Drift Resonant Interactions of Radiation Belt Electrons with ULF waves.

1
Drift Resonant Interactions of Radiation Belt
Electrons with ULF waves.

• L. G. Ozeke, I. R. Mann, A. Degeling, V. Amalraj,
  and I. J. Rae
• University of Alberta
• REPW, August 6th 10th 2007

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Contents
Radiation Belt Dynamics Simulations

• Analytic model of the electric and magnetic
  fields of a high-m guided poloidal FLR.
• 3D adiabatic equations of motion for a charged
  particle in a dipole field plus the ULF magnetic
  and electric field perturbations.
• Pitch-angle dependence on the drift-resonant
  radial transport and energisation of MeV
  electrons in the outer radiation belt.

ULF Wave Power Map

• Development of a ULF wave power map as a function
  of solar wind from the CARISMA magnetometer
  network.

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Guided Poloidal Wave Equation in a Dipole Field
hfEf is the wave electric field, Ef of the guided
poloidal wave multiplied by a dipole scale
factor, hf
W, is an eigenvalue which gives the frequency, w0
of the wave (real only for infinite SP).
z and y indicates the position along
the field line, zcosq and yzz3 The
field-aligned density profile, r, varies
along the field line as,
Polar coordinates
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Guided Poloidal Wave Equation Solution
If the density varies with z as
then
So that
Here X is a constant, C(L) is a gaussian function
with a 180º phase change across the L-shell of
the resonant field line.
Since the wave is Alfvenic we assume the
compressional magnetic field, is zero b 0, so
that
The b- components are obtained from
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Wave E-field Structure
Along the field line
Across L-shell
Ej
En

• FLR at L4 with peak amp of 3mV/m.
• Period of wave 100 seconds.
• Azimuthal wavelength m15.

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Wave b-field Structure
Along the field line
At the ionosphere
b amp nT
bn
b amp nT
bj

• Phase of b changes by 180º at the equatorial
  plane.
• H and D components of b may have the same amp
  on the ground.
• Analytic solution

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Equations of Motion
Taken from, T. G. Northrop, The Adiabatic Motion
of Charged Particles
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Wave Acceleration of Electrons by Drift Resonance
Eastward drifting electrons may be energised by
fundamental guided poloidal mode waves via this
drift resonance mechanism. Here the electrons
azimuthally drift around the Earth at the same
phase speed as the wave. m is the waves
azimuthal wave number w is the angular frequency
of the wave
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Trapping width for low, medium and high
equatorial pitch-angles, aeq (T2500 sec)
aeq 18º aeq 45º
aeq 90º
L-shell
Energy, MeV
wave phase mf-wt
wave phase mf-wt
wave phase mf-wt
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Pitch-Angle Dependence of Electron Transport for
a fixed energy.
Low pitch-angle electrons on trapped orbits,
(T2500 sec). High pitch-angle electrons on open
orbits. Electrons are transported inward onto
the same L-shell with the same energy. If the
wave amplitude decays away the electrons will
remain on the same L and with the same energy
Initial Conditions W0.95MeV, L3.7
20º
L-shell
27º
15º
10º
54º
wave phase mf-wt
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Pitch-Angle Distributions CRRES(taken from Horne
et. al., JGR, 2003)
Enhanced Transport of low equatorial pitch-angle
electrons may help produce these observed
flat-top and butterfly pitch-angle distributions.
Flat-top distribution
Butterfly distribution
L4
L6
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M7.5MeV/µT, freq3mHz
Degeling, et. al., JGR, 2007
Drift-resonant transport with a decaying wave
amplitude producing a peak in phase space density
(90º pitch-angles)
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Summary of Resonant Transport

• Trapping width independent of electrons
  pitch-angle.
• Resonant energy is dependent on the equatorial
  pitch-angle of the electron.
• The lower the electrons equatorial pitch-angle
  the higher the resonant energy.
• Possible for low equatorial pitch-angle electrons
  to be transported inward (and outward) much
  further than higher pitch-angle electrons.
• This may produce energy dependent butterfly
  pitch-angle distributions (or erode them) as the
  wave amplitude decays with time.

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Future Work

• Develop bounce averaged equations of motion from
  these analytic fields.
• Run simulations with distributions of electrons,
  look at the evolution of PSD.

