Transcript Thessa_pres

Kinetic processes in plasmas
A. Mangeney
Observatoire de Paris
Basic processes of turbulent plasmas,
2003
A composite fluctuation spectrum in the Solar wind
Basic processes of turbulent plasmas,
2003
From a system of N particles to "the typical" particle
At the finest level:
N particles of mass
moving according to Newton's law
under the action of forces, both external and internal
with interaction forces
Basic processes of turbulent plasmas,
2003
From a system of N particles to "the typical" particle
:
dimensionnal phase space, {X }
Among functions defined on phase space:
Klimontovich distribution:
"counts" the number of particles in a volume dxdv of
Basic processes of turbulent plasmas,
2003
From a system of N particles to "the typical" particle
How to loose information?
Instead of all the details of the distribution of particles in
consider only a small number of velocity moments:
Density:
Momentum
density:
Kinetic energy
density:
Kinetic energy
flux
etc…
Basic processes of turbulent plasmas,
2003
From a system of N particles to "the typical" particle
How to loose more information?
: random because
-complicated motion of the particles because of mutual
pair wise interactions, even chaotic…
-unknown initial conditions,
-etc….
Basic processes of turbulent plasmas,
2003
A typical particle
At the kinetic level:
The identity of the particles has been lost; a small number
of smooth functions, describing the statistics of the fine grained
distribution
What happens to the interaction forces?
Mean field, resulting from the linear superposition of the fields of
all particles; the discrete character of the particles has been lost
Basic processes of turbulent plasmas,
2003
Collisionnal/collisionless
But the interaction force is also a random quantity,
with fluctuations determined by f2
(Correlation function, retaining part of the memory of particle discreteness)
A) Mean field description - Vlasov equation if
otherwise
B) Collisional fluid - Boltzmann ( Fokker-Planck) equation
Basic processes of turbulent plasmas,
2003
What makes a plasma to differ from another fluid?
For most neutral fluids interparticle forces are short range!
 In a plasma, or a gravitating stellar system, interparticle forces
are long range!
1D, electrostatic:
Particles are actually charged sheets, with charge ±e. A particle
at xi is the source of a piece-wise constant electric field:
E
x
Basic processes of turbulent plasmas,
2003
Thus, total electric field E(x) at a given x depend only on the total
number of particles of both signs at left and at right of x, but not
on their precise location; if these numbers are large, E(x) vary slowly
with x, with little jumps each time a particle is crossed (discreteness
effects)
x
1/n
<E> varying on scales greater than particle separation
on scales comparable to particle separation
Screening effects have to be taken into account
Basic processes of turbulent plasmas,
2003
Charge neutrality
What is really the scale of variation of the average field?
Debye-Huckel (1923)
Electrons move fast to cancel any notable average charge separation
Poisson equation:
Debye length and
electron plasma frequency
Basic processes of turbulent plasmas,
2003
Collisionless plasma
l
1d:
N particles of both signs
~ nl
Charge density fluctuation
Potential fluctuation
in 3d:
Basic processes of turbulent plasmas,
2003
Vlasov (Mean field):
Charged particles move in a self consistent mean electric field
Vlasov
Poisson
distribution functions remain constant along a particle trajectory:
if these trajectories are complicated, the distribution function may
become also very complex (see later)
Stationary states :
Infinite number of invariants
Basic processes of turbulent plasmas,
2003
Collisionnal case
In that case, one has to include the fluctuating electric field
due to discreteness:
When averaging over the fluctuations, one obtains a Fokker planck type
of equation
Particle recoil for sponatneous emission
Random walk in the fluctuating potential
Basic processes of turbulent plasmas,
2003
Lennard Balescu equation:
Not too far from equilibrium i.e.
fluctuation spectrum ~ what is expected from free streaming particles
Still extremely complex due to dielectric effects,
screening, etc…
However, in the absence of external forces, only one
stationary solution, the maxwellian distribution, at
temperature T:
Basic processes of turbulent plasmas,
2003
From a typical particle to fluid-like quantities
How to loose STILL more information?
Moments:
From an infinite number of fields to 3 hydrodynamic fields!
Basic processes of turbulent plasmas,
2003
Infinite hierarchy of equations!
etc…
(for each particle species)
Basic processes of turbulent plasmas,
2003
Closure:
A) Collisions
-Local maxwellian: gaussian random variables in v for all (x,t):
Ideal Euler equations
-ETL: Transport processes, Navier Stokes equations
B) No a priori valid closures for the collisionless case
Several "nested" closure:
- correlations
- moments of f1
Importance of boundary conditions!
Basic processes of turbulent plasmas,
2003
