The Kinetic Ion-Temperature-Gradient-driven Instability and Its Localisation

We construct a description of Ion Temperature Gradient (ITG) driven localisedmodes which retains both wave-particle and magnetic drift resonant effectswhile capturing the field-line dependence of the electrostatic potential. Weexploit the smallness of the magnetic drift and the strong localisation of themode to resolve the problem with a polynomial-gaussian expansion in thefield-following co-ordinate. A simple semi-analytical formula for the spectrumof the mode is shown to capture long wavelength Landau damping, ion-scaleLarmor radius stabilization, weakening of Larmor radius effects atshort-wavelengths and magnetic-drift resonant stabilisation. These elementslead to spectra with multiple maxima as observed in gyrokinetic simulations instellarators. Connections to the transition to extended eigenfunctions andthose localized by less unfavorable curvature regions (hopping solutions) arealso made. The model provides a clear qualitative framework with which tointerpret numerically simulated ITG mode spectra with realistic geometries,despite its limitations for exact quantitative predictions.