Gyrokinetic equations in an extended ordering

A gyrokinetic theory has been developed in an extended ordering in which the small parameter is the ratio of the E×B shearing rate to the gyrofrequency. This allows for long wavelength E×B flows of order of the thermal velocity, instead of the more restrictive standard orderings which either require that the electrostatic potential or the E×B flow velocity be small compared with the thermal levels. This theory generalizes prior work to allow for time dependence in the large long-wavelength component of the electric field and a continuum of scales in the field components rather than just two distinct components. In the new theory, a significant part of the polarization drift now resides in the equations of motion. However, there is still an identifiable polarization density that can be used to solve for the electrostatic potential from a quasineutrality or vorticity equation. The present derivation is carried out for the case of electrostatic perturbations and a slab equilibrium magnetic field, as this is sufficient and most clear for demonstrating the new results and issues associated specifically with the extended ordering.