We describe a new scheme for evolving the equations of force-free electrodynamics, the vanishing-inertia limit of magnetohydrodynamics. This pseudospectral code uses global orthogonal basis function expansions to take accurate spatial derivatives, allowing the use of an unstaggered mesh and the complete force-free current density. The method has low numerical dissipation and diffusion outside of singular current sheets. We present a range of one- and two-dimensional tests, and demonstrate convergence to both smooth and discontinuous analytic solutions. As a first application, we revisit the aligned rotator problem, obtaining a steady solution with resistivity localised in the equatorial current sheet outside the light cylinder.
We present a code for simulations of force-free electrodynamics: phaedra (Pseudospectral High-Accuracy Electro-Dynamics for Relativistic Astrophysics). The systems we study are 'force-free' in the sense that the Lorentz force density vanishes everywhere, because the electromagnetic fields are strong enough that hydrodynamic forces can be neglected, resulting in self-balancing electromagnetic fields. Relativistic force-free electrodynamics has long been recognised as the appropriate limit for describing the magnetospheres of neutron stars and black holes, yet only recently has a concerted effort begun to study it with direct numerical simulation.
This infinite-magnetisation, or vanishing-inertia, limit is the appropriate one for the magnetospheres of pulsars (Goldreich & Julian 1969;Contopoulos, Kazanas, & Fendt 1999;Gruzinov 2006;Spitkovsky 2006), and magnetars, whose persistent and transient high-energy emission may be due to the distortion, reconnection, and dissipation of forcefree fields (Thompson & Duncan 1995;Lyutikov 2006;Beloborodov 2009). Force-free electrodynamics is the standard tool to study the extraction of rotational energy from black holes (Blandford & Znajek 1977;MacDonald & Thorne 1982), where the magnetic field is thought to be supplied by a conducting accretion disc. The natural self-collimation of electromagnetic fields make them attractive candidates for explaining relativistic jets in quasars and active galactic nuclei, whose high Lorentz factors suggest low baryon loading and electromagnetic dominance (Blandford 1976); similar Poynting jets may be responsible for gamma-ray bursts (Mészáros & Rees 1997). An argument can be made that all ultra-relativistic outflows are essentially electromagnetic, rather than gas dynamical (Blandford 2002).
In any case, it is clear that there exists a wealth of challenging problems for the field of electrodynamic numerical simulation. Direct time-dependent simulation is valuable because it permits the study of general realistic initialvalue problems, without the restrictions, like self-similarity or stationarity, that are often necessary in analytical models, and because it naturally tests the stability of field configurations, a question often unanswered by steady-state numerical work.
Several time-dependent force-free electrodynamics codes exist, both those using finite differences (Spitkovsky 2006;Kalapotharakos & Contopoulos 2009;Palenzuela et al. 2010), and those that take a finite volume, or Godunov, approach (Komissarov 2004a;Cho 2005;Asano, Uchida, & Matsumoto 2005;McKinney 2006a;Yu 2011). Our numerical scheme is entirely different, and complementary, being based on orthogonal basis function expansions. Previous codes have large numerical dissipation or diffusion, introduced either because they do not maintain E • B = 0 self-consistently or through the intrinsic diffusivity of the method, while force-free problems often demand long simulations, as the fields may evolve over many wave-crossing times. It is desirable to have a method which can be run for long times without intrinsic dissipation, captures discontinuities, and accurately describes fast dynamics.
The crucial question one asks of a force-free configuration is that of its stability, the onset of instability commonly leading to a dramatic rearrangement of a magnetosphere, sometimes involving explosive reconnection. Spectral calculations tend to have less numerical noise than those of comparable finite-difference or finite-volume (’local’) schemes; this noise can erroneously trigger instability. In a study of Sweet-Parker reconnection, the spectral magnetohydrodynamics (MHD) code is found to be largely immune to the secondary island formation, caused by a tearing-mode instability, that is found using local methods for the same problem (Ng & Ragunathan 2011).
In this paper, we describe a code for axisymmetric simulations, in flat space-time. It has been designed in such a way as to be extensible with minimal restructuring to a fully three-dimensional setting, in curved space-time.
Force-free electrodynamics, and its relation to relativistic MHD, is discussed in Section 2. Section 3 contains a detailed description of the code and its practical implementation, including some background on each of its components. A range of one-and two-dimensional test problems are presented in Section 4, including convergence tests in realistic scenarios. The aligned rotator is examined in Section 5. Finally, we discuss our results in Section 6, and outline some promising areas of future research.
Note that in Section 3 we distinguish between the contravariant, F i , and covariant, Fi, components of a vector field, while when we discuss results in Sections 4 and 5 we refer only to the components in an orthonormal basis, also written Fi.
The system of force-free electrodynamics is the vanishinginertia, or, equivalently, ultra-relativistic, limit of magnetohydrodynamics. The latter can be written as
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