We present a new recursive procedure to find a full f electrostatic gyrokinetic equation correct to first order in an expansion of gyroradius over magnetic field characteristic length. The procedure provides new insights into the limitations of the gyrokinetic quasineutrality equation. We find that the ion distribution function must be known at least to second order in gyroradius over characteristic length to calculate the long wavelength components of the electrostatic potential self-consistently. Moreover, using the example of a steady-state θ-pinch, we prove that the quasineutrality equation fails to provide the axisymmetric piece of the potential even with a distribution function correct to second order. We also show that second order accuracy is enough if a more convenient moment equation is used instead of the quasineutrality equation. These results indicate that the gyrokinetic quasineutrality equation is not the most effective procedure to find the electrostatic potential if the long wavelength components are to be retained in the analysis.
Gyrokinetic treatment of a grazing angle magnetic presheath