Transport in the Radiation Belts and the role of Magnetospheric ULF Waves

S. Elkington, GEM 2003

Transport Transport in thein the Radiation Belts Radiation Belts and the role ofand the role of

Magnetospheric ULF WavesMagnetospheric ULF Waves

Scot R. ElkingtonScot R. ElkingtonLASP, University of ColoradoLASP, 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 2003Snowmass, CO


OutlineOutline

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 Radial transport and the role of ULF

waves.waves. Characteristics of ULF waves, MHD Characteristics of ULF waves, MHD

simulationssimulations Boundary conditions on radial transport.Boundary conditions on radial transport. Conclusion.Conclusion.


The outer zone radiation The outer zone radiation beltsbelts

Trapped particles drifting Trapped particles drifting in orbits encircling Earth.in orbits encircling Earth.

Two spatial populations: Two spatial populations: inner zone and outer zone.inner zone and outer zone.

Energies from ~200 Energies from ~200 keV to > few MeVkeV to > few MeV


Particle motion in the Particle motion in the radiation beltsradiation belts

Trapped particles execute 3 characteristic types of motion:Trapped particles execute 3 characteristic types of motion:

•Gyro: ~ millisecond•Bounce: ~ 0.1-1.0 s•Drift: ~ 1-10 minutes

Characteristic time scales:


Adiabatic invariantsAdiabatic invariants

MM: perpendicular motion: perpendicular motion KK: parallel motion: parallel motion LL: radial distance of eq-: radial distance of eq-

crossing in a dipole field.crossing in a dipole field.

Associated with each motion is a corresponding adiabatic invariant:•Gyro: M=p2/2m0B•Bounce: K•Drift: L

If the fields guiding the particle change slowly compared to the characteristic motion, the corresponding invariant is conserved.


Fluxes in the radiation Fluxes in the radiation beltsbelts

The radiation belts exhibit substantial The radiation belts exhibit substantial variation in time:variation in time: •Storm

commencement: minutes•Storm main phase: hours•Storm recovery: days•Solar rotation: 13-27 days•Season: months•Solar cycle: years


Why study the radiation Why study the radiation belts?belts?

Because they’re Because they’re physically physically interesting!interesting!

Relativistic Relativistic electrons have electrons have been associated been associated with spacecraft with spacecraft ‘anomalies’.‘anomalies’.

Want to try to Want to try to describe how describe how radiation evolves radiation evolves in time at a given in time at a given point in space.point in space.


Describing the radiation Describing the radiation beltsbelts

),,,,,( zyx pppzyxff

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 (px+dpx, py+dpy, pz+dpz ).The flux in a region of space may be related to the distribution function through

2p

jf


The distribution function The distribution function and the adiabatic invariantsand the adiabatic invariantsThe 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.

),,,,,(

),,,,,(

321 LKMf

pppzyxff zyx

),,( LKMff


Transport and Fokker-Transport and Fokker-Planck Planck

Sf

J

fD

Jf

dt

dJ

Jdt

df

jij

ji i

i

i i

,

11

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

f

L

fD

LLL

dt

dfLL

2

2 1

(M. Schulz, AGU Monograph 97, 1996)


Transport in M, K: local Transport in M, K: local heatingheating


Local heating example:Local heating example:resonant interactions with resonant interactions with

VLF wavesVLF 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:


Transport in Transport in LL: radial : radial transporttransport


Energization and ULF Energization and ULF waveswaves

ffDD~mHz … ULF waves!~mHz … ULF waves! Only need consider Only need consider

resonant frequencies resonant frequencies in analysis:in analysis:

Evqdt

dWD

)cos(0 tmEE

Electron moving in a dipole magnetic field with slowly-varying dawn-dusk potential electric field


……and to radial transportand to radial transport

3/10

W

MBL

Nonrelativistically, and in a dipole,

or

0

3

0

2

2 B

WL

Bm

pM

so changing energy (W) while conserving M willnecessarily lead to transport in L.


Observed associations Observed associations between ULF waves and between ULF waves and radiation belt activity?radiation belt activity?

•Baker et al., GRL 25, 2975, 1998 •Rostoker, GRL 25, 3701, 1998•Mathie & Mann, GRL 27, 3621, 2000•O’brien et al., JGR, 2003, in press.


Characterizing ULF Characterizing ULF interactionsinteractions

Start particles at differing Start particles at differing energiesenergiesdiffering drift freqs.differing drift freqs.

Impose waves characterized Impose waves characterized by particular m, by particular m,

Look max energy gain atLook max energy gain at

Poincare plots:Poincare plots: Plot motion in the phase space Plot motion in the phase space

coordinates coordinates (L,(L,),), at snapshots in at snapshots in time differing by 1/time differing by 1/ff..

Particle motion constrained to lines.Particle motion constrained to lines. Drift resonance appears as ‘island’ Drift resonance appears as ‘island’

at appropriate energy (frequency).at appropriate energy (frequency).

Dm


Higher-order resonancesHigher-order resonancesIn the outer zone, the m=1 noon-midnight asymmetry leads to additional resonances: Dm )1(


Multiple frequencies: radial Multiple frequencies: radial diffusiondiffusion


Quantifying effects of Quantifying effects of diffusiondiffusion

f

L

fD

LLL

dt

dfLL

2

2 1

),(60 Dmm

LL mLPLDD

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…

DmmLL mLPL

MDD

)1(,11)1(

0)1(

•What are the D0’s?•What is the ULF power?


