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S. Elkington, GEM 2003 Transport in the Radiation Belts and the role of Magnetospheric ULF Waves Scot R. Elkington LASP, University of Colorado With many thanks to: D. N. Baker, D. H. Brautigam, A. A. Chan, Y. Fei, J. Green, M. K. Hudson, X. Li, K. L. Perry, E. J. Rigler, and M. J. Wiltberger GEM 2003 Snowmass, CO

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S. Elkington, GEM 2003 Outline Introduction to radiation belts. Introduction to radiation belts. Transport in the radiation belts. Transport in the radiation belts. Radial transport and the role of ULF waves. Radial transport and the role of ULF waves. Characteristics of ULF waves, MHD simulations Characteristics of ULF waves, MHD simulations Boundary conditions on radial transport. Boundary conditions on radial transport. Conclusion. Conclusion.

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S. Elkington, GEM 2003 The outer zone radiation belts Trapped particles drifting in orbits encircling Earth. Trapped particles drifting in orbits encircling Earth. Two spatial populations: inner zone and outer zone. Two spatial populations: inner zone and outer zone. Energies from ~200 keV to > few MeV Energies from ~200 keV to > few MeV

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S. Elkington, GEM 2003 Particle motion in the radiation belts Trapped particles execute 3 characteristic types of motion: Gyro: ~ millisecond Bounce: ~ 0.1-1.0 s Drift: ~ 1-10 minutes Characteristic time scales:

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S. Elkington, GEM 2003 Adiabatic invariants M: perpendicular motion M: perpendicular motion K: parallel motion K: parallel motion L: radial distance of eq-crossing in a dipole field. L: radial distance of eq-crossing in a dipole field. Associated with each motion is a corresponding adiabatic invariant: Gyro: M=p 2 /2m 0 B Bounce: K Drift: L If the fields guiding the particle change slowly compared to the characteristic motion, the corresponding invariant is conserved.

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S. Elkington, GEM 2003 Fluxes in the radiation belts The radiation belts exhibit substantial variation in time: Storm commencement: minutes Storm main phase: hours Storm recovery: days Solar rotation: 13-27 days Season: months Solar cycle: years

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S. Elkington, GEM 2003 Why study the radiation belts? Because theyre physically interesting! Because theyre physically interesting! Relativistic electrons have been associated with spacecraft anomalies. Relativistic electrons have been associated with spacecraft anomalies. Want to try to describe how radiation evolves in time at a given point in space.

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S. Elkington, GEM 2003 Describing the radiation belts The radiation belts may be completely characterized at a point in time by its distribution function: Also referred to as the phase space density, f gives the number of particles in a volume (x+dx, y+dy, z+dz), with momenta between (p x +dp x, p y +dp y, p z +dp z ). The flux in a region of space may be related to the distribution function through

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S. Elkington, GEM 2003 The distribution function and the adiabatic invariants The distribution function may equivalently be written in terms of the invariants and corresponding phase: If the distribution is uniform in phase (e.g. uniform 3 no drift bunching in L), then the phase space density taken at a point may be considered the same at all points corresponding to the same M, K, and L.

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S. Elkington, GEM 2003 Transport and Fokker-Planck The evolution of the phase space density is given by the Fokker-Planck equation: Coherent terms (e.g. friction) Stochastic terms (e.g. diffusion) Losses Sources For example, to quantify a process that leads only to diffusion in L, we would write (M. Schulz, AGU Monograph 97, 1996)

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S. Elkington, GEM 2003 Transport in M, K: local heating

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S. Elkington, GEM 2003 Local heating example: resonant interactions with VLF waves Whistler mode chorus at dawn combined with EMIC interactions heat and isotropize particles. Leads to transport in M, K, and L. Summers et al. (JGR 103, 20487, 1998) proposed that resonant interactions with VLF waves could heat particles:

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S. Elkington, GEM 2003 Transport in L: radial transport

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S. Elkington, GEM 2003 Energization and ULF waves f D ~mHz … ULF waves! Only need consider resonant frequencies in analysis: Electron moving in a dipole magnetic field with slowly-varying dawn-dusk potential electric field

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S. Elkington, GEM 2003 …and to radial transport Nonrelativistically, and in a dipole, or so changing energy (W) while conserving M will necessarily lead to transport in L.

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S. Elkington, GEM 2003 Observed associations between ULF waves and radiation belt activity? Baker et al., GRL 25, 2975, 1998 Rostoker, GRL 25, 3701, 1998 Mathie & Mann, GRL 27, 3621, 2000 Obrien et al., JGR, 2003, in press.

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S. Elkington, GEM 2003 Characterizing ULF interactions Start particles at differing energies differing drift freqs. Start particles at differing energies differing drift freqs. Impose waves characterized by particular m, Impose waves characterized by particular m, Look max energy gain at Look max energy gain at Poincare plots: Plot motion in the phase space coordinates (L, ), at snapshots in time differing by 1/f. Plot motion in the phase space coordinates (L, ), at snapshots in time differing by 1/f. Particle motion constrained to lines. Particle motion constrained to lines. Drift resonance appears as island at appropriate energy (frequency). Drift resonance appears as island at appropriate energy (frequency).

