Solar Wind Energy Coupling

Through The CuspRobert Sheldon

NASA/MSFC/NSSTC/XD12

Ted Fritz, Jiasheng Chen, BU

ABSTRACT

• Three variants of solarwind-magnetosphere energy coupling are well-known: the rectified solar wind electric field Ey (Dungey, Akasofu ); the viscous interaction (Axford & Hines); and the shock-driven inductive electric fields (1991 event). We suggest a fourth, intermediate category active during high speed solar wind streams, corresponding to “recurrent magnetic storms”. Such streams do not have a good impedance match to the dipole magnetosphere, and therefore neither supply electric nor viscous energy. However, they are well-matched to the quadrupole cusps, with several promising mode conversion mechanisms available. This may explain the correlation with MeV electrons, highest not for internal (AE, Dst), or external drivers (ram pressure, Ey) but for “mixed” drivers such as Kp and Vsw. We present POLAR data and simulations showing good agreement with statistical studies, diffusive gradients, energetic particle spectra, elemental composition, and dynamical development consistent with a cusp transducer. The major difficulty is the lack of historical data, since few missions excepting POLAR have flown through the cusp with energetic particle instruments. Still, even equatorial s/c such as CRRES or AMPTE or the proposed RBSP, can see the cusp source as a “butterfly” pitchangle distribution diffusing into the equatorial plane.

Transducers: The Oldest Physics Problem

How does point A influence point B?A. 500BC Aristotle: mind, “spooky action-at-a-distance”B. 500BC–2000AD Democritus to Descartes: particlesC. 1690AD Newton: “action-at-a-distance” gravity (tides)D. 1650AD Huygens: wavesE. 1840AD Faraday: fields

How does the Sun transfer energy to Earth?1. Photons+protons (DC equil.): heat,pressure (Chapman)2. Electric+Magnetic fields (AC/DC): currents (Alfven)3. Waves+impulsive events (AC mechanical):

compressional, shocks, viscous (Axford)

SunEarth Transducers None work for MeV electrons!

• Proton pressure Bow shock, hot plasma (100eV electron, 1 kev/nuc ion), thermalized ram energy “Frictional” or “viscous” (V5/2)

• Impulsive SSC, shock acceleration, Fermi, radial diffusion, Kp, “mechanical” (V, V2)

• Fields Polar cap potential, convection, ring current, Dst, AE, “electrical” (V*Bz) [ICME]

• What transducer powers Outer Radiation Belt MeV Electrons (ORBE)? Poor correlation with all of the above! V correlates best all by its lonesome. Why?

Springs & Shock Absorbers:The importance of matching impedances

• Why does a car have BOTH springs & shocks? – Springs are “reversible”, adiabatic, they “bounce back”

ruining the tire tread as the energy dissipates in the tires.– Shock absorbers are “irreversible”, non-adiabatic, they

convert the energy to heat. But with too slow a response.– Springs match the impedances of potholes to shocks

• ORBE/Vsw energy transducer must be irreversible.– Cannot be too “stiff”, ideally it is “critically damped”

• Magnetic fields are “springs”, what are “shocks”?– Something responding to Vsw, yet dissipative…

The Dipole Trap in Lab & Space

• Great Trap

• Poor accelerator

• Best for producing ENA of E >1 keV particles outside trap.

McIlwain 1963

↑Sheldon 2002: Lab magnetosphere with NIB magnet @–400V

Electrons

Ions2.2cm

Quadrupole Trap in the Laboratory

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

2.2cm 2.2cm

TOP

SIDE

separatrix

cusp

Maxwell 1880 Chapman 1930

Maxwell solved the “image dipole” problem, plotting the quadrupoles. Chapman used it 50 years later to explain the magnetosphere.

T87

The 2nd (Cusp) Invariant

|B|N.Ionosphere Equator S.Ionosphere

3 wells

2 wells

Bouncing on a field line without crossing the equatorNear the nose, a single equatorial B-maximum, near both cusps N. & S., a double local B-maxima.

s-distance

CF currents

T96Cusp TopologyDot marks the spot of quadrupole null point as a function of season/UT.“UFO” is ionospheric footprint of null, darker smaller |B|

Solstice16UT

Solstice4UT

Equinox16UT Equinox

16UT,-Bz

Ionospheric Footpoint of the HiLatitude Minima: Tilt vs Press-3.67deg +1.75deg +7.3deg

5dyn

3.3dyn

1.7dyn

Null Point Poleward Minima

Equatorward Minima

Ionospheric Footpoint of HiLatitude Minima: Tilt vs Dst

-3.67deg +1.75deg +7.3deg

-50nT

-30nT

-10nT

Both sunward (positive) tilt and/or high solar wind pressure are needed to produce the poleward “dome” cusp minima.

Ionospheric Footprint of HiLatitude Minima: Press v Dst

-50nT

-30nT

-10nT

1.7dyn 3.3dyn 5dyn

|Dst| alone doesn’t develop the poleward side of the cusp, but it amplifies or magnifies what is already there. (Significant for statistical correlations.)

