3-d resistive MHD simulations of magnetic reconnection and the tearing mode instability in current sheets
aa r X i v : . [ a s t r o - ph ] J un October 29, 2018 16:32 WSPC - Proceedings Trim Size: 11in x 8.5in gmurphy G. C. Murphy ∗ Laboratoire d’Astrophysique de Grenoble, CNRS, Universit´e Joseph Fourier, Grenoble, France ∗ E-mail: [email protected]
R. Ouyed
Department of Physics and Astronomy, University of Calgary,AB, CanadaE-mail: [email protected]
G. Pelletier
Laboratoire d’Astrophysique de Grenoble, CNRS, Universit´e Joseph Fourier, Grenoble, FranceE-mail: [email protected]
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 Poyntingflux in GRBs. The magnetic field acts as a reservoir of energy and can dissipate its energy to thermal and kineticenergy via the tearing mode instability. We have performed 3d nonlinear MHD simulations of the tearing modeinstability in a current sheet. Results from a temporal stability analysis in both the linear regime and weakly nonlinear(Rutherford) regime are compared to the numerical simulations. We observe magnetic island formation, island mergingand oscillation once the instability has saturated. The growth in the linear regime is exponential in agreement withlinear theory. In the second, Rutherford regime the island width grows linearly with time. We find that thermal energyproduced 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 upan energy cascade.
Keywords : MHD-Plasma physics-numerical simulations
1. Introduction
Magnetic reconnection plays a critical role in manyastrophysical processes, e.g. particle acceleration [1],accretion disks [2] and solar flares [3]. It is also impor-tant in laboratory fusion devices such as tokamaks.Magnetic reconnection is a topological change in thefield which violates the frozen-flux condition of idealmagnetohydrodynamics (MHD). If a magnetic fieldcan leak across the plasma it can reach a lower energystate - in the case of a current sheet it can undergo“tearing” into filaments or magnetic islands. A cur-rent layer of thickness a may dissipate on timescalesshorter than the resistive timescale a /η due to thetearing mode instability (hereafter TMI). The TMIwas first discovered in tokamaks and stellerators andhas been extensively studied since the pioneeringworks [4] and [5] (see also [6], [7], [8]). In this paperwe present numerical simulations of the dissipationof current sheets and the formation of magnetic is-lands due to the tearing mode instability. In Section2 we remind the reader of the predictions of the linear and weakly nonlinear analyses of Furth et al. [4] andRutherford [5]. In Section 3 we present our numeri-cal method. In Section 4 we compare the results withlinear and weakly nonlinear theory. In Section 5 wepresent power spectra derived from 3d simulations.In Section 6 we discuss the pitfalls of simulationswhere reconnection is not properly tracked and theimplications for reconnection-driven turbulence.
2. Theory2.1.
The linear regime
In the linear analysis it is assumed that resistivityis only important within the current layer. Separatesolutions may then be derived for the exterior, idealMHD region and the interior, resistive region. Us-ing asymptotic matching [4] derived stability rangeand growth rates for the TMI in the inviscid linearregime, neglecting the effects of compressibility. Therange of the instability is α < , α = ak , where k isthe wave number and a is the current sheet width. ctober 29, 2018 16:32 WSPC - Proceedings Trim Size: 11in x 8.5in gmurphy The growth rate is: γ ∼ α − S /τ R (1)The linear growth rate is a function of the modenumber, k , the current sheet width, a , and theLundquist or magnetic Reynolds number, S . Theequations are valid for α > S − . The system stayslinear for as long as the magnetic island does notexceed the width of the current sheet. Non-linear regime
Rutherford [5] studied the evolution of the TMI inthe weakly nonlinear regime, by considering the ef-fect of second order eddy currents on the currentsheet. [5] found in the nonlinear regime that the is-land width grows as t, instability growth rate slowsdown from exponential to t , and that the criticalamplitude where the linear solution ceases to be validis: B max = √ ηργα (2) Table 1. Growth Rates and Saturation Times forLundquist number 2400 α Linear Growth Rate [ τ − A ] Saturation [ τ A ]Theory Sim Theory Sim0.01 0.0590 0.0528 153.233 1700.04 0.0339 0.0484 394.980 2500.3 0.0151 0.01276 508.370 520 Previous work
More recent work has concentrated on including thephysics of the Hall effect using multifluid or HallMHD and electron inertia, using PIC techniques.Significant increase in reconnection rates is found inthese studies [9]. However in these works the authorsdo not compare against analytical theory nor is the3D power spectrum calculated.
