1. Convective instability of a decelerating relativistic shell: an origin of magnetic fields in the early afterglow phase?
Amir Levinson, Tel Aviv University

2. Evolution: early stage
Shocked ISM
Shocked
ejecta
reverse shock
contact
forward shock

3. Evolution: late stage
Shocked ISM
forward shock
Γ t-3/2

4. Some open questions Origin of magnetic field in shocked ISM ?
Particle acceleration ?
Optical flash from reverse shock ?
Stability of the system ?
Is Blandford-McKee solution an attractor ?

5. Stability
Convective Rayleigh-Taylor instability of the contact is expected due to the strong deceleration of the ejecta. In the relativistic case this is expected in cases where the reverse shock is not ultra-relativistic.
Small structure grows faster, leading to generation of vorticity and magnetic fields in the shocked regions (Gull 73).
In young SNR this can account for the sub-structure often seen.

6. Stability - Non-relativistic case
Linear stability analysis of a self-similar solution (Chevalier et al 92).
Self-similar perturbations were found for all modes. Not so in relativistic case!

7. Full 2 and 3D MHD simulations (Jun+Norman 96)
JN found that R-T and K-H instabilities produce magnetic fields in the shocked ISM region, and can explain the structure of young SNRs.

9. Relativistic case Linear stability analysis
Unperturbed solution: self-similar (Nakamura+Shigeyama 06)
Global analysis

10. unperturbed solution: self-similar solution derived by Nakamura+Shigeyama 06
Unmagnetized ejecta
Shocked ISM
Shocked
ejecta Γc
Γfs
Γrs
Unshocked ejecta
Ambient medium

11. Properties of the solution
Trajectory of the reverse shock:
Conditions at the contact (no flow + pressure balance) give

12. Self-similar solution
n=1.1, k=2
density
Lorentz factor
pressure
ejecta
ISM

13. Relativistic case Linear stability analysis
Unperturbed solution: self-similar (Nakamura+Shigeyama 06)
Linear perturbations of the form Q=Q0Q(t,)Ylm(,) on each side of the contact discontinuity.
The perturbations are coupled through boundary conditions at the contact discontinuity.
There is an analytic solution for spherical perturbations (l=0). This is used as initial condition for the evolution of non-spherical perturbations.

14. Properties of the perturbations
Boundary conditions at forward shock implies:
Likewise at the reverse shock. So in general the perturbations are non self-similar. Only in spherical case

15. Preliminary Results Self-similar spherical (l=0) modes are stable:
Q/Q ts with s<0.
Examples: n=1.1, k=2, s= -0.71
n=1.1, k=0, s= -1.8
Non-spherical perturbations are not self similar. S-m is broken by the boundary conditions at the shock.
System is stable to disturbances of angular scale larger than the causality scale: l(l+1)/Γ2 << 1.
Unstable to modes with l(l+1)/Γ2 > 1.
Preliminary

16. Oscillations of the contact
Log t
Log t

17. Oscillations of the shock fronts
- Forward shock
- Reverse shock
Log t

18. Pressure evolution at the contact
Normalized to initial value
Log t

19. Pressure evolution near shock surfaces
- Forward shock
- Reverse shock
Log t

20. How does the instability evolve at late stages ?

21. Stability of the BM solution Gruzinov 2000
BM solution is stable but non-universal

22. Remarks
Instability may considerably affect the physics of reverse shock. Optical flashes?
Oscillations of the forward shock may lead to generation of vorticity and magnetic fields (MacFadyen + 09).
significant magnetization of the ejecta should suppress the instability.
Likewise if the reverse shock is highly relativistic.
detailed studies require high resolution, 3D MHD simulations!