Recent Results from Theory and Modeling of Radiation Belt Electron Transport, Acceleration, and Loss

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Recent Results from Theory and Modeling of
Radiation Belt Electron Transport, Acceleration,
and Loss

• Anthony Chan, Bin Yu, Xin Tao, Richard Wolf
• Rice University
• Scot Elkington, Seth Claudepierre
• University of Colorado
• Jay Albert
• AFRL
• Michael Wiltberger
• NCAR

REPW, Rarotonga, Cook Islands, August 7, 2007
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OUTLINE

• 1. Radial Diffusion in High-Speed-Stream Storms
• 2. MHD-Particle Simulation of a HSS Storm
• 3. Multidimensional Diffusion Using SDEs

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1. Radial Diffusion in High-Speed-Stream Storms

• Bin Yu, PhD thesis, 2007
• Solve the standard radial diffusion equation,
  with loss term.
• DLL from Brautigam and Albert 2000.
• Loss lifetime from Shprits et al 2004, and
  Meredith et al 2006.
• Dynamic outer boundary Location min(L_GEO,
  0.9L_last-closed)
• Outer boundary value from Li et al 2001
  GEO model.
• Fixed inner boundary L2, value from AE8MIN
• Initial condition from AE8MIN.
• Magnetic field Hilmer and Voigt 1995.
• For comparison Tsyganenko 2001, dipole

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Some Details of the Radial Diffusion Model

• M 20 MeV/G 6000MeV/G, 100 bins
• L 27, 100 bins
• Time Steps 4min. Total time 6 days
• Method Crank-Nicholson implicit method
• Our approach
• Consider a model HSS storm in declining phase of
  the solar cycle
• Compare with a series of HSS events, between
  1995 and 1996, published by Hilmer et al 2000.

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A Typical High-Speed-Stream Storm

• Solar wind parameters and indices for the January
  1995 high-speed
• stream (HSS) storm

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Solar Wind Parameters for a Model
High-Speed-Stream Storm
Schematic illustration of a CIR Pizzo, 1978.
Input parameters for our idealized declining
phase magnetic storm (a) solar wind density n
(cm-3), (b) solar wind velocity V (km/s), (c) IMF
Bz (nT), (d) solar wind ram pressure P (nPa),
(e) Dst index, (f) Kp index, (g) midnight
equatorward boundary of the aurora.
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Electron Lifetime Model

• Plasmapause location
• Lpp 5.6 0.46 Kp Carpenter
  and Anderson, JGR, 1992
• Outside the plasmapause
• Use a Kp-dependent lifetime of electron
  loss from Shprits et al, GRL, 2004.
• I.e., 0.5 day during storm main phase (Kp6), 3
  days under quiet conditions (Kp2), and
  linearly dependent on Kp.
• GEO to GPS is mostly outside the
  plasmapause for HSS events.
• Inside the plasmapause
• Estimate the recovery-phase electron
  lifetime based on CRRES measurements Meredith et
  al., JGR, 2006.
• Assume typical VLF wave amplitudes of
  10pT and 35pT and multiply the lifetime by
  (10/35)2 to get the main-phase lifetime.

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PSD f(R,M,t) Results for the January 1995 Event
Six-hour averages of PSD from observations Hilmer
et al., 2000 for Julian Day 28-34, 1995

• Compare simulation results with observations.
• Middle simulation results exhibit similar shape
  with observations, but diffusion is too fast.
• Lowering DLL by a factor of two gives better
  agreement.

Simulation result using similar solar wind
condition and Brautigam and Albert formula of
DLL.
Simulation result using DLL/2.
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PSD f(R,M,t) Results for the July 1995 Event

• Another high-speed-stream event The July 1995
  storm event
• No growth of phase space density at R 4.2 Re is
  observed
• Average Kp during the recovery phase is about 3
• Again, better agreement with observations is
  obtained if we divide the Brautigam and Albert
  diffusion coefficient by 2.

Observations
DLL
DLL/2
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Reasonable agreement is obtained between measured
and simulated rate-of-increase of PSD at GPS,
using BA DLL divided by 2(0.5).
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Radial Diffusion in High-Speed-Stream Storms
Summary

• Enhancement of MeV electrons at R 4 during
  high-speed-stream storms is well reproduced by
  radial diffusion modeling.
• Diffusion can transport electrons efficiently to
  lower L from a source region near L6.6Re,
  consistent with the GPS data.
• If we artificially divide the Brautigam and
  Albert 200 formula for DLL by a factor of 2,
  the simulation results reproduced the Hilmer et
  al. 2001 observations well.

