Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics

Fluidlike models have long been used to develop qualitative understanding of the drift-wave class of instabilities (such as the ion temperature gradient mode and various trapped-particle modes) which are prime candidates for explaining anomalous transport in plasmas. Here, the fluid approach is improved by developing fairly realistic models of kinetic effects, such as Landau damping and gyroradius orbit averaging, which strongly affect both the linear mode properties and the resulting nonlinear turbulence. Central to this work is a simple but effective fluid model Phys. Rev. Lett. 64, 3019 (1990) of the collisionless phase mixing responsible for Landau damping (and inverse Landau damping). This model is based on a nonlocal damping term with a damping rate ∼ vt‖k∥‖ in the closure approximation for the nth velocity space moment of the distribution function f, resulting in an n-pole approximation of the plasma dispersion function Z. Alternatively, this closure approximation is linearly exact (and therefore physically realizable) for a particular f0 which is close to Maxwellian. ‘‘Gyrofluid’’ equations (conservation laws for the guiding-center density n, momentum mnu∥, and parallel and perpendicular pressures p∥ and p⊥) are derived by taking moments of the gyrokinetic equation in guiding-center coordinates rather than particle coordinates. This naturally yields nonlinear gyroradius terms and an important gyroaveraging of the shear. The gyroradius effects in the Bessel functions are modeled with robust Padé-like approximations. These new fluid models of phase mixing and Landau damping are being applied by others to a broad range of applications outside of drift-wave turbulence, including strong Langmuir turbulence, laser–plasma interactions, and the α-driven toroidicity-induced Alfvén eigenmode (TAE) instability.