"Thermal" noise in the Solar Wind
Here only quietest solar wind state, far from Shocks, etc…
Basic processes of turbulent plasmas,
2003
Collisionless evolution
Phase mixing, Landau damping
Violent relaxation: virialisationattempt to reach mechanical equilibrium
holes in phase space,
observed almost everywhere in space
as soon as time resolution sufficient
Development of microscopic instabilities
Basic processes of turbulent plasmas,
2003
Suprathermal electrons with energies above about 80 eV at 1 AU continually stream
out along magnetic field lines with a velocity distributions, f(v) usually consisting of
• a dominant field-aligned component directed outward from the Sun, the strahl
(found in high speed solar wind)
• a weaker and
more isotropic halo component;
Wind observations
• significant variability of the strahl and/or halo,
• other types of distributions, such as counterstreaming strahls, angular depletions
and enhancements, and sunward streaming conics
Basic processes of turbulent plasmas,
2003
Electric fluctuations at lower frequency
• Quasi thermal noise
(Issautier et al.,1999)
with Gaussian statistics
E 
V ( f ) 10
2
B
(B: bandwidth,
 integration time )
• Intermittent
non
thermal emission
Basic processes of turbulent plasmas,
2003
13.5
13 2
10 V / Hz
Histogram of electric
fluctuations at two
frequencies
At f = 4.27 kHz, non thermal emission
is observed above 5 10-13V2/Hz, with a
power law distribution.
Above 7 kHz, these nonthermal
Emissions disappear.
Basic processes of turbulent plasmas,
2003
At high time resolution:
Langmuir waves
« Ion acoustic waves »
In the «quiet » Solar wind, all events
• Langmuir waves
recorded
Sampler
by
the
Time
Domain
(above a threshold of
~
50mV/m) are coherent waveforms (
Mangeney et al., 1999)
Basic processes of turbulent plasmas,
2003
Weak Double Layers (WDL)
About 30% of these CEW are
Isolated Electrostatic Waveforms
with a measurable net potential
jump:
e  10 4  10 3
k BTe
or
  10- 3Volts
The corresponding electric field is
almost always directed towards the
Earth
Basic processes of turbulent plasmas,
2003
Phase mixing
Basic processes of turbulent plasmas,
2003
Phase mixing
All moment perturbations decrease because of velocity
integration which washes out fine structures developping
in the velocity dependance.
One may even prepare the system to obtain a wave propagating
at an arbitrary velocity
by ajusting the initial
distribution
Damping rate is diminished
Basic processes of turbulent plasmas,
2003
Phase relationships between moments
Suggests closure (non local)
: depending on k,
may be imaginary
Basic processes of turbulent plasmas,
2003
Phase relationships between moments
Suggests closure (non local)
: depending on k,
may be imaginary
Basic processes of turbulent plasmas,
2003
Landau damping and phase mixing
In the free streaming case no restoring force and
no wave modes.
If one retains the electric field, there is now
a restoring force and wave modes; however the
same phenomenon occurs:there are a continuum
of wave modes in phase space, while velocity averages
decrease, now only exponentially (in a stable plasma),
due to a subtler phase mixing (Landau damping).
Landau closures: compare a linearized fluid theory, with
ad-hoc transport coefficient and the "exact" Vlasov linear theory,
and try to fit one theory with the other; leads to non local
transport coefficient
Basic processes of turbulent plasmas,
2003
Example : Heat transport in fluids and collisionless plasmas
Fluids: small deviations from ETL
Collisionless plasmas: apparition of strong electric fields
Some particles travel almost freely: ballistic mixing
while others are strongly affected
Landau closures: attempt to mimic collisionnal theory
with Landau damping
Basic processes of turbulent plasmas,
2003
Nonmaxwellian plasma
Stationary fluid equilibrium
Two maxwellian electron distribution: cold and hot
Cold, at rest:
Hot, speed uh
Basic processes of turbulent plasmas,
2003
Fluid like equilibrium, not Vlasov equilibrium!
1d, open boundary Vlasov simulation (x,v),
electrons and ions, to test Landau closures
(for this summer school)
Basic processes of turbulent plasmas,
2003
t=0
Basic processes of turbulent plasmas,
2003
"Ballistic evolution"
Basic processes of turbulent plasmas,
2003
Ballistic evolution, electric pulse formation and proton acceleration
Basic processes of turbulent plasmas,
2003
Evolution of the electric potential
Basic processes of turbulent plasmas,
2003
Evolution of electron
temperature
Does not seem compatible with a fluid like closure !
Basic processes of turbulent plasmas,
2003
Random forcing (mimic discreteness effects)
A)Full N-body calculation - Heavy!!!
B)Random forcing:
B1) « self consistent » gaussian force leading
to the
Landau equation (Qiang et al, 2000,
for example)
B2) Constant temperature molecular dynamics
method: a random force is introduced to
allow the
system to sample a
canonical or
microcanonical ensemble
B3) Dirty way : artificial random forcing
Here, B3!
(Collaboration F. Califano)
Basic processes of turbulent plasmas,
2003
f p
f p
e 
   f p
v

