A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004...

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A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop

Transcript of A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004...

Page 1: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

A High Latitude Source for Radiation Belt Particles?

Robert Sheldon, NSSTC

October 19, 2004

Huntsville Modelling Workshop

Page 2: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

McIlwain, 1966

Page 3: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Correlations• Highest SW correlation for energetic particles in the

radiation belts is: velocity. R=.7-.8 during high-speed streams)

• V is NOT an energy. Not a density. Nor a Force(mv)• Multiplying by density ram or mechanical

energy, makes the correlation worse.• Multiplying by Bz Electrical energy, makes the

correlation worse.• Whatever the mechanism, it is not energy-starved

(2nd moment) or density starved (0th moment).

Page 4: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Motivation• The origin of radiation belt particles has been

resistant to analysis and modelling for some 40 years since their discovery. Nevertheless, it is a crucial component of Space Weather. To paraphrase Jim Drake on reconnection, “We cannot hold up our heads until we solve this problem”

• Recent satellite discoveries and theories have converged that the cusp might hold the key.

• But, current radiation belt models are equatorial, they do not incorporate high latitude sources.

Page 5: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

OutlineI. Data support for high-latitude source

a) POLAR data

b) Simulations and invariants

II. Comparison of accelerator efficienciesa) Definition of efficiency

b) Dipole / Fermi / Quadrupole

III. Transport from the cuspa) 4D Fokker-Planck

Page 6: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Sheldon et al., GRL 1998

POLAR/ CAMMICE data

1 MeV electron

PSD in outer cusp

Page 7: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Maxwell 1880 Chapman 1930

Page 8: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

QuadrupolarT87 Magnetosphere

• Since Chapman & Ferraro 1937, we’ve known the magnetosphere is a quadrupole.

• All trapping models have assumed a dipole.

• If Quadrupole is source, how does it change the transport?

Page 9: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

The Simulated T96 Quadrupole Trap

• Lousy Trap

• Great Accelerator

• Can be made to trap better though.

Page 10: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

e- Trap Regions of T96 Cusp

Page 11: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Cusp Provisional Invariants• Minimum energy is defined by “separatrix” energy

(ExB = B) ~ 30 keV

• The max energy defined by rigidity.~ 4 MeV e-

• Mapping C-shell limits to the dipole give 5<L<∞. very close to the PSD “bump”

• Mapping Quad Energy limits to the rad belts, give ~ 0-50 keV for protons, and 1-10 MeV for electrons.

• Mapping pitchangles give 0o < < 50o microburst?

• Cusp particles have all the right properties ORBE

Page 12: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Accelerator Efficiency

Why would the cusp accelerate at all? Why not just use standard well-

known accelerators?

Page 13: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Plasma Thermodynamics

• A neglected field—so this discussion is very qualitative. (See Krall & Trivelpiece)

• In ideal gases the Maxwell equilibrium is extremely well maintained by collisions

• In plasmas, turbulence takes the place of collisions. But the long-range Coulomb interaction means that equilibrium is often attained very slowly.

• Therefore one must get accustomed to non-Maxwellian distributions, and non-standard measures of temperature and entropy. E.g. kappas

Page 14: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

What is Acceleration?• Entropy is often defined as S = Q/T. Thus highly

energetic particles have low entropy. Often they are “bump-on-a-tail” distributions, or “power law” greater than Maxwellian. Such low-entropy conditions cannot happen spontaneously, so some other process must increase in entropy. Separating the two mechanisms, we have a classic heat-engine.

Note: the entropy INCREASE of heat flow into the cold reservoir, is counterbalanced by the DECREASE of entropy into the Work.

Page 15: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Why does a Trap help?• Single-stage acceleration mechanisms operate very

far from equilibrium. They therefore have a huge difference in entropy between W and Q1, and are therefore inherently inefficient. Likewise environmental conditions are far from equilibrium, and are thus inherently unlikely. The product of these two situations = low density of accelerated particles.

