A theory of the gyro-average operation in the high-wavenumber limit

An analytic study is conducted on the gyro-average operation of magnetized ions in the asymptotic limit of the high-wavenumber electromagnetic fluctuations. This method is suitable for evaluating the gyro-averaged potential experienced by energetic particles or thermal ions in magnetic fusion plasmas when the fluctuation scale length is significantly shorter than the ion Larmor radius. The methodology of two-dimensional fractional calculus is shown to be essential for the computation of the gyro-average operator in this limit. Making use of the asymptotic expression for the operator, we obtain a new formula expressed by the Riesz potential of order 3/2. We also derive the double gyro-average operator in terms of spatial differential operation. Finally, the same problem is considered from a statistical viewpoint. Under the assumption of the von Mises distribution of fluctuations due to the periodicity of the fluctuation along the orbit, we obtain an expectation value which should be identical to the direct calculation of the gyro-averaged potential represented by the integral transform.