A Numerical Method for General Relativistic Magnetohydrodynamics

Jean-Pierre De Villiers and John F. Hawley
Astronomy Department
University of Virginia
P.O. Box 3818, University Station
Charlottesville, VA 22903-0818

Submitted to Astrohysical Journal

Abstract:

This paper describes the development and testing of a general relativistic magnetohydrodynamic (GRMHD) code to study ideal MHD in the fixed background of a Kerr black hole.  The code is a direct extension of the hydrodynamic code of Hawley, Smarr, and Wilson, and uses Evans and Hawley constrained transport (CT) to evolve the magnetic fields.  Two categories of test cases were undertaken.  A one dimensional version of the code (Minkowski metric) was used to verify code performance in the special relativistic limit.  The tests include Alfven wave propagation, fast and slow magnetosonic shocks, rarefaction waves, and both relativistic and non-relativistic shock tubes.  A series of one- and two-dimensional tests were also carried out in the Kerr metric:  magnetized Bondi inflow, a magnetized inflow test due to Gammie, and two-dimensional magnetized constant-l tori that are subject to the magnetorotational instability.
 

Download PDF version of paper: grmhd.pdf

Movies: (see paper for further information)

The two animations show here are derived from tests involving constant specific angular momentum (l) disks, solutions of the axisymmetric GR hydrodynamic equations described Hawley, Smarr, and Wilson (1984). Here, we add weak poloidal magnetic field loops that overlay the hydrodynamic solution to trigger the Magneto-Rotational Instability (MRI). We quantify the strength of the magnetic field through the beta-parameter, the ratio of the volume-averaged gas pressure to magnetic pressure. For these animations, beta=100 initially.

The animations show the results of axisymmetric (2D) magnetized constant-l torus simulations in the Schwarzschild and Kerr metrics. For the Schwarzschild case (Model SF0), the disk has a specific angular momentum l=4.5, an initial pressure maximum at r = 15.3 M, and an orbital period at the pressure maximum of T(orb) =376 M. For the Kerr case (Model SFP), the black hole is maximally rotating, a=1, and the disk has a prograde specific angular momentum l=4.3, an initial pressure maximum at r = 15.4 M, and an orbital period at the pressure maximum of T(orb) =386 M. The initial state is perturbed with random 1% enthalpy fluctuations. The MRI develops after a few orbits, and is soon fully developed. By the end of the animation (10 orbits), the MRI-induced turbulence has settled down considerably, as is to be expected since the imposition of axisymmetry precludes the development of the azimuthal modes which sustain the MRI.

• Model SFP  prograde l=4.3 torus, Kerr black hole with a = 1.0, animation of log(density) from t=0 to t=10 orbits
• Model SF0   l=4.5 torus, Schwarzschild black hole, animation of log(density) from t=0 to t=10 orbits

2002-10-23