RMHD simulations of the non-resonant streaming instability near relativistic magnetized shocks

1. RMHD simulations of the non-resonant streaming instability near relativistic magnetized shocks Fabien Casse AstroParticule & Cosmologie (APC) - Université Paris Diderot In collaboration with Alexandre Marcowith (LUPM) & Rony Keppens (KUL)

2. Particle acceleration and magnetic amplification near astrophysical shocks • Observations exhibit non thermal high energy emissions near astrophysical shocks (e.g. Cassam-Chenaï et al. 2004). • Thin bright X-ray rims are observed at the location of the forward shock (e.g. Bamba et al 2006). • X-ray rim structure in agreement with a localized magnetic field amplification (~102 BISM, e.g. Parizot et al 2006) • Similar magnetic amplification is likely to occur in external relativistic shocks of GRBs (Li & Waxman 2006). Tycho SNR

3. “Streaming” instability • The presence of cosmic rays leads to a non-neutral thermal plasma prone to an extra Lorentz force.. • Linear calculation leads to a dispersion relation such as • where s stands for the CR feedback on the thermal flow. • Two regimes have been identified • Resonant regime: describes the interaction between Alfvèn waves and CR (e.g. McKenzie & Vôlk 1982) (large wavelength). • Non-resonant: works for small wavelength waves and the CR population can be considered as a passive fluid as a first approximation (Bell 2004, Pelletier et al. 2006).

4. “Streaming” instability vs relativistic shocks • Ultra-relativistic shocks exhibit magnetic structure where the main magnetic field is perpendicular to the shock normal because of relativistic transformation. • The resonant regime cannot work efficiently since Alfvèn waves cannot easily propagate through the upstream region. • The non-resonant regime still work for ultra relativistic shocks (Pelletier et al. 2009). • Linear calculation performed by considering the CR population as passive as a first approximation.

5. RMHD shock and Cosmic Rays • The presence of cosmic rays induces an electomotive force in the RMHD equations  the upstream medium’s equilibrium is modified by this charge density rCR • The balance is achieved by adding a slow motion of the fluid parallel to the shock front and perpendicular to the magnetic field in order to balance the electromotive field. We assume a strong shock so that

6. Linear analysis • Linear analysis of the RMHD set of equations leads to a dispersion relation such that (PLM09) : • Analysis done assuming • 1D instability regime (kz, ky =0) shows two kinds of growthth modes, namely

7. RMHD simulations • The AMRVAC code solves the set of RMHD equations in a conservative way (finite volume type code). • The conservative form of the equations can be written as Conservative variables Primary variables

8. Isothermal SRMHD • In order to use isothermal SRMHD in perpendicularshocks, wedesigned a new procedure to switchfrom conservative to primitive variables (Casse et al. 2012) • In regimewhere D>>B2, the above polynomial ismonotonic for x > gDF/21/2. Newton-Raphsonalgorithmworks fine (quadraticefficiency). Conservative variables Primary variables

9. Finite Volume Codes • Finite volume MHD codes rely on the conservative properties of [R]-MHD equations. • The numerical domain is divided into small cells where the physical quantities stand for cell-avera ged quantities. • All conservation equations can be written as • Green-Ostrogradki theorem enables us to calculate to temporal variation of the physical quantites by estimating the various fluxes occurring through the cell surface: • The numerical methods used to compute the fluxes and the temporal derivatives are code dependent (Approximate Riemann solver).

10. Adaptative Mesh Refinement • The structure of the grid is controlled by an Adaptative Mesh Refinement (AMR) algorithm that locally inforce resolution where needed. • Various criteria are implemented (+user’s defined): one of the best involves the Lohner’s criterion • The mesh is fragmented into several small for each level. The grids are organized using an octree repartition. • Grids are dynamically dispatched using MPI with a Morton load balance (Speed-up ~80% theoretical limit at CINES).

11. RMHD simulations

12. RMHD description of non-resonant “streaming” instability • Considering the cosmic ray fluid as a passive one, one has to consider the effect of the electric charge carried by the CRs  the local thermal plasma is no longer neutral near the shock ! • The external CR charge density has to be included in the RMHD set of equations: Source terms

13. A typical simulation – Initial setup gSH=100; xCR=10% ; Pth~10-8rc2

14. A typical simulation –Setting a perturbation • In the shock frame, the fastmagnetosonicwavespropagatesthrough the thermal plasma withvelocity • Suchwavepropagating in the unperturbedupstream plasma corresponds to velocity, density and magnetic perturbations • Once entered the cosmic ray dominatedregion, the velocity perturbation willbeprone to amplification.

15. A typical simulation -Setting perturbations

16. A typical simulation – Velocity perturbation growth • Linearized RMHD momentum equation provides • During the initial growth stage, only the velocity perturbation is growing

17. A typical simulation – Velocity perturbation growth

18. A typical simulation – Magnetic perturbation growth • The velocity perturbation amplification leads to phase shift betweenvelocity and magnetic perturbations • Magnetic perturbation isthengrowing once the phase shift becomessufficiently large, i.e. when the vertical velocityreachs a threshold • wherelis the perturbation wavelength.

19. A typical simulation – Magnetic perturbation growth

20. A typical simulation – Entering the non linear regime

21. A typical simulation – Entering the non linear regime

22. A typical simulation – The saturation stage

23. A typical simulation – The saturation stage

24. A typical simulation

25. A typical simulation – the spatial growth rate

26. The influence of the wavenumber

27. Exploring the « mildly » relativistic regime

28. Exploring the « mildly » relativistic regime

29. Exploring the « mildly » relativistic regime

30. Exploring the « mildly » relativistic regime

31. Exploring the « mildly » relativistic regime

32. Final stage in the non-relativistic regime

33. Concluding remarks • The 1D version of the non resonant streaming instability can be described using a RMHD code. • Small scale magnetosonic waves are amplified by the cosmic ray charge density effects after an initial amplification of the velocity perturbation. • We recover the predictions provided by the linear analysis • The instability is working through the mildly relativistic regime but with larger growth rates. • Stepping into 2D simulations is the next step in order to include the cosmic ray current effect as well as Alfvèn waves…