Diffusion Characteristics: Diffusion Characteristics: DD00

Start series of particles Start series of particles at one at one L.L.

Let them evolve in a Let them evolve in a compressed dipole compressed dipole under the influence of a under the influence of a constant constant P(L,P(L,).).

Watch spread of Watch spread of LL in in time.time.

Calculate DCalculate DLLLL=<dL=<dL22>/2>/2 Invert to calculate DInvert to calculate D00..


Diffusion Characteristics, Diffusion Characteristics, 09/24-26/1998:09/24-26/1998:

constant P, time-varying constant P, time-varying compressioncompression


Characteristics of ULF Characteristics of ULF waves?waves?

ULF PowerULF Power ULF waves: ULF waves:

decrease with decrease with decreasing Ldecreasing L

Decrease with Decrease with increasing frequencyincreasing frequency

Depend on solar Depend on solar wind velocity wind velocity

May or may not have May or may not have significant local time significant local time dependence.dependence.

Kepko et al., GRL 29(8), 1197, 2002


ULF activity in MHD ULF activity in MHD simulationssimulations

ULF waves can be ULF waves can be described within the described within the framework provided framework provided by MHD.by MHD.

Lyon-Fedder-Mobarry:Lyon-Fedder-Mobarry:•Global, 3dGlobal, 3d•Driven by upstream Driven by upstream conditionsconditions


MHD simulations of ULF MHD simulations of ULF power, 09/24/1998power, 09/24/1998

ULF power in MHD ULF power in MHD shows expected shows expected radial, frequency radial, frequency dependence.dependence.

Azimuthal dependence: Azimuthal dependence: frequently see structure frequently see structure in local time.in local time.


Parametric studies, Parametric studies, perturbationsperturbations

Constant solar wind Constant solar wind velocity 600 km/s.velocity 600 km/s.

Density perturbations Density perturbations in solar wind at 3 mhZ.in solar wind at 3 mhZ.

Spectral analysis in Spectral analysis in magnetosphere shows magnetosphere shows strong peak at 3 mHz, strong peak at 3 mHz, but also peaks at 5, 7, but also peaks at 5, 7, and 10 mHz.and 10 mHz.


Parametric studies, const Parametric studies, const solar windsolar wind

Constant solar wind Constant solar wind velocity 600 km/s.velocity 600 km/s.

Nothing else!Nothing else!

Indicates that 5, 7 Indicates that 5, 7 mHz features are due mHz features are due to K-H waves on the to K-H waves on the flanks.flanks.


Shear waves & particle Shear waves & particle accelerationacceleration

Limited local time: Limited local time: propagating waves dusk propagating waves dusk and counterpropagating and counterpropagating waves dusk still lead to waves dusk still lead to energization. energization.


Mode structure in ULF Mode structure in ULF simulationssimulations

Diffusion coefficients: depend on Diffusion coefficients: depend on P(mP(mDD),), etc., so etc., so must know the power in must know the power in each specific mode numbereach specific mode number..

Main

P

hase

Reco

very

P

hase


Energization through radial Energization through radial transport: a plasma sheet transport: a plasma sheet

source?source? Simplistic considerations put Simplistic considerations put rr00~20 ~20

RREE for for MM corresponding to a corresponding to a 1 MeV1 MeV geosynchronous electrongeosynchronous electron

Conversely, Conversely, W for rW for r00=6.6R=6.6REE is 50 keV. is 50 keV. So is transport/acceleration a viable So is transport/acceleration a viable

mechanism at all? mechanism at all?

N. Tsyganenkohttp://nssdc.gsfc.nasa.gov/space/model/magnetos/data-based/modeling.html


Plasmasheet access to Plasmasheet access to inner magnetosphere: inner magnetosphere:

observationsobservations

Korth et al. [Korth et al. [JGR 104JGR 104, 25047, 1997] found good , 25047, 1997] found good agreement with theory for keV electrons.agreement with theory for keV electrons.

MeV electrons?MeV electrons?


MHD/particle simulations MHD/particle simulations of energetic electron of energetic electron

trappingtrapping

60 keV test electrons, 60 keV test electrons, constant constant MM. .

Started 20 RStarted 20 REE downtail, 15s intervals.downtail, 15s intervals.

Evolves naturally Evolves naturally under MHD under MHD E E and and BB fields.fields.

Removed from Removed from simulation at simulation at magnetopause.magnetopause.

Color coded by energy.Color coded by energy.


Conclusions/SummaryConclusions/SummaryOur understanding of radial transport in the Our understanding of radial transport in the

radiation belts is better, but still will require radiation belts is better, but still will require considerable effort.considerable effort.

Transport equations: DLL proportionalities, Transport equations: DLL proportionalities, energy dependence.energy dependence.

ULF waves:ULF waves: Power spectrum and occurrence characteristics of ULF Power spectrum and occurrence characteristics of ULF

waves.waves. Mode structure of magnetospheric waves.Mode structure of magnetospheric waves. How is power coupled from solar wind to inner How is power coupled from solar wind to inner

magnetosphere?magnetosphere? Boundary conditions: the plasma sheet?Boundary conditions: the plasma sheet?

Determine plasmasheet access.Determine plasmasheet access. Distribution in plasmasheet.Distribution in plasmasheet.

Transport in the Radiation Belts and the role of Magnetospheric ULF Waves

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