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

Contents

• 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.

Radiation Belt Dynamics Simulations

ULF Wave Power Map

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

W, is an eigenvalue which gives the frequency, of the wave (real only for infinite ΣP). z and y indicates the position along the field line, z=cos and y=z+z3

The field-aligned density profile, varies alongthe field line as,

Guided Poloidal Wave Equation in a Dipole Field

0)()31(

)1(22

62

2

2

ΕhzP

z

zW

dy

Εhd

62

22

)1(

)31()(

z

zzP

hE is the wave electric field,

E of the guided poloidal wave multiplied by a dipole scale factor, h Polar coordinates

Guided Poloidal Wave Equation Solution

0|| E

)(exp)cos()( mtiWyLXCΕh

0)()31(

)1(22

62

2

2

ΕhzP

z

zW

dy

Εhd

62

22

)1(

)31()(

z

zzP

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

02

2

ΕWhdy

Εhd

LE

The b┴ components are obtained from ||E

t

b

Wave E-field Structure

•FLR at L=4 with peak amp of 3mV/m.

•Period of wave 100 seconds.

•Azimuthal wavelength m=15.

E

E

Along the field line Across L-shell

Wave b-field Structure

•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 solution0 b

b

b

Along the field line At the ionosphere

b am

p nT

b am

p nT

Equations of Motion

222||

0

||

B

BE

Bq

BMB

B

BB

q

pB

Bm

pvd

B

MqE

B

BB

q

p

B

B

t

p

3||||

EEE bbBB dipole

Taken from, T. G. Northrop, The Adiabatic Motion of Charged Particles

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 numberis the angular frequency of the wave

Wave Acceleration of Electrons by Drift Resonance

m/

Trapping width for low, medium and high equatorial pitch-angles, αeq

(T~2500 sec) αeq =18º αeq = 45º

αeq =90º

wave phase mt wave phase mt wave phase mt

L-s

he

llE

nerg

y, M

eV

Pitch-Angle Dependence of Electron Transport for a

fixed energy.

10º

15º

20º

27º

54º

Initial ConditionsW=0.95MeV, L=3.7

wave phase m-t

Low pitch-angle electrons on trapped orbits, (T~2500 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

L-sh

ell

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

L=6L=4

Drift-resonant transport with a decaying wave amplitude producing a peak in phase space density (90º pitch-angles)

Degeling, et. al., JGR, 2007M=7.5MeV/μT, freq=3mHz

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.

Future Work

• Develop bounce averaged equations of motion from these analytic fields.

• Run simulations with distributions of electrons, look at the evolution of PSD.

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”

ULF Wave Radial Diffusive Transport Models

m

DEmRB

mLL mLPLD

E),(6

81

220

m

DBmRBq

MmLL mLPLD

E),(42

81

20

“MAGNETIC”

Compressional Magnetic Field Power

f

L

fD

LLL

dt

dfLL

2

2 1

“ELECTRIC”

Perpendicular (azimuthal) Electric

Field Power

Both terms are difficult to prescribe with space-based observational data

CARISMA stations TALO to PINA

10 years of data (1994-2003).

L=11.2

L=4.1

Open field lines

Morning Sector Peak ~4mHz

0600 MLT GILL L=6.2

FLR

•Power increases with solar wind speed, all frequencies.

•Power law decay with frequency. Power~frequency-L

GILL (L=6.6)

12 MLT

18 MLT

06 MLT

24 MLT

Evidence of FLR on the day side and morning sector only.

H-component 1-10 mHz(same scale)

•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 L=5-7 in the noon and morning sectors.

LOW SW

HIGH SW

D-component 1-10mHz (same scale)

•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 L=5-7 in the noon and morning sectors.

LOW SW

HIGH SW

D-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

HIGH SW

H-component

LOW SW

HIGH SW

Mapping Guided Alfvén Waves from the Ground to the Ionosphere

iP

i Eb 0

EH

Pg

i

R

hLm

b

b2/1

2

22

)(

4exp

Hughes and Southwood, JGR, 1976 showed that the magnetic field amplitude on the ground, bg is related to that at the ionosphere, bi by

is the latitudinal wavelength at the ionosphere (~half-width of the FLR.m is the azimuthal wavenumber.h is the height of the ionosphere.

At the ionosphere the electric field, Ei , and magnetic field, bi are related by

2

P

H Wallis and Budzinski, JGR, 1981

The Guided Alfvén Wave Equations in a Dipole Field

0)()31(

)1( 222

62

2

EhWZP

Z

Z

Y0)()1( 262

2

EhWZPZ

Z

W, is an eigenvalue which gives the frequency, 0 and damping factor of the wave

W=((0 +i)LRE)2 /Aeq

2 . hand h are dipole scale factors

Guided Toroidal mode EGuided Poloidal mode E

We assume that the electric field varies along the field line in the same way both at the FLR location and away from it.

Fundamental Mode Guided Alfvén Wave Eigenfunctions(solid curves L=4.5, dashed curves L=8.5)

Guided Alfvén Wave Eeq/bi as a Function of L-shell

•Away from the plasmapause, Eeq/bi gets smaller with increasing L-shell.

•FLR’s with the same magnetic field amplitude in the ionosphere, have electric field amplitudes 10 times greater at L=4.5 than at L=8.5.

•Eeq/bi does not depend on the ionospheric

Pedersen conductance P

frequencyb

Ei

eq

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 L=4.

Determine an empirical function to describe the ULF power maps in terms of physical quantities, L, MLT, SW.

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.

• FLR’s with the same magnetic field amplitude in the ionosphere can have electric field amplitudes ~10 times greater at L=4.5 than at L=8.5.

• Consequently, FLR’s which occur on low L-shells may have a significant effect on the energisation and dynamics of radiation belt electrons.

Conjugate Ground and Satellite FLR Observation

T89c Model

Ground-Based Magnetometer Observation Toroidal FLR observed on the ground at GILL, dipole L=6.7.

~75 nT peak amplitude

m=4 azimuthal wavenumber

Latitudinual half-width, ~6o

Period=667 sec

~6º

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 Eeq=1.6 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.