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Relativistic Runaway Electrons Above Thunderstorms

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Relativistic Runaway Electrons Above Thunderstorms 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 20 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 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, ß-particles ... [will] have already acquired energies exceeding those of the fastest known ß-particles from radioactive substances ... Runaway and conventional breakdown fields Fd ~ Nm Et=Fd,min /e 80 60 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= Fd,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) 25 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.
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