Francesco Valentini Francesco Valentini , Pierluigi Veltri , Pierluigi Veltri Dipartimento di Fisica, Università degli Studi della Dipartimento di Fisica, Università degli Studi della Calabria (Italy) Calabria (Italy) André Mangeney André Mangeney Observatoire de Paris-Meudon (France) Observatoire de Paris-Meudon (France) First nonlinear results of First nonlinear results of the cylindric Vlasov- the cylindric Vlasov- Poisson code: the Poisson code: the Bernstein-Landau paradox Bernstein-Landau paradox revisited revisited
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Francesco ValentiniFrancesco Valentini, Pierluigi Veltri, Pierluigi Veltri Dipartimento di Fisica, Università degli Studi della Calabria (Italy)Dipartimento di Fisica, Università degli Studi della Calabria (Italy)
André MangeneyAndré MangeneyObservatoire de Paris-Meudon (France)Observatoire de Paris-Meudon (France)
First nonlinear results of the First nonlinear results of the cylindric Vlasov-Poisson code: cylindric Vlasov-Poisson code: the Bernstein-Landau paradox the Bernstein-Landau paradox
revisitedrevisited
23-28 September 200323-28 September 2003 Basic Processes in Turbulent PlasmasBasic Processes in Turbulent Plasmas
Unmagnetized case: critical initial states Unmagnetized case: critical initial states (Lancellotti and Dorning, 1998)(Lancellotti and Dorning, 1998)
Lancellotti and Dorning showed that there exist “critical initial Lancellotti and Dorning showed that there exist “critical initial states” that mark the transition between the states” that mark the transition between the Landau regime Landau regime (in (in which the wave is definitively damped to zero) ant the which the wave is definitively damped to zero) ant the O’Neil O’Neil regime regime (in which the electric field goes on oscillating around an (in which the electric field goes on oscillating around an approximately constant value)approximately constant value)
The evolution of the wave The evolution of the wave was studied as a was studied as a “bifurcation problem” and “bifurcation problem” and the value of the critical the value of the critical perturbation was calculated perturbation was calculated analytcally. analytcally.
For initial perturbations For initial perturbations greater than the critical greater than the critical amplitude, the Landau amplitude, the Landau damping is stoppeddamping is stopped
Landau regime
O’Neil regime
23-28 September 200323-28 September 2003 Basic Processes in Turbulent PlasmasBasic Processes in Turbulent Plasmas
Magnetized case: the Bernstein-Landau Magnetized case: the Bernstein-Landau paradoxparadox
(Landau regime)(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.
23-28 September 200323-28 September 2003 Basic Processes in Turbulent PlasmasBasic Processes in Turbulent Plasmas
The cylindric Vlasov-Poisson code (1D-2V)The cylindric Vlasov-Poisson code (1D-2V)
1),,(
0])([1
])([1
0
ddvvxfxE
fEv
fEvvv
fB
xf
vtf
x
vx
The basic equations for the temporal evolution of the electron distribution function (the ions cannot partecipate in the high the ions cannot partecipate in the high frequency plasma oscillations and just form a uniform frequency plasma oscillations and just form a uniform background charge)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
yyxx
x
z
evevv
ekk
eBB
ˆˆ
ˆ
ˆ0
x
y
yx
v
v
vvv
arctan
22
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Landau regime Landau regime
B=0
B=0.3
B=0.0629, 0.085,0.125
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Sukhorukov and Stubbe theory (1997)Sukhorukov and Stubbe theory (1997)
tevk
ntn rt
p
thL
i
cos21)(2
22
0
B
t2
)cos(2)(222)8/3(
2
22
0
th
vk
p
th kvevk
ntn th
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
B
t2
forB
t 2
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))
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O’Neil regime, weak magnetic fieldO’Neil regime, weak magnetic field
510
0.0
B
B
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 fieldThe behavior is qualitatively the same
1max 650 pt
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O’Neil regime, stronger and stronger magnetic O’Neil regime, stronger and stronger magnetic field field
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The evolution of the distribution function (2)The evolution of the distribution function (2)
001.0BCase: 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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ConclusionsConclusions
•The nonlinear evolution of electrostatic waves in a The nonlinear evolution of electrostatic waves in a magnetized plasma is investigated, using a cylindric magnetized plasma is investigated, using a cylindric Vlasov-Poisson code, in order to describe the wave-Vlasov-Poisson code, in order to describe the wave-particle interaction in the magnetized case particle interaction in the magnetized case
• In the Landau regime,In the Landau regime, the numerical results are in the numerical results are in agreement with previous analytical and numerical agreement with previous analytical and numerical studiesstudies
• A strange behavior is observed in the O’Neil regime, A strange behavior is observed in the O’Neil regime, where the electric field is damped, in spite of the where the electric field is damped, in spite of the trapping interaction and the magnetic effecttrapping interaction and the magnetic effect
