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Weak Turbulence Theories

Weak Turbulence Theories. Quasilinear Theory. 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. The History of QLT.

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Weak Turbulence Theories

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  1. Weak Turbulence Theories Quasilinear Theory

  2. 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

  3. The History of QLT In 1960~1961 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.

  4. 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.

  5. Linearization of Vlasov Equations

  6. 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.

  7. From linear theory we obtain and the usual dielectric function

  8. Wave Kinetic Equation Introducing the spectral energy density of the wave field the wave kinetic equation may be written

  9. Resonant and Non-resonant Particles • In general we should consider the non-resonant part as well. Then we have • And the resonant part

  10. Most Common Form of QLT

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

  12. 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.

  13. A Summary of QL Theory

  14. More Preferred Form of Kinetic Equation Making use of the preceding results we obtain

  15. Most Frequently Cited Diffusion Coefficient Since thediffusion tensor is a real quantity Usually we emphasize the resonant interaction.

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