II' Plasma Physics Fundamentals

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II. Plasma Physics Fundamentals

• 4. The Particle Picture
• 5. The Kinetic Theory
• 6.

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5. The Kinetic Theory

• 5.1 The Distribution Function
• 5.2 The Kinetic Equations
• 5.3 Relation to Macroscopic Quantities
• 5.4 Landau Damping

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5.4 Landau Damping

• 5.4.1 Electromagnetic Wave Refresher
• 5.4.2 The Physical Meaning of Landau Damping
• 5.4.3 Analysis of Landau Damping

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5.4.1 Electromagnetic Wave Refresher

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Electromagnetic Wave Refresher (II)

• The field directions are constant with time,
  indicating that the wave is linearly polarized
  (plane waves).
• Since the propagation direction is also constant,
  this disturbance may be written as a scalar wave
  E Emsin(kz-wt) B Bmsin(kz-wt)
  k is the wave number, z is the propagation
  direction, w is the angular frequency, Em and Bm
  are the amplitudes of the E and B fields
  respectively.
• The phase constants of the two waves are equal
  (since they are in phase with one another) and
  have been arbitrarily set to 0.

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5.4.2 The Physical Meaning of Landau Damping

• An e.m. wave is traveling through a plasma with
  phase velocity vf
• Given a certain plasma distribution function
  (e.g. a maxwellian), in general there will be
  some particles with velocity close to that of the
  wave.
• The particles with velocity equal to vf are
  called resonant particles

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The Physical Meaning of Landau Damping (II)

• For a plasma with maxwellian distribution, for
  any given wave phase velocity, there will be more
  near resonant slower particles than near
  resonant fast particles
• On average then the wave will loose energy
  (damping) and the particles will gain energy
• The wave damping will create in general a local
  distortion of the plasma distribution function
• Conversely, if a plasma has a distribution
  function with positive slope, a wave with phase
  velocity within that positive slope will gain
  energy

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The Physical Meaning of Landau Damping (III)

• Whether the speed of a resonant particle
  increases or decreases depends on the phase of
  the wave at its initial position
• Not all particles moving slightly faster than the
  wave lose energy, nor all particles moving
  slightly slower than the wave gain energy.
• However, those particles which start off with
  velocities slightly above the phase velocity of
  the wave, if they gain energy they move away from
  the resonant velocity, if they lose energy they
  approach the resonant velocity.

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The Physical Meaning of Landau Damping (IV)

• Then the particles which lose energy interact
  more effectively with the wave
• On average, there is a transfer of energy from
  the particles to the electric field.
• Exactly the opposite is true for particles with
  initial velocities lying just below the phase
  velocity of the wave.

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The Physical Meaning of Landau Damping (V)

• The damping of a wave due to its transfer of
  energy to near resonant particles is called
  Landau damping
• Landau damping is independent of collisional or
  dissipative phenomena it is a mere transfer of
  energy from an electromagnetic field to a
  particle kinetic energy (collisionless damping)

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5.4.3 Analysis of Landau Damping

• A plane wave travelling through a plasma will
  cause a perturbation in the particle velocity
  distribution f(r,v,t) f0(r,v,t) f1(r,v,t)
• If the wave is traveling in the x direction the
  perturbation will be of the form

• For a non-collisional plasma analysis the Vlasov
  equation applies. For the electron species it
  will be

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Analysis of Landau Damping (II)

• A linearization of the Vlasov equation considering

• (since only contributions along v are studied)

• yields

• or, considering the wave along the dimension x,

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Analysis of Landau Damping (III)

• The electric field E1 along x is not due to the
  wave but to charge density fluctuations
• E1 be expressed in function of the density
  through the Gauss theorem (first Maxwell equation)

• or, in this case, considering a perturbed
  density n1 equivalent to the perturbed
  distribution f1

• Finally the density can be expressed in terms of
  the distribution function as

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Analysis of Landau Damping (IV)

• The linearized Vlasov equation for the wave
  perturbation

• can be rewritten, after few manipulations as a
  relation between w, k and know quantities

where
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Analysis of Landau Damping (V)

• For a wave propagation problem a relation between
  w and k is called dispersion relation
• The integral in the dispersion relation

• can be computed in an approximate fashion for a
  maxwellian distribution yielding

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Analysis of Landau Damping (VI)

• For a one-dimensional maxwellian along the x
  direction

• This will cause the imaginary part of the
  expression

to be negative (for a positive wave propagation
direction)
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Analysis of Landau Damping (VII)

• For a wave is traveling in the x direction the of
  the form

a negative imaginary part of w will produce an
attenuation, or damping, of the wave.