Solar Wind Energy Coupling Through The Cusp

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Solar Wind Energy Coupling Through The Cusp

• Robert Sheldon NASA/MSFC/NSSTC/XD12
• Ted Fritz, Jiasheng Chen, BU

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ABSTRACT

• Three variants of solarwind-magnetosphere energy
  coupling are well-known the rectified solar wind
  electric field Ey (Dungey, Akasofu e) 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.

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Transducers The Oldest Physics Problem

• How does point A influence point B?
• 500BC Aristotle mind, spooky action-at-a-distanc
  e
• 500BC2000AD Democritus to Descartes particles
• 1690AD Newton action-at-a-distance gravity
  (tides)
• 1650AD Huygens waves
• 1840AD Faraday fields
• How does the Sun transfer energy to Earth?
• Photonsprotons (DC equil.) heat,pressure
  (Chapman)
• ElectricMagnetic fields (AC/DC) currents
  (Alfven)
• Wavesimpulsive events (AC mechanical)
  compressional, shocks, viscous (Axford)

Can you connect A-E with 1-3?
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Sun?Earth Transducers None work for MeV
electrons!

• Proton pressure ? Bow shock, hot plasma (100eV
  electron, 1 kev/nuc ion), thermalized ram energy
  Frictional or viscous (rV5/2)
• Impulsive ? SSC, shock acceleration, Fermi,
  radial diffusion, Kp, mechanical (rV, rV2)
• Fields ? Polar cap potential, convection, ring
  current, Dst, AE, electrical (VBz) 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?

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Springs Shock AbsorbersThe 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

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The Dipole Trap in Lab Space
Electrons ?

• Great Trap
• Poor accelerator
• Best for producing ENA of E gt1 keV particles
  outside trap.

?2.2cm?
Ions?
?Sheldon 2002 Lab magnetosphere with NIB magnet
_at_400V
?McIlwain 1963
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Quadrupole Trap in the Laboratory(Two, 1T,
parallel NIB magnets, -400V, 50mTorr)
TOP
cusp
separatrix
?2.2cm?
?2.2cm?
SIDE
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Maxwell solved the image dipole problem,
plotting the quadrupoles. Chapman used it 50
years later to explain the magnetosphere.
T87
Maxwell 1880 Chapman 1930
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The 2nd (Cusp) Invariant
Bouncing on a field line without crossing the
equator Near the nose, a single equatorial
B-maximum, near both cusps N. S., a double
local B-maxima.
B
CF currents
N.Ionosphere Equator S.Ionosphere
3 wells
2 wells
s-distance
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T96Cusp TopologyDot marks the spot of quadrupole
null point as a function of season/UT.UFO is
ionospheric footprint of null, darker? smaller
B
Solstice 4UT
Solstice 16UT
Equinox 16UT
Equinox 16UT,-Bz
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Ionospheric Footpoint of the HiLatitude Minima
Tilt vs Press
1.75deg
7.3deg
-3.67deg
5dyn
Null Point
Poleward Minima
3.3dyn
Equatorward Minima
1.7dyn
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Ionospheric Footpoint of HiLatitude Minima Tilt
vs Dst
1.75deg
7.3deg
-3.67deg
-50nT
Both sunward (positive) tilt and/or high solar
wind pressure are needed to produce the poleward
dome cusp minima.
-30nT
-10nT
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Ionospheric Footprint of HiLatitude Minima Press
v Dst
3.3dyn
5dyn
1.7dyn
-50nT
-30nT
Dst alone doesnt develop the poleward side of
the cusp, but it amplifies or magnifies what is
already there. (Significant for statistical
correlations.)
-10nT
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Cusp Equator(min B on fieldline)
Dotted B-field lines
Solid B-mag contours
Trapped particle orbits on several C-shells
Cshell1
Side Front
C1.5
C2
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Tracing in a T96 Quadrupole Trap
Quad null pt
B-field lines
Trapped e- trajectory
Quasi- Chaotic
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H Trapping in T96 Cusp
Hi E cutoff Numerical Roundoff Loss-cone cutoff
Red None GreenQuasi- Blue Yes
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e- Trapping in T96 Cusp
Hi E cutoff Numerical Roundoff Loss-cone cutoff
Red None GreenQuasi- Blue Yes
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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 lt C lt6 Re

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Mapping Cusp to Dipole

• Conserving the 1st invariant, and pitchangle
  scatter the particles into the cusp-loss cone
  (lt40o), 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 5ltLlt8?? very
  close to the PSD bump.
• Mapping pitchangles ? 50o lt a lt 90o at dipole eq?
  
