Abstract
In the past, longtime evolution of an initial perturbation in collisionless Maxwellian plasma (q = 1) has been simulated numerically. The controversy over the nonlinear fate of such electrostatic perturbations was resolved by Manfredi [Phys. Rev. Lett. 79, 2815–2818 (1997)] using longtime simulations up to . The oscillations were found to continue indefinitely leading to BernsteinGreeneKruskal (BGK)like phasespace vortices (from here on referred as “BGK structures”). Using a newly developed, high resolution 1D VlasovPoisson solver based on piecewiseparabolic method (PPM) advection scheme, we investigate the nonlinear Landau damping in 1D plasma described by toy qdistributions for long times, up to . We show that BGK structures are found only for a certain range of qvalues around q = 1. Beyond this window, for the generic parameters, no BGK structures were observed. We observe that for values of where velocity distributions have long tails, strong Landau damping inhibits the formation of BGK structures. On the other hand, for where distribution has a sharp fall in velocity, the formation of BGK structures is rendered difficult due to high wave number damping imposed by the steep velocity profile, which had not been previously reported. Wherever relevant, we compare our results with past work.
The authors would like to thank A. Sen for careful reading of the manuscript. The authors are also grateful to P. K. Kaw for bringing to our attention Ref. 14. We also thank the Anonymous Referee for pointing out Ref. 23.
I. INTRODUCTION
II. THE NUMERICAL CODE
III. SIMULATIONS
A. Benchmarking the VlasovPoisson solver for linear Landau damping
B. Nonlinear Landau damping
1. Case q = 1
2. Case
3. Case
IV. SUMMARY
Key Topics
 Plasma BGK modes
 33.0
 Entropy
 25.0
 Cumulative distribution functions
 19.0
 Rotating flows
 16.0
 Maxwell equations
 13.0
Figures
Comparison of the “analytically” obtained and simulated values of the damping rate γ varying with q (k = 0.4, ).
Comparison of the “analytically” obtained and simulated values of the damping rate γ varying with q (k = 0.4, ).
Plot of damping rate γ as a function of q for values of k = 0.4, 0.7, 1.2. One can see that as k increases, the values of γ are higher than those for a lower k.
Plot of damping rate γ as a function of q for values of k = 0.4, 0.7, 1.2. One can see that as k increases, the values of γ are higher than those for a lower k.
A run corresponding to Run I of Manfredi, ^{ 5 } who had shown the oscillations to continue till t = 1600. We extended the run till t = 5000. The vertical grey line indicates the duration of Manfredi's simulations. Notice the continuation of the oscillations.
A run corresponding to Run I of Manfredi, ^{ 5 } who had shown the oscillations to continue till t = 1600. We extended the run till t = 5000. The vertical grey line indicates the duration of Manfredi's simulations. Notice the continuation of the oscillations.
For q = 1, at t = 5000, the phasespace vortex can be seen around at v = 3.21.
For q = 1, at t = 5000, the phasespace vortex can be seen around at v = 3.21.
Plot of relative entropy S_{rel} with time. The vertical line represents the duration of Manfredi's simulation.
Plot of relative entropy S_{rel} with time. The vertical line represents the duration of Manfredi's simulation.
Plots for the amplitude of the first harmonic of the electric field E _{1} with time for Set I. One can notice that the oscillatory structures are not found for . Also, as damping rate increases, one can notice that the amplitude of oscillations decreases. This is similar to the result obtained by Valentini. ^{ 14 } The vertical line represents the time of Valentini's simulations.
Plots for the amplitude of the first harmonic of the electric field E _{1} with time for Set I. One can notice that the oscillatory structures are not found for . Also, as damping rate increases, one can notice that the amplitude of oscillations decreases. This is similar to the result obtained by Valentini. ^{ 14 } The vertical line represents the time of Valentini's simulations.
Plot of relative entropy S_{rel} with time for till t = 3000. The vertical line represents the time up to which Valentini's simulations were performed.
Plot of relative entropy S_{rel} with time for till t = 3000. The vertical line represents the time up to which Valentini's simulations were performed.
Plot of distribution function for the run with q = 0.85, around , at t = 3000.
Plot of distribution function for the run with q = 0.85, around , at t = 3000.
Plots for the amplitude of the first harmonic of the electric field E _{1} with time. The vertical line represents the time of Valentini's simulations. As we can see, the field has not saturated within this time.
Plots for the amplitude of the first harmonic of the electric field E _{1} with time. The vertical line represents the time of Valentini's simulations. As we can see, the field has not saturated within this time.
Plot of relative entropy S_{rel} with time for . It can be seen that the entropy saturates within t = 3000. The vertical line represents the time up to which Valentini's simulations were performed.
Plot of relative entropy S_{rel} with time for . It can be seen that the entropy saturates within t = 3000. The vertical line represents the time up to which Valentini's simulations were performed.
Plot of distribution function for the run with q = 1.15, around , at t = 3000.
Plot of distribution function for the run with q = 1.15, around , at t = 3000.
Plot of velocity distribution function at t = 3000 comparing cases with .
Plot of velocity distribution function at t = 3000 comparing cases with .
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