Motivation: basic concepts of phase space turbulence

Issues in phase space turbulence

Up to now, our discussion of plasma turbulence has developed by following the two parallel roads shown in Figure 8.1. Following the first, well trodden, path, we have developed the theory of nonlinear mode interaction and turbulence as applied to fundamentally fluid dynamical systems, such as the Navier–Stokes (NS) equation or the quasi-geostrophic (QG) Hasegawa–Mima equation. Along the way, we have developed basic models such as the scaling theory of eddy cascades as in the Kolmogorov theory, the theory of coherent and stochastic wave interactions, renormalized theories of fluid and wave turbulence, the Mori–Zwanzig memory function formalism for elimination of irrelevant variables, and the theory of structure formation in Langmuir turbulence by disparate scale interaction. Following the second, less familiar trail, we have described the theory of kinetic Vlasov turbulence – i.e. turbulence where the fundamental dynamical field is the phase space density f (x, v, t) and the basic equation is the Vlasov equation or one of its gyrokinetic variants.