Numerical simulations of relativistic magnetic reconnection with Galerkin methods

**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c⃝**Copyright Year** Astronomical Society of the Pacific Numerical simulations of relativistic magnetic reconnection with Galerkin methods O. Zanotti,1 and M. Dumbser2 1Max-Planck-Institut f¨ur Gravitationsphysik, Albert Einstein Institut, Golm, Germany, 2Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy Abstract. We present the results of two-dimensional magnetohydrodynamical nu- merical simulations of relativistic magnetic reconnection, with particular emphasis on the dynamics of Petschek-type configurations with high Lundquist numbers, S ∼105 − 108. The numerical scheme adopted, allowing for unprecedented accuracy for this type of calculations, is based on high order finite volume and discontinuous Galerkin meth- ods as recently proposed by Dumbser & Zanotti (2009). The possibility of producing high Lorentz factors is discussed, by studying the effects produced on the dynamics by different magnetization and resistivity regimes. We show that Lorentz factors close to ∼4 can be produced for a plasma magnetization parameter σm = 20. Moreover, we find that the Sweet-Parker layers are unstable, generating secondary magnetic islands, but only for S > S c ∼108, much larger than what is reported in the Newtonian regime. 1. Introduction Relativistic magnetic reconnection is a high-energy process converting magnetic energy into heat and plasma kinetic energy over short timescales. It is supposed to play a fundamental role in the magnetospheres of pulsars (Uzdensky 2003; Gruzinov 2005); at the termination shock of a relativistic striped pulsar wind (P´etri & Lyubarsky 2007); in soft gamma-ray repeaters (Lyutikov 2003, 2006); in gamma-ray burst jets (Drenkhahn & Spruit 2002; Barkov & Komissarov 2010; McKinney & Uzdensky 2010; Rezzolla et al. 2011); in accretion disc coronae (di Matteo 1998; Schopper et al. 1998; Jaroschek et al. 2004). In spite of the initial optimistic expectations that relativistic magnetic reconnection could provide very fast reconnection rates, evidence has emerged over the years that the relativistic Petschek reconnection should not be considered as a mechanism for the di- rect conversion of the magnetic energy into the plasma energy and the reconnection rate would be at most 0.1 the speed of light, contrary to what originally suggested by Blackman & Field (1994). Fundamental progresses in the numerical modeling of rela- tivistic magnetic reconnection have been recently obtained by Watanabe & Yokoyama (2006); Zenitani et al. (2009b,a), However, two major numerical limitations still pre- vent realistic astrophysical applications of numerical schemes specifically devoted to relativistic magnetic reconnection. The first limitation is due to the difficulty in reach- ing sufficiently high magnetization parameters σm, while the second limitation is due to the difficulty in treating physical systems with very high Lundquist numbers S . 1 arXiv:1109.0746v1 [astro-ph.HE] 4 Sep 2011 2 In this work, by adopting the innovative numerical method presented in Dumb- ser & Zanotti (2009), we show the results of numerical simulations in the very high Lundquist numbers regime, S ∼105 −108, showing that that Sweet-Parker current sheets are unstable to super-Alfvenically fast formation of plasmoid chains, but only for S > S c ∼108. 2. Physical set up and numerical approach The initial model that we have considered is built on Harris model, as reported by Kirk & Skjæraasen (2003), and it reproduces a current sheet configuration in the x −y plane. Gas pressure and density are given by p = p0 + σmρ0[p0 cosh2(2x)]−1, ρ = ρ0 + σmρ0[p0 cosh2(2x)]−1, where p0 and ρ0 are the constant values outside the current sheet, whose thickness is δ = 1. The magnetic field changes orientation across the current sheet according to By = B0 tanh(2x), where the value of B0 is given in terms of the magnetization parameter σm = B2 0/(2ρ0Γ2 0). All over the grid there is a small background uniform resistivity ηb, except for a circle of radius rη = 0.8, defining a region of anomalous resistivity of amplitude ηi0 = 1.0. The resistivity can be written as η = ( ηb + ηi0 h 2(r/rη)3 −3(r/rη)2 + 1 i for r ≤rη, ηb for r > rη, (1) where r = p x2 + y2. The velocity field is initially zero, while the electric field is given by Ez = η(∂By/∂x). We have considered the case with p0 = 1, ρ0 = 1. The Lundquist number for every model is S = vAL/ηb, where L is the length of the initial current sheet, while v2 A = B2/(hρ + B2) is the relativistic Alfven velocity. A well known and challenging feature of the relativistic resistive magnetohydro- dynamics equations is that the source terms in the three equations for the evolution of the electric field become stiffin the limit of high conductivity. To cope with this difficulty, we have applied the strategy described by Dumbser & Zanotti (2009), who used the so called high order PNPM methods, which combine high order finit

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