The analytical treatment of plasma kinetic linear instabilities in toroidal geometry is commonly tackled employing a power series expansion of the resonant part of the dispersion relation. This expansion is valid under the assumption that the modulus of the mode frequency is smaller than the magnitude of the frequencies characterising the system (the drift, bounce and transit frequencies for example). We will refer to this approximation as high frequency approximation (HFA). In this paper the linear plasma dispersion relation is derived in the framework of the gyro-kinetic model, for the electrostatic case, in the local limit, in the absence of collisions, for a non rotating plasma, considering adiabatic electrons, in toroidal circular geometry, neglecting the parallel dynamics effect. A systematic analysis of the meaning and limitations of the HFA is performed. As already known, the HFA is not valid for tokamak relevant parameters. A new way to approximate the resonant part of the dispersion relation, called here Improved high frequency approximation (IHFA), is therefore proposed. A quantitative analysis of the ion temperature gradient (ITG) instability is presented. The IHFA is shown to be applicable to the treatment of the ITG instability in tokamaks.
Analysis of global hydromagnetic instabilities driven by strongly sheared toroidal flows in tokamak plasmas