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Kinetic Descriptions of Low Frequency Modes Obtained by Gyroaveraging

Book chapter, 2012

We have, in Chap. 4 studied kinetic descriptions in simple geometries. Characteristic of these has been that inhomogeneities have been assumed to be constant along particle orbits. This can be achieved by representing magnetic curvature by a simple gravity force. Then, it is possible to integrate the Vlasov equation in a magnetized plasma, along the characteristics (linear orbit) for all times. This can be made for arbitrary frequencies and gyroradii, thus including cyclotron resonances and the full Finite Larmor Radius (FLR) effects. This can also be done keeping nonlinear terms although we only did that for the drift kinetic equation which does not involve FLR effects. In the present chapter we will drop the assumption of inhomogeneities that are constant in space and include the full kinetic magnetic drifts [1–20]. In this case we do not know even the unperturbed orbits for all times. This case can still be treated in a reasonably simple form if we restrict our study to low frequency modes which have ω < < Ωc. We can then average over the fast timescale. We will start with the simplest case when FLR effects are small and derive a more general drift kinetic equation than we did in Chap. 4. We also include a brief survey of this area [1–20].

Vlasov Equation

Fluid Theory

Particle Orbit

Background Magnetic Field

Kinetic Description