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OUTLINE

• 1. Radial Diffusion in High-Speed-Stream Storms
• 2. MHD-Particle Simulation of a HSS Storm
• 3. Multidimensional Diffusion Using SDEs

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2. MHD-Particle Simulation of a HSS Storm
Overview

• A. MHD-Particle Simulation
• B. Phase-Space Density Evolution
• C. Radial Diffusion Coefficients
• Summary
• Bin Yu, PhD thesis, 2007

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A. MHD-Particle Simulation

• The LFM global MHD code is driven by solar-wind
  inputs for the Jan 1995 high-speed-stream (HSS)
  storm
• Equatorial particles are traced by solving
  relativistic guiding-center equations of Brizard
  and Chan Phys. Plasmas, 1999.

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MHD-Particle Simulation Results

• Black lines Constant-B contours. Dashed
  circles 3, 5, 7, RE
• Color particle energy, M 2100 MeV/G
• Particle boundaries at 3.5 RE and 10 RE
• Reference for
  MHD-particle method Elkington et al, JASTP, 2002.

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MHD-Particle Simulation Results Snapshots

• From pre-storm to late recovery phase (top L to
  R, bottom L to R)
• Magnetopause loss occurs early in Jan 29 (between
  panels 2 and 3)

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B. Phase-Space Density Evolution

• Overview of Method
• Use Liouvilles theorem, regard GC particles as
  markers.
• Initial PSD f is scaled from AE8 empirical
  model.
• Step markers in time with GC equations of motion.
• PSD f is conserved along each marker trajectory.
• Recalculate PSD f on an equatorial grid using an
  area-weighting scheme Nunn, J. Comp. Phys.,
  1993

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The phase-space density (PSD) weighting scheme
The contribution of each marker to the total
phase-space density is calculated on the grid
using an area-weighting formula
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Advantages of this PSD-evolution algorithm

• Low noise level and efficient use of
  particles/markers.
• The resulting PSD f is always non-negative.
• (Negative values can be a problem in PDE
  solvers.)
• A variety of boundary conditions can be
  implemented.
• E.g., markers at or outside GEO may be assigned
  the observed GEO phase-space density.
• New markers can be added, if needed (but marker
  weights have to be carefully re-normalized)
• A loss lifetime can be used to decrease PSD at
  each grid point, at each time step.

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Phase-Space Density Results I

• Observed (blue) and simulated (red) electron PSD.
• Solid line GEO, dashed lines GPS. M
  2100 MeV/G
• Observations show increase at GEO, followed by
  increase at GPS
• Simulations have free boundary condition and no
  loss lifetime.
• Poor agreement at GEO suggests a source nearby
• Observations from Hilmer et al., JGR, 2000

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Phase-Space Density Results II

• Observed (blue) and simulated (red) electron PSD.
• Solid line GEO, dashed lines GPS. M 2100
  MeV/G
• Simulations now have dynamic outer boundary
  condition (but still no electron lifetime).
• At GPS better agreement, but simulation PSD is
  still too high - this suggests adding electron
  lifetime

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Phase-Space Density Results III

• Observed (blue) and simulated (red) electron PSD.
• Solid line GEO, dashed lines GPS. M 2100
  MeV/G
• Simulations now have dynamic outer boundary
  condition and electron lifetime model Shprits
  et al, GRL, 2004 Meredith et al, JGR, 2006
• Good agreement at GPS!

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Phase-Space Density Results Summary
Simulated (red) and observed (blue) electron
phase-space density (M 2100 MeV/G)

• Free outer boundary condition
• No electron lifetime loss

• Dynamic outer boundary condition
• No electron lifetime loss

• Dynamic outer boundary condition
• Loss lifetime of Shprits et al, 2004

With the dynamic GEO boundary condition and an
electron lifetime model good agreement is
obtained between simulations and observations.
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C. Radial Diffusion Coefficients

• Fourier analysis of MHD fields yields electric
  and magnetic power spectral densities (next talk
  in this session)
• Power spectral densities can be substituted into
  formulae for quasilinear radial diffusion
  coefficients to obtain DLL

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DLL for electromagnetic perturbations

