Undamped electrostatic plasma waves: theory and Vlasov ...

Undamped electrostatic plasma waves: theory and Vlasov simulations

Francesco [email protected]

D. Perrone, F. Califano, F. Pegoraro, P. Veltri, P. J. Morrison, T. M. O'Neil

SIF, XCVIII Congresso Nazionale, Napoli 17-21 Settembre 2012


Outline

- The electrostatic plasma waves

- Collisionless dissipation of plasma waves: Vlasov equation and Landau theory

- Saturation of wave damping: the nonlinear theory by O'Neil

- How to suppress Landau damping and to get stable plasma oscillations, the Electron Acoustic Waves (EAW)

- Numerical simulations and experimental results

- Applications to the design phase of space missions

- Conclusive remarks

A simple view: background of fixed ions+ + + + + + + ++ + + + + + + +

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x

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Electrostatic plasma waves



The kinetic point of view for a collision-free plasma

Electron distribution function

Average number of particles in

Particle density

Landau solution in regime of small amplitude oscillations

Electrostatic fluctuations

L. D. Landau, J. Phys. USSR 10, 25 (1946)

The frequency of Langmuir waves

Vlasov-Poisson equations


The meaning of Landau damping

The resonant particles surf the wave and exchange energy with the wave.

For monotonically decreasing velocity distributions the number of resonant particles slower than the wave is larger than the number of resonant particles faster than the wave

The wave energy is dissipated


Nonlinear regime: particle trapping

T. M. O'Neil, Phys. Fluids 8, 2255 (1965)

Nonlinear saturation of Landau damping

The process of particle trapping producesa velocity diffusion that leads to the generation of a flat region (called plateau) in the particle velocity distribution at the wave phase speed

Trapped particle phase space trajectories

The search for undamped solutions to the Landau problem

➢Assume a plateau in the equilibrium velocity distribution at the wave phase speed thus neglecting the imaginary part of the dielectric function

➢Assume the plateau is so small that the equilibrium velocity distribution can be considered as aMaxwellian

➢Solve for the roots of the real part of the dielectric function

J. P. Holloway and J. J. Dorning, Phys. Rev. A 44, 3856–3868 (1991)

Electron Acoustic Waves (EAW)


The excitation of an EAW in a numerical experiment

F. Valentini, T. M. O'Neil and D. H. Dubin, Phys. Plasmas 13, 052303 (2006)

The strategy:➢We solve numerically the Vlasov-Poisson equations through a Eulerian scheme➢An external driver electric field is applied to an initially Maxwellian plasma of electrons and fixed ions.

The resonance peak

We noticed that:➢The resonance velocity is shifted more and more,with respect to the analytical predictions, as the strength of the external driver increases

➢The width of the resonance peak gets larger and larger,as the strength of the external driver increases

Analogous results can be obtained at lower frequency for the proton branchF. Valentini, F. Califano, D. Perrone, F. Pegoraro and P. Veltri, Phys. Rev. Lett. 106, 165002 (2011), Plasma Phys. Control. Fusion 53, 105017 (2011)


The excitation of the EAW in a real experiment

F. Anderegg, C. F. Driscoll, D. H. E. Dubin, T. M. O’Neil, and F. Valentini, Phys. Plasmas 16, 055705 (2009)

The electron trap at UC San Diego

Wave excitationExperimental thumb curve

F. Anderegg, C.F. Driscoll, D.H.E. Dubin, and T.M. O'Neil, Phys. Rev. Lett. 102, 095001 (2009)


Correction to the theory of undamped plasma waves

F. Valentini, D. Perrone, F. Califano, F. Pegoraro, P. Veltri, P. J. Morrison, and T. M. O’Neil, Phys. Plasmas 19, 092103 (2012)

At variance with previous analytical treatments, here we retain the contribution of the small plateau

As observed in real experiments, undamped waves can be obtained in a region much larger than the thumb diagram of previous treatments

Analytical results Numerical confirmation


Applications to the construction of technological devices

S. Mudaliar and V. I. Sotnikov, IEEE Transaction on Antennas and Propagation 99, 1, (2012)

This region can be covered byElectron Acoustic-like Waves


Conclusive remarks

✔We designed a strategy to get stable plasma waves that can survive Landau damping at a frequency that is a fraction of the electron plasma frequency. The same strategy can be used in numerical as well as in real laboratory experiments

✔By analyzing the roots of the dielectric function retaining the contribution of the plateau region in the particle velocity distribution, one gets unstable plasma waves in a wide range of frequencies and wavenumbers

✔Since the milestone work by Landau in 1946 the collisionless dissipation of plasma waves has been considered a fascinating physical phenomenon. The study of linear and nonlinear Landau damping has nowadays come back powerfully to the attention of the plasma scientific community with new results and implications.

✔Cedric Villani received the Fields Medal in 2010 for his studies on Landau damping

✔This work started as an old-fashioned academic exercise, but then it turned out that it has an unexpected interdisciplinary character, many implications in real experiments and useful applications in technology


Undamped electrostatic plasma waves: theory and Vlasov ...

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