Biancalani, A.

Scott, B.

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Scott, B. D.

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Nonlinear growth acceleration in gyrofluid simulations of collisionless reconnection

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Nonlinear growth acceleration in gyrofluid simulations of collisionlessreconnectionA. Biancalani∗, B. D. ScottMax-Planck-Institut für Plasmaphysik, Euratom Association, D-85748 Garching, GermanyIntroductionMagnetic reconnection is an instability which occurs in magnetized plasmas, associated to abreaking and reconnecting of the magnetic field lines [1]. Characteristic resistive times are toolarge in many situations, like for instance in hot plasmas, where the resistivity is low. In thesesituations, collisionless reconnection, due to finite inertia of the electrons, can provide fasterreconnection rates than resistive reconnection [2, 3]. In ma y cases the ion Larmor radius islarger than the reconnective inertial layer and 3 dimensional nonlinear phenomena can develop.In this paper we want to face these cases, therefore we adopt a3D gyro-fluid model.Two dimensional studies with a gyrofluid model were performed in Ref. [4] with gyrofluidcode REC2, in periodic configuration, with flat temperature and density profiles. Finite Larmorradius effects (FLR) were investigated with REC2 in Ref. [5]and more recently in Ref. [6]:linear growth rates were found consistent with analytical sling with ρi . Three dimensionalstudies were performed with a two-fluid model in Ref. [7], in the large∆′ regime. It was foundthat small perpendicular scales develop in proximity of theisland separatrix [8, 9]. Nonlinearacceleration has been found at the beginning of the nonlinear ph se in Ref. [10]. The valueof the nonlinear growth rate was found to increase for smaller values ofρs/L andde/L. Thenonlinear growth rate was found at values up to twice the linear growth rate.In this paper, we investigate the small∆′ regime, and push the limit of very small values ofρs/L andde/L, for values ofβ in the transition between fusion and space plasmas.Model equations.Reconnecting modes are investigated, evolving in a three dim nsional Harris-pinch configu-ration (By(x) = B̂y tanh(x) , Bz = const.). Flat density and temperature profiles are initialized.A 3D gyrofluid code named GEM (Gyrofluid ElectroMagnetic) is ued for our numericalsimulations [11]. A strong guide fiels is assumed and fast magneto-sonic waves are neglected.Frequencies lower than the ion gyro-frequency are considered. The gyrofluid model equations∗also at Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau, Germany38th EPS Conference on Plasma Physics (2011) O4.409for the ions are:( ∂∂ t+uE ·∇)ñi = −B∇‖ũ‖B(1)β̂∂ Ã‖∂ t+ ε̂( ∂∂ t+uE ·∇)ũ‖ =−∇‖(φ̃G + τi ñi) (2)and for the electrons:( ∂∂ t+vE ·∇)ñe = −B∇‖ṽ‖B(3)β̂∂ Ã‖∂ t− µ̂( ∂∂ t+vE ·∇)ṽ‖ =−∇‖(φ̃ − ñe) (4)where the tilde symbol denotes perturbed quantities,τi = Ti/Te, n, u andv are the fluid den-sity and ion and electron velocities, andφ and A are the scalar and vector potentials. Theparametersβ̂ , ε̂ and µ̂ are defined by:β̂ = βeε̂, ε̂ = (qR/L)2 and µ̂ = µeε̂ = (me/mi)ε̂,1 1.5 2 2.5 3 3.50.0220.0240.0260.0280.030.0320.0340.036ρτ / ρsγ L Lx / csFigure 1: Linear growth rate, normal-ized to the sound speed, versus the gyro-radius to sound gyro-radius avarageρτ ,for ∆′ρs = 0.7, βe = 5.4 ·10−5.whereβe = 4π pe/B2, andqR and L are the scalelength along and perpendicular to the equilibriummagnetic field.Note that all quantities with symbolˆare normal-ized to ε̂. The parallel gradient is calculated in thedirection parallel to the total (equilibrium plus per-turbed) magnetic field, and the gyro-averaged scalarpotential is defined as̃φG = Γ1/20 (φ̃). The advectionvelocitiesvE anduE are theE×B drifts, given byc∇φ̃G×B/B2 (where for the electrons̃φG = φ̃ ).Moreover, the polarization and induction equa-tions are:Γ1/20 ñi +Γ0−1τiφ̃ = ñe , −∇2⊥Ã‖ = J̃‖ = ũ‖− ṽ‖We also use the Padé approximant forms:Γ0 = (1−ρ2i ∇2⊥)−1, Γ1/20 = (1−ρ2i ∇2⊥/2)−1, whereρi is the ion gyro-radius.We consider a plasma regime with∆′ρs = 0.7. We obtain a very thin reconnection layerwith respect of the perpendicular size of the simulation box(ρs/Lx = 0.008). A deuteriumplasma is considered (µe = 0.00027) in our simulations. The ion to electron temperature ratiois considered unitary,τi = 1, unless otherwise specified.A gyro-fluid model is capable of investigating finite Larmor radius effects