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MODELING RELATIVISTIC MAGNETIZED PLASMA
Komissarov Serguei
University of Leeds
UK
RELATIVISTIC IDEAL MHD
Conditions:
Equations:
- perfect conductivity
-stress-energy-momentum of
electromagnetic field
-stress-energy-momentum of matter
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
.
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) .
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
MAGNETODYNAMICS (MD)
Condition:
Equations:
(Komissarov 2002)
Perfect conductivity:
or
MAGNETODYNAMICS is MAGNETOHYDRODYNAMICS
without the HYDRO part
MAGNETODYNAMICS
Advantages:
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:
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;
MAGNETODYNAMICS
Example: Stability of the Blandford-Znajek solution.
Hf
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)
MAGNETODYNAMICS
Breakdowns of the MD approximation. 1D example:
Initial solution
Time evolution
B 2- E 2
Bx
By
E=0
x
x
A need for finite conductivity in order to keep E down !
RESISTIVE ELECTRODYNAMICS
Covariant
3+1 form
Equations:
Constitutive
a, b - lapse function and shift
vector (space-time metric)
RESISTIVE ELECTRODYNAMICS
Ohm’s Law:
(no particle inertia)
- drift current
- anisotropic conductivity
Typical conditions of BH and
pulsar magnetospheres:
In current sheets:
or even
RESISTIVE ELECTRODYNAMICS
Advantages:
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);
Disadvantages:
1) Does not allow adiabatic transfer of energy
between the electromagnetic field and particles;
2) Fast wave speed equals to c.
RESISTIVE ELECTRODYNAMICS
Example: Ergospheric current sheet.
B2- E2
W
Kerr black hole in
uniform at infinity
magnetic field; plasma
version.
(Komissarov, 2004)
RELATIVISTIC IDEAL MHD
with a density floor
Prescription: do not let the particle energy density to slip below a curtain
small fraction of the electromagnetic energy density.
Advantages:
1) All the advantages of Ideal MHD;
2) Allows to get quite close to the MD limit;
Disadvantages:
1) All the disadvantages of Ideal MHD;
2) How to handle current sheets ?
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)
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.