• Cusp particles look like ORBE injections.

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POLAR Oct 12-16, 1996
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Sheldon et al., GRL 1998
POLAR/ CAMMICE data 1 MeV electrons PSD in outer
cusp
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POLAR 4/1/97 Cusp Traversal
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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
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The 1-D Fermi-Trap Accelerator
Waves convecting with the solar wind,
compress trapped ions between the local B
enhancement and the planetary bow shock,
resulting in 1-D compression, or E//
enhancement. Pitchangle diffusion keeps it in.
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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 hT h1 h2 h3 h4 h5 h6

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PROPERTY DIPOLE FERMI QUADRUPOLE
Stochasticity .00111000 s .001gt103gt104 s 0.1110 s
Process Flow rimgtctrgtblocked endgtsidegtdiffus ctrgtrimgtopen
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 SWinternal
e- Max Energy 900MeV_at_10Re 1.8 MeV_at_.1Re 280 MeV_at_3Re
e- Min Energy 45 keV 2.5 keV 30 keV
Trap Volume 1024 m3 1020 m3 1022 m3
Trap Lifetime gt 1013s 104s 109105s
Accel. Time gt 300,000s 8,000s 25,000s
Trap Power lt 5x108W 106W 5x107W
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Model

• Fast solar wind is trapped in the cusp
• 27 day recurrence, non-linear with Vsw
• High Alfvenic turbulence of fast SW heats the
  trap
• Low Q-value, ?compressional, BEN
• 2nd Order Fermi accelerates electrons
• Low energy appear first, then high w/rigidity
  cutoff.
• Trap empties into rad belts simultaneous L4-10
• gentle evaporation, or rapid topology change
  
• Initially butterfly around 70-deg equatorial

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1. Non-Linear Vsw Dependence
30keV
100eV
10keV
1keV
Flux
Vsw
Flux
seed
trap
E
E
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.
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Kolmogorov, Arnold, Moser (applied to Jupiter
perturbation of Earth)
Earth orbit as Perturbed by Jupiter.
Poincaré slice x vs. vX taken along the E-J line.
Earth orbit if Jupiter were 50k Earth masses.
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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.

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Cusp Diamagnetic Cavitiesa.k.a Magnetic Bubbles
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Turbulence, Power, Spectra
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Schematic Cusp Diamagnetic Cavity
POLAR sees thick (1-6 Re) CDC, whereas Cluster
sees thin (lt 1Re). We interpret this as a radial
dependence on the thickness of the CDC.
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Stability of Infinitesimal Dipole
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Stability of Finite Ring
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Cusp Energetic Particles (Ions)
Spectra at 2 times Ratio of
Spectra
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McIlwain, 1966
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ORBE (McIlwain 1966)
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McIlwain 1966
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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.

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Empirical Prediction

• McIlwain 1966 Geo MeV e increases
• Paulikas Blake 1979 Vsw best external
• Nagai 1988 Kp best internal predictor
• Baker 90 LPF, KoonsGorney 90 NN
• DmitrievChao03 Log-Linear
• Ukhorskiy et al., 04 NonLinear

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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 (Bcuspr)2 (B0/Rstag3)Rstag
• m E/B is constant
• EPlanet EEarth(RPBPlanet/REBEarth)2
  (RE-Stag/RP-Stag)4

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Scaled Planetary ORBE
Planet Mercury Earth Mars Jupiter Saturn Uranus Ne
ptune
ERAD 0.66 MeV 5 MeV lt .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 lt 6 430,000
21,000 23,000 14,000
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1996
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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