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How to lock or limit a free ballistic expansion of Energetic Particles?. What could limit the free collisionless expansion?. Dominant Scenario: -Instabilities / to create the “waves (as scatterers)”; -Feedback of growing fluctuations on particles due to the particles – waves interaction;

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what could limit the free collisionless expansion

What could limit the free collisionless expansion?

Dominant Scenario:

-Instabilities / to create the “waves

(as scatterers)”;

-Feedback of growing fluctuations on particles due to the particles – waves interaction;

-Non-Linear saturation of instability / build up of saturation spectrum;

implementation of such scenario
Implementation of such scenario
  • -Instability: Velocity Anisotropy (“Cyclotron Instability” of Alfven waves);
  • -Feedback on particles: Quasilinear Theory of Particles/Cyclotron Waves Interaction;
  • -Non-Linear saturation: Strong MHD / Alfven Waves/ Turbulence;
slide6

Connection of Quasilinear Theory to KAM-Theory:

From Planetary Resonances to Plasma

slide7

V

x

slide8

V

ADD MORE WAVES

X

slide11

Width of resonance

vs.

Distance between

resonances

slide12

much less than

This limit corresponds to KAM (Kolmogoroff-Arnold-Mozer) case.

KAM-Theorem :

As applied to our case of Charged Particle – Wave Packet Interaction –

“Particle preserves its orbit “

slide13

greater than

That means - overlapping of neighboring resonances

Repercussions:

-”collectivization” of particles between neighboring waves;

-particles moving from one resonance to another – “random walk”? And if yes

-what isDiffusion Coefficient ?(in velocity space)

slide15

D=

Repercussions: Quasilinear Theory, Plateau Formation,

Beam + Plasma Instability Saturation etc.

general conclusions
General Conclusions
  • Kolmogoroff: Application of KAM theory to the Dynamics of Planetary System
  • Plasma case: Application to the Dynamics of Charged Particles

more applications:

  • Waves-Particles interaction at Cyclotron Resonance
  • Magnetic Surfaces Splitting? (Trieste, 1966)
  • Advection in Fluids (+20 years)
quasilinear diffusion
Quasilinear Diffusion
  • . The simplified approach to such diffusion is equivalent to a truncation of quasilinear velocity space diffusion similar to tau-approximation form of collision integral in kinetic equation .

Further simplifications:

  • -Plasma pressure is much greater than Magnetic field pressure;
  • -Bulk of plasma particles out of resonance with “waves” (even in strong turbulence definition);
  • -CR particle density too small to produce competitive nonlinear effects by themselves and do not affect waves nonlinear saturation process.
net effect of instabilities
Net effect of Instabilities
  • Both types of Instability together:
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