on the reconnectiondynamics. The linear growth rateγL, is shown to increase with the ion Larmor radius via the38th EPS Conference on Plasma Physics (2011) O4.409Figure 2: Evolution of the growth rates for the run starting with a magnetic island seed (left),and with a white noise (right). The nonlinear acceleration is found in the phase t=350-400L/csfor the case with initial seed, and at t=500-550L/cs for the case with initial white noise.quantityρτ , whereρτ =√ρ2i +ρ2s (see Fig. 1). This is consistent with the analytical theoreticalprediction given in Ref. [3]. No qualitative changes are found in the dynamics of our interestdue to finite larmor radius effects, neither in the linear nori the nonlinear phase.Nonlinear evolution.We perform two kinds of runs: one starting with an island seedand one starting with an initialwhite noise. In both the cases, after a transient phase the growth becomes exponential in time,10−510−410−310−210−410−310−210−1100βeγ L x / c sFigure 3: Growth rates vs beta. Nonlineargrowth rates are in blue; linear growth rates inblack. Note that forβe of the order of 10−3-10−2, the nonlinear growth rate is one order ofmagnitude higher than the linear growth rate.and the growth rate tends to a constant value.This phase where the growth rate is constantin time is named linear phase. At the begin-ning of the nonlinear phase, we have an accel-eration of the reconnection growth, associatedwith a peak in the growth rate (see Fig. 2).The ratio between nonlinear and lineargrowth rates increases with increaing beta,reaching values up to more than one order ofmagnitude atβe of the order of 10−3-10−2.For our normalization convention (all quanti-ties are normalized toρs) increasingβe corre-sponds in keeping constantde and increase allother length scales. After the nonlinear accel-eration we find a saturation of the island, dueto smaller structures forming inside the magnetic island anto secondary instabilities of theKelvin-Helmoltz type, forming at the separatrix. For the whole runs the energy conservation isaccurately checked. In fact, the contribution of the sub-grid dissipation is always calculated and38th EPS Conference on Plasma Physics (2011) O4.409checked to be negative: this gives us the certainty that reconnection is physical and is not drivenby artificial effects.Acceleration in collisionless reconnection growth rates wre studied in literature in Ref. [10]and found to be linked to the presence of narrow current layers. Also, in resistive reconnectionnonlinear acceleration was found to be associated with turbulence forming around the X-point(see e.g. Ref. [12]). In our simulations, the novel result isthat the nonlinear acceleration isreaching values one order of magnitude higher than the linear growth rates, assuming the form ofexplosive reconnection. Current singularities are confirmed to be present at the separatrix duringthe nonlinear phase of reconnection. The study of their sizeand importance in the reconnectiondynamics is still in progress.AcknowledgmentsThis work was carried out in a collaboration with the Max-Planck institute for solar systemresearch at Katlenburg-Lindau, Germany. The authors thankFrancesco Pegoraro, FrancescoCalifano, Daniele Del Sarto and Tiago Ribeiro for enlightening discussions.References[1] H.P. Furth, J. Killeen and M.N. Rosenbluth, Phys. Fluids6, 459 (1963)[2] A.W. Edwardset al., Phys. Rev. Lett.57, 210 (1986)[3] F. Porcelli, Phys. Rev. Lett.66, 425 (1991)[4] B.D. Scott and F. Porcelli, Phys. Plasmas11(12), 5468 (2004)[5] A. Biancalani and B.D. Scott, 37th EPS Conf. on Plasma Physics (Dublin, Ireland, 21-25June 2010) Paper P4.108 http://ocs.ciemat.es/EPS2010PAP/pdf/P4.108.pdf[6] D. Grasso, E. Tassi and F.L. Waelbroeck, Phys. Plasmas17, 082312 (2010)[7] D. Borgogno, D. Grasso, F. Porcelli, F. Califano, F. Pegorar and D. Farina, Phys. Plasmas12, 032309 (2005)[8] M. Ottaviani and F. Porcelli, Phys. Rev. Lett.71, 3802 (1993)[9] D. Grasso, F. Califano, F. Pegoraro and F. Porcelli, Phys. Rev. Lett.86(22), 5051 (2001)[10] A. Bhattacharjee, K. Germaschewski, and C. S. Ng, Phys.Plasmas12, 042305 (2005)[11] B.D. Scott, Pl. Phys. Control. Fus.45, A385 (2003)[12] N.F. Loureiro et al., Mon. Not. R. Astron. Soc.399, L146 (2009)38th EPS Conference on Plasma Physics (2011) O4.409

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Nonlinear growth acceleration in gyrofluid simulations of collisionless reconnection

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