Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach

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Shock Acceleration at an Interplanetary ShockA
Focused Transport Approach

• J. A. le Roux
• Institute of Geophysics Planetary Physics
• University of California at Riverside

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1. The Guiding center Kinetic Equation for
f(x,p,p?,t)
Transform position x to guiding center position
xg (x xgrg) Transform transverse momentum to
plasma frame p? p? VE Smallness parameter ?
m/q ltlt 1 (small gyroradius, large
gyrofrequency) Keep terms to order ?o Average
over gyrophase
where
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Transform (p, p?)?(?, M)
where
Conservation of magnetic moment
Electric field drift
Grad-B drift
Curvature drift
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2. The Focused Transport Equation for f(x,p,?,t)
To get focused transport equation, assume in
guiding center kinetic equation E -U ? B,
whereby VE U? Transform parallel momentum to
moving frame (p p mU) Transform
(p,p?) ? (p, ? )
Convection
Mirroring/focusing
Adiabatic energy changes
Parallel Diffusion
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Advantages
Same mechanisms as standard cosmic-ray transport
equation No restriction on pitch-angle
anisotropy - injection Describe both (i) 1st
order Fermi, and (ii) shock drift acceleration
with or without scattering
Disadvantages
No cross-field diffusion Gradient and curvature
drift effect on spatial convection
ignored Magnetic moment conservation at shocks
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3. Results (a) Proton spectra in SW frame
SW Kappa distribution (? 3) ? 1/r2
SW density C
1 AU
0.7 AU
f(v) ? v-3.2
f(v) ? v-4.2
f(v) ? v-3.7
upstream
downstream
SW density ? 1/r2
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SW Kappa distribution (? 3) ? 1/r2
__0.13 AU
Shift in peak to lower energies
---0.27 AU
0.49 AU
f(v) ? v-5.3
_.._0.71 AU
___1 AU
f(v) ? v-32
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(b) Anisotropies in SW frame at 1 AU
downstream
shock
?25
upstream
?82
100 keV
1 MeV
?9
10 MeV
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(c) Spatial distributions across shock
100 keV
100 MeV
1 MeV
10 MeV
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(d) Time distributions at 1 AU
10 MeV
10 MeV
100 keV
100 MeV
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3. COMPARISON OF STRONG AND WEAK SHOCKS
(a) Accelerated spectra
Weak shock (s3.3?2.2)
f(v) ? v-3.7
Strong shock (s4)
f(v) ? v-7.1
0.7 AU
0.7 AU
Spectrum harder than predicted by steady state DSA
Spectrum softer than predicted by steady state
DSA
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(b) Anisotropies
Strong shock (s4)
Weak shock (s3.3?2.2)
100 keV
100 keV
Pitch-angle anisotropy larger for a strong shock
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1D parallel Alfven wave transport equation at
parallel shock solved
Convection
Convection in wave number space- Compression
shortens wave length
Compression raises wave amplitude
Wave generation by SEP anisotropy
Wave-wave interaction forward cascade in wave
number space in inertial range
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Wave-wave interaction Model (Zhou Matthaeus,
1990)

• Kraichnan cascade I(k) ? k-3/2
• tA ltlt tNL (?U ?VA ltlt VA ) or (?B ltlt
  Bo)

• Kolmogorov cascade I(k) ? k-5/3
• tNL ltlt tA (?U ?VA gtgt VA ) or
  (?B gtgt Bo)

Comments

• Turbulence can be modeled as a nonlinear
  diffusion process in wavenumber space (Dkk)
• Ad-hoc turbulence model based on dimensional
  analysis
• Contrary to weak turbulence theory assume that
  3-wave coupling for Alfven waves is possible
  imply wave resonance broadening momentum and
  energy conservation in 3-wave coupling does not
  hold

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Alfven wave Spectrum (Kolmogorov Cascade)
downstream
Need CK gt 8 for cascading to work
at shock at 0.1 AU
upstream
0.3 AU
0.5 AU
0.7 AU
1 AU
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Kolmogorov or Kraichnan Cascade?
Bastille Day CME event
Kallenbach, Bamert, le Roux et al., 2008
Used Marty Lee (1983) quasi-linear theory
extended to include nonlinear wave cascading via
a diffusion term in wavenumber space. Applied
theory to Bastille Day and Halloween 1 CME
events to calculate kmin for wave growth Find
that Kraichnan-type turbulence gives the correct
kmin Kolmogorov-type turbulence overestimates
kmin by a factor 2-4
Halloween CME event 1
Vasquez et al. , 2007
For cascade rates and proton heating rates to
agree in SW CK 12 needed instead of the
expected CK 1 Kraichnan cascade rate is close
to heating rate at 1 AU
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A Weak Kinetic Turbulence Theory Approach to Wave
Cascading?
Nonlinear wave-wave interaction
Wave Kinetic Equation
Wave generation
Wave decay
Linear wave-particle interaction
Wave-wave interaction term a Boltzmann scattering
term
When assume weak scattering, term can be put in
Fokker-Planck form so that nonlinear diffusion
coefficients for wave cascading Dkk can be derived