WaveParticle Interaction in Collisionless Plasmas: Resonance and Trapping

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Wave-Particle Interaction in Collisionless
Plasmas Resonance and Trapping Zhihong
Lin Department of Physics Astronomy University
of California, Irvine International Summer
School on Plasma Turbulence and
Transport Chengdu, 8/16-8/18, 2007
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Wave-Particle Interaction (WPI)

• In high temperature plasmas, collisional mean
  free path is much longer than wavelength
• Wave-particle energy exchange depends on ratio of
  wave phase velocity to particle velocity kinetic
  effects
• WPI plays key roles in
• Excitation and damping of collective modes
• Diffusion in velocity space thermalization,
  heating, acceleration
• Transport of particle, momentum, and energy
• Studies of WPI
• Coherent WPI resonance, trapping
• Chaos, quasilinear theory
• Weak strong turbulence theory

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Outline

• Linear resonance how does particle responds to a
  given wave
• Linear Landau damping
• Nonlinear trapping
• 1D particle-in-cell code is used for
  illustrations
• http//gk.ps.uci.edu/zlin/zlin/pic1d/

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Particle Motion in a Propagating Wave

• Given an electrostatic 1D plane wave
• Motion of a particle with mass m and charge q
• What is particle energy gain/loss?
• Linearization assume that wave amplitude is
  small use unperturbed orbit when calculating
  particle acceleration
• Lowest order equation of motion
• i represents initial value, 0 represents 0th
  order quantities

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Non-Resonant Particle

• First order velocity perturbation
• For particle with
• Doppler shifted frequency
• Particle sees changing phase of wave
• Response is oscillatory no net energy transfer
  to particle over a complete Doppler shifted wave
  period

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Resonant Particle

• For particle with
• Doppler shifted frequency is zero particle ride
  on the wave
• Particle sees constant phase of wave static
  potential
• Response is secular
  particle gain/lose energy
• Phase space volume of
  resonant particle is zero

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Landau Resonant

• Particle with infinitesimally smaller velocity
  will be accelerated
• If there are more slower particles, particles
  gain energy
• Energy exchange between wave and particle depends
  on the velocity slope at resonant velocity vw/k
  Landau resonant
• Transit time resonance in magnetized plasma
  mirror force
• Resonances in tokamak plasmas
• Cyclotron Resonance wpWc
• Transit resonance wpwt
• Precessional resonance wwp
• Three resonances break three adiabatic
  invariants, respectively
• Nonlinear Landau resonance w2-w1(k2-k1)v

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Outline

• Linear resonance
• Linear Landau damping what is the feed back on
  wave by particle collective response?
• Nonlinear trapping

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Vlasov-Poisson Equations

• 1D electrostatic Vlasov-Poisson equations
  collisionless plasmas
• Summation over species s
• Conservation of probability density function
  (PDF) in phase space
• Time reversible
• Assume uniform, time stationary plasmas
• Small amplitude perturbation at t0 , expansion

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Initial Value Approach

• Causality response of a stable medium occurs
  after the impulse
• Fourier-in-space, Laplace-in-time transformation
• Inverse transformation
• w-integration path C1 lies above any singularity
  so that
• f(w,k) analytic at Im(w)gtg

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Linearized Vlasov Equation

• Perturbed distribution function
• Singular at resonance
• Poisson equation

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Inverse Laplace Transform

• Dielectric constant
• Inverse Laplace transform

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Analytic Continuation

• f(w, k) was originally defined at Im (w)gtg
• For tgt0, need to lower path C1 to C2 so that
• Deform the contour such that no pole is crossed
• f(w, k) is now defined
  on the whole
  w-plane

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Time Asymptotic Solution

• As , f(t) is dominated by
  contributions from poles
• Ballistic modes wkv, continuous spectrum,
• Damped quickly by phase-mixing
• Normal modes D(w,k)0, discrete spectrum,
• nth root nth branch, wnwn(k)

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Phase Mixing of Ballistic Mode

• Ballistic (Van Kampen) modes
• Assuming a smooth initial perturbation
• Initial phase-space perturbation propagates
  without damping
• Perturbed potential decays in t1/we for kl1
• Phase mixing destructive phase interference
• BGK mode finite amplitude Van Kampen modes
  singular df

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Normal Modes

• Long time evolution dominated by normal modes
• As w-contour is lowered from C1 to C2, the pole
  in the complex-v plane cross real-v axis
• To preserve C3 integral, C3 contour needs to be
  deformed into C4
• P is principal value.

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Landau Damping

• Linear dispersion relation
• For weakly damped mode Dr gtgtDi, wrgtgtwi
• 0th order
• 1st order

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Landau Damping of Plasma Oscillation

• Uniform Maxwellian
• Assuming ion fixed
  background
• Dielectric constant
• Dispersion relation
• Damping depends on velocity slope of distribution
  function
• Instability due to inverted shape of distribution
  function

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Outline

• Linear resonance
• Linear Landau damping
• Nonlinear trapping What is the back-reaction on
  distribution function? validity of linear theory?
  Transition to chaos?

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Validity of Linear Theory

• Expansion
• Linearization ignore nonlinear term
• Valid if
• Linear solution of normal modes
• Linear theory breaks down most easily at resonance

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Validity of Linear Theory

• Assuming weakly damped normal modes
• At resonance
• Linear theory valid if
• Bounce frequency of trapped particles

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Nonlinear Trapping

• Given a plane wave
• Transform to wave frame
• Near a potential valley (qflt0)
• For deeply trapped resonant particle
• Simple harmonic oscillation

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Phase Space Island Separatrix

• Hamiltonian
• Passing particle mechanical energy
• Trapped particle
• Phase space trajectory of trapped particle
  closed island
• Passing particle open
• Boundary separatrix
• Assumption of unperturbed orbit invalid when
  trapping occurs upper bound of wave amplitude
  for linear Landau damping

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Phase Space Island Separatrix

• Phase space island separatrix integrable
  system
• Oscillation of wave amplitude
• Small dissipation lead to chaotic region near
  sepatrix non-integrable system

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Multi-modes Island Overlap

• Island size is set by wave amplitude
• Island separation is set by number of modes,
  i.e., mode density
• Islands overlap for densely populated modes
  island size gt separation
• Particles jump between resonances before complete
  a bounce motion
• Large degree of freedom onset of stochasticity
• Quasilinear theory for small amplitude fluctuation

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Quasilinear Theory

• Vlasov equation
• Slow evolution of distribution function, spatial
  average over wavelength and time average over
  wave period
• Use linear solution of perturbed distribution
  function
• Quasilinear diffusion

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Quasilinear Flattening

• Quasilinear diffusions flattening of f0
• Relaxation to marginal stability
• Time irreversible
• HW what approximation in QLT lead to time
  irreversibility?