Clementson, J.

Grulke, O.

Klinger, T.

Lewerentz, L.

Schneider, R.

Sydora, R.

MPG.PuRe

A gyrokinetic model for studies of magnetic reconnection in periodic andbounded plasmasL. Lewerentz1, J. Clementson2, R. Schneider1, R. D. Sydora3, O. Grulke2 and T. Klinger1,21Department of Physics, University of Greifswald, DE-17487 Greifswald, Germany2Max Planck Institute for Plasma Physics, DE-17491 Greifswald, Germany3Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1AbstractA gyrokinetic Particle-in-Cell (PIC) code to investigate magnetic reconnection in the low-βlimit is presented. With a magnetic field geometry similar to the VINETA II experiment [1],the model can be applied to study the kinetic effects of local plasma parameters and plasmaboundary conditions on magnetic reconnection.IntroductionLaboratory plasmas are confined by vessel walls. In addition to infinite model systems,boundary conditions such as the plasma sheath are important for the particle kinetics. Alsofor natural occurrences in space, e.g. many magnetic field lines in the solar atmosphere havebase points on the conducting surface of the sun [2], boundary conditions do play a role.The spatial dimensions vary from small-scale dynamics in the current sheet to large-scale globalboundary conditions which both influence each other [3]. To model such phenomena the hugetime and spatial scales involved in magnetic reconnection pose a problem.Figure 1: A schematic of the VINETA II experiment is shown (left). Theexperiment is mimicked in the slab model which is indicated on the righthand side. The model consists of two sets of coils - one pair to generatethe in-plane x-point topology and one to drive the flux surfaces. Image:courtesy of Hannes Bohlin.Gyrokinetic simulations area basic tool to studylow frequency instabili-ties, turbulence and theiranomalous transport in mag-netized plasmas on largerscales than normal kineticcodes. In order to dis-criminate the influence ofbounded currents on mag-netic reconnection a gy-rokinetic Particle-in-Cell(PIC) code [4] is adapted41st EPS Conference on Plasma Physics P1.144to study driven magnetic reconnection in the strong guide field limit. To conceive magneticreconnection the currents in the system need to be understood. Instead of calculating the fullparticle dynamics a reduced set of equations is used. The fast gyromotion is neglected and onlyfinite Larmor radius (FLR) effects are taken into account. This decreases the temporal and spa-tial demands for many problems so that they become feasible to calculate.The code closely mimics the magnetic field configuration of the VINETA II experiment [5].The setup of the linear, cylindrical device is seen in Fig. 1. The simulation intrinsically containsa strong guide field along the axial direction of the experiment. It contains two axial coils withsteady current in order to produce an x-point configuration. Two additional coils have a sinu-soidal current to perturb the magnetic x-point topology and to drive the reconnection process.The gyrokinetic modelThe PIC code solves the Vlasov-Poisson-Ampère system of equations for a low-β plasma invector potential A and canonical momentum p notation [4]. It follows the characteristics of thenonlinear electrostatic gyrokinetic Vlasov equation.Particle position and velocity of the ith particle of species s are denoted by Rsi and vsi ,respectively. The index z indicates the z-component. Using the canonical momentum psi,z =vsi,z +qsmsAz(Rsi , t) and a generalized potential Ψ(Rsi , t) = Φ̄(Rsi , t)− vsi,zAz(Rsi , t) the equations ofmotion reduce todRsidt= vsz,ib̂−∇Ψ×bB0(1)dpsi,zdt=− qsmsb∇Ψ . (2)The gyroradius is denoted by ρ and the magnetic field B is described in terms of the vectorpotential B = ∇× A. ms, qs and b̂ = b+ (Az× b)/B0 are the mass, the charge and the unitvector along the magnetic field, respectively. The notations Φ̄(R) = 〈∫Φ(r)δ (r−R−ρ)dr〉ϕand 〈〉ϕ =∮dϕ/2π are used.The electrostatic potential is calculated from the gyrokinetic Poisson equation:∇2Φ̄+ω2piω2ci∇2⊥Φ̄ =qeε0Ne∑jδ (R−Rej)+qiε0Ni∑jδ (R−Rij), (3)where ωpi is the ion plasma frequency, ∇⊥ is the gradient perpendicular to the axial magneticfield and δ is the delta distribution. The second term on the left hand side represents the ion41st EPS Conference on Plasma Physics P1.144polarization shielding effects. Ampère’s law becomes in the canonical momentum formulation−δ 2e ∇2⊥Az =1N0[(∑Nepe−∑Nipi)−({∑NeAzδ (R−Rej)}+memi{∑NiAzδ (R−Rij)})].The electron skin depth is denoted by δe = c/ωpe.The code uses three-dimensional cartesian coordinates. The 2D model applies a slab geome-try (x and y direction) with periodic boundaries in the axial z-dimension. The in-plane fields arebounded by conducting walls and particles are reflected. Along the guide field (z-direction) theboundaries are periodic for the fields and the particles leaving the system are entering the simu-lated domain from the other side. Calculations with various particle density profiles are possibleto study the magnetic field penetration [6]. A full 3D model with reflecting or absorbing con-ducting walls and logical sheath boundaries is currently being implemented. Various boundaryconditions are needed in order to calculate the nonlinear effects on the particle kinetics similarto the metal wall end plates in the linear VINETA II experiment. A necessity is a strong axialguide field to have electrons and ions fully magnetized, whereas the in-plane magnetic field isgenerated by currents in external coils parallel to the guide field. This allows to study recon-nection rates and the dynamics of the generated currents as functions of plasma density profilesand different ratios of in-plane to guide magnetic field ratios.The logical sheath modelIn order to calculate the influence of the plasma sheath effects on magnetic reconnection alogical sheath model is used. On earth most laboratory plasmas are confined within a vacuumvessel. Such bounded plasmas are in contact with the surface of a wall resulting in a plasmasheath at the plasma edge. The spatial extent of the sheath is about a few Debye lengths. Astandard electrostatic PIC code must have a grid size below the Debye length in order to resolvethe sheath potential drop. In gyrokinetics the cell size is larger. Thus, the sheath can not becalculated self-consistently. The idea of the logical sheath is not to resolve those spatial scales,but to mimic the equilibrium conditions at the plasma edge [7].If ions and electrons hit a conducting surface they recombine and are lost to the plasma. Suchsurface acts as a perfect sink for the particles. Normally, electrons in a plasma move faster thanions, so that initially the electron flux to the wall exceeds the ion flux by far. The flux imbalancewill charge up the wall negatively and generates a potential drop which cannot be resolved ina gyrokinetic model. This barrier repels electrons and attracts ions until both fluxes are equal.Thereby, the potential drop adjusts and steady-state is achieved. This flux balance is applied to41st EPS Conference on Plasma Physics P1.144all particles exceeding the extents of the simulation. Ions are streaming freely out of the systemand are absorbed if they exceed particle boundaries.Figure 2: Comparison of the logical sheath solution (blue, right axis)with the self-consistent calculation (black, left axis) of a 1D electrostaticPIC code. Shown are the electrostatic potential Φ, the particle density nand the mean velocity of the ions ui. To discriminate both solutions theaxis for the logical sheath is shifted vertically.If Ni is the number ofabsorbed ions per pusherstep, then only the fastest,up to Ni, electrons are ab-sorbed and the remainingslow particles are reflectedas if repelled by the sheathpotential drop.In the case where the num-ber of absorbed ions isgreater than the number ofelectrons hitting the parti-cle boundary all electronsare absorbed and the ex-cessive "charge" is storedfor the next particle push.The potential drop can beestimated from the speed of the fastest repelled electron ∆Φ = m2ev2max.The logical sheath has been tested separately in a one-dimensional electrostatic PIC code [8],see Fig. 2, and is now being implemented into the gyrokinetic model.References[1] H. Bohlin, A. von Stechow, K. Rahbarnia, et al., Rev. Sci. Instrum. 85(2), 023501 (2014)[2] M. Yamada, Rev. Mod. Phys. 82(1), 607 (2010)[3] A. Kuritsyn, H. Ji, et al., Geophys. Res. Lett., 34, L16106, (2007)[4] R.D. Sydora, Phys. Plasmas 8(5), 1929 (2001)[5] A. von Stechow, H. Bohlin, K. Rahbarnia, et al., Europhys. Conf. Abs. 38, P4.111 (2014)[6] J. Clementson, L. Lewerentz, O. Grulke, et al., Europhys. Conf. Abs. 38, P4.118 (2014)[7] S.E. Parker, R.J. Procassini, et al., J. Comput. Phys., 104(2), 41 (1993)[8] K. Matyash, R. Schneider, Contrib. Plasma Phys., 44, (2004)41st EPS Conference on Plasma Physics P1.144

A gyrokinetic model for studies of magnetic reconnection in periodic and bounded plasmas

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