1. Using a combined PIC-MHD code to simulate particle acceleration
Allard Jan van Marle (APC)
Fabien Casse (APC), Alexandre Marcowith (LUPM)

2. Acceleration of cosmic particles
We know astrophysical shocks accelerate particles
This process also influences the local magnetic field and the behaviour of the shocks
For astrophysical shocks: magneto- hydrodynamics (MHD)
Fluid approach
Computationally efficient
Lacks particle physics
For particle acceleration: particle-in-cell (PIC)
Kinetic approach
Can simulate particle physics
Computationally expensive
The challenge: To combine both in a single code
X-ray: Nasa/CXC/Rutgers/K. Eriksen et al.; Optical: DSS

3. Introducing Particles in MHD cells
We split the work:
MHD code (MPI-AMRVAC, van der Holst et al. 2008, describes the thermal plasma, particles represent the supra-thermal component component
The MHD grid serves as cells for the PIC
Lorentz force gives us the effect of electromagnetic field on particles
Effect of supra-thermal particles on thermal plasma can be treated through Ohm’s law (Bai et al. 2015)
Advantages
Can simulate particle acceleration and feedback (unlike MHD)
No need for a huge particle population that represents the thermal gas (unlike PIC)
We don’t solve all Maxwell equations, which reduces numerical noise (no Cherenkov waves)
Disadvantages
Limited regime: nthermal >> nnon-thermal
Some restrictions owing to use of grid-MHD
Need to determine a particle injection rate (obtained from PIC simulations)

4. Ohm’s law For MHD With supra-thermal particles (R = ncr/ntotal)
The Lorentz force
(Bai et al. 2015)

5. The new PI(MHD)C equations
MHD conservation equations
Induction equation
Particle equation of motion
(Bai et al. 2015)

6. PI[MHD]C Move particles Using Boris-pusher and the B and E fields
Update MHD quantities through conservation equations,
including charge and current from particles
Interpolate from particles to determine charge and current in cell centres
Constrained transport ensures div.B=0
MHD cell-centres function as PIC cell-corners

7. Parallel shock Based on tests shown in Bai et al. 2015
Non-relativistic shock with pre-existing magnetic field (MA=30)
Injecting particles (proton equivalent) at the shock (VCR = 3 VS)
Box: 30x240 Rl
Grid: 30x240 (+ 4 levels)
Recovered analytical predictions (Bell ´04) and earlier numerical results. (Caprioli & Spitkovski ´14, Bai et al. ´15)
Upstream: non-resonant streaming instability
Downstream: turbulence
van Marle, Casse & Marcowith (submitted)

8. SED for parallel shock

9. Near-oblique shock
hybrid-PIC simulations (Caprioli & Spitkovsky 2014) NO B-field amplification or particle acceleration occurs at angles > ~60o.
However, Vlasov-Focker-Planck based models do show upstream instabilities very close to the shock (Reville & Bell 2013)
Hybrid PIC simulations show only results during early phase
Their domain is approx. six times smaller along vertical axis
They suffer from relatively low number of particles. Approx. 2x108, or 4 per cell, with only a small fraction non-thermal (Caprioli & Spitkovsky 2014)
Particles are accelerated by repeated shock crossings (Decker 1988)
Small instabilities cause shock corrugation
Shock corrugation causes variation in upstream current
Upstream current disturbs the upstream magnetic field
Disturbed field increases shock corugation
van Marle, Casse & Marcowith (submitted)

10. Particle SED for oblique shock
Diffusive shock acc.
Shock drift acc.

11. Summary
PI[MHD]C can reliably reproduce hybrid-PIC models and analytical predictions for the parallel shock case
The increased computational efficiency of this method has allowed us to improve on previous results for the near-oblique shock situation
Next step:
Extend 2-D models to realistic astrophysical scenarios (Stellar wind shocks, supernova remnants, cluster shocks, etc.)
3-D models (again, possible because of computational efficiency)
Combining relativistic MHD with PIC (early supernova explosions, GRBs, AGNs)