Cusp Equator(min |B| on fieldline)

Dotted B-field lines

Solid|B|-mag contours

Cshell=1

C=2

C=1.5

Trapped particle orbits on several C-shells

Side Front

Tracing in a T96 Quadrupole Trap

B-field linesQuad null pt

Trapped e- trajectory

Quasi-Chaotic

H+ Trapping in T96 Cusp

Hi E cutoff Numerical Roundoff Loss-cone cutoff

Red= None

Green=Quasi-

Blue= Yes

Hi E cutoff Numerical Roundoff Loss-cone cutoff

Red= None

Green=Quasi-

Blue= Yes

e- Trapping in T96 Cusp

Cusp Provisional Invariant Limits

• Energy Limits (1st invariant at 100nT)– Minimum energy, Emin, is defined by cusp “separatrix”

energy (ExB = B) ~ 30 keV in the dipole?– Max energy, Emax, defined by rigidity.~ 4 MeV e-

(20keV H+)– Consequently, no protons are expected to be trapped.

• Pitchangles locally 40-90o, (2nd invariant)

• Low C-shells are empty below 1 Re for all energy, with a high-Cshell cutoff ~6 inversely dependent on Energy. 1 < C <~6 Re

Mapping Cusp to Dipole

• Conserving the 1st invariant, and pitchangle scatter the particles into the cusp-loss cone (<40o), then the particles can appear in the dipole trap, or radiation belts. What would their distribution look like?– Energy limits to the rad belts, give ~ 0-100 keV for

protons, and 1-15 MeV for electrons.– C-shell limits to the dipole give ~5<L<∞? very close

to the PSD “bump”.– Mapping pitchangles 50o < < 90o at dipole eq?

• Cusp particles look like ORBE injections.

POLAR: Oct 12-16, 1996

POLAR/ CAMMICE data

1 MeV electrons

PSD in outer cusp

Sheldon et al., GRL 1998

POLAR 4/1/97 Cusp Traversal

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 1-D 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 2-D Quadrupole Trap

• A quadrupole is simply the sum of two dipoles. • Quadrupoles have “null-points” which stably trap charged

particles (eg. Paul trap)• Motion of the dipoles results in a 2D constriction of the

volume. This is just a generalization of 1D Fermi-acceleration to 2D.

• 1D Fermi acceleration increases E//, violating the 2nd invariant.

• 2D betatron acceleration increases E┴ , violating the 1st & 3rd invariants

• Efficiency Product: T = 1 2 3 4 5 6…

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 MeV@.1Re 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

Model1. Fast solar wind is trapped in the cusp

– 27 day recurrence, non-linear with Vsw

2. High Alfvenic turbulence of fast SW heats the trap– Low Q-value, compressional, BEN

3. 2nd Order “Fermi” accelerates electrons– Low energy appear first, then high w/rigidity cutoff.

4. Trap empties into rad belts simultaneous L=4-10 – “gentle” evaporation, or “rapid” topology change – Initially “butterfly” around 70-deg equatorial

1. Non-Linear Vsw Dependence

E

Flux

EFlux

30keV100eV 1keV 10keV

Vsw

seed trap seed trap

The Reason that Vsw interacts non-linearly is that it does several things at once. It heats the seed population,while also making the trap deeper.

Kolmogorov, Arnol’d, Moser (applied to Jupiter perturbation of

Earth)………Earth orbit as Perturbed byJupiter.

Earth orbit if Jupiter were 50k Earth masses.

Poincaré slicex vs. vX takenalong the E-J

line.

Real Life• Up to this point, we have developed the theory of

cusp trapping and acceleration in an ideal, vacuum quadrupole.

• However, real life is far more interesting. POLAR data, which triggered this investigation, shows trapped ion flux and a highly modified magnetic field, which we argue is a Cusp Diamagnetic Cavity.

• The positive feedback between the quadrupole and trapped ions, suggests that CDC are ubiquitous and important.

Cusp Diamagnetic Cavitiesa.k.a Magnetic Bubbles

Turbulence, Power, Spectra…

Schematic Cusp Diamagnetic Cavity

POLAR sees thick (1-6 Re) CDC, whereas Cluster sees thin (< 1Re). We interpret this as a radial dependence on the thickness of the CDC.

Stability of Infinitesimal Dipole

Stability of Finite Ring

Cusp Energetic Particles (Ions)

Spectra at 2 times Ratio of Spectra

McIlwain, 1966

ORBE (McIlwain 1966)

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.• There is a Dst signature with ORBE, but magnitudes

are uncorrelated, only occurrence.

Empirical Prediction

• McIlwain 1966: Geo MeV e increases

• Paulikas & Blake 1979: Vsw best external

• Nagai 1988: Kp best internal predictor

• Baker 90 LPF, Koons&Gorney 90 NN

• Dmitriev&Chao03 Log-Linear

• Ukhorskiy et al., 04 NonLinear

Cusp 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

EPlanet~ EEarth(RPBPlanet/REBEarth)2 (RE-Stag/RP-Stag)4

Scaled Planetary ORBE

Planet

Mercury

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

ERAD

0.66 MeV

5 MeV

< .5 eV

7.1 MeV

1.6 MeV

0.81 MeV

0.12 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

1996

Conclusions• The quadrupole is a nearly universal trap and cosmic

accelerator more efficient than Fermi (and shocks).

• The quadrupole cusp has ideal properties to couple AC mechanical energy from SW into the magnetosphere.

• The peculiar correlations of ORBE with SW can be explained by requiring an intermediate stage of the non-linear cusp.

• A test of the mechanism using comparative magnetospheres shows the correct energy scaling.

Soli Deo Gloria