Francesco Valentini, Pierluigi Veltri

1
First nonlinear results of the cylindric
Vlasov-Poisson code the Bernstein-Landau paradox
revisited

• Francesco Valentini, Pierluigi Veltri
• Dipartimento di Fisica, Università degli Studi
  della Calabria (Italy)
• André Mangeney
• Observatoire de Paris-Meudon (France)

2
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
ONeil regime (in which the electric field goes
on oscillating around an approximately constant
value)
ONeil regime
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
3
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
The cylindric Vlasov-Poisson code (1D-2V)
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
Landau regime
B0
B0.3
B0.0629, 0.085,0.125
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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
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
Electron plasma frequency
Electron Debye lenght
Electron thermal velocity
8
ONeil 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
ONeil regime, stronger and stronger magnetic
field
B0.001
Strange behavior DAMPED OSCILLATIONS
B0.03
Strange behavior ISOLATED ELECTROSTATIC
STRUCTURES
B0.18
Strong magnetic field UNDAMPED OSCILLATIONS
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The evolution of the distribution function (1)
Case
damped wave
t100 (a),150 (b), 200 (c), 400 (d), 600 (e),
800 (f)
The function rotates under the effect of the
magnetic field, but the perturbation in the
resonant zone become smaller and smaller, during
the rotation
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The evolution of the distribution function (2)
t100 (a),150 (b), 200 (c), 400 (d), 600 (e),
800 (f)
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
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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 case
• In the Landau regime, the numerical results are
  in agreement with previous analytical and
  numerical studies
• A strange behavior is observed in the ONeil
  regime, where the electric field is damped, in
  spite of the trapping interaction and the
  magnetic effect