Revisiting Gyro-Kinetics as a Transport Theory, I of II

A derivation of the non-linear Gyro-kinetic equation using Method of Multiple Scales is presented. Results are obtained for slab and toroidal field geometries. This approach assumes three disparate frequency scales: the ion cyclotron frequency, turbulent frequency, and transport rate. Also assumed are two important spatial scales: the ion larmor radius ($\rho_i$) and plasma size ($L \sim n/|\nabla n|$). The fundamental expansion parameter is $\epsilon = \rho_i/L$. The Fokker-Planck equation is expanded in orders of $\epsilon$, up to the Gyro-kinetic equation. Finally, a physical interpretation of Gyro-kinetic equation is given.