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 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.

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.

Linearization of Vlasov Equations

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.

From linear theory we obtain and the usual dielectric function

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

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

Most Common Form of QLT

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

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.

A Summary of QL Theory

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

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