1. Electron behaviour in three-dimensional collisionless magnetic reconnection A. Perona 1, D. Borgogno 2, D. Grasso 2,3 1 CFSA, Department of Physics, University of Warwick, UK 2 Burning Plasma Research Group, Politecnico di Torino, Italy 3 Istituto dei Sistemi Complessi-CNR, Roma, Italy

2. Aim of this project, still in progress, is to investigate the behaviour of an electron population during the evolution of a spontaneous collisionless magnetic reconnection event reproduced by a fluid formulation in a three-dimensional geometry. In this 3D setting the magnetic field lines become stochastic when islands with different helicities are present 1. The reconstruction of the test electron momenta, in particular, can assess the small scale behaviour shown by the fluid vorticity and by the current density during the nonlinear phase of the reconnection process even in the presence of chaoticity. We present here preliminary results of the numerical tool developed on this purpose. 1 D. Borgogno et al., “Aspects of three-dimensional magnetic reconnection”, Phys. Plasmas, Vol. 12, 032309, (2005). Abstract 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011

3. The fluid model for collisionless magnetic reconnection 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011  We consider a fluid model that describes drift-Alfvén perturbations in a plasma immersed in a strong, uniform, externally imposed magnetic field.  Effects related to the electron temperature, through the sonic Larmor radius, and to the electron density, through the electron inertia, are retained.  Modes with different helicities can be present. They evolve independently during the linear phase, while a strong interaction occurs during the nonlinear stage of the process.  The set of equations is closed by assuming constant temperatures for both the ions and the electrons.

4. 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 The fluid model for collisionless magnetic reconnection sound Larmor radiuselectron skin depth B = B 0 e z +  ( x,y,z,t)  e z magnetic field with B 0 =const. and | B 0 |>>| B  | (  = 8  p 2 / B 2 <<1),  magnetic flux function,  stream function. Two scale lengths: Fluid velocity: v  = e z +  T. J. Schep, F. Pegoraro, B. N. Kuvshinov, Phys. Plasmas, Vol. 1, 2843, (1994)

5. We adopt a static, linearly unstable magnetic equilibrium Single-helicity initial perturbation The magnetic reconnection model: simulation of the reconnection process 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 Island width vs time linear nonlinear 

6. Single-helicity case (k z  0): the ‘fluid’ parallel electric field 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 Linear phase: monopole structure peaked at the X-point of the magnetic island.

7. The electron model 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 In order to verify whether the parallel electric field generated during the reconnection leads to suprathermal energetic generation, a relativistic Hamiltonian formulation* of the electron guiding-center dynamics has been chosen. The unperturbed relativistic guiding-centre phase-space Lagrangian can be written in terms of the guiding-centre coordinates as where * A. J. Brizard, A. A. Chan, Phys. Plasmas 6, 4548 (1999)

8. The electron model 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 magnetic moment: The relativistic Hamiltonian is The equations of motion follow from the Hamiltonian in the usual manner where the conjugate momenta to the y and z coordinates are given by

9. The electron equations 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011

10. The  f approach  A  f approach has been adopted in order to reduce the number of particles required while resolving small fluctuations in the electron distribution function.  The distribution function is decomposed in an analytically described background component and the remaining component In each cell labeled by i in the real space, we calculate the electron density the current density and the mean longitudinal kinetic energy of electrons 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011

11. Kinetic simulations  The electron equations have been implemented in a 3-D code, which reads at each time step the fluid fields provided by the reconnection code 1.  The kinetic code evolves the spatial coordinates, the parallel velocity and the change in the weight of the markers according to the fluid fields.  We assume periodic boundary conditions along y and z for the flux and for the stream function.  The results presented have been obtained loading 2.5x10 6 electrons in the 5-D phase space ( x,y,z,p | |, p  ).  The code has been parallelized by distributing markers among processers using Message Passing Interface (MPI) librairies. 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 1 A. Perona, L-G Eriksson, D. Grasso, Phys. Plasmas 17, 042104 (2010)

12. Kinetic simulations: single particle trajectory 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 electron electron trajectory magnetic surface The Poincaré plot of a single test electron crossing the X -point maps the magnetic island as expected. ( k z =0 case)

13. Numerical simulations: kinetic results (k z  0) 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 Linear phase: Kinetic current density Fluid current density During the linear phase the electron and fluid current density evolve in good agreement, while the velocity distribution becomes thinner and the amount of electrons in the tails increases.

14. Numerical simulations: kinetic current density 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011 fluid current * kinetic current Linear growth rate: Linear phase Nonlinear phase Good agreement with the toroidal current of small Tokamaks such as T-3 (a ≈ 10 -1 m, I t =100 kA).

15. Further steps  The behaviour of the electron population will be followed during the nonlinear phase of the single-helicity case ( k z  0) in order to confirm the agreement with the fluid results, as already observed in the 2D case.  The test electron distribution function will be reconstructed in the presence of multiple-helicity fluid fields. This will allow us to assess the structures of the electron and current density in the stochastic fields that develop during the nonlinear phase of the reconnection process.  In order to analyse the influence of the magnetic chaoticity on the electron transport we plan to compare the test electron distribution and the magnetic barriers detected through the analysis of the finite-time Lyapunov exponent ridges 1. 1 D. Borgogno et al., “Barriers in the transition to global chaos in collisionless magnetic reconnection”, in press Phys. Plasmas (2011). 14 th European Fusion Theory Conference, Frascati, September 26-29, 2011