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S. Elkington, GEM 2003 Higher-order resonances In the outer zone, the m=1 noon- midnight asymmetry leads to additional resonances:

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S. Elkington, GEM 2003 Multiple frequencies: radial diffusion

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S. Elkington, GEM 2003 Quantifying effects of diffusion Effect of diffusion on radiation belts can be quantified by solving the diffusion equation with appropriate boundary conditions: To do this, we need to know the appropriate diffusion coefficients… What are the D 0 s? What is the ULF power?

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S. Elkington, GEM 2003 Diffusion Characteristics: D 0 Start series of particles at one L. Start series of particles at one L. Let them evolve in a compressed dipole under the influence of a constant P(L, ). Let them evolve in a compressed dipole under the influence of a constant P(L, ). Watch spread of L in time. Watch spread of L in time. Calculate D LL = /2 Calculate D LL = /2 Invert to calculate D 0. Invert to calculate D 0.

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S. Elkington, GEM 2003 Diffusion Characteristics, 09/24-26/1998: constant P, time-varying compression

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S. Elkington, GEM 2003 Characteristics of ULF waves? ULF Power ULF Power ULF waves: decrease with decreasing L ULF waves: decrease with decreasing L Decrease with increasing frequency Decrease with increasing frequency Depend on solar wind velocity Depend on solar wind velocity May or may not have significant local time dependence. May or may not have significant local time dependence. Kepko et al., GRL 29(8), 1197, 2002

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S. Elkington, GEM 2003 ULF activity in MHD simulations ULF waves can be described within the framework provided by MHD. ULF waves can be described within the framework provided by MHD. Lyon-Fedder-Mobarry: Global, 3dGlobal, 3d Driven by upstream conditionsDriven by upstream conditions

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S. Elkington, GEM 2003 MHD simulations of ULF power, 09/24/1998 ULF power in MHD shows expected radial, frequency dependence. ULF power in MHD shows expected radial, frequency dependence. Azimuthal dependence: frequently see structure in local time. Azimuthal dependence: frequently see structure in local time.

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S. Elkington, GEM 2003 Parametric studies, perturbations Constant solar wind velocity 600 km/s. Constant solar wind velocity 600 km/s. Density perturbations in solar wind at 3 mhZ. Density perturbations in solar wind at 3 mhZ. Spectral analysis in magnetosphere shows strong peak at 3 mHz, but also peaks at 5, 7, and 10 mHz. Spectral analysis in magnetosphere shows strong peak at 3 mHz, but also peaks at 5, 7, and 10 mHz.

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S. Elkington, GEM 2003 Parametric studies, const solar wind Constant solar wind velocity 600 km/s. Constant solar wind velocity 600 km/s. Nothing else! Nothing else! Indicates that 5, 7 mHz features are due to K-H waves on the flanks. Indicates that 5, 7 mHz features are due to K-H waves on the flanks.

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S. Elkington, GEM 2003 Shear waves & particle acceleration Limited local time: propagating waves dusk and counterpropagating waves dusk still lead to energization. Limited local time: propagating waves dusk and counterpropagating waves dusk still lead to energization.

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S. Elkington, GEM 2003 Mode structure in ULF simulations Diffusion coefficients: depend on P(m D ), etc., so must know the power in each specific mode number. Diffusion coefficients: depend on P(m D ), etc., so must know the power in each specific mode number. Main Phase Recovery Phase

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S. Elkington, GEM 2003 Energization through radial transport: a plasma sheet source? Simplistic considerations put r 0 ~20 R E for M corresponding to a 1 MeV geosynchronous electron Simplistic considerations put r 0 ~20 R E for M corresponding to a 1 MeV geosynchronous electron Conversely, W for r 0 =6.6R E is 50 keV. Conversely, W for r 0 =6.6R E is 50 keV. So is transport/acceleration a viable mechanism at all? So is transport/acceleration a viable mechanism at all? N. Tsyganenko http://nssdc.gsfc.nasa.gov/space/http://nssdc.gsfc.nasa.gov/space/model/magnet os/data-based/modeling.html

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S. Elkington, GEM 2003 Plasmasheet access to inner magnetosphere: observations Korth et al. [JGR 104, 25047, 1997] found good agreement with theory for keV electrons. Korth et al. [JGR 104, 25047, 1997] found good agreement with theory for keV electrons. MeV electrons? MeV electrons?

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S. Elkington, GEM 2003 MHD/particle simulations of energetic electron trapping 60 keV test electrons, constant M. 60 keV test electrons, constant M. Started 20 R E downtail, 15s intervals. Started 20 R E downtail, 15s intervals. Evolves naturally under MHD E and B fields. Evolves naturally under MHD E and B fields. Removed from simulation at magnetopause. Removed from simulation at magnetopause. Color coded by energy. Color coded by energy.

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S. Elkington, GEM 2003 Conclusions/Summary Our understanding of radial transport in the radiation belts is better, but still will require considerable effort. Transport equations: DLL proportionalities, energy dependence. Transport equations: DLL proportionalities, energy dependence. ULF waves: ULF waves: Power spectrum and occurrence characteristics of ULF waves. Power spectrum and occurrence characteristics of ULF waves. Mode structure of magnetospheric waves. Mode structure of magnetospheric waves. How is power coupled from solar wind to inner magnetosphere? How is power coupled from solar wind to inner magnetosphere? Boundary conditions: the plasma sheet? Boundary conditions: the plasma sheet? Determine plasmasheet access. Determine plasmasheet access. Distribution in plasmasheet. Distribution in plasmasheet.

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