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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