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 x g (x = x g +r g ) Transform transverse momentum to plasma frame p  = p ’  + V E Smallness parameter  = m/q << 1 (small gyroradius, large gyrofrequency) Keep terms to order  o Average over gyrophase where

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

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

5. Advantages Same mechanisms as standard cosmic-ray transport equation No restriction on pitch-angle anisotropy - injection Describe both (i) 1 st 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. 1 AU downstream upstream SW Kappa distribution (  = 3)  1/r 2 3. Results : (a) Proton spectra in SW frame SW density = C SW density  1/r 2 0.7 AU f(v)  v -4.2 f(v)  v -3.7 f(v)  v -3.2

7. __0.13 AU ---0.27 AU ___1 AU _.._0.71 AU …0.49 AU SW Kappa distribution (  = 3)  1/r 2 Shift in peak to lower energies f(v)  v -5.3 f(v)  v -32

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

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

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

11. Strong shock (s=4) Weak shock (s=3.3  2.2)f(v)  v -3.7 f(v)  v -7.1 3. COMPARISON OF STRONG AND WEAK SHOCKS: 0.7 AU Spectrum harder than predicted by steady state DSA Spectrum softer than predicted by steady state DSA (a) Accelerated spectra

12. (b) Anisotropies 100 keV Strong shock (s=4) Weak shock (s=3.3  2.2) 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. Turbulence can be modeled as a nonlinear diffusion process in wavenumber space (D kk ) 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 Wave-wave interaction Model (Zhou & Matthaeus, 1990): Kolmogorov cascade I(k)  k -5/3 t NL > V A ) or (  B >> B o ) Kraichnan cascade I(k)  k -3/2 t A << t NL (  U =  V A << V A ) or (  B << B o ) Comments

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

16. Kolmogorov or Kraichnan Cascade? 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 k min for wave growth Find that Kraichnan-type turbulence gives the correct k min Kolmogorov-type turbulence overestimates k min by a factor 2-4 Kallenbach, Bamert, le Roux et al., 2008 Bastille Day CME event Halloween CME event #1 Vasquez et al., 2007 For cascade rates and proton heating rates to agree in SW – C K = 12 needed instead of the expected C K ~1 Kraichnan cascade rate is close to heating rate at 1 AU

17. A Weak Kinetic Turbulence Theory Approach to Wave Cascading? Wave Kinetic Equation 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 D kk can be derived Linear wave- particle interaction Nonlinear wave-wave interaction Wave generationWave decay