3. Numerical setup
We perform first 2D and then 3D simulations in re-sistive MHD using the astrophysical code PLUTO.As in any numerical code some numerical resis-tivity is present. In the linear analysis numerical re-sistivity has been assumed to be small in comparisonwith the physical resistivity. Our initial mean field is a Harris current sheet of the form B y = tanh( x ).We initialise the field in the magnetic vector po-tential A z = ln | cosh(x) | and take the curl to de-rive B . To this we add a perturbation of the formcos(kx) in 2d and cos(kx)cos(ky) in 3d. Free param-eters are α , β = pB and the resistivity η . Zero-gradient Neumann-like “outflow” boundary condi-tions often produce subsonic reflection for compress-ible flows [10, 11]. We use a coarse mesh boundarywhich reduces this effect.
4. Simulation Results4.1. Fig. 1. Formation of magnetic islands: magnetic field linesand density colourmap are shown at times: 245, 365, 490.Fig. 2. Same as Figure 1 but for α = 0 . We performed 2d and 3d simulations of cur-rent sheets for different initial perturbations k =0 . , . , . , .
6. We tracked the growth rate of thecross-sheet magnetic field B x . In the simulations aninitial transient period, t transient ∼ τ A roughlycorresponding to half the crossing time of the do-main ( t = 112 τ A ) was observed before the linearmode was established. As the tearing mode growslinearly, field lines reconnect, new separatrices areformed and magnetic islands grow in size, eventuallyexceeding the size of the original current sheet. InFigures 1 and 2 the time evolution of magnetic is-lands for α = 0 . , . ctober 29, 2018 16:32 WSPC - Proceedings Trim Size: 11in x 8.5in gmurphy units of the Alfv´en crossing time, τ A . Fig. 3. Growth of log of maximum B x plotted against time.The linear growth rate is plotted and the critical value forsaturation.Fig. 4. Same as Fig. 3 but for α = 0 . Agreement with linear analysis
In Figures 3 and 4 the growth rate in cross-sheetmagnetic field, B x for different exciting modes ofthe TMI is plotted against time. In all cases a lin- ear regime is found with slopes near the analyticalpredictions of Furth et al. [4]. Fig. 6. Magnetic island width evolution plotted against timein units of τ A for α = 0 .
01. The width is estimated using thefull-width half maximum of the thermal pressure. It remainsapproximately constant in the linear phase and transfers to alinear growth in t once in the Rutherford regime.Fig. 7. Same as 6 but for α = 0 . Agreement with nonlinear analysis
We find plotting the island width against time thatit remains small in the linear regime and grows as t in the Rutherford regime. We can estimate the timeof transition from linear to Rutherford from both the B x plots and the island width plots, however the is-land width plots provide a more accurate estimate.The time at which the system becomes nonlinear istabulated in Table 1. The magnetic island width canbe derived from eqn 18. in [5, 6] and can be writtenin approximate form as w I ∼ ˜ t − t d S , (3)where t d accounts for the total delay (transient andtime spent in linear regime) in our simulations beforeentering the Rutherford regime. ctober 29, 2018 16:32 WSPC - Proceedings Trim Size: 11in x 8.5in gmurphy
5. 3D Power spectra
In Figure 8 we plot the power spectrum for β = 23d current sheet. The slope of P ( k ) is − .
8, withinthe range of values found by [12] for β = 1 and β = ∞ . [12] note that in their driven, supersonicturbulence simulation they have a constant beta anda constant mass-to-flux ratio “modulo the numeri-cal reconnection effect”. We obtain the same powerspectrum scaling without any added velocity pertur-bations. Fig. 8. 3d kinetic power spectrum at time t = 295 τ A withoverplotted k − .
6. Discussion and Conclusion6.1.
Consequences for driven-turbulencesimulations
Our results show that even without either a driventurbulence mechanism, or an initially turbulent ve-locity spectrum it is possible for the vortices gener-ated by the TMI to mimic the power spectrum seenin simulations of turbulence. This can have seriousconsequences for MHD simulations where the resis-tivity is not explicitly constrained (i.e. numerical re-sistivity plays a role), since there will be inevitably besome numerical-reconnection driven vortices presentin the simulation. However using an explicit resistiv-ity, the contribution from reconnection can be esti-mated using the formulae in [4, 5].Finally, we propose as a benchmark for resis-tive MHD codes the tearing mode instability test,which can be compared with both analytical resultsfor both the linear and nonlinear regimes, as well as with laboratory experiments. The test may alsoprove useful as a way of quantifying the effects ofnumerical resistivity in an MHD code.
Acknowledgements
G.C.M. would like to acknowledge funding fromthe Agence Nationale de Recherche, the Universityof Calgary, and the Dublin Institute for AdvancedStudies.
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