1. DSA in the non-linear regime Hui Li Department of Astronomy, Nanjing University

2. Test-Particle approximation Bell 1978

3. Relationship! Thermal particles CRs Alfven wave DSA Heating, ionizing; Back Pressure Streaming instabilities Confine and promote acc Heating and pre-heating MHD turbulence and instabilities ISM

4. Non-linear effects: I Shock structure: precursor and CD; RT Temperature Concaved CR spectrum ……

5. Non-linear effects: II MFA produced by CRs streaming instability. CR energy  wave energy

6. Numerical treatments Two-fluid model. (Drury&Voelk 1981 a,b) Kinetic theory: a. fully numerical simulation (Berezhko & Voelk 1997; Kang & Jones 1997, 2005, 2006, 2007; Zirakashvili & Aharonian 2010) b. Monte Carlo simulation (Ellison & Eichler 1984,1985; Vladimirov et al. 2006) c. quasi-stationary semi-analytical solution (Blasi et al. 2007; Caprioli et al 2009)

7. Two-fluid Model Treat the accelerated particles as a second fluid with energy density E_c and pressure P_c, but negligible mass density. Drury & Voelk 1981

8. Two-fluid Model Useful to study the dynamics of a shock modified by the presence of acc CRs. Can’t provide any information about the spectrum of energetic particles.

9. Fully numerical simulation Forward shock injection Reverse shock injection Kang & Jones 2006 Zirakashvili & Aharonian 2010

10. Fully numerical simulation 1-D HD instabilities ignored Can’t use mature MHD codes. Zirakashvili & Aharonian 2010

11. Semi-analytical solution Guess solution: static-state solution Iterate until stable!! Blasi 2002; 2004; 2005; Caprioli et al. 2010

12. Comparison Caprioli et al. 2010

14. A new way?? HD simulation+CR modification Instabilities? Inhomogenous environments? Warren et al. 2005

15. How to deal with CR feedback? HD simulation output: a. shock information: V_s, M_s. b. injection recipe: c. cutoff recipe: p_max Semi-analytical approximation: γ_eff= γ_eff(V_s,M_s,p_inj,p_max) Re-input γ_eff into the HD simulation

16. Blondin & Ellison 2001 1-D 2-D Growth of instab. suppressed by CR acc

17. 3-D? Ferrand et al. 2009 Pure HD case With back reaction Instability level?

18. Where are we now? Observational Shock Dynamic? Shock structure: precursor and CD Helder et al. 2009; Warren et al. 2010 RCW86 SN1006 Tycho

19. Lower temperature? Concaved CR spectrum ？ SN 1006 J1713 Berezhko et al 2002; 2009 Ellison et al. 2010

20. MFA? Thin filaments in X-ray? Time scale? X-ray knots? The extreme cooling or fast variation of MF? Why radio filaments? Time scale of instability? Which instability dominates?

21. Where are we now? Theoretical Know little about DSA!! Self-consistent ways of injections of super- thermal particles? Thermal leakage? Self-consistent ways of escaping of relativistic CRs? Maximum energy and free escaping boundary? Self-consistent ways of wave excitation and magnetic field amplification? … PIC may help, but ….

22. What can we do? HD Simulation? CR influence on instability; SNR-MC; MHD simulation; MFA? Filament width vs. shock velocity? Observational constrain on non-linear filaments micro-structure; cut-off energy; line emission on lower temperature; ion temperature?

23. Thank you!