1. Francesco Valentini, Pierluigi Veltri Dipartimento di Fisica, Università degli Studi della Calabria (Italy) Dipartimento di Fisica, Università degli Studi della Calabria (Italy) André Mangeney Observatoire de Paris-Meudon (France) First nonlinear results of the cylindric Vlasov-Poisson code: the Bernstein-Landau paradox revisited

2. 23-28 September 2003 Basic Processes in Turbulent Plasmas Unmagnetized case: critical initial states (Lancellotti and Dorning, 1998) Lancellotti and Dorning showed that there exist “critical initial states” that mark the transition between the Landau regime (in which the wave is definitively damped to zero) ant the O’Neil regime (in which the electric field goes on oscillating around an approximately constant value) The evolution of the wave was studied as a “bifurcation problem” and the value of the critical perturbation was calculated analytcally. For initial perturbations greater than the critical amplitude, the Landau damping is stopped Landau regime O’Neil regime

3. 23-28 September 2003 Basic Processes in Turbulent Plasmas Magnetized case: the Bernstein-Landau paradox (Landau regime) The essence of the paradox: Electrostatic waves in unmagnetized plasma conlisionless Landau damping Bernstein modes in magnetized plasma (perpendicular to the magnetic field) exactly undamped, indipendent of the strength of the magnetic field.

4. 23-28 September 2003 Basic Processes in Turbulent Plasmas The cylindric Vlasov-Poisson code (1D-2V) the ions cannot partecipate in the high frequency plasma oscillations and just form a uniform background charge) The basic equations for the temporal evolution of the electron distribution function (the ions cannot partecipate in the high frequency plasma oscillations and just form a uniform background charge) : The cylindric geometry is used in the velocity space to describe the rotation of the particles, around the direction of the magneti field 

5. 23-28 September 2003 Basic Processes in Turbulent Plasmas Landau regime B=0 B=0.3 B=0.0629, 0.085,0.125

6. 23-28 September 2003 Basic Processes in Turbulent Plasmas Sukhorukov and Stubbe theory (1997) They obtained an analitical solution for perturbations perpendicular to the magnetic field, which is a generalizzation of the well-known Landau work to magnetized plasmas. In the approximation of large wave length, they obtained: for They showed that each cyclotron period the magnetic field raises the electron density oscillations, and at large time these are completely undamped (the results are in agreement with Baldwin and Rowlands (1966))

7. 23-28 September 2003 Basic Processes in Turbulent Plasmas Electron plasma frequency Electron Debye lenght Electron thermal velocity

8. 23-28 September 2003 Basic Processes in Turbulent Plasmas O’Neil regime, weak magnetic field In the case of weak magnetic field, we expect to observe a behavior similar to the unmagnetized case. In the first box (a), in the unmagnetized case, we observe trapping oscillations, due to the nonlinear wave- particle interaction. In the second one (b), it is visible a weak magnetic effect on the evolution of the electric field The behavior is qualitatively the same

9. 23-28 September 2003 Basic Processes in Turbulent Plasmas O’Neil regime, stronger and stronger magnetic field B=0.001 B=0.03 B=0.18 Strong magnetic field: UNDAMPED OSCILLATIONS Strange behavior: DAMPED OSCILLATIONS Strange behavior: ISOLATED ELECTROSTATIC STRUCTURES

10. 23-28 September 2003 Basic Processes in Turbulent Plasmas The evolution of the distribution function (1) Case:damped wave The function rotates under the effect of the magnetic field, but the perturbation in the resonant zone become smaller and smaller, during the rotation t=100 (a),150 (b), 200 (c), 400 (d), 600 (e), 800 (f)

11. 23-28 September 2003 Basic Processes in Turbulent Plasmas The evolution of the distribution function (2) Case:damped wave During the rotation, the shape of the distribution becomes maxwellian; there is not wave-particle interaction any more, and the trapping is not able to sustains the oscillations t=100 (a),150 (b), 200 (c), 400 (d), 600 (e), 800 (f)

12. 23-28 September 2003 Basic Processes in Turbulent Plasmas Conclusions The nonlinear evolution of electrostatic waves in a magnetized plasma is investigated, using a cylindric Vlasov-Poisson code, in order to describe the wave- particle interaction in the magnetized caseThe nonlinear evolution of electrostatic waves in a magnetized plasma is investigated, using a cylindric Vlasov-Poisson code, in order to describe the wave- particle interaction in the magnetized case In the Landau regime,the numerical results are in agreement with previous analytical and numerical studies In the Landau regime, the numerical results are in agreement with previous analytical and numerical studies A strange behavior is observed in the O’Neil regime, where the electric field is damped, in spite of the trapping interaction and the magnetic effect A strange behavior is observed in the O’Neil regime, where the electric field is damped, in spite of the trapping interaction and the magnetic effect