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Global ULF Wave Power Maps

• ULF waves may play a critical role in the
  energisation of the inner magnetosphere and
  radiation belts.
• Characterise ULF Wave power as a function of
  measurable parameters in the magnetosphere and
  solar wind.
• Provide the input to radial-diffusion driven
  radiation belt models.
• May provide evidence (or lack thereof) of
  magnetospheric magic frequencies

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ULF Wave Radial Diffusive Transport Models
MAGNETIC
ELECTRIC
Compressional Magnetic Field Power
Perpendicular (azimuthal) Electric Field Power
Both terms are difficult to prescribe with
space-based observational data
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CARISMA stations TALO to PINA10 years of data
(1994-2003).
Open field lines
L11.2
L4.1
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Morning Sector Peak 4mHz
FLR
0600 MLT GILL L6.2

• Power increases with solar wind speed, all
  frequencies.
• Power law decay with frequency. Powerfrequency-L

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GILL (L6.6)
12 MLT
18 MLT
06 MLT
24 MLT
Evidence of FLR on the day side and morning
sector only.
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H-component 1-10 mHz(same scale)
LOW SW

• Red high power purple low power (log scale)
• Integrated power 1 mHz to 10 mHz
• Power increases across all L-shell with
  increasing solar wind.
• Clear evidence of FLR, enhanced power between
  L5-7 in the noon and morning sectors.

HIGH SW
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D-component 1-10mHz (same scale)
LOW SW

• Red high power purple low power (log scale)
• Integrated power 1 mHz to 10 mHz
• Power increases across all L-shell with
  increasing solar wind.
• No clear enhanced power between L5-7 in the noon
  and morning sectors.

HIGH SW
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D-component
H-component

• Independently scaled (linear)
• H and D clearly have different structures.
• H-power domanated by morning sector FLRs.
• D-power night time (substorms).
• Morning to dayside?

LOW SW
LOW SW
HIGH SW
HIGH SW
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Mapping Guided Alfvén Waves from the Ground to
the Ionosphere
Hughes and Southwood, JGR, 1976 showed that the
magnetic field amplitude on the ground, bg is
related to that at the ionosphere, bi by
Dq is the latitudinal wavelength at the
ionosphere (half-width of the FLR). m is the
azimuthal wavenumber. h is the height of the
ionosphere.
Wallis and Budzinski, JGR, 1981
At the ionosphere the electric field, Ei , and
magnetic field, bi are related by
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The Guided Alfvén Wave Equations in a Dipole Field
Guided Poloidal mode Ej
Guided Toroidal mode En
W, is an eigenvalue which gives the frequency, w0
and damping factor g of the wave W((w0 ig)LRE)2
/Aeq2 . hj and hn are dipole scale factors
We assume that the electric field varies along
the field line in the same way both at the FLR
location and away from it.
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Fundamental Mode Guided Alfvén Wave
Eigenfunctions(solid curves L4.5, dashed
curves L8.5)
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Guided Alfvén Wave Eeq/bi as a Function of L-shell

• Away from the plasmapause, Eeq/bi gets smaller
  with increasing L-shell.
• FLRs with the same magnetic field amplitude in
  the ionosphere, have electric field amplitudes 10
  times greater at L4.5 than at L8.5.
• Eeq/bi does not depend on the ionospheric
  Pedersen conductance SP

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Summary Global ULF wave maps

• ULF diffusive transport (and coherent transport)
  theories require equatorial electric field
  amplitudes.
• However, in-situ observations of the waves
  electric field are rare in comparison to
  ground-based measurements of the magnetic field
  amplitude.
• Here we have characterized the ULF wave power on
  the ground, showing the influence of FLRs and SW
  speed on the amount of wave power.
• By numerically solving the guided toroidal (and
  guided poloidal) wave equations we showed how bg
  on the ground can be used to estimate Eeq.

Future Work
Extend the Global ULF power maps below
L4. Determine an empirical function to describe
the ULF power maps in terms of physical
quantities, L, MLT, SW.
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Summary

• Drift resonance theories show that the equatorial
  electric field amplitude determines the amount of
  energy an electron can receive from a single ULF
  wave.
• However, in-situ observations of the waves
  electric field are rare in comparison to
  ground-based measurements of the magnetic field
  amplitude.
• By numerically solving the guided poloidal wave
  equation we showed how bg on the ground can be
  converted to Eeq.
• FLRs with the same magnetic field amplitude in
  the ionosphere can have electric field amplitudes
  10 times greater at L4.5 than at L8.5.
• Consequently, FLRs which occur on low L-shells
  may have a significant effect on the energisation
  and dynamics of radiation belt electrons.

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Conjugate Ground and Satellite FLR Observation
T89c Model
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Ground-Based Magnetometer Observation
Toroidal FLR observed on the ground at GILL,
dipole L6.7. 75 nT peak amplitude m4
azimuthal wavenumber Latitudinual half-width,
Dq6o Period667 sec
6º
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In-situ Satellite Observation

• Using our mapping model the FLR observed on the
  ground with a peak amplitude of 75 nT will result
  in an equatorial electric field amplitude of
  Eeq1.6 mV/m
• From the conjugate satellite observation it
  appears that the peak electric field amplitude
  Eeq is 1.5 mV/m.
• Consequently, the mapping of the ground-based
  magnetic field to the equatorial electric field
  is in excellent agreement with the observation.