Gyro-averaging operators with magnetic field inhomogeneity

We derive expressions for the gyro-averaging operator that is applicable to electrostatic fluctuations in a spatially inhomogeneous magnetic field. Both low and high wavenumber limits are considered. The gyro-averaging operator for the former case is represented by sums of Bessel functions with different orders. A simplified expression is provided as a Pade approximant in the low wavenumber limit. This form could be used in practical computations based on the gyrofluid formulation. In the high wavenumber limit, we find that the operator naturally involves fractional derivatives whose physical interpretations are yet to be explored. Discussions are made of a potential impact of this asymptotic expression in the high wavenumber limit. Published under an exclusive license by AIP Publishing.