Equivalence of two different approaches to global $\delta f$ gyrokinetic simulations

A set of flux tube gyrokinetic equations that includes the effect of the spatial variation of the density, temperature and rotation gradients on the turbulence is derived. This new set of equations uses periodic boundary conditions. In the limit $l_\bot/L \ll 1$, where $l_\bot$ is the characteristic perpendicular length of turbulent structures and $L$ is the characteristic size of the device, this new set of flux tube gyrokinetic equations is shown to be equivalent to the traditional global $\delta f$ gyrokinetic equations to an order higher in $l_\bot/L$ than the usual flux tube formulations.