Department of Meteorology DIFFUSION DUE TO WHISTLER WAVES

Department of Meteorology DIFFUSION DUE TO WHISTLER WAVES USING PIC SIMULATIONS: CLASSICAL DIFFUSION OR ELSE? 28 MARCH 2018 O. Allanson & C. E. J. Watt (both Reading); H. Ratcliffe (Warwick) http: //www. personal. reading. ac. uk/~as 915849/ o. allanson@reading. ac. uk This research was funded in part by the Natural Environment Research Council (NERC) Highlight Topic Grant #NE/P 01738 X/1 (Rad-Sat). Image: R. V. Hilmer, Air Force Research Laboratory Copyright University of Reading 1 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

RADIATION BELT DYNAMICS • Particles confined by Earth’s magnetic field: Periodic motion (t~ms) ~3 -6+ RE (t~s) (t~ks) NASA Ames Research Center ~1 -3 RE Adiabatic Invariants System gradients > Timescales • Resulting periodic motion → adiabatic invariants, Ji = ( μ, J, Φ) • Write evolution (Fokker-Planck) equation for plasma wrt Ji Adiabatic ‘breathing’ → no diffusion! 2 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

WHAT WAVES CAN DO • • Solar wind forcing & geomagnetic activity → spectrum of plasma waves Different wave modes violate different adiabatic invariants Usually think of plasma as a function of E, α, and L =req/RE (or L*) To ‘first order’ … Waves Freq. Broken Primary effect Invariant(s) And then … ULF <3 Hz Φ Radial transport in L=r/RE Energisation & magnetopause loss ELF VLF 3 Hz -3 k. Hz 3 KHz-30 KHz µ, J and Φ Local scattering in E, α Energisation and ionospheric precipitation 3 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

WAVE-PARTICLE INTERACTIONS Vphase B 0 F ~ e E → Acceleration → wave-particle energy exchange • • Different waves (ω, k and vph) pick out different plasma populations i. e when vparticle ~ vph for n=0 n = 0 Landau resonance n = ± 1 Fundamental cyclotron resonance 4 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

SEGUE 1: QUASILINEAR DIFFUSION • • • Background field with waves on top, and assumptions on waves … Random phase (incoherent) Broadband Small amplitudes → Hence no particle trapping within the waves a. k. a “Weak turbulence” theory Classical Diffusion in • Energy • Pitch angle • (And L – radial diffusion) 5 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

WHAT THIS LOOKS LIKE Classical diffusion / Random walk / Stochastic e. g. coin flipping, Heads=+1, Tails=-1 Score ~ t 1/2 Running Score t 6 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

SEGUE 2: WHISTLER MODE WAVES • ELF/VLF and Right-Hand polarised wave with ω/Ω ce < 1 • Waves can be generated… Ø Locally due to kinetic instabilities Ø or can be anthropogenic (Naval Radio Transmitters, (k. Hz, MW)) • ω/Ω ce < 1 → whistlers prime candidates for (E, α) scattering via wave-particle interactions (n=0 and n=± 1 dominant) Tsurutani & Lakhina 1997 • Particularly important for acceleration (E) and loss (α) 7 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

Line of Quasilinearliness WHISTLER = QUASILINEAR? Less quasilinear: Coherent Monochromatic Large(r) amplitude? To what extent should/can we use QL theory for whistlers. Might be two very different things. Doing “should” first. More quasilinear: Incoherent Broadband Small(er) Amplitude? Figure credit: Clare Watt 8 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

1 D PIC EXPERIMENTS • • Open-source, explicit, parallelised (~>90%), relativistic Easily customised with a variety of different boundary conditions. c=c, mi/me = REAL VALUE. http: //www. ccpp. ac. uk/epoch_user. pdf Demonstrated utility for the study of whistler-mode waves (Ratcliffe, H. and C. E. J. Watt (2017), doi: 10. 1002/2017 JA 024399) Stationary & almost entirely cold prototypical L~2. 5 population 24 k. Hz @ L~2. 5 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

MISSION STATEMENT • Classical diffusion based on ‘randomness’ • Wave-particle interactions are not! • ‘Value of coin toss’ depends on E and α at least • (When) Is QL assumption valid/invalid and how much does it matter? • If diffusive then normal, super- or sub-diffusion? • Identify importance and effect of non-diffusive (nonlinear) behaviour: e. g. phase trapping 10 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

