ABSTRACT Magnetic reconnection plays a critical role in many astrophysical processes where high energy emission is observed, e.g. particle acceleration, relativistic accretion powered outflows, pulsar winds and probably in dissipation of Poynting flux in GRBs. The magnetic field acts as a reservoir of energy and can dissipate its energy to thermal and kinetic energy in the tearing mode instability. We have performed 3d non-linear MHD simulations of the tearing mode instability in a current sheet. Results from a temporal stability analysis in both the linear regime and weakly non-linear (Rutherford) regime are compared to the numerical simulations. We observe magnetic island formation, island merging and oscillation once the instability has saturated. The growth in the linear regime is exponential in agreement with linear theory. In the second, Rutherford stage the island width grows linearly with time. We find that thermal energy produced in the current sheet strongly dominates the kinetic energy. Finally preliminary analysis indicates a P(k) ~ -4.8 power law for the power spectral density which suggests that the tearing mode vortices play a role in setting up an energy cascade. Reconnection in Current Sheets G. C. Murphy 1 , R. Ouyed 2 , G. Pelletier 1 1 Laboratoire d’Astrophysique de Grenoble, CNRS, Université Joseph Fourier, Grenoble, France 2 Dept of Physics and Astronomy, University of Calgary, AB, Canada INTRODUCTION Magnetic reconnection is a topological change in the field which violates the frozen- flux condition of ideal magnetohydrodynamics (MHD). If a magnetic field can leak across a separatrix it can reach a lower energy state - in the case of a current sheet it can undergo “tearing” into filaments or magnetic islands. A current layer may dissipate on time scales shorter than the resistive time scale due to the tearing mode instability (hereafter TMI). REFERENCES Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963, Phys. Fluids 6, 459 Rutherford, P. H. 1973, Phys. Fluids, 16, 1903 Vestuto, J. G., Ostriker, E. C. & Stone, J. M. 2003, ApJ, 590, 858 DISCUSSION AND CONCLUSIONS It is well established that current sheets are unstable to the tearing mode. We show that even in compressible MHD it is possible to predict with some accuracy the growth rate of the tearing mode. In addition using Rutherford theory can also estimate the saturation time of the instability. We find that magnetic islands tend to increase independent of mode number depending mainly on the Lundquist number. Finally we derive a power spectrum for a 3d sheet and show its slope is similar to those generated by driven MHD turbulence suggesting a link between the two processes. ACKNOWLEDGEMENTS The work presented here has supported by the Agence Nationale de la Recherche (ANR) and the University of Calgary. LINEAR AND NONLINEAR GROWTH Using asymptotic matching Furth et al. (1963) derived stability range and growth rates for the TMI in the inviscid linear regime: where a is the current sheet width, S is the Lundquist number, τ r is the resistive timescale. Rutherford (1973) found in the non-linear regime that the island width grows linearly in time, instability growth rate slows down from exponential to t2 and that the critical amplitude where the linear solution ceases to be valid is:  R =  −2 5  S  2 5  B =  2 / GROWTH RATES COMPARED WITH THEORY Figures 3 and 4. The growth of cross-sheet magnetic field compared with Furth et al prediction of the growth rate and the Rutherford prediction of transition to non-linear regime for alpha=0.01 and 0.3. POWER SPECTRUM OF 3D CURRENT SHEET We plot the power spectrum for 3d current sheet. The slope of P(k) is -4.8, within the range of values found by Vestuto et al (2003) for β = 1 and β= ∞. Vestuto et al. (2003) note that in their driven, supersonic turbulence simulation they have a constant beta and a constant mass-to-flux ratio “modulo the numerical reconnection effect”. Figure 1. Density colour map and selected magnetic field lines of magnetic island formation for alpha=0.3 Figure 2. Density colour map and selected magnetic field lines of magnetic island formation for alpha=0.01 RESISTIVE MHD EQUATIONS We solve the resistive MHD equations using the numerical code PLUTO: ∂ ρ ∂ t   ∇⋅ρ u =0 ∂ ρ u ∂ t   ∇⋅  ρ uu +p I  BB  = 0 ∂ B ∂ t   ∇×  B × u+ J η  = 0 ∂ E ∂ t   ∇⋅ [  E+p tot  u − u . B  B  +B × J η ] = 0 E= 1 2 u ρ 2  p γ −1  1 2 B 2 p tot =p+ 1 2 B 2 ISLAND WIDTH Figures 5 and 6 The FWHM of the magnetic island plotted vs. time. In the linear regime the island width is almost constant and then increases linearly in the Rutherford regime, followed by oscillations. = ka 1