Energy Source * Efficiency * Probability = Power• A trap allows small impulses to be cumulative. The

pulses are close to equilibrium and are also likely. Thus a trap = higher density of accelerated particles.

Page 16: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

The Dipole Trap Accelerator

• The dipole trap has a positive B-gradient that causes particles to trap, by B-drift in the equatorial plane.

Three symmetries to the Dipole each with its own “constant of the motion”1)Gyromotion around B-field Magnetic moment, “”; 2) Reflection symmetry about equator Bounce invariant “J”; 3) Cylindrical symmetry about z-axis Drift invariant “L”

Betatron acceleration by E┴ compression, violation of 3rd invariant, L-shell

Page 17: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

The Fermi-Trap Accelerator

Waves convecting withthe solar wind, compresstrapped ions between the local |B| enhancementand the planetary bow shock, resulting in 1-Dcompression, or E// enhancement. Pitchangle diffusion keeps it in.

Page 18: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

The Quadrupole Trap• A quadrupole is simply the sum of two dipoles.

– Dipole moving through a magnetized plasma, heliosphere, magnetosphere, galactosphere. Maxwell showed how conducting plane (plasma) “reflects” image

– A binary system of magnetized objects –binary stars– A distributed current system-Earth’s core, supernovae

• Quadrupoles have “null-points” which stably trap charged particles (eg. Paul trap used in atomic physics produced 3 mass-spectrometer designs and a several Nobel prizes.)

• Betatron acceleration by E┴ compression, violation of 1st & 3rd invariants

Page 19: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Fermi I,II vs Alfven I,II • 1-D compression, E//

• Upstream Alfven waves impinging on barrier

• Scattering inside trap due to waves

• Max Energy due to scale size of barrier (curvature) becoming ~ gyroradius

• Acceleration time exponentially increasing

• Critically depends on angle of SW B-field with shock

• 2-D compression, E┴

• Compressional waves impinging on cusp

• Scattering inside trap due to quadrupole null point

• Max energy due to gyroradius larger than cusp radius (rigidity).

• Acceleration time exponentially increasing

• Critically depends on cusp angle with SW

Page 20: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

PROPERTY DIPOLE FERMI QUADRUPOLE

Stochasticity .001:1:1000 s .001:>103:>104 s 0.1:1:10 s

Process Flow rim>ctr>blocked end>side>diffus ctr>rim>open

Wave Coupling hi E weak all E same hi E best

Accel. in trap Traps Detraps Trap/Release

Diffusion Essential Helpful Neutral

Adiabatic Heat 2D pancake 1D cigar 2D pancake

Energy Source SW compress SW Alfven SW+internal

e- Max Energy 900MeV@10Re 1.8 [email protected] 280 MeV@3Re

e- Min Energy 45 keV 2.5 keV 30 keV

Trap Volume 1024 m3 1020 m3 1022 m3

Trap Lifetime >1013s 104s 109:105s

Accel. Time >300,000s 8,000s 25,000s

Trap Power <5x108W 106W 5x107W

Page 21: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Cusp Feedback

But the cusp is turbulent! How can the REAL cusp trap anything?

It doesn’t.

Usually.

Page 22: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Quadrupole Trap in the Laboratory

(Two 1-T magnets, -400V, 50mTorr)

Page 23: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Cusp Diamagnetic Cavities

Page 24: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Diamagnetic Effect of Cold Plasma

Page 25: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Time-Energy Dispersed Events

Page 26: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Cusp ORBE Scaling Laws

• Maximum energy from rigidity cutoffs, scaled by distance of planetary cusp to surface of planet.

• Assuming:– Brad ~ Bsurface= B0

– Bcusp ~ B0/Rstag3

– Erad= 5 MeV for Earth

– Ecusp ~ v2perp~ (Bcusp)2 ~ [(B0/Rstag

3)Rstag]

• E/B is constant

Erad-planet~(Rstag-Earth/Rstag-planet)(B0-planet/B0-Earth)2Erad-Earth

Page 27: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Scaled Planetary ORBE

Planet

Mercury

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

ERAD

4 keV

5 MeV

< 1.5 eV

150 MeV

1.2 MeV

1.4 MeV

0.42 MeV

R STAG

1.4

10.4

1.25

65

20

20

25

B0 (nT)

330

31,000

< 6

430,000

21,000

23,000

14,000

Page 28: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

High Latitude Transport

If energetic particles are trapped in the cusp, how do they get to the

dipole radiation belts?

