Models of FLR effects for Full-F Continuum Gyrokinetic Codes

To speed development, the Gkeyll code at first is focussing on the long-wavelength limit of gyrokinetics, retaining FLR-related effects only in the ion polarization term in the gyrokinetic Poisson equation. This is sufficient to include long wavelength ITG/drift wave types of turbulence (which are FLR effects relative to MHD), and generically gives drift wave frequencies that are proportional to $k_y$ for small wavenumber $k_y$ but fall for $k_y \rho_s > 1$ and so includes some FLR effects. These long wavelength gyrokinetic equations have the same structure of the kinetic equation and field equations as full gyrokinetics, and so are useful for initial development, but we eventually want to upgrade this to a more complete treatment of FLR. We will present a simplified model of FLR effects that is easy to implement as a first step, to allow a quick test of their importance, and then a complete rigorous implementation of FLR effects using N-point averaging (as implemented in early gyrokinetics by W.W. Lee) with filtering and Pad\’e approximations. We document advantages of using values of $N$ that are prime or that minimize the number of factors of low primes like 2.