1. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Francesco Valentini Dipartimento di Fisica, Università della Calabria Rende (CS) – Italy valentin@fis.unical.it Nonlinear wave-particle interaction in collisionless plasmas: unmagnetized and magnetized case In collaboration with: P. Veltri, V. Carbone, A. Mangeney

2. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Outline Nonlinear unmagnetized case: Landau damping saturation, asymptotic behavior of the electrostatic oscillations Numerical simulations versus O’Neil theory and Isichenko analytical results Particle dynamics: phase space flights and Lyapunov exponents of the phase space trajectories Magnetized case: trapping oscillation and cyclotron motion Velocity space evolution of the distribution function and isolated electrostatic structures Conclusions

3. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Unmagnetized case: Vlasov-Poisson 1D-1V Initial conditions Temporal evolution of the electric field Envelope oscillation: T=100

4. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 In nonlinear regime, the damping rate of the wave starts oscillating about zero on the trapping time scale with a decreasing amplitude. O’Neil theory Isichenko conjecture Asymptotic algebraic damping law: the motion in the potential well is not simply oscillatory-like O’Neil theory (1965) and Isichenko results (1996) Escaped particle Trapped particle

5. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Lagrangian study of the phase space trajectories We integrate the motion equations for 500-4000 test particles: The electric field is evaluated by solving numerically the Vlasov- Poisson system of equations Phase space trajectories Lyapunov exponents The system dynamics is followed up to Poincaré sections

6. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Phase space flights We observe two kinds of trajectories in the resonant region: A particle trapped in the potential well A particle which escapes from the potential well Phase space flight

7. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Poincaré sections and Lyapunov exponents

8. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Statistical study of the phase space flights Two significant peaks are clearly visible in the spatial and in the temporal histogram. The phase space flights are of limited extent and are not able to dissipate the energy of the wave The second peak is around t=100 Trapped particles Re-trapped particles

9. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Physical interpretation of the envelope oscillation

10. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Magnetized case: Vlasov-Poisson (1D-2V) The cylindrical geometry is adopted in the velocity space in order to describe the rotation of the particles, around the direction of the magnetic field

11. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Linear analysis: Sukhorukov and Stubbe (1997) Using the Landau approach, in the approximation of large wave length, they obtained: for At large time the oscillations are completely undamped (the results are in agreement with Baldwin and Rowlands (1966)) K=0.4, B=0.04, 0.06, 0.1K=1, B=0.25

12. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Trapping time versus cyclotron period Trapping oscillation and cyclotron motion y X Wavelenght Larmor radius Wave-particle interaction No Wave-particle interaction ??

13. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Weak magnetic field

14. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Stronger and stronger magnetic field Strong magnetic field: UNDAMPED OSCILLATIONS DAMPED OSCILLATIONS ISOLATED ELECTROSTATIC STRUCTURES  

15. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 The evolution of the distribution function t=100 (a),150 (b), 200 (c), 400 (d), 600 (e), 800 (f)

16. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 The evolution of the distribution function t=800

17. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Electrostatic structures in the solar wind

18. IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 The introduction of an external uniform magnetic field strongly modifies the wave-particle interaction. From our simulations the ratio between the trapping time and the cyclotron period seems to determine the evolution of the electrostatic oscillations Unexpected damped oscillations and isolated electrostatic structures appear for intermediate values of the external field, as a result of the competition between the trapping motion and the magnetic effect Conclusions Nonlinear wave-particle interaction is analyzed using a lagrangian approach. Statistical analysis of the particle phase space trajectories allows us to conclude that there exists a zone, at the border of the resonant region, in which the system dynamics becomes chaotic. Resonant particles can escape from the potential well and perform flights in the phase space of limited extent, whose characteristic length is strictly related to the long time behavior of the solution