Page 29: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Warm PlasmasheetTransport • Why is it important whether particles are transported or in

situ accelerated, do empirical models care? In a word: prediction.

•Example: Williams, Frank & Shelley peak seen in ISEE-3 data (35 keV protons). Local acceleration or transport?

Page 30: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Fokker-Planck Equation• dfk/dt= ∂fk/∂t+ C∂f/∂y+ Dz∂2fk/∂z2+ ∂/∂xi(Dij ∂fk/∂xj)

– i,j= index for (M,K) or (E,); k = quad or dipole trap– 1st term is the source + loss terms

– 2nd term is convective velocity in (U,Bm)-space

– 3rd term is diffusion in (U,Bm)-space

– 4th term is remaining diffusion in (E,a)-space

• Mixed parabolic & elliptic PDE solved with standard finite element techniques. Operator splitting, mapping between ƒ1 ƒ2 (Fast,

efficient, not numerically diffusive).

Page 31: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

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• If coordinates are not separable, ==>new physics:– Vortices, convection

– Anomalous, or enhanced diffusion, “migration”

• Convection + diffusion = confusion

Diffusive-Convective Transport

• If coordinates are separable, then D = D1 * D2 ==> diagonally dominant, easily generalized from 1-D.

• DLL, DD ==> radial diffusion models (Schulz 74)

Is there a coordinate system which is separable?

Page 32: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

3D Coordinate Diffusion (Roederer 1970)

Basis set 1 bounce-reson

2longitud. avg.

3 J~0 only

Distribution Function Radial Asymm

Pitch-angle

Symm Pitch-angle

Energy Loss

,J,M ƒ(X,,J,M,t) ,J,M1 J,M1 J,M

L,K,M ƒ(X,L,K,M,t) L L,K,M1 K,M1 M

L, Bm,T ƒ(X,L,Bm,t) L,T L, Bm Bm T

L, T ƒ(L,,T,t) L,T L, 2 T

L, ƒ(L,,,t) L L, 3 3

Diffusion

Page 33: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

4-D UBK Hamiltonian • The Quadrupole trap is located at a specific MLT as

well as high-latitude. Thus we need to keep the 3rd invariant phase4-D

• We use two, energy conserving, 4D coordinate systems:– ƒ1(M,K,U,Bm,n,t): “radial” diffusion + convection. FLUXONS

– ƒ2(T,,X,Y,t): Coulomb collisions + pitchangle diffusion

– Where “n” separates quadrupole & dipole trapping

• Operator splitting enables us to carry out radial transport on ƒ1, and pitchangle+energy loss on ƒ2.

Page 34: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Summarize 4D Transport • The UBK transform has the unique property of

solving several requirements at once:– Separates convection from diffusion (MLT-dependence)– Permits Hamilton-Jacobi solution to equations of motion– Treats both Quadrupole & Dipole Regions (hi-latitudes)– Permits rapid calculation of diffusion coefficients– Can use operator splitting to diagonalize Diffusion tensor

• Cannot handle time- or spatial- scales that violate 1st invariant. E.g. rare shock acceleration events (1991).

• It should indicate whether a hi-latitude source exists.

Page 35: A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop.

Conclusions• Cusp source term may explain the mysterious origin

of outer radiation belt particles.

• We need to develop the theory of both the acceleration and transport of hi-latitude sources. The generalization of Fermi acceleration applied to cusp, followed by a diffusive transport to radiation belts.

• Such a theory compares favorably with radiation belts in other magnetospheres.

• It may even solve Fermi’s problem of the origin of the galactic cosmic rays.

SDG