Download The first results of the cilindric Vlasov

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)
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)
O’Neil regime The evolution of the wave was
studied as a “bifurcation
problem” and the value of the
critical
perturbation
was
calculated analytcally.
Landau regime
23-28 September 2003
For initial perturbations
greater than the critical
amplitude,
the
Landau
damping is stopped
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.
23-28 September 2003
Basic Processes in Turbulent Plasmas
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) :
f
f
f
1 
1 
 vx
 B0

[v Ev ( ) f ] 
[ E ( ) f ]  0
t
x
 v v
v 

E x

x

f ( x, v ,  )dv d  1
The cylindric geometry is
used in the velocity space
to describe the rotation
of the particles, around
the direction of the
magneti field
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
B  B 0 eˆ z
v   v x2  v 2y

k  keˆ x
vy 
  arctan  

v  v x eˆ x  v y eˆ y
 vx 
Basic Processes in Turbulent Plasmas

Landau regime
B=0
B=0.3
B=0.0629, 0.085,0.125
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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:
2 2 

k
vth  i t

nL (t )  n0 1  2 2 e cosr t 



p


n(t )  2n0
k 2vth2
2
p
e
2 2
 (3 / 8) k 2 vth

for
t
2
B
cos(kvth ) for t  2   t  2
B
B
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))
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.
B  0.0
t max  650 p1
B  10  5
In the first box (a), in the
unmagnetized
case,
we
observe trapping oscillations,
due to the nonlinear waveparticle 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
23-28 September 2003
Basic Processes in Turbulent Plasmas
O’Neil regime, stronger and stronger magnetic
field
B=0.001
Strange behavior:
DAMPED
OSCILLATIONS
B=0.03
Strange behavior:
ISOLATED
ELECTROSTATIC
STRUCTURES
B=0.18
Strong magnetic field:
UNDAMPED
OSCILLATIONS
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Basic Processes in Turbulent Plasmas
The evolution of the distribution function (1)
Case:
B  0.001
damped wave
t=100 (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
23-28 September 2003
Basic Processes in Turbulent Plasmas
The evolution of the distribution function (2)
Case:
B  0.001
damped wave
t=100 (a),150 (b), 200 (c),
400 (d), 600 (e), 800 (f)
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
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 waveparticle 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 O’Neil regime,
where the electric field is damped, in spite of the
trapping interaction and the magnetic effect
23-28 September 2003
Basic Processes in Turbulent Plasmas