The free energy balance equation applied to gyrokinetic instabilities, the effect of the charge flux constraint, and application to simplified kinetic models

The free energy balance equation for gyrokinetic fluctuations is derived and applied to instabilities. An additional term due to electromagnetic sources is included. This can provide a simpler way to compute the free energy balance in practical applications, and is also conceptually clarifying. The free energy balance, by itself, is not sufficient to determine an eigenfrequency. The preceding results are derived in general geometry. The charge flux constraint in gyrokinetics can provide a necessary additional relation, and the combination of these two can be equivalent to a dispersion relation. The charge flux constraint can prevent the appearance of an unstable eigenmode even though the free energy balance would allow strongly growing fluctuations. The application of these concepts to simplified kinetic models in simplified geometry is also indicated.