Weak Turbulence Theories

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Weak Turbulence Theories

• Quasilinear Theory

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Primary Objective of Early Efforts

• To understand the saturation of bump-on-tail
  instability
• To study the effect of growing Langmuir waves on
  unperturbed distribution function

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The History of QLT

• In 19601961 quasilinear theories of a beam
  instability were proposed by U.S. and Soviet
  scientists independently.
• W. Drummond D. Pines on the U.S. side
• A. Vedenov, E. Velikhov, R. Sagdeev on the
  Soviet side.

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Basic Notion

• In linearized Vlasov theories the unperturbed
  distribution function is usually treated as a
  quantity that is not time dependent.
• This approximation would break down at a time
  scale comparable to the wave growth time.
• When waves grow, they tend to affect the
  unperturbed distribution.

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Linearization of Vlasov Equations

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Quasilinear Equations

• The reaction of the waves on the unperturbed
  distribution function is described by
• This is called the quasilinear kinetic equation
  since the perturbation quantities obey linear
  theory.

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From linear theory we obtain

• and the usual dielectric function

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Wave Kinetic Equation

• Introducing the spectral energy density of the
  wave field
• the wave kinetic equation may be written

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Resonant and Non-resonant Particles

• In general we should consider the non-resonant
  part as well. Then we have
• And the resonant part

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Most Common Form of QLT
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Problems

• Momentum and energy are not conserved.
• Usually the importance of non-resonant
  interaction is overlooked.

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Shortcomings

• Emphasizing growth rate
• Leaving an impression that QLT is only for the
  study of unstable plasmas.
• Conceptually it is not clear whether
  non-resonant interactions are important for a
  stable plasma.

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A Summary of QL Theory
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More Preferred Form of Kinetic Equation

• Making use of the preceding results we obtain

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Most Frequently Cited Diffusion Coefficient

• Since the diffusion tensor is a real quantity
• Usually we emphasize the resonant interaction.