions ¥ negative ions ¥ positive cluster ions [Glukhov et al., 1992]

4 10

3 10 80 km 2 10 -3 , cm e 1 N 10

0 10 70 km

-1 10 0 20 40 60 80 100 Time, sec Conclusions

¥ We calculated uniform runaway electron avalanche rates in constant electric and magnetic fields using a 3D Monte Carlo model and compared them to previously done work.

¥ We modelled a runaway breakdown due to a positive return stroke using (a) Cartesian coordinates for a laterally extensive thundercloud and tilted geomagnetic field and (b) cylindrical coordinates for vertical geomagnetic field.

¥ The geomagnetic field controls the motion of runaways at >35km at mid- latitudes, where most sprites are observed, and close to equatorial region, where the terrestrial γ-ray flashes are observed.

¥ The optical emissions associated with relativistic electrons (the contribution to sprites) are small compared to conventional breakdown emissions.

¥ For sufficiently large discharge values, the runaway electron-produced γ- rays flux values agree with BATSE data [Fishman et al., 1994].

¥ Runaway electrons leave the atmosphere and travel to the conjugate region, causing detectable optical emissions, gamma ray emissions and ionization.

¥ Part of the electrons leaving the atmosphere becomes the radiation belt electrons by forming energetic electron curtains