A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004...
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Transcript of A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004...
A High Latitude Source for Radiation Belt Particles?
Robert Sheldon, NSSTC
October 19, 2004
Huntsville Modelling Workshop
McIlwain, 1966
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).
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.
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
Sheldon et al., GRL 1998
POLAR/ CAMMICE data
1 MeV electron
PSD in outer cusp
Maxwell 1880 Chapman 1930
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?
The Simulated T96 Quadrupole Trap
• Lousy Trap
• Great Accelerator
• Can be made to trap better though.
e- Trap Regions of T96 Cusp
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
Accelerator Efficiency
Why would the cusp accelerate at all? Why not just use standard well-
known accelerators?
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
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.
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.
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
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.
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
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
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
Cusp Feedback
But the cusp is turbulent! How can the REAL cusp trap anything?
It doesn’t.
Usually.
Quadrupole Trap in the Laboratory
(Two 1-T magnets, -400V, 50mTorr)
Cusp Diamagnetic Cavities
Diamagnetic Effect of Cold Plasma
Time-Energy Dispersed Events
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
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
High Latitude Transport
If energetic particles are trapped in the cusp, how do they get to the
dipole radiation belts?
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?
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).
31
• 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?
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
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.
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.
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