EXPERIMENTS TO TEST ASSUMPTION • Aim: Test the validity of QL theory (classical diffusion) for whistler-mode waves • Run fully self-consistent PIC experiments • Previous work has focussed on test-particle approach (no plasma)… • …and/or conditions that are most ‘diffusive’ e. g. Tao et al 2011 Track individual particles • Directly track diffusion in E, α space • “Use y=mxa” to empirically measure the power of Δt and directly construct diffusion coefficients DEE etc 11 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

SCATTER PLOTS E (1000 tracers, 1 m. V/m laser) 12 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

CLASSICAL WHEN YOU BIN DATA? α • Some structure! • a (“the power”) = 1 is exception not rule 13 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

BASIC CONCLUSIONS & FUTURE WORK • (EPOCH) PIC simulations are a powerful tool for testing a cornerstone theory for radiation belt diffusion. • Reasonable first experiments: When does classical diffusion hold? • [Prelim. ] Classical diffusion seems to hold on aggregate. • [Prelim. ] Classical diffusion exception once you bin in E, α space. • Examine dependence on w. p. int. ~ Landau & Cyclotron resonance • Extend parameter space ~ How does diffusion vary with B, n, T, wave properties • Extend time of simulation ~ Saturation/turbulent timescales? • Extend to 2 D ~ Effect of curvature/oblique waves • Other types of waves: Hiss next • Increase number of tracers 10^3 -> 10^6 ~ better statistics http: //www. personal. reading. ac. uk/~as 915849/ o. allanson@reading. ac. uk 14 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

EXTRA SLIDES 15 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

SCATTER PLOTS (1000 tracers, 1 m. V/m laser) E 16 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

WHISTLER AMPLITUDES Wilson III et al 2011 17 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

18 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

SCATTER PLOTS 19 LIMITLESS POTENTIAL | LIMITLESS OPPORTUNITIES | LIMITLESS IMPACT

Basic simulation parameters • Equatorial Bx~ 2000 n. T @ L~2. 5 => w/wce~0. 4 • IC: Driven experiment, not an instability. • BC: Open. . Ok since large domain • Wave travels 1/100 box: => L~600 km ~ 8. 5 Wavelengths p/point ~<5% leave box • Tworld=20 tce =3 x 10 -4 s : Twall-time~2 hrs on 10 cores !

The plasma populations in the box 200 or 1000 tracers Can/will be increased e: 1 e. V cold @ 98. 9% i: 1 e. V cold @ 100% Tracers on hot plasma population at 100 ke. V (well samples phase space) Tracers population narrowly focussed on bulk flow: vph from cold plasma DR (Stix) e: 10 ke. V warm @ 1% e: 100 ke. V hot @ 0. 1% 1 0. 001 • n ~ 103 cm-3: fixed by wpe/wce= 8 (from CRRES data) • Resonant (“res”) tracers: T ~ 20 e. V (narrowly focussed) on vph/c ~ 0. 06 7/15

9/15 Tracer statistics (better than they sound) Tracers: Cut up data -> ~T(T+1)/2 * n data points Statistics at each step • 66 * n data points • (T-t)*n at each delta t • ~(T(T+1)/2 * n/n_bin^2) in each E, alpha bin • In each bin these samples are distributed ~ as above

Why PIC? • Why kinetic: wave-particle interactions in principle require fully relativistic kinetic theory (i. e. with up to six dimensions in phase space), timescales range from microseconds to tens of seconds. Non-linear studies of diffusion in whistler-mode waves have so far focussed on the test-particle approach [e. g. Tao and Bortnik, 2010]. Natural extension is to use PIC. • Why not Vlasov: Problems with velocity-space filamentation over long domains: mitigating lamentation requires high resolution in phase space which is computationally expensive in 1 -D and prohibitively so in 2 -D. • Why not hybrid: (e. g. Katoh and Omura, 2013) treat lower energy populations of electrons and ions as fluid, and the fast electron population as particles (electron timescales cannot be neglected). However, there are known problems in the modelling of short (grid scale comparable) wavelength whistlers in electron fluid hybrid schemes, leading to unphysical energy build up. Furthermore, evolution of the higher energy tails of the lower energy distributions can in fact be important for large wave amplitudes, and so a fully kinetic treatment is required.