Particle acceleration in a turbulent electric field produced by 3D reconnection

1. University of Thessaloniki Particle acceleration in a turbulent electric field produced by 3D reconnection Marco Onofri

2. Presentation of the work An MHD numerical code is used to produce a turbulent electric field form magnetic reconnection in a 3D current sheet The fields are frozen during the motion of the particles Particles are injected in the fields obtained from the MHD simulation at different times

3. Incompressiblecartesiancode We study the magnetic reconnection in an incompressible plasma in three-dimensional slab geometry. Different resonant surfaces are simultaneously present in different positions of the simulation domain and nonlinear interactions are possible not only on a single resonant surface, but also between adjoining resonant surfaces. The nonlinear evolution of the system is different from what has been observed in configurations with an antiparallel magnetic field.

4. Geometry The MHD incompressible equations are solved to study magnetic reconnection in a current layer in slab geometry: Periodic boundary conditions along y and z directions Dimensions of the domain: -lx < x < lx, 0 < y < 2ly, 0 < z < 2lz

5. Description of the simulations: equations and geometry Incompressible, viscous, dimensionless MHD equations: B is the magnetic field, V the plasma velocity and P the kinetic pressure. and are the magnetic and kinetic Reynolds numbers.

6. Description of the simulations: the initial conditions Equilibrium field: plane current sheet (a= c.s. width) Incompressible perturbations superposed:

7. Description of the simulations: the numerical code Boundary conditions: • periodic boundaries along y and z directions • in the x direction: Numerical method: • FFT algorithms for the periodic directions (y and z) • fourth-order compact difference scheme in the inhomogeneous direction (x) • third order Runge-Kutta time scheme • code parallelized using MPI directives

8. Numerical results: characteristics of the runs Magnetic reconnection takes place on resonant surfaces defined by the condition: where is the wave vector of the perturbation. The periodicity in y and z directions imposes the following conditions:

9. Testing the code: growth rates The numerical code has been tested by comparing the growth rates calculated in the linear stage of the simulation with the growth rates predicted by the linear theory. Two-dimensional modes

10. Testing the code: energy conservation The conservation of energy has been tested according to the following equation:

11. Numerical results: instability growth rates Parameters of the run: Resonant surfaces on both sides of the current sheet Perturbed wavenumbers: -4  m  4, 0  n  12

12. Numerical results: B field lines and current at y=0

13. Numerical results: time evolution of the spectra

14. Spectrum anisotropy where and are the r.m.s. of the wave vectors weighted by the specrtal energy: The energy spectrum is anisotropic, developping mainly in one specific direction in the plane, identified by a particular value of the ratio, which increases with time. This can be expressed by introducing an anisotropy angle:

15. Spectrum anisotropy

16. Spectrum anisotrpy Contour plots of the total energy spectrum at different times. The straight lines have a slope corresponding to the anisotropy angle.

17. Three-dimensional structure of the electric field Isosurfaces of the current at t=400

18. Time evolution of the electric field The surfaces are drawn for E=0.005 from t=200 to t=300

19. Time evolution of the electric field The surfaces are drawn for E=0.01 from t=300 to t=400

20. Distribution function of the electric field P(E) t=300 t=200 t=400 E

21. Fractal dimension of the electric field

22. Particle acceleration Relativistic equations of motions: The equations are solved with a fourth-order Runge Kutta adaptive step-size scheme. The electric and magnetic field are interpolated with local 3D interpolation to provide the field values where they are needed

23. Parameters of the run: Number of particles: 10000 Maximum time: 0.2 s Magnetic field: 100 G Particle density: Size of the box: Particle temperature: 100 eV

24. Kinetic energy as a function of time <E> (erg) t=400 t=300 Magnetic energy: Total final energy of particles: t (s)

25. Trajectories of particles in the xz plane The trajectories of the particles are simpler near the edges than in the center, where they are accelerated by the turbulent electric field

26. Initial and final distribution function of particles for different electric fields t=300 t=400 P(E) E (erg) E (erg)

27. Summary • We observe coalescence of magnetic islands in the center of the current sheet and prevalence of small scale structures in the lateral regions • We have developped a MHD numerical code to study the evolution of magnetic reconnection in a 3D current sheet • The spectrum of the fluctuations is anisotropic, it develops mainly in one specific direction, which changes in time • The electric field is fragmented and its fractal dimesion increases with time • The particles injected in the fields obtained from the MHD simulation are strongly accelerated • The results are not reliable when the energy gained by the particles becomes bigger than a fraction of the energy contained in the fields