• For general electromagnetic perturbations (for
  equatorial particles)
• where and are power
  spectral densities of compressional magnetic and
  azimuthal electric fields, evaluated at
  
• Brizard and Chan, Phys. Plasmas, 2004 Fei et
  al, JGR, 2006
• In the nonrelativistic, limit, and
  with , the above result agrees
  with of Falthammar 1968

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Results Main-phase DLL values

• Dominated by the magnetic power term for L lt 6
• Proportional to L5.8 (Compare with L10
  Brautigam and Albert, JGR, 2000)
• 2-3 orders of magnitude larger than pre-storm
  values

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MHD-Particle Simulation of a HSS Storm Summary

• We have developed an improved algorithm for
  evolving PSD f in MHD-particle simulations.
• We have simulated the Jan 1995 HSS storm and
  compared to spacecraft MeV electron data at GEO
  and GPS.
• With both the dynamic GEO boundary condition and
  an electron lifetime model we obtain good
  agreement with observations.
• During the main phase, DLL calculated from MHD
  power is
• Proportional to L5.8
• 2-3 orders of magnitude larger than pre-storm
  values
• What is the role of VLF/ELF local acceleration in
  HSS storms?

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OUTLINE

• 1. Radial Diffusion in High-Speed-Stream Storms
• 2. MHD-Particle Simulation of a HSS Storm
• 3. Multidimensional Diffusion Using SDEs

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3. Multidimensional Diffusion Using SDEs

• Xin Tao (PhD thesis research), Anthony Chan, Jay
  Albert
• Cyclotron resonances give coupled pitch-angle and
  energy/momentum diffusion.
• Radiation belt diffusion may be described by
  Fokker-Planck diffusion equations in (J1,J2,J3)
  coordinates, or (pitch-angle, momentum, L)
  coordinates,
• Standard finite-difference methods fail for
  non-diagonal diffusion tensors.
• Albert and Young 2005 transform to coordinates
  which diagonalize the 2D equatorial-pitch-angle-mo
  mentum diffusion tensor.
• We have developed a new method for solving RB
  diffusion eqs

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Fokker-Planck Equations and SDEs

• It can be shown that every Fokker-Planck equation
  is mathematically equivalent to a set of
  stochastic differential equations (SDEs).
• A 1D SDE has the form
• dX b dt s dW
• where dX is a change in a stochastic
  variable associated with a time increment dt, dW
  sqrt(dt) N(0,1) is called a Wiener process
  (here N(0,1) is a Gaussian normal random
  variable), and b and s are regular scalar
  functions.
• For an n-dimensional diffusion equation there are
  n coupled SDEs of the above form, but b and dW
  are vectors and s is a matrix.
• The coefficients b and s are directly related to
  the diffusion tensor of the corresponding
  Fokker-Planck equation.

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Advantages of the SDE Methods

• Generalization of the SDE methods to higher
  dimensions is straightforward.
• SDE methods have no difficulty with off-diagonal
  diffusion tensors and they always yield
  non-negative phase-space densities.
• SDE methods are more efficient than
  finite-difference methods when applied to
  high-dimensional problems.
• SDE codes are easy to parallelize.
• SDE methods provide an exciting new numerical
  method for solving RB
• diffusion equations!

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2D SDE Results I Comparison with the Albert and
Young 2005 Diagonalization Method

• Electron flux vs. equatorial pitch angle, 0.5
  MeV, L4.5, chorus wave parameters, 4.5º
  loss-cone angle.
• Solid line Rice SDE solver, dashed line
  Albert and Young 2005.
• Note the excellent agreement!

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2D SDE Results II Comparison of fluxes for
Albert and Young 2005 vs. Summers 2005
coefficients
t 1 day
t 0.1 day

• Electron flux vs. equatorial pitch angle, 0.5
  MeV, L4.5, chorus wave
• parameters, 4.5º loss-cone angle.
• Dashed lines Summers 2005 Parallel waves,
  neglect off-diagonal terms
• Solid lines Albert and Young 2005 Oblique
  waves, retain off-diagonal terms
• Results agree near 90º, but Summers 2005
  overestimates at small angles

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Multidimensional Diffusion Using SDEs Summary

• We have developed and tested a new method for
  solving RB diffusion equations.
• SDE methods have some advantages over
  finite-difference methods (but we need both!)
• First 2D results are encouraging and extension to
  3D is straightforward.

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