How to lock or limit a free ballistic expansion of Energetic Particles?

L eff

Steady State Pedestal ?

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;

Add Fermi Acceleration

L eff

Fermi=Add multiple front crossings (for the most“lucky” CR particles)

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;

Feedback on particles: Quasilinear Theory of Particles/Cyclotron Waves Interaction

k

v

vz

plateau

v + v2

2 2 v

k= const

If 1

tkviEedt

dVm ii exp

Connection of Quasilinear Theory to KAM-Theory:

From Planetary Resonances to Plasma

x

V

1k

21

me

kV

V

X

1k

2k

ADD MORE WAVES

tk

x

kV

tk

x

kV

21

me

kV

nn kk

1

Width of resonance

Distance between

resonances

vs.

21

me

nn kk

1much 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 “

21

me

nn kk

1greater 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 is Diffusion Coefficient ?(in velocity space)

tkviEedt

dVm ii exp

kvitkviEmeV i exp

kvitvkvkiEEm

e jiji exp*2

2

VXdV/dt =

DtV 2

D= )(/222 kvEme

dkk 2

1

Repercussions: Quasilinear Theory, Plateau Formation,

Beam + Plasma Instability Saturation etc.

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

• . 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.

Add Fermi Acceleration

)z

fD(

zp

fp

z

V

3

1

z

fV

t

f

Truncate Quasilinear Eqn

Simplified approach

• Spatial Diffusion approximation is valid:

-QL estimate of eff

B

( B) 2

B 2

L ; eff C

eff

pedestal

Leff

C

U

U – shock velocity

Nonlinear Saturation Conjecture

Saturation Amplitude of Driven MHD as function of the Growth Rate

V

B

Re1

MHD + Expanding Cloud of Energetic Particles + “Return

CurrentMHD Waves modified:

(1) 222BIzaz VkVk

(2)

21

2222

z

x

BIzaz

k

k

VkVk

Net effect of Instabilities

• Both types of Instability together:

(3) 0

22

n

n

V

uiVk CR

CRBBIz

kV

kVVVV

2)(Im