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Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach

Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach. J. A. le Roux Institute of Geophysics & Planetary Physics University of California at Riverside. 1. The Guiding center Kinetic Equation for f ( x , p || , p ’  , t ).

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Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach

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  1. Shock Acceleration at an Interplanetary Shock:A Focused Transport Approach J. A. le Roux Institute of Geophysics & Planetary Physics University of California at Riverside

  2. 1. The Guiding center Kinetic Equation for f(x,p||,p’,t) Transform position x to guiding center position xg (x = xg+rg) Transform transverse momentum to plasma frame p = p’ + VE Smallness parameter  = m/q << 1 (small gyroradius, large gyrofrequency) Keep terms to order o Average over gyrophase where

  3. Transform (p||, p’)(’, M) where Conservation of magnetic moment Electric field drift Grad-B drift Curvature drift

  4. 2. The Focused Transport Equation for f(x,p’,’,t) To get focused transport equation, assume in guiding center kinetic equation E = -U B, wherebyVE = U Transform parallel momentum to moving frame (p|| = p’|| + mU||) Transform (p’||,p’)  (p’, ’ ) Convection Mirroring/focusing Adiabatic energy changes Parallel Diffusion

  5. 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

  6. 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

  7. 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

  8. (b) Anisotropies in SW frame at 1 AU downstream shock =25% upstream =82% 100 keV 1 MeV =9% 10 MeV

  9. (c) Spatial distributions across shock 100 keV 100 MeV 1 MeV 10 MeV

  10. (d) Time distributions at 1 AU 10 MeV 10 MeV 100 keV 100 MeV

  11. 3. COMPARISON OF STRONG AND WEAK SHOCKS: (a) Accelerated spectra f(v)  v-3.7 Weak shock (s=3.32.2) Strong shock (s=4) 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

  12. (b) Anisotropies Strong shock(s=4) Weak shock(s=3.32.2) 100 keV 100 keV Pitch-angle anisotropy larger for a strong shock

  13. 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

  14. Wave-wave interaction Model (Zhou & Matthaeus, 1990): • 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 energyconservation in 3-wave coupling does not hold Kraichnan cascade I(k)  k-3/2 tA << tNL(U = VA << VA) or (B << Bo) • Kolmogorov cascade I(k)  k-5/3 tNL << tA (U = VA >> VA) or (B >> Bo) Comments

  15. Alfven wave Spectrum (Kolmogorov Cascade) downstream NeedCK > 8 for cascading to work at shock at 0.1 AU upstream 0.3 AU 0.5 AU 0.7 AU 1 AU

  16. 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 kminfor 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 rateat 1 AU

  17. 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

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