IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Francesco Valentini...

IV Congresso Italiano di Fisica del Plasma

Firenze, 12-13 Gennaio 2004

Francesco ValentiniDipartimento di Fisica, Università della Calabria

Rende (CS) – [email protected]

Nonlinear wave-particle interaction in collisionless plasmas: unmagnetized and

magnetized case

Francesco Valentini Francesco Valentini Dipartimento di Fisica, Dipartimento di Fisica,

Università della CalabriaUniversità della Calabria

In collaboration with:

P. Veltri, V. Carbone, A. Mangeney

mailto:[email protected]




Università della CalabriaUniversità della Calabria Outline

• Nonlinear unmagnetized case: Landau damping saturation, asymptotic behavior of the electrostatic oscillations

• Numerical simulations versus O’Neil theory and Isichenko analytical results

• Particle dynamics: phase space flights and Lyapunov exponents of the phase space trajectories

• Magnetized case: trapping oscillation and cyclotron motion

• Velocity space evolution of the distribution function and isolated electrostatic structures

• Conclusions



)2/exp(2

1)(

)]cos(1)[()0,,(

22thM

M

vvvf

kxAvftvxf

th

Dx

Dx

vvA

A

Lk

L

2.3

5.41

05.0

4.02

5.15

1


Università della CalabriaUniversità della CalabriaUnmagnetized case: Vlasov-Poisson 1D-1V

th

D

pe

v

vv

xx

tt

1

Initial conditions

Temporal evolution of the electric field

Envelope oscillation: T=100

)(4

0

0 fdvnex

E

v

f

m

eE

x

fv

t

f



1A



In nonlinear regime, the In nonlinear regime, the damping rate of the wave starts damping rate of the wave starts oscillating about zero on the oscillating about zero on the trapping time scaletrapping time scale with a with a decreasing amplitudedecreasing amplitude..

O’Neil theory

Isichenko conjecture Asymptotic algebraic Asymptotic algebraic dampingdamping law: the motion in the potential law: the motion in the potential well is not simply oscillatory-well is not simply oscillatory-likelike

O’Neil theory (1965) and Isichenko results (1996)

2A

Escaped particle

Trapped particle



Lagrangian study of the phase space trajectories

m

txeE

dt

dv

vdt

dx

),(

We integrate the motion equations for 500-4000 test We integrate the motion equations for 500-4000 test particles:particles:

The electric field is evaluated The electric field is evaluated by solving numerically the by solving numerically the Vlasov-Poisson system of Vlasov-Poisson system of equationsequations

Phase space trajectoriesPhase space trajectories

Lyapunov exponentsLyapunov exponents



The system dynamics is followed up to

11200 pet

Poincaré sectionsPoincaré sections



Phase space flights

We observe two kinds We observe two kinds of trajectories in the of trajectories in the resonant region: resonant region:

A particle trapped A particle trapped in the potential in the potential wellwell

A particle which A particle which escapes from escapes from the potential the potential wellwell

Phase space flight






Università della CalabriaUniversità della Calabria Poincaré sections and Lyapunov exponents



Statistical study of the phase space flights

Two significant peaks Two significant peaks are clearly visible in the are clearly visible in the spatial and in the spatial and in the temporal histogram.temporal histogram.

The phase space flights The phase space flights are of limited extent are of limited extent and are not able to and are not able to dissipate the energy of dissipate the energy of the wavethe wave

The second peak The second peak is around is around t=100t=100

Trapped particles Re-trappedparticles






Università della CalabriaUniversità della CalabriaPhysical interpretation of the envelope oscillation



Magnetized case: Vlasov-Poisson (1D-2V)

1),,(

0])([1

])([1

0

ddvvxfxE

fEv

fEvvv

fB

xf

vtf

x

vx

The cylindrical geometry is adopted in the velocity space in order to describe the rotation of the particles, around the direction of the magnetic field

x

y

yx

v

v

vvv

arctan

22



yyxx

x

z

vv

k

B

eev

ek

eB

0



Linear analysis: Sukhorukov and Stubbe (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

Using the Landau approach, in the approximation of large wave length, they obtained:

for

B

t2

forB

t 2

At large time the oscillations are completely undamped (the results are in agreement with Baldwin and Rowlands (1966))



K=0.4, B=0.04, 0.06, 0.1 K=1, B=0.25



BT Trapping time versus cyclotron period


Università della CalabriaUniversità della Calabria Trapping oscillation and cyclotron motion

y

XWavelenght

Larmor radiusWave-particle interaction

No Wave-particle interaction

??

1

1

1

0

A

B

TB



Weak magnetic field

00.00 B

1max 650 pet



550 10610 B

th

Dx

Dx

vvA

A

Lk

L

2.3

8.51

03.0

4.02

5.15

1



Stronger and stronger magnetic field

04.118.00 B Strong magnetic field: UNDAMPED OSCILLATIONS

DAMPED OSCILLATIONS

ISOLATED ELECTROSTATIC STRUCTURES



006.0001.00 B

17.003.00 B




Università della CalabriaUniversità della Calabria Electrostatic structures in the solar wind

DD 5010 D 5.15



The introduction of an external uniform magnetic field strongly modifies the wave-particle interaction. From our simulations the ratio between the trapping time and the cyclotron period seems to determine the evolution of the electrostatic oscillations

Unexpected damped oscillations and isolated electrostatic structures appear for intermediate values of the external field, as a result of the competition between the trapping motion and the magnetic effect


Università della CalabriaUniversità della Calabria Conclusions

Nonlinear wave-particle interaction is analyzed using a lagrangian approach. Statistical analysis of the particle phase space trajectories allows us to conclude that there exists a zone, at the border of the resonant region, in which the system dynamics becomes chaotic.

Resonant particles can escape from the potential well and perform flights in the phase space of limited extent, whose characteristic length is strictly related to the long time behavior of the solution

IV Congresso Italiano di Fisica del Plasma Firenze, 12-13 Gennaio 2004 Francesco Valentini...

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