Kinetic processes in plasmas

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Kinetic processes in plasmas

• A. Mangeney
• Observatoire de Paris

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A composite fluctuation spectrum in the Solar wind
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From a system of N particles to "the typical"
particle
N particles of mass
At the finest level
moving according to Newton's law
under the action of forces, both external and
internal
with interaction forces
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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
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From a system of N particles to "the typical"
particle
How to loose information?
Density
Momentum density
Kinetic energy density
Kinetic energy flux
etc
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From a system of N particles to "the typical"
particle
How to loose more information?
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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
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Collisionnal/collisionless
A) Mean field description - Vlasov equation if
otherwise
B) Collisional fluid - Boltzmann ( Fokker-Planck)
equation
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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
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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
ltEgt varying on scales greater than particle
separation

on scales comparable to particle separation
Screening effects have to be taken into account
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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
Debye length and electron plasma frequency
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Collisionless plasma
1d
N particles of both signs nl
Charge density fluctuation
Potential fluctuation
in 3d
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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
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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
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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
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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!
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Infinite hierarchy of equations!
etc (for each particle species)
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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!
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"Thermal" noise in the Solar Wind
Here only quietest solar wind state, far from
Shocks, etc
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Collisionless evolution
Phase mixing, Landau damping Violent relaxation
virialisation- attempt to reach mechanical
equilibrium holes in phase space, observed
almost everywhere in space as soon as time
resolution sufficient Development of microscopic
instabilities
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• 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

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Electric fluctuations at lower frequency

• Quasi thermal noise
• (Issautier et al.,1999)
• with Gaussian statistics

(B bandwidth, t integration time )

• Intermittent non thermal emission

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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.
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At high time resolution
Langmuir waves
 Ion acoustic waves 
In the quiet  Solar wind, all events recorded
by the Time Domain Sampler (above a threshold of
50mV/m) are coherent waveforms ( Mangeney et
al., 1999)

• Langmuir waves

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Weak Double Layers (WDL)
About 30 of these CEW are Isolated Electrostatic
Waveforms with a measurable net potential jump
The corresponding electric field is almost always
directed towards the Earth
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Phase mixing
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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
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Phase relationships between moments
Suggests closure (non local)
depending on k, may be imaginary
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Phase relationships between moments
Suggests closure (non local)
depending on k, may be imaginary
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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 occursthere
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
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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
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Nonmaxwellian plasma
Stationary fluid equilibrium
Two maxwellian electron distribution cold and hot
Cold, at rest
Hot, speed uh
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Fluid like equilibrium, not Vlasov equilibrium!
1d, open boundary Vlasov simulation (x,v),
electrons and ions, to test Landau closures
(for this summer school)
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t0
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"Ballistic evolution"
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Ballistic evolution, electric pulse formation and
proton acceleration
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Evolution of the electric potential
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Evolution of electron temperature
Does not seem compatible with a fluid like
closure !
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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)
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(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
h0 forcing only on protons 1 forcing both
on protons and electrons
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Random forcing
I-transient compressions or expansions
c(x) spatial profile (compression/expansion) q(
t) time profile
l
t

• (xj, tj) independant random points and times

• (sj , lj ,tj) randomly distributed around
  typical values s, l, t

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II- random charge fluctuations
When the forcing concerns both electrons and
protons (h1), 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
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However,  thermal charge fluctuations related
to particle discreteness have a spectral density
while the random charges used here have very
different space time properties, and smaller
level!
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Two sets of 1D runs
(I) Nx512, Nv401, L1000 lDe
RUN A h 0, F 0, Y ? 0, l10 RUN B
h 1, F ? 0, Y 0, l10
(II) Nx2048, Nv501, L5000 lDe
RUN C h 1, F ? 0, Y ? 0, l100 RUN D
h 1, F ? 0, Y 0, l100
Quasineutrality random forcing only on the
protons
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• Two runs with same amplitude of forcing
• (A) forcing only on the protons, h0
• (B) forcing on electrons and protons, h1

dq2
(A)
dnp2
(B)

dne2
A
B
A) electric neutrality maintained at all times
B) smaller density fluctuations but much larger
charge fluctuations at forcing times!
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Forcing on protons (Y?0) leads to formation of
long lived, small scale, stuctures
Life time 2000 tpegtgt t20 spatial scale 50
lDeltlt 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)
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• 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

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Proton density variation
Langmuir wave power density
Long lived coherent density cavities generated
by LF proton forcing trap Langmuir waves
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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
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Electron distribution function Phase space
modulation for vgt0 and vlt0
Proton distribution function

• slow phase mixing on the proton distribution
  function
• no significant proton heating
• strong interaction of tail electrons with
  Langmuir waves

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Formation of a symmetric  halo  electron
population

• when the forcing includes
• a LF forcing on protons

t0
Space averaged electron distibution function

• and not when there is no
• LF forcing on protons

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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