E  


0


t
x M
x
x v
fe
fe e 
  f e
v
  E 

0


t
x m
x v
E e
 np  ne  n p   dvf p (x,v,t)
x  0
ne 
 f (x,v,t)
e
(1) External force acting only on the
protons, deriving from potential Y(x,t)
(2) Random « external » electric potential: F(x,t)
acting on electrons and protons
0 : forcing only on protons
= 1: forcing both on protons and electrons
Basic processes of turbulent plasmas,
2003
Random forcing:
I-transient compressions or expansions
(x,t) 
s
j
(x  x j ) (t  t j )
j
(x) : spatial profile
(compression/expansion)
(t) : time profile
l
• (xj, tj): independant random points and times
• (sj , lj ,j): randomly distributed around typical values s*, l*, *
Basic processes of turbulent plasmas,
2003

II- random charge fluctuations
 2
qext   2
x
When the forcing concerns both electrons and protons (1),
it is equivalent to the introduction of external charges
t
x
Space - time distribution of
random charges
Spatial profile
« Discreteness » introduced by random external charges
Basic processes of turbulent plasmas,
2003
However, « thermal »charge fluctuations related to particle discreteness
have a spectral density
qdisc2 (k,  )  2e 2   dv  (  kv) f (v)

while the random charges used
here have very different space
time properties, and smaller
level!
Basic processes of turbulent plasmas,
2003
Two sets of 1D runs:
(I) Nx=512, Nv=401, L=1000 lDe
RUN A:
RUN B:
  0,   0,  ≠ 0, l*=10
  1,  ≠ 0,  0, l*=10
(II) Nx=2048, Nv=501, L=5000 lDe
RUN C:
RUN D:
  1,  ≠ 0,  ≠ 0, l*=100
  1,  ≠ 0,   0, l*=100
Quasineutrality
random forcing only on the protons
Basic processes of turbulent plasmas,
2003
Two runs with same amplitude of forcing:
• (A) forcing only on the protons, 0
• (B) forcing on electrons and protons, 1
q2
(A)
np2
(B)
ne2
A
B
A) electric neutrality maintained at all times
4
q 10 np
2
B) smaller density fluctuations but much larger charge fluctuations
at forcing times!
Basic processes of turbulent plasmas,
2003
2
Forcing on protons (≠0) leads to formation of long lived, small scale,
stuctures
Life time ≥ 2000 tpe>> *~20 ; spatial scale ~ 50 lDe<< l
evolving time scale comparable with proton phase mixing time
(If forcing sufficiently strong: formation of electron holes with their
associated bipolar electric field signature; not considered here)
Basic processes of turbulent plasmas,
2003
• When the forcing is only on protons
• Heating of protons
• Generation of electron plasma waves and electron heating
but no halo formation
• If some external charge fluctuations are added
• Heating of protons
• Generation of plasma waves
• Formation of a stationary halo for large t
Basic processes of turbulent plasmas,
2003
Proton density
variation
Langmuir wave
power density
Long lived coherent
density cavities
generated by LF proton forcing
trap Langmuir waves
Basic processes of turbulent plasmas,
2003
Spectral electric density integrated
in a band around the electron plasma frequency
Run C
LF proton forcing produces
a broader k-spectrum of
Langmuir wave
Run D
Basic processes of turbulent plasmas,
2003
Electron distribution function:
Phase space modulation
for v>0 and v<0
Proton distribution function
• slow phase mixing on the proton distribution function
• no significant proton heating
• strong interaction of tail electrons with Langmuir waves
Basic processes of turbulent plasmas,
2003
Formation of a symmetric « halo » electron population
• when the forcing includes
a LF forcing on protons
Space averaged electron distibution function
• and not when there is no
LF forcing on protons
Basic processes of turbulent plasmas,
2003
t=0
Conclusion
On the basis of 1d, electrostatic Vlasov-Poisson simulations,
including
• random forcing of the proton component, modelling the
influence of large scale nonlinearities,
•random charge fluctuations, modelling discreteness effects
we show that
both effects are necessary to obtain the formation of a
suprathermal halo on the electron distribution function
- the low frequency forcing on protons create density depletions
- these depletions trap and enhance Langmuir waves
- the Langmuir waves tend to reach an equilibrium with the halo
electrons
Basic processes of turbulent plasmas,
2003