1. MODELING RELATIVISTIC MAGNETIZED PLASMA Komissarov Serguei University of Leeds UK

2. RELATIVISTIC IDEAL MHD Conditions: Equations: -stress-energy-momentum of electromagnetic field -stress-energy-momentum of matter - perfect conductivity

3. RELATIVISTIC IDEAL MHD Advantages: 1)Allows adiabatic transfer of energy and momentum between the electromagnetic field and particles; 2)Allows dissipation at shocks; 3)All wave speeds below c. Disadvantages: 1)Complexity; 2)Difficult to implement if.

4. RELATIVISTIC IDEAL MHD Godunov-type schemes: 1)Robust and simple Lax-Friedrichs-type Riemann solvers; 2)More accurate and complex linear Riemann solver (contacts, shears; Komissarov 1999); 3)No exact Riemann solvers so far - too expensive; 4)A number of ways to handle the “divB-problem”; (i) constrained transport (Evans & Hawley 1988); (ii) generalised Lagrange multiplier (Dedner et al. 2002); (iii) smoothing operator (Toth 2000).

5. RELATIVISTIC IDEAL MHD Example: - X-ray image of the Inner Crab Nebula based on 2D relativistic MHD simulations (Komissarov & Lyubarsky 2003) Chandra image of the real Crab Nebula

6. MAGNETODYNAMICS (MD) Condition: Equations: Perfect conductivity: or MAGNETODYNAMICS is MAGNETOHYDRODYNAMICS without the HYDRO part (Komissarov 2002)

7. MAGNETODYNAMICS Advantages: 1)Does not allow adiabatic transfer of energy and momentum transfer between the electromagnetic field and particles; 2) Does not allow dissipation; 3)Fast wavespeed equals to c; 4)Often breaks down; 1)Simple hyperbolic system of conservation laws (linearly degenerate fast and Alfven modes); 2)Perfectly describes force-free magnetospheres of black holes and neutron stars; Disadvantages:

8. MAGNETODYNAMICS Example: Stability of the Blandford-Znajek solution. Analitical and numerical solutions for a black hole with a=0.1, 0.5, and 0.9 at r =10 and t=120. (Komissarov 2001) H 

9. MAGNETODYNAMICS B B y x B - E 22 x x E=0 Breakdowns of the MD approximation. 1D example: Initial solution Time evolution A need for finite conductivity in order to keep E down !

10. RESISTIVE ELECTRODYNAMICS Equations: Covariant 3+1 form Constitutive  lapse function and shift vector (space-time metric)

11. RESISTIVE ELECTRODYNAMICS Ohm’s Law: - drift current - anisotropic conductivity (no particle inertia) Typical conditions of BH and pulsar magnetospheres : In current sheets : or even

12. RESISTIVE ELECTRODYNAMICS Advantages: 1)Does not allow adiabatic transfer of energy between the electromagnetic field and particles; 2)Fast wave speed equals to c. Disadvantages: 1)Simplicity; 2)Drives solutions towards the force-free state; 3) Allows dissipation in current sheets (transfer of energy between the electromagnetic field and radiation);

13. RESISTIVE ELECTRODYNAMICS Example: Ergospheric current sheet. B - E 22  Kerr black hole in uniform at infinity magnetic field; plasma version. (Komissarov, 2004)

14. RELATIVISTIC IDEAL MHD with a density floor Advantages: 1)All the disadvantages of Ideal MHD; 2)How to handle current sheets ? Disadvantages: 1)All the advantages of Ideal MHD; 2)Allows to get quite close to the MD limit; Prescription: do not let the particle energy density to slip below a curtain small fraction of the electromagnetic energy density.

15. RELATIVISTIC IDEAL MHD with a density floor Example: Inertial effects in the Blandford-Znajek problem. Lorentz factor (colour) and the critical surfaces at t=170; a=0.9. Analytical and numerical solutions for a Kerr black hole with a=0.1. (Komissarov, 2001)

16. CONSLUSIONS 1)These have been few first steps in exploring ways of modelling relativistic magnetized plasma; 2)A number of important results have been obtained; 3)There is still a long way to go and a promise of new important results in near future.