Kinetic theory of flowing, magnetized plasma

This work derives a drift-kinetic equation—that is, a kinetic equation for guiding-center motion—for a magnetized plasma whose flow velocity is comparable to its thermal speed. Such rapidly flowing plasmas play an increasingly important role in several areas of plasma physics research. For example, they occur in a number of astrophysical phenomena, such as galactic jets; they play a key role in various laboratory plasmas, such as the “centrifugal confinement” device; and they underly studies of novel plasma equilibria, such as Ref. 5. Finally, rapid plasma flow can have dramatic effects on plasma confinement in tokamak devices. Most previous studies of rapidly flowing, magnetized plasmas are based on fluid models. However, in many applications the collision frequency is relatively small, allowing significant departure from Maxwellian distribution functions and requiring kinetic analysis. There are a number of kinetic treatments, but these are not sufficiently general to treat some plasmas of recent interest. In particular the previous studies assume the lowest-order distribution function to be Maxwellian and thus isotropic in velocity space. This assumption simplifies the kinetic equation and its derivation, but it may not apply to the low collisionality regimes associated with astrophysical jets, to laboratory plasmas in nontoroidal geometry, or to toroidal plasmas that are strongly driven. The main objective of this work is to remove the isotropy assumption and derive a drift-kinetic that is fully general with regard both to isotropy and plasma geometry. In deriving the new result, we also attempt to improve upon previous literature in other ways. First, we have tried to make the derivation as transparent and systematic as possible, and to express the result in a convenient form. Second, we demonstrate its invariance to velocity-coordinate rotations in the plane normal to the magnetic field, by expressing all the drift-kinetic coefficients in term of the local magnetic field and its gradients. Third, we relate the new result to the well-known kinetic equation used in “kinetic magnetohydrodynamics !MHD",” showing in the process that both new and previous results conserve phase space. More generally we include detailed comparisons of our result to previous kinetic descriptions, including Refs. 7, 11, and 13. Our work is similar in some respects to Ref. 10, in which the authors carried out a derivation of the gyrokinetic equation, in a small gyroradius expansion limit. They assumed a flow which did not need to be small compared with the thermal speed, but they assumed that this large flow was perpendicular to the magnetic field. That is, any parallel component of the flow was assumed to be much smaller than the thermal speed. In contrast, we have assumed that the flow has an arbitrary parallel component, which need not be small. Our results are expected to be more useful in the treatment of magnetically confined plasmas, where the perpendicular and parallel flow components may be comparable. Also, the general discussion of phase-space conservation given in Ref. 10 applies directly to the zeroth-order terms in our drift-kinetic equation, but it is not clear how to apply it to the first-order !drift" terms, because the operator ! defined in Sec. III appears twice in Eqs. !10" and !11". We have included a direct proof of phase-space conservation for the first-order terms, in Sec. VI, which is independent of the discussion in Ref. 10. The following section presents our notational and geometrical conventions. The heart of the analysis is displayed in Sec. III, and the results given in Sec. IV. A discussion of the zero-gyroradius limit and its relation to MHD is presented in Sec. V. Phase-space conservation through first order in gyroradius is demonstrated in Sec. VI. Our conclusions are summarized in Sec. VII, which displays a relatively self-contained statement of the drift-kinetic equation.