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
Fermi=Add multiple front
crossings (for the most
“lucky” CR particles)
L eff
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
v
plateau
v2+ v
2
2
k
vz

v
k
= const
If

1
Connection of Quasilinear Theory to
KAM-Theory:
From Planetary Resonances to
Plasma
dV
m
 e Ei exp i i  kv t
dt
V
e




V  

k  m
1
2
 k 1
x
V
ADD MORE
WAVES
 k 2
 k 1
X
V

k
x

k
t
V

k
x

k
t
e




V


k  m
1
2
Width of
resonance
vs.
 k  k

n 1


n
Distance between
resonances
 e 
 m
1
2
much less than
   



k n1
k
This limit corresponds to KAM (KolmogoroffArnold-Mozer) case.
KAM-Theorem :
As applied to our case of Charged Particle –
Wave Packet Interaction –
“Particle preserves its orbit “
n
 e 
 m
1
2
greater than
   



k n1
k
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
Diffusion Coefficient ?(in
velocity space)
- what is
n
dV
m
 e Ei exp i i  kv t
dt
Ve
E exp i  kvt

m
i  kv
i
VXdV/dt =
e
2
m
2
 EE
*
exp i i   j  ki v  k j v t
V  Dt
2
i  kv
D=
e / m
2

k
2
E
 (kv   )
2
1

dk

2
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
f
 f 1 V f 
f
V 
p  (D )
t
 z 3 z p z z
Truncate Quasilinear Eqn
Simplified approach
• Spatial Diffusion
approximation is
valid:
-QL estimate of
 eff B
L
eff

C
 eff
U – shock
velocity
;

2
( B)
B2
pedestal
L
C
eff
U
Nonlinear Saturation Conjecture
Saturation Amplitude of Driven MHD as
function of the Growth Rate
V
B
1

Re 
MHD + Expanding Cloud of
Energetic Particles + “Return
Current
MHD Waves modified:
  k V  k zVI  B
2
2 2
z a
k zVI  B
 k V 
2
kx
1
k2z
2
2 2
z a
(1)
(2)
Net effect of Instabilities
• Both types of Instability together:
u nCR
  k zVI  B  i
VCR n0
2
2
B
Im V  (V)V  kV
V

k
2
(3)