Non-local gyrokinetic model of linear ion-temperature-gradient modes

A theory of non-local linear ion-temperature-gradient (ITG) drift modes while retaining non-adiabatic electrons is presented, extending the previous work [S. Moradi, et al {\em Phys. Plasmas} {\bf 18}, 062106 (2011)]. A dispersion relation is derived to quantify the effects of the fractional velocity operator in the Fokker-Planck equation modified by temperature gradients and non-adiabatic electrons on the real frequency and growth rate. Solving the dispersion relation, it is shown here that as the plasma becomes more turbulent, it deviates from a Maxwellian distribution and becomes L\'{e}vy distributed. The resulting L\'{e}vy distribution of the plasma may thus significantly alter the transport. The relative effect of the fractional derivative is larger on the real frequency than on the growth rate of the ITG mode.