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)

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 (~10 2 B ISM, e.g. Parizot et al 2006) Similar magnetic amplification is likely to occur in external relativistic shocks of GRBs (Li & Waxman 2006). Tycho SNR

“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  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).

“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.

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  CR 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

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

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

Isothermal SRMHD In order to use isothermal SRMHD in perpendicular shocks, we designed a new procedure to switch from conservative to primitive variables (Casse et al. 2012) In regime where D>>B 2, the above polynomial is monotonic for  >  DF/2 1/2. Newton-Raphson algorithm works fine (quadratic efficiency). Conservative variables Primary variables

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).

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).

RMHD simulations

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

A typical simulation – Initial setup  SH =100;  CR =10% ; P th ~10 -8  c 2

A typical simulation –Setting a perturbation In the shock frame, the fast magnetosonic waves propagates through the thermal plasma with velocity Such wave propagating in the unperturbed upstream plasma corresponds to velocity, density and magnetic perturbations Once entered the cosmic ray dominated region, the velocity perturbation will be prone to amplification.

A typical simulation -Setting perturbations

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

A typical simulation – Velocity perturbation growth

A typical simulation – Magnetic perturbation growth The velocity perturbation amplification leads to phase shift between velocity and magnetic perturbations Magnetic perturbation is then growing once the phase shift becomes sufficiently large, i.e. when the vertical velocity reachs a threshold where is the perturbation wavelength.

A typical simulation – Magnetic perturbation growth

A typical simulation – Entering the non linear regime

A typical simulation – The saturation stage

A typical simulation

A typical simulation – the spatial growth rate

The influence of the wavenumber

Exploring the « mildly » relativistic regime

Final stage in the non-relativistic regime

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…